CHAPTER_13Introduction_to_Optimization_ModelingINVENTORY

CHAPTER 13 Introduction to Optimization Modeling

INVENTORY OPTIMIZATION AT GM The key to selling automobiles in the United States is the relationship between manufacturers, dealers, and custom- ers. When customers want to make a purchase, they almost always visit a dealer and purchase a vehicle from the lot. If their vehicle of choice is not on the lot, that dealer can make a request from a nearby dealer who might have the requested vehicle. The manufacturer keeps track of pur- chases at dealers and supplies them with new automobiles as necessary.

The article by Inman et al. (2017) describes how General Motors (GM) developed two optimization models

to determine new-vehicle inventory at its dealers. The first model finds the optimal number of vehicles to build for each dealer. The second model finds the optimal vehicle configurations each dealer should stock. These models differ from the traditional way GM had determined the number of vehicles to stock and their configurations. In the past, the standard approach of finding the level of inventory necessary to achieve a given fill rate, such as meeting 98% of customer demand with on-hand inventory, was used to determine the stock level. The configurations to stock were determined by ranking configurations by demand and stocking those with the highest rankings. Inman and his team used a different approach. For the number of vehicles to stock, they maximized variable profit (revenue minus variable cost) minus carrying costs. For the configurations, they used a “set- covering” approach to find a set of configurations that would cover the observed variety of customer demands.

In the first model, determining the optimal number to stock on dealers’ lots helps GM make better production decisions (overtime, assembly line rate, and number of shifts) and marketing decisions (rebates and advertising). To optimize this number, Inman’s team rejected the argument that carrying costs are incurred only by the dealers and therefore should not be a concern of GM management. Instead, they took a total supply chain view- point, with GM and its dealers considered a single entity. This led them to optimizing variable profit minus carrying costs. Besides, they argue that carrying costs go beyond the traditional costs of inventory such as floor space, insurance, and cost of capital. Customers typically want the newest model vehicle, so the longer vehicles remain on the dealers’ lots, the more heavily the dealers must discount their prices to sell them. This type of “carrying cost” hurts both GM and the dealers, and it provides an incentive to hold less inventory. Their model also considers diversions, where if a customer’s first choice is not in stock, the customer might divert to their second choice and hence still purchase a GM vehicle.

The second model, determining the optimal set of configurations to stock, is possibly even more challenging. A “full” configuration specifies every option possible: color, body style, powertrain, and a host of others. Determining which full configurations to stock would not only be virtually impossible (because of the vast number of configurations) but also pointless. Most customers are looking for a few key features, such as color, and they don’t really care about others. Therefore, Inman’s team concentrated on “partial” configurations, the sets of features that appear to be in highest demand at any given dealer. This greatly

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13-2 Introduction to Optimization 5 7 7

limits the number of decision variables in the optimization model. In addition, they rejected the standard procedure of stocking only the partial configurations in highest demand. To them, it made more sense to “cover” the range of configurations customers wanted. As a simple example, even if data indicate that black and white vehicles are most popular, it still doesn’t make sense to stock all black and white vehicles. At least a few customers will want blue, red, or green vehicles, so at least a few of these should be on dealers’ lots.

The result of the team’s models is an inventory-balancing report tool. For each model vehicle the dealer stocks, the report shows a column for each partial configuration. These partial configurations account for all the dealer’s sales. The report provides details familiar to dealers, who can then use their judgment to fine-tune their ordering decisions. After developing the tool, more than 800 dealers piloted it for six months. These dealers aver- aged a three to five percent increase in sales and revenue compared to a control group of about 7000 dealers not using the tool.

Inman’s team’s models have also helped GM to reduce overall retail inventory. Tra- ditionally, GM held more retail inventory than its competitors, but with the help of the optimization models, GM’s 2015 year-end inventory was 61 days-supply (the number of days to deplete supply at typical customer demand rates), down 14% from 2014 and sub- stantially lower than Ford’s 79 days-supply and Fiat Chrysler’s 81 days-supply.

13-1 Introduction In this chapter, we introduce spreadsheet optimization, one of the most powerful and flexible methods of quantitative analysis. The specific type of optimization we will discuss here is linear programming (LP). LP is used in all types of organizations, often on a daily basis, to solve a wide variety of problems. These include problems in labor scheduling, inventory management, selection of advertising media, bond trading, management of cash balances, operation of an electrical utility’s hydroelectric system, routing of delivery vehicles, blend- ing in oil refineries, hospital staffing, and many others. The goal of this chapter is to intro- duce the basic elements of LP: the types of problems it can solve, how LP problems can be modeled in Excel®, and how Excel’s Solver add-in can be used to find optimal solutions. Then in the next chapter we will examine a variety of LP applications, and we will also look at applications of integer and nonlinear programming, two important extensions of LP.

13-2 Introduction to Optimization We first discuss optimization in general. All optimization problems have several common elements. They all have decision variables, the variables whose values the decision maker is allowed to choose. Either directly or indirectly, the values of these variables determine such outputs as total cost, revenue, and profit. Essentially, they are the variables a com- pany or organization must know to function properly; they determine everything else. All optimization problems have an objective function (objective, for short) to be optimized— maximized or minimized. Finally, most optimization problems have constraints that must be satisfied. These are usually physical, logical, or economic restrictions, depending on the nature of the problem. In searching for the values of the decision variables that opti- mize the objective, only those values that satisfy all the constraints are allowed.

Excel uses its own terminology for optimization, and we will use it as well. Excel refers to the decision variables as the decision variable cells.1 These cells must contain numbers that are allowed to change freely; they are not allowed to contain formulas.

1 In Excel 2007 and previous versions, Excel’s Solver add-in referred to these as “changing cells.” Starting with Excel 2010, it refers to them as “decision variable cells” (or simply “variable cells”), so we will use the newer terminology.

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Excel refers to the objective as the objective cell. There can be only one objective cell, which could contain profit, total cost, total distance traveled, or others, and it must be related through formulas to the decision variable cells. When the decision variable cells change, the objective cell should change accordingly.

The decision variable cells contain the values that can be changed to optimize the objective. The objective cell contains the quantity to be minimized or maximized. The constraints impose restrictions on the values in the decision variable cells.

Finally, there must be appropriate cell formulas that operationalize the constraints. For example, one constraint might indicate that the amount of labor used can be no more than the amount of labor available. In this case, there must be cells for each of these two quantities, and typically at least one of them (probably the amount of labor used) will be related through formulas to the decision variable cells. Constraints can come in a variety of forms. One very common form is nonnegativity. This type of constraint states that decision variable cells must have nonnegative (zero or positive) values. Nonnegativity constraints are usually included for physical reasons. For example, it is impossible to pro- duce a negative number of automobiles.

Nonnegativity constraints imply that decision variable cells must contain non- negative values.

There are basically two steps in solving an optimization problem. The first step is to develop the model. Here you decide what the decision variables are, what the objective is, which constraints are required, and how everything is related. If you are developing an algebraic model, you must derive the correct algebraic expressions. If you are developing a spreadsheet model, the focus of this book, you must relate all variables with appropriate cell formulas. In particular, you must ensure that your model contains formulas that relate the decision variable cells to the objective cell and formulas that operationalize the con- straints. This model development step is where most of your effort goes.

The second step in any optimization model is to optimize. This means that you must systematically choose the values of the decision variables that make the objective as large (for maximization) or small (for minimization) as possible and satisfy all the constraints. Some terminology is useful here. Any set of values of the decision variables that satisfies all of the constraints is called a feasible solution. The set of all feasible solutions is called the feasible region. In contrast, an infeasible solution is a solution that violates at least one constraint. Infeasible solutions are not allowed. The desired feasible solution is the one that provides the best value—minimum for a minimization problem, maximum for a maximization problem—of the objective. This solution is called the optimal solution.

Typically, most of your effort goes into the development of the model.

A feasible solution is a solution that satisfies all the constraints. The feasible region is the set of all feasible solutions. An infeasible solution violates at least one of the constraints and is not allowed. The optimal solution is the feasible solution that optimizes the objective.

Although most of the effort typically goes into the model development step, much of the published research in optimization has been about the optimization step. Algorithms have been devised for searching through the feasible region to find the optimal solution.

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13-3 a two-Variable product Mix Model 5 7 9

One such algorithm is called the simplex method. It is used for linear models. There are other more complex algorithms used for other types of models (those with integer decision variables and/or nonlinearities).

We will not discuss the details of these algorithms. They have been programmed into Excel’s Solver add-in. All you need to do is develop the model and then tell Solver what the objective cell is, what the decision variable cells are, what the constraints are, and what type of model (linear, integer, or nonlinear) you have. Solver then finds the best feasible solution with the appropriate algorithm. You should appreciate that if you used a trial-and-error procedure, even a clever and fast one, it could take hours, weeks, or even years to complete. However, by using the appropriate algorithm, Solver typically finds the optimal solution in a matter of seconds.

There is really a third step in the optimization process: sensitivity analysis. You typi- cally choose values of input variables, such as unit costs, forecasted demands, and resource availabilities, and then find the optimal solution for these particular input values. This pro- vides a single “answer.” However, in any realistic situation, it is wishful thinking to believe that all the input values you use are exactly correct. Therefore, it is useful—indeed, man- datory in most applied studies—to follow up the optimization step with what-if questions. What if the unit costs increased by 5%? What if forecasted demands were 10% lower? What if resource availabilities could be increased by 20%? What effects would such changes have on the optimal solution? This type of sensitivity analysis can be done in an informal manner or it can be highly structured. Fortunately, as with the optimization step itself, good soft- ware allows you to obtain answers to various what-if questions quickly and easily.

13-3 A Two-Variable Product Mix Model We begin with a very simple two-variable example of a product mix problem. This is a type of problem frequently encountered in business where a company must decide its product mix—how much of each of its potential products to produce—to maximize its net profit. You will see how to model this problem algebraically and then how to model it in Excel. You will also see how to find its optimal solution with Solver. Next, because it con- tains only two decision variables, you will see how it can be solved graphically. Although this graphical solution is not practical for most problems, it provides useful insights into general LP models. The final step is then to ask a number of what-if questions about the completed model.

An algorithm is a prescription for carrying out the steps required to achieve some goal, such as finding an optimal solution. An algorithm is typically translated into a computer program that performs the work.

EXAMPLE

13.1 ASSEMBLING AND TESTING COMPUTERS AT PC TECH The PC Tech company assembles and then tests two models of computers, Basic and XP. For the coming month, the company wants to decide how many of each model to assemble and then test. No computers are in inventory from the previous month, and because these models are going to be changed after this month, the company doesn’t want to hold any inventory after this month. It believes the most it can sell this month are 600 Basics and 1200 XPs. Each Basic sells for $300 and each XP sells for $450. The cost of component parts for a Basic is $150; for an XP it is $225. Labor is required for assembly and testing. There are at most 10,000 assembly hours and 3000 testing hours available. Each labor hour for assembling costs $11 and each labor hour for testing costs $15. Each Basic requires five hours for assembling and one hour for testing, and each XP requires six hours for assembling and two hours for testing. PC Tech wants to know how many of each model it should produce (assemble and test) to maximize its net profit, but it cannot use more labor hours than are available, and it does not want to produce more than it can sell.

Objective To use LP to find the best mix of computer models that stays within the company’s labor availability and maximum sales constraints.

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Solution The essence of spreadsheet modeling is transforming a “story problem” into an Excel model. Based on our teaching experience, a “bridge” between the two is often needed, especially for complex models. Therefore, in the next few chapters, most examples in the book will start with a “big picture” diagram to help you understand the model—what the key elements are and how they are related—and get you ready for the eventual spreadsheet model.2 Each diagram is in its own Excel file, such as Product Mix 1 Big Picture.xlsx for this example. (These big picture files are available, just like the example files.) A screenshot of this big picture appears in Figure 13.1.

2 We have created these diagrams with Palisade’s BigPicture add-in, part of the DecisionTools Suite.

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Figure 13.1 Big Picture for Product Mix Model

Labor hours per unit

Labor hours used

Labor hours available

Cost per labor hour

Selling price

Maximum sales

Maximize profit

Number produced <=

Cost of component parts

Unit margin

Playing a slide show

If you load the BigPicture add-in (from the Palisade group of programs) and then open the big picture file, you can see more than this static diagram. First, each of the shapes in the diagram can have a “note,” much like an Excel cell comment. When you move the cursor over the shape, the note appears. Second, the software allows you to create slide shows. We have done this for all of the big pictures in the book. This lets you see how the model “evolves,” and each slide is accompanied by a “pop-up” text box explanation to help you understand the model even better. To run the slide show, click the Play button on the BigPicture ribbon and then the Next Slide button for each new slide. When you are finished, click the Stop button.

BigPicture Tip

We have adopted a color-coding/shape convention for these big pictures.

Our Big Picture Conventions • Blue rectangles indicate given inputs. • Red ovals indicate decision variables. • Green rectangles with rounded tops indicate uncertain quantities (relevant for Chapters 15 and 16). • Yellow rounded rectangles indicate calculated quantities. • Shapes with thin gray borders indicate bottom line outputs or quantities to optimize. • Arrows indicate that one quantity helps determine another. However, if an arrow includes an inequality or equality sign, as

you will often see in the optimization chapters, the arrow indicates a constraint.

The decision variables in this product mix model are straightforward. The company must decide how many Basics to produce and how many XPs to produce. Once these are known, they can be used with the problem inputs to calculate the num- ber of computers sold, the labor used, and the revenue and cost. However, as you will see with other models in this chapter and the next chapter, determining the decision variables is not always this obvious.

Pictures such as this one bridge the gap between the problem statement and the ultimate spreadsheet (or algebraic) model.

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Algebraic Model In the traditional algebraic solution method, you first identify the decision variables.3 In this small problem they are the num- bers of computers to produce. We label these x1 and x2, although any other labels would do. The next step is to write expres- sions for the total net profit and the constraints in terms of the x’s. Finally, because only nonnegative amounts can be produced, explicit constraints are added to ensure that the x’s are nonnegative. The resulting algebraic model is

Maximize 80x1 1 129x2

subject to: 5x1 1 6x2 # 10000

x1 1 2x2 # 3000

x1 # 600

x2 # 1200

x1, x2 $ 0

To understand this model, consider the objective first. Each Basic produced sells for $300, and the total cost of producing it, including component parts and labor, is 150 1 5(11) 1 1(15) 5 $220, so the profit margin is $80. Similarly, the profit margin for an XP is $129. Each profit margin is multiplied by the number of computers produced, and these products are then summed over the two computer models to obtain the total net profit.

The first two constraints are similar. For example, each Basic requires five hours for assembling and each XP requires six hours for assembling, so the first constraint says that the total hours required for assembling is no more than the number avail- able, 10,000. The third and fourth constraints are the maximum sales constraints for Basics and XPs. Finally, negative amounts cannot be produced, so nonnegativity constraints on x1 and x2 are included.

For many years, all LP problems were modeled this way in textbooks. In fact, many com- mercial LP computer packages are still written to accept LP problems in essentially this for- mat. Since around 1990, however, a more intuitive method of expressing LP problems has become popular. This method takes advantage of the power and flexibility of spreadsheets. Actually, LP problems could always be modeled in spreadsheets, but now with the addition of Excel’s Solver add-in, spreadsheets have the ability to solve—that is, optimize—LP problems as well. We use Excel’s Solver for all examples in this book.4

Graphical Solution When there are only two decision variables in an LP model, as there are in this product mix model, you can solve the problem graphically. Although this graphical solution approach is not practical in most realistic optimization models—where there are many more than two decision variables—the graphical procedure illustrated here still yields important insights for general LP models.

In general, if the two decision variables are labeled x1 and x2, then the steps of the method are to express the constraints and the objective in terms of x1 and x2, graph the constraints to find the feasible region [the set of all pairs (x1, x2) satisfying the constraints, where x1 is on the horizontal axis and x2 is on the vertical axis], and then move the objective through the feasible region until it is optimized.

To do this for the product mix problem, note that the constraint on assembling labor hours can be expressed as 5x1 1 6x2 # 10000. To graph this, consider the associated equality (replacing # with 5 ) and find where the associated line crosses the axes. Specifically, when x1 5 0, then x2 5 10000>6 5 1666.7; and when x2 5 0, then x1 5 10000>5 5 2000. This produces the line labeled “assembling hour constraint” in Figure 13.2. It has slope 25>6 5 20.83. The set of all points that satisfy the assembling hour constraint includes the points on this line plus the points below it, as indicated by the arrow drawn from the line. [The feasible points are below the line because the point (0, 0) is obviously below the line, and (0, 0) clearly satisfies the assembly hour constraint.] Similarly, the testing hour and maximum sales constraints are shown in the figure. The points that satisfy all three of these constraints and are nonnegative comprise the feasible region, which is below the heavier lines in the figure.

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3 This is not a book about algebraic models; the main focus is on spreadsheet modeling. However, we present algebraic models of the examples in this chapter for comparison with the corresponding spreadsheet models. 4 The Solver add-in built into Excel was developed by a third-party software company, Frontline Systems. This company develops much more powerful versions of Solver for commercial sales, but its standard version built into Microsoft Excel suffices for us. More information about Solver software offered by Frontline can be found at www.solver.com.

Many commercial optimi- zation packages require, as input, an algebraic model of a problem. If you ever use one of these packages, you will have to think algebraically.

This graphical approach works only for problems with two decision variables. Recall from algebra that any line of the form ax1 + bx2 = c has slope −a/b. This is because it can be put into the slope − intercept form x2 = c/b − (a/b)x1.

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Figure 13.2 Graphical Solution to Two-Variable Product Mix Problem

30002000

1500

1666.7

600

1200

Feasible region (below dark lines)

Testing hour constraint

Assembling hour constraint

Basic sales constraint

XP sales

Optimal solution

constraint Isoprofit lines (dotted)

XPs produced

Basics produced

To the left and below the dark line is the feasible region. As the dotted objective line is pushed as far up to the right as possible, the last feasible point it hits is the one shown. In general, the corner point that is optimal depends on the relative slopes of the lines.

To see which feasible point maximizes the objective, it is useful to draw a sequence of lines where, for each, the objective is constant. A typical line is of the form 80x1 1 129x2 5 c, where c is a constant. Any such line has slope 280>129 5 20.620, regardless of the value of c. This line is steeper than the testing hour constraint line (slope 20.5), but not as steep as the assem- bling hour constraint line (slope 20.83). Then the idea is to move a line with this slope up and to the right, making c larger, until it just barely touches the feasible region. The last feasible point it touches is the optimal point.

Several lines with slope 20.620 are shown in Figure 13.2. The middle dotted line is the one with the largest net profit that still touches the feasible region. The associated optimal point is clearly the point where the assembling hour and XP maximum sales lines intersect. You will eventually find (from Solver) that this point is (560,1200), but even if you didn’t have the Solver add-in, you could find the coordinates of this point by solving two equations (the ones for assembling hours and XP maximum sales) in two unknowns.

Again, the graphical procedure illustrated here can be used only for the simplest of LP models, those with two decision variables. However, the type of behavior pictured in Figure 13.2 generalizes to all LP problems. In general, all feasible regions are (the mul- tidimensional versions of) polygons. That is, they are bounded by straight lines (actually hyperplanes) that intersect at several corner points. There are five corner points in Figure 13.2, three of which are on the axes. [One of them is (0,0).] When the dot- ted objective line is moved as far as possible toward better values, the last feasible point it touches is one of the corner points. The actual corner point it last touches is determined by the slopes of the objective and constraint lines. Because there are only a finite number of corner points, it suffices to search among this finite set, not the infinite number of points in the entire feasible region.5 This insight is largely responsible for the efficiency of the simplex method for solving LP problems.

Although limited in use, the graphical approach yields the important insight that the optimal solution to any LP model is a corner point of a polygon. This limits the search for the optimal solution and makes the simplex method possible.

5 This is not entirely true. If the objective line is exactly parallel to one of the constraint lines, there can be multiple optimal solutions—a whole line segment of optimal solutions. Even in this case, however, at least one of the optimal solutions is a corner point.

Geometry of Lp Models and the Simplex Method

The feasible region in any LP model is always a multidimensional version of a polygon, and the objective is always a hyperplane, the multidimensional version of a straight line. The objective should always be moved as far as possible in the maximizing or minimizing direction until it just touches the edge of the feasible region. Because of this geometry, the optimal solution is always a corner point of the polygon. The simplex method for LP works so well because it can search through the finite number of corner points extremely efficiently and recognize when it has found the best corner point. This rather simple insight, plus its clever implementation in software packages, has saved companies many, many millions of dollars in the past 50 years.

Fundamental Insight

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13-3 a two-Variable product Mix Model 5 8 3

Spreadsheet Model We now turn our focus to spreadsheet modeling. There are many ways to develop an LP spreadsheet model. Everyone has his or her own preferences for arranging the data in the various cells. We do not provide exact prescriptions, but we do present enough examples to help you develop good habits. The common elements in all LP spreadsheet models are the inputs, decision variable cells, objective cell, and constraints.

• Inputs. All numeric inputs—that is, all numeric data given in the statement of the problem—should appear somewhere in the spreadsheet. Our convention is to color all of the input cells blue. We also try to put most of the inputs in the upper left section of the spreadsheet. However, we sometimes violate this convention when certain inputs fit more naturally some- where else.

• Decision variable cells. Instead of using variable names, such as x, spreadsheet models use a set of designated cells for the decision variables. The values in these cells can be changed to optimize the objective. The values in these cells must be allowed to vary freely, so there should not be any formulas in the decision variable cells. To designate them clearly, our convention is to color them red.

• Objective cell. One cell, called the objective cell, contains the value of the objective. Solver systematically varies the values in the decision variable cells to optimize the value in the objective cell. This cell must be linked, either directly or indirectly, to the decision variable cells by formulas. Our convention is to color the objective cell gray.

Our coloring conventions

Color all input cells blue. Color all of the decision variable cells red. Color the objective cell gray.

• Constraints. Excel does not show the constraints directly on the spreadsheet. Instead, they are specified in a Solver dialog box, to be discussed shortly. For example, a set of related constraints might be specified by

B16:C16*5B18:C18

This implies two separate constraints. The value in B16 must be less than or equal to the value in B18, and the value in C16 must be less than or equal to the value in C18. We will always assign range names to the ranges that appear in the con- straints. Then a typical constraint might be specified as

Number_to_produce*5Maximum_sales

This is much easier to read and understand. (If you don’t want to use range names, you don’t have to. Solver models work fine with cell addresses only.)

• Nonnegativity. Normally, the decision variables—that is, the values in the decision variable cells—must be nonnegative. These constraints do not need to be written explicitly; you simply check an option in the Solver dialog box to indicate that the decision variable cells should be nonnegative. Note, however, that if you want to constrain any other cells to be nonneg- ative, you must specify their constraints explicitly.

Overview of the Solution Process As mentioned previously, the complete solution of a problem involves three stages. In the model development stage you enter all of the inputs, trial values for the decision variable cells, and formulas relating these in a spreadsheet. This stage is the most crucial because it is here that all of the ingredients of the model are included and related appropriately. In particular, the spreadsheet must relate the objective to the decision variable cells, either directly or indirectly, so that if the values in the deci- sion variable cells vary, the objective value varies accordingly. Similarly, the spreadsheet must include formulas for the various constraints (usually their left sides) that are related directly or indirectly to the decision variable cells.

After the model is developed, you can proceed to the second stage—invoking Solver. At this point, you formally designate the objective cell, the decision variable cells, the constraints, and selected options, and you tell Solver to find the optimal solu- tion. If the first stage has been done correctly, the second stage is usually very quick and straightforward.

The third stage is sensitivity analysis. Here you see how the optimal solution changes (if at all) as selected inputs are var- ied. This often provides important insights about the behavior of the model.

We now illustrate this procedure for the product mix problem in Example 13.1.

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Where Do the Numbers Come From? There are a variety of inputs in PC Tech’s problem, some easy to find and others more difficult. Here are some ideas on how they might be obtained.

• The unit costs in rows 3, 4, and 10 should be easy to obtain. (See Figure 13.3.) These are the going rates for labor and the component parts. Note, however, that the labor costs are probably regular-time rates. If the company wants to consider over- time hours, then the overtime rate (and labor hours availability during overtime) would be necessary, and the model would need to be modified.

Figure 13.3 Two-Variable Product Mix Model with an Infeasible Solution

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

GFEDCBA Assembling and testing computers Range names used:

Hours_available =Model!$D$21:$D$22 Cost per labor hour assembling $11

$15

$150 $300

5 1

$225 $450

6 2

Labor hours for assembly Basic XP

Basic

Hours used 10200

3000 10000

3000

$48,000 $154,800 $202,800

<= <=

Basic XP Total

Hours available

Cost of component parts Selling price Unit margin

Assembling, testing plan (# of computers)

Number to produce

Maximum sales

Constraints (hours per month)

Labor hours for testing

Hours_used =Model!$B$21:$B$22 Maximum_sales =Model!$B$18:$C$18 Number_to_produce =Model!$B$16:$C$16

=Model!$D$25Total_profit

Cost per labor hour testing

Labor availability for assembling Labor availability for testing

Net profit ($ this month)

Inputs for assembling and testing a computer

$129

1200

$80

600

1200600

• The resource usages in rows 8 and 9, often called technological coefficients, should be available from the production depart- ment. These people know how much labor it takes to assemble and test these computer models.

• The unit selling prices in row 11 have actually been chosen by PC Tech’s management, probably in response to market pres- sures and the company’s own costs.

• The maximum sales values in row 18 are probably forecasts from the marketing and sales department. These people have some sense of how much they can sell, based on current outstanding orders, historical data, and the prices they plan to charge.

• The labor hour availabilities in rows 21 and 22 are probably based on the current workforce size and possibly on new work- ers who could be hired in the short run. Again, if these are regular-time hours and overtime is possible, the model would have to be modified to include overtime.

Developing the Spreadsheet Model The spreadsheet model appears in Figure 13.3. (See the file Product Mix 1 Finished.xlsx.) To develop this model, use the following steps.

1. Inputs. Enter all of the inputs from the statement of the problem in the blue cells as shown. 2. Range names. Create the range names shown in columns E and F. Our convention is to enter enough range names, but

not to go overboard. Specifically, we enter enough range names so that the setup in the Solver dialog box, to be explained

Developing the Product Mix 1 Model

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shortly, is entirely in terms of range names. Of course, you can add more range names if you like (or you can omit them altogether). The following tip indicates a quick way to create range names.

Shortcut for Creating Range Names

Select a range such as A16:C16 that includes nice labels in column A and the range you want to name in columns B and C. Then, from the Formulas ribbon, select Create from Selection and accept the default. You automatically get the labels in cells A16 as the range name for the range B16:C16. (Note that if the label contains spaces or other “illegal” characters, they are replaced by underscores in the range name.) This shortcut illustrates the usefulness of adding concise but informative labels next to ranges you want to name.

Excel Tip

3. Unit margins. Enter the formula

5B112B8*$B$32B9*$B$42B10

in cell B12 and copy it to cell C12 to calculate the unit profit margins for the two models. (Enter relative/absolute addresses that allow you to copy whenever possible.)

4. Decision variable cells. Enter any two values for the decision variable cells in the Number_ to_produce range. Any trial values can be used initially; Solver will eventually find the opti- mal values. Note that the two values shown in Figure 13.3 cannot be optimal because they use more assembling hours than are available. However, you do not need to worry about satisfying constraints at this point; Solver will take care of this later on.

5. Labor hours used. To operationalize the labor availability constraints, you must calculate the amounts used by the production plan. To do this, enter the formula

5SUMPRODUCT(B8:C8,Number_to_produce)

in cell B21 for assembling and copy it to cell B22 for testing. This formula is a shortcut for the following fully written out formula:

5B8*B161C8*C16

The SUMPRODUCT function is very useful in spreadsheet models, especially LP models, and you will see it often. Here, it multiplies the number of hours per computer by the number of computers for each model and then sums these products over the two models. When there are only two products in the sum, as in this example, the SUMPRODUCT formula is not really any simpler than the written-out formula. However, imagine that there are 50 models. Then the SUMPRODUCT formula is much simpler to enter (and read). For this reason, you should use it whenever possible. Note that each range in this function, B8:C8 and Number_to_produce, is a one-row, two-column range. It is important in the SUMPRODUCT function that the two ranges be exactly the same size and shape.

6. Net profits. Enter the formula

5B12*B16

in cell B25, copy it to cell C25, and sum these to get the total net profit in cell D25. This latter cell is the objective to maximize. Note that if you didn’t care about the net profits for the two individual models, you could calculate the total net profit with the formula

5SUMPRODUCT(B12:C12,Number_to_produce)

As you see, the SUMPRODUCT function appears once again. It and the SUM function are the most used functions in LP models.

Experimenting with Possible Solutions The next step is to specify the decision variable cells, the objective cell, and the constraints in a Solver dialog box and then instruct Solver to find the optimal solution. However, before you do this, it is instructive to try a few guesses in the decision

At this stage, it is pointless to try to outguess the optimal solution. Any values in the decision variable cells will suffice.

The “linear” in linear programming is all about sums of products. Therefore, the SUMPRODUCT function is natural and should be used whenever possible.

13-3 a two-Variable product Mix Model 5 8 5

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variable cells. There are two reasons for doing so. First, by entering different sets of values in the decision variable cells, you can confirm that the formulas in the other cells are working correctly. Second, this experimentation can help you to develop a better understanding of the model.

For example, the profit margin for XPs is much larger than for Basics, so you might suspect that the company should produce only XPs. The most it can produce is 1200 (maximum sales), and this uses fewer labor hours than are available. This solution appears in Figure 13.4. However, you can probably guess that it is far from optimal. There are still many labor hours available, so the company could use them to produce some Basics and make more profit.

You can continue to try different values in the decision variable cells, attempting to get as large a total net profit as possi- ble while staying within the constraints. Even for this small model with only two decision variable cells, the optimal solution is not totally obvious. You can only imagine how much more difficult it is when there are hundreds or even thousands of decision variable cells and many constraints. This is why software such as Excel’s Solver is required. Solver uses a quick and efficient algorithm to search through all feasible solutions (or more specifically, all corner points) and eventually find the optimal solu- tion. Fortunately, it is quite easy to use, as we now explain.

Figure 13.4 Two-Variable Product Mix Model with a Suboptimal Solution

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

GFEDCBA Assembling and testing computers Range names used:

Hours_available =Model!$D$21:$D$22 Cost per labor hour assembling $11

$15

$150 $300

5 1

$225 $450

6 2

Labor hours for assembly Basic XP

Basic

Hours used 7200 2400

10000 3000

$0 $154,800 $154,800

<= <=

Basic XP Total

Hours available

Cost of component parts Selling price Unit margin

Assembling, testing plan (# of computers)

Number to produce

Maximum sales

Constraints (hours per month)

Labor hours for testing

Hours_used =Model!$B$21:$B$22 Maximum_sales =Model!$B$18:$C$18 Number_to_produce =Model!$B$16:$C$16

=Model!$D$25Total_profit

Cost per labor hour testing

Labor availability for assembling Labor availability for testing

Net profit ($ this month)

Inputs for assembling and testing a computer

$129

1200

$80

1200600

Using Solver To invoke Excel’s Solver, select Solver from the Data ribbon. (If there is no such item on your PC, you need to load Solver. To do so, click the File button, then Options, then Add-Ins, and then Go at the bottom of the dialog box. This displays the list of available add-ins. If there is a Solver Add-in item in the list, check it to load Solver. If there is no such item, you need to rerun the Microsoft Office installer and elect to install Solver. It should be included in a typical install, but some people elect not to install it the first time around.) The dialog box in Figure 13.5 appears.6 It has three important sec- tions that you must fill in: the objective cell, the decision variable cells, and the constraints. For the product mix problem, you can fill these in by typing cell references or you can point, click, and drag the appropriate ranges in the usual way. Better yet, if there are any named ranges, these range names appear instead of cell addresses when you drag the ranges. In fact, for reasons of readability, our convention is to use only range names, not cell addresses, in this dialog box.

6 This is the Solver dialog box since Excel 2010. It is more convenient than similar dialog boxes in previous versions because the typical settings now all appear in a single dialog box. In previous versions you had to click the Options button to complete the typical settings.

The Solver item is under the Tools menu on the Mac.

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1. Objective. Select the Total_profit cell as the objective cell, and click the Max option. (Actually, the default option is Max.) 2. Decision variable cells. Select the Number_to_produce range as the decision variable cells. 3. Constraints. Click the Add button to bring up the dialog box in Figure 13.6. Here you specify a typical constraint by

entering a cell reference or range name on the left, the type of constraint from the dropdown list in the middle, and a cell reference, range name, or numeric value on the right. Use this dialog box to enter the constraint

Number_to_produce*5Maximum_sales

(Note: You can type these range names into the dialog box, or you can drag them in the usual way. If you drag them, the cell addresses shown in the figure eventually change into range names if range names exist.) Then click the Add button and enter the constraint

Hours_used*5Hours_available

Figure 13.5 Solver Dialog Box (since Excel 2010)

Range Names in Solver Dialog Box

Our usual procedure is to use the mouse to select the relevant ranges for the Solver dialog box. Fortunately, if these ranges have already been named, the range names will automatically replace the cell addresses.

Excel Tip

Figure 13.6 Add Constraint Dialog Box

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Inequality and Equality Labels in Spreadsheet Models

The 6= signs in cells B17:C17 and C21:C22 (see Figure 13.3 or Figure 13.4) are not a necessary part of the Excel model. They are entered simply as labels in the spreadsheet and do not substitute for entering the con- straints in the Add Constraint dialog box. However, they help to document the model, so we include them in all the examples. In fact, you should try to plan your spreadsheet models so that the two sides of a constraint are in nearby cells, with “gutter” cells in between where you can attach labels like 6=, 7=, or =. This convention tends to make the resulting spreadsheet models more readable.

Excel Tip

Entering Constraints in Groups

Constraints typically come in groups. Beginners often enter these one at a time, such as B16 6= B18 and C16 6= C18, in the Solver dialog box. This can lead to a long list of constraints, and it is time-consuming. It is better to enter them as a group, as in B16:C16 6= B18:C18. This is not only quicker, but it also takes advantage of range names you have created. For example, this group ends up as Number_to_produce6=Maximum_Sales.

Solver Tip

Then click OK to get back to the Solver dialog box. The first constraint says to produce no more than can be sold. The second constraint says to use no more labor hours than are available. 4. Nonnegativity. Because negative production quantities make no sense, you must tell

Solver explicitly to make the decision variable cells nonnegative. To do this, check the Make Unconstrained Variables Non-Negative option shown in Figure 13.5. This automat- ically ensures that all decision variable cells are nonnegative.

5. Linear model. There is one last step before clicking the Solve button. As stated previously, Solver uses one of several numer- ical algorithms to solve various types of models. The models discussed in this chapter are all linear models. (We will discuss the properties that distinguish linear models shortly.) Linear models can be solved most efficiently with the simplex method. To instruct Solver to use this method, make sure Simplex LP is selected in the Select a Solving Method dropdown list in Figure 13.5.

6. Optimize. Click the Solve button in the dialog box in Figure 13.5. At this point, Solver does its work. It searches through a number of possible solutions until it finds the optimal solution. (You can watch the progress on the lower left of the screen, although for small models the process is virtually instantaneous.) When it finishes, it displays the message shown in Figure 13.7. You can then instruct it to return the values in the decision variable cells to their original (probably nonop- timal) values or retain the optimal values found by Solver. In most cases you should choose the latter. For now, click OK to keep the Solver solution. You should see the solution shown in Figure 13.8.

Checking the Non-Negative option ensures only that the decision variable cells will be nonnegative.

Figure 13.7 Solver Results Message

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Figure 13.8 Two-Variable Product Mix Model with the Optimal Solution

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

GFEDCBA Assembling and testing computers Range names used:

Hours_available =Model!$D$21:$D$22 Cost per labor hour assembling $11

$15

$150 $300

5 1

$225 $450

6 2

Labor hours for assembly Basic XP

Basic

Hours used 10000

2960 10000

3000

$44,800 $154,800 $199,600

<= <=

Basic XP Total

Hours available

Cost of component parts Selling price Unit margin

Assembling, testing plan (# of computers)

Number to produce

Maximum sales

Constraints (hours per month)

Labor hours for testing

Hours_used =Model!$B$21:$B$22 Maximum_sales =Model!$B$18:$C$18 Number_to_produce =Model!$B$16:$C$16

=Model!$D$25Total_profit

Cost per labor hour testing

Labor availability for assembling Labor availability for testing

Net profit ($ this month)

Inputs for assembling and testing a computer

$129

1200

$80

560

1200600

Discussion of the Solution This solution indicates that PC Tech should produce 560 Basics and 1200 XPs. This plan uses all available labor hours for assembling, has a few leftover labor hours for testing, produces as many XPs as can be sold, and produces a few less Basics than could be sold. No plan can provide a net profit larger than this one—that is, without violating at least one of the constraints.

The solution in Figure 13.8 is typical of solutions to optimization models in the following sense. Of all the inequality con- straints, some are satisfied exactly and others are not. In this solution the XP maximum sales and assembling labor constraints are met exactly. Each of these is called a binding constraint. However, the Basic maximum sales and testing labor constraints do not hold as equalities. Each of these is called a nonbinding constraint. You can think of the binding constraints as bottle- necks. They are the constraints that prevent the objective from being improved. If it were not for the binding constraints on maximum sales and labor, PC Tech could obtain an even larger net profit.

An inequality constraint is binding if the solution makes it an equality. Otherwise, it is nonbinding.

In a typical optimal solution, you should pay particular attention to two aspects of the solution. First, you should check which of the decision variables are positive (as opposed to 0). Generically, these are the “activities” that are done at a positive level. In a product mix model, they are the products included in the optimal mix. Second, you should check which of the constraints are binding. Again, these represent the bottlenecks that keep the objective from improving.

13-3 a two-Variable product Mix Model 5 8 9

Messages from Solver

The message in Figure 13.7 is the one you hope for. However, in some cases Solver is not able to find an optimal solution, in which case one of several other messages appears. We discuss two of these later in the chapter.

Solver Tip

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13-4 Sensitivity Analysis Having found the optimal solution, it might appear that the analysis is complete. But in real LP applications the solution to a single model is hardly ever the end of the analysis. It is almost always useful to perform a sensitivity analysis to see how (or if) the optimal solu- tion changes as one or more inputs vary. We illustrate systematic ways of doing so in this section. Actually, we discuss two approaches. The first uses an optional sensitivity report that Solver offers. The second uses an add-in called SolverTable that Albright developed.

13-4a Solver’s Sensitivity Report When you run Solver, the dialog box in Figure 13.7 offers you the option to obtain a sen- sitivity report.7 This report is based on a well-established theory of sensitivity analysis in optimization models, especially LP models. This theory was developed around algebraic models that are arranged in a “standardized” format. Essentially, all such algebraic mod- els look alike, so the same type of sensitivity report applies to all of them. Specifically, they have an objective function of the form c1x1 1 g 1 cnxn, where n is the number of decision variables, the c’s are constants, and the x’s are the decision variables, and each constraint can be expressed as a1x1 1 g 1 anxn # b, a1x1 1 g 1 anxn $ b, or a1x1 1 g 1 anxn 5 b, where the a’s and b’s are constants. Solver’s sensitivity report performs two types of sensitivity analysis: (1) on the coefficients of the objective, the c’s, and (2) on the right sides of the constraints, the b’s.

We illustrate the typical analysis by looking at the sensitivity report for PC Tech’s product mix model in Example 13.1. For convenience, the algebraic model is repeated here.

Maximize 80x1 1 129x2

subject to: 5x1 1 6x2 # 10000

x1 1 2x2 # 3000

x1 # 600

x2 # 1200

x1, x2 $ 0

On this Solver run, a sensitivity report is requested in Solver’s final dialog box. (See Figure 13.7.) The sensitivity report appears on a new worksheet, as shown in Figure 13.9.8

Many analysts view the “finished” model as a starting point for many what-if questions. We agree.

Sensitivity for the Product Mix 1 Model

Binding and Nonbinding Constraints

Most optimization models contain constraints expressed as inequalities. In an optimal solution, each such constraint is either binding (holds as an equality) or nonbinding. It is important to identify the binding constraints because they are the constraints that prevent the objective from improving. A typical constraint is on the availability of a resource. If such a constraint is binding, the objective could typically improve by having more of that resource. But if such a resource con- straint is nonbinding, more of that resource would not improve the objective at all.

Fundamental Insight

7 It also offers Answer and Limits reports. We don’t find these particularly useful, so we will not discuss them here. 8 If your table looks different from ours, make sure you chose the simplex method. Otherwise, Solver uses a nonlinear algorithm and produces a different type of sensitivity report.

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13-4 Sensitivity analysis 5 9 1

Figure 13.9 Solver Sensitivity Results

A B C D E F G H 6 Variable Cells

Constraints

$B$16 $C$16

560 0 80 27.5 80 1200

10000 16 02960

10000 200 2800 3000

33 129

1E+30

33 Number to produce Basic Number to produce XP

Labor availability for assembling Hours used Labor availability for testing Hours used

$B$21 $B$22

Cell Name Final Value

Reduced Cost

Objective Coefficient

Allowable Increase

Allowable Decrease

Cell Name Final Value

Shadow Price

Constraint R.H. Side

Allowable Increase

Allowable Decrease

7 8 9

10 11 12 13 14 15 16

It contains two sections. The top section is for sensitivity to changes in the two coeffi- cients, 80 and 129, of the decision variables in the objective. Each row in this section indi- cates how the optimal solution changes if one of these coefficients changes. The bottom section is for the sensitivity to changes in the right sides, 10000 and 3000, of the labor constraints. Each row of this section indicates how the optimal solution changes if one of these availabilities changes. (The maximum sales constraints represent a special kind of constraint—upper bounds on the decision variable cells. Upper bound constraints are han- dled in a special way in the Solver sensitivity report, as described shortly.)

In the first row of the top section, the allowable increase and allowable decrease indi- cate how much the coefficient of profit margin for Basics in the objective, currently 80, could change before the optimal product mix would change. If the coefficient of Basics stays within this allowable range, from 0 (decrease of 80) to 107.5 (increase of 27.5), the optimal product mix—the set of values in the decision variable cells—does not change at all. However, outside of these limits, the optimal mix between Basics and XPs might change.

To see what this implies, change the selling price in cell B11 from 300 to 299, so that the profit margin for Basics decreases to $79. This change is well within the allowable decrease of 80. If you rerun Solver, you will obtain the same values in the decision vari- able cells, although the objective value will decrease. Next, change the value in cell B11 to 330. This time, the profit margin for Basics increases by 30 from its original value of $300. This change is outside the allowable increase, so the solution might change. If you rerun Solver, you will indeed see a change—the company now produces 600 Basics and fewer than 1200 XPs.

The reduced costs in the second column indicate, in general, how much the objective coefficient of a decision variable that is currently 0 or at its upper bound must change before that variable changes (becomes positive or decreases from its upper bound). The interesting variable in this case is the number of XPs, currently at its upper bound of 1200. The reduced cost for this variable is 33, meaning that the number of XPs will stay at 1200 unless the profit margin for XPs decreases by at least $33. Try it. Starting with the original inputs, change the selling price for XPs to $420, a change of less than $33. If you rerun Solver, you will find that the optimal plan still calls for 1200 XPs. Then change the selling price to $410, a change of more than $33 from the original value. After rerunning Solver, you will find that fewer than 1200 XPs are in the optimal mix.

The reduced cost for any decision variable with value 0 in the optimal solution indicates how much better that coefficient must be before that variable enters at a positive level. The reduced cost for any decision variable at its upper bound in the optimal solution indicates how much worse its coefficient must be before it will decrease from its upper bound. The reduced cost for any variable between 0 and its upper bound in the optimal solution is irrelevant.

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Now turn to the bottom section of the report in Figure 13.9. Each row in this section corresponds to a constraint, although upper bound constraints on decision variable cells are omitted in this section. To have this part of the report make economic sense, the model should be developed as has been done here, where the right side of each constraint is a numeric constant (not a formula). Then the report indicates how much these right-side constants can change before the optimal solution changes. To understand this more fully, the concept of a shadow price is required. A shadow price indicates the change in the objective when a right-side constant changes.

The term shadow price is an economic term. It indicates the change in the optimal value of the objective when the right side of some constraint changes by one unit.

A shadow price is reported for each constraint. For example, the shadow price for the assembling labor constraint is 16. This means that if the right side of this con- straint increases by one hour, from 10000 to 10001, the optimal value of the objective will increase by $16. It works in the other direction as well. If the right side of this con- straint decreases by one hour, from 10000 to 9999, the optimal value of the objective will decrease by $16. However, as the right side continues to increase or decrease, this $16 change in the objective might not continue. This is where the reported allowable increase and allowable decrease are relevant. As long as the right side increases or decreases within its allowable limits, the same shadow price of 16 still applies. Beyond these limits, how- ever, a different shadow price might apply.

You can prove this for yourself. First, increase the right side of the assembling labor constraint by 200 (exactly the allowable increase), from 10000 to 10200, and rerun Solver. (Don’t forget to reset other inputs to their original values if you have made changes.) You will see that the objective indeed increases by 16(200)=$3200, from $199,600 to $202,800. Now increase this right side by one more hour, from 10200 to 10201 and rerun Solver. You will observe that the objective doesn’t increase at all. This means that the shadow price beyond 10200 is less than 16; in fact, it is zero. This is typical. When a right side increases beyond its allowable increase, the new shadow price is typically less than the original shadow price (although it doesn’t always fall to zero, as in this example).

resource availability and Shadow prices

If a resource constraint is binding in the optimal solution, the company is willing to pay up to some amount, the shadow price, to obtain more of the resource. This is because the objective improves by having more of the resource. However, there is typically a decreasing marginal effect: As the company owns more and more of the resource, the shadow price tends to decrease. This is usually because other constraints become binding, which causes extra units of this resource to be less useful (or not useful at all).

Fundamental Insight

The idea is that a constraint “costs” the company by keeping the objective from being better than it would be. A shadow price indicates how much the company would be will- ing to pay (in units of the objective) to “relax” a constraint. In this example, the company would be willing to pay $16 for each extra assembling hour. This is because such a change would increase the net profit by $16. But beyond a certain point—200 hours in this exam- ple—further relaxation of the constraint does no good, and the company is not willing to pay for any further increases.

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13-4 Sensitivity analysis 5 9 3

The constraint on testing hours is slightly different. It has a shadow price of zero. In fact, the shadow price for a nonbinding constraint is always zero, which makes sense. If the right side of this constraint is changed from 3000 to 3001, nothing at all happens to the optimal product mix or the objective value; there is just one more unused testing hour. However, the allowable decrease of 40 indicates that something does change when the right side reaches 2960. At this point, the constraint becomes binding—the number of testing hours used equals the number of testing hours available—and beyond this, the optimal product mix starts to change. By the way, the allowable increase for this constraint, shown as 11E30, means that it is essentially infinite. The right side of this con- straint can be increased above 3000 indefinitely, and nothing will change in the optimal solution.

the effect of Constraints on the Objective

If a constraint is added or an existing constraint becomes more constraining (for example, less of some resource is available), the objective can only get worse; it can never improve. The easiest way to understand this is to think of the feasi- ble region. When a constraint is added or an existing constraint becomes more constraining, the feasible region shrinks, so some solutions that were feasible before, maybe even the optimal solution, are no longer feasible. The opposite is true if a constraint is deleted or an existing constraint becomes less constrain- ing. In this case, the objective can only improve; it can never get worse. Again, the idea is that when a constraint is deleted or an existing constraint becomes less constraining, the feasible region expands. In this case, all solutions that were feasible before are still feasible, and there are some additional feasible solutions available.

Fundamental Insight

13-4b SolverTable Add-In The reason Solver’s sensitivity report makes sense for the product mix model is that the spreadsheet model is virtually a direct translation of a standard algebraic model. However, given the flexibility of spreadsheets, this is not always the case. We have seen many per- fectly good spreadsheet models—and have developed many ourselves—that are structured quite differently from their standard algebraic-model counterparts. In these cases, we have found Solver’s sensitivity report to be more confusing than useful. Therefore, Albright developed an Excel add-in called SolverTable. SolverTable allows you to ask sensitivity questions about any of the input variables, not just coefficients of the objective and right sides of constraints, and it provides straightforward answers.

The SolverTable add-in is on this textbook’s website.9 To install it, simply copy the SolverTable files to a folder on your hard drive. These files include the add-in itself (the .xlam file) and the online help file. To load SolverTable, you can proceed in either of two ways:

1. Open the SolverTable.xlam file just as you open any other Excel file. 2. Go to the add-ins list in Excel (click the File button, then Options, then Add-Ins, then Go)

and check the SolverTable item. If it isn’t in the list, Browse for the SolverTable.xlam file.

Solver’s sensitivity report is almost impossible to unravel for some models. In these cases SolverTable is pref- erable because of its easily interpreted results.

We haven’t been able to program a version of Solver Table for the Mac.

9 It is also available from the Free Downloads link on the authors’ website at www.kelley.iu.edu/albrightbooks.

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The advantage of the second option is that if SolverTable is checked in the add-ins list, it will open automatically every time you open Excel (and you can always uncheck it if you don’t want it to open automatically).

The SolverTable add-in was developed to mimic Excel’s built-in data table tool. Recall that data tables allow you to vary one or two inputs in a spreadsheet model and see instantaneously how selected outputs change. SolverTable is similar except that it runs Solver for every new input (or pair of inputs), and the current version also provides auto- matic charts of the results. There are two ways it can be used.

• One-way table. A one-way table means that there is a single input cell and any number of output cells. That is, there can be a single output cell or multiple output cells.

• Two-way table. A two-way table means that there are two input cells and one or more output cells. (You might recall that an Excel two-way data table allows only one output. SolverTable allows more than one. It creates a separate table for each selected output as a function of the two inputs.)

We illustrate some of the possibilities for the product mix example. Specifically, we check how sensitive the optimal production plan and net profit are to (1) changes in the selling price of XPs, (2) the number of labor hours of both types available, and (3) the maximum sales of the two models.

We assume that the model has been formulated and optimized, as shown in Figure 13.8, and that the SolverTable add-in has been loaded. To run SolverTable, click the Run SolverTable button on the SolverTable ribbon. You will be asked whether there is a Solver model on the active sheet. (Note that the active sheet at this point should be the sheet containing the model. If it isn’t, click Cancel and then activate this sheet.) You are then given the choice between a one-way or a two-way table. For the first sensitivity question, choose the one-way option. You will see the dialog box in Figure 13.10. For the sensitivity analysis on the XP selling price, fill it in as shown. Note that ranges can be entered as cell addresses or range names. Also, multiple ranges in the Outputs box must be separated by commas.

We chose the input range from 350 to 550 in increments of 25, but you can choose any desired range of input values.

Selecting Multiple Ranges

If you need to select multiple output ranges, the trick is to keep your finger on the Ctrl key as you drag the ranges. This automatically enters the separat- ing comma(s) for you. Actually, the same trick works for selecting multiple decision variable cell ranges in Solver’s dialog box. It even works for entering multiple range arguments in any Excel function.

Excel Tip

When you click OK, Solver solves a separate optimization problem for each of the nine rows of the table and then reports the requested outputs (number produced and net profit) in the table, as shown in Figure 13.11. It can take a while, depending on the speed of your computer and the complexity of the model, but everything is automatic. However, if you want to update this table—by using different XP selling prices in column A, for example—you must repeat the procedure. Note that if the requested outputs are included in named ranges, the range names are used in the SolverTable headings. For example, the label Number_to_produce_1 indicates that this output is the first cell in the Number_to_produce range. The label Total_profit indicates that this output is the only cell in the Total_profit range. If a requested output is not part of a named range, its cell address is used as the label in the SolverTable results.

On the Mac, keep your finger on the command key.

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13-4 Sensitivity analysis 5 9 5

The outputs in this table show that when the selling price of XPs is relatively low, the company should make as many Basics as it can sell and a few less XPs, but when the selling price is relatively high, the company should do the opposite. Also, the net profit increases steadily through this range. You can calculate these changes (which are not part of the SolverTable output) in column E. The increase in net profit per every extra $25 in XP selling price is close to, but not always exactly equal to, $30,000.

SolverTable also produces the chart in Figure 13.12. There is a dropdown list in cell K4 where you can choose any of the SolverTable outputs. (We selected the total profit, cell D25.) The chart then shows the data for that column from the table in Figure 13.11. Here there is a steady increase (slope about $30,000) in net profit as the XP selling price increases.

Figure 13.10 SolverTable One-Way Dialog Box

Figure 13.11 SolverTable Results for Varying XP Price

1 2

7 8 9

10 11 12 13

B C D E F G

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Selling Price XP (cell $C$11) values along side, output cell(s) along top

1166.667 1166.667 1166.667

1200 1200 1200 1200 1200 1200

$81,833 $111,000 $140,167

600 600 600 560 560 560 560 560 560

$169,600 $199,600 $229,600 $259,600 $289,600

$350 $375 $400 $425 $450 $475 $500 $525 $550 $319,600

Oneway analysis for Solver model in Model worksheet

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Figure 13.12 Associated SolverTable Chart for Total Profit

50000

100000

150000

200000

250000

300000

350000

$350 $375 $400 $425 $450 $475 $500 $525 $550 Selling Price XP ($C$11)

Sensitivity of Total_profit to Selling Price XP

81833.33

Data for chart

K L M N O P Q R

111000

140166.7

169600

199600

229600

259600

289600 319600

27 28

To ta

l_ pr

of it

When you select an output from the dropdown list in cell $K$4, the chart will adapt to that output.

The second sensitivity question asks you to vary two inputs, the two labor avail- abilities, simultaneously. This requires a two-way SolverTable, so you should fill in the SolverTable dialog box as shown in Figure 13.13. Here two inputs and two input ranges are specified, and multiple output cells are again allowed. An output table is generated for each of the output cells, as shown in Figure 13.14. For example, the top table shows how the optimal number of Basics varies as the two labor availabilities vary. Comparing the columns of this top table, it is apparent that the optimal number of Basics becomes increasingly sensitive to the available assembling hours as the number of available testing hours increases. The SolverTable output also includes two charts (not shown here) that let you graph any row or any column of any of these tables.

The third sensitivity question, involving maximum sales of the two models, reveals the flexibility of SolverTable. Instead of letting these two inputs vary independently in a two- way SolverTable, it is possible to let both of them vary according to a single percentage change. For example, if this percentage change is 10%, both maximum sales increase by 10%. The trick is to modify the model so that one percentage-change cell drives changes in both maximum sales. The modified model appears in Figure 13.15. Starting with the origi- nal model, enter the original values, 600 and 1200, in new cells, E18 and F18. (Do not copy the range B18:C18 to E18:F18. This would make the right side of the constraint E18:F18,

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13-4 Sensitivity analysis 5 9 7

Figure 13.14 Two-Way SolverTable Results

Figure 13.13 SolverTable Two-Way Dialog Box

A B C D E F G H I J 3 Assembling hours (cell $D$21) values along side, Testing hours (cell $D$22) values along top, output cell in corner

4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Number_to_produce_1 8000

2000 600 600 600 600 600 600 600 600 600

700 700 700 700 700 700 700 700 700

8500 9000 9500

10000 10500 11000 11500 12000

Number_to_produce_2 8000 8500 9000 9500

10000 10500 11000 11500 12000

Total_profit 8000 8500 9000 9500

10000 10500 11000 11500 12000

2500 3000 3500 4000 4500 5000

2000 2500 1125 1200

1200 1200 1200 1200 1200 1200 1200 1200

1200

1200 1200 1200 1200 1200 1200 1200 1200

1200

1200 1200 1200 1200 1200 1200 1200 1200

1200

1200 1200 1200 1200 1200 1200 1200 1200

1200

1200 1200 1200 1200 1200 1200 1200 12001000

950 950 950 950 950 950 950

3000 3500 4000 4500 5000

2000 2500 3000 3500 4000 4500 5000

250 160 160 160 160 160 500 260 260 260 260 260 600 360 360 360 360 360 600 460 460 460 460 460 600 560 560 560 560 560 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600

$138,300 $165,125 $167,600 $167,600 $138,300 $169,000 $175,600 $175,600 $138,300 $170,550 $183,600 $183,600 $138,300 $170,550 $191,600 $191,600 $138,300 $170,550 $199,600 $199,600 $138,300 $170,550 $202,800 $202,800 $138,300 $170,550 $202,800 $202,800 $138,300 $170,550 $202,800 $202,800 $138,300 $170,550 $202,800 $202,800

$167,600 $175,600 $183,600 $191,600 $199,600 $202,800 $202,800 $202,800 $202,800

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which is not the desired behavior.) Then enter any percentage change in cell G18. Finally, enter the formula

5E18*(1 1 $G$18)

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

HGFEDCBA Assembling and testing computers

Cost per labor hour assembling $11 $15

$150 $300

5 1

$225 $450

6 2

Labor hours for assembly Basic XP

Basic

Hours used 10000

2960 10000

3000

$44,800 $154,800 $199,600

<= <= Original values % change in both

<= <=

Basic XP Total

Hours available

Cost of component parts Selling price Unit margin

Assembling, testing plan (# of computers)

Number to produce

Maximum sales

Constraints (hours per month)

Labor hours for testing

Cost per labor hour testing

Labor availability for assembling Labor availability for testing

Net profit ($ this month)

Inputs for assembling and testing a computer

$129

1200

$80

560

1200600 1200 0%600

The trick here is to let the single value in cell G18 drive both values in cells B18 and C18 from their original values.

Figure 13.15 Modified Model for Simultaneous Changes

in cell B18 and copy it to cell C18. Now a one-way SolverTable can be used with the percent- age change in cell G18 to drive two different inputs simultaneously. The SolverTable dia- log box should be set up as in Figure 13.16, with the corresponding results in Figure 13.17.

Figure 13.16 SolverTable One- Way Dialog Box

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13-4 Sensitivity analysis 5 9 9

Figure 13.17 Sensitivity to Percentage Change in Maximum Sales

4 5 6 7 8 9

10 11

A B C D E F G

% change in max sales (cell $G$18) values along side, output cell(s) along top

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$B $1

–30% –20% –10%

420 840 $141,960 $80 480 960 $162,240 $80 540 1080 $182,520 $80

0% 560 1200 $199,600 $80 10% 500 1250 $201,250 $80 20% 500 1250 $201,250 $80 30% 500 1250 $201,250 $80

Oneway analysis for Solver model in Modified Model worksheet

You should always scan these sensitivity results to see if they make sense. For exam- ple, if the company can sell 20% or 30% more of both models, it makes no more profit than if it can sell only 10% more. The reason is labor availability. By this point, there isn’t enough labor to produce the increased demand.

It is always possible to run a sensitivity analysis by changing inputs manually in the spreadsheet model and rerunning Solver. The advantages of SolverTable are that it enables you to perform a systematic sensitivity analysis for any selected inputs and outputs, and it keeps track of the results in a table and associated chart(s). You will see other applications of this useful add-in later in this chapter and in the next chapter.

13-4c A Comparison of Solver’s Sensitivity Report and SolverTable Sensitivity analysis in optimization models is extremely important, so it is important that you understand the pros and cons of the two tools in this section. Here are some points to keep in mind.

• Solver’s sensitivity report focuses only on the coefficients of the objective and the right sides of the constraints. SolverTable allows you to vary any of the inputs.

• Solver’s sensitivity report provides useful information through its reduced costs, shadow prices, and allowable increases and decreases. This same information can be obtained with SolverTable, but it requires a little more work and some experimentation with the appropriate input ranges.

• Solver’s sensitivity report is based on changing only one objective coefficient or one right side at a time. This one-at-a-time restriction prevents you from answering some questions directly. SolverTable is more flexible in this respect.

• Solver’s sensitivity report is based on a well-established mathematical theory of sensitivity analysis in linear programming. If you lack this mathematical background— as many users do—the outputs can be difficult to understand, especially for somewhat “nonstandard” spreadsheet formulations. In contrast, SolverTable’s outputs are straightforward. You can vary one or two inputs and see directly how the optimal solution changes.

• Solver’s sensitivity report is not even available for integer-constrained models, and its interpretation for nonlinear models is more difficult than for linear models. SolverTable’s results have the same interpretation for any type of optimization model.

• Solver’s sensitivity report comes with Excel. SolverTable is a separate add-in that is not included with Excel—but it is included with this book and is freely available from the Free Downloads link at the authors’ website, www.kelley.iu.edu/albrightbooks.

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In summary, each of these tools can be used to answer certain questions. We tend to favor SolverTable because of its flexibility, but in the optimization examples in this chap- ter and the next chapter, we will illustrate both tools.

13-5 Properties of Linear Models Linear programming is an important subset of a larger class of models called mathemati- cal programming models.10 All such models select the levels of various activities that can be performed, subject to a set of constraints, to maximize or minimize an objective such as total profit or total cost. In PC Tech’s product mix example, the activities are the numbers of PCs to produce, and the purpose of the model is to find the levels of these activities that maximize the total net profit subject to specified constraints.

In terms of this general setup—selecting the optimal levels of activities—there are three important properties that LP models possess that distinguish them from general mathematical programming models: proportionality, additivity, and divisibility.

Proportionality means that if the level of any activity is multiplied by a constant fac- tor, the contribution of this activity to the objective, or to any of the constraints in which the activity is involved, is multiplied by the same factor. For example, suppose the produc- tion of Basics is cut from its optimal value of 560 to 280—that is, it is multiplied by 0.5. Then the amounts of labor hours from assembling and from testing Basics are both cut in half, and the net profit contributed by Basics is also cut in half.

Proportionality is a perfectly valid assumption in the product mix model, but it is often violated in certain types of models. For example, in various blending models used by petroleum companies, chemical outputs vary in a nonlinear manner as chemical inputs are varied. For example, if a chemical input is doubled, the resulting chemical output is not necessarily doubled. This type of behavior violates the proportionality property, and it requires nonlinear optimization, which we discuss in the next chapter.

The additivity property implies that the sum of the contributions from the various activities to any constraint equals the total contribution to that constraint. For example, if the two PC models use, respectively, 560 and 2400 testing hours (as in Figure 13.8), then the total number used in the plan is the sum of these amounts, 2960 hours. Similarly, the additivity property applies to the objective. That is, the value of the objective is the sum of the contributions from the various activities. In the product mix model, the net profits from the two PC models sum to the total net profit. The additivity property implies that the contribution of any decision variable to the objective or to any constraint is independent of the levels of the other decision variables.

The divisibility property simply means that both integer and noninteger levels of the activities are allowed. In the product mix model, we got integer values in the opti- mal solution, 560 and 1200, just by luck. For slightly different inputs, they could eas- ily have been fractional values. In general, if you want the levels of some activities to be integer values, there are two possible approaches: (1) You can solve the LP model without integer constraints, and if the solution turns out to have fractional values, you can attempt to round them to integer values; or (2) you can explicitly constrain certain decision variables to have integer values. However, the latter approach requires integer programming, which we discuss in the next chapter. At this point, we simply state that in general, integer-constrained problems are much more difficult to solve than problems without integer constraints.

So far, the discussion of these three properties, especially proportionality and additiv- ity, is fairly abstract. How can you recognize whether a model satisfies proportionality and

10 The word programming in linear programming or mathematical programming has nothing to do with com- puter programming. It originated with the British term programme, which is essentially a plan or a schedule of operations.

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13-5 properties of Linear Models 6 0 1

additivity? This is easy if the model is described algebraically. In this case the objective must be of the form

a1x1 1 a2x2 1 g1 anxn

where n is the number of decision variables, the a’s are constants, and the x’s are decision variables. This expression is called a linear combination of the x’s. Also, each constraint must be equivalent to a form where the left side is a linear combination of the x’s and the right side is a constant. For example, the following is a typical linear constraint:

3x1 1 7x2 2 2x3 # 50

It is not quite so easy to recognize proportionality and additivity—or the lack of them—in a spreadsheet model, because the logic of the model is typically embedded in a series of cell formulas. However, the ideas are the same. First, the objective cell must ultimately (possibly through a series of formulas in intervening cells) be a sum of products of constants and decision variable cells, where a “constant” means that it does not depend on decision variable cells. Second, each side of each constraint must ultimately be either a constant or a sum of products of constants and decision variable cells. This explains why linear models contain so many SUM and SUMPRODUCT functions.

It is usually easier to recognize when a model is not linear. Two particular situations that lead to nonlinear models are when (1) there are products or quotients of expressions involving decision variable cells or (2) there are nonlinear functions, such as squares, square roots, or logarithms, that involve decision variable cells. These are typically easy to spot, and they guarantee that the model is nonlinear.

Whenever you model a real problem, you usually make some simplifying assump- tions. This is certainly the case with LP models. The world is frequently not linear, which means that an entirely realistic model typically violates some or all of the three properties in this section. However, numerous successful applications of LP have demonstrated the usefulness of linear models, even if they are only approximations of reality. If you suspect that the violations are serious enough to invalidate a linear model, you should use an inte- ger or nonlinear model, as we illustrate in the next chapter.

In terms of Excel’s Solver, if the model is linear—that is, if it satisfies the propor- tionality, additivity, and divisibility properties—you should check the Simplex LP option. Then Solver uses the simplex method, a very efficient method for a linear model, to solve the problem. Actually, you can check the Simplex LP option even if the divisibility prop- erty is violated—that is, for linear models with integer-constrained variables—but Solver then embeds the simplex method in a more complex algorithm in its solution procedure.

Linear Models and Scaling11

In some cases you might be sure that a model is linear, but when you check the Simplex LP option and then solve, you get a Solver message to the effect that the conditions for linearity are not satisfied. This can indicate a logical error in your formulation, so that the propor- tionality and additivity conditions are indeed not satisfied. However, it can also indicate that Solver erroneously thinks the linearity conditions are not satisfied, which is typically due to roundoff error in its calculations—not any error on your part. If the latter occurs and you are convinced that the model is correct, you can try not using the simplex method to see whether that works. If it does not, you should consult your instructor. It is possible that the non- simplex algorithm employed by Solver simply cannot find the solution to your problem.

In any case, it always helps to have a well-scaled model. In a well-scaled model, all numbers are roughly the same magnitude. If the model contains some very large num- bers—100,000 or more, say—and some very small numbers—0.001 or less, say—it is poorly scaled for the methods used by Solver, and roundoff error is more likely to be an issue, not only in Solver’s test for linearity conditions but in all its algorithms.

Real-life problems are almost never exactly linear. However, linear approxima- tions often yield very useful results.

11 This section might seem overly technical. However, when you develop a model that you are sure is linear and Solver then tells you it doesn’t satisfy the linear conditions, you will appreciate this section.

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If you believe your model is poorly scaled, there are three possible remedies. The first is to check the Use Automatic Scaling option in Solver. (It is found by clicking on the Options button in the main Solver dialog box.) This might help and it might not; we have had mixed success. (Frontline Systems, the company that develops Solver, has told us that the only drawback to checking this box is that the solution procedure can be slower.) The second option is to redefine the units in which the various quantities are defined. Finally, you can change the Precision setting in Solver’s Options dialog box to a larger number, such 0.00001 or 0.0001. (The default has five zeros.)

You can decrease the chance of getting an incorrect “Conditions for Assume Linear Model are not satisfied” message by changing Solver’s Precision setting.

Rescaling a Model

Suppose you have a range of input values expressed, say, in dollars, and you would like to reexpress them in thousands of dollars, that is, you would like to divide each value by 1000. There is a simple copy/paste way to do this. Enter the value 1000 in some unused cell and copy it. Then select the range you want to rescale, and from the Paste dropdown menu, select Paste Special and then the Divide option. No formulas are required; your original values are automatically rescaled (and you can then delete the 1000 cell). You can use this same method to add, subtract, or multiply by a constant.

Excel Tip

13-6 Infeasibility and Unboundedness In this section we discuss two things that can go wrong when you invoke Solver. Both of these might indicate that there is a mistake in the model. Therefore, because mistakes are common in LP models, you should be aware of the error messages you might encounter.

The first problem is infeasibility. Recall that a solution is feasible if it satisfies all the constraints. Among all the feasible solutions, you are looking for the one that optimizes the objective. However, it is possible that there are no feasible solutions to the model. There are generally two reasons for this: (1) there is a mistake in the model (an input was entered incorrectly, such as a # symbol instead of $) or (2) the problem has been so constrained that there are no solutions left. In the former case, a careful check of the model should find the error. In the latter case, you might need to change, or even eliminate, some of the constraints.

To show how an infeasible problem could occur, suppose in PC Tech’s product mix problem you change the maximum sales constraints to minimum sales constraints (and leave everything else unchanged). That is, you change these constraints from # to $. If Solver is then used, the message in Figure 13.18 appears, indicating that Solver cannot find a feasible solution. The reason is clear: There is no way, given the constraints on labor hours, that the company can produce these minimum sales values. The company’s only choice is to set at least one of the minimum sales values lower. In general, there is no fool- proof way to remedy the problem when a “no feasible solution” message appears. Careful checking and rethinking are required.

A second problem is unboundedness. In this case, the model has been formulated in such a way that the objective is unbounded—that is, it can be made as large (or as small, for minimization problems) as you like. If this occurs, you have probably entered a wrong input or forgotten some constraints. To see how this could occur in the product mix problem, sup- pose that you change all constraints to be # instead of $. Now there is no upper bound on available labor hours or the number of PCs the company can sell. If you make these changes in the model and then use Solver, the message in Figure 13.19 appears, stating that the objective cell does not converge. In other words, the total net profit can grow without bound.

Infeasibility and Unboundedness

A perfectly reasonable model can have no feasible solutions because of too many constraints.

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13-6 Infeasibility and Unboundedness 6 0 3

Figure 13.18 No Feasible Solution Message

Figure 13.19 Unbounded Solution Message

Infeasibility and unboundedness are quite different in a practical sense. It is quite possible for a reasonable model to have no feasible solutions. For example, the market- ing department might impose several constraints, the production department might add some more, the engineering department might add even more, and so on. Together, they might constrain the problem so much that there are no feasible solutions left. The only way out is to change or eliminate some of the constraints. An unboundedness problem is quite different. There is no way a realistic model can have an unbounded solution. If you get the message shown in Figure 13.19, then you must have made a mistake: You entered an input incorrectly, you omitted one or more constraints, or there is a logical error in your model.

Except in very rare situ- ations, if Solver informs you that your model is unbounded, you have made an error.

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Problems Solutions for problems whose numbers appear within a colored box can be found in the Student Solution Files.

Level A 1. Other sensitivity analyses besides those discussed could

be performed on the PC Tech product mix model. Use SolverTable to perform each of the following. In each case keep track of the values in the decision variable cells and the objective cell, and discuss your findings. a. Let the selling price for Basics vary from $220 to

$350 in increments of $10. b. Let the labor cost per hour for assembling vary from

$5 to $20 in increments of $1. c. Let the labor hours for testing a Basic vary from 0.5 to

3.0 in increments of 0.5. d. Let the labor hours for assembling and testing an XP

vary independently, the first from 4.5 to 8.0 and the second from 1.5 to 3.0, both in increments of 0.5.

2. In PC Tech’s product mix model, assume there is another PC model, the VXP, that the company can produce in addition to Basics and XPs. Each VXP requires eight hours for assembling, three hours for testing, $275 for component parts, and sells for $560. At most 50 VXPs can be sold. a. Modify the spreadsheet model to include this

new product, and use Solver to find the optimal prod- uct mix.

b. You should find that the optimal solution is not inte- ger-valued. If you round the values in the decision variable cells to the nearest integers, is the resulting solution still feasible? If not, how might you obtain a feasible solution that is at least close to optimal?

3. Continuing the previous problem, perform a sensitivity analysis on the selling price of VXPs. Let this price vary from $500 to $650 in increments of $10, and keep track of the values in the decision variable cells and the objec- tive cell. Discuss your findings.

4. Again continuing problem 2, suppose that you want to force the optimal solution to be integers. Do this in Solver by adding a new constraint. Select the decision variable cells for the left side of the constraint, and in the middle dropdown list, select the “int” option (for “integer”). How does the optimal integer solution com- pare to the optimal noninteger solution in problem 2? Are the decision variable cell values rounded versions of

those in problem 2? Is the objective value more or less than in problem 2?

5. If all inputs in PC Tech’s product mix model are non- negative (as they should be for any realistic version of the problem), are there any input values such that the resulting model has no feasible solutions? (Refer to the graphical solution.)

6. There are five corner points in the feasible region for the PC Tech product mix model. We identified the coordi- nates of one of them: (560, 1200). Identify the coordi- nates of the others. a. Only one of these other corner points has positive

values for both decision variable cells. Discuss the changes in the selling prices of either or both mod- els that would be necessary to make this corner point optimal.

b. Two of the other corner points have one decision vari- able cell value positive and the other zero. Discuss the changes in the selling prices of either or both models that would be necessary to make either of these corner points optimal.

Level B 7. Using the graphical solution of the PC Tech product mix

model as a guide, suppose there are only 2800 testing hours available. How do the answers to the previous problem change? (Is the previous solution still optimal? Is it still feasible?)

8. Again continuing problem 2, perform a sensitivity analysis where the selling prices of Basics and XPs simultaneously change by the same percentage, but the selling price of VXPs remains at its original value. Let the percentage change vary from 225% to 50% in increments of 5%, and keep track of the values in the decision variable cells and the total profit. Discuss your findings.

9. Consider the graphical solution to the PC Tech product mix model. Now imagine that another constraint—any constraint—is added. Which of the following three things are possible: (1) the feasible region shrinks; (2) the feasible region stays the same; (3) the feasible region expands? Which of the following three things are possible: (1) the optimal value in objective cell decreases; (2) the optimal value in objective cell stays the same; (3) the optimal value in objective cell increases? Explain your answers. Do they hold just for this particular model, or do they hold in general?

13-7 A Larger Product Mix Model The problem we examine in this section is a direct extension of the product mix model in the previous section. There are two modifications. First, the company makes eight com- puter models, not just two. Second, testing can be done on either of two lines, and these two lines have different characteristics.

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13-7 a Larger product Mix Model 6 0 5

EXAMPLE

13.2 PRODUCING COMPUTERS AT PC TECH WITH TWO TESTING LINES As in the previous example, PC Tech must decide how many of each of its computer models to assemble and test, but there are now eight available models, not just two. Each computer must be assembled and then tested, but there are now two lines for testing. The first line tends to test faster, but its labor costs are slightly higher, and each line has a certain number of hours available for testing. Any computer can be tested on either line. The inputs for the model are same as before: (1) the hourly labor costs for assembling and testing, (2) the required labor hours for assembling and testing any computer model, (3) the cost of component parts for each model, (4) the selling prices for each model, (5) the maximum sales for each model, and (6) labor availabilities. These input values are listed in the file Product Mix 2.xlsx. As before, the company wants to determine the product mix that maximizes its total net profit.

Objective To use LP to find the mix of computer models that maximizes total net profit and stays within the labor hour availability and maximum sales constraints.

Where Do the Numbers Come From? The same comments as in Example 13.1 apply here.

Solution The diagram shown in Figure 13.20 (see the file Product Mix 2 Big Picture.xlsx) is similar to the one for the smaller product mix model, but it is not the same. You must choose the number of computers of each model to produce on each line, the sum of which cannot be larger than the maximum that can be sold. This choice determines the labor hours of each type used and all revenues and costs. No more labor hours can be used than are available.

Labor hours per unit

Labor hours used

Labor hours available

Cost per labor hour

Selling price

Maximum sales

Maximize profit

Total computers produced

Number computers tested on each line

Cost of component parts

Unit margin

Figure 13.20 Big Picture for Larger Product Mix Model

It might not be immediately obvious what the decision variable cells are for this model (at least not before you look at Figure 13.20). You might think that the company simply needs to decide how many computers of each model to produce. How- ever, because of the two testing lines, this is not enough information. The company must also decide how many of each model to test on each line. For example, suppose they decide to test 100 model 4’s on line 1 and 300 model 4’s on line 2. This means they will need to assemble (and ultimately sell) 400 model 4’s. In other words, given the detailed plan of how many to test on each line, everything else is determined. But without the detailed plan, there is not enough information to complete the model. This is the type of reasoning you must use to determine the appropriate decision variables for any LP model.

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Algebraic Model We will not spell out the algebraic model for this expanded version of the product mix model because it is so similar to the two-variable product mix model. However, we will say that it is larger, and hence probably more intimidating. Now we need decision variables of the form xij, the number of model j computers to test on line i, and the total net profit and each labor availability constraint will include a long SUMPRODUCT formula involving these variables. Instead of focusing on these algebraic expressions, we turn directly to the spreadsheet model.

Developing the Spreadsheet Model The spreadsheet in Figure 13.21 illustrates the solution procedure for PC Tech’s product mix prob- lem. (See the file Product Mix 2 Finished.xlsx.) The first stage is to develop the spreadsheet model step by step.

1. Inputs. Enter the various inputs in the blue ranges. Again, remember that our convention is to color all input cells blue. Enter only numbers, not formulas, in input cells. They should always be numbers directly from the problem statement. (In this case, we supplied them in the spreadsheet template.)

2. Range names. Name the ranges indicated. According to our convention, there are enough named ranges so that the Solver dialog box contains only range names, no cell addresses. Of course, you can name additional ranges if you like. (You can again use the range-naming shortcut explained in the Excel tip for the previous example. That is, you can take advantage of labels in adjacent cells, except for the Profit cell.)

3. Unit margins. Note that two rows of these are required, one for each testing line, because the costs of testing on the two lines are not equal. To calculate them, enter the formula

5B$13-$B$3*B$9-$B4*B10-B$12

in cell B14 and copy it to the range B14:I15.

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Developing the Product Mix 2 Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

JIHGFEDCBA Assembling and testing computers

Cost per labor hour assembling $11 Cost per labor hour testing, line 1 Cost per labor hour testing, line 2

$19 $17

Inputs for assembling and testing a computer Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

Labor hours for assembly Labor hours for tes�ng, line 1 Labor hours for tes�ng, line 2

5 5.5 5.5 5.5 6 1.5

4 5 5 2 2 2 2.5 2.5 2.5 3

2 2.5 2.5 2.5 3 3 3.5 3.5 Cost of component parts $150 $225 $225 $225 $250 $250 $250 $300 Selling price $600$530$525$500$470$460$450$350 Unit margin, tested on line 1 Unit margin, tested on line 2

$128 $132 $142 $152 $142 $167 $172 $177 $122 $128 $138 $148 $139 $164 $160 $175

50000 1250

0 0

Number tested on line 1 Number tested on line 2

1000 800 0 0 0 0

Total computers produced 0 0 0 1250 0 500 1000 800 <= <= <= <= <= <= <= <=

Maximum sales 1250 1000 1000 1000 800125012501500

Constraints (hours per month) Hours used Hours available Labor availability for assembling 19300 <= 20000 Labor availability for tes�ng, line 1 Labor availability for tes�ng, line 2

6150 <= 5000 3125 <= 6000

Net profit ($ per month) Totals $0$0$0$0$0Tested on line 1

Tested on line 2 $172,000$83,500

$184,375 $141,600 $397,100

$184,375 $581,475

$0$0$0$0$0$0$0

Assembling, testing plan (# of computers)

Figure 13.21 Larger Product Mix Model with Infeasible Solution

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4. Decision variable cells. As discussed above, the decision variable cells are the red cells in rows 19 and 20. You do not have to enter the values shown in Figure 13.21. You can use any trial values initially; Solver will eventually find the optimal values. Note that the four values shown in Figure 13.21 cannot be optimal because they do not satisfy all the constraints. Specifically, this plan uses more labor hours for assembling than are available. However, you do not need to worry about satisfying constraints at this point; Solver will take care of this later.

5. Labor used. Enter the formula

5SUMPRODUCT(B9:E9,Total_computers_produced)

in cell B26 to calculate the number of assembling hours used. Similarly, enter the formulas

5SUMPRODUCT(B10:I10,Number_tested_on_line_1)

and

5SUMPRODUCT(B11:I11,Number_tested_on_line_2)

in cells B27 and B28 for the labor hours used on each testing line.

Copying Formulas with Range Names

When you enter a range name in an Excel formula and then copy the formula, the range name reference acts like an absolute reference. Therefore, it wouldn’t work to copy the formula in cell B27 to cell B28. However, this would work if range names hadn’t been used. This is one potential disadvantage of range names that you should be aware of.

Excel Tip

6. Revenues, costs, and profits. The area from row 30 down shows the summary of monetary values. Actually, only the total profit in cell J33 is needed, but it is also useful to calculate the net profit from each computer model on each testing line. To obtain these, enter the formula

5B14*B19

in cell B31 and copy it to the range B31:I32. Then sum these to obtain the totals in column J. The total in cell J33 is the objective to maximize.

Experimenting with Other Solutions Before going any further, you might want to experiment with other values in the decision variable cells. However, it is a real challenge to guess the optimal solution. It is tempting to fill up the decision variable cells corresponding to the largest unit margins. However, this totally ignores their use of the scarce labor hours. If you can guess the optimal solution to this model, you are better than we are!

Using Solver The Solver dialog box should be filled out as shown in Figure 13.22. (Again, note that there are enough named ranges so that only range names appear in this dialog box.) Except that this model has two rows of decision variable cells, the Solver setup is identical to the one in Example 13.1.

Discussion of the Solution When you click Solve, you obtain the optimal solution shown in Figure 13.23. The optimal plan is to produce computer models 1, 4, 6, and 7 only, some on testing line 1 and others on testing line 2. This plan uses all the available labor hours for assembling and testing on line 1, but about 1800 of the testing line 2 hours are not used. Also, maximum sales are achieved only for computer models 1, 6, and 7. This is typical of an LP solution. Some of the constraints are met exactly— they are binding—whereas others contain a certain amount of “slack.” The binding constraints prevent PC Tech from earning an even higher profit.

You typically gain insights into a solution by checking which constraints are binding.

13-7 a Larger product Mix Model 6 0 7

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Figure 13.22 Solver Dialog Box

Roundoff Error

Because of the way numbers are stored and calculated on a computer, the optimal values in the decision variable cells and elsewhere can contain small roundoff errors. For example, the value that really appears in cell E20 on one of our PCs is 475.000002015897, not exactly 475. For all practical purposes, this number can be treated as 475, and we have formatted it as such in the spreadsheet. (It appears that roundoff in Solver results are less of a problem in recent versions of Excel.)

Excel Tip

Sensitivity Analysis If you want to experiment with different inputs to this problem, you can simply change the inputs and then rerun Solver. The second time you use Solver, you do not have to specify the objective and decision variable cells or the constraints. Excel remembers these settings and saves them when you save the file.

You can also use SolverTable to perform a more systematic sensitivity analysis on one or more input variables. One possibility appears in Figure 13.24, where the number of available assembling labor hours is allowed to vary from 18,000 to 25,000 in increments of 1000, and the numbers of computers produced and profit are designated as outputs. There are several ways to interpret the output from this sensitivity analysis. First, you can look at columns B through I to see how the product mix changes as more assembling labor

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hours become available. For assembling labor hours from 18,000 to 23,000, the only thing that changes is that more model 4’s are produced. Beyond 23,000, however, the company starts to produce model 3’s and produces fewer model 7’s. Second, you can see how extra labor hours add to the total profit. Note exactly what this increased profit means. For example, when labor hours increase from 20,000 to 21,000, the model requires that the company must pay $11 apiece for these extra hours (if it uses them). But the net effect is that profit increases by $29,500, or $29.50 per extra hour. In other words, the labor cost increases by $11,000 3=$11(1000)4, but this is more than offset by the increase in revenue that comes from having the extra labor hours.

As column J illustrates, it is worthwhile for the company to obtain extra assembling labor hours, even though it has to pay for them, because its profit increases. However, the increase in profit per extra labor hour—the shadow price of assembling labor hours—is not constant. In the top part of the table, it is $29.50 (per extra hour), but it then decreases to $20.44 and then $2.42. The accompanying SolverTable chart of column J illustrates this decreasing shadow price through its decreasing slope.

Figure 13.23 Optimal Solution to Larger Product Mix Model

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Assembling, testing plan (# of computers) Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

Number tested on line 1 Number tested on line 2

1500 0 0 125 0 0 0 000475000

Total computers produced 1500 0 0 600 0 0 <= <= <= <= <= <= <= <=

80010001000

1000

1000 1000

10001250125012501500Maximum sales

Constraints (hours per month) Hours used Hours available Labor availability for assembling 20000 <= 20000 Labor availability for testing, line 1 Labor availability for testing, line 2

5000 <= 5000 4187.5 <= 6000

Net profit ($ per month) Totals $191,250

$0 $0 $0

$0 $0

$0 $0 $0

$0 $0$0 $163,500

$172,000Tested on line 1 Tested on line 2

$19,000 $382,250 $70,063 $233,563

$615,813

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

JIHGFEDCBA Assembling and testing computers

Cost per labor hour assembling $11 Cost per labor hour testing, line 1 Cost per labor hour testing, line 2

$19 $17

Inputs for assembling and testing a computer Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

Labor hours for assembly Labor hours for testing, line 1 Labor hours for testing, line 2

5 5.5 5.5 5.5 6 1.5

4 5 5 2 2 2 2.5 2.5 2.5 3

2 2.5 2.5 2.5 3 3 3.5 3.5 Cost of component parts $150 $225 $225 $225 $250 $250 $250 $300 Selling price $600$530$525$500$470$460$450$350 Unit margin, tested on line 1 Unit margin, tested on line 2

$128 $132 $142 $152 $142 $167 $172 $177 $122 $128 $138 $148 $139 $164 $160 $175

Charts and Roundoff

As SolverTable performs its Solver runs, it reports and then charts the values found by Solver. These can include small roundoff errors and slightly misleading charts. For example, the chart in Figure 13.25 shows one pos- sibility, where we varied the cost of testing on line 2 and charted the assembling hours used. Throughout the range, this output value was 20,000, but because of slight roundoff in two of the cells (19999.9999999292 and 20000.0000003259 on our PC), the chart doesn’t appear to be flat. If you see this behavior, you can change the chart manually by modifying its vertical scale.

SolverTable Technical Tip

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Figure 13.24 Sensitivity to Assembling Labor Hours

1 2

4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

A B C D E F G H I J

Assembling labor (cell $D$26) values along side, output cell(s) along top

To ta

l_ co

m pu

te rs

_p ro

du ce

d_ 1

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te rs

_p ro

du ce

d_ 2

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te rs

_p ro

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d_ 3

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m pu

te rs

_p ro

du ce

d_ 4

To ta

l_ co

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te rs

_p ro

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d_ 5

To ta

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_p ro

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d_ 6

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of it

18000 1500 0 0 200 0 1000 0 $556,813 19000 1500 0 0 400 0 1000 0 $586,313 20000 1500 0 0 600 0 1000 0 $615,813 21000 1500 0 0 800 0 1000 0 $645,313 22000 1500 0 0 1000 0 1000 0 $674,813 23000 1500 0 0 1200 0 1000 0 $704,313 24000 1500 0 700 1250 0 1000 0 $724,750 25000 1500 0 1250 1250 0 1000

1000 1000 1000 1000 1000 1000

500 60 0 $727,170

Oneway analysis for Solver model in Model worksheet

800000

700000

600000

500000

400000

300000

200000

100000

0 18000 19000 20000 21000 22000 23000 24000 25000

Assembling labor ($D$26)

Sensitivity of Total_profit to Assembling labor

20000

20000 $10 $11 $12 $13 $14 $15 $16 $17 $18 $19 $20 $21 $22 $23 $24 $25

Testing cost 2 ($B$5)

Sensitivity of Hours_used_1 to Testing cost 2Figure 13.25 A Misleading SolverTable Chart

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6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

HGFEDCBA Variable Cells

Final Value

Reduced Cost

Allowable Increase

Allowable DecreaseNameCell

1500 0 127.5 1E+30 2.125 0 –20 0 –10

132 20 1E+30 142 10 1E+30

125 0 152 2.833333333 1.7 0 –25.875 142 25.875 1E+30 0 –2.125

0 –6.75 0 –2.125 0 –20 0 –10

0 –23.75

0 –6.375

$B$19 Number tested on line 1 Model 1 $C$19 $D$19

0 –2.5

167 2.125 1E+30 1000 0 172 1E+30 4.125

177 6.75 1E+30 122 2.125 1E+30

127.5 20 1E+30 137.5 10 1E+30

475 0 147.5 1.136363636 2.083333333 138.5 23.75 1E+30

1000 0 163.5 1E+30 1.25 160 6.375 1E+30

174.5 2.5 1E+30

Constraints Final Value

Shadow Price

Constraint R.H. Side

Allowable Increase

Allowable DecreaseNameCell

20000 29.5 20000 3250 2375

4187.5 0 6000 1E+30 1812.5 5000 2.25 5000 950 250

1500 6.125 1500 166.6666667 812.5 1E+30 1E+30 1E+30 1E+30

1E+30

1250 1250 1250

00 1250 1250

00 600 0 650

10001000 1000 1000

00 1000 1.25 431.8181818 590.9090909 1000 4.125 100 590.9090909

$B$26 Labor availability for assembling Hours used $B$27 Labor availability for testing, line 1 Hours used

Labor availability for testing, line 2 Hours used$B$28 $B$21 Total computers produced Model 1

Total computers produced Model 2 Total computers produced Model 3 Total computers produced Model 4 Total computers produced Model 5 Total computers produced Model 6 Total computers produced Model 7 Total computers produced Model 8

$C$21 $D$21 $E$21 $F$21 $G$21 $H$21 $I$21 80080000

Objective Coefficient

$E$19 $F$19 $G$19 $H$19 $I$19 $B$20 $C$20 $D$20 $E$20 $F$20 $G$20 $H$20 $I$20

Number tested on line 1 Model 2 Number tested on line 1 Model 3 Number tested on line 1 Model 4 Number tested on line 1 Model 5 Number tested on line 1 Model 6 Number tested on line 1 Model 7 Number tested on line 1 Model 8 Number tested on line 2 Model 1 Number tested on line 2 Model 2 Number tested on line 2 Model 3 Number tested on line 2 Model 4 Number tested on line 2 Model 5 Number tested on line 2 Model 6 Number tested on line 2 Model 7 Number tested on line 2 Model 8

Figure 13.26 Solver’s Sensitivity Report

Finally, you can gain additional insight from Solver’s sensitivity report, shown in Figure 13.26. However, you have to be careful in interpreting this report. Unlike Example 13.1, there are no upper bound (maximum sales) constraints on the decision variable cells. The maximum sales constraints are on the total computers produced (row 21 of the model), not the decision variable cells. Therefore, the only nonzero reduced costs in the top part of the table are for decision variable cells currently at zero (not those at their upper bounds as in the previous example). Each nonzero reduced cost indicates how much the profit margin for this activity would have to change before this activity would be profitable. Also, there is a row in the bottom part of the table for each constraint, including the maximum sales constraints. The interesting values are again the shadow prices. The first two indicate the amount the company would pay for an extra assembling or line 1 testing labor hour. (Does the 29.5 value look familiar? Compare it to the SolverTable results above.) The shadow prices for all binding maximum sales constraints indi- cate how much more profit the company could make if it could increase its demand by one computer of that model.

The information in this sensitivity report is all relevant and definitely provides some insights if studied carefully. However, this really requires you to know the rules Solver uses to create this report. That is, it requires a fairly in-depth knowledge of the theory behind LP sensitivity analysis, more than we have provided here. Fortunately, we believe the same basic information— and more—can be obtained in a more intuitive way by creating appropriate SolverTable reports.

13-7 a Larger product Mix Model 6 1 1

Problems

Level A Solutions for problems whose numbers appear within a colored box can be found in the Student Solution Files.

10. Modify PC Tech’s two-line product mix model so that there is no maximum sales constraint. (This is easy to

do in the Solver dialog box. Just highlight the constraint and click on the Delete button.) Does this make the problem unbounded? Does it change the optimal solu- tion at all? Explain its effect.

11. In PC Tech’s two-line product mix model it makes sense to change the maximum sales constraint to a “minimum sales” constraint, simply by changing the direction of the inequality. Then the input values in row 23 can be con- sidered customer demands that must be met. Make this

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change and rerun Solver. What do you find? What do you find if you run Solver again, this time making the values in row 23 one-quarter of their current values?

12. In PC Tech’s two-line product mix model, use Solver Table to run a sensitivity analysis on the cost per assembling labor hour, letting it vary from $5 to $20 in increments of $1. Keep track of the computers produced in row 21, the hours used in the range B26:B28, and the total profit. Discuss your findings. Are they intuitively what you expected?

13. Create a two-way SolverTable for PC Tech’s two-line product mix model, where total profit is the only output and the two inputs are the testing line 1 hours and testing line 2 hours available. Let the former vary from 4000 to 6000 in increments of 500, and let the latter vary from 3000 to 5000 in increments of 500. Discuss the changes in profit you see as you look across the various rows of the table. Discuss the changes in profit you see as you look down the various columns of the table.

14. In PC Tech’s two-line product mix model, model 8 has fairly high profit margins, but it isn’t included at all in the optimal mix. Use SolverTable, along with some experimentation on the correct range, to find the (approximate) selling price required for model 8 before it enters the optimal product mix.

Level B 15. Suppose you want to increase all three resource avail-

abilities in PC Tech’s two-line product mix model

simultaneously by the same percentage. You want this percentage to vary from 225% to 50% in increments of 5%. Modify the spreadsheet model slightly so that this sensitivity analysis can be performed with a one-way SolverTable, using the percentage change as the single input. Keep track of the computers produced in row 21, the hours used in the range B26:B28, and the total profit. Discuss the results.

16. Some analysts complain that spreadsheet models are difficult to resize. You can be the judge of this. Sup- pose the current PC Tech two-line product mix problem is changed so that there is an extra resource, packaging labor hours, and two additional PC models, 9 and 10. What additional input data are required? What modifi- cations are necessary in the spreadsheet model (includ- ing range name changes)? Make up values for any extra required input data and incorporate these into a modified spreadsheet model. Then optimize with Solver. Do you conclude that it is easy to resize a spreadsheet model? (By the way, algebraic models are typically much easier to resize.)

17. In Solver’s sensitivity report for PC Tech’s two- line product mix model, the allowable decrease for available assembling hours is 2375. This means that something happens when assembling hours fall to 20,000 2 2375 5 17,625. See what this means by first running Solver with 17,626 available hours and then again with 17,624 available hours. Explain how the two solutions compare to the original solution and to each other.

13-8 A Multiperiod Production Model The product mix examples illustrate a very important type of LP model. However, LP models come in many forms. For variety, we now present a different type of model that can also be solved with LP. (In the next chapter we provide other examples, linear and oth- erwise.) The distinguishing feature of the following model is that it relates decisions made during several time periods. This type of problem occurs when a company must make a decision now that will have ramifications in the future. The company does not want to focus completely on the short run and ignore the long run.

EXAMPLE

13.3 PRODUCING FOOTBALLS IN MULTIPLE TIME PERIODS Pigskin Company produces footballs. Pigskin must decide how many footballs to produce each month. The company has decided to use a six-month planning horizon. The forecasted monthly demands for the next six months are 10,000, 15,000, 30,000, 35,000, 25,000, and 10,000. Pigskin wants to meet these demands on time, knowing that it currently has 5000 footballs in inventory and that it can use a given month’s production to help meet the demand for that month. (For simplicity, we assume that production occurs during the month, and demand occurs at the end of the month.) During each month there is enough production capacity to produce up to 30,000 footballs, and there is enough storage capacity to store up to 10,000 footballs at the end of the month,

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after demand has occurred. The forecasted production costs per football for the next six months are $12.50, $12.55, $12.70, $12.80, $12.85, and $12.95, respectively. The holding cost per football held in inventory at the end of any month is figured at 5% of the production cost for that month. (This cost includes the cost of storage and also the cost of money tied up in inven- tory.) The selling price for footballs is not considered relevant to the production decision because Pigskin will satisfy all cus- tomer demand exactly when it occurs—at whatever the selling price is. In other words, total revenue for the planning horizon is fixed, regardless of production decisions. Therefore, Pigskin wants to determine the production schedule that minimizes the total production and holding costs.

Objective To use LP to find the production schedule that meets demand on time and minimizes total production and inventory holding costs.

Where Do the Numbers Come From? The input values for this problem are not all easy to find. Here are some thoughts on where they might be obtained.

• The initial inventory should be available from the company’s database system or from a physical count.

• The unit production costs would probably be estimated in two steps. First, the company might ask its cost accountants to estimate the current unit production cost. Then it could examine historical trends in costs to estimate inflation factors for future months.

• The holding cost percentage is typically difficult to determine. Depending on the type of inventory being held, this cost can include storage and handling, rent, property taxes, insurance, spoilage, and obsolescence. It can also include capital costs— the cost of money that could be used for other purposes.

• The demands are probably forecasts made by the marketing and sales department. They might be “seat-of-the-pants” fore- casts, or they might be the result of a formal quantitative forecasting procedure as discussed in Chapter 12. Of course, if there are already some orders on the books for future months, these are included in the demand figures.

• The production and storage capacities are probably supplied by the production department. They are based on the size of the workforce, the available machinery, availability of raw materials, and physical space.

Solution The variables for this model appear in Figure 13.27. There are two keys to relating these variables. First, the months cannot be treated independently. This is because the ending inventory in one month is the beginning inventory for the next month. Sec- ond, to ensure that demand is satisfied on time, the amount on hand after production in each month must be at least as large as the demand for that month. This constraint must be included explicitly in the model.

Production capacity

Production cost

Ending inventory

Holding cost

Holding cost percentage

Unit production cost

Minimize total cost

Units produced

Storage capacity<=<=

Available inventory

Forecasted demand

Initial inventory >=

Figure 13.27 Big Picture for Production Planning Model

13-8 a Multiperiod production Model 6 1 3

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When you model this type of problem, you must be very specific about the timing of events. Depending on the assumptions you make, there can be a variety of potential models. For example, when does the demand for footballs in a given month occur: at the beginning of the month, at the end of the month, or continually throughout the month? The same question can be asked about production in a given month. The answers to these two questions indicate how much of the production in a given month can be used to help satisfy the demand in that month. Also, are the maximum storage constraint and the holding cost based on the ending inventory in a month, the average amount of inventory in a month, or the maximum inventory in a month? Each of these possibilities is reasonable and could be implemented.

To simplify the model, we assume that (1) all production occurs at the beginning of the month, (2) all demand occurs after production, so that all units produced in a month can be used to satisfy that month’s demand, and (3) the storage constraint and the holding cost are based on ending inven- tory in any month. (You are asked to modify these assumptions in the problems.)

Algebraic Model In the traditional algebraic model, the decision variables are the production quantities for the six months, labeled P1 through P6. It is also convenient to let I1 through I6 be the corresponding end-of-month inventories (after demand has occurred).12 For example, I3 is the number of footballs left over at the end of month 3. Therefore, the obvious constraints are on production and inventory storage capacities: Pj # 30000 and Ij # 10000 for 1 # j # 6.

In addition to these constraints, balance constraints that relate the P’s and I’s are necessary. In any month the inventory from the previous month plus the current production equals the current demand plus leftover inventory. If Dj is the forecasted demand for month j, the balance equation for month j is

Ij 2 1 1 Pj 5 Dj 1 Ij

The balance equation for month 1 uses the known beginning inventory, 5000, for the previous inventory (the Ij 2 1 term). By putting all variables (P’s and I’s) on the left and all known values on the right (a standard LP convention), these balance con- straints can be written as

P1 2 I1 5 1000 2 5000

I1 1 P2 2 I2 5 15000

I2 1 P3 2 I3 5 30000

I3 1 P4 2 I4 5 35000

I4 1 P5 2 I5 5 25000

I5 1 P6 2 I6 5 10000

(13.1)

As usual, there are nonnegativity constraints: all P’s and I’s must be nonnegative. What about meeting demand on time? This requires that in each month the inventory from the preceding month plus the

current production must be at least as large as the current demand. But take a look, for example, at the balance equation for month 3. By rearranging it slightly, it becomes

I3 5 I2 1 P3 2 30000

Now, the nonnegativity constraint on I3 implies that the right side of this equation, I2 1 P3 2 30000, must also be nonnega- tive. But this implies that demand in month 3 is covered—the beginning inventory in month 3 plus month 3 production is at least 30000. Therefore, the nonnegativity constraints on the I’s automatically guarantee that all demands will be met on time, and no other constraints are needed. Alternatively, the constraint can be written directly as I2 1 P3 $ 30000. In words, the amount on hand after production in month 3 must be at least as large as the demand in month 3. The spreadsheet model takes advantage of this interpretation.

Finally, the objective to minimize is the sum of production and holding costs. It is the sum of unit production costs multi- plied by P’s, plus unit holding costs multiplied by I’s.

Developing the Spreadsheet Model The spreadsheet model of Pigskin’s production problem is shown in Figure 13.28. (See the file Production Scheduling Finished.xlsx.) The main feature that distinguishes this model from the product mix model is that some of the constraints, namely, Equations (13.1), are built into the spreadsheet itself by means of

By modifying the timing assumptions in this type of model, alternative—and equally realistic— models with very different solutions can be obtained.

12 This example illustrates a subtle difference between algebraic and spreadsheet models. It is often convenient in algebraic models to define “decision variables,” in this case the I’s, that are really determined by other decision variables, in this case the P’s. In spreadsheet models, however, we typically define the decision variable cells as the smallest set of variables that must be chosen—in this case the production quantities. Then values that are determined by these decision variable cells, such as the ending inventory levels, can be calculated with spreadsheet formulas.

Developing the Production Planning Model

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1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

A B C D E F G H I J K Multiperiod production model

Input data Initial inventory 5000 Holding cost as % of prod cost 5%

Month Month 1 Month 2 Month 3 Month 4 Month 5 Month 6 Production cost/unit $12.50 $12.55 $12.70 $12.80 $12.85 $12.95

Production plan Month Units produced 100002500030000300001500015000

Production capacity 30000 30000 30000 30000 30000 30000

On hand after production 20000 25000 40000 40000 30000 15000

100002500035000300001500010000Demand

Ending inventory 500050005000100001000010000

<= <= <= <= <= <=

>= >= >= >= >= >=

<= <= <= <= <= <= Storage capacity 100001000010000100001000010000

Summary of costs Month Month 1 Month 2 Month 3 Month 4 Month 5 Month 6 Totals Production costs $187,500 $188,250 $381,000 $384,000 $129,500 $1,591,500 Holding costs $6,350 $3,200 $3,213 $3,238 $28,525

$1,620,025$324,463 $132,738$387,200

$321,250

$387,350 $6,250 $6,275

$194,525$193,750Totals

1 2 3 4 5 6

Range names used Demand

On_hand_after_production Production_capacity

Ending_inventory

Storage_capacity Total_Cost Units_produced

=Model!$B$20:$G$20 =Model!$B$18:$G$18

=Model!$B$16:$G$16 =Model!$B$14:$G$14 =Model!$B$22:$G$22

=Model!$B$12:$G$12 =Model!$H$28

Figure 13.28 Production Planning Model with a Suboptimal Solution

formulas. This means that the only decision variable cells are the production quantities. The ending inventories shown in row 20 are determined by the production quantities and Equations (13.1). As you see, the decision variables in an algebraic model (the P’s and I’s) are not necessarily the same as the decision variable cells in an equivalent spreadsheet model. (The only deci- sion variable cells in the spreadsheet model correspond to the P’s.)

To develop the spreadsheet model in Figure 13.28, proceed as follows.

1. Inputs. Enter the inputs in the blue cells. Again, these are all entered as numbers directly from the problem statement. (Unlike some spreadsheet modelers who prefer to put all inputs in the upper left corner of the spreadsheet, we enter the inputs wherever they fit most naturally. Of course, this takes some planning before diving in.)

2. Name ranges. Name the ranges indicated. Note that all but one of these (Total_cost) can be named easily with the range-naming shortcut, using the labels in column A.

3. Production quantities. Enter any values in the range Units_produced as production quantities. As always, you can enter values that you believe are good, maybe even optimal. This is not crucial, however, because Solver eventually finds the optimal production quantities.

4. On-hand inventory. Enter the formula

5B41B12

in cell B16. This calculates the first month’s on-hand inventory after production (but before demand). Then enter the typical formula

5B201C12

for on-hand inventory after production in month 2 in cell C16 and copy it across row 16. 5. Ending inventories. Enter the formula

5B16-B18

for ending inventory in cell B20 and copy it across row 20. This formula calculates ending inventory in the current month as on-hand inventory before demand minus the demand in that month.

6. Production and holding costs. Enter the formula

5B8*B12

in cell B26 and copy it across to cell G26 to calculate the monthly production costs. Then enter the formula

5$B$5*B8*B20

in cell B27 and copy it across to cell G27 to calculate the monthly holding costs. Note that these are based on monthly ending inventories. Finally, calculate the cost totals in column H with the SUM function.

In multiperiod prob- lems, there is often one formula for the first period and a slightly different (copyable) formula for all other periods.

13-8 a Multiperiod production Model 6 1 5

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Using Solver To use Solver, fill out the main dialog box as shown in Figure 13.29. The logic behind the constraints is straightforward. The constraints are that (1) the production quantities cannot exceed the production capacities, (2) the on-hand inventories after pro- duction must be at least as large as the demands, and (3) the ending inventories cannot exceed the storage capacities. Check the Non-Negative option and select the Simplex LP method, and then click Solve.

Figure 13.29 Solver Dialog Box for Production Planning Model

Discussion of the Solution The optimal solution from Solver appears in Figure 13.30. The solution can be interpreted best by comparing production quan- tities to demands. In month 1, Pigskin should produce just enough to meet month 1 demand (taking into account the initial inventory of 5000). In month 2, it should produce 5000 more footballs than month 2 demand, and then in month 3 it should produce just enough to meet month 3 demand, while still carrying the extra 5000 footballs in inventory from month 2 produc- tion. In month 4, Pigskin should finally use these 5000 footballs, along with the maximum production amount, 30,000, to meet

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

A B C D E F G H I J K Multiperiod production model

Input data Initial inventory 5000 Holding cost as % of prod cost 5%

Month Month 1 Month 2 Month 3 Month 4 Month 5 Month 6 Production cost/unit $12.50 $12.55 $12.70 $12.80 $12.85 $12.95

Production plan Month Units produced 10000250003000030000200005000

Production capacity 30000 30000 30000 30000 30000 30000

On hand after production 10000 20000 35000 35000 25000 10000

100002500035000300001500010000Demand

Ending inventory 000500050000

<= <= <= <= <= <=

>= >= >= >= >= >=

<= <= <= <= <= <= Storage capacity 100001000010000100001000010000

Summary of costs Month Month 1 Month 2 Month 3 Month 4 Month 5 Month 6 Totals Production costs $62,500 $251,000 $381,000 $384,000 $129,500 $1,529,250 Holding costs $3,175 $0 $0 $0 $6,313

$1,535,563$321,250 $129,500$384,000

$321,250

$384,175 $0 $3,138

$254,138$62,500Totals

1 2 3 4 5 6

Range names used Demand

On_hand_after_production Production_capacity

Ending_inventory

Storage_capacity Total_Cost Units_produced

=Model!$B$20:$G$20 =Model!$B$18:$G$18

=Model!$B$16:$G$16 =Model!$B$14:$G$14 =Model!$B$22:$G$22

=Model!$B$12:$G$12 =Model!$H$28

Figure 13.30 Optimal Solution for Production Planning Model

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month 4 demand. Then in months 5 and 6 it should produce exactly enough to meet these months’ demands. The total cost is $1,535,563, most of which is production cost.

Could you have guessed this optimal solution? Upon reflection, it makes perfect sense. Because the monthly holding costs are large relative to the differences in monthly production costs, there is little incentive to produce footballs before they are needed to take advantage of a “cheap” production month. Therefore, Pigskin Company produces foot balls in the month when they are needed—when possible. The only exception to this rule is the 20,000 footballs produced during month 2 when only 15,000 are needed. The extra 5000 footballs produced in month 2 are needed, however, to meet the month 4 demand of 35,000, because month 3 production capacity is used entirely to meet the month 3 demand. Thus month 3 capacity is not available to meet the month 4 demand, and 5000 units of month 2 capacity are used to meet the month 4 demand.

You can often improve your intuition by trying to reason why Solver’s solution is indeed optimal.

Multiperiod Optimization problems and Myopic Solutions

Many optimization problems are of a multiperiod nature, where a sequence of decisions must be made over time. When making the first of these decisions, the one for this week or this month, say, it is usually best to include future decisions in the model, as has been done here. If you ignore future periods and make the initial decision based only on the first period, the resulting decision is called myopic (short-sighted). Myopic decisions are occa- sionally optimal, but not very often. The idea is that if you act now in a way that looks best in the short run, it might lead you down a strategically unattractive path for the long run.

Fundamental Insight

Sensitivity Analysis SolverTable can now be used to perform a number of interesting sensitivity analyses. We illustrate two possibilities. First, note that the most inventory ever carried at the end of a month is 5000, although the storage capacity each month is 10,000. Perhaps this is because the holding cost percent- age, 5%, is fairly large. Would more ending inventory be carried if this holding cost percentage were lower? Or would even less be carried if it were higher? You can check this with the SolverTable out- put shown in Figure 13.31. Now the single input cell is cell B5, and the single output is the maximum ending inventory ever held, which you can calculate in cell B31 with the formula

5MAX(Ending_inventory)

As the SolverTable results indicate, the storage capacity limit is reached only when the holding cost percentage falls to 1%. (This output doesn’t indicate which month or how many months the ending inventory is at the upper limit.) On the other hand, even when the holding cost percentage reaches 10%, the company still continues to hold a maximum ending inventory of 5000.

If you want Solver Table to keep track of a quantity that is not in your model, you need to create it with an appropriate formula in a new cell.

Figure 13.31 Sensitivity of Maximum Inventory to Holding Cost

7 8 9

10 11 12 13 14

B C D E F G

2 1

$B $3

Holding cost pct (cell $B$5) values along side, output cell(s) along top

Oneway analysis for Solver model in Model worksheet

10000 5000 5000 5000 5000 5000 5000 5000 5000

1% 2% 3% 4% 5% 6% 7% 8% 9%

10% 5000

Sometimes you’d like to use SolverTable on an output that isn’t explicitly part of the model. In that case, just calculate it in a new cell (as in cell B31 in the Model sheet) and then use SolverTable.

13-8 a Multiperiod production Model 6 1 7

A second possible sensitivity analysis is suggested by the way the optimal production schedule would probably be imple- mented. The optimal solution to Pigskin’s model specifies the production level for each of the next six months. In reality,

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however, the company would probably implement the model’s recommendation only for the first month. Then at the beginning of the second month, it would gather new forecasts for the next six months, months 2 through 7, solve a new six-month model, and again implement the model’s recommendation for the first of these months, month 2. If the company continues in this manner, we say that it is following a six-month rolling planning horizon.

The question, then, is whether the assumed demands (really, forecasts) toward the end of the planning horizon have much effect on the optimal production quantity in month 1. We would hope not, because these forecasts could be quite inaccurate. The two-way Solver table in Figure 13.32 shows how the optimal month 1 production quantity varies with the forecasted demands in months 5 and 6. As you can see, if the errors in the forecasted demands for months 5 and 6 remain fairly small, the optimal month 1 production quantity remains at 5000. This is good news. It means that the optimal production quantity in month 1 is fairly insensitive to the possibly inaccurate forecasts for months 5 and 6.

Figure 13.32 Sensitivity of Month 1 Production to Demand in Months 5 and 6

B C D E F G H I J

2 1

Month 5 demand (cell $F$18) values along side, Month 6 demand (cell $G$18) values along top, output cell in corner

Twoway analysis for Solver model in Model worksheet

Units_produced_1 10000 20000 30000

10000 5000 5000 5000

20000 5000 5000 5000

30000 5000 5000 5000

Solver’s sensitivity report for this model appears in Figure 13.33. The bottom part of this report is fairly straightforward to interpret. The first six rows are for sensitivity to changes in the demand, whereas the last six are for sensitivity to changes in the storage capacity. (There are no rows for the production capacity constraints because these are simple upper-bound

6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Variable Cells

$B$12 $C$12 $D$12 $E$12 $F$12 $G$12

Cells

Units produced Month 1 Units produced Month 2 Units produced Month 3 Units produced Month 4 Units produced Month 5 Units produced Month 6

Name

5000 20000 30000 30000 25000 10000

Final

Value

0 0

�0.477500009 �1.012500019

0 0

Reduced

Cost

16.31750006 15.74250005 15.26500004 14.73000003 14.14000002 13.59750001

Objective

Coefficient

1E+30 0.575000009 0.477500009 1.012500019 1.602500028

0.54250001

Allowable

Increase

0.575000009 0.477500009

1E+30 1E+30

0.54250001 13.59750001

Allowable

Decrease

Constraints

$B$16 $C$16 $D$16 $E$16 $F$16 $G$16 $B$20 $C$20 $D$20 $E$20 $F$20 $G$20

Cells

On hand after production <= On hand after production <= On hand after production <= On hand after production <= On hand after production <= On hand after production <= Ending inventory >= Ending inventory >= Ending inventory >= Ending inventory >= Ending inventory >= Ending inventory >=

Name

10000 20000 35000 35000 25000 10000

0 5000 5000

0 0 0

Final

Value

0.575000009 0 0

1.602500028 0.54250001

13.59750001 0 0 0 0 0 0

Shadow

Price

10000 15000 30000 35000 25000 10000 10000 10000 10000 10000 10000 10000

Constraint

R.H. Side

10000 5000 5000 5000 5000

10000 1E+30 1E+30 1E+30 1E+30 1E+30 1E+30

Allowable

Increase

5000 1E+30 1E+30

5000 20000 10000 10000

5000 5000

10000 10000 10000

Allowable

Decrease

A B C D E F G H

Figure 13.33 Solver Sensitivity Report for Production Planning Model

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constraints on the decision variables. Recall that Solver’s sensitivity report handles this type of constraint differently from “normal” constraints.) In contrast, the top part of the report is virtually impossible to unravel. This is because the objective coefficients of the decision variables are each based on multiple inputs. (Each is a combination of unit production costs and the holding cost percentage.) Therefore, if you want to know how the solution will change if you change a single unit production cost or the holding cost percentage, this report does not answer your question. This is one case where a sensitivity analysis with SolverTable is much more straightforward and intuitive. It allows you to change any of the model’s inputs and directly see the effects on the solution.

Modeling Issues We assume that Pigskin uses a six-month planning horizon. Why six months? In multiperiod models such as this, the company has to make forecasts about the future, such as the level of customer demand. Therefore, the length of the planning horizon is usually the length of time for which the company can make reasonably accurate forecasts. Here, Pigskin evidently believes that it can forecast up to six months from now, so it uses a six-month planning horizon.

13-9 a Comparison of algebraic and Spreadsheet Models 6 1 9

Modify the Pigskin model with this new assumption, and use Solver to find the optimal solution. How does this change the optimal production schedule? How does it change the optimal total cost?

Level B 22. Modify the Pigskin spreadsheet model so that except for

month 6, demand does not have to be met on time. The only requirement is that all demand must be met even- tually by the end of month 6. How does this change the optimal production schedule? How does it change the optimal total cost?

23. Modify the Pigskin spreadsheet model so that demand in any of the first five months must be met no later than a month late, whereas demand in month 6 must be met on time. For example, the demand in month 3 can be met partly in month 3 and partly in month 4. How does this change the optimal production schedule? How does it change the optimal total cost?

24. Modify the Pigskin spreadsheet model in the following way. Assume that the timing of demand and produc- tion are such that only 70% of the production in a given month can be used to satisfy the demand in that month. The other 30% occurs too late in that month and must be carried as inventory to help satisfy demand in later months. How does this change the optimal production schedule? How does it change the optimal total cost? Then use SolverTable to see how the optimal production schedule and optimal cost vary as the percentage of pro- duction usable for this month’s demand (now 70%) is allowed to vary from 20% to 100% in increments of 10%.

Problems

Level A 18. Can you guess the results of a sensitivity analysis on the

initial inventory in the Pigskin model? See if your guess is correct by using SolverTable and allowing the ini- tial inventory to vary from 0 to 10,000 in increments of 1000. Keep track of the values in the decision variable cells and the objective cell.

19. Modify the Pigskin model so that there are eight months in the planning horizon. You can make up reasonable values for any extra required data. Don’t forget to mod- ify range names. Then modify the model again so that there are only four months in the planning horizon. Do either of these modifications change the optimal produc- tion quantity in month 1?

20. As indicated by the algebraic formulation of the Pigskin model, there is no real need to calculate inventory on hand after production and constrain it to be greater than or equal to demand. An alternative is to calculate ending inventory directly and constrain it to be nonnegative. Modify the cur- rent spreadsheet model to do this. (Delete rows 16 and 17, and calculate ending inventory appropriately. Then add an explicit nonnegativity constraint on ending inventory.)

21. In one modification of the Pigskin model, the maximum storage constraint and the holding cost are based on the average inventory (not ending inventory) for a given month, where the average inventory is defined as the sum of beginning inventory and ending inventory, divided by 2, and beginning inventory is before production or demand.

13-9 A Comparison of Algebraic and Spreadsheet Models To this point you have seen algebraic optimization models and corresponding spreadsheet models. How do they differ? If you review the two product mix examples in this chap- ter, we believe you will agree that (1) the algebraic models are quite straightforward and (2) the spreadsheet models are almost direct translations into Excel of the algebraic models.

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In particular, each algebraic model has a set of x’s that corresponds to the decision variable cell range in the spreadsheet model. In addition, each objective and each left side of each constraint in the spreadsheet model corresponds to a linear expression involving x’s in the algebraic model.

However, the Pigskin production planning model is quite different. The spread- sheet model includes one set of decision variable cells, the production quantities, and everything else is related to these through spreadsheet formulas. In contrast, the alge- braic model has two sets of variables, the P’s for the production quantities and the I’s for the ending inventories, and together these constitute the decision variables. These two sets of variables must then be related algebraically, which is done through a series of balance equations.

This is a typical situation in algebraic models, where one set of variables (the produc- tion quantities) corresponds to the real decision variables, and other sets of variables, along with extra equations or inequalities, are introduced to capture the logic. We believe—and this belief is reinforced by years of teaching experience—this extra level of abstraction makes algebraic models much more difficult for typical users to develop and comprehend. It is the primary reason we have decided to focus almost exclusively on spreadsheet mod- els in this book.

13-10 A Decision Support System If your job is to develop an LP spreadsheet model to solve a problem such as Pigskin’s production problem, you will be considered the “expert” in LP. Many people who need to use such models, however, are not experts. They might understand the basic ideas behind LP and the types of problems it is intended to solve, but they will not know the details. In this case it is useful to provide these users with a decision support system (DSS) that can help them solve problems without having to know technical details.

We will not teach you in this book how to build a full-scale DSS, but we will show you what a typical DSS looks like and what it can do.13 (We consider only DSSs built around spreadsheets. There are many other platforms for developing DSSs that we will not consider.) Basically, a spreadsheet-based DSS contains a spreadsheet model of a problem, such as the one in Figure 13.26. However, as a user, you will probably never even see this model. Instead, you will see a front end and a back end. The front end allows you to select input values for your particular problem. The user interface for this front end can include several features, such as buttons, dialog boxes, toolbars, and menus—the things you are used to seeing in Windows applications. The back end will then produce a report that explains the solution in nontechnical terms.

We illustrate a DSS for a slight variation of the Pigskin problem in the file Decision Support.xlsm. This file has three worksheets. When you open the file, you see the Expla- nation sheet shown in Figure 13.34. It contains two buttons, one for setting up the prob- lem (getting the user’s inputs) and one for solving the problem (running Solver). When you click the Set Up Problem button, you are asked for the inputs: the initial inventory, the forecasted demands for each month, and others. An example appears in Figure 13.35. These input boxes should be self-explanatory, so that all you need to do is enter the values you want to try. (To speed up the process, the inputs from the previous run are shown by default.) After you have entered these inputs, you can view the Model sheet. This sheet contains a spreadsheet model similar to the one in Figure 13.30 but with the inputs you just entered.

Developing a Decision Support System

13 For readers interested in learning more about this DSS, this textbook’s essential resource website includes notes about its development in the file Developing the Decision Support application.docx, under Chapter 13 Example Files, and the accompanying videos provide details for developing a slightly less complex DSS for a product mix model. If you are interested in learning more about spreadsheet DSSs in general, Albright has written the book VBA for Modelers, now in its fifth edition. It contains a primer on the VBA language and presents many applications and instructions for creating DSSs with VBA.

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13-10 a Decision Support System 6 2 1

Now go back to the Explanation sheet and click the Find Optimal Solution button. This automatically sets up the Solver dialog box and runs Solver. There are two possibilities. First, it is possible that there is no feasible solution to the problem with the inputs you entered. In this case you see a message to this effect, as in Figure 13.36. In most cases, however, the problem has a feasible solution. In this case you see the Report sheet, which summarizes the optimal solution in nontechnical terms. Part of one sample output appears in Figure 13.37.

Figure 13.34 Explanation Sheet for DSS

Pigskin Production Scheduling

Set Up Problem Find Optimal Solution

This application solves a 6-month production planning model similar to the example in the chapter. The only difference is that the production capacity and storage capacity are allowed to vary by month. To run the application, click the left button to enter inputs. Then click the right button to run Solver and obtain a solution report.

Figure 13.35 Dialog Box for Obtaining User Inputs

Figure 13.36 Indication of No Feasible Solutions

After studying this report, you can then click the Solve Another Problem button, which takes you back to the Explanation sheet so that you can solve a new problem. All this is done automatically with Excel macros. These macros use Microsoft’s Visual Basic for Applications (VBA) programming language to automate various tasks. In most profes- sional applications, nontechnical people can just enter inputs and view reports. Therefore, the Model sheet and VBA code will most likely be hidden and protected from users.

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13-11 Conclusion This chapter has provided a good start to LP modeling—and to optimization modeling in general. You have learned how to develop three basic LP spreadsheet models, how to use Solver to find their optimal solutions, and how to perform sensitivity analyses with Solver’s sensitivity reports or the SolverTable add-in. You have also learned how to recognize whether a mathematical programming model satisfies the linear assumptions. In the next chapter you will see a variety of other optimization models, but the three basic steps of model development, Solver optimization, and sensitivity analysis remain the same.

Summary of Key Terms

Figure 13.37 Optimal Solution Report

Monthly schedule

Month 1

Month 2

Units Dollars

Production cost

Holding cost

Dollars

Start with 5000 5000

10000 0

Produce Demand is End with

Units Start with 0

15000 15000

Produce Demand is End with

Month 3 Units Start with 0

30000 30000

Produce Demand is End with

$62,500

Production cost

Holding cost

$189,750

Dollars

Production cost

Holding cost

$382,500

TERM EXPLANATION EXCEL PAGES Linear programming Refers to optimization models with a linear

objective and linear constraints, often abbreviated as LP

600

Objective The value, such as profit, to be optimized in an optimization model

601

Constraints Conditions that must be satisfied in an optimization model

601

Decision variable cells Cells that contain the values of the decision variables

Specify in Solver dialog box 601

Objective cell Cell that contains the value of the objective Specify in Solver dialog box 601

Nonnegativity constraints Constraints that require the decision variables to be nonnegative, usually for physical reasons

601

Feasible solution A solution that satisfies all constraints 602

Feasible region The set of all feasible solutions 602

Infeasible solution A solution that doesn’t satisfy all constraints 602

Optimal solution The feasible solution that has the best value of the objective

602

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13-11 Conclusion 6 2 3

TERM EXPLANATION EXCEL PAGES

Solver Add-in that ships with Excel for performing optimization, developed by Frontline Systems

Solver on Data ribbon 602

Simplex method An efficient algorithm for finding the optimal solution in a linear programming model

602

Sensitivity analysis Seeing how the optimal solution changes as selected input values change

602

Algebraic model A model that expresses the constraints and the objective algebraically

604

Graphical solution Shows the constraints and objective graphically so that the optimal solution can be identified; useful only when there are two decision variables

605

Spreadsheet model A model that uses spreadsheet formulas to express the logic of the model

607

Binding constraint A constraint that holds as an equality 615

Nonbinding constraint An inequality constraint where there is a difference between the two sides of the inequality

615

Solver’s sensitivity report Report available from Solver that shows sensitivity to objective coefficients and right sides of constraints

Available in Solver dialog box after Solver runs

616

Reduced cost Amount the objective coefficient of a variable currently equal to zero must change before it is optimal for that variable to be positive, or the amount the objective of a variable currently at its upper bound must change before that variable decreases from its upper bound

617

Shadow price The change in the objective for a change in the right side of a constraint; indicates amount a company would pay for more of a scarce resource

617

SolverTable Add-in developed by Albright that performs sensitivity analysis to any inputs and reports results in tabular and graphical form

SolverTable tab 619

Selecting multiple ranges Useful when decision variable cells, e.g., are in noncontiguous ranges

Pressing Ctrl key, drag ranges, one after the other

620

Mathematical programming model

Any optimization model, whether linear, integer, or nonlinear

626

Proportionality, additivity, divisibility

Properties of optimization model that result in a linear programming model

627

Infeasibility Condition where a model has no feasible solutions

629

Unboundedness Condition where there is no limit to the objective; almost always a sign of an error in the model

629

Rolling planning horizon Multiperiod model where only the decision in the first period is implemented, and then a new multiperiod model is solved in succeeding periods

647

Decision support system System where a user can enter inputs to a model and see outputs, but need not be concerned with technical details

650

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C.9. In a production scheduling problem like Pigskin’s, suppose the company must produce several products to meet customer demands. Would it suffice to solve a separate model for each product, as we did for Pigskin, or would one big model for all products be necessary? If the latter, discuss what this big model might look like.

C.10. In any optimization model such as those in this chap- ter, we say that the model is unbounded (and Solver will indicate as such) if there is no limit to the value of the objective. For example, if the objective is profit, then for any dollar value, no matter how large, there is a feasible solution with profit at least this large. In the real world, why are there never any unbounded models? If you run Solver on a model and get an “unbounded” message, what should you do?

Level A 25. A chemical company manufactures three chemicals: A,

B, and C. These chemicals are produced via two produc- tion processes: 1 and 2. Running process 1 for an hour costs $400 and yields 300 units of A, 100 units of B, and 100 units of C. Running process 2 for an hour costs $100 and yields 100 units of A and 100 units of B. To meet customer demands, at least 1000 units of A, 500 units of B, and 300 units of C must be produced daily. a. Use Solver to determine a daily production plan that

minimizes the cost of meeting the company’s daily demands.

b. Confirm graphically that the daily production plan from part a minimizes the cost of meeting the company’s daily demands.

c. Use SolverTable to see what happens to the decision variables and the total cost when the hourly processing cost for process 2 increases in increments of $0.50. How large must this cost increase be before the deci- sion variables change? What happens when it contin- ues to increase beyond this point?

26. A furniture company manufactures desks and chairs. Each desk uses four units of wood, and each chair uses three units of wood. A desk contributes $400 to profit, and a chair contributes $250. Marketing restrictions require that the number of chairs produced be at least twice the number of desks produced. There are 2000 units of wood available. a. Use Solver to maximize the company’s profit. b. Confirm graphically that the solution in part a maxi-

mizes the company’s profit. c. Use SolverTable to see what happens to the decision

variables and the total profit when the availability of wood varies from 1000 to 3000 in 100-unit incre- ments. Based on your findings, how much would the company be willing to pay for each extra unit of wood over its current 2000 units? How much profit would the company lose if it lost any of its current 2000 units?

Problems

Conceptual Questions C.1. Suppose you use Solver to find the optimal solution to

a maximization model. Then you remember that you omitted an important constraint. After adding the con- straint and running Solver again, is the optimal value of the objective guaranteed to decrease? Why or why not?

C.2. Consider an optimization model with a number of resource constraints. Each indicates that the amount of the resource used cannot exceed the amount available. Why is the shadow price of such a resource constraint always zero when the amount used in the optimal solu- tion is less than the amount available?

C.3. If you add a constraint to an optimization model, and the previously optimal solution satisfies the new con- straint, will this solution still be optimal with the new constraint added? Why or why not?

C.4. Why is it generally necessary to add nonnegativity constraints to an optimization model? Wouldn’t Solver automatically choose nonnegative values for the deci- sion variable cells?

C.5. Suppose you have a linear optimization model where you are trying to decide which products to produce to maximize profit. What does the additive assumption imply about the profit objective? What does the pro- portionality assumption imply about the profit objec- tive? Be as specific as possible. Can you think of any reasonable profit functions that would not be linear in the amounts of the products produced?

C.6. In a typical product mix model, where a company must decide how much of each product to produce to max- imize profit, discuss possible situations where there might not be any feasible solutions. Could these be realistic? If you had such a situation in your company, how might you proceed?

C.7. In a typical product mix model, where a company must decide how much of each product to produce to max- imize profit, there are sometimes customer demands for the products. We used upper-bound constraints for these: Don’t produce more than you can sell. Would it be realistic to have lower-bound constraints instead: Produce at least as much as is demanded? Would it be realistic to have both (where the upper bounds are greater than the lower bounds)? Would it be realistic to have equality constraints: Produce exactly what is demanded?

C.8. In a typical production scheduling model like Pig- skin’s, if there are no production capacity constraints— the company can produce as much as it needs in any time period—but there are storage capacity constraints and demand must be met on time, is it possible that there will be no feasible solutions? Why or why not?

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13-11 Conclusion 6 2 5

27. A farmer in Iowa owns 450 acres of land. He is going to plant each acre with wheat or corn. Each acre planted with wheat yields $2000 profit, requires three workers, and requires two tons of fertilizer. Each acre planted with corn yields $3000 profit, requires two workers, and requires four tons of fertilizer. There are currently 1000 workers and 1200 tons of fertilizer available. a. Use Solver to help the farmer maximize the profit

from his land. b. Confirm graphically that the solution from part a

maximizes the farmer’s profit from his land. c. Use SolverTable to see what happens to the decision

variables and the total profit when the availability of fertilizer varies from 200 tons to 2200 tons in 100 -ton increments. When does the farmer discontinue producing wheat? When does he discontinue produc- ing corn? How does the profit change for each 10-ton increment?

28. During the next four months, a customer requires, respectively, 500, 650, 1000, and 700 units of a com- modity, and no backlogging is allowed (that is, the cus- tomer’s requirements must be met on time). Production costs are $50, $80, $40, and $70 per unit during these months. The storage cost from one month to the next is $20 per unit (assessed on ending inventory). It is esti- mated that each unit on hand at the end of month 4 can be sold for $60. Assume there is no beginning inventory. a. Determine how to minimize the net cost incurred in

meeting the demands for the next four months. b. Use SolverTable to see what happens to the decision

variables and the total cost when the initial inventory varies from 0 to 1000 in 100-unit increments. How much lower would the total cost be if the company started with 100 units in inventory, rather than none? Would this same cost decrease occur for every 100 -unit increase in initial inventory?

29. A company faces the following demands during the next three weeks: week 1, 2000 units; week 2, 1000 units; week 3, 1500 units. The unit production costs during each week are as follows: week 1, $130; week 2, $140; week 3, $150. A holding cost of $20 per unit is assessed against each week’s ending inventory. At the beginning of week 1, the company has 500 units on hand. In reality, not all goods produced during a month can be used to meet the current month’s demand. To model this fact, assume that only half of the goods produced during a week can be used to meet the current week’s demands. a. Determine how to minimize the cost of meeting the

demand for the next three weeks. b. Revise the model so that the demands are of the form

Dt 1 kCt, where Dt is the original demand (from above) in month t, k is a given factor, and Ct is an amount of change in month t demand. Develop the model in such a way that you can use SolverTable to analyze changes in the amounts produced and the total cost when k varies from 0 to 10 in 1-unit increments,

for any fixed values of the C s. For example, try this when C1 5 200, C2 5 500, and C3 5 300. Describe the behavior you observe in the table. Can you find any reasonable C’s that induce positive production levels in week 3?

30. Maggie Stewart loves desserts, but due to weight and cholesterol concerns, she has decided that she must plan her desserts carefully. There are two possible desserts she is considering: snack bars and ice cream. After read- ing the nutrition labels on the snack bar and ice cream packages, she learns that each serving of a snack bar weighs 37 grams and contains 120 calories and 5 grams of fat. Each serving of ice cream weighs 65 grams and contains 160 calories and 10 grams of fat. Maggie will allow herself no more than 450 calories and 25 grams of fat in her daily desserts, but because she loves desserts so much, she requires at least 120 grams of dessert per day. Also, she assigns a “taste index” to each gram of each dessert, where 0 is the lowest and 100 is the high- est. She assigns a taste index of 95 to ice cream and 85 to snack bars (because she prefers ice cream to snack bars). a. Use Solver to find the daily dessert plan that stays

within her constraints and maximizes the total taste index of her dessert.

b. Confirm graphically that the solution from part a maximizes Maggie’s total taste index.

c. Use a two-way Solver table to see how the optimal dessert plan varies when the calories per snack bar and per ice cream vary. Let the former vary from 80 to 200 in increments of 10, and let the latter vary from 120 to 300 in increments of 10.

31. For a telephone survey, a marketing research group needs to contact at least 600 wives, 480 husbands, 400 single adult males, and 440 single adult females. It costs $3 to make a daytime call and (because of higher labor costs) $5 to make an evening call. The file P13_31.xlsx lists the results that can be expected. For example, 30% of all daytime calls are answered by a wife, 15% of all evening calls are answered by a single male, and 40% of all daytime calls are not answered at all. Due to lim- ited staff, at most 40% of all phone calls can be evening calls. a. Determine how to minimize the cost of completing

the survey. b. Use SolverTable to investigate changes in the unit cost

of either type of call. Specifically, investigate changes in the cost of a daytime call, with the cost of an eve- ning call fixed, to see when (if ever) only daytime calls or only evening calls will be made. Then repeat the analysis by changing the cost of an evening call and keeping the cost of a daytime call fixed.

32. A furniture company manufactures tables and chairs. Each table and chair must be made entirely out of oak or entirely out of pine. A total of 15,000 board feet of oak and 21,000 board feet of pine are available. A table

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b. Use SolverTable to investigate the effects of increases in the minimal reductions required by the state. Spe- cifically, see what happens to the amounts of waste processed at the three factories and the total cost if both requirements (currently 30 and 40 tons, respec- tively) are increased by the same percentage. Revise your model so that you can use SolverTable to investi- gate these changes when the percentage increase var- ies from 10% to 100% in increments of 10%. Do the amounts processed at the three factories and the total cost change in a linear manner?

Level B 35. A company manufactures two types of trucks. Each

truck must go through the painting shop and the assembly shop. If the painting shop were completely devoted to painting type 1 trucks, 800 per day could be painted, whereas if the painting shop were com- pletely devoted to painting type 2 trucks, 700 per day could be painted. If the assembly shop were com- pletely devoted to assembling truck 1 engines, 1500 per day could be assembled, whereas if the assembly shop were completely devoted to assembling truck 2 engines, 1200 per day could be assembled. It is possi- ble, however, to paint both types of trucks in the paint- ing shop. Similarly, it is possible to assemble both types in the assembly shop. Each type 1 truck contributes $1000 to profit; each type 2 truck contributes $1500. Use Solver to maximize the company’s profit. (Hint: One approach, but not the only approach, is to try a graphical procedure first and then deduce the constraints from the graph.)

36. A company manufactures mechanical heart valves from the heart valves of pigs. Different heart operations require valves of different sizes. The company purchases pig valves from three different suppliers. The cost and size mix of the valves purchased from each supplier are given in the file P13_36.xlsx. Each month, the company places an order with each supplier. At least 500 large, 300 medium, and 300 small valves must be purchased each month. Because of the limited availability of pig valves, at most 500 valves per month can be purchased from each supplier. a. Use Solver to determine how the company can mini-

mize the cost of acquiring the needed valves. b. Use SolverTable to investigate the effect on total cost

of increasing its minimal purchase requirements each month. Specifically, see how the total cost changes as the minimal purchase requirements of large, medium, and small valves all increase from their original val- ues by the same percentage. Revise your model so that SolverTable can be used to investigate these changes when the percentage increase varies from 2% to 20% in increments of 2%. Explain intuitively what happens when this percentage is at least 16%.

requires either 17 board feet of oak or 30 board feet of pine, and a chair requires either 5 board feet of oak or 13 board feet of pine. Each table can be sold for $800, and each chair for $300. a. Determine how the company can maximize its

revenue. b. Use SolverTable to investigate the effects of simul-

taneous changes in the selling prices of the products. Specifically, see what happens to the total revenue when the selling prices of oak products and the sell- ing prices of pine products are allowed to vary (inde- pendently) by as much as plus or minus 30%, in increments of 10%, from their original values. Revise your model from the previous problem so that you can use SolverTable to investigate these changes. Can you conclude that total revenue changes linearly within this range?

33. A manufacturing company makes two products. Each product can be made on either of two machines. The time (in hours) required to make each product on each machine is listed in the file P13_33.xlsx. Each month, 500 hours of time are available on each machine. Each month, customers are willing to buy up to the quantities of each product at the prices also given in the same file. The company’s goal is to maximize the revenue obtained from selling units during the next two months. a. Determine how the company can meet this goal.

Assume that it will not produce any units in a month that it cannot sell in that month.

b. Use SolverTable to see what happens if customer demands for each product in each month simultane- ously change by as much as plus or minus 30%, in increments of 10%, from their current values. Revise the model so that you can use SolverTable to investi- gate the effect of these changes on total revenue. Does revenue change in a linear manner over this range? Can you explain intuitively why it changes in the way it does?

34. There are three factories on the Momiss River. Each emits two types of pollutants, labeled P1 and P2, into the river. If the waste from each factory is processed, the pollution in the river can be reduced. It costs $1500 to process a ton of factory 1 waste, and each ton processed reduces the amount of P1 by 0.10 ton and the amount of P2 by 0.45 ton. It costs $1000 to process a ton of fac- tory 2 waste, and each ton processed reduces the amount of P1 by 0.20 ton and the amount of P2 by 0.25 ton. It costs $2000 to process a ton of factory 3 waste, and each ton processed reduces the amount of P1 by 0.40 ton and the amount of P2 by 0.30 ton. The state wants to reduce the amount of P1 in the river by at least 30 tons and the amount of P2 by at least 40 tons. a. Use Solver to determine how to minimize the cost of

reducing pollution by the desired amounts. Are the LP proportionality and additivity assumptions reasonable in this problem?

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13-11 Conclusion 6 2 7

b. Use SolverTable to see how sensitive the total cost is to the 16 mpg requirement. Specifically, let this requirement vary from 14 mpg to 18 mpg in incre- ments of 0.25 mpg. Explain intuitively what happens when the requirement is greater than 17 mpg.

39. A textile company produces shirts and pants. Each shirt requires two square yards of cloth, and each pair of pants requires three square yards of cloth. During the next two months the following demands for shirts and pants must be met (on time): month 1, 1000 shirts and 1500 pairs of pants; month 2, 1200 shirts and 1400 pairs of pants. During each month the following resources are available: month 1, 9000 square yards of cloth; month 2, 6000 square yards of cloth. In addition, cloth that is available during month 1 and is not used can be used during month 2. During each month it costs $8 to pro- duce an article of clothing with regular-time labor and $16 with overtime labor. During each month a total of at most 2500 articles of clothing can be produced with regular-time labor, and an unlimited number of articles of clothing can be produced with overtime labor. At the end of each month, a holding cost of $3 per article of clothing is incurred. a. Determine how to meet demands for the next two

months (on time) at minimum cost. Assume that 100 shirts and 200 pairs of pants are already in inventory at the beginning of month 1.

b. Use a two-way SolverTable to investigate the effect on total cost of two simultaneous changes. The first change is to allow the ratio of overtime to regu- lar-time production cost (currently $16>$8 5 2) to decrease from 20% to 80% in increments of 20%, while keeping the regular time cost at $8. The sec- ond change is to allow the production capacity each month (currently 2500) to decrease by 10% to 50% in increments of 10%. The idea here is that less reg- ular-time capacity is available, but overtime becomes relatively cheaper. Is the net effect on total cost posi- tive or negative?

40. Each year, a shoe manufacturing company faces demands (which must be met on time) for pairs of shoes as shown in the file P13_40.xlsx. Employees work three consecutive quarters and then receive one quarter off. For example, a worker might work during quarters 3 and 4 of one year and quarter 1 of the next year. During a quarter in which an employee works, he or she can produce up to 500 pairs of shoes. Each worker is paid $5000 per quarter. At the end of each quarter, a holding cost of $10 per pair of shoes is incurred. a. Determine how to minimize the cost per year (labor

plus holding) of meeting the demands for shoes. To simplify the model, assume that at the end of each year, the ending inventory is 0. (You can assume that a given worker gets the same quarter off during each year.)

37. A company that builds sailboats wants to determine how many sailboats to build during each of the next four quarters. The demand during each of the next four quarters is as follows: first quarter, 160 sailboats; sec- ond quarter, 240 sailboats; third quarter, 300 sailboats; fourth quarter, 100 sailboats. The company must meet demands on time. At the beginning of the first quarter, the company has an inventory of 40 sailboats. At the beginning of each quarter, the company must decide how many sailboats to build during that quarter. For simplicity, assume that sailboats built during a quarter can be used to meet demand for that quarter. During each quarter, the company can build up to 160 sailboats with regular-time labor at a total cost of $1600 per sail- boat. By having employees work overtime during a quarter, the company can build additional sailboats with overtime labor at a total cost of $1800 per sailboat. At the end of each quarter (after production has occurred and the current quarter’s demand has been satisfied), a holding cost of $80 per sailboat is incurred. a. Determine a production schedule to minimize the sum

of production and inventory holding costs during the next four quarters.

b. Use SolverTable to see whether any changes in the $80 holding cost per sailboat could induce the com- pany to carry more or less inventory. Revise your model so that SolverTable can be used to investi- gate the effects on ending inventory during the four- quarter period of systematic changes in the unit holding cost. (Assume that even though the unit hold- ing cost changes, it is still constant over the four-quar- ter period.) Are there any (nonnegative) unit holding costs that would induce the company to hold more inventory than it holds when the holding cost is $80 ? Are there any unit holding costs that would induce the company to hold less inventory than it holds when the holding cost is $80?

38. During the next two months an automobile manufac- turer must meet (on time) the following demands for trucks and cars: month 1, 400 trucks and 800 cars; month 2, 300 trucks and 300 cars. During each month at most 1000 vehicles can be produced. Each truck uses two tons of steel, and each car uses one ton of steel. During month 1, steel costs $700 per ton; during month 2, steel is projected to cost $800 per ton. At most 2500 tons of steel can be purchased each month. (Steel can be used only during the month in which it is purchased.) At the beginning of month 1, 100 trucks and 200 cars are in the inventory. At the end of each month, a holding cost of $200 per vehicle is assessed. Each car gets 20 miles per gallon (mpg), and each truck gets 10 mpg. During each month, the vehicles produced by the company must average at least 16 mpg. a. Determine how to meet the demand and mileage

requirements at minimum total cost.

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42. A pharmaceutical company manufactures two drugs at Los Angeles and Indianapolis. The cost of manufactur- ing a pound of each drug depends on the location, as indicated in the file P13_42.xlsx. The machine time (in hours) required to produce a pound of each drug at each city is also shown in this table. The company must produce at least 1000 pounds per week of drug 1 and at least 2000 pounds per week of drug 2. It has 500 hours per week of machine time at Indianapolis and 400 hours per week at Los Angeles. a. Determine how the company can minimize the cost of

producing the required drugs. b. Use SolverTable to determine how much the company

would be willing to pay to purchase a combination of A extra hours of machine time at Indianapolis and B extra hours of machine time at Los Angeles, where A and B can be any positive multiples of 10 up to 50.

43. A company manufactures two products on two machines. The number of hours of machine time and labor depends on the machine and product as shown in the file P13_43.xlsx. The cost of producing a unit of each product depends on which machine produces it. These unit costs also appear in the same file. There are 200 hours available on each of the two machines, and there are 400 labor hours available total. This month at least 200 units of product 1 and at least 240 units of product 2 must be produced. Also, at least half of the product 1 requirement must be produced on machine 1, and at least half of the product 2 requirement must be produced on machine 2. a. Determine how the company can minimize the cost of

meeting this month’s requirements. b. Use SolverTable to see how much the “at least half”

requirements are costing the company. Do this by changing both of these requirements from “at least half” to “at least x percent,” where x can be any multi- ple of 5% from 0% to 50%.

b. Suppose the company can pay a flat fee for a train- ing program that increases the productivity of all of its workers. Use SolverTable to see how much the com- pany would be willing to pay for a training program that increases worker productivity from 500 pairs of shoes per quarter to P pairs of shoes per quarter, where P varies from 525 to 700 in increments of 25.

41. A small appliance manufacturer must meet (on time) the following demands: quarter 1, 3000 units; quarter 2, 2000 units; quarter 3, 4000 units. Each quarter, up to 2700 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlim- ited number of units can be produced with overtime labor, at a cost of $60 per unit. Of all units produced, 20% are unsuitable and cannot be used to meet demand. Also, at the end of each quarter, 10% of all units on hand spoil and cannot be used to meet any future demands. After each quarter’s demand is satisfied and spoilage is accounted for, a cost of $15 per unit in ending inventory is incurred. a. Determine how to minimize the total cost of meeting

the demands of the next three quarters. Assume that 1000 usable units are available at the beginning of quarter 1.

b. The company wants to know how much money it would be worth to decrease the percentage of unsuit- able items and/or the percentage of items that spoil. Write a short report that provides relevant informa- tion. Base your report on three uses of SolverTable: (1) where the percentage of unsuitable items decreases and the percentage of items that spoil stays at 10%, (2) where the percentage of unsuitable items stays at 20% and the percentage of items that spoil decreases, and (3) where both percentages decrease. Does the sum of the separate effects on total cost from the first two tables equal the combined effect from the third table? Include an answer to this question in your report.

CASE 13.1 Shelby Shelving Shelby Shelving is a small company that manufactures two types of shelves for grocery stores. Model S is the stan- dard model; model LX is a heavy-duty version. Shelves are manufactured in three major steps: stamping, forming, and assembly. In the stamping stage, a large machine is used to stamp (i.e., cut) standard sheets of metal into appropri- ate sizes. In the forming stage, another machine bends the metal into shape. Assembly involves joining the parts with a combination of soldering and riveting. Shelby’s stamping and forming machines work on both models of shelves. Sep- arate assembly departments are used for the final stage of production.

The file C13_01.xlsx contains relevant data for Shelby. (See Figure 13.38.) The hours required on each machine for

each unit of product are shown in the range B5:C6 of the Accounting Data sheet. For example, the production of one model S shelf requires 0.25 hour on the forming machine. Both the stamping and forming machines can operate for 800 hours each month. The model S assembly department has a monthly capacity of 1900 units. The model LX assem- bly department has a monthly capacity of only 1400 units. Currently Shelby is producing and selling 400 units of model S and 1400 units of model LX per month.

Model S shelves are sold for $1800, and model LX shelves are sold for $2100. Shelby’s operation is fairly small in the industry, and management at Shelby believes it cannot raise prices beyond these levels because of the competition. However, the marketing department believes that Shelby can

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sell as much as it can produce at these prices. The costs of production are summarized in the Accounting Data sheet. As usual, values in blue cells are given, whereas other values are calculated from these.

Management at Shelby just met to discuss next month’s operating plan. Although the shelves are selling well, the overall profitability of the company is a concern. The plant’s engineer suggested that the current production of model S shelves be cut back. According to him, “Model S shelves are sold for $1800 per unit, but our costs are $1839. Even though we’re selling only 400 units a month, we’re losing money on each one. We should decrease production of model S.” The controller disagreed. He said that the problem was the model S assembly department trying to absorb a large over- head with a small production volume. “The model S units are making a contribution to overhead. Even though produc- tion doesn’t cover all of the fixed costs, we’d be worse off with lower production.”

Your job is to develop an LP model of Shelby’s prob- lem, then run Solver, and finally make a recommendation to Shelby management, with a short verbal argument support- ing the engineer or the controller.

Notes on Accounting Data Calculations The fixed overhead is distributed using activity-based cost- ing principles. For example, at current production levels, the forming machine spends 100 hours on model S shelves and 700 hours on model LX shelves. The forming machine is used 800 hours of the month, of which 12.5% of the time is spent on model S shelves and 87.5% is spent on model LX shelves. The $95,000 of fixed overhead in the forming department is distributed as $11,875(= 95,000 3 0.125) to model S shelves and $83,125(= 95,000 3 0.875) to model LX shelves. The fixed overhead per unit of out- put is allocated as $29.69(= 11,875>400) for model S and $59.38(= 83,125>1400) for model LX. In the cal- culation of the standard overhead cost, the fixed and variable costs are added together, so that the overhead cost for the forming department allocated to a model S shelf is $149.69(= 29.69 1 120, shown in cell G20 rounded up to $150). Similarly, the overhead cost for the forming department allocated to a model LX shelf is $229.38(= 59.38 1 170, shown in cell H20 rounded down to $229).

Figure 13.38 Data for Shelby Case

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23

A B C D E F G H I Shelby Shelving Data for Current Production Schedule

Machine requirements (hours per unit) Given monthly overhead cost data Model S Model LX Fixed Variable S Variable LX

$90$80$125,0000.30.3Stamping $170$120$95,000

$80,000 $85,000

0.5

Available 800 8000.25Forming

Model S Assembly $165 $0 Model S Model LX Model LX Assembly $0 $185

Current monthly production 400 1400

Hours spent in departments Model S Model LX Model S Model LX Totals Direct materials $1,000 $1,200

Direct labor:540420120Stamping 800700100Forming

Stamping Forming

$35 $35 $60 $90

Percentages of time spent in departments Assembly $80 $85 Model S Model LX Total direct labor $175 $210

Overhead allocation 77.8%22.2% 87.5%12.5% $149 $159

$150 $229 Unit selling price $1,800 $2,100 Assembly $365 $246

Total overhead $664 $635 Assembly capacity 1900 1400 Total cost $1,839 $2,045

Standard costs of the shelves -- based on the current production levels

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CHAPTER 14 Optimization Models

OPTIMIZATION OF WORK CENTER LOCATIONS AT VERIZON Verizon, the telecommunications giant, must continu- ally install, maintain, and expand its infrastructure. This requires many technicians and vehicles located at garage work centers (GWCs). Each GWC serves as a home base for technicians, it provides parts, supplies, and tools, and it provides parking for vehicles. The article by Allen et al. (2017) describes how a team at Verizon used optimization to determine where to locate GWCs and how to assign tech- nicians to GWCs to provide appropriate service levels at minimum cost.

The analysis involved approximately 500 GWCs and 23,000 technicians at Verizon. Each technician is assigned to a GWC and is then dispatched from the GWC to a service region called a wire center (WC) as the need arises. Therefore, the decisions on the num- ber and locations of GWCs, which WCs are served by which GWCs, and which techni- cians are assigned to which GWCs, have a direct impact on productivity and operating costs. Verizon’s problem is a classical optimization problem in management science. If there are two few, or poorly placed, GWCs, travel costs will be large and the level of service will suffer. In addition, technicians will spend too much time traveling, time that could be used for actual work. If there are too many GWCs, overhead costs such as leas- ing costs will be large. Therefore, an appropriate trade-off had to be found.

The optimization model the analysts developed is a mixed-integer programming model, where some decision variables are continuous and others are integer-valued, often binary (0-1). For example, for each potential GWC site has an associated binary variable: 1 if the site is used and 0 otherwise. If a potential GWC was an existing site, a binary value of 0 would mean closing it and reassigning its technicians to another GWC. If a potential GWC was a new site, a binary value of 1 would mean building a new GWC. In addition, each GWC-WC combination has an associated binary variable: 1 if the WC is assigned to the GWC and 0 otherwise. Because of Verizon policy, the model assumed that that at most two GWCs can serve a single WC.

A significant part of the effort involved the collection and verification of the many required data inputs. These include: distances between potential GWCs and WCs; the capacity of each potential GWC to support technicians; the operating cost of each potential GWC; and the demand at each WC, expressed as the number of technician hours required per day. The data came from multiple systems and databases, so they had to be checked for consistency and accuracy. It was important to develop a standardized data process so that the model could be run in the future to adapt to changes in demand and operations.

The Verizon analysts fortunately discovered a simplification for their large mixed- integer optimization model. The study involved 12 states where Verizon operates. How- ever, due to union rules or public utility commission restrictions, no interactions normally occur across state lines. In fact, two large states, Pennsylvania and New York, could each be divided into two parts that had little interaction. This allowed the analysts to decom- pose one large model into 14 smaller models. Even so, some states’ models were still quite large, with hundreds of thousands of decision variables and hundreds of thousands of constraints. This is much too large to be solved in Excel, the focus in this book, but the

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14-1 Introduction 6 3 1

optimization software package used in the study, CPLEX, was able to solve each model in a few minutes.

Naturally, once the model was up and running, the stakeholders asked a number of “what if” questions. To help answer these, the analysts made several runs, each time fixing the number of GWCs closed at a value ranging from zero to the point where no feasible solutions were possible. They found that when no GWCs were closed, significant sav- ings in labor and vehicle costs were achieved by optimally reassigning WCs to current GWCs. These savings continued to increase when only a few GWCs were closed. How- ever, beyond some number of GWC closures, the total of all costs started to increase. The optimal number of closures is where the total cost stops decreasing and begins to increase.

Verizon estimates that this model has resulted in savings of $18 million annually, and the company plans to continue using the model, on even a larger scale, in the future.

14-1 Introduction In a survey of Fortune 500 firms, 85% of those responding said that they use optimization. In this chapter, we discuss some of the optimization models that are most often applied to real-world applications. Some typical examples include:

• scheduling bank clerks for check encoding • optimizing the operation of an oil refinery • planning dairy production at a creamery • scheduling production of products at a fiberglass manufacturer • optimizing a Wall Street firm’s bond portfolio.

There are two basic goals in this chapter. The first is to illustrate some of the many real applications that can take advantage of optimization. You will see that these applications cover a wide range, from oil production to worker scheduling to cash management. The second goal is to increase your facility in modeling optimization problems on a spreadsheet. We present a few principles that will help you model a wide variety of problems. The best way to learn, however, is to see many examples and work through numerous problems. In short, mastering the art of spreadsheet optimization modeling takes hard work and practice. You will have plenty of opportunity to do both with the material in this chapter.

Although a wide variety of problems can be formulated as linear programming mod- els, there are some that cannot. Either they require integer variables or they are nonlin- ear in the decision variables. We include examples of integer programming and nonlinear programming models in this chapter, just to give you a taste of what is involved.1 You will see that the modeling process for these types of problems is not much different than for linear optimization problems. Once the models are developed, Excel’s Solver can be used to solve them. Then SolverTable can be used to perform sensitivity analysis. How- ever, these integer and nonlinear models are inherently more difficult to solve. Solver must use more complex algorithms and is not always guaranteed to find an optimal solution. Nevertheless, you will see that Solver provides the power to solve a great variety of realis- tic business problems.

Although there is a tremendous amount of theory behind the algorithms that solve these problems, the modeling process itself is fairly straightforward, and you can learn it best by seeing a variety of examples. Therefore, we proceed in this chapter by mod- eling (and then solving) a diverse class of problems that arise in business. The exercises

1 Besides the nonlinear models discussed in this chapter, which can be solved with Solver’s GRG nonlinear algorithm, there is an even more difficult class of nonlinear models called nonsmooth models. Although we will not discuss nonsmooth models, we can recommend Solver’s Evolutionary algorithm for these difficult models. These nonsmooth problems can also be solved with the Evolver add-in, part of Palisade’s DecisionTools® Suite, but we won’t discuss Evolver in this book.

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6 3 2 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

throughout the chapter provide even more examples of how optimization models can be applied.

All these models can benefit from sensitivity analysis, done formally with the SolverTable add-in or informally by changing one or more inputs and rerunning Solver. For reasons of space, we present only a few of the many possible sensitivity analyses. However, we stress that in real applications, model development is just the beginning of the overall analysis. It is then usually followed by extensive sensitivity analysis.

14-2 Employee Scheduling Models Many organizations must determine how to schedule employees to provide adequate ser- vice. The following example illustrates how to use linear programming (with integer con- straints) to schedule employees on a daily basis.

EXAMPLE

14.1 SCHEDULING EMPLOYEES AT BRIGGS Briggs, a small business company, requires different numbers of full-time employees on different days of the week. The num- ber of full-time employees required each day is given in Table 14.1. Union rules state that each full-time employee must work five consecutive days and then receive two days off. For example, an employee who works Monday to Friday must be off on Saturday and Sunday. Briggs wants to meet its daily requirements using only full-time employees. Its objective is to minimize the number of full-time employees on its payroll.

Table 14.1 Employee Requirements

Day of Week Minimum Number of Employees Required

Monday 17

Tuesday 13

Wednesday 15

Thursday 19

Friday 14

Saturday 16

Sunday 11

Objective To develop an optimization model that relates five-day shift schedules to daily numbers of employees available, and to use Solver to find a schedule that uses the fewest number of employees and meets all daily workforce requirements.

Where Do the Numbers Come From? The only inputs needed for this problem are the minimum employee requirements in Table 14.1, but these are not easy to obtain. They would probably be obtained through a combination of two quantitative techniques: forecasting (Chapter 12) and queueing analysis (not covered in this book). The company would first use historical data to forecast customer arrival patterns throughout a typical week. It would then use queueing analysis to translate these arrival patterns into employee requirements on a daily basis. Actually, we have kept the problem relatively simple by considering only daily requirements. In a realistic setting, the organization might forecast employee requirements on an hourly or even a 15-minute basis.

Solution A diagram of this model appears in Figure 14.1. (See the file Employee Sched- uling Big Picture.xlsx.) The trickiest part is identifying the appropriate decision variables. You might think that the decision variables are the numbers of employees working on the various days of the week. Clearly, these values must eventually be

In real employee-scheduling problems, much of the work involves forecasting and queueing analysis to obtain employee requirements. This must be done before an optimal schedule can be found.

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14-2 Employee Scheduling Models 6 3 3

determined. However, it is not enough to specify, for example, that 18 employees are working on Monday. The problem is that this doesn’t indicate when these 18 employees start their five-day shifts. Without this knowledge, it is impossible to implement the five-consecutive-day, two-day-off requirement. (If you don’t believe this, try developing your own model with the wrong decision variables. You will eventually reach a dead end.)

Figure 14.1 Big Picture for Employee Scheduling Model

Employees from each shi� available

Total employees available

Employees required

Employees star�ng 5-day shi�

Minimize total employees

The trick is to define the decision variables as the numbers of employees working each of the seven possible five-day shifts. By knowing these values, the other output variables can be calcu- lated. For example, the number working on Thursday is the sum of those who begin their five-day shifts on Sunday, Monday, Tuesday, Wednesday, and Thursday.

The key to this model is choosing the correct decision variables.

Choosing the Decision Variables

The decision variables should always be chosen so that their values determine all required outputs in the model. In other words, their values should tell the company exactly how to run its business. Sometimes the choice of deci- sion variables is obvious, but in many cases (as in this employee scheduling model), the proper choice of decision variables takes some deeper thinking about the problem. An improper choice of decision variables typically leads to a dead end, where their values do not supply enough information to calculate required outputs or implement certain constraints.

Fundamental Insight

Note that this is a “wraparound” problem. We assume that the daily requirements in Table 14.1 and the employee sched- ules continue week after week. So, for example, the employees assigned to the Thursday through Monday shift always wrap around from one week to the next on their five-day shift.

Developing the Spreadsheet Model The spreadsheet model for this problem is shown in Figure 14.2. (See the file Employee Scheduling Finished. xlsx.) To develop this model, proceed as follows.

1. Inputs and range names. Enter the number of employees needed on each day of the week (from Table 14.1) in the blue cells, and create the range names shown.

2. Employees beginning each day. Enter any trial values for the number of employees beginning work on each day of the week in the Employees_starting range. These beginning days determine the possible five-day shifts. For example, the employees in cell B4 work Monday through Friday.

3. Employees on hand each day. The key to this solution is to realize that the numbers in the Employees_starting range— the decision variable cells—do not represent the number of employees who will show up each day. As an example, the number in cell B4 represent those who start on Monday work Monday through Friday. Therefore, enter the formula

5$B$4

in cell B14 and copy it across to cell F14. Proceed similarly for rows 15–20, being careful to take “wraparounds” into account. For example, the workers starting on Thursday work Thursday through Sunday, plus Monday. Then calculate the total number who are available on each day by entering the formula

5SUM(B14:B20)

in cell B23 and copying it across row 23.

Developing the Employee Scheduling Model

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Figure 14.2 Employee Scheduling Model with Optimal (Non-Integer) Solution

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

A B C D E F G H I J K Employee scheduling model Range names used

Employees_available

Employees_Star�ng Decision variables: number of employees star�ng their five-day shi� on various days Employees_required

1.33Mon 3.33Tue 2.00Wed 7.33Thu 0.00Fri 3.33

3.33 3.33 3.33 3.33 3.33

Sat 5.00

3.33 3.33 3.33 3.33 5.00 5.00 5.00 5.00

Sun

Mon Tue Wed Thu Fri Sat Sun

Result of decisions: number of employees working on various days (along top) who started their shi� on various days (along side) Mon Tue Wed Thu Fri Sat Sun 1.33 1.33 1.33 1.33 1.33

2.002.002.002.00 7.33 7.33 7.33 7.33 0.00 0.00 0.00 0.00

3.33

2.00 7.33 0.00

5.00

Constraint on employee availabili�es Employees available

Employees required

17.00 13.00 15.00 19.00 14.00 16.00 17.67 >= >= >= >= >= >= >= 17 13 15 19 14 16 11

Objec�ve to maximize Total employees

Total_employees

22.33

=Model!$B$4:$B$10 =Model!$B$28

=Model!$B$25:$H$25 =Model!$B$23:$H$23

Ctrl 1 Enter Shortcut You often enter a typical formula in a cell and then copy it. One way to do this efficiently is to select the entire range, here B23:H23. Then enter the typical formula, here 5SUM(B14:B20), and press Ctrl1Enter. This has the same effect as copying, but it is slightly quicker.

Excel Tip

4. Total employees. Calculate the total number of employees in cell B28 with the formula

5SUM(Employees_starting)

Note that there is no double-counting in this sum. For example, the employees in cells B4 and B5 are not the same people.

At this point, you might want to experiment with the numbers in the decision variable cell range to see whether you can guess an optimal solution (without looking at Figure 14.2). It is not that easy. Each employee who starts on a given day works the next four days as well, so when you find a solution that meets the minimal requirements for the various days, you usually have a few more employees available on some days than are needed.

Using Solver Invoke Solver and fill out its main dialog box as shown in Figure 14.3. (You don’t need to include the integer constraint yet. This will be discussed shortly.) Make sure you check the Non-Negative option and select the Simplex LP method.

Discussion of the Solution The optimal solution shown in Figure 14.2 has one drawback: It requires the number of employees starting work on some days to be a fraction. Because part-time employees are not allowed (an assumption of the model), this solution is unrealistic. How- ever, it is simple to add an integer constraint on the decision variable cells. This integer constraint appears in Figure 14.3. (To create this integer constraint in Solver’s Add Constraint dialog box, select the Employees_starting for the left side, and select “int” in the middle dropdown list. The word “integer” will automatically appear in the right side of the constraint.) With this integer constraint, the optimal solution appears in Figure 14.4.

6 3 4 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

The shortcut on the Mac is command+return.

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14-2 Employee Scheduling Models 6 3 5

Figure 14.3 Solver Dialog Box for Employee Scheduling Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

A B C D E F G H I J K Employee scheduling model Range names used

Employees_available

Employees_Star�ng Decision variables: number of employees star�ng their five-day shi� on various days Employees_required

2Mon 3Tue 3Wed 7Thu 0Fri 4

3 3 3 3 3

Sat 4

4 4 4 4 4 4 4 4

Sun

Mon Tue Wed Thu Fri Sat Sun

Result of decisions: number of employees working on various days (along top) who started their shi� on various days (along side) Mon Tue Wed Thu Fri Sat Sun

2 2 2 2 2

3333 7 7 7 7 0 0 0 0

3 7 0

Constraint on employee availabili�es Employees available

Employees required

17 13 16 19 15 17 18 >= >= >= >= >= >= >= 17 13 15 19 14 16 11

Objec�ve to maximize Total employees

Total_employees

=Model!$B$4:$B$10 =Model!$B$28

=Model!$B$25:$H$25 =Model!$B$23:$H$23

Figure 14.4 Optimal Integer Solution to Employee Scheduling Model

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6 3 6 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

The decision variable cells in the optimal solution indicate the numbers of workers who start their five-day shifts on the various days. You can then look at the columns of the B14:H20 range to see which employees are working on any given day. This optimal solution is typical in scheduling problems. Due to a labor constraint—each employee must work five consecutive days and then have two days off—it is typically impossible to meet the minimum employee requirements exactly. To ensure that there are enough employees available on busy days, it is necessary to have more than enough on hand on light days.

Another interesting aspect of this problem is that when you solve it, you might get a different schedule that is still opti- mal—that is, a solution that still uses a total of 23 employees and meets all constraints. This is a case of multiple optimal solutions, not at all uncommon in linear optimization problems. In fact, it is typically good news for a manager, who can then choose among the optimal solutions using other, possibly nonquantitative criteria.

Sensitivity Analysis The most obvious type of sensitivity analysis in this example is to analyze the effect of employee requirements on the optimal solution. Specifically, let’s suppose the number of employees needed on each day of the week increases by two, four, or six. How does this change the total number of employees needed? You can answer this with SolverTable, but you must first modify the model slightly, as shown in Figure 14.5. The problem is that we want to increase each of the daily minimal required values by the same amount. The trick is to enter the original require- ments in row 12, enter a trial value for the extra number required per day in cell K12, enter the formula 5B121$K$12 in cell B27, and then copy this formula across to cell H27. Then you can use the one-way SolverTable option, using the Extra cell as the single input, letting it vary from 0 to 6 in increments of 2, and specifying the Total_employ- ees cell as the single output cell.

The results appear in Figure 14.6. When the requirement increases by two each day, only two extra employees are neces- sary (scheduled appropriately). However, when the requirement increases by four each day, more than four extra employees are necessary. The same is true when the requirement increases by six each day. This might surprise you at first, but there is an intuitive reason: Each extra worker works only five days of the week.

We did not use Solver’s sensitivity report here for two reasons. First, Solver does not offer a sensitivity report for models with integer constraints. Second, even if the integer constraints are deleted, Solver’s sensitivity report does not answer ques- tions about multiple input changes, as we have asked here. It is used for questions about one-at-a-time changes to inputs, such as a change to Thursday’s worker requirement. In this sense, SolverTable is a more flexible tool.

Modeling Issues • The employee scheduling example is called a static scheduling model because we assume that the company faces the same

situation each week. In reality, demands change over time, employees take vacations in the summer, and so on, so the com- pany does not face the same situation each week. A dynamic scheduling model (not covered here) is necessary for such problems.

Solver Integer Optimality Setting When working with integer constraints, you should be aware of Solver’s Integer Optimality setting. The idea is as follows. As Solver searches for the best integer solution, it is often able to find a “good” integer solution fairly quickly, but it often has to spend a lot of time finding slightly better solutions. A nonzero setting allows it to quit early. The default setting is 1 (percent). (It used to be 5, which you still might see, depending on your version of Excel.) This means that if Solver finds a feasible integer solution that is guaranteed to have an objective value no more than 1% from the optimal value, it will quit and report this good solution (which might even be the optimal solution). Therefore, if you keep this default setting, your integer solutions will sometimes not be optimal, but they will be close. If you want to ensure that you get an optimal solution, you can change the Solver setting to zero. (Click the Options button, and then under the All Methods tab, uncheck Ignore Integer Constraints and enter a value in the Integer Optimality (%) box.)

Technical Tip Set Solver’s Integer Optimality to zero to ensure that you get the optimal integer solution. Be aware, however, that this can incur significant extra computing time for larger models.

Multiple optimal solutions have different values in the decision variable cells, but they all have the same objective value.

To run some sensitivity analyses with SolverTable, you need to modify the original model slightly to incorporate the effect of the input being varied.

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14-2 Employee Scheduling Models 6 3 7

• In a weekly scheduling model for a supermarket or a fast-food restaurant, the number of decision variables can grow quickly and optimization software such as Solver will have dif- ficulty finding an optimal solution. In such cases, heuristic methods (essentially clever tri- al-and-error algorithms) have been used to find good solutions to the problem. For example, Love and Hoey (1990) indicate how this was done for a particular staff scheduling problem.

• Our model can easily be expanded to handle part-time employees, the use of overtime, and alternative objectives such as maximizing the number of weekend days off received by employees. You are asked to explore such extensions in the problems.

Figure 14.5 Modified Employee Scheduling Model

1 2 3 4 5 6 7 8 9

10 11

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

A B C D E F G H I J K Employee scheduling model Range names used

Employees_available

Employees_Star�ng Decision variables: number of employees star�ng their five-day shi� on various days Employees_required

2Mon 3Tue 3Wed 7Thu 0Fri 4

3 3 3 3 3

Sat 4

4 4 4 4 4 4 4 4

Sun

Employees required (original values) Extra required each day 0

Mon Tue Wed Thu Fri Sat Sun

Result of decisions: number of employees working on various days (along top) who started their shi� on various days (along side) Mon Tue Wed Thu Fri Sat Sun

2 2 2 2 2

3333 7 7 7 7 0 0 0 0

17 13 15 19 14 1116

7 0

Constraint on employee availabili�es Employees available

Employees required

17 13 16 19 15 17 18 >= >= >= >= >= >= >= 17 13 15 19 14 16 11

Objec�ve to maximize Total employees

Total_employees

12 13

=Model!$B$4:$B$10 =Model!$B$28

=Model!$B$25:$H$25 =Model!$B$23:$H$23

Note how the original model has been modified so that the extra value in cell K12 drives all of the requirements in row 27.

Figure 14.6 Sensitivity to Number of Extra Employees Required per Day

3 2 1

4 5 6 7 8 9

10 11

A B C D E F G H I

Extra required (cell $K$12) values along side, output cell(s) along top

To ta

l_ em

pl oy

ee s

0 23 2 25 4 28 6 31

0 2 4 6 Extra required ($K$12)

Sensi�vity of Total_employees to Extra required

Oneway analysis for Solver model in Model Sensi�vity worksheet

Heuristic solutions are often close to optimal, but they are never guaranteed to be optimal.

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6 3 8 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

14-3 Blending Models In many situations, various inputs must be blended to produce desired outputs. In many of these situations, blending models can be used to find the optimal combination of outputs as well as the mix of inputs that are used to produce the desired outputs. The following are some typical examples of blending problems.

Inputs Outputs

Meat, filler, water Different types of sausage Various types of oil Heating oil, gasolines, aviation fuels Carbon, iron, molybdenum Different types of steels Different types of pulp Different kinds of recycled paper

Example 14.2 illustrates how to model a typical blending problem in Excel. Although this example is small relative to blending problems in real applications, it is still probably too complex for you to guess the optimal solution.

Problems Solutions for problems whose numbers appear within a colored box can be found in the Student Solution Files.

Level A 1. Modify the Briggs employee scheduling model so that

employees are paid $10 per hour on weekdays and $15 per hour on weekends. Change the objective so that you now minimize the weekly payroll. (You can assume that each employee works eight hours per day.) Is the previ- ous optimal solution still optimal?

2. How much influence can the employee requirements for one, two, or three days have on the weekly schedule in the Briggs employee scheduling example? You are asked to explore this in the following questions. a. Let Monday’s requirements change from 17 to 25 in

increments of 1. Use SolverTable to see how the total number of employees changes.

b. Suppose the Monday and Tuesday requirements can each, independently of one another, increase from 1 to 8 in increments of 1. Use a two-way SolverTable to see how the total number of employees changes.

c. Suppose the Monday, Tuesday, and Wednesday requirements each increase by the same amount, where this increase can be from 1 to 8 in increments of 1. Use a one-way SolverTable to investigate how the total number of employees changes.

3. In the Briggs employee scheduling example, suppose each full-time employee works eight hours per day. Thus, Monday’s requirement of 17 workers can be viewed as a requirement of 8(17) 5 136 hours. The company can meet its daily labor requirements by using both full-time and part-time employees. During each week a full-time employee works eight hours a day for five consecutive days, and a part-time employee works four hours a day

for five consecutive days. A full-time employee costs the company $15 per hour, whereas a part-time employee (with reduced fringe benefits) costs the company only $10 per hour. Union requirements limit part-time labor to 25% of weekly labor requirements. a. Modify the model as necessary, and then use Solver to

minimize the post office’s weekly labor costs. b. Use SolverTable to determine how a change in the

part-time labor limitation (currently 25%) influences the optimal solution.

Level B 4. In the Briggs employee scheduling example, suppose

the employees want more flexibility in their schedules. They want to be allowed to work five consecutive days followed by two days off or to work three consecutive days followed by a day off followed by two consecutive days followed by another day off. Modify the original model (with integer constraints) to allow this flexibility. Might this be a good deal for management as well as labor? Explain.

5. In the Briggs employee scheduling example, suppose the company can force employees to work one day of over- time each week on the day immediately following this five-day shift. For example, an employee whose regular shift is Monday to Friday can also be required to work on Saturday. Each employee is paid $100 a day for each of the first five days worked during a week and $135 for the overtime day (if any). Determine how the post office can minimize the cost of meeting its weekly work requirements.

6. In the Briggs employee scheduling example, suppose the company has 28 full-time employees and is not allowed to fire any of them or hire more. Determine a schedule that maximizes the number of weekend days off received by these employees.

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14-3 Blending Models 6 3 9

EXAMPLE

14.2 BLENDING AT CHANDLER OIL Chandler Oil has 5000 barrels of crude oil 1 and 10,000 barrels of crude oil 2 available. Chandler sells gasoline and heating oil. These products are produced by blending the two crude oils together. Each barrel of crude oil 1 has a “quality level” of 10 and each barrel of crude oil 2 has a quality level of 5.2 Gasoline must have an average quality level of at least 8, whereas heating oil must have an average quality level of at least 6. Gasoline sells for $75 per barrel, and heating oil sells for $60 per barrel. In addition, if any barrels of the crude oils are left over, they can be sold for $65 and $50 per barrel, respectively. We assume that demand for heating oil and gasoline is unlimited, so that all of Chandler’s production can be sold. Chandler wants to maximize its revenue from selling gasoline, heating oil, and any leftover crude oils.

Objective To develop an optimization model for finding the revenue-maximizing plan that meets quality constraints and stays within limits on crude oil availabilities.

Where Do the Numbers Come From? Most of the inputs for this problem should be easy to obtain.

• The selling prices for outputs are dictated by market pressures.

• The availabilities of inputs are based on crude supplies from the suppliers.

• The quality levels of crude oils are known from chemical analysis, whereas the required quality levels for outputs are speci- fied by Chandler, probably in response to competitive or regulatory pressures.

Solution The variables and constraints required for this blending model are shown in Figure 14.7. (See the file Blending Oil Big Picture.xlsx.). The key is the selection of the appropriate decision variables. You might think it is sufficient to specify the amounts of the two crude oils used and the amounts of the two products produced. However, this is not enough. The problem is that this information doesn’t tell Chandler how to make the outputs from the inputs. The company instead needs to have a blending plan: how much of each input to use in the production of a barrel of each output. Once you understand that this blending plan is the basic decision, all other output variables follow in a straightforward manner.

In typical blending problems, the correct decision variables are the amounts of each input blended into each output.

Figure 14.7 Big Picture for Oil Blending Model Inputs used

Outputs sold

Quality obtained in outputs

Required quality levels of outputs

Inputs le� over and soldQuality levels of

inputs

Selling prices of outputs

Values of inputs

Inputs available

Blending plan

>= Maximize total revenue

A secondary, but very important, issue in typical blending models is how to implement the quality constraints. (The constraints here are in terms of a generic “quality.” In other blending problems they are often expressed in terms of percentages of some ingredient(s). For example, a typical quality constraint is that some output can contain no more than 2% sulfur.) When we explain how to develop the spreadsheet model, we will discuss the preferred way of implementing the quality constraints.

2 To avoid being overly technical, we use the generic term quality level. In real oil blending, qualities of interest might be octane rating, viscosity, and others.

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6 4 0 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

The gasoline quality constraint is then

AQ in gasoline $ 8 * Gasoline sold (14.1)

Similarly, the heating oil quality constraint is

AQ in heating oil $ 6 * Heating oil sold (14.2)

Developing the Spreadsheet Model The spreadsheet model for this problem appears in Figure 14.8. (See the file Blending Oil Finished.xlsx.) To develop it, proceed as follows.

1. Inputs and range names. Enter the unit selling prices, quality levels for inputs, required quality levels for outputs, and availabilities of inputs in the blue cells. Then name the ranges as indicated.

2. Inputs blended into each output. The quantities Chandler must specify are the barrels of each input used to produce each output. Enter any trial values for these quantities in the Blending_plan range. For example, the value in cell B13 is the amount of crude oil 1 used to make gasoline and the value in cell C13 is the amount of crude oil 1 used to make heat- ing oil. The Blending_plan range contains the decision variable cells.

3. Inputs used and outputs sold. Calculate the row sums (in column D) and column sums (in row 15) of the Blending_plan range. There is a quick way to do this. Select both the row and column where the sums will go (select one, then hold down the Ctrl key and select the other), and click the AutoSum (S) button on the Home ribbon. This creates SUM for- mulas in the selected cells. Then calculate the leftover barrels of each crude oil in column G by subtracting the amount used from the amount available.

4. Quality achieved. Keep track of the quality level of gasoline and heating oil in the Quality_ obtained range as follows. Begin by calculating for each output the average quality (AQ) in the inputs used to produce this output:

AQ in gasoline 5 10 * Oil 1 in gasoline 1 5 * Oil 2 in gasoline

AQ in heating oil 5 10 * Oil 1 in heating oil 1 5 * Oil 2 in heating oil

From here on, the solutions shown are optimal. However, remember that you can start with any solution. It doesn’t even have to be feasible.

Figure 14.8 Oil Blending Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

G HFEDCBA Oil blending model Range names used

Available Barrels_sold Blending_plan Le�over Quality_obtained Quality_required Total_revenue Used

=Model!$F$13:$F$14 =Model!$B$15:$C$15 =Model!$B$13:$C$14 =Model!$G$13:$G$14 =Model!$B$19:$C$19 =Model!$B$21:$C$21 =Model!$B$24 =Model!$D$13:$D$14

Proper�es of crude oil inputs

Proper�es of outputs Selling price per barrel Required quality level

Blending plan (barrels of crude in each output)

Crude oil 1 Crude oil 2 Barrels sold

Quality constraints with cleared denominators Quality constraints in "intui�ve" form

Quality obtained

Quality required

Objec�ve to maximize Total revenue

Crude oil 1 Crude oil 2

Gasoline 3000 2000 5000

Hea�ng oil 2000 8000

10000

Used 5000

10000

Le�over 0 0

<= <=

Gasoline 40000

>= 40000

$975,000

Hea�ng oil 60000

>= 60000

Gasoline 8

>= 8

Hea�ng oil 6

>= 6

Gasoline $75

Hea�ng oil $60

Value per barrel $65 $50

Quality level 10

Available 5000

10000

Developing the Blending Model

Hold down the command key on the Mac.

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To implement Inequalities (14.1) and (14.2), calculate the AQ for gasoline in cell B19 with the formula

5SUMPRODUCT(B13:B14,$C$4:$C$5)

and copy this formula to cell C19 to generate the AQ for heating oil.

5. Quality required. Calculate the required average quality for gasoline and heating oil in cells B21 and C21. Specifically, determine the required average quality for gasoline in cell B21 with the formula

5B9*B15

and copy this formula to cell C21 for heating oil.

6. Revenue. Calculate the total revenue in cell B24 with the formula

5SUMPRODUCT(B15:C15,B8:C8) 1 SUMPRODUCT(G13:G14,B4:B5)

Using Solver Fill out the main Solver dialog box as shown in Figure 14.9. As usual, check the Non-Negative option and select the Simplex LP method before optimizing. You should obtain the optimal solution shown in Figure 14.8.

Figure 14.9 Solver Dialog Box for Blending Model

Discussion of the Solution The optimal solution implies that Chandler should make 5000 barrels of gasoline with 3000 barrels of crude oil 1 and 2000 barrels of crude oil 2. The company should also make 10,000 barrels of heating oil with 2000 barrels of crude oil 1 and 8000 barrels of crude oil 2. With this blend, Chandler will obtain a revenue of $975,000, all from selling gasoline and heating oil.

14-3 Blending Models 6 4 1

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In the second sensitivity analysis, we vary the availability of crude 1 from 2000 barrels to 20,000 barrels in increments of 1000 barrels. The resulting SolverTable output appears in Figure 14.11. These results make sense if you analyze them carefully. First, the revenue increases, but at a decreasing rate, as more crude 1 is available. This is a common occurrence in LP models. As more of a resource is made available, revenue can only increase or remain the same, but each extra unit of the resource produces less (or at least no more) revenue than the previous unit. Second, the amount of gasoline produced increases, whereas the amount of heating oil produced decreases. Here’s why: Crude 1 has a higher quality than crude 2, and gasoline requires higher quality. Gasoline also sells for a higher price. Therefore, as more crude 1 is available, Chandler can produce more gasoline, receive more revenue, and still meet quality standards. However, that there is one exception to this, when only 2000 barrels of crude oil 1 are available. In this case, no gasoline is sold and leftover crude oil 2 is sold instead.

A Caution about Blending Constraints Before concluding this example, we discuss why the model is linear. The key is the implementation of the quality constraints, shown in Inequalities (14.1) and (14.2). To keep a model linear, each side of an inequality constraint must be a constant, the product of a constant and a variable, or a sum of such products. If the quality constraints are implemented as in Inequalities (14.1) and (14.2), the constraints are indeed linear. However, it is arguably more natural to rewrite this type of constraint by dividing through by the amount sold. For example, the modified gasoline constraint becomes

AQ in gasoline

Gasoline sold $ 8 (14.3)

As stated previously, this problem is sufficiently complex to defy intuition. Clearly, gasoline is more profitable per barrel than heating oil, but given the crude availability and the quality constraints, it turns out that Chandler should sell twice as much heating oil as gasoline. This would have been very difficult to guess ahead of time.

This solution uses all of the inputs to produce outputs; no crude oils are left over to sell. However, if you change the value of crude oil 2 to $55 and rerun Solver, you will see a much different solution, where no heating oil is produced and a lot of crude oil 2 is left over for sale. (Try it to convince yourself.) Why would the cheaper crude oil 2 be sold rather than the more expensive crude oil 1? The reason is quality. Gasoline requires a higher quality, and crude oil 1 is able to deliver it.

Sensitivity Analysis We perform two typical sensitivity analyses on this blending model. In each, we see how revenue and the amounts of the inputs and outputs sold vary. In the first analysis, we use the unit selling price of gasoline as the input and let it vary from $50 to $90 in increments of $5. The SolverTable results appear in Figure 14.10. Two things are of interest. First, as the price of gasoline increases from $55 to $65, Chandler starts producing gasoline and less heating oil, exactly as you would expect. Second, when the price of gasoline gets to $80 or more, no heating oil is produced, and leftover crude oil 2 is sold instead. Third, the revenue can only increase or stay the same, as the changes in column G (calculated manually) indicate.

Figure 14.10 Sensitivity to the Selling Price of Gasoline 1

2 3

4 Ba rr

el s_

so ld

Ba rr

el s_

so ld

Le �o

ve r_

1 5 $50

$55 $60 $65 $70 $75 $80 $85 $90

0 0 0

5000 5000 5000

8333.333 8333.333 8333.333

12500 12500 12500 10000 10000 10000

0 0 0

Le �o

ve r_

2500 2500 2500

0 0 0 0 0 0

To ta

l_ re

ve nu

0 0 0 0 0 0

6666.667 6666.667 6666.667

$912,500 $912,500 $912,500 $925,000 $950,000 $975,000

$1,000,000 $1,041,667 $1,083,333

In cr

ea se

$0 $0

$12,500 $25,000 $25,000 $25,000 $41,667 $41,667

6 7 8 9

10 11 12 13

A B C D E F G Oneway analysis for Solver model in Model worksheet

Selling price gasoline (cell $B$8) values along side, output cell(s) along top

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14-3 Blending Models 6 4 3

Modeling Issues In reality, a company using a blending model would run the model periodically and set production on the basis of the current inventory of inputs and the current forecasts of demands and prices. Then the forecasts and the input levels would be updated, and the model would be run again to determine the next period’s production.

Although this form of the constraint is perfectly valid—and is possibly more natural to many people—it has two draw- backs. First, it makes the model nonlinear. This is because the left side is no longer a sum of products; it involves a quotient. We prefer linear models whenever possible. Second, suppose it turns out that Chandler’s optimal solution calls for no gasoline to be sold. Then Inequality (14.3) involves division by zero, and this causes an error in Excel. Because of these two drawbacks, it is best to “clear denominators” in all such blending constraints.

A B C D E F G

2 1

4 5 6 7 8 9

10 11 12 13 14 15 300015 16 17 18 19 20 21 22 23

2000 3000 4000 5000 6000 7000 8000 9000

10000 11000 12000 13000 14000 15000 16000 17000 18000 19000 20000

0 1000 3000 5000 7000 9000

11000 13000 15000 17000 19000 21000 23000 25000 26000 27000 28000 29000 30000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10000 $700,000 $795,000 $885,000 $975,000

$1,065,000 $1,155,000 $1,245,000 $1,335,000 $1,425,000 $1,515,000 $1,605,000 $1,695,000 $1,785,000 $1,875,000 $1,950,000 $2,025,000 $2,100,000 $2,175,000 $2,250,000

12000 $95,000 $90,000 $90,000 $90,000 $90,000 $90,000 $90,000 $90,000 $90,000 $90,000 $90,000 $90,000 $90,000 $75,000 $75,000 $75,000 $75,000 $75,000

11000 10000

9000 8000 7000 6000 5000 4000

2000 1000

0 0 0 0 0 0

Crude oil 1 available (cell $F$13) values along side, output cell(s) along top

Ba rr

el s_

so ld

Ba rr

el s_

so ld

Le �o

ve r_

Le �o

ve r_

To ta

l_ re

ve nu

In cr

ea se

Oneway analysis for Solver model in Model worksheet

Figure 14.11 Sensitivity to the Availability of Crude 1

Clearing Denominators

Some constraints, particularly those that arise in blending models, are most naturally expressed in terms of ratios. For example, the percentage of sulfur in a product is a ratio: (amount of sulfur in product)/(total amount of prod- uct).This ratio could then be constrained to be less than or equal to 6%, for example. This is a perfectly valid way to express the constraint, but it has the undesirable effect of making the model nonlinear. The fix is simple: multiply through by the denominator of the ratio. This has the added benefit of ensuring that division by zero will not occur.

Fundamental Insight

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Problems

Level A 7. Use SolverTable in the Chandler blending model to see

whether, by increasing the selling price of gasoline, you can get an optimal solution that produces only gaso- line, no heating oil. Then use SolverTable again to see whether, by increasing the selling price of heating oil, you can get an optimal solution that produces only heat- ing oil, no gasoline.

8. Use SolverTable in the Chandler blending model to find the shadow price of crude oil 1—that is, the amount Chandler would be willing to spend to acquire more crude oil 1. Does this shadow price change as Chandler keeps getting more of crude oil 1? Answer the same questions for crude oil 2.

9. How sensitive is the Chandler optimal blending solution (barrels of each output sold and profit) to the required quality levels? Answer this by running a two-way SolverTable with these three outputs. You can choose the values of the two inputs to vary.

10. In the Chandler blending model, suppose there is a chemical ingredient called C1 that both gaso- line and heating oil need. At least 3% of every bar- rel of gasoline must be C1, and at least 5% of every barrel of heating oil must be C1. Suppose that 4%

of all crude oil 1 is C1 and 6% of all crude oil 2 is C1. Modify the blending model to incorporate the constraints on C1, and then optimize. Don’t forget to clear denominators.

11. In the current version of the Chandler blending model, a barrel of any input results in a barrel of output. However, in a real blending problem there can be losses. Suppose a barrel of input results in only a fraction of a barrel of output. Specifically, each barrel of either crude oil used for gasoline results in only 0.95 barrel of gasoline, and each barrel of either crude used for heating oil results in only 0.97 barrel of heating oil. Modify the model to incorporate these losses and then find the optimal solu- tion.

Level B 12. We warned you about clearing denominators in the qual-

ity constraints. This problem indicates what happens if you don’t do so. a. Implement the quality constraints in the Chandler

blending model as indicated in Inequality (14.3). Then run Solver with the simplex method. What hap- pens? What if you run Solver with the GRG nonlinear method?

b. Repeat part a, but increase the selling price of heating oil to $120 per barrel. What happens now?

14-4 Logistics Models In many situations a company produces products at locations called origins and ships these products to customer locations called destinations. Typically, each origin has a lim- ited capacity that it can ship, and each destination must receive a required quantity of the product. Logistics models can be used to determine the minimum-cost shipping method for satisfying customer demands.

14-4a Transportation Models We begin by assuming that the only possible shipments are those directly from an origin to a destination. That is, no shipments between origins or between destinations are allowed. Such a problem has traditionally been called a transportation problem.

EXAMPLE

14.3 SHIPPING CARS AT GRAND PRIX AUTOMOBILE The Grand Prix Automobile Company manufactures automobiles in three plants and then ships them to four regions of the country. The plants can supply the amounts listed in the right column of Table 14.2. The customer demands by region are listed in the bottom row of this table, and the unit costs of shipping an automobile from each plant to each region are listed in the middle of the table. Grand Prix wants to find the lowest-cost shipping plan for meeting the demands of the four regions without exceeding the capacities of the plants.

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14-4 Logistics Models 6 4 5

Objective To develop an optimization model for finding the least-cost way of shipping the automobiles from plants to regions, staying within plant capacities and meeting regional demands.

Where Do the Numbers Come From? A typical transportation problem requires three sets of numbers: capacities (or supplies), demands (or requirements), and unit shipping (and possibly production) costs.

• The capacities indicate the most each plant can supply in a given amount of time—a month, say—under current operating conditions. In some cases it might be possible to increase the “base” capacities, by using overtime, for example. In such cases the model could be modified to determine the amounts of additional capacity to use (and pay for).

• The customer demands are typically estimated from some type of forecasting model (as discussed in Chapter 12). The fore- casts are often based on historical customer demand data.

• The unit shipping costs come from a transportation cost analysis—what does it really cost to send a single automobile from any plant to any region? This is not an easy question to answer, and it requires an analysis of the best mode of transportation (such as railroad, ship, or truck). However, companies typically have the required data. Actually, the unit “shipping” cost can also include the unit production cost at each plant. However, if this cost is the same across all plants, as we are tacitly assuming here, it can be omitted from the model.

Solution The variables and constraints required for this model are shown in Figure 14.12. (See the file Transportation Big Picture. xlsx.) The company must choose the number of autos to send from each plant to each region—a shipping plan. Then it can calculate the total number of autos sent out of each plant and the total number received by each region.

Region 1 Region 2 Region 3 Region 4 Capacity

Plant 1 131 218 266 120 450

Plant 2 250 116 263 278 600

Plant 3 178 132 122 180 500

Demand 450 200 300 300

Table 14.2 Input Data for Grand Prix Example

Figure 14.12 Big Picture for Transportation Model

Amount shipped

Total shipped out Plant capacity<=

Total shipped in Region demand

Unit shipping cost

Minimize total cost

Representing Transportation in a Network Model A network diagram of this model appears in Figure 14.13. This diagram is typical of net- work models. It consists of nodes and arcs. A node, indicated by an oval, generally rep- resents a geographical location. In this case the nodes on the left correspond to plants, and the nodes on the right correspond to regions. An arc, indicated by an arrow, generally represents a route for getting a product from one node to another. Here, the arcs all go from a plant node to a region node—from left to right.

The problem data fit nicely on such a diagram. The capacities are placed next to the plant nodes, the demands are placed next to the region nodes, and the unit shipping costs are placed on the arcs. The decision variables are usually called flows. They represent the amounts shipped on the various arcs. Sometimes (although not in this problem), there are upper limits on the flows on some or all of the arcs. These upper limits, called arc capacities, can also be shown on the diagram.3

In a transportation problem, all flows go from left to right— from origins to destinations. You will see more complex network structures in the next subsection.

3 There can even be lower limits, other than zero, on certain flows, but we don’t consider any such constraints here.

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Developing the Spreadsheet Model The spreadsheet model appears in Figure 14.14. (See the file Transportation Finished.xlsx.) To develop this model, perform the following steps.

1. Inputs.4 Enter the unit shipping costs, plant capacities, and region demands in the blue cells. 2. Shipping plan. Enter any trial values for the shipments from plants to regions in the Shipping_plan

range. These are the decision variable cells. Note that this rectangular range is exactly the same shape as the range where the unit shipping costs are entered. This is a natural model design, and it simplifies the formulas in the following steps.

3. Numbers shipped from plants. To calculate the amount shipped out of each plant in the range G13:G15, select this range and click the AutoSum (S) button.

Figure 14.13 Network Representation of Transportation Model

Developing the Transportation Model

Figure 14.14 Transportation Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21

A B C D E F G H I J K Grand Prix transporta�on model Range names used:

Capacity =Model!$I$13:$I$15 Unit shipping costs Demand =Model!$C$18:$F$18

=Model!$C$13:$F$15=Shipping_PlanTo Region 1 Region 2 Region 3 Region 4 =Model!$B$21Total_cost

From Plant 1 =Model!$C$16:$F$16 =Model!$G$13:$G$15

Total_received$120$266$218$131 Plant 2 Total_shipped$278$263$116$250 Plant 3 $178 $132 $122 $180

Shipping plan, and constraints on supply and demand To

Region 1 Region 2 Region 3 Region 4 Total shipped Capacity From Plant 1 150 0 0 300 450 <= 450

Plant 2 100 200 0 0 300 <= 600 Plant 3 200 0 300 0 500 <= 500 Total received 450 200 300 300

>= >= >= >= Demand 450 200 300 300

Objective to minimize Total cost $176,050

4. Amounts received by regions. Similarly, calculate the amount shipped to each region in the range C16:F16 by selecting the range and clicking the AutoSum button.

5. Total shipping cost. Calculate the total cost of shipping power from the plants to the regions in the Total_cost cell with the formula

5SUMPRODUCT(C6:F8,Shipping_plan)

This formula sums all products of unit shipping costs and amounts shipped. You now see the benefit of placing unit ship- ping costs and amounts shipped in similar-size rectangular ranges—you can then use the SUMPRODUCT function.

4 From here on, we might not remind you about creating range names, but we will continue to list our suggested range names on the spreadsheets.

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Using Solver Invoke Solver with the settings shown in Figure 14.15. As usual, check the Non-Negative option and select the Simplex LP method before optimizing.

Figure 14.15 Solver Dialog Box for Transportation Model

Discussion of the Solution The Solver solution appears in Figure 14.14 and is illustrated graphically in Figure 14.16. The company incurs a total shipping cost of $176,050 by using the shipments listed in Figure 14.16. Except for the six routes shown, no other routes are used. Most of the ship- ments occur on the low-cost routes, but this is not always the case. For example, the route

It is typical in transportation models, especially large models, that only a small number of the possible routes are used.

Figure 14.16 Graphical Representation of Optimal Solution

14-4 Logistics Models 6 4 7

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The bottom part of this report is useful because of its shadow prices. For example, plants 1 and 3 are currently shipping at capacity, so the company would benefit from having more capacity at these plants. In particular, the report indicates that each extra unit of capacity at plant 1 is worth $119, and each extra unit of capacity at plant 3 is worth $72. However, because the allowable increase for each of these is 100, you know that after an increase in capacity of 100 at either plant, further increases will probably be worth less than the current shadow prices.

One interesting analysis that cannot be performed with Solver’s sensitivity report is to keep shipping costs and capac- ities constant and allow all demands to change by a certain percentage (positive or negative). To perform this analysis, use SolverTable, with the varying percentage as the single input. Then keep track of the total cost and any amounts shipped of interest. The key to doing this correctly is to modify the model slightly, as illustrated in the previous chapter and Example 14.1, before running SolverTable. The appropriate modifications appear in the third sheet of the Transportation Finished.xlsx file. Then run SolverTable, allowing the percent- age change in all demands to vary from 220% to 30% in increments of 5%, and keep track of total cost. As the table in Figure 14.18 shows, the total shipping cost increases at an increasing rate as the

from plant 2 to region 1 is relatively expensive, and it is used. On the other hand, the route from plant 3 to region 2 is relatively cheap, but it is not used. A good shipping plan tries to use cheap routes, but it is constrained by capacities and demands.

Note that the available capacity is not all used. The reason is that total capacity is 1550, whereas total demand is only 1250. Even though the demand constraints are of the “$” type, there is clearly no reason to send the regions more than they request because it only increases shipping costs. Therefore, the optimal plan sends them the minimal amounts they request and no more. In fact, the demand constraints could have been modeled as “=” constraints, and Solver would have found exactly the same solution.

Sensitivity Analysis There are many sensitivity analyses you could perform on the basic transportation model. For example, you could vary any one of the unit shipping costs, capacities, or demands. The effect of any such change in a single input is captured nicely in Solver’s sensitivity report, shown in Figure 14.17. The top part indicates the effects of changes in the unit shipping costs. The results here are typical. For all routes with positive flows, the corresponding reduced cost is zero, whereas for all routes not currently being used, the reduced cost indicates how much less the unit shipping cost would have to be before the company would start shipping along that route. For example, if the unit shipping cost from plant 2 to region 3 decreased by more than $69, this route would become attractive.

Final Value

Reduced Cost

Allowable Increase

Allowable DecreaseCell Name

Objec�ve Coefficient

6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

A B C D E F G H Variable Cells

$C$13 Plant 1 Region 1 150 0 131 119 13 $D$13 Plant 1 Region 2 0 221 218 1E+30 221 $E$13 Plant 1 Region 3 0 191 266 1E+30 191 $F$13 Plant 1 Region 4 300 0 120 13 239 $C$14 Plant 2 Region 1 100 0 250 39 72 $D$14 Plant 2 Region 2 200 0 116 88 116 $E$14 Plant 2 Region 3 0 69 263 1E+30 69 $F$14 Plant 2 Region 4 0 39 278 1E+30 39 $C$15 Plant 3 Region 1 200 0 178 13 69 $D$15 Plant 3 Region 2 0 88 132 1E+30 88 $E$15 Plant 3 Region 3 300 0 122 69 194 $F$15 Plant 3 Region 4 0 13 180 1E+30 13

Constraints Final Value

Shadow Price

Constraint R.H. Side

Allowable Increase

Allowable DecreaseCell Name

$G$13 Plant 1 Total shipped 450 –119 450 100 150 $G$14 Plant 2 Total shipped 300 0 600 1E+30 300 $G$15 Plant 3 Total shipped 500 –72 500 100 200 $C$16 Total received Region 1 450 250 450 300 100 $D$16 Total received Region 2 200 116 200 300 200 $E$16 Total received Region 3 300 194 300 200 100 $F$16 Total received Region 4 300 239 300 150 100

Figure 14.17 Solver’s Sensitivity Report for Transportation Model

The key to this sensitivity analysis is to modify the model slightly before running SolverTable.

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14-4 Logistics Models 6 4 9

demands increase. However, at some point the problem has no feasible solutions. As soon as the total demand is greater than the total capacity, it is impossible to meet all demand.

3 2 1

5 6 7 8 9

10 11 12 13 14 15

A B C D E F G H

Change in demands (cell $I$10) values along side, output cell(s) along top

Oneway analysis for Solver model in Modified Model for Sensi�vity worksheet

To ta

l_ co

–20% $130,850

–15% $140,350 –10% $149,850

–5% $162,770 0% $176,050 5% $189,330

10% $202,610 15% $215,890 20% $229,170 25% 30%

Not feasible Not feasible

Figure 14.18 Sensitivity Analysis to Percentage Changes in All Demands

An Alternative Model The transportation model in Figure 14.14 is a very natural one. In the graphical representation in Figure 14.13, all arcs go from left to right, that is, from plants to regions. Therefore, the rectangular range of shipments allows you to calculate shipments out of plants as row sums and shipments into regions as column sums. In anticipation of later models in this chapter, however, where the graphical network can be more complex, we present an alternative model of the transportation problem. (See the file Transportation Alternative Finished.xlsx.)

First, it is useful to introduce some additional network terminology. Recall that flows are the amounts shipped on the var- ious arcs. The direction of the arcs indicates which way the flows are allowed to travel. An arc pointed into a node is called an inflow, whereas an arrow pointed out of a node is called an outflow. In the basic transportation model, all outflows originate from suppliers, and all inflows go toward demanders. However, general networks can have both inflows and outflows for any given node.

With this general structure in mind, the typical network model has one decision variable cell per arc. It indicates how much (if any) to send along that arc in the direction of the arrow. There- fore, it is often useful to model network problems by listing all arcs and their corresponding flows in one long list. Then constraints can be indicated in a separate section of the spreadsheet. Specif- ically, for each node in the network, there is a flow balance constraint. These flow balance con- straints for the basic transportation model are the supply and demand constraints already discussed, but they can be more general for other network models, as will be discussed in the next subsection.

The alternative model of the Grand Prix problem appears in Figure 14.19. The plant and region indexes and the associated unit shipping costs are entered manually in the range A5:C16. Each row in this range corresponds to an arc in the network. For example, row 12 corresponds to the arc from plant 2 to region 4, with unit shipping cost $278. Then the decision variable cells for the flows are in column D. (If there were arc capacities, they could be placed to the right of the flows.)

The flow balance constraints are conceptually straightforward. Each cell in the Outflow and Inflow ranges in column G contains the appropriate sum of flows. For example, cell G6, the outflow from plant 1, represents the sum of cells D5 through D8, whereas cell G12, the inflow to plant 1, represents the sum of cells D5, D9, and D13. Fortunately, there is an easy way to enter these summation formulas.5 The trick is to use Excel’s built-in SUMIF function (see explanation below). For example, the formula in cell G6 is

5SUMIF(Origin,F6,Flow)

This formula compares the plant number in cell F6 to the Origin range in column A and sums all flows where they are equal— that is, it sums all flows out of plant 1. This formula can be copied down to cell G8 to obtain the flows out of the other plants. For flows into regions, the similar formula in cell G12 for the flow into region 1 is

5SUMIF(Destination,F12,Flow)

Although this model is possibly less natural than the original model, it generalizes better to other logistics models.

5 Try entering these formulas as simple sums even for a 3 3 4 transportation model, and you will see why the SUMIF function is so handy.

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and this can be copied down to cell G15 for flows into the other regions. In general, the SUMIF function finds all cells in the first argument that satisfy the criterion in the second argument and then sums the corresponding cells in the third argument. It is a very handy function—and not just for network modeling.

Figure 14.19 Alternative Form of Transportation Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

A B C D E F G H I J K L M Grand Prix transporta�on model: a more general network formula�on Range names used:

Capacity =Model!$I$6:$I$8 Network structure and flows Flow balance constraints Demand =Model!$I$12:$I$15

Origin Des�na�on Unit cost Flow Capacity constraints =Model!$B$5:$B$16Des�na�on 1 1 131 150 Plant Ou�low Capacity Flow =Model!$D$5:$D$16 1 2 218 0 1 450 <= 450 Inflow =Model!$G$12:$G$15 1 3 266 0 2 300 <= 600 Origin =Model!$A$5:$A$16 1 4 120 300 3 500 <= 500 Ou�low =Model!$G$6:$G$8

=Model!$B$19Total_Cost10025012 2 2 116 200 Demand constraints 2 3 263 0 Region Inflow Demand 2 4 278 0 1 450 >= 450 3 1 178 200 2 200 >= 200 3 2 132 0 3 300 >= 300 3 3 122 300 4 300 >= 300 3 4 180 0

Objec�ve to minimize Total Cost $176,050

SUMIF

The SUMIF function is useful for summing values in a certain range if cells in a related range satisfy a given condi- tion. It has the syntax 5SUMIF(compareRange,criterion,sumRange), where compareRange and sumRange are similar-size ranges. This formula checks each cell in compareRange to see whether it satisfies the criterion. If it does, it adds the corresponding value in sumRange to the overall sum. For example, =SUMIF(A12:A13,1,D12:D23) sums all values in the range D12:D23 where the corresponding cell in the range A12:A23 has the value 1.

Excel Function

This use of the SUMIF function, along with the list of origins, destinations, unit costs, and flows in columns A through D, is the key to the model. The rest is straightforward. The total cost is a SUMPRODUCT of unit costs and flows, and the Solver dialog box is set up as shown in Figure 14.20.

This alternative model generalizes nicely to other network problems. Essentially, it shows that all network models look alike. There is an additional benefit from this alternative model. Suppose that flows from certain plants to certain regions are not allowed. (Maybe no roads exist.) It is not easy to disallow such routes in the original model. One option is to allow the “disallowed” routes but to impose extremely large unit shipping costs on them. This works, but it is wasteful because it adds decision variable cells that do not really belong in the model. However, the alternative network model simply omits arcs that are not allowed. For example, if the route from plant 2 to region 4 is not allowed, you simply omit the data in the range A12:D12. This creates a model with exactly as many decision variable cells as allowable arcs. This additional benefit can be very valuable when the number of potential arcs in the network is huge—even though the vast majority of them are disallowed—which is the situation in many large network models.

We do not necessarily recommend this more general network model for simple transportation problems. In fact, it is prob- ably less natural than the original model in Figure 14.14. However, it paves the way for the more complex network problems discussed next.

Modeling Issues • The customer demands in typical transportation problems can be handled in one of two ways. First, you can think of these

forecasted demands as minimal requirements that must be sent to the customers. This is how regional demands were treated here. Alternatively, you could consider the demands as maximal sales quantities, the most each region can sell. Then you would constrain the amounts sent to the regions to be less than or equal to the forecasted demands. Whether the demand

The alternative network model not only accommodates more general networks, but it is more efficient because it has fewer decision variable cells.

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14-4b More General Logistics Models The objective of many real-world network models is to ship goods from one set of loca- tions to another at minimum cost, subject to various constraints. There are many variations of these models. The simplest models include a single product that must be shipped via

14-4 Logistics Models 6 5 1

constraints are expressed as “ $” or “ #” (or even “=”) constraints depends on the context of the problem—do the dealers need at least this many, do they need exactly this many, or can they sell only this many?

• If all the supplies and demands for a transportation problem are integers, the optimal Solver solu- tion automatically has integer-valued shipments. Explicit integer constraints are not required. (This might not be obvious, but it has been proved mathematically.) This is a very important benefit. It means that the “fast” simplex method can be used rather than much slower integer algorithms.

• Shipping costs are often nonlinear (and “nonsmooth”) due to quantity discounts. For example, if it costs $3 per item to ship up to 100 items between locations and $2 per item for each additional item, the proportionality assumption of LP is violated and the resulting transportation model is nonlinear. Shipping problems that involve quantity discounts are generally more difficult to solve.

• Excel’s Solver uses the simplex method to solve transportation problems. There is a streamlined version of the simplex method, called the transportation simplex method, that is much more efficient than the ordinary simplex method for trans- portation problems. Large transportation problems are usually solved with the transportation simplex method. See Winston (2003) for a discussion of the transportation simplex method.

Figure 14.20 Solver Dialog Box for Alternative Transportation Model

Depending on how you treat the demand constraints, you can get several variations of the basic transpor- tation model.

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6 5 2 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

one mode of transportation (truck, for example) in a particular period of time. More com- plex models—and much larger ones—can include multiple products, multiple modes of transportation, and/or multiple time periods. We discuss one such problem in this section.

Basically, the general logistics problem is like the transportation problem except for two possible differences. First, arc capacities are often imposed on some or all of the arcs. These become simple upper-bound constraints in the model. Second and more signifi- cant, there can be inflows and outflows associated with any node. Nodes are generally categorized as origins, destinations, and transshipment points. An origin is a location that starts with a certain supply (or a capacity for supplying). A destination is the opposite; it requires a certain amount to end up there. A transshipment point is a location where goods simply pass through.

The best way to think of these categories is in terms of net inflow and net outflow. The net inflow for any node is defined as total inflow minus total outflow for that node. The net outflow is the negative of this, total outflow minus total inflow. Then an origin is a node with positive net outflow, a destination is a node with positive net inflow, and a transshipment point is a node with net outflow (and net inflow) equal to 0. It is important to realize that inflows are sometimes allowed to origins, but their net outflows are positive. Similarly, outflows from destinations are sometimes allowed, but their net inflows are pos- itive. For example, if Cincinnati and Memphis are manufacturers (origins) and Dallas and Phoenix are retail locations (destinations), it is possible that flow could go from Cincinnati to Memphis to Dallas to Phoenix.

There are typically two types of constraints in logistics models (besides nonnegativity of flows). The first type represents the arc capacity constraints, which are simple upper bounds on the arc flows. The second type represents the flow balance constraints, one for each node. For an origin, this constraint is typically of the form Net Outflow 5 Capacity or possibly Net Outflow " Capacity. For a destination, it is typically of the form Net Inflow # Demand or possibly Net Inflow 5 Demand. For a transshipment point, it is of the form Net Inflow 5 0 (which is equivalent to Net Outflow 5 0).

It is easy to visualize these constraints in a graphical representation of the network by simply examining the flows on the arrows leading into and out of the various nodes. We illustrate a typical logistics model in Example 14.4.

Flow Balance Constraints

All network optimization models have some form of flow balance constraints at the various nodes of the network. This flow balance relates the amount that enters the node to the amount that leaves the node. In many network models, the simple structure of these flow balance constraints guarantees that the optimal solutions have integer values. It also enables specialized network versions of the simplex method to solve the huge network models typically encountered in real logistics applications.

Fundamental Insight

EXAMPLE

14.4 PRODUCING AND SHIPPING TOMATO PRODUCTS AT REDBRAND

RedBrand Company produces a tomato product at three plants. This product can be shipped directly to the company’s two customers or it can first be shipped to the company’s two warehouses and then to the customers. Figure 14.21 is a network representation of RedBrand’s problem. Nodes 1, 2, and 3 represent the plants (these are the origins, denoted by S for supplier),

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14-4 Logistics Models 6 5 3

nodes 4 and 5 represent the warehouses (these are the transshipment points, denoted by T), and nodes 6 and 7 represent the customers (these are the destinations, denoted by D). Note that some shipments are allowed among plants, among warehouses, and among customers. Also, some arcs have arrows on both ends. This means that flow is allowed in either direction.

Figure 14.21 Graphical Representation of Logistics Model

The cost of producing the product is the same at each plant, so RedBrand is concerned with minimizing the total ship- ping cost incurred in meeting customer demands. The production capacity of each plant (in tons per year) and the demand of each customer are shown in Figure 14.21. For example, plant 1 (node 1) has a capacity of 200, and customer 1 (node 6) has a demand of 400. In addition, the cost (in thousands of dollars) of shipping a ton of the product between each pair of locations is listed in Table 14.3, where a blank indicates that RedBrand cannot ship along that arc. We also assume that at most 200 tons of the product can be shipped along any arc. This is the common arc capacity. RedBrand wants to determine a minimum-cost shipping schedule.

Table 14.3 Shipping Costs for RedBrand Example (in $1000s)

To node

From node 1 2 3 4 5 6 7

1 5.0 3.0 5.0 5.0 20.0 20.0

2 9.0 9.0 1.0 1.0 8.0 15.0

3 0.4 8.0 1.0 0.5 10.0 12.0

4 1.2 2.0 12.0

5 0.8 2.0 12.0

6 1.0

7 7.0

Objective To develop an optimization model for finding the minimum-cost way to ship the tomato product from suppliers to customers, possibly through warehouses, so that customer demands are met and supplier capacities are not exceeded.

Where Do the Numbers Come From? The network configuration itself would come from geographical considerations—which routes are physically possible (or sensible) and which are not. The numbers would be derived as in the Grand Prix automobile example. (See Example 14.3 for further discussion.)

Solution The variables and constraints for this logistics model are shown in Figure 14.22. (See the file RedBrand Logistics Big Picture. xlsx.) The key to the model is handling the flow balance constraints. You will see exactly how to implement these when we give step-by-step instructions for developing the spreadsheet model. However, it is not enough, for exam- ple, to specify that the flow out of plant 2 is less than or equal to the capacity of plant 2. The reason is that there might also be flow into plant 2 (from another plant). Therefore, the correct flow balance constraint for plant 2 is that its outflow must be less than or equal to its capacity plus its inflow. Equiv- alently, the net outflow from plant 2 must be less than or equal to its capacity.

Other than arc capacity constraints, the only constraints are flow balance constraints.

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Developing the Spreadsheet Model To set up the spreadsheet model, proceed as follows. (See Figure 14.23 and the file RedBrand Logistics Finished.xlsx. Also, refer to the network in Figure 14.21.)

1. Origins and destinations. Enter the node numbers (1 to 7) for the origins and destinations of the various arcs in the range A8:B33. Note that the disallowed arcs are not entered in this list.

2. Input data. Enter the unit shipping costs (in thousands of dollars), the common arc capacity, the plant capacities, and the customer demands in the blue cells. Again, only the nonblank entries in Table 14.3 are used to fill the column of unit shipping costs.

3. Flows on arcs. Enter any initial values for the flows in the range D8:D33. These are the decision variable cells. 4. Arc capacities. To indicate a common arc capacity for all arcs, enter the formula

5$B$4

in cell F8 and copy it down column F.

Figure 14.22 Big Picture for Logistics Model Amounts sent

on routes Route capacity

Plant net ou�low Plant capacity

Customer net inflow

Customer demand

Warehouse net ou�low (or inflow) 0

Minimize total costUnit shipping cost

Figure 14.23 Logistics Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

KJIHGFEDCBA RedBrand shipping model

Inputs Common arc capacity

Network structure, flows, and arc capacity constraints Node balance constraints Origin Arc Capacity Plant constraints

Plant net ou�lowNode200<=1 <=1801200<=1 <=3002200<=1 <=1003200<=1

1 <= 200 1 <= 200 Warehouse constraints

Warehouse net ou�lowNode200<=2 =04200<=2 =05200<=2

2 <= 200 2 <= 200 Customer constraints

Customer net inflowNode200<=2 >=4006200<=3 >=1807200<=3

3 <= 200 3 <= 200 Range names used 3 <= 200 Arc_Capacity =Model!$F$8:$F$33 3 <= 200 Customer_demand =Model!$K$20:$K$21 4 <= 200 Customer_net_inflow =Model!$I$20:$I$21 4 <= 200 =Model!$B$8:$B$33

=Model!$D$8:$D$33Flow Des�na�on

200<=4 5 <= 200 Origin =Model!$A$8:$A$33 5 <= 200 =Model!$K$9:$K$11 5 <= 200 =Model!$I$9:$I$11 6 <= 200 Total_cost =Model!$B$36 7 <= 200 Unit_Cost =Model!$C$8:$C$33

Warehouse_net_ou�low

Plant_net_ou�low Plant_capacity

=Model!$I$15:$I$16

Total cost Objec�ve to minimize

Des�na�on

200

Unit Cost Flow 052 Plant capacity

18033 054 055

6 20 0 7 20 0

091 Required 0093 012014

5 1 0 6 8 180

0157 Customer demand 00.41 082

4 1 80 5 0.5 200 6 10 0 7 12 0 5 1.2 0 6 2 200

0127 4 0.8 0 6 2 200 7 12 0 7 1 180 6 7 0

$3,260

200 300 100

400 180

Developing the Logistics Model

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14-4 Logistics Models 6 5 5

5. Flow balance constraints. Nodes 1, 2, and 3 are supply nodes, nodes 4 and 5 are transship- ment points, and nodes 6 and 7 are demand nodes. Therefore, set up the left sides of the flow balance constraints appropriately for these three cases. Specifically, enter the net outflow for node 1 in cell I9 with the formula

5SUMIF(Origin,H9,Flow)-SUMIF(Destination,H9,Flow)

and copy it down to cell I11. This formula subtracts flows into node 1 from flows out of node 1 to obtain net outflow for node 1. Next, copy this same formula to cells I15 and I16 for the warehouses. (Remember that, for transshipment nodes, the left side of the constraint can be net outflow or net inflow, whichever you prefer. The reason is that if net outflow is zero, net inflow must also be zero.) Finally, enter the net inflow for node 6 in cell I20 with the formula

5SUMIF(Destination,H20,Flow)-SUMIF(Origin,H20,Flow)

and copy it to cell I21. This formula subtracts flows out of node 6 from flows into node 6 to obtain the net inflow for node 6.

6. Total shipping cost. Calculate the total shipping cost (in thousands of dollars) in cell B36 with the formula

5SUMPRODUCT(Unit_cost,Flow)

Using Solver The Solver dialog box should be set up as in Figure 14.24. The objective is to minimize total shipping costs, subject to the three types of flow balance constraints and the arc capacity constraints.

We generally prefer positive numbers on the right sides of constraints. This is why we calculate net outflows for origins and net inflows for destinations.

Figure 14.24 Solver Dialog Box for Logistics Model

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Discussion of the Solution The optimal solution in Figure 14.23 indicates that RedBrand’s customer demand can be satisfied with a shipping cost of $3,260,000. This solution appears graphically in Figure 14.25. Note in particular that plant 1 produces 180 tons (under capac- ity) and ships it all to plant 3, not directly to warehouses or customers. Also, note that all shipments from the warehouses go directly to customer 1. Then customer 1 ships 180 tons to customer 2. We purposely chose unit shipping costs (probably unre- alistic ones) to produce this type of behavior, just to show that it can occur. As you can see, the costs of shipping from plant 1 directly to warehouses or customers are relatively large compared to the cost of shipping directly to plant 3. Similarly, the costs of shipping from plants or warehouses directly to customer 2 are prohibitive. Therefore, RedBrand ships to customer 1 and lets customer 1 forward some of its shipment to customer 2.

Figure 14.25 Optimal Flows for Logistics Model

Sensitivity Analysis How much effect does the arc capacity have on the optimal solution? Currently, three of the arcs with positive flow are at the arc capacity of 200. You can use SolverTable to see how sensitive this number and the total cost are to the arc capacity.6 In this case the single input cell for SolverTable is cell B4, which is varied from 150 to 300 in increments of 25. Two quantities are designated as outputs: total cost and the number of arcs at arc capacity. As before, if you want to keep track of an output that does not already exist, you can create it with an appropriate formula in a new cell before running SolverTable. Specifically, you can enter the formula 5COUNTIF(Flow,B4) in an unused cell. This formula counts the arcs with flow equal to arc capacity. (See the finished version of the file for a note about this formula.)

The SolverTable output in Figure 14.26 is what you would expect. As the arc capacity decreases, more flows bump up against it, and the total cost increases. But even when the arc capacity is increased to 300, two flows are constrained by it. In this sense, even a large arc capacity can cost RedBrand money.

COUNTIF

The COUNTIF function counts the number of values in a given range that satisfy some criterion. The syntax is 5COUNTIF(range, criterion). For example, the formula 5COUNTIF(D8:D33,150) counts the number of cells in the range D8:D33 that contain the value 150. This formula could also be entered as 5COUNTIF(D8:D33,“=150”). Similarly, the formula 5COUNTIF(D8:D33,“>=100”) counts the number of cells in this range with values greater than or equal to 100.7

Excel Function

6 Solver’s sensitivity report would not answer our question. This report is useful only for one-at-a-time changes in inputs, and here we are simultaneously changing the upper limit for each flow. However, this report (its bottom section) can be used to assess the effects of changes in plant capacities or custom- er demands. 7 The COUNTIF and SUMIF functions are limited in that they allow only one condition, such as “7=10”. For this reason, Microsoft added two new functions in Excel 2007, COUNTIFS and SUMIFS, that allow multiple conditions. You can learn about them in online help.

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14-4 Logistics Models 6 5 7

Modeling Issues • There are many variations of this basic logistics model. Two variations are illustrated in the files RedBrand Logistics Mul-

tiple Products Finished.xlsx and RedBrand Logistics Shrinkage Finished.xlsx. In the first variation, two products com- pete for the same arc capacity. In the second, there is shrinkage at the warehouses due to spoilage.

• Excel’s Solver uses the simplex method to solve logistics models. However, the simplex method can be simplified dramati- cally for these types of models. The simplified version of the simplex method, called the network simplex method, is much more efficient than the ordinary simplex method. Specialized computer codes have been written to implement the network simplex method, and all large logistics problems are solved by using the network simplex method. This is fortunate because real logistics models tend to be extremely large. See Winston (2003) for a discussion of this method.

• If the given supplies and demands for the nodes are integers and all arc capacities are integers, the logistics model always has an optimal solution with all integer flows. Again, this is very fortunate for large problems—you get integer solutions “for free” without having to use an integer programming algorithm. However, this “integers for free” benefit is guaranteed only for the basic logistics model, as in the original RedBrand model. When the model is modified in certain ways, such as by adding a shrinkage factor, the optimal solution is no longer guaranteed to be integer-valued.

Figure 14.26 Sensitivity to Arc Capacity

3 2 1

A B C D E F G

Common arc capacity (cell $B$4) values along side, output cell(s) along top

To ta

l_ co

Ar cs

_a t_

ca pa

ci ty

150 $4,120 5

175 $3,643 6

200 $3,260 3

225 $2,998 3

250 $2,735 3

275 $2,473 3

300 $2,320 2

Oneway analysis for Solver model in Model worksheet

Problems

Level A 13. In the original Grand Prix example, the total capacity of

the three plants is 1550, well above the total customer demand. Would it help to have 100 more units of capac- ity at plant 1? What is the most Grand Prix would be willing to pay for this extra capacity? Answer the same questions for plant 2 and for plant 3. Explain why extra capacity can be valuable even though the company already has more total capacity than it requires.

14. The optimal solution to the original Grand Prix prob- lem indicates that with a unit shipping cost of $132, the route from plant 3 to region 2 is evidently too expensive—no autos are shipped along this route. Use SolverTable to see how much this unit shipping cost

would have to be reduced before some autos would be shipped along this route.

15. In the RedBrand example, suppose the plants cannot ship to each other and the customers cannot ship to each other. Modify the model appropriately, and rerun Solver. How much does the total cost increase because of these disallowed routes?

16. Modify the RedBrand example so that all flows must be from plants to warehouses and from warehouses to customers. Disallow all other arcs. How much does this restriction cost RedBrand, relative to the original opti- mal shipping cost?

17. In the RedBrand example, the costs for shipping from plants or warehouses to customer 2 were purposely made high so that it would be optimal to ship to cus- tomer 1 and then let customer 1 ship to customer 2. Use SolverTable appropriately to do the following. Decrease

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the unit shipping costs from plants and warehouses to customer 1, all by the same amount, until it is no longer optimal for customer 1 to ship to customer 2. Describe what happens to the optimal shipping plan at this point.

18. In the RedBrand example, the arc capacity is the same for all allowable arcs. Modify the model so that each arc has its own arc capacity. You can make up the arc capacities.

19. Continuing the previous problem, make the problem even more general by allowing upper bounds (arc capacities) and lower bounds for the flows on the allowable arcs. Some of the upper bounds can be very large numbers, effectively indicating that there is no arc capacity for these arcs, and the lower bounds can be zero or positive. If they are positive, they indicate that some positive flow must occur on these arcs. Modify the model appropri- ately to handle these upper and lower bounds. You can make up the upper and lower bounds.

20. Suppose in the original Grand Prix example that the routes from plant 2 to region 1 and from plant 3 to region 3 are not allowed. (Perhaps there are no railroad lines for these routes.) How would you modify the original model (Figure 14.14) to rule out these routes? How would you modify the alternative model (Figure 14.19) to do so? Discuss the pros and cons of these two approaches.

21. The RedBrand model in the file RedBrand Logistics Multiple Product Finished.xlsx assumes that the unit shipping costs are the same for both products. Modify the model so that each product has its own unit shipping costs. You can assume that the original unit shipping costs apply to product 1, and you can make up new unit shipping costs for product 2.

Level B 22. Here is a problem to challenge your intuition. In the

original Grand Prix example, reduce the capacity of plant 2 to 300. Then the total capacity is equal to the total demand. Rerun Solver on the modified model. You should find that the optimal solution uses all capac- ity and exactly meets all demands with a total cost of $176,050. Now increase the capacity of plant 1 and the demand at region 2 by one automobile each, and opti- mize again. What happens to the optimal total cost? How can you explain this “more for less” paradox?

23. Continuing the previous problem (with capacity 300 at plant 2), suppose you want to see how much extra capacity and extra demand you can add to plant 1 and region 2 (the same amount to each) before the total shipping cost stops decreasing and starts increasing. Use SolverTable appropriately to find out. (You will probably need to use some trial and error on the range of input values.) Can you explain intuitively what causes the total cost to stop decreasing and start increasing?

24. Modify the original Grand Prix example by increasing the demand at each region by 200, so that total demand is well above total plant capacity. However, now interpret

these “demands” as “maximum sales,” the most each region can accommodate, and change the “demand” constraints to become “ #” constraints, not “ $” con- straints. How does the optimal solution change? Does it make realistic sense? If not, how might you change the model to obtain a realistic solution?

25. Modify the original Grand Prix example by increasing the demand at each region by 200, so that total demand is well above total plant capacity. This means that some demands cannot be supplied. Suppose there is a unit “penalty” cost at each region for not supplying an automobile. Let these unit penalty costs be $600, $750, $625, and $550 for the four regions. Develop a model to minimize the sum of shipping costs and penalty costs for unsatisfied demands. (Hint: Introduce a fourth plant with plenty of capacity, and set its unit shipping costs to the regions equal to the unit penalty costs. Then inter- pret an auto shipped from this fictional plant to a region as a unit of demand not satisfied.)

26. How difficult is it to expand the RedBrand model? Answer this by adding a new plant, two new ware- houses, and three new customers, and modify the spreadsheet model appropriately. You can make up the required input data. Would you conclude that these types of spreadsheet models scale easily?

27. In the RedBrand model in the file RedBrand Logistics Shrinkage Finished.xlsx, change the assumptions. Now instead of assuming that there is some shrinkage at the warehouses, assume that there is shrinkage in delivery along each route. Specifically, assume that a certain percentage of the units sent along each arc perish in transit—from faulty refrigeration, for example—and this percentage can differ from one arc to another. Modify the model appropriately to take this type of behavior into account. You can make up the shrinkage factors, and you can assume that arc capacities apply to the amounts originally shipped, not to the amounts after shrinkage. (Make sure your input data permit a feasible solution. After all, if there is too much shrinkage, it will be impossible to meet demands with available plant capacity. Increase the plant capacities if necessary.)

28. Consider a modification of the RedBrand model where there are N plants, M warehouses, and L custom- ers. Assume that the only allowable arcs are from plants to warehouses and from warehouses to customers. If all such arcs are allowable—all plants can ship to all ware- houses and all warehouses can ship to all customers— how many decision variable cells are in the spreadsheet model? Keeping in mind that Excel’s Solver can handle at most 200 decision variable cells, provide some combina- tions of N, M, and L that barely stay within Solver’s limit.

29. Continuing the previous problem, develop a sam- ple model with your own choices of N, M, and L that barely stay within Solver’s limit. You can make up any input data. The important point here is the layout and formulas of the spreadsheet model.

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14-5 Aggregate Planning Models 6 5 9

14-5 Aggregate Planning Models In this section, we extend the production planning model discussed in Example 13.3 of the previous chapter to include a situation where the number of workers available influences the possible production levels. We allow the workforce level to be modified each period through the hiring and firing of workers. Such models, where we determine workforce levels and production schedules for a multiperiod time horizon, are called aggregate plan- ning models. There are many variations of aggregate planning models, depending on the detailed assumptions made. We consider a fairly simple version and then ask you to mod- ify it in the problems.

EXAMPLE

14.5 AGGREGATE PLANNING AT SURESTEP During the next four months SureStep Company must meet (on time) the following demands for pairs of shoes: 3000 in month 1; 5000 in month 2; 2000 in month 3; and 1000 in month 4. At the beginning of month 1, 500 pairs of shoes are on hand, and SureStep has 100 workers. A worker is paid $1500 per month. Each worker can work up to 160 hours a month before he or she receives overtime. A worker can work up to 20 hours of overtime per month and is paid $13 per hour for overtime labor. It takes four hours of labor and $15 of raw material to produce a pair of shoes. At the beginning of each month, workers can be hired or fired. Each hired worker costs $1600, and each fired worker costs $2000. At the end of each month, a holding cost of $3 per pair of shoes left in inventory is incurred. All production in a given month can be used to meet that month’s demand. SureStep wants to determine its optimal production schedule and labor policy.

Objective To develop an optimization model that relates workforce and production decisions to monthly costs, and to find the mini- mum-cost solution that meets forecasted demands on time and stays within limits on overtime hours and production capacity.

Where Do the Numbers Come From? There are a number of required inputs for this type of problem. Some, including initial inventory, holding costs, and demands, are similar to requirements for Example 13.3 in the previous chapter, so we won’t discuss them again here. Others might be obtained as follows.

• The data on the current number of workers, the regular hours per worker per month, the regular hourly wage rates, and the overtime hourly rate, should be well known. The maximum number of overtime hours per worker per month is probably either the result of a policy decision by management or a clause in the workers’ contracts.

• The costs for hiring and firing a worker are not trivial. The hiring cost includes training costs and the cost of decreased pro- ductivity due to the fact that a new worker must learn the job. The firing cost includes severance costs and costs due to loss of morale. Neither the hiring nor the firing cost would be simple to estimate accurately, but the human resources department should be able to estimate their values.

• The unit production cost is a combination of two inputs: the raw material cost per pair of shoes and the labor hours per pair of shoes. The raw material cost is the going rate from the supplier(s). The labor per pair of shoes represents the “production function”—the average labor required to produce a unit of the product. The operations managers should be able to supply this number.

Solution A diagram for this model appears in Figure 14.27. (See the file Aggregate Planning Big Picture. xlsx.) It is divided into three parts: a section for workers, a section for shoes, and a section relating workers to shoe production. As you see, there are many variables to keep track of. In fact, the most difficult aspect of modeling this problem is knowing which variables the company gets to choose— the decision variables—and which variables are determined by these decisions. It should be clear that the company gets to choose the number of workers to hire and fire and the number of shoes to produce. Also, because manage- ment sets only an upper limit on overtime hours, it gets to decide how many overtime hours to use within this limit. But once

The key to this model is choosing the decision variables that determine the required outputs.

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it decides the values of these variables, everything else is determined. We will show how these are determined through detailed cell formulas, but you should mentally go through the yellow calculated values (those in rounded rectangles) and deduce how they are determined by the decision variables. Also, you should convince yourself that the three constraints listed are the ones, and the only ones, that are required.

Developing the Spreadsheet Model The spreadsheet model appears in Figure 14.28. (See the file Aggregate Planning 1 Finished.xlsx.) It can be developed as follows.

1. Inputs and range names. Enter the input data and create the range names listed. 2. Production, hiring, and firing plan. Enter any trial values for the number of pairs of shoes produced each month, the

overtime hours used each month, the workers hired each month, and the workers fired each month. These four ranges, in rows 18, 19, 23, and 30, comprise the decision variable cells.

3. Workers available each month. In cell B17 enter the initial number of workers available with the formula

5B5

Because the number of workers available at the beginning of any other month (before hiring and firing) is equal to the number of workers from the previous month, enter the formula

5B20

in cell C17 and copy it to the range D17:E17. Then calculate the number of workers available in month 1 (after hiring and firing) in cell B20 with the formula

5B171B18-B19

and copy this formula to the range C20:E20 for the other months. 4. Overtime capacity. Because each available worker can work up to 20 hours of overtime in a month, enter the formula

5$B$7*B20

in cell B25 and copy it to the range C25:E25.

Figure 14.27 Big Picture for Aggregate Planning Model

Minimize total cost (sum of costs in gray)

Workers

Workers to shoe production

Shoe production

Hiring cost per worker

Total hiring cost

Workers from previous month

Workers available after hiring and firing

Initial number of workers

Workers hired

Firing cost per worker

Total firing cost

Regular hours per worker per month

Regular-time hours available

Total hours for production

Labor hours per pair of shoes

Total overtime wages

Total regular- time wages

Maximum overtime labor hours available

Overtime wage rate per hour

Regular wages per worker per month

Maximum overtime hours per worker per month

Workers fired

Shoes produced <=

Overtime labor hours used

Raw material cost per pair of shoes

Total raw material cost

Production capacity

Holding cost per pair of shoes per month

Total holding cost

Ending inventory

Initial inventory of shoes

Inventory after production >= Forecasted

demand

Developing the Basic Aggregate Planning Model

This is common in multiperiod problems. You usually have to relate a beginning value in one period to an ending value from the previous period.

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14-5 Aggregate Planning Models 6 6 1

5. Production capacity. Because each worker can work 160 regular-time hours per month, calculate the regular-time hours available in month 1 in cell B22 with the formula

5$B$6*B20

and copy it to the range C22:E22 for the other months. Then calculate the total hours available for production in cell B27 with the formula

5SUM(B22:B23)

and copy it to the range C27:E27 for the other months. Finally, because it takes four hours of labor to make a pair of shoes, calculate the production capacity in month 1 with the formula

5B27/$B$12

in cell B32 and copy it to the range C32:E32. 6. Inventory each month. Calculate the inventory after production in month 1 (which is available to meet month 1 demand)

with the formula

5B41B30

in cell B34. For any other month, the inventory after production is the previous month’s ending inventory plus that month’s production, so enter the formula

5B371C30

in cell C34 and copy it to the range D34:E34. Then calculate the month 1 ending inventory in cell B37 with the formula

5B34-B36

and copy it to the range C37:E37.

In Example 13.3 from the previous chapter, production capacities were given inputs. Now they are based on the size of the workforce, which itself is a decision variable.

Figure 14.28 Aggregate Planning Model

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

IHGFEDCBA SureStep aggregate planning model

Input data Range names used: Ini�al inventory of shoes Forecasted_demand500 Ini�al number of workers Inventory_a�er_produc�on100 Regular hours/worker/month Maximum_over�me_labor_hours_available160 Maximum over�me hours/worker/month Over�me_labor_hours_used20 Hiring cost/worker Produc�on_capacity$1,600 Firing cost/worker Shoes_produced$2,000 Regular wages/worker/month Total_cost$1,500 Over�me wage rate/hour Workers_fired$13 Labor hours/pair of shoes =Model!$B$18:$E$18Workers_hired4 Raw material cost/pair of shoes $15 Holding cost/pair of shoes in inventory/month $3

Worker plan Month 1 Month 2 Month 3 Month 4 Workers from previous month 509394100 Workers hired 0 0 0 0 Workers fired 04316 Workers available a�er hiring and firing 94 93 50 50

Regular-�me hours available 800080001488015040 Over�me labor hours used 00800

<= <= <= <= Maximum over�me labor hours available 1880 1860 1000 1000

Total hours for produc�on 800080001496015040

Produc�on plan Month 1 Month 2 Month 3 Month 4 Shoes Produced 1000200037403760

Produc�on capacity 2000200037403760

Inventory a�er produc�on 1000200050004260

<= <= <= <=

>= >= >= >= Forecasted demand 1000200050003000 Ending inventory 0001260

Monetary outputs Month 1 Month 2 Month 3 Month 4 Totals Hiring cost $0$0$0$0$0 Firing cost $100,000$0$86,000$2,000$12,000 Regular-�me wages $430,500$75,000$75,000$139,500$141,000 Over�me wages $1,040$0$0$1,040$0 Raw material cost $157,500$15,000$30,000$56,100$56,400 Holding cost $3,780$0$0$0$3,780

$692,820$90,000$191,000$198,640$213,180Totals Objec�ve to minimize

=Model!$B$19:$E$19 =Model!$F$46 =Model!$B$30:$E$30 =Model!$B$32:$E$32 =Model!$B$23:$E$23 =Model!$B$25:$E$25 =Model!$B$34:$E$34 =Model!$B$36:$E$36

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6 6 2 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

Using Solver The Solver dialog box should be filled in as shown in Figure 14.29. Note that the decision variable cells include four separate named ranges. To enter these in the dialog box, drag the four ranges, keeping your finger on the Ctrl key. (Alternatively, you can drag a range, type a comma, drag a second range, type another comma, and so on.) As usual, you should also check the Non-Negative option and select the Simplex LP method before optimizing.

Note that there are integer constraints on the numbers hired and fired. You could also constrain the numbers of shoes produced to be integers. However, integer constraints typically require longer solution times. Therefore, it is often best to omit such constraints, especially when the optimal values are fairly large, such as the production quantities in this model. If the solution then has noninteger values, you can usually round them to integers for a solution that is at least close to the optimal integer solution.

Discussion of the Solution The optimal solution is given in Figure 14.28. Observe that SureStep should never hire any workers, and it should fire six work- ers in month 1, one worker in month 2, and 43 workers in month 3. Eighty hours of overtime are used, but only in month 2. The company produces over 3700 pairs of shoes during each of the first 2 months, 2000 pairs in month 3, and 1000 pairs in month 4. A total cost of $692,820 is incurred. The Solver solution will recommend overtime hours only when regular-time production capacity is exhausted. This is because overtime labor is more expensive.

Again, you would probably not force the number of pairs of shoes produced each month to be an integer. It makes little difference whether the company produces 3760 or 3761 pairs of shoes during a month, and forcing each month’s shoe produc- tion to be an integer can greatly increase the time Solver needs to find an optimal solution. On the other hand, it is somewhat

more important to ensure that the numbers of workers hired and fired each month are inte- gers, given the relatively small numbers of workers involved.

Finally, if you want to ensure that Solver finds the optimal solution in a problem where some or all of the decision variable cells must be integers, you should go into Options (in the Solver dialog box) and make sure the Integer Optimality is set to zero. Otherwise, Solver might stop when it finds a solution that is only close to optimal.

7. Monthly costs. Calculate the various costs shown in rows 40 through 45 for month 1 by entering the formulas

5$B$8*B18

5$B$9*B19

5$B$10*B20

5$B$11*B23

5$B$13*B30

5$B$14*B37

in cells B40 through B45. Then copy the range B40:B45 to the range C40:E45 to calculate these costs for the other months.

8. Totals. In row 46 and column F, use the SUM function to calculate cost totals, with the value in F46 being the overall total cost to minimize.

Calculating Row and Column Sums with AutoSum

A common operation in spreadsheet models is to calculate row and column sums for a rectangular range, as we did for costs in step 8. There is a very quick way to do this. Select the row and column where the sums will go (remem- ber to press the Ctrl key to select nonadjacent ranges) and click the AutoSum (S) button. This enters all of the sums automatically. It even calculates the “grand sum” in the corner (cell F46 in the example) if this cell is part of the selection.

Excel Tip Press the command key on the Mac.

Press the command key on the Mac.

Because integer constraints make a model more difficult to solve, use them sparingly—only when they are really needed.

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14-5 Aggregate Planning Models 6 6 3

Sensitivity Analysis There are many possible sensitivity analyses for this SureStep model. We illustrate one of them with SolverTable, where we see how the overtime hours used and the total cost vary with the overtime wage rate.8 The results appear in Figure 14.30. When the wage rate is really low, the company uses considerably more overtime hours, whereas when it is sufficiently large, the com- pany uses no overtime hours. It is not surprising that the company uses much more overtime when the overtime rate is $7 or $9 per hour. The regular-time wage rate is $9.375 per hour (= 1500>160). Of course, the company would never pay less per hour for overtime than for regular time.

The Rolling Planning Horizon Approach In reality, an aggregate planning model is usually implemented via a rolling planning horizon. To illustrate, we assume that SureStep works with a four-month planning horizon. To implement the SureStep model in the rolling planning horizon context, we view the demands as forecasts and solve a four-month model with these forecasts. However, the company would implement only the month 1 production and work scheduling recommendation. Thus (assuming that the numbers of workers hired and fired in a month must be integers) the company would hire no workers, fire six workers, and produce 3760 pairs of shoes with regular-time labor in month 1. Next, the company would observe month 1’s actual demand. Suppose it is 2950. Then SureStep would begin month 2 with 1310 (= 4260 2 2950) pairs of shoes and 94 workers. It would now enter 1310 in cell B4 and 94 in cell B5 (referring to Figure 14.28). Then it would replace the demands in the Demand range with the updated forecasts for the next four months. Finally, SureStep would rerun Solver and use the produc- tion levels and hiring and firing recommendations in column B as the production level and workforce policy for month 2.

Figure 14.29 Solver Dialog Box for Aggregate Planning Model

The term “backlogging” means that the customer’s demand is met at a later date. The term “back- ordering” means the same thing.

8 Solver’s sensitivity report isn’t even available here because of the integer constraints.

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Model with Backlogging Allowed In many situations, backlogging of demand is allowed—that is, customer demand can be met at a later date. We now show how to modify the SureStep model to include the option of backlogging demand. We assume that at the end of each month a cost of $20 is incurred for each unit of demand that remains unsatisfied at the end of the month. This is easily modeled by allowing a month’s ending inventory to be negative. For example, if month 1’s ending inventory is 210, a shortage cost of $200 (and no inventory holding cost) is incurred. To ensure that SureStep produces any shoes at all, we constrain the ending inventory in month 4 to be nonnegative. This implies that all demand is eventually satisfied by the end of the four-month planning horizon. We now need to modify the monthly cost calculations to incorporate costs due to backlogging.

There are actually several modeling approaches to this backlogging problem. We show the most natural approach in Figure 14.31. (See the file Aggregate Planning 2 Finished.xlsx.) To begin, enter the per-unit monthly shortage cost in cell B15. (A new row was inserted for this cost input.) Note in row 38 how the ending inventory in months 1 through 3 can be posi- tive (leftovers) or negative (shortages). You can account correctly for the resulting costs with IF functions in rows 46 and 47. For holding costs, enter the formula

5IF(B38+0,$B$14*B38,0)

in cell B46 and copy it across. For shortage costs, enter the formula

5IF(B38*0,2$B$15*B38,0)

in cell B47 and copy it across. (The minus sign makes this a positive cost.)

Figure 14.30 Sensitivity to Overtime Wage Rate

3 2 1

4 5 6 7 8 9

10 11 12

A B C D E F G

Over me rate (cell $B$11) values along side, output cell(s) along top

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$7 1620 1660 0 0 $684,755 $9 80 1760 0 0 $691,180

$11 0 80 0 0 $692,660 $13 0 80 0 0 $692,820 $15 0 80 0 0 $692,980 $17 0 80 0 0 $693,140 $19 0 0 0 0 $693,220 $21 0 0 0 0 $693,220

Oneway analysis for Solver model in Model worksheet

Figure 14.31 Nonlinear Aggregate Planning Model Using IF Functions

Developing the Aggregate Planning Backlogging Model

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Although these formulas calculate holding and shortage costs accurately, the IF functions make the objective cell a nonlinear function of the decision variable cells, and Solver’s GRG nonlinear algorithm must be used, as indicated in Figure 14.32.9 (How do you know the model is nonlinear? Although there is a mathematical reason, it is easier to try running Solver with the simplex algorithm. Solver will then inform you that the model is nonlinear.)

9 GRG stands for generalized reduced gradient. This is a technical term for the mathematical algorithm used. The other algorithm available in Solver (start- ing with Excel 2010) is the Evolutionary algorithm. It can handle IF functions, but we will not discuss this algorithm here.

Figure 14.32 Solver Dialog Box for the GRG Nonlinear Algorithm

We ran Solver with this setup from a variety of initial solutions in the decision variable cells, and it always found the optimal solution. But we were lucky. When certain functions, including IF, MIN, MAX, and ABS, are used to relate the objective cell to the decision variable cells, the resulting model becomes not only nonlinear but nonsmooth. Essentially, nonsmooth functions can have sharp edges or discontinuities. Solver’s GRG nonlinear algorithm can handle “smooth” nonlinearities, but it has trouble with nonsmooth functions. Sometimes it gets lucky, as it did here, and other times it finds a nonoptimal solution that is not even close to the optimal solution. For example, we changed the unit shortage cost from $20 to $40 and reran Solver. Starting from a solution where all decision variable cells contain zero, Solver stopped at a solution with total cost $726,360, even though the optimal solution has total cost $692,820. So we weren’t so lucky this time.

The moral is that you should avoid these nonsmooth functions in optimization models if at all possible. If you do use them, as we have done here, you should run Solver several times, starting from different initial solutions. There is still no guar- antee that you will get the optimal solution, but you will see more evidence of how Solver is progressing. (Alternatively, you can use the Evolutionary Solver, which became a part of Excel’s Solver in Excel 2010. You can also use the Evolver add-in, part of Palisade’s DecisionTools® Suite.)

14-5 Aggregate Planning Models 6 6 5

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6 6 6 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

Problems

Level A 30. Extend SureStep’s original (no backlogging) aggregate

planning model from four to six months. Try several dif- ferent values for demands in months 5 and 6, and run Solver for each. Is your optimal solution for the first four months the same as the one in the example?

31. The current solution to SureStep’s no-backlogging aggregate planning model does quite a lot of firing. Run a one-way SolverTable with the firing cost as the input variable and the numbers fired as the outputs. Let the firing cost increase from its current value to double that value in increments of $400. Do high firing costs eventu- ally induce the company to fire fewer workers?

32. SureStep is currently getting 160 regular-time hours from each worker per month. This is actually calculated from 8 hours per day times 20 days per month. For this, they are paid $9.375 per hour (= 1500/160). Suppose workers can change their contract so that they have to work only 7.5 hours per day regular time—everything

above this becomes overtime—and their regular-time wage rate increases to $10 per hour. They will still work 20 days per month. Does this change the optimal no-backlogging solution?

33. Suppose SureStep could begin a machinery upgrade and training program to increase its worker productiv- ity. This program would result in the following values of labor hours per pair of shoes over the next four months: 4, 3.9, 3.8, and 3.8. How much would this new program be worth to SureStep, at least for this four-month plan- ning horizon with no backlogging? How might you eval- uate the program’s worth beyond the next four months?

Level B 34. In the current no-backlogging problem, SureStep

doesn’t hire any workers, and it uses almost no over- time. This is evidently because of low demand. Change the demands to 6000, 8000, 5000, and 3000, and rerun Solver. Is there now any hiring and/or over- time? With this new demand pattern, explore the trade- off between hiring and overtime by running a two-way SolverTable. As inputs, use the hiring cost per worker

6 6 6 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

There are sometimes alternatives to using IF, MIN, MAX, and ABS functions that make a model linear. Unfortunately, these alternatives are often far from intuitive, and we will not cover them here. (If you are interested, we have included the “linearized” version of the backlogging model in the file Aggregate Planning 3 Finished.xlsx.)

Nonsmooth Functions

There is nothing inherently wrong with using IF, MIN, MAX, ABS, and other nonsmooth functions in spreadsheet optimization models. The problem is that Solver’s GRG nonlinear algorithm cannot handle these functions in a predictable manner.

Solver Tip

Nonsmooth Functions and Solver

Excel’s Solver, as well as most other commercial optimization software pack- ages, has trouble with nonsmooth nonlinear functions. These nonsmooth func- tions typically have sharp edges or discontinuities that make them difficult to handle in optimization models, and (in Excel) they typically contain functions such as IF, MAX, MIN, ABS, and a few others. There is nothing wrong with using such functions to implement complex logic in Excel optimization models. The problem is that Solver cannot handle models with these functions predict- ably. This is not really the fault of Solver. Such problems are inherently difficult.

Fundamental Insight

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14-6 Financial Models 6 6 7

and the maximum overtime hours allowed per worker per month, varied over reasonable ranges. As outputs, use the total number of workers hired over the four months and the total number of overtime hours used over the four months. Discuss the results.

35. In the SureStep no-backlogging problem, change the demands so that they become 6000, 8000, 5000, and 3000. Also, change the problem slightly so that newly hired workers take six hours to produce a pair of shoes during their first month of employment. After that, they take only four hours per pair of shoes. Modify the model appropriately, and use Solver to find the optimal solution.

36. You saw that the “natural” way to model SureStep’s backlogging problem, with IF functions, leads to a non- smooth model that Solver has difficulty handling. There

is another version of the problem that is also difficult for Solver. Suppose SureStep wants to meet all demands on time (no backlogging), but it wants to keep its employ- ment level as constant over time as possible. To induce this, it charges a cost of $1000 each month on the abso- lute difference between the beginning number of work- ers and the number after hiring and firing—that is, the absolute difference between the values in rows 17 and 20 of the original spreadsheet model. Implement this extra cost in the model in the “natural” way, using the ABS function. Using demands of 6000, 8000, 5000, and 3000, see how well Solver does in solving this non- smooth model. Try several initial solutions, and see whether Solver gets the same optimal solution from each of them.

14-6 Financial Models The majority of optimization examples described in management science textbooks are in the area of operations: scheduling, blending, logistics, aggregate planning, and others. This is probably warranted, because many of the most successful management science applications in the business world have been in these areas. However, optimization and other management science methods have also been applied successfully in a number of financial areas, and they deserve recognition. In this section, we begin the discussion with two typical applications of optimization in finance. The first involves investment strategy. The second involves pension fund management.

EXAMPLE

14.6 FINDING AN OPTIMAL INVESTMENT STRATEGY AT BARNEY-JONES

At the present time, the beginning of year 1, Barney-Jones Investment Corporation has $100,000 to invest for the next four years. There are five possible investments, labeled A through E. The timing of cash outflows and cash inflows for these invest- ments is somewhat irregular. For example, to take part in investment A, cash must be invested at the beginning of year 1, and for every dollar invested, there are returns of $0.50 and $1.00 at the beginnings of years 2 and 3. Information for the other investments follows, where all returns are per dollar invested:10

• Investment B: Invest at the beginning of year 2, receive returns of $0.50 and $1.00 at the beginnings of years 3 and 4

• Investment C: Invest at the beginning of year 1, receive return of $1.20 at the beginning of year 2

• Investment D: Invest at the beginning of year 4, receive return of $1.90 at the beginning of year 5

• Investment E: Invest at the beginning of year 3, receive return of $1.50 at the beginning of year 4

We assume that any amounts can be invested in these strategies and that the returns are the same for each dollar invested. However, to create a diversified portfolio, Barney-Jones wants to limit the amount put into any investment to $75,000. The company wants an investment strategy that maximizes the amount of cash on hand at the beginning of year 5. At the beginning of any year, it can invest only cash on hand, which includes returns from previous investments. Any cash not invested in any year can be put in a short-term money market account that earns 3% annually.

10 You might criticize this model for assuming known returns in future years. If the returns are actually uncertain with given probability distributions, the RISKOptimizer tool in @RISK (part of Palisade’s DecisionTools Suite) can be used to find the investment strategy that maximizes the expected return. However, we won’t discuss this possibility here.

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Objective To develop an optimization model that relates investment decisions to total ending cash, and to use Solver to find the strategy that maximizes ending cash and invests no more than a given amount in any one investment.

Where Do the Numbers Come From? There is no mystery here. We assume that the terms of each investment are spelled out, so that Barney-Jones knows exactly when money must be invested and what the amounts and timing of returns will be. Of course, this would not be the case for many real-world investments, such as money put into the stock market, where considerable uncertainty is involved. We consider one such example of investing with uncertainty when we study portfolio optimization in Section 14-8.

Solution The variables and constraints for this investment model are shown in Figure 14.33. (See the file Investing Big Picture.xlsx.) On the surface, this problem appears to be very straightforward. You must decide how much to invest in the available invest- ments at the beginning of each year, using only the cash available. If you try modeling this problem without our help, however, we suspect that you will have some difficulty. It took us a few tries to get a model that is easy to read and generalizes to other similar investment problems. Note that the second constraint in the table can be expressed in two ways. It can be expressed as shown, where the cash on hand after investing is nonnegative, or it can be expressed as “cash invested in any year must be less than or equal to cash on hand at the beginning of that year.” These are equivalent. The one you choose is a matter of taste.

There are often multiple equivalent ways to state a constraint. You can choose the one that is most natural for you.

Figure 14.33 Big Picture for Investment Model Dollars invested

Cash invested

Cash a�er inves�ng

Maximize final cash

Return from investmentsBeginning cash

Ini�al amount to invest

Interest rate on cash

Maximum per investment

Cash return per dollar invested

Cash outlay per dollar invested

>= 0

Developing the Spreadsheet Model The spreadsheet model for this investment problem appears in Figure 14.34. (See the file Investing Finished.xlsx.) To set up this spreadsheet, proceed as follows.

1. Inputs and range names. As usual, enter the given inputs in the blue cells and name the ranges indicated. Pay particular attention to the two shaded tables. This is probably the first model you have encountered where model development is affected significantly by the way you enter the inputs, specifically, the information about the investments. We suggest separating cash outflows from cash inflows, as shown in the two ranges B11:F14 and B19:F23. The top table indicates when investments can be made, where a blank (equivalent to a 0) indicates no possible investment, and $1.00 indicates a dollar of investment. The bottom table then indicates the amounts and timing of returns per dollar invested.

Developing the Investment Model

The two input tables allow you to create copyable SUMPRODUCT formulas for cash outflows and inflows. Careful spreadsheet planning can often greatly simplify the necessary formulas.

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14-6 Financial Models 6 6 9

2. Investment amounts. Enter any trial values in the Dollars_invested range. This range contains the decision variable cells. Also put a link to the maximum investment amount per investment by entering the formula

5$B$5

in cell B28 and copying it across. 3. Cash balances and flows. The key to the model is the section in rows 32 through 36. For each year, you need to calculate

the beginning cash held from the previous year, the returns from investments that are due in that year, the investments made in that year, and cash balance after investments. Begin by entering the initial cash in cell B32 with the formula

5B4

Moving across, calculate the return due in year 1 in cell C32 with the formula

5SUMPRODUCT(B19:F19,Dollars_invested)

Admittedly, no returns come due in year 1, but this formula can be copied down column C for other years. Next, calculate the total amount invested in year 1 in cell D32 with the formula

5SUMPRODUCT(B11:F11,Dollars_invested)

Now find the cash balance after investing in year 1 in cell E32 with the formula

5B321C32-D32

The only other required formula is the formula for the cash available at the beginning of year 2. Because any cash not invested earns 3% interest, enter the formula

5E32*(11$B$6)

in cell B33. This formula, along with those in cells C32, D32, and E32, can now be copied down. (The zeros in column G are entered manually as a reminder of the nonnegativity constraint on cash after investing.)

Figure 14.34 Investment Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38

Investments with irregular timing of returns Range names used

Inputs Initial amount to invest Maximum per investment

$100,000 $75,000

Interest rate on cash 3%

Cash outlays on investments (all incurred at beginning of year) Investment

EDCBAYear $1.00$1.001

$1.002 $1.003

$1.004

Cash returns from investments (all incurred at beginning of year) Investment

EDCBAYear

$0.50 1

$1.20 $1.00

2 $0.503

$1.50$1.004 $1.905

Investment decisions Dollars invested $75,000$75,000$35,714$75,000$64,286

<= <= <= <= <= Maximum per investment

Constraints on cash balance

Year Beginning cash Returns from investments Cash invested

Cash after investing

0>=$0$100,000$0$100,0001 0>=–$0$75,000$75,000$02 0>=$26,786$75,000$101,786–$03 0>=$140,089$75,000$187,500$27,5894

$142,500$144,2925

Final cash $286,792 Objective to maximize: final cash at beginning of year 5

A B C D E F G H I J

=Model!$E$32:$E$35 Dollars_invested Cash_after_investing

=Model!$B$26:$F$26 Final_cash =Model!$B$38 Maximum_per_investment =Model!$B$28:$F$28

$75,000 $75,000 $75,000 $75,000 $75,000

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4. Ending cash. The ending cash at the beginning of year 5 is the sum of the amount in the money market and any returns that come due in year 5. Calculate this sum with the formula

5SUM(B36:C36)

in cell B38. (Note: Here is the type of error to watch out for. We originally failed to calculate the return in cell C36 and mistakenly used the beginning cash in cell B36 as the objective cell. We realized our error when the optimal solution called for no money in investment D, which is clearly an attractive investment. The moral is that you can often catch errors by looking at the plausibility of the results.)

Review of the Model Take a careful look at this model and how it has been set up. There are undoubtedly alternative ways to model this problem, but the attractive feature of this model is the way the tables of inflows and outflows in rows 11 through 14 and 19 through 23 cre- ate copyable formulas for returns and investment amounts in columns C and D of rows 32 through 35. This same model setup, with only minor modifications, will work for any set of investments, regardless of the timing of investments and their returns. This is a quality you should strive for in your spreadsheet models: generalizability.

Using Solver To find the optimal investment strategy, fill in the main Solver dialog box as shown in Figure 14.35. Note that the explicit nonnegativity constraint is necessary, even though the Non-Negative option is checked. Again, this is because the Non-Nega- tive option covers only the decision variable cells. If you want other output cells to be nonnegative, you must constrain them explicitly.

Always look at the Solver solution for signs of implausibility. This can often help you spot an error in your model.

Figure 14.35 Solver Dialog Box for Investment Model

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14-6 Financial Models 6 7 1

Discussion of the Results The optimal solution appears in Figure 14.34. Let’s follow the cash. The company spends all its cash in year 1 on the two avail- able investments, A and C ($64,286 in A, $35,714 in C). A total of $75,000 in returns from these investments is available in year 2, and it is all invested in investment B. At the beginning of year 3, a total of $101,786 is available from investment A and B returns, and $75,000 of this is invested in investment E. This leaves $26,786 for the money market, which grows to $27,589 at the beginning of year 4. In addition, returns totaling $187,500 from investments B and E come due in year 4. Of this total cash of $215,089, $75,000 is invested in investment D, and the rest, $140,089, is put in the money market. The return from investment D, $142,500, plus the money available from the money market, $144,292, equals the final cash in the objective cell, $286,792.

Sensitivity Analysis A close look at the optimal solution in Figure 14.34 indicates that Barney-Jones is penalizing itself by imposing a maximum of $75,000 per investment. This upper limit is forcing the company to put cash into the money market fund, despite this fund’s low rate of return. Therefore, a natural sensitivity analysis is to see how the optimal solution changes as this maximum value changes. You can perform this sensitivity analysis with a one-way SolverTable, shown in Figure 14.36.11 The maximum in cell B5 is the input cell, varied from $75,000 to $225,000 in increments of $25,000, and the optimal decision variable cells and objective cell are outputs. As you can see, the final cash (column G) grows steadily as the maximum allowable investment amount increases. This is because the company can take greater advantage of the attractive investments and put less in the money market account.

Figure 14.36 Sensitivity of Optimal Solution to Maximum Investment Amount

4 5 6 7 8 9

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A B C D E F G

Max per investment (cell $B$5) values along side, output cell(s) along top 2 1 Oneway analysis for Solver model in Model worksheet

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l_ ca

$75,000 $64,286 $75,000 $35,714 $75,000 $75,000 $286,792 $100,000 $61,538 $76,923 $38,462 $100,000 $100,000 $320,731 $125,000 $100,000 $50,000 $0 $125,000 $125,000 $353,375 $150,000 $100,000 $50,000 $0 $150,000 $125,000 $375,125 $175,000 $100,000 $50,000 $0 $175,000 $125,000 $396,875 $200,000 $100,000 $50,000 $0 $200,000 $125,000 $418,625 $225,000 $100,000 $50,000 $0 $225,000 $125,000 $440,375

11 Because Solver’s sensitivity reports do not help answer our specific sensitivity questions in this example or the next example, we discuss only SolverTable results.

You can go one step further with the two-way SolverTable in Figure 14.37. Now both the maximum investment amount and the money market rate are inputs, and the maxi- mum amount ever put in the money market fund is the single output. Because this latter amount is not calculated in the spreadsheet model, you need to calculate it with the for- mula 5MAX(Cash_after_investing) in an unused cell before using it as the output cell for SolverTable. In every case, even with a large maximum investment amount and a low money market rate, the company puts some money into the money market account. The reason is simple. Even when the maximum investment amount is $225,000, the company

To perform sensitivity on an output variable not calculated explicitly in your spreadsheet model, calculate it in some unused portion of the spreadsheet before running SolverTable.

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6 7 2 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

The following example illustrates a common situation where fixed payments are due in the future and current funds must be allocated and invested so that their returns are suf- ficient to make the payments. We place this in a pension fund context.

A B C D E F G H I

Interest on cash (cell $B$6) values along side, Max per investment (cell $B$5) values along top, output cell in corner

4 5 6 7

Maximum_in_money_market $75,000 $100,000 $125,000 $150,000 $175,000 $200,000 $225,000 0.5% $139,420 $126,923 $112,500 $87,500 $62,500 $37,500 $12,500 1.0% $139,554 $126,923 $112,500 $87,500 $62,500 $37,500 $12,500 1.5% $139,688 $126,923 $112,500 $87,500 $62,500 $37,500 $12,500

$ $ $ $ $ $ $8 9

10 11 12 13

2.0% $139,821 $126,923 $112,500 $87,500 $62,500 $37,500 $12,500 2.5% $139,955 $126,923 $112,500 $87,500 $62,500 $37,500 $12,500 3.0% $140,089 $126,923 $112,500 $87,500 $62,500 $37,500 $12,500 3.5% $140,223 $126,923 $112,500 $87,500 $62,500 $37,500 $12,500 4.0% $140,357 $126,923 $112,500 $87,500 $62,500 $37,500 $12,500 4.5% $140,491 $126,923 $112,500 $87,500 $62,500 $37,500 $12,500

2 1 Twoway analysis for Solver model in Model worksheet

Figure 14.37 Sensitivity of Maximum in Money Market to Two Inputs

evidently has more cash than this to invest at some point (probably at the beginning of year 4). Therefore, it will have to put some of it in the money market.

EXAMPLE

14.7 MANAGING A PENSION FUND AT ARMCO James Judson is the financial manager in charge of the company pension fund at Armco Incorporated. James knows that the fund must be sufficient to make the payments listed in Table 14.4. Each payment must be made on the first day of each year. James is going to finance these payments by purchasing bonds. It is currently the beginning of year 1, and three bonds are available for immediate purchase. The prices and coupons for the bonds are as follows. (All coupon payments arrive in time to meet the pension payments for the year in which they arrive.)

• Bond 1 costs $980 and yields a $60 coupon in years 2 through 5 and a $1060 payment on maturity in year 6.

• Bond 2 costs $970 and yields a $65 coupon in years 2 through 11 and a $1065 payment on maturity in year 12.

• Bond 3 costs $1050 and yields a $75 coupon in years 2 through 14 and a $1075 payment on maturity in year 15.

Year Payment Year Payment Year Payment

1 $11,000 6 $18,000 11 $25,000

2 $12,000 7 $20,000 12 $30,000

3 $14,000 8 $21,000 13 $31,000

4 $15,000 9 $22,000 14 $31,000

5 $16,000 10 $24,000 15 $31,000

Table 14.4 Payments for Pension Example

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14-6 Financial Models 6 7 3

James must decide how much cash to allocate (from company coffers) to meet the initial $11,000 payment and buy enough bonds to make future payments. He knows that any excess cash on hand can earn an annual rate of 4% in a fixed-rate account. How should he proceed?

Objective To develop an optimization model that relates initial allocation of money and bond purchases to future cash availabilities, and to minimize the initialize allocation of money required to meet all future pension fund payments.

Where Do the Numbers Come From? As in the previous financial example, the inputs are fairly easy to obtain. A pension fund has known liabilities that must be met in future years, and information on bonds and fixed-rate accounts is widely available.

Solution The variables and constraints required for this pension fund model are shown in Figure 14.38. (See the file Pension Fund Management Big Picture.xlsx.) When modeling this problem, there is a new twist that involves the money James must allo- cate now for his funding problem. It is clear that he must decide how many bonds of each type to purchase now (note that no bonds are purchased in the future), but he must also decide how much money to allocate from company coffers. This allocated money has to cover the initial pension payment this year and the bond purchases. In addition, James wants to find the mini- mum allocation that will suffice. Therefore, this initial allocation serves two roles in the model. It is a decision variable and it is the objective to minimize. In terms of spreadsheet modeling, it is perfectly acceptable to make the objective cell one of the decision variable cells, and this is done here. You will not see this in many models—because the objective typically involves a linear combination of several decision variables—but it is occasionally the most natural way to proceed.

Cash allocated to purchase bonds and make pension payments - also

the objec ve to minimize

Cash available from bonds and interest

Pension cash requirements

Interest rateCost of bonds Income from bonds

Number of bonds to purchase

Figure 14.38 Big Picture for Pension Fund Management Model

The Objective as a Decision Variable Cell

In all optimization models, the objective cell has to be a function of the decision variable cells, that is, the objective value should change as values in the decision variable cells change. It is perfectly consistent with this requirement to have the objective cell be one of the decision variable cells. This doesn’t occur in very many optimization mod- els, but it is sometimes useful, even necessary.

Fundamental Insight

Developing the Spreadsheet Model The completed spreadsheet model is shown in Figure 14.39. (See the file Pension Fund Manage- ment Finished.xlsx.) You can create it with the following steps.

1. Inputs and range names. Enter the given data and name the ranges as indicated. Note that the bond costs in the range B5:B7 have been entered as positive quantities. Some financial analysts might prefer that they be entered as negative numbers, indicating outflows. It doesn’t really mat- ter, however, as long as you are careful with the Excel formulas later on.

2. Cash allocated and bonds purchased. As discussed previously, the cash allocated in year 1 and the numbers of bonds purchased are both decision variables, so enter any values for these in the

Developing the Pension Fund Model

Always document your spreadsheet conventions as clearly as possible.

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Cash_allocated and Bonds_purchased ranges. Note that the color-coding convention for the Cash_allocated cell has to be modified. Because it is both a decision variable cell and the objective cell, we colored it red but added a note to emphasize that it is the objective to minimize.

3. Cash available to make payments. In the current year, the only cash available is the money initially allocated minus cash used to purchase bonds. Calculate this quantity in cell B20 with the formula

5Cash_allocated-SUMPRODUCT(Bonds_purchased,B5:B7)

For all other years, the cash available comes from two sources: excess cash invested at the fixed interest rate the year before and payments from bonds. Calculate this quantity for year 1 in cell C20 with the formula

5(B20-B22)*(11$B$9)1SUMPRODUCT(Bonds_purchased,C5:C7)

and copy it across row 20 for the other years.

As you can see, this model is fairly straightforward to develop once you understand the role of the amount allocated in cell B16. However, we have often given this problem as an assignment to our students, and many fail to deal correctly with the amount allocated. (They usually forget to make it a decision variable cell.) So make sure you understand what we have done, and why we have done it this way.

Using Solver The main Solver dialog box should be filled out as shown in Figure 14.40. Once again, notice that the Cash_allocated cell is both the objective cell and one of the decision variable cells.

Discussion of the Solution The optimal solution appears in Figure 14.39. You might argue that the numbers of bonds purchased should be constrained to inte- ger values. We tried this and the optimal solution changed very little: The optimal numbers of bonds to purchase changed to 74, 79, and 27, and the optimal money to allocate increased to $197,887. With this integer solution, shown in Figure 14.41, James sets aside $197,887 initially. Any less than this would not work—he couldn’t make enough from bonds to meet future pension payments. All but $20,387 of this (see cell B20) is spent on bonds, and of the $20,387, $11,000 is used to make the current pension payment. After this, the amounts in row 20, which are always sufficient to make the payments in row 22, are com- posed of returns from bonds and cash, with interest, from the previous year. Even more so than in previous examples, there is no way to guess this optimal solution. The timing of bond returns and the irregular pension payments make a spreadsheet optimization model absolutely necessary.

Figure 14.39 Pension Fund Management Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

A B C D E F G H I J K L M N O P Pension fund management

Costs (year 1) and income (in other years) from bonds Year 15Year 14Year 13Year 12Year 11Year 10Year 9Year 8Year 7Year 6Year 5Year 4Year 3Year 2Year 1

Bond 1 $1,060$60$60$60$980 Bond 2 $1,065$65$65$65$65$65$65$65$65$65$970 Bond 3 $1,050 $75 $75 $75

$60 $65 $75 $75 $75 $75 $75 $75 $75 $75 $75 $75 $1,075

Interest rate 4.00%

Number of bonds (allowing frac�onal values) to purchase now Bond 1 73.69 Bond 2 77.21 Bond 3 28.84

Cash allocated $197,768 Objec�ve to minimize, also a decision variable cell

Constraints to meet payments Year 15Year 14Year 13Year 12Year 11Year 10Year 9Year 8Year 7Year 6Year 5Year 4Year 3Year 2Year 1

$20,376 $21,354 $21,332 $19,228 $16,000 $85,298 $77,171 $66,639 $54,646 $41,133 $25,000 $84,390 $58,728 $31,000 $31,000 >= >= >= >= >= >= >= >= >= >= >= >= >= >= >=

Cash required

Cash available

$11,000 $12,000 $14,000 $15,000 $16,000 $18,000 $20,000 $21,000 $22,000 $24,000 $25,000 $30,000 $31,000 $31,000 $31,000

Range names used: Bonds_purchased Cash_allocated Cash_available Cash_required

=Model!$B$12:$B$14 =Model!$B$16 =Model!$B$20:$P$20 =Model!$B$22:$P$22

The value in cell B16 is the money allocated to make the current payment and buy bonds now. It is both a decision variable cell and the objec�ve cell to minimize.

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14-6 Financial Models 6 7 5

Figure 14.40 Solver Dialog Box for Pension Fund Model

Figure 14.41 Optimal Integer Solution for Pension Fund Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22

A B C D E F G H I J K L M N O P Pension fund management

Costs (year 1) and income (in other years) from bonds Year 15Year 14Year 13Year 12Year 11Year 10Year 9Year 8Year 7Year 6Year 5Year 4Year 3Year 2Year 1Year

Bond 1 $1,060$60$60$60$980 Bond 2 $1,065$65$65$65$65$65$65$65$65$65$970 Bond 3 $1,050 $75 $75 $75

$60 $65 $75 $75 $75 $75 $75 $75 $75 $75 $75 $75 $1,075

Interest rate 4.00%

Number of bonds (allowing frac�onal values) to purchase now Bond 1 74.00 Bond 2 79.00 Bond 3 27.00

Cash allocated $197,887 Objec�ve to minimize, also a decision variable cell

Constraints to meet payments Year 15Year 14Year 13Year 12Year 11Year 10Year 9Year 8Year 7Year 6Year 5Year 4Year 3Year 2Year 1Year

$20,387 $21,363 $21,337 $19,231 $16,000 $85,600 $77,464 $66,923 $54,919 $41,396 $25,252 $86,422 $60,704 $32,917 $31,019 >= >= >= >= >= >= >= >= >= >= >= >= >= >= >=

Cash required

Cash available

$11,000 $12,000 $14,000 $15,000 $16,000 $18,000 $20,000 $21,000 $22,000 $24,000 $25,000 $30,000 $31,000 $31,000 $31,000

Sensitivity Analysis Because the bond information and pension payments are fixed, there is only one obvious direction for sensitivity analysis: on the fixed interest rate in cell B9. We tried this, allowing this rate to vary from 2% to 6% in increments of 0.5% and keeping track of the optimal decision variable cells, including the objective cell. The results appear in Figure 14.42 (without the integer constraints). They indicate that as the interest rate increases, James can get by with fewer bonds of types 1 and 2, and he can allocate less money for the problem. The reason is that he is making more interest on excess cash.

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Problems

Level A 37. Modify the Barney-Jones investment model so that there

is a minimum amount that must be put into any invest- ment, although this minimum can vary by investment. For example, the minimum amount for investment A might be $0, whereas the minimum amount for invest- ment D might be $50,000. These minimum amounts should be inputs; you can make up any values you like. Run Solver on your modified model.

38. In the Barney-Jones investment model, increase the maximum amount allowed in any investment to $150,000. Then run a one-way sensitivity analysis to the money market rate on cash. Capture one output variable: the maximum amount of cash ever put in the money market account. You can choose any reasonable range for varying the money market rate.

39. We claimed that our model for Barney-Jones is gener- alizable. Try generalizing it to the case where there are two more potential investments, F and G. Investment F requires a cash outlay in year 2 and returns $0.50 in each of the next four years. Investment G requires a cash out- lay in year 3 and returns $0.75 in each of years 5, 6, and 7. Modify the model as necessary, making the objective the final cash after year 7.

40. In our version of the Barney-Jones model, we ran invest- ments across columns and years down rows. Some finan- cial analysts prefer the opposite. Modify the spreadsheet model so that years go across columns and investments

go down rows. Run Solver to ensure that your modified model is correct. (We suggest two possible ways to do this, and you can experiment to see which you prefer. First, you could start over on a blank worksheet. Sec- ond, you could use Copy and then Paste Special with the Transpose option.)

41. In the Armco pension fund model, suppose there is a fourth bond, bond 4. Its unit cost in year 1 is $1020, it returns coupons of $70 in years 2 through 7 and a pay- ment of $1070 in year 8. Modify the model to incorpo- rate this extra bond, and reoptimize. Does the solution change—that is, should James purchase any of bond 4?

42. In the Armco pension fund model, suppose there is an upper limit of 60 on the number of bonds of any par- ticular type that can be purchased. Modify the model to incorporate this extra constraint and then optimize. How much more money does James need to allocate initially?

43. In the Armco pension fund model, suppose James has been asked to see how the optimal solution will change if the required payments in years 8 through 15 all increase by the same percentage, where this percentage could be anywhere from 5% to 25%. Use an appropriate one-way SolverTable to help him out, and write a memo describ- ing the results.

44. Our version of the Armco pension fund model is streamlined, perhaps too much. It does all of the cal- culations concerning cash flows in row 20. James decides he would like to break these out into several rows of calculations: Beginning cash (for year 1, this is the amount allocated; for other years, it is the unused cash, plus interest, from the previous year), Amount spent on bonds (positive in year 1 only),

4 5 6 7 8 9

10 11 12 13 14 15

A B C D E F G

Interest rate (cell $B$9) values along side, output cell(s) along top 2 1 Oneway analysis for Solver model in Model with Integers worksheet

Bo nd

s_ pu

rc ha

se d_

Bo nd

s_ pu

rc ha

se d_

Bo nd

s_ pu

rc ha

se d_

Ca sh

_a llo

ca te

2.00% 2.40% 2.80% 3.20% 3.60% 4.00% 4.40% 4.80% 5.20% 5.60% 6.00%

77.00 77.00 76.00 75.00 75.00 74.00 73.00 73.00 72.00 71.00 71.00

80.00 80.00 80.00 80.00 79.00 79.00 79.00 77.00 77.00 78.00 76.00

28.00 27.00 27.00 27.00 27.00 27.00 27.00 28.00 28.00 27.00 28.00

$202,219 $201,372 $200,469 $199,605 $198,769 $197,887 $197,060 $196,208 $195,369 $194,574 $193,787

Figure 14.42 Sensitivity to Fixed Interest Rate

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14-7 Integer Optimization Models 6 7 7

Amount received from bonds (positive for years 2 through 15 only), Cash available for making pension fund payments, and, below the Pension payment row, Cash left over (amount invested in the fixed interest rate). Modify the model by inserting these rows, enter the appropriate formulas, and run Solver. You should obtain the same result, but you get more detailed information.

Level B 45. Suppose the investments in the Barney-Jones model

sometimes require cash outlays in more than one year. For example, a $1 investment in investment B might require $0.25 to be spent in year 1 and $0.75 to be spent in year 2. Does the current model easily accommodate such investments? Try it with some cash outlay data you can make up, run Solver, and interpret your results.

46. In the Armco pension fund model, you know that if the amount of money allocated initially is less than

the amount found by Solver, James will not be able to meet all of the pension fund payments. Use the current model to demonstrate that this is true. To do so, enter a value less than the optimal value in cell B16. Then run Solver, but remove the Cash_allocated cell as a deci- sion variable cell and as the objective cell. (If there is no objective cell, Solver simply tries to find a solution that satisfies all of the constraints.) What do you find?

47. Continuing the previous problem in a slightly different direction, continue to use the Cash_allocated cell as a decision variable cell, but add a constraint that it must be less than or equal to any value, such as $195,000, that is less than its current optimal value. With this constraint, James will again not be able to meet all of the pension fund payments. Create a new objective cell to minimize the total amount of payments not met. The easiest way to do this is with IF functions. Unfortunately, this makes the model nonsmooth, and Solver might have trouble finding the optimal solution. Try it and see.

14-7 Integer Optimization Models In this section you will learn how to model some problems by using 0–1 variables (and possibly other integer variables) as decision variables. A 0–1 variable, or binary vari- able, is a variable that must equal 0 or 1. Usually a 0–1 variable corresponds to an activity that is or is not undertaken. If the 0–1 variable corresponding to the activity equals 0, the activity is not undertaken; if it equals 1, the activity is undertaken.

Optimization models in which some or all of the variables must be integers are known as integer programming (IP) models. You have already seen examples of integer con- straints in the discussion of scheduling employees, aggregate planning, and pension fund management. This section illustrates methods that are needed to formulate IP models of complex situations. You should be aware that Solver typically has a much harder time solving an IP problem than an LP problem. In fact, Solver is unable to solve some IP problems, even when they have an optimal solution. The reason is that these problems are inherently difficult, no matter which software package is used. However, as you will see in this section, your ability to model complex problems increases tremendously when you are able to use IP, particularly with 0–1 variables.

Difficulty of Integer Programming Models

You might suspect that IP models would be easier to solve than LP models. After all, there are only a finite number of feasible integer solutions in an IP model, whereas there are infinitely many feasible (integer and noninteger) solutions in an LP model. However, exactly the opposite is true. IP models are much more difficult than LP models. All IP algorithms try to perform an efficient search through the typically huge number of feasible integer solutions. General-purpose algorithms such as branch and bound can be very effective for modest-size prob- lems, but they can fail (or require extremely long computing times) on the large problems often faced in real applications. In such cases, analysts must develop special-purpose optimization algorithms, or perhaps even heuristics, to find “good,” but not necessarily optimal, solutions.

Fundamental Insight

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14-7a Capital Budgeting Models Perhaps the simplest types of IP models are capital budgeting models. Example 14.8 perfectly illustrates the go/no-go decisions inherent in many IP models.

EXAMPLE

14.8 CAPITAL BUDGETING AT TATHAM The Tatham Company is considering seven investments. The cash required for each investment and the net present value (NPV) each investment adds to the firm are listed in Table 14.5. Each NPV is based on a stream of future revenues, and it includes the cash requirement, which is incurred right away. The table also lists the return of investment (ROI) for each invest- ment, defined as the ratio of NPV to cash required, minus 1. The budget for investment is $15 million. Tatham wants to find the investment policy that maximizes its total NPV. The crucial assumption here is that if Tatham wants to take part in any of these investments, it must go all the way. It cannot, for example, go halfway in investment 1 by investing $2.5 million and realizing an NPV of $2.8 million. In fact, if partial investments were allowed, you wouldn’t need IP; you could use LP.

Investment Cash Required NPV ROI

1 $5.0 $5.6 12.0%

2 $2.4 $2.7 12.5%

3 $3.5 $3.9 11.4%

4 $5.9 $6.8 15.3%

5 $6.9 $7.7 11.6%

6 $4.5 $5.1 13.3%

7 $3.0 $3.3 10.0%

Table 14.5 Data for the Capital Budgeting Example ($ millions)

Objective To use binary IP to find the set of investments that stays within budget and maximizes total NPV.

Where Do the Numbers Come From? The initial required cash and the available budget are easy to obtain. Obtaining the NPV for each investment is more difficult. A time sequence of anticipated revenues from the investments and a discount factor are required. In any case, financial analysts should be able to estimate the required NPVs.

Solution The variables and constraints required for this model appear in Figure 14.43. (See the file Capital Budgeting Big Picture. xlsx.) The most important part is that the decision variables must be binary, where a 1 means that an investment is chosen and a 0 means that it isn’t. These variables cannot have fractional values such as 0.5, because partial investments are not allowed. When you set up the Solver dialog box, you must add explicit binary constraints in the Constraints section.

Maximize total NPV

Total cost of investments

Whether to invest

Investment NPV

Investment cost

Budget<=

Figure 14.43 Big Picture for Capital Budgeting Model

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14-7 Integer Optimization Models 6 7 9

Developing the Spreadsheet Model To form the spreadsheet model, which is shown in Figure 14.44, proceed as follows. (See the file Capital Budgeting Finished.xlsx.)

1. Inputs. Enter the initial cash requirements, the NPVs, and the budget in the blue ranges. The ROIs aren’t absolutely required, but you can calculate them in row 7.

2. 0–1 values for investments. Enter any trial 0–1 values for the investments in row 10. Actually, you can even enter fractional values such as 0.5 in these cells. Solver’s binary constraints will eventually force them to be 0 or 1.

3. Cash invested. Calculate the total cash invested in cell B14 with the formula

5SUMPRODUCT(B5:H5,Decisions)

Developing the Capital Budgeting Model

A SUMPRODUCT formula, where one of the ranges has 0–1 values, just sums the values in the other range that correspond to the I’s.

Figure 14.44 Capital Budgeting Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17

A B C D E F G H I KJ L Capital budge�ng model Range names used

Budget Decisions Total_cost Total_NPV

=Model!$D$14 =Model!$B$10:$H$10 =Model!$B$14 =Model!$B$17

Input data on poten�al investments ($ millions) Investment Cost NPV ROI

1 if yes, 0 if no

$16.7

Total cost $14.9

Budget $15

Decisions: whether to invest

Budget constraint

Objec�ve to maximize Total NPV

1 $5.0 $5.6

12.0%

2 $2.4 $2.7

12.5%

3 $3.5 $3.9

11.4%

5 $6.9 $7.7

11.6%

6 $4.5 $5.1

13.3%

7 $3.0 $3.3

10.0%

4 $5.9 $6.8

15.3%

0 0 01

Note that this formula sums the costs only for those investments with 0–1 variables equal to 1. To see this, think how the SUMPRODUCT function works when one of its ranges is a 0–1 range. It effectively sums the cells in the other range corresponding to the 1’s.

4. NPV contribution. Calculate the NPV contributed by the investments in cell B17 with the formula

5SUMPRODUCT(B6:H6,Decisions)

Again, this sums only the NPVs of the investments with 0–1 variables equal to 1.

Using Solver The Solver dialog box appears in Figure 14.45. The objective is to maximize the total NPV while staying within the budget. However, the decision variable cells must be constrained to be 0–1. Solver makes this simple, as shown in Figure 14.46. You add a constraint with the decision variable cells in the left box and choose the bin option in the middle box. The word “binary” in the right box is then added automatically. Note that if all decision variable cells are binary, the Non-Negative option is optional (because 0 and 1 are certainly nonnegative), but you should still choose the Simplex LP method if the model is linear, as it is here.12 Finally, in the Solver Options dialog box, you should make sure the Ignore Integer Constraints option is not checked.

Discussion of the Solution The optimal solution in Figure 14.44 indicates that Tatham can obtain a maximum NPV of $16.7 million by selecting invest- ments 3, 5, and 6. These three investments consume only $14.9 million of the available budget, with $100,000 left over. However, this $100,000 is not nearly enough—because of the “investing all the way” requirement—to invest in any of the remaining investments.

Solver makes it easy to specify binary constraints, by choosing the bin option.

12 All the models in this section satisfy two of the three properties of linear models in Chapter 13: proportionality and additivity. Even though they clearly violate the third assumption, divisibility, which precludes integer constraints, they are still considered linear by Solver. Therefore, you should still choose the Simplex LP method.

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If Tatham’s investments are ranked on the basis of ROI (see row 7 of Figure 14.44), the ranking from best to worst is 4, 6, 2, 5, 1, 3, 7. Using your economic intuition, you might expect the investments to be chosen in this order—until the budget runs out. However, the optimal solution does not do this. It selects the second-, fourth-, and sixth-best investments, and it ignores the best. To understand why it does this, imagine investing in the order from best to worst, according to row 7, until the budget allows no more. By the time you have chosen investments 4, 6, and 2, you will have consumed $12.8 million of the budget, and the remainder, $2.2 million, is not sufficient to invest in any of the rest. This strategy provides an NPV of only $14.6 million. A smarter strategy, the optimal solution from Solver, gains you an extra $2.1 million in NPV.

Sensitivity Analysis SolverTable can be used on models with binary variables exactly as you have used it in previous models.13 Here you can use it to see how the total NPV varies as the budget increases. Select the Budget cell as the single input cell, allow it to vary from $15 million to $25 million in increments of $1 million, and keep track of the total investment cost, the total NPV, and the binary variables. The results are shown in Figure 14.47. Clearly, Tatham can achieve a larger NPV with a larger budget, but as the

Figure 14.45 Solver Dialog Box for Capital Budgeting Model

Figure 14.46 Specifying a Binary Constraint

13 As mentioned earlier, Solver’s sensitivity report is not even available for models with integer constraints because the mathematical theory behind the report changes significantly when variables are constrained to be integers.

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14-7 Integer Optimization Models 6 8 1

Figure 14.47 Sensitivity to Budget

3 2 1

4 5 6 7 8 9

10 11 12 13 14 15

A B C D E F G H I J

Budget (cell $D$14) values along side, output cell(s) along top

De ci

sio ns

De ci

sio ns

De ci

sio ns

De ci

sio ns

De ci

sio ns

De ci

sio ns

De ci

sio ns

To ta

l_ co

To ta

l_ N

In cr

ea se

in N

$15 0 0 1 0 1 1 0 $14.9 $16 1 0 1 0 0 1 1 $1.2

$1.2 $1.1 $1.2 $0.9 $1.2 $1.2 $0.9 $1.2 $1.1

$17 0 0 1 1 0 1 1 $18 1 1 0 1 0 1 0 $19 1 0 1 1 0 1 0 $20 1 1 1 1 0 0 1 $21 1 1 0 1 0 1 1 $22 1 0 1 1 0 1 1 $23 1 0 1 0 1 1 1 $24 0 0 1 1 1 1 1 $25 1 1 0 1 1 1 0

$16.7 $16.0 $17.9 $16.9 $19.1 $17.8 $20.2 $18.9 $21.4 $19.8 $22.3 $20.8 $23.5 $21.9 $24.7 $22.9 $25.6 $23.8 $26.8 $24.7 $27.9

Oneway analysis for Solver model in Model worksheet

increases in column K indicate, each extra $1 million of budget does not have the same effect on total NPV. Note also how the selected investments vary as the budget increases. This somewhat strange behavior is due to the “lumpiness” of the inputs and the all-or-nothing nature of the problem.

Effect of Solver Integer Optimality Setting To illustrate the effect of the Solver Integer Optimality setting, compare the SolverTable results in Figure 14.48 with those in Figure 14.47. Each is for the Tatham capital budgeting model, but Figure 14.48 uses Solver’s (old) default setting of 5%, whereas Figure 14.47 uses a setting of 0%. The four shaded cells in Figure 14.48 indicate lower total NPVs than the corresponding cells in Figure 14.47. In these three cases, Solver stopped short of finding the true optimal solutions because it found solutions within the 5% of the optimal objective value and then quit.

When the Integer Optimality setting is 5% instead of 0%, Solver’s solution might not be optimal, but it will be close.

Figure 14.48 Results with Integer Optimality at 5%

3 2 1

4 5 6 7 8 9

10 11 12 13 14 15

A B C D E F G H I J

Budget (cell $D$14) values along side, output cell(s) along top

De ci

sio ns

De ci

sio ns

De ci

sio ns

De ci

sio ns

De ci

sio ns

De ci

sio ns

To ta

l_ co

To ta

l_ N

$15 0 0 1 0 1 1 0 $14.9 $16 0 1 0 1 0 1 1 $17 0 1 1 1 0 1 0 $18 1 1 0 1 0 1 0 $19 1 0 1 1 0 1 0 $20 0 1 1 1 0 1 1 $21 1 1 0 1 0 1 1 $22 1 1 1 1 0 1 0 $23 0 1 0 1 1 1 1 $24 0 1 1 1 1 1 0 $25 1 1 0 1 1 1 0

De ci

sio ns

$16.7 $15.8 $17.9 $16.3 $18.5 $17.8 $20.2 $18.9 $21.4 $19.3 $21.8 $20.8 $23.5 $21.3 $24.1 $22.7 $25.6 $23.2 $26.2 $24.7 $27.9

Oneway analysis for Solver model in Model worksheet

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6 8 2 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

• If Tatham could choose a fractional amount of an investment, you could maximize its NPV by deleting the binary constraint. The optimal solution to the resulting LP model has a total NPV of $17.064 million. All of investments 2, 4, and 6, and 44% of investment 1 are chosen.14 However, there is no way to round the decision variable cell values from this LP solution to obtain the optimal IP solution. Sometimes the solution to an IP model without the integer constraints bears little resemblance to the optimal IP solution.

• Any IP model involving 0–1 variables with only one constraint is called a knapsack problem. Think of the problem faced by a hiker going on an overnight hike. For example, suppose that the hiker’s knapsack can hold only 35 pounds, and she must choose which of several available items to take on the hike. The benefit derived from each item is analogous to the NPV of each project, and the weight of each item is analogous to the cash required by each investment. The single constraint is analogous to the budget constraint—that is, only 35 pounds can fit in the knapsack. In a knapsack problem, the goal is to get the most value in the knapsack without overloading it.

Modeling Issues • Capital budgeting models with cash requirements in multiple time periods can also be handled. Figure 14.49 shows one

possibility. (See the Capital Budgeting Multiple Period Finished.xlsx file.) The costs in rows 5 and 6 are both incurred if any given investment is selected. Now there are two budget constraints, one in each year, but otherwise the model is exactly as before. Note that some investments can have a zero cost in year 1 and a positive cost in year 2. This effectively means that these investments are undertaken in year 2 rather than year 1. Also, it is easy to modify the model to incorporate costs in years 3, 4, and so on.

Figure 14.49 Two-Period Capital Budgeting Model 1

2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

Capital budgeting model with costs in multiple periods

Input data on potential investments ($ millions) 7654321tnemtsevnI

Year 1 $0.0$1.5$3.9$3.0$1.0$2.0$4.0 $3.0$3.0$3.0$2.9$2.5$0.4$1.0 $3.3$5.1$7.7$6.8$3.9$2.7$5.6

10.0%13.3%11.6%15.3%11.4%12.5%12.0%

tsoc Year 2 tsoc

VPN ROI

Decisions: whether to invest 1 if yes, 0 if no

Year 1 Year 2

0001011

Budget constraint Total cost Budget

$9.0 $9 $4.3

Objective to maximize Total NPV $15.1

<= <= $6

A B C D E F G H

14-7b Fixed-Cost Models Fixed-cost models are used when a fixed cost is incurred if an activity is undertaken at any positive level. This cost is independent of the level of the activity and is known as a fixed cost (or fixed charge). Here are three examples of fixed costs:

• Construction of a warehouse incurs a fixed cost that is the same whether the warehouse is used at partial or full capacity.

• A cash withdrawal from a bank incurs a fixed cost, independent of the size of the withdrawal.

• A machine that is used to make several products must be set up for the production of each product. Regardless of the number of units of a product the company produces, the same fixed cost (lost production due to the setup time) is incurred.

In these examples a fixed cost is incurred if an activity is undertaken at any positive level, and zero fixed cost is incurred if the activity is not undertaken at all. Although it might

14 If you try this with the Capital Budgeting.xlsx file, delete the binary constraint, but don’t forget to constrain the decision variables to be nonnegative and less than or equal to 1.

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14-7 Integer Optimization Models 6 8 3

Binary Variables for Modeling

Binary variables are often used to transform a non-linear model into a lin- ear (integer) model. For example, a fixed cost is not a linear function of the level of some activity; it is either incurred or it isn’t incurred. This type of all-or-nothing behavior is difficult for nonlinear algorithms to handle. How- ever, this behavior can often be handled easily by using binary variables to make the model linear. Still, large models with many binary variables can be difficult to solve. One approach is to solve the model without inte- ger constraints and then round fractional values to the nearest integer (0 or 1). Unfortunately, this approach is typically not very good because the rounded solution is often infeasible. Even if it is feasible, its objective value can be con- siderably worse than the optimal objective value.

Fundamental Insight

EXAMPLE

14.9 TEXTILE MANUFACTURING AT GREAT THREADS Great Threads Company is capable of manufacturing shirts, shorts, pants, skirts, and jackets. Each type of clothing requires Great Threads to acquire the appropriate type of machinery. The machinery needed to manufacture each type of clothing must be rented at the weekly rates shown in Table 14.6. This table also lists the amounts of cloth and labor required per unit of clothing, as well as the selling price and the unit variable cost for each type of clothing. There are 4000 labor hours and 4500 square yards (sq yd) of cloth available in a given week. The company wants to find a solution that maximizes its weekly profit.

Rental Cost Labor Hours Cloth (sq yd) Selling Price Unit Variable Cost

Shirts $1500 2.0 3.0 $35 $20

Shorts $1200 1.0 2.5 $40 $10

Pants $1600 6.0 4.0 $65 $25

Skirts $1500 4.0 4.5 $70 $30

Jackets $1600 8.0 5.5 $110 $35

Table 14.6 Data for Great Threads Example

Objective To develop a linear model with binary variables that can be used to maximize the company’s profit, correctly accounting for fixed costs and staying within resource availabilities.

Where Do the Numbers Come From? Except for the fixed costs, this is the same basic problem as the product mix problem (Examples 13.1 and 13.2) in the previous chapter. Therefore, the same discussion there about input variables applies here. As for the fixed costs, they are the given rental rates for the machinery.

Solution The variables and constraints required for this model are shown in Figure 14.50. (See the file Fixed Cost Manufacturing Big Picture.xlsx.) Note that the cost of producing x shirts during a week is 0 if x 5 0, but it is 1500 1 20x if x 7 0. This cost structure violates the proportionality assumption (discussed in the previous chapter) that is needed for a linear model.

not be obvious, this feature makes the problem inherently nonlinear, which means that a straightforward application of LP is not possible. However, Example 14.9 illustrates how a clever use of binary variables results in a linear model.

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If proportionality were satisfied, the cost of making, say, 10 shirts would be double the cost of making five shirts. However, because of the fixed cost, the total cost of making five shirts is $1600, and the cost of making 10 shirts is only $1700. This violation of proportionality requires you to use binary variables to obtain a linear model.

Maximize profit

Total fixed cost of equipment

Fixed cost of equipment

Resources (labor and cloth) used

Whether to produce any

Units produced Effec�ve capacity

Total variable cost

Total revenue

Variable cost per unit

Resources (labor and cloth) per unit

Resources (labor and cloth) available

Selling price per unit

Figure 14.50 Big Picture for Fixed-Cost Manufacturing Model

Developing the Spreadsheet Model The spreadsheet model, shown in Figure 14.51, can now be developed as follows. (See the file Fixed Cost Manufacturing Finished.xlsx.)

Figure 14.51 Fixed-Cost Manufacturing Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

KJIGFEDCBA Great Threads fixed cost clothing model Range names used:

Effec�ve_capacity =Model!$B$18:$F$18 Input data on products Rent_equipment =Model!$B$14:$F$14

Shirts Shorts Pants Skirts Jackets Profit =Model!$B$29 =Model!$D$22:$D$23Resource_available84612Labor hours/unit

Cloth (sq. yd.)/unit =Model!$B$22:$B$23Resource_used5.54.542.53 Units_produced =Model!$B$16:$F$16

Selling price/unit $110$70$65$40$35 Variable cost/unit $35$30$25$10$20 Fixed cost for equipment $1,500 $1,200 $1,600 $1,500 $1,600

Produc�on plan, constraints on capacity Shirts Shorts Pants Skirts Jackets

Rent equipment 10010

Units produced 379.3100965.520 <= <= <= <= <=

Effec�ve capacity 500.000.000.001800.000.00

Constraints on resources Resource used Available

Labor hours 4000<=4000.00 4500<=4500.00Cloth

Monetary outputs

Variable cost Fixed cost for equipment

Revenue

Profit

$80,345 $22,931

$2,800 $54,614 Objec�ve to maximize

Developing the Fixed-Cost Manufacturing Model

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14-7 Integer Optimization Models 6 8 5

1. Inputs. Enter the given inputs. 2. Binary values for clothing types. Enter any trial values for the binary variables for the various clothing types in the

Rent_equipment range. For example, a 1 in cell C14 implies that the machinery for making shorts is rented and its fixed cost is incurred.

3. Production quantities. Enter any trial values for the numbers of the various clothing types produced in the Units_pro- duced range. At this point you could enter “illegal” values, such as 0 in cell B14 and a positive value in cell B16. This is illegal because it implies that the company produces some shirts but doesn’t incur the fixed cost of the machinery for shirts. However, Solver will eventually disallow such illegal combinations.

4. Labor and cloth used. In cell B22 enter the formula

5SUMPRODUCT(B5:F5,Units_produced)

to calculate total labor hours, and copy this to cell B23 for cloth. 5. Effective capacities. Here is the tricky part of the model. You need to ensure that if any of a given type of clothing is

produced, then its binary variable equals 1. This ensures that the model incurs the fixed cost of renting the machine for this type of clothing. You could easily implement these constraints with IF statements. For example, to implement the con- straint for shirts, you could enter the following formula in cell B14:

5IF(B16+0,1,0)

However, Solver is unable to deal with IF functions predictably. Therefore, the fixed-cost constraints are modeled in a different way, as follows:

Shirts produced # Maximum capacity 3 (091 variable for shirts) (14.4)

There are similar inequalities for the other types of clothing. Here is the logic behind Inequality (14.4). If the 0–1 variable for shirts is 0, the right side of the inequality is 0, which

means that the left side must be 0—no shirts can be produced. That is, if the binary variable for shirts is 0, so that no fixed cost for shirts is incurred, then Inequality (14.4) does not allow Great Threads to “cheat” and produce a positive number of shirts. On the other hand, if the binary variable for shirts is 1, the inequality is certainly true and is essentially redundant. It simply states that the number of shirts produced must be no greater than the maximum number that could be produced. Inequality (14.4) rules out the one case that needs to be ruled out—namely, that Great Threads produces shirts but avoids the fixed cost.

To implement Inequality (14.4), a maximum capacity is required. To obtain this, suppose the company puts all of its resources into producing shirts. Then the number of shirts that can be produced is limited by the smaller of

Available labor hours

Labor hours per shirt and

Available square yards of cloth

Square yards of cloth per shirt

Therefore, the smaller of these—the most limiting—can be used as the maximum needed in Inequality (14.4). To imple- ment this logic, calculate the effective capacity for shirts in cell B18 with the formula

5B14*MIN($D$22/B5,$D$23/B6)

Then copy this formula to the range C16:F16 for the other types of clothing.15 By the way, this MIN formula causes no problems for Solver because it does not involve decision variable cells, only input cells.

6. Monetary values. Calculate the total sales revenue and the total variable cost by entering the formula

5SUMPRODUCT(B8:F8, Units_produced)

in cell B26 and copying it to cell B27. Then calculate the total fixed cost in cell B28 with the formula

5SUMPRODUCT(B10:F10, Rent_equipment)

This formula sums the fixed costs only for those products with binary variables equal to 1. Finally, calculate the total profit in cell B29 with the formula

5B26-B27-B28

15 Why not set the upper limit on shirts equal to a huge number like 1,000,000? The reason is that Solver works most efficiently when the upper limit is as tight—that is, as low—as possible. A tighter upper limit means fewer potential feasible solutions for Solver to search through.

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Although Solver finds the optimal solution automatically, you should understand the effect of the logical upper-bound constraint on production. It rules out a solution such as the one shown in Figure 14.53. This solution calls for a positive production level of pants but does not incur the fixed cost of the pants equipment. The logical upper-bound constraint rules this out because it prevents a positive value in row 16 if the corresponding binary value in row 14 is 0. In other words, if the company wants to produce some pants, the constraint in Inequality (14.4) forces the associated binary variable to be 1, thus incurring the fixed cost for pants.

Inequality (14.4) does not rule out the situation you see for skirts, where the binary value is 1 and the production level is 0. However, Solver will never choose this type of solution as optimal. Solver recognizes that the binary value in this case can be changed to 0, so that the fixed cost for skirt equipment is not incurred.

Discussion of the Solution The optimal solution appears in Figure 14.51. It indicates that Great Threads should produce about 966 shorts and 379 jackets, but no shirts, pants, or skirts. The total profit is $54,614. The binary variables for shirts, pants, and skirts are all 0, which forces production of these products to be 0. However, the binary variables for shorts and jackets, the products that are produced, are 1. This ensures that the fixed cost of producing shorts and jackets is included in the total cost.

Using Solver The Solver dialog box is shown in Figure 14.52. The goal is to maximize profit, subject to using no more labor hours or cloth than are available, and ensure that production is less than or equal to effective capacity. The key is that this effective capacity is zero if none of a given type of clothing is produced. As usual, check the Non-Negative option, and set the Integer Optimality to zero (under the Options button). By the clever use of binary decision variables, the resulting model is linear, which means that the simplex algorithm can be used.

Figure 14.52 Solver Dialog Box for Fixed-Cost Model

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14-7 Integer Optimization Models 6 8 7

It might be helpful to think of this solution as occurring in two stages. In the first stage Solver determines which prod- ucts to produce—in this case, shorts and jackets only. Then in the second stage, Solver decides how many shorts and jackets to produce. If you knew that the company plans to produce shorts and jackets only, you could then ignore the fixed costs and determine the best production quantities with the same types of product mix models discussed in the previous chapter. How- ever, these two stages—deciding which products to produce and how many of each to produce—are interrelated, and Solver considers both of them in its solution process.

The Great Threads management might not be very excited about producing shorts and jackets only. Suppose the company wants to ensure that at least three types of clothing are produced at positive levels. One approach is to add another constraint—namely, that the sum of the binary values in row 14 is greater than or equal to 3. You can check, however, that when this constraint is added and Solver is rerun, the binary variable for skirts becomes 1, but no skirts are produced. Shorts and jackets are more profitable than skirts, so only shorts and jackets are produced. (See Figure 14.54.) The new constraint forces Great Threads to rent an extra piece of machinery (for skirts), but it doesn’t force the company to use it. To force the company to produce some skirts, you would also need to add a constraint on the value in E16, such as E16 7 5 100. Any of these additional constraints will cost Great Threads money, but if, as a matter of policy, the company wants to produce more than two types of clothing, this is its only option.

Sensitivity Analysis Because the optimal solution currently calls for only shorts and jackets to be produced, an interesting sensitivity analysis is to see how much incentive is required for other products to be produced. One way to model this is to increase the selling price for a non- produced product such as skirts in a one-way SolverTable. The results of this, keeping track of all binary variables and profit, are shown in Figure 14.55. When the selling price for skirts is $85 or less, the company continues to produce only shorts and jackets. However, when the selling price is $90 or greater, the company stops producing shorts and jackets and produces only skirts. You can check that the optimal production quantity of skirts is 1000 when the selling price of skirts is any value $90 or above. The only reason that the profits in Figure 14.55 increase from row 9 down is that the revenues from these 1000 skirts increase.

A Model with IF Functions In case you are still not convinced that the binary variable approach is required, and you think IF functions could be used instead, take a look at the last sheet in the finished version of the file. The resulting model looks the same as in Figure 14.51, but it incorporates the following changes:

• The binary range is no longer part of the decision variable cells range. Instead, the formula 5IF(B16+0,1,0) is entered in cell B14 and copied across to cell F14. Logically, this probably appears more natural. If a production quantity is positive, a 1 is entered in row 14, which means that the fixed cost is incurred.

Figure 14.53 An Illegal (and Nonoptimal) Solution 1

2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

FEDCBA Great Threads fixed cost clothing model

Input data on products Shirts Shorts Pants Skirts Jackets

84612Labor hours/unit Cloth (sq. yd.)/unit 5.54.542.53

Selling price/unit $110$70$65$40$35 Variable cost/unit $35$30$25$10$20 Fixed cost for equipment $1,500 $1,200 $1,600 $1,500 $1,600

Produc�on plan, constraints on capacity Shirts Shorts Pants Skirts Jackets

Rent equipment 11010

Units produced 100.000500.00 450.000 <= <= <= <= <=

Effec�ve capacity 500.001000.000.001800.000.00

Constraints on resources Resource used Available

Labor hours 4000<=4000.00 4500<=3600.00Cloth

Monetary outputs

Variable cost Fixed cost for equipment

Revenue

Profit

$60,250 $19,750

$4,300 $36,200 Objec�ve to maximize

As always, adding constraints can only make the objective worse. In this case, it means decreased profit.

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Figure 14.54 Fixed-Cost Model with Extra Constraint 1

2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

F G H IEDCBA Great Threads fixed cost clothing model

Input data on products Shirts Shorts Pants Skirts Jackets

84612Labor hours/unit Cloth (sq. yd.)/unit 5.54.542.53

Selling price/unit $110$70$65$40$35 Variable cost/unit $35$30$25$10$20 Fixed cost for equipment $1,500 $1,200 $1,600 $1,500 $1,600

Produc�on plan, constraints on capacity Shirts Shorts Pants Skirts Jackets Sum Required

Rent equipment 1 3 >= 31010

Units produced 379.310965.52 00 <= <= <= <= <=

Effec�ve capacity 500.001000.000.001800.000.00

Constraints on resources Resource used Available

Labor hours 4000<=4000.00 4500<=4500.00Cloth

Monetary outputs

Variable cost Fixed cost for equipment

Revenue

Profit

$80,345 $22,931

$4,300 $53,114 Objec�ve to maximize

Figure 14.55 Sensitivity of Binary Variables to Selling Price of Skirts

4 5 6 7 8 9

10 11

A B C D E F G

Selling price skirts (cell $E$8) values along side, output cell(s) along top

Re nt

_e qu

ip m

en t_

Re nt

_e qu

ip m

en t_

Re nt

_e qu

ip m

en t_

Re nt

_e qu

ip m

en t_

Re nt

_e qu

ip m

en t_

Pr ofi

t $70 0 1 0 0 1 $54,614 $75 0 1 0 0 1 $54,614 $80 0 1 0 0 1 $54,614 $85 0 1 0 0 1 $54,614 $90 0 0 0 1 0 $58,500 $95 0 0 0 1 0 $63,500

$100 0 0 0 1 0 $68,500

2 1 Oneway analysis for Solver model in Model worksheet

• The effective capacities are calculated in row 18 with IF functions. Specifically, the formula 5IF(B16+0,MIN ($D$22/B5,$D$23/B6),0) is entered in cell B18 and copied across to cell F18.

• The Solver dialog box is modified as shown in Figure 14.56. The Rent_equipment range is not part of the decision variable cells range, and there is no binary constraint. However, the simplex method cannot be used because the IF functions make the model nonlinear.

When we ran Solver on this modified model, we found inconsistent results, depending on the initial production quantities entered in row 16. For example, when we entered initial values all equal to 0, the Solver solution was exactly that—all 0’s. Of course, this solution is terrible because it leads to a profit of $0. However, when we entered initial production quantities all equal to 100, Solver found the correct optimal solution, the same as in Figure 14.51. Was this just lucky? To check, we tried another initial solution, where the production quantities for shorts and jackets were 0, and the production quantities for shirts, pants, and skirts were all 500. In this case Solver found a solution where only skirts are produced. Of course, we know this is not optimal.

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14-7 Integer Optimization Models 6 8 9

Actually, the problem with using the GRG Nonlinear method indicated in Figure 14.56 is that this model is “nonsmooth,” and the GRG Nonlinear method doesn’t work well on nonsmooth models. Starting in Excel 2010, there is another option—the Evolutionary method. This method works well on nonsmooth models, but it guarantees only an approximately optimal solu- tion, and it is relatively slow. It is not discussed further in this book.

In any case, the IF-function approach is not the way to go. Its success depends on the initial values in the decision variable cells, and this requires good (or lucky) guesses. The binary approach ensures that Solver finds the correct solution.

Figure 14.56 Solver Dialog Box When IF Functions Are Used

14-7c Set-Covering Models In a set-covering model, each member of a given set (set 1) must be “covered” by an acceptable member of another set (set 2). The objective in a set-covering problem is to minimize the number of members in set 2 necessary to cover all the members in set 1. For example, set 1 might consist of all the cities in a county and set 2 might consist of the cities in which a fire station is located. A member of set 2 covers, or handles the needs of, a city in set 1 if the fire station is located within, say, 10 minutes of the city. The goal is to minimize the number of fire stations needed to cover all cities. Set-covering models have been applied to areas as diverse as airline crew scheduling, truck dispatching, polit- ical redistricting, and capital investment. The following example illustrates a typical set- covering model.

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EXAMPLE

14.10 HUB LOCATION AT WESTERN AIRLINES Western Airlines has decided that it wants to design a hub system in the United States. Each hub is used for connecting flights to and from cities within 1000 miles of the hub. Western runs flights among the following cities: Atlanta, Boston, Chicago, Denver, Houston, Los Angeles, New Orleans, New York, Pittsburgh, Salt Lake City, San Francisco, and Seattle. The company wants to determine the smallest number of hubs it will need to cover all of these cities, where a city is “covered” if it is within 1000 miles of at least one hub. Table 14.7 lists the cities that are within 1000 miles of other cities.

Table 14.7 Data for Western Airlines Set-Covering Example

Cities Within 1000 Miles

Atlanta (AT) AT, CH, HO, NO, NY, PI

Boston (BO) BO, NY, PI

Chicago (CH) AT, CH, NY, NO, PI

Denver (DE) DE, SL

Houston (HO) AT, HO, NO

Los Angeles (LA) LA, SL, SF

New Orleans (NO) AT, CH, HO, NO

New York (NY) AT, BO, CH, NY, PI

Pittsburgh (PI) AT, BO, CH, NY, PI

Salt Lake City (SL) DE, LA, SL, SF, SE

San Francisco (SF) LA, SL, SF, SE

Seattle (SE) SL, SF, SE

Objective To develop a binary model to find the minimum number of hub locations that can cover all cities.

Where Do the Numbers Come From? Western has evidently made a policy decision that its hubs will cover cities within a 1000-mile radius. Then the cities covered by any hub location can be found from a map. In a later sensitivity analysis, we explore how the solution changes when the allowable coverage distance varies.

Solution The variables and constraints for this set-covering model appear in Figure 14.57. (See the file Locating Hubs Big Picture. xlsx.) The model is straightforward. There is a binary variable for each city to indicate whether a hub is located there. Then the number of hubs that cover each city is constrained to be at least one. There are no monetary costs in this version of the prob- lem. The goal is to minimize the number of hubs.

Minimize number of

City used as hub

Number of hubs city is covered by 1Matrix of ci�es covered

by poten�al hubs

City to city distance matrix

Figure 14.57 Big Picture for Hub Location Model

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14-7 Integer Optimization Models 6 9 1

1. Inputs. Enter the information from Table 14.7 in the input cells. A 1 in a cell indicates that the column city covers the row city, whereas a 0 indicates that the column city does not cover the row city. For example, the three 1’s in row 7 indicate that Boston, New York, and Pittsburgh are the only cities within 1000 miles of Boston.

2. Binary values for hub locations. Enter any trial values of 0’s or 1’s in the Use_as_hub range to indicate which cities are used as hubs. These are the decision variable cells.

3. Cities covered by hubs. Calculate the total number of hubs within 1000 miles of Atlanta in cell B25 with the formula

5SUMPRODUCT(B6:M6,Use_as_hub)

For any binary values in the decision variable cells range, this formula sums the number of hubs that cover Atlanta. Then copy this to the rest of the Hubs_covered_by range. Note that a value in the Hubs_covered_by range can be 2 or greater. This indicates that a city is within 1000 miles of multiple hubs.

4. Number of hubs. Calculate the total number of hubs used in cell B39 with the formula

5SUM(Use_as_hub)

Using Solver The completed Solver dialog box is shown in Figure 14.59. The goal is to minimize the total number of hubs, subject to cover- ing each city by at least one hub and ensuring that the decision variable cells are binary.

Developing the Spreadsheet Model The spreadsheet model for Western is shown in Figure 14.58. (See the file Locating Hubs Finished.xlsx.) It can be developed as follows. Developing the Hub

Location Model

Figure 14.58 Hub Location Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Western Airlines hub loca�on model

Input data: which ci�es are covered by which poten�al egnaRsbuh names used: Poten�al hub

93$B$!ledo=sbuh_latoTESFSLSIPYNONALOHEDHCOBTAytiC B$!ledo=Used_as_hub000111010101TA $21:$M$21

000110000010OB 000111000101HC 001000001000ED 000001010001OH 011000100000AL 000001010101ON 000110000111YN 000110000111IP 111000101000LS 111000100000FS 111000000000ES

Decisions: which ci�es to use as hubs

Used as hub AT BO CH DE HO LA NO NY PI SL SF SE

0 0 0 0 1 0 0 1 0 1 0 0

Constraints that each city must be covered by at least one hub City Hubs covered by Required

1=>2TA 1=>1OB 1=>1HC 1=>1ED 1=>1OH 1=>1AL 1=>1ON 1=>1YN 1=>1IP 1=>1LS 1=>1FS 1=>1ES

Objec�ve to minimize Total 3sbuh

A B C D E F G H I J K L M N O P Q

Hubs_covered_by =Model!$B$25:$B$36 M M

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Discussion of the Solution Figure 14.60 is a graphical representation of the optimal solution, where the double ovals indicate hub locations and the large circles indicate ranges covered by the hubs. (These large circles are not drawn to scale. In reality, they should be circles of

Figure 14.59 Solver Dialog Box for Hub Location Model

Figure 14.60 Graphical Solution to Hub Location Model

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radius 1000 miles centered at the hubs.) Three hubs—in Houston, New York, and Salt Lake City—are needed.16 The Hous- ton hub covers Houston, Atlanta, and New Orleans. The New York hub covers Atlanta, Pittsburgh, Boston, New York, and Chicago. The Salt Lake City hub covers Denver, Los Angeles, Salt Lake City, San Francisco, and Seattle. Atlanta is the only city covered by two hubs; it can be serviced by New York or Houston.

Sensitivity Analysis An interesting sensitivity analysis for Western’s problem is to see how the solution is affected by the mile limit. Currently, a hub can service all cities within 1000 miles. What if the limit were 800 or 1200 miles, say? To answer this question, you must first collect data on actual distances among all of the cities. Once you have a table of these distances, you can build the binary table, corresponding to the range B6:M17 in Figure 14.58, with IF functions. The modified model appears in the file Locating Hubs with Distances Finished.xlsx (not shown here). You can check that the typical formula in B24 is 5IF(B8*5$B$4,1,0), which is then copied to the rest of the B24:M35 range.17 You can then run SolverTable, selecting cell B4 as the single input cell, letting it vary from 800 to 1200 in increments of 100, and designating the hub locations and the number of hubs as out- puts. The SolverTable results in Figure 14.61 show the effect of the mile limit. When this limit is lowered to 800 miles, four hubs are required, but when it is increased to 1100 or 1200, only two hubs are required. Note that the solution shown for the 1000-mile limit is different from the previous solution in Figure 14.58, but it still requires three hubs. (This is a case of multi- ple optimal solutions.)

Figure 14.61 Sensitivity to Mile Limit

3 2 1

4 5 6 7 8 9

A B C D E F G H I J K L M N

Mile limit (cell $B$4) values along side, output cell(s) along top

U se

d_ as

_h ub

U se

d_ as

_h ub

U se

d_ as

_h ub

U se

d_ as

_h ub

U se

d_ as

_h ub

U se

d_ as

_h ub

U se

d_ as

_h ub

U se

d_ as

_h ub

U se

d_ as

_h ub

U se

d_ as

_h ub

_1 0

U se

d_ as

_h ub

_1 1

U se

d_ as

_h ub

_1 2

To ta

l_ hu

800 1 1 0 0 0 0 0 0 0 1 0 1 4 900 1 1 0 0 0 0 0 0 0 1 0 0 3

1000 1 1 0 0 0 0 0 0 0 1 0 0 3 1100 0 0 1 0 0 0 0 0 0 1 0 0 2 1200 0 0 1 0 0 1 0 0 0 0 0 0 2

Oneway analysis for Solver model in Model worksheet

16 There are multiple optimal solutions for this model, all requiring three hubs, so you might obtain a different solution from ours. 17 We have warned you about using IF functions in Solver models. However, the current use affects only the inputs to the problem, not quantities that depend on the decision variable cells. Therefore, it causes no problems.

14-7 Integer Optimization Models 6 9 3

Problems

Level A 48. Solve the following modifications of the Tatham capital

budgeting model. (Solve each part independently of the others.) a. Suppose that at most two of projects 1 through 5 can

be selected. b. Suppose that if investment 3 is selected, then invest-

ment 1 must also be selected. c. Suppose that at least one of investments 1 and 2 must

be selected. d. Suppose that investment 5 can be selected only if both

investments 2 and 3 are selected. 49. In the Tatham capital budgeting model we supplied the

NPV for each investment. Suppose instead that you are given only the streams of cash inflows from each investment

shown in the file P14_49.xlsx. This file also shows the cash requirements and the budget. You can assume that (1) all cash outflows occur at the beginning of year 1; (2) all cash inflows occur at the ends of their respective years; and (3) the company uses a 8% discount rate for calculating its NPVs. Which investments should the company make?

50. Solve the previous problem using the input data in the file P14_50.xlsx.

51. Solve Problem 49 with the extra assumption that the investments can be grouped naturally as follows: 194, 598, 9912, 13916, and 17920. a. Find the optimal investments when at most one invest-

ment from each group can be selected. b. Find the optimal investments when at least one invest-

ment from each group must be selected. (If the budget isn’t large enough to permit this, increase the budget to a larger value.)

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52. In the Tatham capital budgeting model, investment 4 has the largest ROI, but it is not selected in the optimal solu- tion. How much NPV is lost if Tatham is forced to select investment 4? Answer by solving a suitably modified model.

53. As it currently stands, investment 7 in the Tatham capi- tal budgeting model has the lowest ROI, 10%. Keeping this same ROI, can you change the cash requirement and NPV for investment 7 in such a way that it is selected in the optimal solution? Does this lead to any general insights? Explain.

54. Expand the Tatham capital budgeting model so that there are now 20 possible investments. You can make up the data on cash requirements, NPVs, and the budget. However, use the following guidelines: • The cash requirements and NPVs for the various

investments can vary widely, but the ratio of NPV to cost should be between 1.05 and 1.25 for each investment.

• The budget should allow somewhere between 5 and 10 of the investments to be selected.

55. Suppose in the Tatham capital budgeting model that each investment requires $100,000 during year 2 and only $300,000 is available for investment during year 2. a. Assuming that available money uninvested at the end

of year 1 cannot be used during year 2, what combina- tion of investments maximizes NPV?

b. Suppose that any uninvested money at the end of year 1 can be used for investment in year 2. Does your answer to part a change?

56. How difficult is it to expand the Great Threads fixed-cost model to accommodate another type of clothing? Answer by assuming that the company can also produce sweat- shirts. The rental cost for sweatshirt equipment is $1100, the variable cost per unit and the selling price are $15 and $45, respectively, and each sweatshirt requires one labor hour and 3.5 square yards of cloth.

57. Referring to the previous problem, if it is optimal for the company to produce sweatshirts, use SolverTable to see how much larger the fixed cost of sweatshirt machinery would have to be before the company would not pro- duce any sweatshirts. However, if the solution to the pre- vious problem calls for no sweatshirts to be produced, use SolverTable to see how much lower the fixed cost of sweatshirt machinery would have to be before the com- pany would start producing sweatshirts.

58. In the Great Threads fixed-cost model, the production quantities in row 16 were not constrained to be inte- gers. Presumably, any fractional values could be safely rounded to integers. See whether this is true. Constrain these quantities to be integers and then run Solver. Are the optimal integer values the same as the rounded frac- tional values in Figure 14.51?

59. In the optimal solution to the Great Threads fixed-cost model, the labor hour and cloth constraints are both binding—the company is using all it has. a. Use SolverTable to see what happens to the optimal

solution when the amount of available cloth increases from its current value. (You can choose the range of input values to use.) Capture all of the decision vari- able cells, the labor hours and cloth used, and the profit as outputs. The real issue here is whether the company can profitably use more cloth when it is already constrained by labor hours.

b. Repeat part a, but reverse the roles of labor hours and cloth. That is, use the available labor hours as the input for SolverTable.

60. In the optimal solution to the Great Threads fixed-cost model, no pants are produced. Suppose Great Threads has an order for 300 pairs of pants that must be pro- duced. Modify the model appropriately and use Solver to find the new optimal solution. (Is it enough to put a lower bound of 300 on the production quantity in cell D16? Will this automatically force the binary value in cell D14 to be 1? Explain.) How much profit does the company lose because of having to produce pants?

61. In the original Western Airlines set-covering model, we assumed that each city must be covered by at least one hub. Suppose that for added flexibility in flight routing, Western requires that each city must be covered by at least two hubs. How do the model and optimal solution change?

62. In the original Western Airlines set-covering model, we used the number of hubs as the objective to minimize. Suppose instead that there is a fixed cost of locating a hub in any city, where these fixed costs can vary across cities. Make up some reasonable fixed costs, modify the model appropriately, and use Solver to find the solution that minimizes the sum of fixed costs.

63. Set-covering models such as the original Western Airlines model often have multiple optimal solutions. See how many alternative optimal solutions you can find. Of course, each must use three hubs because we know this is optimal. (Hint: Use various initial values in the decision variable cells and then run Solver repeatedly.)18

64. How hard is it to expand a set-covering model to accom- modate new cities? Answer this by modifying the sec- ond set-covering model. (See the file Locating Hubs with Distances Finished.xlsx.) Add several cities that must be served: Memphis, Dallas, Tucson, Philadelphia, Cleveland, and Buffalo. You can look up the distances from these cities to each other and to the other cities on the Web, or you can make up approximate distances. a. Modify the model appropriately, assuming that these

new cities must be covered and are candidates for hub locations.

18 One of our colleagues at Indiana University, Vic Cabot, now deceased, worked for years trying to develop a general algorithm (not just trial and error) for finding all alternative optimal solutions to optimization models. It turns out that this is a very difficult problem—and one that Vic never completely solved.

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14-8 Nonlinear Optimization Models 6 9 5

b. Modify the model appropriately, assuming that these new cities must be covered but are not candidates for hub locations.

Level B 65. The models in this section are often called combinato-

rial models because each solution is a combination of the various 0’s and 1’s, and there are only a finite num- ber of such combinations. For the Tatham capital bud- geting model, there are seven investments, so there are 27 5 128 possible solutions (some of which are infea- sible). This is a fairly large number, but not too large. Solve the model without Solver by listing all 128 solu- tions. For each, calculate the total cash requirement and total NPV for the model. Then manually choose the one that stays within the budget and has the largest NPV.

66. Make up an example, as described in Problem 54, with 20 possible investments. However, do it so that the ratios of NPV to cash requirement are in a very tight range, such as from 1.0 to 1.1. Then use Solver to find the optimal solution when the Solver Integer Opti- mality is set to its default value of 5%, and record the solution. Next, solve again with the Integer Optimal- ity set to zero. Do you get the same solution? Try this on a few more instances of the model, where you keep tinkering with the inputs. The question is whether the

Integer Optimality setting matters in these types of nar- row-range problems.

67. In the Great Threads fixed-cost model, we found an upper bound on production of any clothing type by cal- culating the amount that could be produced if all of the resources were devoted to this clothing type. a. What if you instead use a very large value such

as 1,000,000 for this upper bound? Try it and see whether you get the same optimal solution.

b. Explain why any such upper bound is required. Exactly what role does it play in the model?

68. In the last sheet of the finished version of the Great Threads file, we illustrated one way to model the Great Threads problem with IF functions, but saw that this approach doesn’t work. Try a slightly different approach here. Eliminate the binary variables in row 14 altogether, and eliminate the upper bounds in row 18 and the corre- sponding upper bound constraints in the Solver dialog box. (The only constraints are now on resource avail- ability.) However, use IF functions to calculate the total fixed cost of renting equipment, so that if the amount of any clothing type is positive, then its fixed cost is added to the total fixed cost. Is Solver able to handle this model? Does it depend on the initial values in the decision variable cells? (You will have to use Solver’s nonlinear algorithm, not the simplex method.)

14-8 Nonlinear Optimization Models In many optimization models the objective and/or the constraints are nonlinear functions of the decision variables. Such an optimization model is called a nonlinear programming (NLP) model. In this section we discuss how to use Excel’s Solver to find optimal solu- tions to NLP models. We then discuss two interesting applications, including the import- ant portfolio optimization model.

14-8a Difficult Issues in Nonlinear Optimization When you solve an LP model with Solver, you are guaranteed that the solution obtained is an optimal solution. When you solve an NLP model, however, it is very possible that Solver will obtain a suboptimal solution. This is because a nonlinear function can have local optimal solutions that are not the global optimal solution. A local optimal solution is one that is better than all nearby points, whereas the global optimum is the one that beats all points in the entire feasible region. If there are one or more local optimal solu- tions that are not globally optimal, then it is entirely possible that Solver will stop at one of them. Unfortunately, this is not what you want; you want the global optimum.

There are mathematical conditions that guarantee the Solver solution is indeed the global optimum. However, these conditions are often difficult to check, and they aren’t always satisfied. A much simpler approach is to run Solver several times, each time with different starting values in the decision variable cells. In general, if Solver obtains the same optimal solution in all cases, you can be fairly confident—but still not absolutely sure—that Solver has found the global optimal solution. On the other hand, if you try dif- ferent starting values for the decision variable cells and obtain several different solutions, you should keep the best solution found so far. That is, you should keep the solution with

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the lowest objective value (for a minimization problem) or the highest objective value (for a maximization problem).

14-8b Managerial Economics Models Many problems in economics are nonlinear but can be solved with Solver. The following example illustrates a nonlinear pricing model.

Local Optimal Solution Versus Global Optimal Solution

Nonlinear objective functions can behave in many ways that make them diffi- cult to optimize. In particular, they can have local optimal solutions that are not globally optimal, and nonlinear optimization algorithms can stop at such local optimal solutions. The important property of linear objectives that makes the sim- plex method so successful—namely, that the optimal solution is a corner point— doesn’t hold for nonlinear objectives. Now any point in the feasible region can conceivably be optimal. This not only makes the search for the optimal solution more difficult, but it also makes it much more difficult to recognize whether a promising solution (a local optimum) is indeed the global optimum. This is why researchers have spent so much effort trying to obtain conditions that, when true, guarantee that a local optimum must be a global optimum. Unfortunately, these conditions are often difficult to check, and they aren’t always satisfied.

Fundamental Insight

EXAMPLE

14.11 ELECTRICITY PRICING AT FLORIDA POWER AND LIGHT

Florida Power and Light (FPL) faces demands during both on-peak and off-peak times. FPL must determine the price per megawatt hour (mWh) to charge during both on-peak and off-peak periods. The monthly demand for power during each period (in millions of mWh) is related to price as follows:

Dp 5 2.253 2 0.013Pp 1 0.003Po (14.5)

Do 5 1.142 2 0.015 Po 1 0.005Pp (14.6)

Here, Dp and Pp are demand and price during on-peak times, whereas Do and Po are demand and price during off-peak times. Note from the signs of the coefficients that an increase in the on-peak price decreases the demand for power during the on-peak period but increases the demand for power during the off-peak period. Similarly, an increase in the price for the off-peak period decreases the demand for the off-peak period but increases the demand for the on-peak period. In economic terms, this implies that on-peak power and off-peak power are substitutes for one another. In addition, it costs FPL $75 per month to maintain one mWh of capacity. The company wants to determine a pricing strategy and a capacity level that maximize its monthly profit.

Objective To use a nonlinear model to determine prices and capacity when there are two different daily usage patterns: on-peak and off-peak.

The positive coefficients of prices in these demand equations indicate substitute behavior. A larger price for one product tends to induce customers to demand more of the other.

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14-8 Nonlinear Optimization Models 6 9 7

Developing the Spreadsheet Model The spreadsheet model appears in Figure 14.63. (See the file Electricity Pricing Finished.xlsx.) It can be developed as follows:

1. Inputs. Enter the parameters of the demand functions and the cost of capacity in the light blue ranges.

2. Prices and capacity level. Enter any trial prices (per mWh) for on-peak and off-peak power in the Prices range, and enter any trial value for the capacity level in the Capacity cell. These are the three values FPL has control over, so they become the decision variable cells.

3. Demands. Calculate the demand for the on-peak period by substituting into Equation (14.5). That is, enter the formula

5B61SUMPRODUCT(Prices,C6:D6)

in cell B19. Similarly, enter the formula

5B71SUMPRODUCT(Prices,C7:D7)

in cell C19 for the off-peak demand.

Where Do the Numbers Come From? As usual, a cost accountant should be able to estimate the unit cost of capacity. The real difficulty here is estimating the demand functions in Equations (14.5) and (14.6). This requires either sufficient historical data on prices and demands (for both on-peak and off-peak periods) or educated guesses from management.

Solution The variables and constraints for this model are shown in Figure 14.62. (See the file Electricity Pricing Big Picture.xlsx.) The company must decide on two prices and the amount of capacity to maintain. Because this capacity level, once determined, is relevant for on-peak and off-peak peri- ods, it must be large enough to meet demands for both periods. This is the reasoning behind the constraints.

Due to the relationships between the demand and price variables, it is not obvious what FPL should do. The pricing decisions determine demand, and larger demand requires larger capacity, which costs money. In addition, revenue is price multiplied by demand, so it is not clear whether price should be low or high to increase revenue.

The capacity must be at least as large as the on-peak and off-peak demands. There is no incentive for the capacity to be larger than the maximum of these two demands.

Figure 14.62 Big Picture for Electricity Pricing Model

Unit cost of capacity

Cost of capacityOff-peak demand

On-peak demand

Revenue Parameters of demand func�ons

Capacity

On-peak price

Off-peak price

Maximize profit

Developing the Electricity Pricing Model

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4. Copy capacity. To indicate the capacity constraints, enter the formula

5Capacity

in cells B21 and C21. The reason for creating these links is that the two demand cells in row 19 need to be paired with two capacity cells in row 21 so that the Solver constraints can be specified appropriately. (Solver doesn’t allow a “two versus one” constraint such as B19:C19 6 5 B15.)

5. Monetary values. Calculate the daily revenue, cost of capacity, and profit in the corresponding cells with the formulas

5SUMPRODUCT(Demands,Prices)

5Capacity*B9

and

5B24@B25

Using Solver The Solver dialog box should be filled in as shown in Figure 14.64. The goal is to maximize profit by setting appropriate prices and capacity and ensuring that demand never exceeds capacity. You should also check the Non-Negative option (prices and capacity cannot be negative), and you should select the GRG Nonlinear method. Again, this is because prices are multiplied by demands, which are functions of prices, so that profit is a nonlinear function of the prices.

Discussion of the Solution The Solver solution in Figure 14.63 indicates that FPL should charge $137.57 per mWh during the on-peak period and $75.85 during the off-peak-load period.19 These prices generate demands of 0.692 million mWh in both periods, which is therefore the capacity required. The cost of this capacity is $51.908 million. When this is subtracted from the revenue of $147.712 million, the monthly profit becomes $95.804 million.

Figure 14.63 Electricity Pricing Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Electricity pricing model A B C D E F G H

Input egnaRatad names used: Coefficients of demand 51$B$!=ModelyticapaCsnoitcnuf

Common_Capacity =Model!$B$21:$C$21 On-peak demand- =Model!$B$19:$C$19 Off-peak demand Prices =Model!$B$13:$C$13

Profit =Model!$B$26 Cost of capacity/mWh $75

Decisions Off-peak

Price per mWh

Capacity (millions of mWh)

Constraints on demand (in million of mWh) On-peak

On-peak

On-peak price Off-peak price

Off-peak Demand

<= <= Capacity 0.692

Monetary summary ($ millions) Revenue 147.712 Cost of capacity 51.908 Profit 95.804

0.692 0.692

0.692

$137.57

1.142 2.253

$75.85

Demands Constant

0.005 �0.013 0.003

�0.015

0.692

19 With these prices, if a typical customer uses 1.5 mWh in a month, two-thirds of which is on-peak usage, the bill for the month will be $137.57 1 $75.85/25$175.50. This seems reasonable.

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14-8 Nonlinear Optimization Models 6 9 9

To gain some insight into this solution, consider what happens if FPL changes the peak- load price slightly from its optimal value of $137.57. If FPL decreases the price to $137, say, you can check that the on-peak demand increases to 0.700 and the off-peak demand decreases to 0.689. The net effect is that revenue increases slightly, to $148.118. However, the on-peak demand is now greater than capacity, so FPL must increase its capacity from 0.692 to 0.700. This increases the cost of capacity to $52.500, which more than offsets the increase in reve- nue. A similar chain of effects occurs if FPL increases the on-peak price to $138. In this case, on-peak demand decreases, off-peak demand increases, and total revenue decreases. Although FPL can get by with lower capacity, the net effect is slightly less profit. Fortunately, Solver evaluates these trade-offs for you when it finds the optimal solution.

Is the Solver Solution Optimal? It is not difficult to show that the constraints for this model are linear and the objective is a concave function. This is enough to guarantee that there are no local maxima that are not globally optimal. In short, this guarantees that the Solver solution is optimal.

Sensitivity Analysis To gain even more insight, you can use SolverTable to see the effects of changing the unit cost of capacity, allowing it to vary from $60 to $80 in increments of $2. The results appear in Figure 14.65. They indicate that as the cost of capacity increases, the on-peak and off-peak prices both increase, the capacity decreases, and profit decreases. The latter two effects are probably intuitive, but we challenge you to explain the effects on price.

Figure 14.64 Solver Dialog Box for the Electricity Pricing Model

Varying the values of the decision variables slightly from their optimal values sometimes provides insight into the optimal solution.

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14-8c Portfolio Optimization Models Given a set of investments, how do financial analysts determine the portfolio that has the lowest risk and yields a high expected return? This question was answered by Harry Mar- kowitz in the 1950s. For his work on this and other investment topics, he received the Nobel Prize in economics in 1991. The ideas discussed in this section are the basis for most methods of asset allocation used by Wall Street firms. For example, portfolio opti- mization models are used to determine the percentage of assets to invest in stocks, gold, and Treasury bills. Before proceeding, however, you need to learn about some important formulas involving the expected value and variance of sums of random variables.

Weighted Sums of Random Variables Let Ri be the (random) return earned during a year on a dollar invested in investment i. For example, if Ri 5 0.10, a dollar invested at the beginning of the year grows to $1.10 by the end of the year, whereas if Ri 5 20.20, a dollar invested at the beginning of the year decreases in value to $0.80 by the end of the year. We assume that n investments are available. Let xi be the fraction of our money invested in investment i. We assume that x1 1 x2 1 g 1 xn 5 1, so that all of our money is invested. (To prevent shorting a stock—that is, selling shares we don’t own—we assume that xi $ 0.) Then the annual return on our investments is given by the random variable Rp, where

Rp 5 R1x1 1 R2x2 1 g 1 Rnxn

(The subscript p on Rp stands for “portfolio.”) Let mi be the expected value (also called the mean) of Ri , let s2

i be the variance of Ri (so that si is the standard deviation of Ri), and let rij be the correlation between Ri and Rj. To do any work with investments, you must understand how to use the following formulas, which relate the data for the individual investments to the expected return and the variance of return for a portfolio of investments.

Figure 14.65 Sensitivity to Cost of Capacity

3 2 1

4 5 6 7 8 9

10 11 12 13 14 15

A B C D E F G

Cost of capacity (cell $B$9) values along side, output cell(s) along top

Pr ic

es _1

Pr ic

es _2

Ca pa

ci ty

Pr of

$60 $62

$133.82 $72.10 0.730 106.466 $134.32 $72.60 0.725 105.012

$64 $134.82 $73.10 0.720 103.568 $66 $135.32 $73.60 0.715 102.134 $68 $135.82 $74.10 0.710 100.710 $70 $136.32 $74.60 0.705 99.295 $72 $136.82 $75.10 0.700 97.891 $74 $137.32 $75.60 0.695 96.497 $76 $137.82 $76.10 0.690 95.113 $78 $138.32 $76.60 0.685 93.738 $80 $138.82 $77.10 0.680 92.374

Oneway analysis for Solver model in Model worksheet

Expected value of Rp 5 m1x1 1 m2 x2 1 g 1 mnxn (14.7)

Variance of Rp 5 s2 1 x

2 1 1 s2

2 x 2 2 1 g 1 s2

n x 2 n 1Sij rij si sj xi

x j (14.8)

The latter summation in Equation (14.8) is over all pairs of investments. The quantities in Equations (14.7) and (14.8) are important in portfolio selection because of the risk– return trade-off investors need to make. Investors want to choose portfolios with high return, measured by the expected value in Equation (14.7), but they also want portfolios with low risk, usually measured by the variance in Equation (14.8).

7 0 0 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

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14-8 Nonlinear Optimization Models 7 0 1

Equation (14.8) can be rewritten slightly by using covariances instead of correlations. The covariance between two stock returns is another measure of the relationship between the two returns, but unlike a correlation, it is not scaled to be between 21 and 11. This is because covariances are affected by the units in which the returns are measured. Although a covariance is a somewhat less intuitive measure than a correlation, it is used so frequently by financial analysts that we use it here as well. If cij is the estimated covariance between stocks i and j, then cij 5 rijsisj. (Here, r is an estimated correlation, and s is an estimated standard deviation.) Using this equation and the fact that the correlation between any stock and itself is 1, we can also write cii 5 si

2 for each stock i. Therefore, an equivalent form of Equation (14.8) is the following.

Estimated variance of Rp 5 Si, j cij xi xj (14.9)

This allows you to calculate the portfolio variance with Excel’s matrix functions, as explained next.

Matrix Functions in Excel Equation (14.9) for the variance of portfolio return looks intimidating, particularly if there are many potential investments. Fortunately, two built-in Excel matrix functions, MMULT and TRANSPOSE, simplify the calculation. In this subsection we illustrate how to use these two functions. Then in the next subsection we use them in the portfolio selection model.

A matrix is a rectangular array of numbers. The matrix is an i 3 j matrix if it consists of i rows and j columns. For example,

A 5 a1 2 3

4 5 6 b

is a 2 3 3 matrix, and

B 5 ° 1 2

3 4

5 6

is a 3 3 2 matrix. If the matrix has only a single row, it is called a row vector. Similarly, if it has only a single column, it is called a column vector.

If matrix A has the same number of columns as matrix B has rows, it is possible to calculate the matrix product of A and B, denoted AB. The entry in row i, column j of the product AB is obtained by summing the products of the values in row i of A with the cor- responding values in column j of B. If A is an i 3 k matrix and B is a k 3 j matrix, the product AB is an i 3 j matrix.

For example, if

A 5 a1 2 3

2 4 5 b

and

B 5 ° 1 2

3 4

5 6

then AB is the following 2 3 2 matrix:

AB 5 a1(1) 1 2(3) 1 3(5) 1(2) 1 2(4) 1 3(6)

2(1) 1 4(3) 1 5(5) 2(2) 1 4(4) 1 5(6) b 5 a22 28

39 50 b

The Excel MMULT function performs matrix multiplication in a single step. The spreadsheet in Figure 14.66 indicates how to multiply matrices of different sizes. (See the file Matrix Multiplication Finished.xlsx.) For example, to multiply matrix 1 by matrix 2

Matrix Multiplication in Excel

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7 0 2 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

(which is possible because matrix 1 has three columns and matrix 2 has three rows), select the two-row, two-column range B13:C14, type the formula

5MMULT(B4:D5,B7:C9)

and press Ctrl1Shift1Enter (all three keys at once). You should select a range with two rows and two columns because matrix 1 has two rows and matrix 2 has two columns.

The matrix multiplication in cell B24 indicates that (1) it is possible to multiply three matrices together by using MMULT twice, and (2) the TRANSPOSE function can be used to convert a column vector to a row vector (or vice versa), if necessary. Here, you want to multiply Column 1 by the product of Matrix 3 and Column 1. However, Column 1 is 3 3 1, and Matrix 3 is 3 3 3, so Column 1 multiplied by Matrix 3 doesn’t work. Instead, you must transpose Column 1 to make it 1 3 3. Then the result of multiplying all three together is a 1 3 1 matrix (a number). It can be calculated by selecting cell B24, typing the formula

5MMULT(TRANSPOSE(I4:I6),MMULT(B17:D19,I4:I6))

and pressing Ctrl1Shift1Enter. This formula uses MMULT twice because MMULT can multiply only two matrices at a time.

Figure 14.66 Examples of Matrix Multiplication in Excel

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Matrix mul�plica�on in Excel

Typical mul�plica�on of two matrices Mul�plica�on of a matrix and a column Matrix 1 Column 1321

3542 4

Matrix 2 Matrix 1 �mes Column 1, with formula =MMULT(B4:D5,I4:I6)43 Select range with 2 rows, 1 column, enter formula, press Ctrl+Shi�+Enter65

20 Matrix 1 �mes Matrix 2, with formula =MMULT(B4:D5,B7:C9) 63 Select range with 2 rows, 2 columns, enter formula, press Ctrl+Shi�+Enter.

Mul�plica�on of a row and a matrix woR5039

Mul�plica�on of a quadra�c form (row �mes matrix �mes column)

Quadra�c form

Row 1 �mes Matrix 1, with formula =MMULT(I14:J14,B4:D5) Matrix 3 Select range with 1 row, 3 columns, enter formula, press Ctrl+Shi�+Enter312

1 –1 3728140

Mul�plica�on of a row and a column

Transpose of Column 1 �mes Matrix 3 �mes Column 1 Row 2

1 2

22 28 1 4 5

43 0

1 6 3

Formula is =MMULT(TRANSPOSE(I4:I6),MMULT(B17:D19,I4:I6)) Select range with 1 row, 1 column, enter formula, press Ctrl+Shi�+Enter

Row 2 �mes Column 1, with formula =MMULT(I22:K22,I4:I6)

123 Select range with 1 row, 1 column, enter formula, press Ctrl+Shi�+Enter

A B C D E F G H I J K L M N

MMULT

The MMULT and TRANSPOSE functions are useful for matrix operations. They are called array functions because they operate on an entire range, not just a single cell. The MMULT function multiplies two matrices and has the syntax 5MMULT(range1,range2), where range1 must have as many col- umns as range2 has rows. To use this function, select a range that has as many rows as range1 and as many columns as range2, type the formula, and press Ctrl1Shift1Enter. The resulting formula will have curly brackets around it in the Excel formula bar. You should not type these curly brackets. Excel enters them automatically to remind you that this is an array formula.

Excel Function

Press control 1 shift 1 return on a Mac.

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14-8 Nonlinear Optimization Models 7 0 3

The Portfolio Selection Model Most investors have two objectives in forming portfolios: to obtain a large expected return and to obtain a small variance (to minimize risk). The problem is inherently nonlinear because the portfolio variance is nonlinear in the investment amounts. The most common way of handling this two-objective problem is to specify a minimal required expected return and then minimize the variance subject to the constraint on the expected return. The following example illustrates how to do this.

EXAMPLE

14.12 PORTFOLIO OPTIMIZATION AT PERLMAN & BROTHERS

The investment company Perlman & Brothers intends to invest a given amount of money in three stocks. From past data, the means and standard deviations of annual returns have been estimated as shown in Table 14.8. The correlations between the annual returns on the stocks are listed in Table 14.9. The company wants to find a minimum-variance portfolio that yields an expected annual return of at least 0.12 (that is, 12%).

Table 14.8 Estimated Means and Standard Deviations of Stock Returns

Table 14.9 Estimated Correlations between Stock Returns

Stock Mean Standard Deviation

1 0.14 0.20

2 0.11 0.15

3 0.10 0.08

Combination Correlation

Stocks 1 and 2 0.6

Stocks 1 and 3 0.4

Stocks 2 and 3 0.7

Objective To use NLP to find the portfolio that minimizes the risk, measured by portfolio variance, subject to achieving an expected return of at least 12%.

Where Do the Numbers Come From? Financial analysts typically estimate the required means, standard deviations, and correlations for stock returns from historical data. However, you should be aware that there is no guarantee that these estimates, based on historical return data, are relevant for future returns. If analysts have new information about the stocks, they should incorporate this new information into their estimates.

Solution The variables and constraints for this model appear in Figure 14.67. (See the file Portfolio Selection Big Picture.xlsx.) One interesting aspect of this model is that it is not necessary to specify the amount of money invested—it could be $100, $1000, $1,000,000, or any other amount. The model determines the fractions of this amount to invest in the various stocks, and these fractions are relevant for any investment amount. The only requirement is that the fractions should sum to 1, so that all of the money is invested. Besides this, the fractions are constrained to be nonnegative to prevent shorting stocks.20 Finally, the expected portfolio return is constrained to be at least as large as a specified expected return, such as 12%.

Developing the Spreadsheet Model The individual steps are now listed. (See Figure 14.68 and the file Portfolio Selection Finished.xlsx.)

1. Inputs. Enter the inputs in the input cells. These include the estimates of means, standard deviations, and correlations, as well as the required expected return.

Developing the Portfolio Selection Model

20 If you want to allow shorting, do not check the Non-Negative option in the Solver dialog box.

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7 0 4 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

2. Fractions invested. Enter any trial values in the Investment_weights range for the fractions of Perlman’s money placed in the three investments. Then sum these with the SUM function in cell B19.

3. Expected annual return. Use Equation (14.7) to calculate the expected annual return in cell B23 with the formula

5SUMPRODUCT(B5:D5,Investment_weights)

4. Covariance matrix. Equation (14.9) is used to calculate the portfolio variance. To do this, you must first calculate a matrix of covariances. Using the general formula for covariance, cij 5 rijsisj (which holds even when i 5 j, because rii 5 1), these can be calculated from the inputs by using lookups. Specifically, enter the formula

5HLOOKUP($F9,$B$4:$D$6,3)*B9*HLOOKUP(G$8,$B$4:$D$6,3)

Figure 14.67 Big Picture for Portfolio Selection Model

Standard devia�ons of returns

Correla�ons between returns

Mean returns

Actual mean por�olio return

Sum of investment weights

Required mean por�olio return

Investment weights

Minimize variance (or standard devia�on) of

por�olio return1=

Figure 14.68 Portfolio Selection Model

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2 27 28 29 30 31 32 33 34

A B C D E F G H I Por�olio selec�on model

Range names used:

Mean_por�olio_return =Model!$B$23 Investment_weights =Model!$B$15:$D$15

Stock input data

Por�olio_stdev =Model!$B$26

Stock 1 Stock 2 Stock 3

Por�olio_variance =Model!$B$25

Mean return

=Model!$D$23Required_mean_return

0.10.110.14 StDev of return

=Model!$B$19Total_weights

0.080.152.0

Stock 1Correla�ons Stock 2 Stock 3 Covariances Stock 1 Stock 2 Stock 3 Stock 1 Stock 10.40.61 0.04 0.018 0.0064 Stock 2 Stock 20.710.6 0.018 0.0225 0.0084 Stock 3 Stock 310.70.4 0.0064 0.0084 0.0064

Investment decisions Stock 1 Stock 2 Stock 3

Investment weights 0.500 0.000 0.500

Constraint on inves�ng everything Total weights Required value

1=1.00

Constraint on expected por�olio return Mean por�olio return Required mean return

0.120 >= 0.120

Por�olio variance 0.0148 Por�olio stdev 0.1217

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14-8 Nonlinear Optimization Models 7 0 5

in cell G9, and copy it to the range G9:I11. (B9 captures the relevant correlation. The two HLOOKUP terms capture the appropriate standard deviations.)

5. Portfolio variance. Although the mathematical details are not presented here, it can be shown that the summation in Equation (14.9) is the product of three matrices: a row of fractions invested multiplied by the covariance matrix multiplied by a column of fractions invested. To calculate it, enter the formula

5MMULT(Investment_weights,MMULT(G9:I11,TRANSPOSE(Investment_weights)))

in cell B25 and press Ctrl1Shift1Enter. (Remember that Excel puts curly brackets around this formula. You should not type these curly brackets.) This formula uses two MMULT functions. Again, this is because MMULT can multiply only two matrices at a time. The formula first multiplies the last two matrices and then multiplies this product by the first matrix.

6. Portfolio standard deviation. Most financial analysts talk in terms of portfolio variance. However, it is probably more intuitive to talk about portfolio standard deviation because it is in the same units as the returns. Calculate the standard deviation in cell B26 with the formula

5SQRT(Portfolio_variance)

Actually, either cell B25 or B26 can be used as the objective cell to minimize. Minimizing the square root of a function is equivalent to minimizing the function itself.

Using Solver The completed Solver dialog box is shown in Figure 14.69. The constraints specify that the expected return must be at least as large as the minimal required expected return, and all the company’s money must be invested. The decision variable cells are constrained to be nonnegative (to avoid short selling), and because of the squared terms in the variance formula, you must select the GRG Nonlinear method.

Figure 14.69 Solver Dialog Box for Portfolio Selection Model

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7 0 6 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

Discussion of the Solution The solution in Figure 14.68 indicates that the company should put half of its money in each of stocks 1 and 3, and it should not invest in stock 2 at all. This might be somewhat surprising, given that the ranking of riskiness of the stocks is 1, 2, 3, with stock 1 being the most risky but also having the highest expected return. However, the correlations play an important role in portfolio selection, so you can usually not guess the optimal portfolio on the basis of the means and standard deviations alone.

The portfolio standard deviation of 0.1217 can be interpreted in a probabilistic sense. If stock returns are approximately normally distributed, the actual portfolio return will be within one standard deviation of the expected return with probability about 0.68, and the actual portfolio return will be within two standard deviations of the expected return with probability about 0.95. Given that the expected return is 12%, this implies a lot of risk—two standard deviations below this mean is a negative return (or loss) of slightly more than 12%.

Is the Solver Solution Optimal? The constraints for this model are linear, and it can be shown that the portfolio variance is a convex function of the investment fractions. This is sufficient to guarantee that the Solver solution is indeed optimal.

Sensitivity Analysis This model begs for a sensitivity analysis on the minimum required return. When the company requires a larger expected return, it must assume a larger risk. This behavior is illustrated in Figure 14.70, where SolverTable has been used with cell D23

4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

A B C D E F G H I

Required return (cell $D$23) values along side, output cell(s) along top 2 1 Oneway analysis for Solver model in Model worksheet

Po r�

ol io

_s td

In ve

st m

en t_

w ei

gh ts

In ve

st m

en t_

w ei

gh ts

In ve

st m

en t_

w ei

gh ts

M ea

n_ po

r� ol

io _r

et ur

0.100 0.105 0.110 0.115 0.120 0.125 0.130 0.135 0.140

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1.000 0.875 0.750 0.625 0.500 0.375 0.250 0.125 0.000

0.100 0.105 0.110 0.115 0.120 0.125 0.130 0.135 0.140

0.0800 0.0832 0.0922 0.1055 0.1217 0.1397 0.1591 0.1792 0.2000

0.150

0.140

0.130

0.110

0.120

0.100

0.090

0.080 0.0500 0.0700 0.0900 0.1100 0.1300 0.1500 0.1700 0.1900 0.2100

M ea

n po

r� ol

io re

tu rn

Standard devia�on of por�olio return (risk)

Efficient Fron�er

Figure 14.70 The Efficient Frontier

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14-8 Nonlinear Optimization Models 7 0 7

as the single input cell, varied from 0.10 to 0.14 in increments of 0.005. Note that values outside this range are of little interest. Stock 3 has the minimum expected return, 0.10, and stock 1 has the highest expected return, 0.14, so no portfolio can have an expected return outside of this range.

The results indicate that the company should put more and more into risky stock 1 as the required return increases—and stock 2 continues to be unused. The accompanying scatter chart (with the option to “connect the dots”) shows the risk–return trade-off. As the company assumes more risk, as measured by portfolio standard deviation, the expected return increases, but at a decreasing rate.

The curve in this chart is called the efficient frontier. Points on the efficient frontier can be achieved by appropriate port- folios. Points below the efficient frontier can be achieved, but they are not as good as points on the efficient frontier because they have a lower expected return for a given level of risk. In contrast, points above the efficient frontier are unachievable—the company cannot achieve this high an expected return for a given level of risk.

Modeling Issues • Typical real-world portfolio selection problems involve a large number of potential investments, certainly many more than

three. This admittedly requires more input data, particularly for the correlation matrix, but the basic model does not change at all. In particular, the matrix formula for portfolio variance is exactly the same. This shows the power of using Excel’s matrix functions. Without them, the formula for portfolio variance would be a long, involved sum.

• If Perlman is allowed to short a stock, the fraction invested in that stock is allowed to be negative. To implement this, you can eliminate the nonnegativity constraints on the decision variable cells.

• An alternative objective might be to minimize the probability that the portfolio loses money. This possibility is illustrated in one of the problems.

demand is always equal to capacity for at least one of the two periods of the day?

71. For each of the following, answer whether it makes sense to multiply the matrices of the given sizes. In each case where it makes sense, demonstrate an example in Excel, where you can make up the numbers.

a. AB, where A is 3 3 4 and B is 4 3 1

b. AB, where A is 1 3 4 and B is 4 3 1

c. AB, where A is 4 3 1 and B is 1 3 4

d. AB, where A is 1 3 4 and B is 1 3 4

e. ABC, where A is 1 3 4, B is 4 3 4, and C is 4 3 1

f. ABC, where A is 3 3 3, B is 3 3 3, and C is 3 3 1

g. ATB, where A is 4 3 3 and B is 4 3 3, and AT

denotes the transpose of A 72. Add a new stock, stock 4, to the Perlman portfolio

optimization model. Assume that the estimated mean and standard deviation of return for stock 4 are 0.125 and 0.175, respectively. Also, assume the correlations between stock 4 and the original three stocks are 0.3, 0.5, and 0.8. Run Solver on the modified model, where the required expected portfolio return is again 0.12. Is stock 4 in the optimal portfolio? Then run SolverTable as in the example. Is stock 4 in any of the optimal portfolios on the efficient frontier?

73. In the Perlman portfolio optimization model, stock 2 is not in the optimal portfolio. Use SolverTable to see whether it ever enters the optimal portfolio as its cor- relations with stocks 1 and 3 vary. Specifically, use a two-way SolverTable with two inputs, the correlations

Problems

Level A 69. In the FPL electricity pricing model, the demand func-

tions have positive and negative coefficients of prices. The negative coefficients indicate that as the price of a product increases, demand for that product decreases. The positive coefficients indicate that as the price of a product increases, demand for the other product increases. a. Increase the magnitudes of the negative coefficients

from 20.013 and 20.015 to 20.018 and 20.023, and rerun Solver. Are the changes in the optimal solution intuitive? Explain.

b. Increase the magnitudes of the positive coefficients from 0.005 and 0.003 to 0.007 and 0.005, and rerun Solver. Are the changes in the optimal solution intu- itive? Explain.

c. Make the changes in parts a and b simultaneously and rerun Solver. What happens now?

70. In the FPL electricity pricing model, we assumed that the capacity level is a decision variable. Assume now that capacity has already been set at 0.65 million of mWh. (Note that the cost of capacity is now a sunk cost, so it is irrelevant to the decision problem.) Change the model appropriately and run Solver. Then use SolverT- able to see how sensitive the optimal solution is to the capacity level, letting it vary over some relevant range. Does it appear that the optimal prices will be set so that

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7 0 8 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

between stock 2 and stocks 1 and 3, each allowed to vary from 0.1 to 0.9. Capture as outputs the three decision variable cells. Discuss the results. (Note: You will have to change the model slightly. For example, if you use cells B10 and C11 as the two SolverTable input cells, you will have to ensure that cells C9 and D10 change accordingly. This is easy. Just enter formulas in these lat- ter two cells.)

74. The stocks in the Perlman portfolio optimization model are all positively correlated. What happens when they are negatively correlated? Answer for each of the fol- lowing scenarios. In each case, two of the three correla- tions are the negatives of their original values. Discuss the differences between the optimal portfolios in these three scenarios. a. Change the signs of the correlations between stocks

1 and 2 and between stocks 1 and 3. (Here, stock 1 tends to go in a different direction from stocks 2 and 3.)

b. Change the signs of the correlations between stocks 1 and 2 and between stocks 2 and 3. (Here, stock 2 tends to go in a different direction from stocks 1 and 3.)

c. Change the signs of the correlations between stocks 1 and 3 and between stocks 2 and 3. (Here, stock 3 tends to go in a different direction from stocks 1 and 2.)

75. The file P14_75.xlsx contains historical monthly returns for 28 companies. For each company, calculate the esti- mated mean return and the estimated variance of return. Then calculate the estimated correlations between the companies’ returns. Note that “return” here means monthly return.

76. The file P14_76.xlsx includes contains the data from the previous problem. It also contains fractions in row 3 for creating a portfolio. These fractions are currently all equal to 1/28, but they can be changed to any values you like, so long as they continue to sum to 1. For any such fractions, find the estimated mean, variance, and stan- dard deviation of the resulting portfolio return.

Level B 77. Continuing the previous problem, find the portfolio that

achieves an expected monthly return of at least 0.01(1%) and minimizes portfolio variance. Then use SolverTable to sweep out the efficient frontier. Create a chart of this efficient frontier from your SolverTable results. What are the relevant lower and upper limits on the required expected monthly return?

78. In many cases you can assume that the portfolio return is at least approximately normally distributed. Then you can use Excel’s NORM.DIST function as in Chapter 5 to calculate the probability that the portfolio return is negative. The relevant formula is 5NORM.DIST(0,mean,stdev,TRUE) , where mean and stdev are the expected portfolio return and standard deviation of portfolio return, respectively. a. Modify the Perlman portfolio optimization model

slightly, and then run Solver to find the portfolio that achieves at least a 12% expected return and minimizes the probability of a negative return. Do you get the same optimal portfolio as before? What is the probabil- ity that the return from this portfolio will be negative?

b. Using the model in part a, create a chart of the effi- cient frontier. However, this time put the probability of a negative return on the horizontal axis.

14-9 Conclusion This chapter has led you through spreadsheet optimization models of many diverse problems. No standard procedure can be used to model all problems. However, there are several keys to most models.

• First, determine the decision variables. For example, in blending problems it is important to realize that the decision variables are the amounts of inputs used to produce outputs, and in employee scheduling problems, it is important to realize that the decision variables are the number of employees who start their five-day shift each day of the week.

• Set up the model so that you can easily calculate what you want to maximize or minimize (usually profit or cost). For example, in the aggregate planning model it is a good idea to calculate total cost by calculating the monthly cost of the various activities in separate rows and then summing the subtotals.

• Set up the model so that the relationships between the cells in the spreadsheet and the constraints of the problem are readily apparent. For example, in the employee scheduling model it is convenient to calculate the number of people working each day of the week adjacent to the minimum required number of people for each day of the week.

• Optimization models do not always fall into ready-made categories. A model might involve a combination of the ideas discussed in the production scheduling, blending, and aggregate planning examples. In fact, many real applications are not strictly analogous to any of the models we have discussed. However, exposure to the models in this chapter should give you the insights you need to solve a wide variety of interesting problems.

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14-9 Conclusion 7 0 9

Summary of Key Terms TERM EXPLANATION PAGES Employee scheduling models Models for choosing the staffing levels to meet workload requirements 663

Multiple optimal solutions Situation where several solutions obtain the same optimal objective value 668

Blending models Models where inputs must be mixed in the right proportions to produce outputs

670

Logistics models Models where goods must be shipped from one set of locations to another at minimal cost

676

Flow balance constraint Constraint that relates the flow into a node and the flow out of the node 682

Aggregate planning models Models where workforce levels and production levels must be set to meet customer demand

693

Integer programming (IP) models Models where at least some of the decision variables must be integers 714

Binary variable Integer variable that must be 0 or 1; used to indicate whether an activity takes place

714

Capital budgeting models Models where a subset of investment activities is chosen from a set of possible activities

714

Fixed-cost models Models where fixed costs are incurred for various activities if they are done at any positive level

720

Set-covering models Models where members of one set must be selected to cover services to members of another set

729

Nonlinear programming (NLP) models

Models where either the objective function or the constraints (or both) are nonlinear functions of the decision variables

735

Global optimum Solution that is the best in the entire feasible region 735

Local optimal solution Solution that is better than all nearby solutions (but might not be optimal globally)

735

Portfolio optimization models Models that attempt to find the portfolio of securities that achieves the best balance between risk and return

740

Problems

Conceptual Exercises C.1. The employee scheduling model in this chapter was

purposely made small (only seven decision variable cells). What would make a similar problem for a com- pany like McDonald’s much harder? What types of constraints would be required? How many decision variable cells (approximately) might there be?

C.2. Explain why it is problematic to include a constraint such as the following in an LP model for a blending problem:

Total octane in gasoline 1 blend

Barrels of gasoline 1 blended daily $ 10

C.3. “It is essential to constrain all shipments in a transpor- tation problem to have integer values to ensure that the optimal LP solution consists entirely of integer-valued shipments.” Is this statement true or false? Why?

C.4. What is the relationship between transportation models and more general logistics models? Explain how these two types of linear optimization models are similar and how they are different.

C.5. Unlike the small logistics models presented here, real- world logistics problems can be huge. Imagine the global problem a company like FedEx faces each day. Describe as well as you can the types of decisions and constraints it has. How large (number of decision variables, number of constraints) might such a prob- lem be?

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7 1 0 C H A P T E R 1 4 O p t i m i z a t i o n M o d e l s

C.6. Suppose you develop and solve an integer program- ming model with a cost-minimization objective. Assume the optimal solution yields an objective cell value of $500,000. Now, consider the same linear opti- mization model without the integer restrictions. That is, suppose you drop the requirement that the decision variable cells be integer-valued and reoptimize with Solver. How does the optimal objective cell value for this modified model (called the LP relaxation of the IP model) compare to the original total cost value of $500,000? Explain your answer.

C.7. The portfolio optimization model presented here is the standard model: minimize the variance (or standard deviation) of the portfolio, as a measure of risk, for a given required level of expected return. In general, the goal is to keep risk low and expected return high. Can you think of other ways to model the problem to achieve these basic goals? Is high variability all bad risk?

Level A 79. A bus company believes that it will need the follow-

ing numbers of bus drivers during each of the next five years: 60 drivers in year 1; 70 drivers in year 2; 50 driv- ers in year 3; 65 drivers in year 4; 75 drivers in year 5. At the beginning of each year, the bus company must decide how many drivers to hire or fire. It costs $4000 to hire a driver and $2000 to fire a driver. A driver’s salary is $45,000 per year. At the beginning of year 1 the com- pany has 50 drivers. A driver hired at the beginning of a year can be used to meet the current year’s requirements and is paid full salary for the current year. a. Determine how to minimize the bus company’s salary,

hiring, and firing costs over the next five years. b. Use SolverTable to determine how the total number

hired, total number fired, and total cost change as the unit hiring and firing costs each increase by the same percentage.

80. A pharmaceutical company produces the drug NasaMist from four chemicals. Today, the company must produce 1000 pounds of the drug. The three active ingredients in NasaMist are A, B, and C. By weight, at least 8% of NasaMist must consist of A, at least 4% of B, and at least 2% of C. The cost per pound of each chemical and the amount of each active ingredient in one pound of each chemical are given in the file P14_80.xlsx. It is necessary that at least 100 pounds of chemical 2 and at least 450 pounds of chemical 3 be used. a. Determine the cheapest way of producing today’s

batch of NasaMist. b. Use SolverTable to see how much the percentage of

requirement of A is really costing the company. Let the percentage required vary from 6% to 12%.

81. A bank is attempting to determine where to invest its assets during the current year. At present, $500,000

is available for investment in bonds, home loans, auto loans, and personal loans. The annual rates of return on each type of investment are known to be the following: bonds, 6%; home loans, 8%; auto loans, 5%; personal loans, 10%. To ensure that the bank’s portfolio is not too risky, the bank’s investment manager has placed the fol- lowing three restrictions on the bank’s portfolio:

• The amount invested in personal loans cannot exceed the amount invested in bonds.

• The amount invested in home loans cannot exceed the amount invested in auto loans.

• No more than 25% of the total amount invested can be in personal loans.

Help the bank maximize the annual return on its investment portfolio.

82. A fertilizer company blends silicon and nitrogen to pro- duce two types of fertilizers. Fertilizer 1 must be at least 40% nitrogen and sells for $70 per pound. Fertilizer 2 must be at least 70% silicon and sells for $40 per pound. The company can purchase up to 8000 pounds of nitro- gen at $15 per pound and up to 10,000 pounds of silicon at $10 per pound. a. Assuming that all fertilizer produced can be sold,

determine how the company can maximize its profit. b. Use SolverTable to explore the effect on profit

of changing the minimum percentage of nitrogen required in fertilizer 1.

c. Suppose the availabilities of nitrogen and silicon both increase by the same percentage from their current values. Use SolverTable to explore the effect of this change on profit.

83. Optimization models are used by many Wall Street firms to select a desirable bond portfolio. The following is a simplified version of such a model. A company is con- sidering investing in four bonds; $1 million is available for investment. The expected annual return, the worst- case annual return on each bond, and the duration of each bond are given in the file P14_83.xlsx. (The dura- tion of a bond is a measure of the bond’s sensitivity to interest rates.) The company wants to maximize the expected return from its bond investments, subject to three constraints:

• The worst-case return of the bond portfolio must be at least 8%.

• The average duration of the portfolio must be at most 6. For example, a portfolio that invests $600,000 in bond 1 and $400,000 in bond 4 has an average duration of 3600,000(3) 1 400,000(9)4/1,000,000 5 5.4.

• Because of diversification requirements, at most 40% of the total amount invested can be invested in a single bond.

Determine how the company can maximize the expected return on its investment.

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84. At the beginning of year 1, you have $10,000. Invest- ments A and B are available; their cash flows are shown in the file P14_84.xlsx. Assume that any money not invested in A or B earns interest at an annual rate of 2%. a. Determine how to maximize your cash on hand at the

beginning of year 4. b. Use SolverTable to determine how a change in the

year 2 return for investment A changes the optimal solution to the problem.

c. Use SolverTable to determine how a change in the year 3 return of investment B changes the optimal solution to the problem.

85. An oil company produces two types of gasoline, G1 and G2, from two types of crude oil, C1 and C2. G1 is allowed to contain up to 4% impurities, and G2 is allowed to contain up to 3% impurities. G1 sells for $48 per barrel, whereas G2 sells for $72 per barrel. Up to 4200 barrels of G1 and up to 4300 barrels of G2 can be sold. The cost per barrel of each crude, their availability, and the level of impurities in each crude are listed in the file P14_85.xlsx. Before blending the crude oil into gas, any amount of each crude can be “purified” for a cost of $3.00 per barrel. Purification eliminates half of the impurities in the crude oil. a. Determine how to maximize profit. b. Use SolverTable to determine how an increase in the

availability of C1 affects the optimal profit. c. Use SolverTable to determine how an increase in the

availability of C2 affects the optimal profit. d. Use SolverTable to determine how a change in the

profitability of G2 changes profitability and the types of gas produced.

86. The government is auctioning off oil leases at two sites: 1 and 2. At each site 10,000 acres of land are to be auc- tioned. Cliff Ewing, Blake Barnes, and Alexis Pickens are bidding for the oil. Government rules state that no bidder can receive more than 40% of the land being auc- tioned. Cliff has bid $10,000 per acre for site 1 land and $20,000 per acre for site 2 land. Blake has bid $9000 per acre for site 1 land and $22,000 per acre for site 2 land. Alexis has bid $11,000 per acre for site 1 land and $19,000 per acre for site 2 land. a. Determine how to maximize the government’s

revenue. b. Use SolverTable to see how changes in the govern-

ment’s rule on 40% of all land being auctioned affect the optimal revenue. Why can the optimal revenue not decrease if this percentage required increases? Why can the optimal revenue not increase if this percentage required decreases?

87. An automobile company produces cars in Los Angeles and Detroit and has a warehouse in Atlanta. The com- pany supplies cars to customers in Houston and Tampa. The costs of shipping a car between various points are listed in the file P14_87.xlsx, where a blank means that a shipment is not allowed. Los Angeles can produce

up to 1100 cars, and Detroit can produce up to 2900 cars. Houston must receive 2400 cars, and Tampa must receive 1500 cars. a. Determine how to minimize the cost of meeting

demands in Houston and Tampa. b. Modify the answer to part a if shipments between Los

Angeles and Detroit are not allowed. c. Modify the answer to part a if shipments between

Houston and Tampa are allowed at a cost of $5 per car. 88. An oil company produces oil from two wells. Well 1 can

produce up to 150,000 barrels per day, and well 2 can produce up to 200,000 barrels per day. It is possible to ship oil directly from the wells to the company’s cus- tomers in Los Angeles and New York. Alternatively, the company could transport oil to the ports of Mobile and Galveston and then ship it by tanker to New York or Los Angeles, respectively. Los Angeles requires 160,000 barrels per day, and New York requires 140,000 barrels per day. The costs of shipping 1000 barrels between var- ious locations are shown in the file P14_88.xlsx, where a blank indicates shipments that are not allowed. Deter- mine how to minimize the transport costs in meeting the oil demands of Los Angeles and New York.

89. Based on Bean et al. (1987). Boris Milkem’s firm owns six assets. The expected selling price (in millions of dol- lars) for each asset is given in the file P14_89.xlsx. For example, if asset 1 is sold in year 2, the firm receives $20 million. To maintain a regular cash flow, Milkem must sell at least $20 million of assets during year 1, at least $30 million worth during year 2, and at least $35 million worth during year 3. Determine how Milkem can maximize his total revenue from assets sold during the next three years.

90. Based on Sonderman and Abrahamson (1985). In treat- ing a brain tumor with radiation, physicians want the maximum amount of radiation possible to bombard the tissue containing the tumors. The constraint is, however, that there is a maximum amount of radiation that normal tissue can handle without suffering tissue damage. Phy- sicians must therefore decide how to aim the radiation so as to maximize the radiation that hits the tumor tissue subject to the constraint of not damaging the normal tis- sue. As a simple example, suppose there are six types of radiation beams (beams differ in where they are aimed and their intensity) that can be aimed at a tumor. The region containing the tumor has been divided into six regions: three regions contain tumors and three contain normal tissue. The amount of radiation delivered to each region by each type of beam is shown in the file P14_90. xlsx. If each region of normal tissue can handle at most 60 units of radiation, which beams should be used to maximize the total amount of radiation received by the tumors?

91. A leading hardware company produces three types of computers: Pear computers, Apricot computers, and Orange computers. The relevant data are given in the

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file P14_91.xlsx. The equipment cost is a fixed cost; it is incurred if any of this type of computer is produced. A total of 30,000 chips and 12,000 hours of labor are avail- able. The company wants to produce at least two types of computers. a. Determine how the company can maximize its profit. b. For any computer type not in the optimal product mix,

use SolverTable to find how much larger its unit mar- gin would have to be before it would enter the optimal product mix.

92. A food company produces tomato sauce at five differ- ent plants. The tomato sauce is then shipped to one of three warehouses, where it is stored until it is shipped to one of the company’s four customers. All of the inputs for the problem are given in the file P14_92.xlsx, as follows:

• The plant capacities (in tons)

• The cost per ton of producing tomato sauce at each plant and shipping it to each warehouse

• The cost of shipping a ton of sauce from each ware- house to each customer

• The customer requirements (in tons) of sauce

• The fixed annual cost of operating each plant and warehouse.

The company must decide which plants and warehouses to open, and which routes from plants to warehouses and from warehouses to customers to use. All customer demand must be met. A given customer’s demand can be met from more than one warehouse, and a given plant can ship to more than one warehouse. a. Determine the minimum-cost method for meeting

customer demands. b. Use SolverTable to see how a change in the capacity

of plant 1 affects the total cost. c. Use SolverTable to see how a change in the customer

2 demand affects the total cost. 93. You are given the following means, standard devia-

tions, and correlations for the annual return on three potential investments. The means are 0.12, 0.15, and 0.20. The standard deviations are 0.20, 0.30, and 0.40. The correlation between stocks 1 and 2 is 0.65, between stocks 1 and 3 is 0.75, and between stocks 2 and 3 is 0.41. You have $100,000 to invest and can invest no more than half of your money in any single invest- ment. Determine the minimum-variance portfolio that yields an expected annual return of at least 0.14.

94. You have $50,000 to invest in three stocks. Let Ri be the random variable representing the annual return on $1 invested in stock i. For example, if Ri 5 0.12, then $1 invested in stock i at the beginning of a year is worth $1.12 at the end of the year. The means are E(R1) 5 0.14, E(R2) 5 0.11, and E(R3) 5 0.10. The variances are Var R1 5 0.20 , Var R2 5 0.08 , a n d Var R3 5 0.18. The correlations are r12 5 0.8, r13 5 0.7, and r23 5 0.9.

Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12.

Level B 95. The risk index of an investment can be obtained by tak-

ing the absolute values of percentage changes in the value of the investment for each year and averaging them. Suppose you are trying to determine the percent- ages of your money to invest in stocks, 3-month Treasury bills, and 10-year Treasury bonds. The file P14_95.xlsx lists the annual returns (percentage changes in value) for these investments since 1970. Let the risk index of a portfolio be the weighted average of the risk indices of these investments, where the weights are the fractions of the portfolio assigned to the investments. Suppose the amount of each investment must be between 20% and 50% of the total invested. You would like the risk index of your portfolio to equal 0.12, and your goal is to max- imize the expected return on your portfolio. Determine the maximum expected return on your portfolio, subject to the stated constraints. Use the average return earned by each investment during these years as your estimate of expected return.

96. Broker Sonya Wong is currently trying to maximize her profit in the bond market. Four bonds are available for purchase and sale at the bid and ask prices shown in the file P14_96.xlsx. Sonya can buy up to 1000 units of each bond at the ask price or sell up to 1000 units of each bond at the bid price. During each of the next three years, the person who sells a bond will pay the owner of the bond the cash payments listed in the same file. Son- ya’s goal is to maximize her revenue from selling bonds minus her payment for buying bonds, subject to the con- straint that after each year’s payments are received, her current cash position (due only to cash payments from bonds and not purchases or sales of bonds) is nonnega- tive. Note that her current cash position can depend on past coupons and that cash accumulated at the end of each year earns 2.5% annual interest. Determine how to maximize net profit from buying and selling bonds, sub- ject to the constraints previously described. Why do you think we limit the number of units of each bond she can buy or sell?

97. A financial company is considering investing in three projects. If it fully invests in a project, the realized cash flows (in millions of dollars) will be as listed in the file P14_97.xlsx. For example, project 1 requires a cash out- flow of $3 million today and returns $5.5 million three years from now. The company currently has $2 million in cash. At each time point (0, 6, 12, 18, 24, and 30 months from now), the company can, if desired, borrow up to $2 million at 3.5% interest (per six months). Left- over cash earns 3% interest (per six months). For exam- ple, if after borrowing and investing at the current time, the company has $1 million, it will receive $30,000 in

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interest six months from now. The company’s goal is to maximize cash on hand after cash flows three years from now are accounted for. What investment and bor- rowing strategy should it use? Assume that the com- pany can invest in a fraction of a project. For example, if it invests in one-half of project 3, it has cash outflows of 2$1 million now and six months from now.

98. You are a CFA (chartered financial analyst). An over- extended client has come to you because she needs help paying off her credit card bills. She owes the amounts on her credit cards listed in the file P14_98 .xlsx. The client is willing to allocate up to $5000 per month to pay off these credit cards. All cards must be paid off within 36 months. The client’s goal is to min- imize the total of all her payments. To solve this prob- lem, you must understand how interest on a loan works. To illustrate, suppose the client pays $5000 on Saks during month 1. Then her Saks balance at the beginning of month 2 is 20,000 2 35000 2 0.005(20,000)4 . This follows because she incurs 0.005(20,000) in inter- est charges on her Saks card during month 1. Help the client solve her problem. Once you have solved this problem, give an intuitive explanation of the solution found by Solver.

99. A food company produces two types of turkey cutlets for sale to fast-food restaurants. Each type of cutlet consists of white meat and dark meat. Cutlet 1 sells for $4 per pound and must consist of at least 70% white meat. Cutlet 2 sells for $3 per pound and must consist of at least 60% white meat. At most 6000 pounds of cutlet 1 and 2000 pounds of cutlet 2 can be sold. The two types of turkey used to manufacture the cutlets are purchased from a turkey farm. Each type 1 turkey costs $10 and yields five pounds of white meat and two pounds of dark meat. Each type 2 turkey costs $8 and yields three pounds of white meat and three pounds of dark meat. Determine how the company can maximize its profit.

100. Each hour from 10 a.m. to 7 p.m., a bank receives checks and must process them. Its goal is to process all checks the same day they are received. The bank has 13 check processing machines, each of which can process up to 500 checks per hour. It takes one worker to operate each machine. The bank hires both full-time and part-time workers. Full-time workers work 10 a.m. to 6 p.m., 11 a.m. to 7 p.m., or noon to 8 p.m. and are paid $160 per day. Part-time workers work either 2 p.m. to 7 p.m. or 3 p.m. to 8 p.m. and are paid $75 per day. The numbers of checks received each hour are listed in the file P14_100. xlsx. In the interest of maintaining continuity, the bank believes that it must have at least three full-time workers under contract. Develop a work schedule that processes all checks by 8 p.m. and minimizes daily labor costs.

101. An oil company has oil fields in San Diego and Los Angeles. The San Diego field can produce up to 500,000 barrels per day, and the Los Angeles field can

produce up to 400,000 barrels per day. Oil is sent from the fields to a refinery, either in Dallas or in Houston. (Assume that each refinery has unlimited capacity.) To refine 100,000 barrels costs $700 at Dallas and $900 at Houston. Refined oil is shipped to customers in Chicago and New York. Chicago customers require 400,000 bar- rels per day, and New York customers require 300,000 barrels per day. The costs of shipping 100,000 barrels of oil (refined or unrefined) between cities are listed in the file P14_101.xlsx. a. Determine how to minimize the total cost of meeting

all demands. b. If each refinery had a capacity of 380,000 barrels per

day, how would you modify the model in part a? 102. An electrical components company produces capaci-

tors at three locations: Los Angeles, Chicago, and New York. Capacitors are shipped from these locations to public utilities in five regions of the country: northeast (NE), northwest (NW), midwest (MW), southeast (SE), and southwest (SW). The cost of producing and ship- ping a capacitor from each plant to each region of the country is given in the file P14_102.xlsx. Each plant has an annual production capacity of 100,000 capaci- tors. Each year, each region of the country must receive the following number of capacitors: NE, 55,000; NW, 50,000; MW, 60,000; SE, 60,000; SW, 45,000. The company believes that shipping costs are too high, and it is therefore considering building one or two more production plants. Possible sites are Atlanta and Hous- ton. The costs of producing a capacitor and shipping it to each region of the country are given in the same file. It costs $3 million (in current dollars) to build a new plant, and operating each plant incurs a fixed cost (in addition to variable shipping and production costs) of $50,000 per year. A plant at Atlanta or Houston will have the capacity to produce 100,000 capacitors per year. Assume that future demand patterns and pro- duction costs will remain unchanged. If costs are dis- counted at a rate of 12% per year, how can the company minimize the net present value (NPV) of all costs asso- ciated with meeting current and future demands?

103. Based on Bean et al. (1988). The owner of a shop- ping mall has 10,000 square feet of space to rent and wants to determine the types of stores that should occupy the mall. The minimum number and max- imum number of each type of store and the square footage of each type are given in the file P14_103 .xlsx. The annual profit made by each type of store depends on the number of stores of that type in the mall. This dependence is given in the same file, where all profits are in units of $10,000. For example, if there are two department stores in the mall, each department store will earn $210,000 profit per year. Each store pays 5% of its annual profit as rent to the owner of the mall. Determine how the owner of the mall can maximize its rental income.

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104. A city (labeled C for convenience) is trying to sell municipal bonds to support improvements in recre- ational facilities and highways. The face values (in thousands of dollars) of the bonds and the due dates (years from now) at which principal comes due are listed in the file P14_104.xlsx. An underwriting com- pany (U) wants to underwrite C’s bonds. A proposal to C for underwriting this issue consists of the follow- ing: (1) an interest rate, 3%, 4%, 5%, 6%, or 7%, for each bond, where coupons are paid annually, and (2) an up-front premium paid by U to C. U has determined the set of fair prices (in thousands of dollars) for the bonds listed in the same file. For example, if U under- writes bond 2 maturing two years from now at 5%, it will charge C $444,000 for that bond. U is constrained to use at most three different interest rates. U wants

to make a profit of at least $46,000, where its profit is equal to the sale price of the bonds minus the face value of the bonds minus the premium U pays to C. To maximize the chance that U will get C’s business, U wants to minimize the total cost of the bond issue to C, which is equal to the total interest on the bonds minus the premium paid by U. For example, if bond 1 is issued at a 4% rate, then C must pay two years of coupon interest: 2(0.04)($700,000) 5 $56,000. What assignment of interest rates to each bond and up-front premiums ensure that U will make the desired profit (assuming it gets the contract) and maximize the chance of U getting C’s business? To maximize this chance, you can assume that U minimizes the net cost to C, that is, the cost of its coupon payments minus the premium from U to C.

CASE 14.1 Giant Motor Company This problem deals with strategic planning issues for a large company.21 The main issue is planning the company’s pro- duction capacity for the coming year. At issue is the overall level of capacity and the type of capacity—for example, the degree of flexibility in the manufacturing system. The main tool used to aid the company’s planning process is a mixed integer programming model. A mixed integer program has both integer and continuous variables.

Problem Statement Giant Motor Company (GMC) produces three lines of cars for the domestic (U.S.) market: Lyras, Libras, and Hydras. The Lyra is a relatively inexpensive subcompact car that appeals mainly to first-time car owners and to households using it as a second car for commuting. The Libra is a sporty compact car that is sleeker, faster, and roomier than the Lyra. Without any options, the Libra costs slightly more than the Lyra; additional options increase the price further. The Hydra is the luxury car of the GMC line. It is significantly more expensive than the Lyra and Libra, and it has the high- est profit margin of the three cars.

Retooling Options for Capacity Expansion Currently GMC has three manufacturing plants in the United States. Each plant is dedicated to producing a single line of cars. In its planning for the coming year, GMC is consider- ing the retooling of its Lyra and/or Libra plants. Retooling either plant would represent a major expense for the com- pany. The retooled plants would have significantly increased production capacities. Although having greater fixed costs, the retooled plants would be more efficient and have lower marginal production costs—that is, higher marginal profit contributions. In addition, the retooled plants would be

flexible: They would have the capability of producing more than one line of cars.

The characteristics of the current plants and the retooled plants are given in Table 14.10. The retooled Lyra and Libra plants are prefaced by the word new. The fixed costs and capacities in Table 14.10 are given on an annual basis. A dash in the profit margin section indicates that the plant can- not manufacture that line of car. For example, the new Lyra plant would be capable of producing both Lyras and Libras but not Hydras. The new Libra plant would be capable of producing any of the three lines of cars. Note, however, that the new Libra plant has a slightly lower profit margin for producing Hydras than the Hydra plant does. The flexible new Libra plant is capable of producing the luxury Hydra model but is not quite as efficient as the current Hydra plant that is dedicated to Hydra production.

The fixed costs are annual costs that are incurred by GMC independent of the number of cars that are produced by the plant. For the current plant configurations, the fixed costs include property taxes, insurance, payments on the loan that was taken out to construct the plant, and so on. If a plant is retooled, the fixed costs will include the previous fixed costs plus the additional cost of the renovation. The addi- tional renovation cost will be an annual cost representing the cost of the renovation amortized over a long period.

Demand for GMC Cars Short-term demand forecasts have been very reliable in the past and are expected to be reliable in the future. (Lon- ger-term forecasts are not so accurate.) The demand for GMC cars for the coming year is given in Table 14.11.

A quick comparison of plant capacities and demands in Tables 14.10 and 14.11 indicates that GMC is faced with

21 The idea for this case came from Eppen, Martin, and Schrage, “A Scenario Approach to Capacity Planning.” Operations Research 37, no. 4 (July–August 1989): 517–527.

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insufficient capacity. Partially offsetting the lack of capacity is the phenomenon of demand diversion. If a potential car buyer walks into a GMC dealer showroom wanting to buy a Lyra but the dealer is out of stock, frequently the salesperson can convince the customer to purchase the better Libra car, which is in stock. Unsatisfied demand for the Lyra is said to be diverted to the Libra. Only rarely in this situation can the salesperson convince the customer to switch to the luxury Hydra model.

From past experience GMC estimates that 30% of unsatisfied demand for Lyras is diverted to demand for Libras and 5% to demand for Hydras. Similarly, 10% of unsatisfied demand for Libras is diverted to demand for Hydras. For example, if the demand for Lyras is 1,400,000 cars, then the unsatisfied demand will be 400,000 if no capacity is added. Out of this unsatisfied demand, 120,000 (= 400,000 3 0.3) will materialize as demand for Libras, and 20,000 (= 400,000 3 0.05) will materialize as demand for Hydras. Similarly, if the demand for Libras is

1,220,000 cars (1,100,000 original demand plus 120,000 demand diverted from Lyras), then the unsatisfied demand for Lyras would be 420,000 if no capacity is added. Out of this unsatisfied demand, 42,000 (= 420,000 3 0.1) will materialize as demand for Hydras. All other unsatisfied demand is lost to competitors. The pattern of demand diver- sion is summarized in Table 14.12.

Lyra Libra Hydra New Lyra New Libra

Capacity (in 1000s) 1000 800 900 1600 1800

Fixed cost (in $millions) 2000 2000 2600 3400 3700

Profit Margin by Car Line (in $1000s)

Lyra 2 — — 2.5 2.3

Libra — 3 — 3.0 3.5

Hydra — — 5 — 4.8

Table 14.10 Plant Characteristics

Table 14.11 Demand for GMC Cars

Demand (in 1000s)

Lyra 1400

Libra 1100

Hydra 800

Lyra Libra Hydra

Lyra NA 0.3 0.05

Libra 0 NA 0.10

Hydra 0 0.0 NA

Table 14.12 Demand Diversion Matrix

Questions GMC wants to decide whether to retool the Lyra and Libra plants. In addition, GMC wants to determine its production plan at each plant in the coming year. Based on the previ- ous data, develop a mixed integer programming model (some variables integer-constrained, some not) for solving GMC’s production planning–capacity expansion problem for the coming year. According to the optimal solution, what should GMC do? How sensitive is the optimal solution to key inputs? The file C14_01.xlsx gets you started.

CASE 14.2 GMS Stock Hedging Kate Torelli, a security analyst for LionFund, has identified a gold-mining stock (ticker symbol GMS) as a particularly attractive investment. Torelli believes that the company has invested wisely in new mining equipment. Furthermore, the company has recently purchased mining rights on land that has high potential for successful gold extraction. Torelli notes that gold has underperformed in the stock market for the last decade and believes that the time is ripe for a large increase in gold prices. In addition, she reasons that condi- tions in the global monetary system make it likely that inves- tors may once again turn to gold as a safe haven in which

to park assets. Finally, supply and demand conditions have improved to the point where there could be significant upward pressure on gold prices.

GMS is a highly leveraged company, so it is quite a risky investment by itself. Torelli is mindful of a passage from the annual report of a competitor, Baupost, which has an extraor- dinarily successful investment record: “Baupost has man- aged a decade of consistently profitable results despite, and perhaps in some respect due to, consistent emphasis on the avoidance of downside risk. We have frequently carried both high cash balances and costly market hedges. Our results are

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particularly satisfying when considered in the light of this sustained risk aversion.” She would therefore like to hedge the stock purchase—that is, reduce the risk of an investment in GMS stock.

Currently GMS is trading at $100 per share. Torelli has constructed seven scenarios for the price of GMS stock one month from now. These scenarios and corresponding proba- bilities are shown in Table 14.13.

To hedge an investment in GMS stock, Torelli can invest in other securities whose prices tend to move in the direction opposite to that of GMS stock. In particular, she is consid- ering over-the-counter put options on GMS stock as poten- tial hedging instruments. The value of a put option increases as the price of the underlying stock decreases. For example, consider a put option with a strike price of $100 and a time to expiration of one month. This means that the owner of the put has the right to sell GMS stock at $100 per share one month in the future. Suppose that the price of GMS falls to $80 at that time. Then the holder of the put option can exercise the option and receive $20 (= 100 2 80). If the price of GMS falls to $70, the option would be worth $30 (= 100 2 70). However, if the price of GMS rises to $100 or more, the option expires worthless.

Torelli called an options trader at a large investment bank for quotes. The prices for three European-style put options are shown in Table 14.14. (A European put can be exercised only at the expiration date, not before.) Torelli wishes to invest $10 million in GMS stock and put options.

Questions 1. Based on Torelli’s scenarios, what is the expected return

of GMS stock? What is the standard deviation of the return of GMS stock?

2. After a cursory examination of the put option prices, Torelli suspects that a good strategy is to buy one put option A for each share of GMS stock purchased. What are the mean and standard deviation of return for this strategy?

3. Assuming that Torelli’s goal is to minimize the standard deviation of the portfolio return, what is the optimal port- folio that invests all $10 million? (For simplicity, assume that fractional numbers of stock shares and put options can be purchased. Assume that the amounts invested in each security must be nonnegative. However, the number of options purchased need not equal the number of shares of stock purchased.) What are the expected return and standard deviation of return of this portfolio? How many shares of GMS stock and how many of each put option does this portfolio correspond to?

4. Suppose that short selling is permitted—that is, the nonnegativity restrictions on the portfolio weights are removed. Now what portfolio minimizes the standard deviation of return?

(Hint: A good way to attack this problem is to create a table of security returns, as indicated in Table 14.15, where only a few of the table entries are shown. To correctly calcu- late the standard deviation of portfolio return, you will need to incorporate the scenario probabilities. If ri is the portfolio return in scenario i, and pi is the probability of scenario i, then the standard deviation of portfolio return is

Åa7

i5 1 Pi(ri 2 m)2

where m 5 g 7 i 5 1 piri is the expected portfolio return.)

Table 14.13 Scenarios and Probabilities for GMS Stock in 1 Month

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7

Probability 0.05 0.10 0.20 0.30 0.20 0.10 0.05

GMS stock price($) 150 130 110 100 90 80 70

Table 14.14 Put Option Prices (Today) for GMS Case Study

Put Option A Put Option B Put Option C

Strike Price ($) 90 100 110

Option Price ($) 2.20 6.40 12.50

Table 14.15 Table of Security Returns

GMS Stock Put Option A Put Option B Put Option C

Scenario 1 –100%

2 30% f

7 220%

09953_ch14_ptg01_630-716.indd 716 04/03/19 10:49 PM

Fordham University

Chapter Title: Transpacific Entanglements Chapter Author(s): Yên Lê Espiritu, Lisa Lowe and Lisa Yoneyama

Book Title: Flashpoints for Asian American Studies Book Editor(s): Cathy J. Schlund-Vials Published by: Fordham University. (2018) Stable URL: https://www.jstor.org/stable/j.ctt1xhr6h7.13

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P a r t I I I

Remapping Asia, Recalibrating Asian Amer i ca

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175

In our collaborative contribution, we examine the geopo liti cal, military, and epistemological entanglements between U.S. wars in Asia, U.S. racial capitalism, and U.S. empire and argue that U.S. empire and militarism in Asia and the Pacific Islands have been critical, yet underrecognized, parts of the genealogy of the con temporary condition of U.S. neoliberalism. We emphasize that U.S. neoliberalism mediates itself through the U.S. national security state, which is si mul ta neously a racial state and a settler state; this is expressed not merely in the racialization of Asian and Pacific Islander peoples but significantly in the erasure of historical and ongoing settler colonialism and, furthermore, in a racial social order that si mul ta neously pronounces antiblackness and Islamophobia. In our elaboration of “trans- pacific entanglements,” historical and ongoing settler logics of invasion, re- moval, and seizure continuously articulate with other forms of appropriation and subjugation: This U.S. settler logic intersects with racialized capitalism and overseas empire asserts itself— often through the collaborative networks of the U.S.- backed, patriarchally or ga nized, subimperial Asian “client- states”—in transpacific arrangements such as: export pro cessing zones in the Philippines, U.S. military bases in Okinawa and Guam, nuclear test sites

C h a p t e r 1 0

Transpacific Entanglements

Yên Lê Espiritu, Lisa Lowe, and Lisa Yoneyama

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176 Yên Lê Espiritu, Lisa Lowe, and Lisa Yoneyama

in the Marshall Islands, the exportation of nuclear power plants through- out Asia, the partition of Korea, and the joint military operations that demonstrate and secure the empire’s reach. In this way, we conceptualize U.S. empire in Asia and the Pacific Islands as at once a settler- colonial formation, a racial and sexualized cap i tal ist formation, and a military proj- ect that permits overseas dominance, appropriation, and exploitation. Em- pire works si mul ta neously, yet differentially, to naturalize U.S. presence and possession in Asia and the Pacific Islands through the imperatives of national security and war time necessity, to racialize the peoples it captures, occupies, kills, and governs, and to disavow the historical and ongoing dispossessions of Indigenous peoples.

In Immigrant Acts: On Asian American Cultural Politics, Lisa Lowe situated racialized Asian immigrant labor within the history of U.S. capitalism, expanding its global reach through wars in Asia; she noted that U.S. im- migration exclusion acts and naturalization laws managed and produced Asian American racial formation through both exclusion and inclusion in relation to the history of U.S. wars in Asia. As Yên Lê Espiritu has argued in her recent book Body Counts: The Vietnam War and Militarized Refuge(es), and Lisa Yoneyama makes evident in Cold War Ruins: Transpacific Critique of American Justice and Japa nese War Crimes, American exceptionalism has rationalized U.S. military and cap i tal ist interventions in Asia and the Pa- cific Islands as necessary for the “national security” of the United States and for the humanitarian “rescue” of Asian peoples. The global portrait of the United States as triumphant and humanitarian liberator of Asia, and to a similar extent the Pacific Islands, has buttressed the military buildup against a broad range of Amer i ca’s “enemies,” variously named in twentieth- century history as “fascists” during the Second World War, “underdevel- opment” in the U.S.- modeled modernization proj ects of postcolonial third world countries, “communists” in the Cold War, and “terrorists” in the War on Terror. These links indicate that U.S. militarism and empire in Asia and the Pacific Islands displaces and racializes Asian and Pacific mi- grants and refugees and that the discourse of the United States as libera- tor of Asia is employed in turn to explain the necessary expansion of militarism in the Philippines, Guam, Hawaii, Okinawa, Korea, and Viet- nam and to justify the current wars in Iraq, Af ghan i stan, Syria, and else- where. In that pro cess, moreover, the settler logic of U.S. imperial nation and its “military- security- academic” regime si mul ta neously discipline and regulate the class, race, sexual, and other uneven social relations and iden- tities within and along the national border of the U.S. mainland. It is im- perative, therefore, that any attempts at transforming the American pres ent

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Transpacific Entanglements 177

must take into account the geohistorical ramifications of U.S. settler colo- nialism and its ties to specific transpacific conditions in Asia and the Pacific Islands.

At this moment of reinvigorated U.S. imperialism and globalized mili- tarization, we suggest it is impor tant to interrogate anew the public recol- lections of the U.S. war in Vietnam. By most accounts, Vietnam was the site of one of the most brutal and destructive wars between Western im- perial powers and the people of Asia, Africa, and Latin Amer i ca. U.S. mil- itary policies cost Vietnam at least three million lives, the maiming of countless bodies, the poisoning of its water, land, and air, the razing of its countryside, and the devastation of most of its infrastructure. Indeed, more explosives were dropped on Vietnam, a country two- thirds the size of California, than in all of World War II. Yet post-1975 public discussions of the Vietnam War in the United States often skip over this devastating his- tory. This “skipping over” of the Vietnam War constitutes an or ga nized and strategic forgetting of a controversial and unsuccessful war, enabling Americans to continue to push military intervention as key in Amer i ca’s self- appointed role as liberators— protectors of democracy, liberty, and equality, both at home and abroad.

As a controversial and unsuccessful war, the Vietnam War has the po- tential to unsettle the master narratives of “rescue and liberation” and re- focus attention on the troubling rec ord of U.S. military aggression. And yet, as demonstrated by the recent wars in West and South Asia, the U.S. loss in Vietnam has not curbed the United States’ crusade to remake the world by military force. Instead, the United States appears to have been able to fold the Vietnam War into its list of “good wars.” The narrative of the “good refugee,” deployed by the larger U.S. society and by Viet nam- ese Americans themselves, has been crucial in enabling the United States to turn the Vietnam War into a good war. Other wise absent in U.S. pub- lic discussions of Vietnam, Viet nam ese refugees become most vis i ble and intelligible to Americans as successful, assimilated, and anticommunist newcomers to the American “melting pot.” Represented as the grateful beneficiary of U.S.- style freedom, Viet nam ese in the United States be- come the featured evidence of the appropriateness of the U.S. war in Vietnam: that the war, no matter the cost, was ultimately necessary, just, and successful. Having been deployed to “rescue” the Vietnam War for Americans, Viet nam ese refugees thus constitute a solution rather than a prob lem for the United States, as often argued.

The good war narrative requires the production not only of the good refugee but also of the United States as the good refuge. The making of

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178 Yên Lê Espiritu, Lisa Lowe, and Lisa Yoneyama

the “good refuge” was launched in April 1975 as U.S. media and officials extolled and sensationalized the U.S. airlifting of approximately 130,000 Viet nam ese out of the city in the final days before the Fall of Saigon and subsequently encamping the refugees at military bases throughout the Pacific archipelago. With the Defense Department coordinating transpor- tation and the Joint Chiefs of Staff- Pacific Command in charge of the military moves necessary for the evacuation, Viet nam ese were airlifted from Saigon on U.S. military aircrafts, transferred to U.S. military bases on (neo)colonized spaces such as the Philippines, Guam, Wake Island, and Hawaii, and delivered to yet another set of military bases throughout the United States: Camp Pendleton in California, Fort Chaffee in Arkansas, Eglin Air Force Base in Florida, or Fort Indiantown Gap in Pennsylvania. While these efforts have been widely covered by the media and scholars alike as “rescuing missions,” we need to expose them instead as colonial and militarized ventures.

Moving from one U.S. military base to another, Viet nam ese refugees witnessed firsthand the reach of the U.S. empire in the Asia- Pacific region. Far from confirming U.S. benevolence, the U.S. evacuation of Viet nam ese refugees made vis i ble the legacy of U.S. colonial and military expansion into the Asia Pacific region. The fact that the majority of the first- wave refugees were routed through the Philippines and Guam revealed the lay- ering of U.S. past colonial and ongoing militarization practices on these islands. It was the region’s (neo)colonial dependence on the United States that turned the Philippines and Guam, U.S. former and current colonial territories respectively, into the “ideal” receiving centers of the U.S. res- cuing proj ect; and it was the enormity of the U.S. military buildup in the Pacific that uniquely equipped U.S. bases there to handle the large- scale refugee rescue operation. As such, U.S. evacuation efforts were not a slap- dash response to an emergency situation that arose in Vietnam in 1975 but rather part and parcel of the long- standing militarized histories and cir- cuits that connected Vietnam, the Philippines, and Guam, dating back to 1898.

The U.S. initial designation of Clark Air Force Base as a refugee stag- ing point was intimately linked to, and a direct outcome of, U.S. colonial subordination and militarization of the Philippines. Soon after, when President Ferdinand Marcos refused to accept any more Viet nam ese refu- gees, U.S. officials moved the premier refugee staging area from the Phil- ippines to Guam. As an unincorporated or ga nized territory of the United States under the jurisdiction of the Department of the Interior, Guam— specifically, its U.S. air and naval bases, which took up one- third of the

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Transpacific Entanglements 179

island— became the “logical” transit camps for the pro cessing of evacuees. With total land area of about two hundred square miles, and meager local resources, Guam was hardly an ideal location for the large- scale refugee operation. That it became the major refugee staging point in the Pacific had more to do with the U.S. militarization of Guam than with U.S. hu- manitarianism. The U.S. decision to designate Guam the primary staging ground for refugees, even when the island’s resources were severely stretched and its inhabitants adversely affected, repeats the long- standing belief that indigenous land is essentially “empty land”— that is, land empty of its Indigenous population. The refugee situation on Guam thus bespeaks the intertwined histories of U.S. settler colonialism and U.S. military co- lonialism on Guam and its war in Vietnam: It was the militarization of the colonized island and its Indigenous inhabitants that turned Guam into an “ideal” dumping ground for the unwanted Viet nam ese refugees, the dis- cards of U.S. war in Vietnam. At the same time, as Jana Lipman argues, the refugee presence bore witness not only to the tenacity but also to the limits of U.S. empire, critically juxtaposing “the United States’ nineteenth- century imperial proj ect with its failed Cold War objectives in Southeast Asia.”1

From Guam, many Viet nam ese refugees journeyed to Marine Corps Base Camp Pendleton, a 125,000- acre amphibious training base on the Southern California coast, in San Diego County. It was here, at a military base, that the largest Viet nam ese population outside of Vietnam got its start in the United States. Like Clark and Andersen Air Force Bases, Camp Pendleton emerged out of a history of conquest: It is located in the tradi- tional territory of the Juaneño, Luiseño, and Kumeyaay Tribes, which had been “discovered” by Spanish padres and voyagers who traveled to South- ern California in the late eigh teenth century, “owned” by unscrupulous Anglo- American settlers for about a century as the California state legis- lature repeatedly blocked federal ratification of treaties with Native com- munities, and ultimately “acquired” by the U.S. Marine Corps in 1942 in order to establish a West Coast base for combat training of Marines. Camp Pendleton’s prized land— its varied topography, which combines a breath- takingly beautiful seventeen- mile shoreline and diverse maneuver areas, making it ideal for combat training environment—is thus what Richard Carrico called “stolen land,” an occupied territory like Guam.2

The material and ideological conversion of U.S. military bases into a place of refuge— a place that resolves the refugee crisis, promising peace and protection— discursively transformed the United States from violent aggressors in Vietnam to benevolent rescuers of its people. This “make over”

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180 Yên Lê Espiritu, Lisa Lowe, and Lisa Yoneyama

obscures the violent roles that these military bases— these purported places of refuge— played in the Vietnam War, which spurred the refugee exodus in the first place; the construction of military bases as “refuges” also ob- scures the historical and ongoing settler- colonial occupation of indigenous land, as well as the dispossession and displacement of Indigenous peoples. In the Philippines, from 1965 to 1975, Clark Air Force Base, as the largest overseas U.S. military base in the world, became the major staging base for U.S. involvement in Southeast Asia, providing crucial logistical sup- port for the Vietnam War. In Guam, Andersen Air Force Base played a “legendary” role in the Vietnam War, launching devastating bombing mis- sions over North and South Vietnam for close to a de cade. As Robert Rogers documents, Andersen rapidly became the United States’ largest base for B-52 bombers. In 1972, Andersen was the site of the most massive buildup of airpower in history, with more than fifteen thousand crews and over 150 B-52s lining all available flight line space— about five miles long. At its peak, Andersen housed about 165 B-52s.3 The U.S. air war, launched from Guam, decisively disrupted life on the island, underscoring once again the total disregard for the island’s Indigenous inhabitants. Fi nally, as the Department of Defense’s busiest training installation, California’s Camp Pendleton, the refugees’ first home in the United States, trains more than 40,000 active- duty and 26,000 reserve military personnel each year for combat. Camp Pendleton is also the home base of the illustrious 1st Ma- rine Regiment, whose battalions participated in some of the most fero- cious battles of the war. As such, the Pacific military bases, Clark and Andersen Air Force Bases, and California’s Marine Corps Base Camp Pendleton, credited and valorized for resettling Viet nam ese refugees in 1975, were the very ones responsible for inducing the refugee displacement. The massive tonnage of bombs, along with the ground fighting provided by Marine units like the Camp Pendleton’s 1st Marines, displaced some twelve million people in South Vietnam— almost half the country’s total population at the time— from their homes.

The transvaluation of U.S. military and colonial vio lence into a benev- olent act of rescue, liberation, and rehabilitation finds even deeper geneal- ogy in the racialized constitution of U.S. modernity, humanism, and liberalism, all of which continue to shore up what Richard H. Immerman called the “empire for liberty,” or what Oscar V. Campomanes, following William Appleman Williams, in a similar sense called the “anticolonial empire.”4 Williams famously characterized the United States’ Open Door Policy since the nineteenth century as “Amer i ca’s version of the liberal pol- icy of informal empire or free trade imperialism,” which was at the same

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Transpacific Entanglements 181

time driven by “the benevolent American desire to reform the world in its own image.”5 In other words, the United States developed as a colonizing empire even as it disavowed its histories of colonialism and military take- over through the liberal tenets of freedom, consensus, private property, and self- determination. After the Second World War, the genealogy of liberal empire culminated in the Cold War ascendancy of the United States as a leader of the free world who claimed to have replaced the nineteenth- century colonial order in Asia and the Pacific Islands.

The American image of itself as the benevolent liberator and reformer of the people and land it subjugates is sustained by the discursive force of liberal humanism and humanitarianism for which the notion of “debt” is the operative term. In tracking the emergence of new relations of subju- gation around the idea of freedom in the U.S. post- Reconstruction era, Saidiya V. Hartman argues that the trope of debt was deployed to bind the newly emancipated enslaved peoples to a new system of bondage and in- dentureship. Hartman writes: “Emancipation instituted indebtedness. . . . The emancipated were introduced to the cir cuits of exchange through the figurative deployment of debt. . . . The transition from slavery to freedom introduced the free agent to the cir cuits of exchange through this construc- tion of already accrued debt, an abstinent pres ent, and a mortgaged future. In short, to be free was to be a debtor— that is, obliged and duty- bound to others.”6 Though firmly situated in the specific geohistorical con- text of the failure of American Reconstruction, Hartman’s observation of the “figurative deployment of debt” lays out the constitutive contradiction of post- Enlightenment humanism, helping us understand how the Ameri- can notions of freedom, emancipation, liberty, and property owner ship have developed by producing as well as disciplining modernity’s others. This contradiction fundamental to the modern ideals of liberty and freed subjects underpins what we also consider to function as the American myth of rescue, liberation, and rehabilitation, a discursive economy of geopoli- tics that consolidated the racialized and heterosexualized logic of subjuga- tion and bondage in the U.S. relationship to Asia throughout the twentieth century.

The American myth of rescue, liberation, and rehabilitation then leaves the indelible marks on the liberated of inferiority, subordination, and be- latedness (to freedom, democracy, and property) but also of indebtedness. It assigns “the already accrued debt” to the liberated. Once marked as “the liberated” and therefore “the indebted,” one cannot easily enter into a re- ciprocal relationship with the liberators. This myth, which pres ents both vio lence and liberation as “gifts for the liberated,” has serious implications

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182 Yên Lê Espiritu, Lisa Lowe, and Lisa Yoneyama

for the redressability of U.S. military vio lence, as Yoneyama observes of the American aggression in the Iraq War.7 According to this myth, the losses and damages brought on by U.S. military vio lence are deemed “pre- paid debts” incurred by those liberated by American intervention. The injured and violated bodies of the liberated do not seem to require redress according to this discourse of indebtedness, for their liberation has already served as the payment/reparation that supposedly precedes the vio lence inflicted upon them. Furthermore, American myths of rescue, liberation, and rehabilitation have seriously threatened the way that postcolonial sov- ereignty might be conceived. As Chungmoo Choi notes, the dominant historiography of U.S.- Korea relations posits the post– World War II “liberation as a gift of the allied forces, especially of U.S.A.”8 This dele- gitimized the agency of Koreans in nation- building in the aftermath of Japa nese colonial rule, which was followed immediately by the Korean War— the “forgotten” war that consolidated the military- security- industrial- academic complex— and the ongoing Cold War partition. Mimi Nguyen similarly discusses how the U.S. “gift of freedom” to refugees of the war in Vietnam, imposes indebtedness while binding them to the colonial histories that deemed them “unfree.”9

The predicament of U.S. bases in Okinawa urgently demonstrates the intersection of the settler logic of invasion, removal, and seizure and the American imperialist myth of rescue, liberation, and rehabilitation. As a number of critics remind us, there is not one piece of land in Okinawa’s main island that was willingly offered to the U.S. military for use. And yet, under the U.S.- Japan Security Treaty, 75 percent of the entire U.S. base facilities in Japan is accommodated in Okinawa prefecture, which consti- tutes less than 1 percent of Japa nese soil. Okinawa’s subjection to Japan began in the seventeenth century when the Ryukyu kingdom came under the rule of the Shimazu domain, followed by its integration as a prefecture into the modern Meiji state that was then emerging into the nineteenth century international and interimperial order. After the Second World War, the United States “liberated” Okinawa from the many centuries of Japa nese rule. At the same time, the United States insisted on the posses- sion of Okinawa in exchange for the heavy American military sacrifice in the Pacific battlefronts. Okinawa thus remained under U.S. military oc- cupation for nearly three de cades, until its “reversion” to Japan (1945– 1972). During the occupation, the United States formally recognized Japan’s “residual sovereignty” over Okinawa, thereby pre- empting any Okinawan Indigenous claim to local sovereignty. It also insisted that the United States would maintain control over Okinawa until its eventual

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Transpacific Entanglements 183

approval as a U.N. trusteeship under U.S. authority. As Toriyama Atsushi points out, the two concepts, “residual sovereignty” and “pending” trustee- ship, allowed the United States’ exclusive management of Okinawa while at the same time helped to contain the Indigenous demands for self- determination and Okinawa’s in de pen dence. What is impor tant to our discussion is that the American recognition of Japan’s “residual sover- eignty” rationalized the United States’ de facto control over the Okinawan land, the ocean, and the airspace, without contradicting its claims to be the supreme leader of freedom and liberty in the postcolonial world.10 The on- going settler logics of invasion, removal, and seizure thus sustains and is in turn secured by the historically rooted and still power ful exceptionalist discourse of the U.S. liberal empire. Yet, it is equally impor tant to remem- ber that this pro cess is far from complete. As Yoneyama argues in her ear- lier work on the politics of memory inspired by Walter Benjamin, what we hope to offer below is a method of unlearning the “universal history”— Marxist, liberal, or other wise—in such a way that would enable us to dis- cern and connect the “missed opportunities and unfulfilled promises in history, as well as unrealized events that might have led to a dif fer ent pres- ent.”11 Such a method calls upon us to link apparently separate subjects, con- texts, and issues whose connections have been rendered unavailable by existing geo graph i cal, po liti cal, and disciplinary bound aries.

Although American exceptionalism, humanitarianism, and national se- curity with re spect to Asia are more often understood as discourses that address international relations, we wish to emphasize that they are critical parts of a neoliberal racial social ordering within the United States as well, which naturalizes white settlement and perpetuates assaults on Black, Muslim, and poor communities of color. In the nineteenth century until World War II, Chinese, Japa nese, Filipino, and South Asians were re- cruited as racialized noncitizen labor and barred from the po liti cal and cultural spheres in the development of the national economy; today, the racialization of Asian immigrants within the United States is no longer ex- clusively that of noncitizen immigrant labor. The U.S. state responded to challenges by radical social movements of the 1960s and decolonization movements worldwide with aggressive incorporative mea sures, from the 1964 Civil Rights Acts and the 1965 Immigration and Nationality Act, to the large- scale capital investment in the development and rapid “integra- tion” of Asia into global capitalism. Within these conditions, the racial- ization of Asian Americans as the model minority has been crucial to the pernicious fictions of multiculturalism and “postracial” inclusion, just as modernization discourse has in an analogous fashion “racialized” the newly

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184 Yên Lê Espiritu, Lisa Lowe, and Lisa Yoneyama

industrialized Asian nations to be what Takashi Fujitani calls “model” mo- dernity nations.12 Sociologist Hijin Park discusses this neoliberal position- ing of the “East Asian miracle” as the discourse of “rising Asia.”13 The “rising Asia” discourse opposes newly industrialized Asia to less “devel- oped” countries in Africa, Latin Amer i ca, Central Asia; it is comple- mented by the fiction of Asian “culture” as traditionally patriarchal and heteronormative, a fiction employed, as Chandan Reddy argues, to disci- pline black, brown, and queer of color communities, as “backward” and threatening.14

These new racial formations signal forms of governance that represent “Asians” as included in both the U.S. nation and in global capitalism, even as Asian and Pacific Islander peoples are forcibly dislocated from sites in which the United States has occupied, interned, and conducted imperial war. In other words, violent exclusion from national belonging has char- acterized the historical emergence of the United States from its beginnings: justifying the dispossession of Indigenous peoples, the enslavement of Africans and Jim Crow segregation, the stolen labors of indentured and immigrant workers, and the losses of life as the United States waged wars in Latin Amer i ca, East, Northeast, and Southeast Asia, Central Asia, and West Asia. Yet within the formation we call “neoliberalism,” vio lence accompanies not only exclusion from the nation but also inclusion into it: as “emancipated” slave, indebted poor, grateful immigrant, or rehabilitated inmate. The vio lence of inclusion, an operation that proposes to convert subjugated others into normative humanity and multicultural citizenship, is clearly also a pro cess of racial governance. Yên Espiritu elaborates this concept as “differential inclusion,” to counter the myth of “voluntary” im- migration and to emphasize the pro cesses through which dif fer ent groups are “included” yet si mul ta neously legally subordinated, eco nom ically ex- ploited, or culturally degraded, often in relation to one another.15 The rela- tive inclusions of Asians and Pacific Islander peoples by postwar racial liberalism, which appeared to legitimize an official antiracism, provided the United States moral legitimacy as it sought to gain Cold War hege- mony against the Soviet Union.16 It built upon the fiction of the U.S. state as the guarantor of civil rights and suppressed radical challenges— from Black Power to third world decolonization movements— even as it ex- tended U.S. militarism and neoliberal capitalism globally.

Con temporary neoliberalism naturalizes the market, buttresses the pri- macy of the deserving individual, and instantiates the superiority of the private over the public; it moreover affirms “colorblindness” that disavows the per sis tence of antiblackness, promotes a discourse of “rising Asia” that

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masks the longer history of North Atlantic empires, and it denies ongoing settler colonialism while justifying imperial war and military basing that occupies, dispossesses, and displaces Indigenous communities. Neoliber- alism erases the longer history in which “emancipation,” “ free wage labor,” and “ free trade” have enabled the development of what Cedric Robinson calls “racial capitalism”— built upon settler colonialism, slavery, indenture, and unfree immigrant labor.17 Grace Hong characterizes the current mo- ment of neoliberal restructuring as an epistemological structure of dis- avowal, a means of claiming that racial and gendered vio lence has been overcome, while disavowing the continuing assaults on inassimilable poor communities of color.18 We suggest that where many theorists of neolib- eralism, whether inspired by Marx, Weber, or Foucault, observe the de- fining feature of neoliberalism to be the generalization of the market logic of exchange to all spheres of human life, this definition of neoliberalism of- ten leaves unchallenged such disavowal, and we argue that the critique of neoliberalism must be nuanced in relation to the much longer histories of racial capitalism, colonialism, and militarism.19 In light of the commod- ification of human life within slavery, colonialism, as well as in con temporary global capitalism, we insist that what is currently theorized as a “new” fi- nancialization of human life under neoliberalism occurred brutally and routinely and continues to occur throughout the course of modern em- pires. Indeed, what Lowe discusses elsewhere as the “colonial division of humanity” is a signature feature of long- standing liberal modes of distinc- tion that privilege par tic u lar lives as “ human” and treat other lives as the laboring, replaceable, or disposable contexts that constitute that “human- ity.”20 In their recent work on neoliberal forms of “surrogate humanity”— whether gestational surrogacy in India, life extension through medical technologies, or the advanced automation of labor by Amazon robotics— Neda Atanasoski and Kalindi Vora argue that even “post- human” con- cepts and practices continue to reproduce racial, gender, and geopo liti cal hierarchies of social difference.21

The national security imperatives developed during the Cold War have been critical to the neoliberal reor ga ni za tion of the state and economy, with consequences for the domestic racial order, which involves the withdrawal of the state from its traditional purpose of social welfare to investment in the repressive arms of the U.S. state: policing, drug enforcement, immi- gration and border patrol, and prisons— all of which racialize and divide in the distribution of punishments. The privatization of traditional public “goods,” such as schooling, health care, and social securities further con- tributes to the precarity of poor communities of color, while the buildup

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186 Yên Lê Espiritu, Lisa Lowe, and Lisa Yoneyama

of policing, detention, and prisons targets and suppresses U.S. black, brown, and Muslim communities. The links between U.S. militarism abroad and policing at home is nowhere more evident than in the Israeli training of police serving in Ferguson, Missouri, who suppressed the community pro- testing Michael Brown’s death with military surplus equipment such as aircraft, armored vehicles, assault rifles, grenades, vests, and tear gas.22 Re- becca Bohrman and Naomi Murakawa ironically term this refortification of the repressive arms of the state the “remaking big government,” while Ruth Wilson Gilmore calls this a transformation from the “welfare state” to the “warfare state.”23 The national security discourse exonerates the neoliberal state for its violent antiblackness and Islamophobia.

The “Asian American” has often been made to serve as the sign that me- diates this disavowal of the racial ordering called “postraciality” or “col- orblindness,” the deployment of which governs this neoliberal restructuring of culture, society, economy, and politics. Whether expressed through Asian immigrant exclusion or inclusion, the U.S. state has produced itself as a global power through the formation of the “Asian American” as a means to resolve the contradictions of the U.S. racial capitalism and its im- perial military proj ect. Yet we wish to emphasize that for this reason, now, as before, the “Asian American” is a critical mediating figure for diagnosing racial power and geopo liti cal ordering. Rather than stabilizing binary categories of white/nonwhite, settler/native, or developed/underdeveloped, the “Asian American” continues still to mediate racial relationality or to triangulate these terms. Asian racial formation continues to shed light on how race is not fixed or essential but a shifting designation within what Stuart Hall terms society “structured- in- dominance.”24 Asian racial formations—as noncitizen labor, as model minority, as threatening cap i- tal ist rival— devalue Asians and mea sure and mediate geopo liti cal and na- tional transformations. In our current moment, Asianness as “model minority/model modernity” is an index of the pres ent U.S. settler racial order that reduces and dehumanizes blackness as surplus population, con- structs Indigenous peoples as extinct or vanishing, and frames Muslims as threatening vio lence.

In conclusion, we conceive U.S. racial empire as an open pro cess, an unfolding of dynamic, multiple relations of rule and re sis tance. Alyosha Goldstein discusses U.S. colonialism as “a continuously failing, perpetu- ally incomplete, proj ect that labors to find a workable means of resolution to sustain its logic of possession by disavowing ongoing contestation.”25 Like Goldstein, we understand U.S. racial empire to be continuously fail- ing and incomplete, which suggests that its destruction, exploitation, and

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Transpacific Entanglements 187

administration is never total, that its cruelty and cunning produces con- tingency and excess that we cannot entirely know or anticipate. In Body Counts, Espiritu frames the possibility of anti- imperial collaborations through critical juxtaposition, which she elaborates by interpreting the ge- ography of Viet nam ese refugee flight in 1975: The refugees were taken by military aircraft and routed through U.S. bases, from Vietnam to the Phil- ippines to Guam and then to California. This itinerary connects Viet- nam ese postwar displacement to that of settler colonialism and militarism that has dispossessed Filipino, Chamorro, and Native American peoples and gives rise to an antimilitarist association of incommensurable yet linked groups. This is what Yoneyama describes, in Cold War Ruins, as transpa- cific cultures of reckoning and redress in the face of unredressable war and vio lence. The related, yet often illegible, strug gles against these inter- locking formations are the pos si ble po liti cal itineraries that Lowe evokes in Intimacies as “connections that could have been, but were lost, and are thus, not yet.”26 Thus, in our piece, we take a comparative relational ap- proach and employ the method of critical juxtaposition— the deliberate bringing together of seemingly dif fer ent historical events in an effort to reveal what would other wise remain invisible—in this case, to examine the contours, contents, and limits of U.S. imperialism, wars, and genocide in the Asia- Pacific region and racialization on the U.S. mainland. We gesture toward, and call into being, what Yoneyama calls the competing “yearnings” for “justice beyond judicialization” across unlike, asymmetri- cal, yet linked terrains.27 We can and must always make “Asia” and “Asian American” signify more than American exceptionalism.

Notes

1. Jana K. Lipman, “ ‘Give Us a Ship’: The Viet nam ese Repatriate Move- ment on Guam, 1975,” American Quarterly 64 (2012): 3–4 (emphasis added). 2. Richard Carrico, Strangers in a Stolen Land: Indians of San Diego County from Prehistory to the New Deal (Newcastle, Calif.: Sierra Oaks, 1987). 3. Robert F. Rogers, Destiny’s Landfall: A History of Guam (Honolulu: University of Hawaii Press, 1995). 4. Oscar V. Campomanes, “1898 and the Nature of the New Empire,” Radical History Review 73 (Winter 1999): 130–46. See also Richard H. Immerman, Empire for Liberty: A History of American Imperialism from Benjamin Franklin to Paul Wol fo witz (Prince ton, N.J.: Prince ton University Press, 2010). 5. William Appleman Williams, The Tragedy of American Diplomacy (Cleveland, Ohio: World Publishing, 1959), 67, 47.

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188 Yên Lê Espiritu, Lisa Lowe, and Lisa Yoneyama

6. Saidiya V. Hartman, Scenes of Subjection: Terror, Slavery, and Self- Making in Nineteenth- Century Amer i ca (Oxford: Oxford University Press, 1997), 131. 7. Lisa Yoneyama, “Traveling Memories, Contagious Justice: Ameri- canization of Japa nese War Crimes at the End of the Post- Cold War,” Journal of Asian American Studies (February 2003): 57–63. 8. Chungmoo Choi, “The Discourse of Decolonization and Popu lar Memory: South Korea,” Positions: East Asia Cultures Critique 1, no. 1 (1993): 80. 9. Mimi Nguyen, The Gift of Freedom: War, Debt, and Other Refugee Passages (Durham, N.C.: Duke University Press, 2014). 10. Atsushi Toriyama, “Okinawa’s ‘Postwar’: Some Observations on the Formation of American Military Bases in the Aftermath of Terrestrial Warfare,” trans. David Buist, in The Inter- Asia Cultural Studies Reader, ed. Kuan Hsing Chen and Beng H. Chua, 267–88 (London: Routledge, 2007), 273. 11. Lisa Yoneyama, Hiroshima Traces: Time, Space and the Dialectics of Memory (Berkeley: University of California Press, 1999), 19. 12. Takashi Fujitani, Race for Empire: Koreans as Japa nese and Japa nese as Americans during World War II (Berkeley: University of California Press, 2011). 13. Hijin Park, “Neoliberalizing Differential Racialization: Asian Uplift, Indigenous Death, and Blackness as Surplus,” pre sen ta tion, McMaster University, October 2012. 14. Chandan Reddy, “Asian Diasporas, Neoliberalism, and Family: Reviewing the Case for Homosexual Asylum in the Context of Family Rights,” Social Text 84–85, vol. 23, nos. 3–4 (Fall– Winter 2005): 101–19. 15. Yên Lê Espiritu, Home Bound: Filipino Lives across Cultures (Berkeley: University of California Press, 2003), 47–48. 16. On “racial liberalism” and the postwar incorporation of racialized minorities, see especially Jodi Melamed, Represent and Destroy: Rationalizing Vio lence in the New Racial Capitalism (Minneapolis: University of Minnesota Press, 2011). 17. Cedric Robinson, Black Marxism: The Making of the Black Radical Tradition (Chapel Hill: University of North Carolina Press, 1983). 18. Grace Kyungwon Hong, Death beyond Disavowal: The Impossible Politics of Difference (Minneapolis: University of Minnesota Press, 2015). 19. See for example, Thomas Lemke (“The Birth of Bio- politics: Michel Foucault’s Lectures at the Collège de France on Neo- liberal Governmental- ity,” Economy and Society 30, no. 2 [May 2001]: 190–207), who argues that neoliberalism is characterized by the collapse of the distinction between liberal economy and liberal governance and the withdrawal of the state in favor of the apotheosis of the economy such that neoliberal subjects are

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Transpacific Entanglements 189

controlled precisely through their “freedom.” Wendy Brown explains that neoliberalism involves “extending and disseminating market values to all institutions and social action”; “all dimensions of human life are cast in terms of a market rationality”; and the state ceases to be “the Hegelian constitutional state conceived as the universal repre sen ta tion of the people.” Wendy Brown, “Neo- liberalism and the End of Liberal Democracy,” in Edgework: Critical Essays on Knowledge and Politics (Prince ton, N.J.: Prince ton University Press, 2005), 40–41. See also David Harvey, A Brief History of Neoliberalism (Oxford: Oxford University Press, 2005). 20. Lisa Lowe, The Intimacies of Four Continents (Durham, N.C.: Duke University Press, 2015), 107. 21. Kalindi Vora and Neda Atanasoski, “Surrogate Humanity: Posthu- man Networks and the (Racialized) Obsolescence of Labor,” Catalyst: Feminism, Theory, Technoscience 1, no. 1 (2015): 1–40. 22. Kristian Davis Bailey, “The Ferguson/Palestine Connection,” Ebony, August 19, 2014, http:// www . ebony . com / news - views / the - fergusonpalestine - connection - 403; Tom Giratikanon, Alicia Parlapiano, and Jeremy White, “Mapping the Spread of the Military’s Surplus Gear,” New York Times, August 15, 2014, http:// www . nytimes . com / interactive / 2014 / 08 / 15 / us / surplus - military - equipment - map . html. 23. Rebecca Bohrman and Naomi Murakawa, “Remaking Big Govern- ment,” in Global Lockdown: Race, Gender, and the Prison- Industrial Complex, ed. Julia Sudbury (New York: Routledge, 2005); Ruth Wilson Gilmore, “Global- ization and U.S. Prison Growth: From Military Keynesianism to Post- Keynesian Militarism,” Race & Class 40, nos. 2–3 (1998–1999): 171–88. 24. Stuart Hall, “Race, Articulation, and Socie ties Structured in Domi- nance,” in Black British Cultural Studies, ed. Houston A. Baker and Manthia Diawara (Chicago: University of Chicago Press, 1996). 25. Alyosha Goldstein, ed., Formations of United States Colonialism (Durham, N.C.: Duke University Press, 2013), 3. 26. Lowe, Intimacies of Four Continents, 174. 27. Yoneyama, Cold War Ruins, 205.

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The Prison Abolitionist Imagination:

A Conversation

[An outstretched hand offers you a wisp of hair]

and yet, I cannot give you what outruns us both:

this text

which you will lose, as all are lost

This I know: what I cannot lay claim to

[the joy of a power

that rises and returns,

which no one owns,

because it cannot be appropriated]1

I will these words be with you

as a connective tissue

conjoining [us]:

unassailable creatures

endlessly in process

searching for the tiny miracle

of encountering each other here …

The Prison Abolitionist Imagination / 297

The late Mark Fisher once famously said that it’s easier to imagine the end of the world than it is to imagine the end of capitalism. The same could be said about prisons: it is easier to imagine the end of the world than it is to imagine a world without prisons. And yet the modern prison as it currently exists in the United States is a fairly recent inven- tion. Although penological debates about competing systems of punishment and rehabilitation raged in the North in the early nineteenth century, by the end of the Civil War, physical penitentiaries were uncommon in some frontier states. Florida— which now has one of the largest prison systems in the U.S.—had no physical penitentiaries at the end of the Civil War and had to create its penal system from scratch.

Yet at this historical juncture prisons have become thoroughly naturalized. Imagining and working toward a world without prisons—which is the project of prison abolition—would not only require us to fundamentally rethink the role of the state in society, but it would also require us to work toward the total transformation of all social rela- tions. A project as lofty and ambitious as this is easy to dismiss as unrealistic, utopian, impractical, naive—an unrealizable dream. But what if— instead of reacting to these charges with counter- arguments that persuasively demonstrate that the abolitionist position is the only sensible position—

298 / Carceral Capitalism

we instead strategically use these charges them- selves as a point of departure to show how the prison itself is a problem for thought that can only be unthought using a mode of thinking that does not capitulate to the realism of the Present? Can the re- enchantment of the world be an instrument that we use to shatter the realism of the prison?

What follows is a series of questions—conversa- tions with revolutionaries, dead and alive, on death, dreams, the struggle, and the phenomeno- logical experience of freedom.

There are moments I want to enter. Will you follow me there, to the place where the breathing walls quietly exhale a low freedom song?

* Inside a dark cell, the revolutionary Rosa Luxemburg retreats into her mind. Outside, World War I is raging. “We’re in a tomb.” Outside, peo- ple are creating memories. Inside, she relives old ones. While everyone sleeps, she incubates a secret—journeys to the place where the mystery is audible. As a guard stands watch over the night, she sees beyond the walls into a flowering meadow she once knew, or only knew, in a dream. From where does this small song emanate? If only …

If only we knew how to listen properly, and to brandish our incandescence to the lie that is a lock.

The Prison Abolitionist Imagination / 299

A DOZEN ROSES VS. THE POLICE STATE

In the hours after [Mike] Brown’s body was finally moved, residents erected a makeshift memorial of teddy bears and memorabilia on the spot where police had left his body. When the police arrived with a canine unit, one officer let a dog urinate on the memorial. Later, when Brown’s mother, Lesley McSpadden, laid out rose petals in the form of his initials, a police cruiser whizzed by, crushing the memorial and scattering the flowers. The next evening, McSpadden and other friends and family went back to the memorial site and laid down a dozen roses. Again, a police cruiser came through and destroyed the flowers. Later that night, the uprising began. —Keeanga-Yamahtta Taylor, From #BlackLivesMatter to Black Liberation 2

I think about how the people gathered after Mike Brown was killed—how they made a makeshift memorial on the bloodstained spot in the road where he had been murdered by the police state. What do I see in this encounter? The will of the people butting up against the police’s desire to destroy—to crush all public expressions of grief. The police’s show of force is unnecessary, compensatory. They want us to believe that police cars will always crush rose petals. They tell themselves that their uniform and the power that backs it makes them invulnerable—not like the rose petals arranged in

300 / Carceral Capitalism

the shape of MB. They tried to erase the name “Mike Brown,” but it will forever be seared into the minds of the people of Ferguson. Erase the memo- rials, erase the flowers—the people will still rise up.

That night, an uprising bloomed out of the ground where the memorial flowers had been crushed.

* I once read an article about the dreams of dying people. There was a former cop who couldn’t stop having nightmares about the people he had violated. He told a hospice nurse that on the job he had “done bad stuff.” Tormented by his dreams, he gets “stabbed, shot, or can’t breathe.”3

Eric Garner’s last “I can’t breathe” circles in time to haunt the officers who take the air out of the world. The cop died with so much regret.

The conscious mind of the police officer may be

sure of its correctness,

but the unconscious mind knows it has done terrible

things.

The trampling of the memorial flowers is an act of

repression.

But whatever you try to blot out and refuse to

integrate

returns with greater vigor.

If I ever met the officers, I would tell them:

The Prison Abolitionist Imagination / 301

Before you die you will encounter

the lives you took and violated.

You, driving around in your

steel-enclosed fantasy of invincibility.

You who must desecrate memorials

to prove to yourself you are strong—

to hide this weakness of imagination:

a police cruiser scattering rose petals.

What was it you tried to crush there—

was it a way to blot out awareness

of your own death?

And yet every time you tried to destroy the memorial

the people returned, with objects that bore

the memory of Mike Brown.

You tried to force the people of Ferguson to forget.

The people returned

with a will to carry

the memory into the streets.

* Yesterday I saw a tweet that said: Remember: We lost in Ferguson. We lost in Standing Rock.

Over and over again, the ecstatic moment of revolt was met with repression even greater than what we had anticipated.

The fissure was not a place where we could live. We could not hold on to the new social forms we invented in the process of revolt. The establishment

302 / Carceral Capitalism

leaders were sent to neutralize the protesters. We were told to go home. We failed to make the revo- lution our permanent home.

But the spark is kept alive, underground, waiting for the right conditions.

The specter of Attica The specter of Wounded Knee The specter of Ferguson The specter of Harpers Ferry The specter of Haiti

THE PRISON IS OUR SHADOW

Neither a prisoner nor a free man, because prison is density. No one has spent a night in it without spending the whole night rubbing the muscles of freedom, sore from loitering so frequently on sidewalks, exposed, naked, and hungry. Here you are embracing it from every side, free and liberated from the burden of proof. How small it is, how simple, and so swift to respond to the agility of a mirage. It is in you, within reach of the hand with which you knock at the walls of the cell. It is in you, borrowing the bird’s example, in the falling of rain, the blowing of winds, the laughter of light upon a forgotten rock, in the pride of a beggar who reprimands his benefactors when they are stingy, in an unequal dialogue with your jailer when you say to him:

The Prison Abolitionist Imagination / 303

You, not I, are the loser. He who lives on depriving others of light drowns in the darkness of his own shadow. You will never be free of me unless my freedom is generous to a fault. Then it would teach you peace and guide you home. You, not I, are afraid of what the cell is doing to me. You who guard my sleep, dream, and a delirium mined with signs. I have the vision and you have the tower, the heavy key chain, and a gun trained on a ghost. I have sleepiness, with its silky touch and essence. You have to stay up watching over me lest sleepiness take the weapon from your hand before your eye can see it. Dreaming is my profession while yours is pointless eavesdropping on an unfriendly conversation between my freedom and me. —Mahmoud Darwish, In the Presence of Absence4

Although the guard may gloat his psyche is harassed by the glut of ghosts who bark and moan beneath the light of the moon.

The poet-prisoner haunts the guard, who becomes a prisoner of his paranoia. The profession of the poet is dreaming. The profession of the jailer is to contain. The poet is the one who makes the light. The guard is the one who takes it. He who lives on depriving others of light drowns in the darkness of his own shadow. Will the ones who built the nightmare also drown in it?

304 / Carceral Capitalism

The prisoner knows the true meaning of freedom while the guard knows only how to police this freedom.

What does the jailer give up when he becomes an instrument of the state?

Does the jailer remember what it means to love, to grieve, to rub the muscles of freedom or borrow the bird’s example?

They cannot annihilate what we carry in our hearts and minds: This vision of an elsewhere, or the memory of a bird. How many poets and revolu- tionaries discovered freedom in a cell?

ENTOMBED FLOWERS

Yesterday I lay awake for a long time—these days I can’t fall asleep before 1 a.m., but I have to go to bed at 10, because the light goes out then, and then I dream to myself about various things in the dark. Last night this is what I was thinking: how odd it is that I’m constantly in a joyful state of exaltation—without any particular reason. For example, I’m lying here in a dark cell on a stone-hard mattress, the usual silence of a church ceme- tery prevails in the prison building, it seems as though we’re in a tomb; on the ceiling can be seen reflections

The Prison Abolitionist Imagination / 305

coming through the window from the lanterns that burn all night in front of the prison. From time to time one hears, but only in quite a muffled way, the distant rumbling of a train passing by or quite nearby under the windows the whispering of the guards on duty at night, who take a few steps slowly in their heavy boots to relieve their stiff legs. The sand crunches so hopelessly under their heels that the entire hopeless wasteland of existence can be heard in this damp, dark night. I lie there quietly, alone, wrapped in these many-layered black veils of darkness, boredom, lack of freedom, and winter—and at the same time my heart is racing with an incomprehensible, unfamiliar inner joy as though I were walking across a flowering meadow in radiant sunshine. And in the dark I smile at life, as if I knew some sort of magical secret that gives the lie to every- thing evil and sad and changes it into pure light and happiness. And all the while I’m searching within myself for some reason for this joy, I find nothing and must smile to myself again—and laugh at myself. I believe that the secret is nothing other than life itself; the deep darkness of night is so beautiful and as soft as velvet, if one only looks at it the right way; and in the crunching of the damp sand beneath the slow, heavy steps of the sentries a beautiful small song of life is being sung—if one only knows how to listen properly. At such moments I think of you and I would like so much to pass on this magical key to you, so that always and in all situations you would be aware of the beautiful and

306 / Carceral Capitalism

the joyful, so that you too would live in a joyful euphoria as though you were walking across a multi-colored meadow. I am certainly not thinking of foisting off on you some sort of asceticism or made-up joys. I don’t begrudge you all the real joys of the senses that you might wish for yourself. In addition, I would only like to pass on to you my inexhaustible inner cheerfulness, so that I could be at peace about you and not worry, so that you could go through life wearing a cloak covered with stars, which would protect you against everything petty and trivial and everything that might cause alarm. —Rosa Luxemburg (To Sophie Liebknecht, Breslau, before December 24, 1917)5

In the dark of the night you traveled to a colorful meadow, and with your powerful imagination wove that meadow into a cloak of stars that you imparted to your comrade Sophie—to wear as a shield against everything terrible. What bloomed in your mind that night as you lay quietly listening to the boots of the sentries crunch the sand? You were sharpening your perceptive faculties so you could tune in to the exalted frequency. You were sensitized by your cell, by the boredom weighing you down, until the pres- sure of the darkness gave way to an understanding of the deepest mysteries of what it means to be alive— of the connection between desire and politics.

I think of your fate, of George Jackson’s fate, of Fred Hampton’s fate—the state must know when

The Prison Abolitionist Imagination / 307

the universe gives birth to a true revolutionary—it must see in them a light it must extinguish, lest their spark find and set alight the divine spark in us all, which would spread until the world as we know it has been upended.

Alone in your cell, your body became pure nerve. You were perceiving everything. It made you giddy, the inner joy you felt against the bleak back- drop of the Breslau prison.

I imagine how you passed the time there— studying political economy and botany, writing letters to your comrades, assembling your herbaria, preparing for the revolution, getting lost in the flowers of your imagination.

You were the secret. You were the principle of life itself. You were a tree they had to cut down.

to unspeakable wonder to freedom that blooms on stumps

—Édouard Glissant 6

THE STARS SEEN FROM PRISON

In September 1971 the prisoners of Attica rose up, took the prison, and carved out a small space of freedom: a temporary liberated zone from which they could observe the stars.

308 / Carceral Capitalism

Despite the sense of foreboding, there were moments of levity and, for some, even a feeling of unexpected joy as men who hadn’t felt the fresh air of night for years reveled in this strange freedom. Out in the dark, music could be heard—“drums, a guitar, vibes, flute, sax, [that] the brothers were playing.” This was the lightest many of the men had felt since being processed into the maximum security facility. That night was in fact a deeply emotional time for all of them. Richard Clark watched in amazement as men embraced each other, and he saw one man break down into tears because it had been so long since he had been “allowed to get close to someone.” Carlos Roche watched as tears of elation ran down the withered face of his friend “Owl,” an old man who had been locked up for decades. “You know,” Owl said in wonderment, “I haven’t seen the stars in twenty-two years.” As Clark later described this first night of the rebellion, while there was much trepidation about what might occur next, the men in D Yard also felt wonderful, because “no matter what happened later on, they couldn’t take this night away from us.” —Heather Ann Thompson, Blood in the Water: The Attica Prison Uprising of 1971 and Its Legacy7

In the cracks of the prison, something bloomed. A field of wildflowers imposed on a night sky. Blood was coming. Joy and dread mingled there, infusing the air with a powerful sense of rapture and uncertainty.

The Prison Abolitionist Imagination / 309

What exalted frequency was discovered that night, then lost, when Governor Nelson Rockefeller ordered the police to put down the uprising?

Blood was coming. The new world never arrived. How terrible it must have been for W. E. B. Du Bois to realize he had mistaken dusk for dawn, that darkness would follow and not the radiance of a new day—his people’s strivings rendered crepus- cular. The dream of liberation collapsed in a heap of bloodstained rubble.

Blood was coming. The drumming would not last. The prisoners would be punished for daring to glimpse the stars.

Will those who have constructed this Hell ever wonder—What was it all for? The subordination of all life to these systems that hem us in. Why cover the sky?

The Atacama Desert in Chile is so dry that dead bodies are preserved for posterity, and traces of ancient human communities remain unscathed, as though immortalized in amber. Because of its high elevation and lack of moisture, the skies above the Atacama Desert are completely clear, allowing for an unobstructed view of the stars. Over the years, scien- tists and astronomers have converged on this region to build powerful telescopes to observe the cosmos.

Years after Attica, on another continent, politi- cal prisoners banished to the Atacama Desert by

310 / Carceral Capitalism

the U.S.-sponsored dictator Augusto Pinochet were observing the same stars from the confines of a prison camp.

Patricio Guzmán [documentary director]: What did you feel watching the stars whilst in prison?

Luís Henríquez [Chacabuco concentration camp survivor]: We all had a feeling ... … of great freedom. Observing the sky and the stars, marveling at the constellations, ... we felt completely free.

The military banned the astronomy lessons. They were convinced that the prisoners could escape ... ... guided by the constellations.

Guzmán: Luis’s dignity lies in his memory. He wasn’t able to escape, but, by communicating with the stars, he managed to preserve his inner freedom. —Nostalgia for the Light, 20108

The Prison Abolitionist Imagination / 311

I return to the stars—

to the question of why people feel free when

looking up at the stars.

Is it because, when we are communing with the

stars, we become

part of the Whole?

The whole of Life—

we feel ourselves as recycled matter and energy

congealed in a temporary form

a form that will not hold

that will one day fall apart.

What did they feel when they looked up at the

night sky?

Did the vastness produce a feeling of freedom?

Did they remember—there is a world beyond the

walls of this prison.

Were they transported to their childhoods, to the

mystery,

to the first time they contemplated their

place in the Whole?

In his autobiography Dusk of Dawn, Du Bois wrote about race as a prison—one that could only be abolished through a material and spiritual revo- lution. Anticipating the arc of my book, he wrote that the immediate problem of his people was “the question of securing existence, of labor and income, of food and home, of spiritual independence and

312 / Carceral Capitalism

democratic control of the industrial process” but that it would not do to “concenter all effort on eco- nomic well-being”—that his people “must live and eat and strive, and still hold unfaltering commerce with the stars.”9

THE DEATH THAT IS NOT DEATH, BUT THE BIRTH OF EVERYTHING POSSIBLE

What is prison? It is immobility. “Free man, you will always cherish the sea!” (Baudelaire). It is becoming more and more obvious that mobility is one of the signs of our times. To restrict a man for eleven years to sur- veying the same four or five square meters—which in the end become several thousand meters within the same four walls opened up by the imagination—would justify a young man if he wanted to go … where, for example? To China perhaps, and perhaps on foot. Jackson was this man and this imagination, and the space he tra- versed was quite real, a space from which he brought back observations and conclusions that strike a death blow to white America (by “America” I mean Europe too, and the world that strips all the rest, reduces it to the status of a disrespected labor force—yesterday’s colonies, today’s neocolonies). Jackson said this. He said it several thousand times and throughout the entire world. It still remained for him to say truths unbearable for our consciences. The better to silence him, the

The Prison Abolitionist Imagination / 313

California police …. But what am I saying? Jackson’s book goes far beyond the reach of this police. —Jean Genet on George Jackson10

I can only be executed once. —George Jackson, Blood in My Eye11

Language has no body. The message is a virus.

The message cannot be killed.

A REMIX OF A STATEMENT BY HUEY P. NEWTON, SERVANT OF THE PEOPLE, BLACK PANTHER PARTY AT THE REVO- LUTIONARY MEMORIAL SERVICE FOR GEORGE JACKSON:

A revolutionary example cannot be killed

The soldier and his spirit a living thing

His spirit says, George’s body goes

Although fallen

See

His ideas live

In young bodies

Our children are saying

It’s true

There will be revolution

And on he will go to the next legacy

We believe George’s immortality

As generation upon generation advance

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We know the people

We believe the people

Into immortality we win

Go on

No matter how still

How wrongly done

The love no matter how wrongly

This is pain giving up

No pain in giving up

And why he felt his life

For his people

Violence sorts spurs and contracts

Every alive state costs someone the death course

If it could give itself the semblance of executioners

—We don’t

We don’t have the kind of violence the police have

We deliver to them the struggle of everything possible

The audacity to accept the right to do everything

To preserve George

I see George growing in our suffering

In thirty seconds there will be pain

The prison order killing our stories won’t make our

suffering die

We say there will be pain

But in all of us a strength growing

For us

An incredible will living in the pain we know

I see two kinds of death

One death is not death

The Prison Abolitionist Imagination / 315

The other is death

George died in a way not-death

For in all of us there is George

In our suffering there is George

I see us die the not-death

The day George fell is not his death

The future will now know the way we will die

Revolutionary death

The way his mind determined the people’s name

To change them wholly or else be a feather

We’ll name people THE PEOPLE

We’ll support the name

In the name of the people, ALL POWER

TO THE PEOPLE ALL POWER

IMAGINATIONS HELD CAPTIVE

First of all, I would say that prison is an accurate name for our contemporary culture, and prison as culture presumes a certain set of problems and reinforces a dominant reaction in our imaginations. Sylvia Wynter talks about reservation—which is also an accurate name for our contemporary culture—meaning that at the same moment indigenous people are confined to reservations by the state, our imaginations are also con- fined. All of us. And, I would also say that the moments in which prisons became a dominant feature of the U.S., our imaginations (for all, not just those of us

316 / Carceral Capitalism

disproportionately imprisoned) also became imprisoned. The way we imagine work, our relationships, the future, family, everything, is locked down. —Alexis Pauline Gumbs12

Everywhere I look I see sleepwalkers under the spell of the prison.

What counter-spell is powerful enough to break the prison’s stranglehold on our imaginations?

But the spell is never total. The intensification of the desire for life undermines the prison’s capacity to structure our mental lives.

Imagination is excess, is that which could never be contained by the prison, that which will always exceed it.

What night endeavors must we embrace to enter that hidden frequency—that special vibration, the one Sun Ra believed would set us free.

THE DIALECTIC OF DREAMING

The imagination is constitutive ... It’s not just unworldly, detached from the world spinning off the refusal of things, rather it’s constitutive in the sense that

The Prison Abolitionist Imagination / 317

the imagination becomes so intense and embedded that it becomes real through its intensification and articu- lation. That puts theory in the realm of prophecy, but not prophecy in the realm of saying what’s going to happen. Instead, it’s the fostering of the imagination, the encouraging of that power to recognize that life can be, and in some ways already is, different. —Michael Hardt13

Dreams and reality are opposites. Action synthesizes them. —Assata Shakur14

Before Assata Shakur was liberated from prison, her grandmother and family came to visit her, bearing a dream: “You’re coming home soon,” her grandmother said. “I don’t know when it will be, but you’re coming home. You’re getting out of here. It won’t be too long, though.” She went on: “I dreamed we were in our old house in Jamaica … i was dressing you … putting your clothes on.” Assata’s grandmother was known for her prophetic dreams—they came when they were needed, but it was ultimately the responsibility of the recipients of the visions to make them real, not only by believing in the veracity of the prophecies, but by acting so as to give them flesh.

When Assata returned to her prison cell, she could not help but dance and sing. She writes, “No amount of scientific, rational thinking could

318 / Carceral Capitalism

diminish the high that i felt. A tingly, giddy excite- ment had caught hold of me. I had gotten drunk on my family’s arrogant, carefree optimism. I literally danced in my cell, singing, ‘Feet, don’t fail me now.’ I sang the ‘feet’ part real low, so i guess the guards must have thought i was bugging out, stomping around my cage singing ‘feet,’ ‘feet.’”

When we act in accordance with the prophetic dream, the dream comes to directly constitute reality.

THE POLITICS OF DREAMING

We are building a reality that we have never seen before. We are asking people to flex their visioning and dreaming skills, something that is not readily supported in our society. —Mia Mingus15

I think there is an inherent danger in conflating mili- tant reform and human rights strategies with the underlying logic of anti-prison radicalism, which con- ceives of the ultimate eradication of the prison as a site of state violence and social repression. What is required, at least in part, is a new vernacular that enables this kind of political dream. How does prison abolition necessitate new political language, teachings, and organizing strategies? —Dylan Rodriguez16

The Prison Abolitionist Imagination / 319

Sometimes I don’t know what to tell you, or how to end.

For some time I have been thinking about how to convey the message of police and prison abolition to you, but I know that as a poet, it is not my job to win you over with a persuasive argument, but to impart to you a vibrational experience that is capable of awakening your desire for another world.

A couple of years ago I saw the Black Arts Movement poet and activist Sonia Sanchez speak. I was moved by the way she paused whenever she experienced vertigo and spontaneously started singing as a way to find her rhythm after nearly passing out.

In a haiku Sonia writes:

without your residential breath i lose my timing.17

Our bodies are not closed loops. We hold each other and keep each other in time by marching, singing, embracing, breathing.

We synchronize our tempos so we can find a rhythm through which the urge to live can be expressed, collectively.

And in this way, we set the world into motion. In this way, poets become the timekeepers of

the revolution.

320 / Carceral Capitalism

PLANTING THE DREAM

What shall we build on the ashes of a nightmare? —Robin D. G. Kelley18

I won’t propose much more since the design and reali- zation of such a space ought to be the product of a collective imagination shaped and reshaped by the very process of turning rubble and memory into the seeds of a new society. —Robin D. G. Kelley19

I see

I see our shadow in the trees

Watching the wheel unfold

I see our one shadow on the wall

I see your restless hand in the spider’s thread

I am the ice cave and there is water,

deep blue and white, a light at the bottom

I am equal to my love for you

Let down your hair, willow

in the moonlight: the river

lulls us into the dream. Nightmares

jostle branches in our eyes. I long

for the world that is before you,

the plate you set on the slate

of tomorrow. Your fingers flutter

to feel for the grass

The Prison Abolitionist Imagination / 321

between the valley,

where one foot follows the other

into the flaming creek.

We don’t know what name to give

the throbbing stone

perched atop the hill.

From here, I see for you

Look at what I lost

when you were lost

and I could only hear

the call of the stones

A body, returned

floats down the river

dressed in candles

I send you the secret

while you are asleep

The nights you carried in the length of a strand of hair—

The unforgiving flash of his teeth—

I stroke your cheek to unlock your jaw

and release the rose you carry in your mouth

Your tongue is raw

and your mouth

is filling with blood

Dear

Dear,

322 / Carceral Capitalism

Forgive us for having fallen so far

from where you planted the seed:

At the bottom of the sea, waiting

for the body to ride the stream

back to where the rubble

gave birth to the first

dream

The egg cracks, night

wanders seaward

barefoot in her evening

slip

And by this sadness you are shown

the path to the holding sea, a trail

burned by a herd of somnambulant turtles

who folded, one by one, in their grief

until a single remained

to carry the breath of time

back

to the seed.

7. The Prison Abolitionist Imagination: A Conversation

A DOZEN ROSES VS. THE POLICE STATE
THE PRISON IS OUR SHADOW
ENTOMBED FLOWERS
THE STARS SEEN FROM PRISON
THE DEATH THAT IS NOT DEATH, BUT THE BIRTH OF EVERYTHING POSSIBLE
A REMIX OF A STATEMENT BY HUEY P. NEWTON, SERVANT OF THE PEOPLE, BLACK PANTHER PARTY AT THE REVOLUTIONARY MEMORIAL SERVICE FOR GEORGE JACKSON:
IMAGINATIONS HELD CAPTIVE
THE DIALECTIC OF DREAMING
THE POLITICS OF DREAMING
PLANTING THE DREAM

The “Comfort” of Critical Consolidation: Pedagogy, Ethical Alienation, and Asian American Literary Studies

Tina Chen

Abstract This article proposes ethical alienation as a critical pedagogical practice by analyzing the neoteric critical consolidation of Asian American literary and cultural studies, a consolidation marked by the recent (and near simultaneous) publication of four major compendiums—The Routledge Companion to Asian American and Pacific Islander Literature (Lee 2014), Keywords for Asian American Studies (Schlund-Vials, Vo, and Wong 2015), The Cambridge Companion to Asian American Literature (Parikh and Kim 2015), and The Cambridge History of Asian Ameri- can Literature (Srikanth and Song 2015). The publication of these four compendiums within the span of a year signals a pivotal moment in Asian American literary and cultural studies, one that both recognizes the institutional gains of the field and addresses the (in)compatibilities between the field’s theoretical developments and its pedagogical practices. This article considers the centrality of ethics to a postidentity Asian American studies, suggesting how ethical alienation—signaled both by the figuration of “comfort women”/military sex slaves/halmoni and our own estrangement from it—can create productive classroom and metapedagogical practices in the study of Asian American literature. By attending to the field’s consolidations around a set of pedagogical and scholarly imperatives and analyzing that critical solidification in relation to “comfort women”/military sex slaves/halmoni both as a figuration of “complex per- sonhood” (Gordon 1997, 4–5) and as an interdisciplinary “term of analysis and history rather than personhood” (Chuh 2003a, 9), this article argues that ethical alienation as pedagogical practice can lead us to a differently ordered set of disciplinary priorities. Keywords pedagogy, ethics, ethical alienation, Asian American studies, comfort women

Even from the historically myopic vantage point of the present, it is easy to see why 2014–15 is an institutionally signifi- cant moment in Asian American literary and cultural studies. The pub- lication of four major compendiums—The Routledge Companion to Asian American and Pacific Islander Literature (Lee 2014), Keywords

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for Asian American Studies (Schlund-Vials, Vo, and Wong 2015), The Cambridge Companion to Asian American Literature (Parikh and Kim 2015), and The Cambridge History of Asian American Literature (Srikanth and Song 2015)—within the span of a year constitutes an opportunity not only to recognize the institutional gains of the field but also to address the (in)compatibilities between the field’s theo- retical developments and its pedagogical practices. Although the four volumes are conceptualized quite differently, they all share an aware- ness of the importance of disseminating developing critical, theoreti- cal, and disciplinary knowledge via the classroom. In other words, even as each volume draws on the scholarship generated by the field of Asian American studies, each is also shaped by the institutional imperative of the field’s (re)production in pedagogical terms. Tar- geted to students and teachers as well as more generally to critics, the pedagogical impetus behind each project significantly contours its organizational logic and approach—something rendered explicit at the end of Rachel Lee’s introduction to The Routledge Companion. Lee groups keywords according to the various course rubrics governing the study of Asian American and Pacific Islander literatures and cul- tures, her concluding assessments thus laying bare some of the con- nections between disciplinary knowledge formation and pedagogical practice undergirding all these publications. Crucially, field consolida- tion offers us as critics and teachers a unique chance to engage with the discipline’s coherences and contradictions. Two aspects rendered especially palpable in this moment of criti-

cal consolidation are the shifting relationship between margins and centers effected by the professionalization of the field and the problems and possibilities of establishing shared terms. To make vis- ible the dynamics of such consolidation while also exploring the ways in which the teaching of literature can generatively respond to the epistemological and institutional aspects of field formation, I focus this article on comfort women—figures who have some- how remained relatively unaccounted for in these handbooks despite having been important subjects of critical attention in Asian Ameri- can studies, particularly in the early 2000s. Specifically, I offer the unwieldy, awkward, and nonstandard term “comfort women”/ military sex slaves/halmoni as a wedge into the homogenizing effects created by the critical consolidation of keywords, and I delve into the ways that pedagogical practices centered on such figures

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can refine and complicate the intellectual and critical assumptions of the field of Asian American literary and cultural studies.

The plight of “comfort women”/military sex slaves/halmoni— women who were coerced into sexual slavery by the Japanese mili- tary during World War II—has gained much international attention in the last twenty-five years.1 Since 1991, when Kim Hak-sun publicly testified that she had been forced by the Japanese military to be a “comfort woman” in response to assertions made by the Japanese gov- ernment that such women were prostitutes who voluntarily serviced soldiers during the war, evidence has surfaced confirming that more than 100,000 women from all parts of Asia (although the majority of these women were Korean) had been forced into sexual service by the Japanese military.2 “Comfort woman” is the English translation of the Japanese ianfu and the Korean wianbu, and as scholars have pointed out, the term is “a travesty” (Watanabe 1994, 4) and has itself “played a role in concealing and normalizing the violence used against these women” (O’Brien 2000, 213). Even so, many scholars and activ- ists still employ this term, albeit often marked by scare quotes to sig- nal a simultaneous disavowal of it, because it is the name by which the women are most commonly known. “Military sex slaves” marks a radical shift from the euphemism of “comfort women” and has been adopted by both the Korean Council for the Women Drafted for Mili- tary Sexual Slavery by Japan and by the United Nations. Some of the former Korean victims have suggested, though, that both “comfort women” and “military sex slaves” are distasteful, and they would pre- fer being called halmoni, which is the Korean word for “grandmother” and is an honorific accorded all older women in Korea. Additionally, there are a number of other terms in circulation, most notably Chong- shindae, which means “corps volunteering their bodies” in Korean, and is often used by Korean Americans.

The tripartite nomenclature of “comfort women”/military sex slaves/ halmoni alerts us to contested vocabularies, consolidating critical imperatives, and shifting notions of marginality and centrality as well as evolving, overlapped, and often incongruent historical conscious- nesses. Inspired by Laura Hyun Yi Kang’s (2003, 27) insistence on the importance of “dispel[ling] the wishful trajectory in which a more intimate identification with the Korean ‘comfort women’ leads to bet- ter representations of the ‘comfort women,’ which in turn secures greater justice for these women,” I draw attention to how such a term

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encourages us to consider the ways in which identification—of art- ist with character, of critic with victim, of student with subaltern subject—might be suspect or problematic in efforts to create ethical work and to teach ethically. As a term, “comfort women”/military sex slaves/halmoni foregrounds the ways in which identities are created and maintained via particular regimes of knowledge (in this case denoted by the very different terminologies in play) as well as to make visible the discontinuities that productively trouble the hermeneutic desire to fix the identities of such subjects in any easy or uncompli- cated way. Clearly what is at stake in these multiple designations, both the ones used and the ones whose titular absence makes apparent the partiality of the term’s construction, concerns not only the ways in which the survivors of the Japanese military comfort stations repre- sent themselves or have been represented by activists, artists, and aca- demics who have become invested in their cause but also the vexed reality of how identification and identity structure our ways of under- standing the subjects and practices that make up Asian American stud- ies. Even as practicality dictates a strategically essentialist choice between the many naming options available, I think the deliber- ately defamiliarizing usage of “comfort women”/military sex slaves/ halmoni will help enact as well as explore the issue of ethical alien- ation on which this article elaborates. In directing our attention to the figure of “comfort women”/military sex slaves/halmoni, I suggest that such a figure enables us to question the critical investments of Asian American literary studies by revealing how ethical alienation might counter the field’s own “wishful trajectory” as it manifests in an identification-based alignment with the marginalized and oppressed. In suggesting that we not privilege identification when considering

“comfort women”/military sex slaves/halmoni, Kang foregrounds the critical possibilities of what I term ethical alienation—an interpretive position and pedagogical philosophy that, while not discounting iden- tification and the valorization of sameness on which such a psycholog- ical structure usually depends, insist also on a valuation of difference and disjunction in order to create an ethical orientation to the subjects being represented and reviewed. More concretely, an ethical alien- ation acknowledges the critical investments we bring to any study of alterity and encourages us both to disidentify with (while not disavow- ing) victims and to acknowledge affiliations with victimizers. As such, ethical alienation becomes an inordinately useful pedagogical practice,

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one that actively encourages us to engage with the fraught issues raised by the seemingly unproblematic embrace of “comfort women”/ military sex slaves/halmoni—a plural subject whose uneven treat- ment in US academia signals both the possibilities and limits of the “beyond nation” turn in American studies.

In order to make my arguments about the pedagogical possibilities of ethical alienation, I briefly analyze the neoteric critical consoli- dation of Asian American literary and cultural studies marked by the recent (and near simultaneous) publication of four major compendi- ums to frame how the institutional gains of the field have produced (in)compatibilities between the field’s theoretical developments and its pedagogical practices. Then I consider the centrality of eth- ics to a postidentity Asian American studies, suggesting how ethi- cal alienation—signaled both by the figuration of “comfort women”/ military sex slaves/halmoni and by our own estrangement from it— can create productive classroom and metapedagogical practices in the study of Asian American literature. Finally, I identify some critical issues—the dangers of empathy and the field’s shifting alignments with dominance and repression—raised by the literary representa- tion and pedagogy of “comfort women”/military sex slaves/halmoni. By attending to the field’s consolidations around a set of pedagogical and scholarly imperatives and analyzing that critical solidification in relation to “comfort women”/military sex slaves/halmoni both as a figuration of “complex personhood” (Gordon 1997, 4–5) and as an interdisciplinary “term of analysis and history rather than person- hood” (Chuh 2003a, 9), this article argues that ethical alienation as pedagogical practice can lead us to a differently ordered set of disci- plinary priorities.

Critical Consolidation, Active Interference, and Ethical Alienation

Four major compendiums published within the span of a year can mean many things. For some, the publication of these volumes sig- nals that Asian American studies has come of age; in echoing the pri- macy of Bildung, this reading emphasizes the ways in which the field has developed the intellectual and academic wherewithal to support the creation of substantive pedagogical materials. For others, these volumes mark the field’s power to manufacture and legitimate cultural capital through the educational system, a reflection of the ways in

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which Asian American literary studies sanctifies its own “authority” via what John Guillory (1993, 56) has identified as a process of creden- tialization by which the school not only “produces a specific relation to culture . . . [but also] reproduces social relations.” For still others, these volumes suggest a critical opportunity for the field to manage its various (dis)alignments—aesthetic and political, theoretical and pedagogical, institutional and critical. Without privileging any one of these possible interpretations—

indeed, by considering them collectively—I begin with the assump- tion that the simultaneity of these publications suggests both the significant institutional gains made by the field and the pedagogical challenges resulting from those gains. Although the volumes are structured quite differently, reading them as part of a collective field- consolidating movement highlights four major takeaways.

1. There is detailed attention to developing and interpreting a common vocabulary; even when the volumes are not explicitly organized by keywords, key terms—such as immigration, law, identity/postidentity, and war, for example—emerge as part of the process of field interpretation.

2. Despite being organized to elucidate specific disciplinary practices, there is a greater awareness of the increasingly interdisciplinary, multidisciplinary, and transdisciplinary resonances of the key- words being used to organize academic knowledge production.

3. The shared investment in identifying the critical concerns unify- ing the field of Asian American literary and cultural studies also results in the inevitable privileging of certain pedagogical perspec- tives and approaches, varyingly determined by discipline, method- ology, history, and the attendant conceptual and theoretical vocabularies they generate.

4. The pedagogical consequences of Asian American literature’s institutional success must be theorized in relation to canonicity, a “profoundly problematic attainment for a body of writing that drew its strength from being at the margins and challenging the com- plicity of the center” (Srikanth and Song 2015, 27), and Asian American studies’ institutionalization as an interdiscipline.

In their assertion that it is “odd . . . to consider Asian America as refer- ring to anything but a margin,” Rajini Srikanth and Min Hyoung Song (2015, 27) recognize that the transition of Asian American literary

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studies from emergence to establishment brings with it new chal- lenges and opportunities. In other words, field consolidation offers us as critics and teachers a unique chance to engage with the discipline’s coherences and contradictions. Viet Thanh Nguyen (2015, 296) notes in his analysis of this transitional period that as Asian American literary criticism is disciplined, it has shifted from an “ideologically driven criticism” to an “institutionally driven criticism.” Nguyen does not argue this movement in either simplistic or developmental terms, instead encouraging us to concentrate on the costs of critical consolidation.

Crucially, although “comfort women”/military sex slaves/halmoni have been important figures of critical attention in Asian American studies, especially in the early 2000s, they do not appear in any signifi- cant way in any of the compendiums. As such, they offer a unique opportunity to self-reflexively assess the less visible consequences of interdisciplinary field formation. Specifically, I argue that focusing on such figures—on their naming, their literary representation, and especially on the kinds of pedagogical insights such figures enable— can help us highlight and complicate some of the critical tendencies generated as part of field-making and field-interpreting efforts.

As ongoing discussions of “Asian American” itself as a term by crit- ics such as Susan Koshy (1996), Kandice Chuh (2003a, 2003b), and Colleen Lye (2005) have highlighted, establishing a critical lexicon is a necessary step for making intellectual work possible. Pedagogically speaking, however, such instantiation engenders an unavoidably homogenizing effect that immediately begins rendering invisible the core assumptions and priorities driving its usage. If ethical alien- ation is in part about countering those erasures and acknowledging the possibility of destabilizing critical consolidations before they con- cretize into unquestioned prominence, utilizing a term like “comfort women”/military sex slaves/halmoni commits to a pedagogy and crit- ical practice of active interference. Gregory Jay (1991, 271) persua- sively argues in his article about the teaching of a multicultural Ameri- can literature that “undoing the canon[ical] doesn’t just mean adding on previously excluded figures; it requires a disturbance of the inter- nal security of the classics themselves.” Although Jay was writing at the beginning of the 1990s and as such against a narrowly defined canon of male- and white-authored texts, I employ his adjuration as a still-meaningful response to the challenges of the specific historical

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moment in which I write this article, when the successful institutional- ization of the minority literatures fighting for curricular recognition has resulted in the establishment of alternative canons. Interestingly enough, ethics is not an explicitly identified keyword

in any of the four compendiums surveyed in the first section of this article. In this case, I argue that such an absence ironically signals the fundamental importance of ethics to the field. More concretely, atten- tion to ethics as an embodied practice, an intellectual question, and a theoretical concern is so ingrained in the field’s identity that it becomes an assumed foundation undergirding not just the field’s obvi- ous social and political commitments (to social justice, protest and resistance, and human rights) but also to its less overtly ideological concerns (aesthetics, comparativism, and self-critique). In practical and pedagogical terms, ethics is crucial to the continued development of the field beyond identity politics paradigms because it encourages us to embrace a problematic Asian American agency—one that identi- fies not just with victims but also with victimizers. What does an ethically complicated, problematic Asian Ameri-

can agency look like? To address this issue, I return to the figure of Korean “comfort women”/military sex slaves/halmoni in order to consider how the discourse and classroom practices surrounding these figures make visible various kinds of ethical imperatives and considerations. In the introduction by Margaret Stetz and Bonnie B. C. Oh to Legacies of the Comfort Women of World War II (2001, xiii), such women become the occasion for producing work that encour- ages, perhaps even requires, “acts of fusion and of crossing over”:

Indeed, the challenges raised by these war victims are so wide- reaching—challenges not only to the concepts of imperialism, mili- tarism, sexism, classism, and racism, but also to the ways in which history itself has traditionally been recorded and written—that they can only be addressed through multi-faceted approaches. Thus, in confronting, exploring, understanding, and taking up these challenges, academics are finding themselves moved to political activism, activists are turning to mediums of artistic representation, and artists are performing the duties of scholarly researchers and analysts.

In this formulation, “comfort women”/military sex slaves/halmoni provide an unprecedented opportunity for academics, activists, and

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artists to remake themselves in interdisciplinary ways. By moving out- side of the comfort zones created by vocational training and avo- cational affiliation, those of us who study and remember “comfort women”/military sex slaves/halmoni are taking advantage of what Lisa Lowe, in her article titled “The International within the National: American Studies and Asian American Critique,” has identified as the value of interdisciplinary study; in Lowe’s (2002, 85) assessment, inter- disciplinary sites (and subjects) mediate contradiction to the degree that they can (1) disrupt empiricist paradigms separating the scholar- subject and the object of study; (2) challenge a developmentalist his- toricism that requires assimilation of “primitive,” nonmodern, and racialized knowledges to the terms of Western rationalism; (3) refuse univocality, totalization, and scholarly indifference; and (4) argue for the inseparability of the nonequivalent determinations of race, class, and gender. Such end results are, although not described explicitly as such, ethical goals; they can be understood as part of a larger and at times rather diffuse set of academic practices that operate in pursuit of what Chuh (2003b, 8) identifies as the “(im)possibility of justice”—which understands justice “not as the achievement of a determinate end, but rather as an endless project of searching out the knowledge and material apparatuses that extinguish some (Other) life ways and that hoard economic and social opportunities only for some.”

And yet, even as the figure of “comfort women”/military sex slaves/ halmoni has “emerge[d] as an especially and even perhaps ideally interdisciplinary and even trans-disciplinary subject, . . . such repre- sentational and epistemological mobility . . . must alert us to the hidden discursive and ideological work for which this newly visible subject is enlisted” (Kang 2003, 42). Thus while taking up “comfort women”/military sex slaves/halmoni in relation to an ethics of knowl- edge seems to argue the obvious, it behooves us to ask what it means for us to treat such subject matter ethically as Asian Americanists, both critically and pedagogically. How might we look to our pedagogy to respond to, refine, and complicate the scholarship we produce? According to legal theorist Leti Volpp, a “condemnatory reaction” to what is perceived as problematic behavior does not, in and of itself, necessarily constitute an ethical response. For Volpp (2000, 115), con- demnation, which “distances the observer from the practice and defi- nes the observer as the antithesis of that practice, relies upon and per- petuates a failure to see subordinating practices in our own culture.”

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Significantly, Volpp (114) understands condemnation to “unreflectively say ‘that is not me’” just as a laudatory response would essentially articulate “that is me.” In delineating the position of ethical alienation this article explores, neither of these underlying utterances—“that is not me” or “that is me”—is adequate since ethical alienation is pre- mised on a measured disidentification that can occur only after a rec- ognition of nondifference. Conceptually and performatively, then, ethi- cal alienation inhabits neither the “that is me” nor “that is not me” of Volpp’s formulation but rather the vexed space of Rhonda Blair’s (1993, 303) litotic notion of contrary performance, what she calls the “not-me/not-not-me.” In the classroom, a position of ethical alienation means exposing

what has traditionally been valorized in a Western educational frame- work as effective reading practice (empathetic identification as a way of incorporating the representation of difference into a humanistic framework that does not make explicit its own assumptions about cul- tural authority). It is critical to treat such identificatory desire skepti- cally, since even as “learning and teaching to read . . . in ways that acknowledge difference can be the most basic training for the demo- cratic imagination,” all too often students are taught to embrace “one- sided approaches that ask them to identify with favorite characters, without also asking what interferes in the process” (Sommer 1999, 8).3 Margaret Stetz (2003), in an essay in Radical Teacher, highlights the process by which this operates in her recounting of the experi- ences of Michiko Hase, a Japanese-born feminist scholar. Hase often experiences student anger and resistance when trying to discuss “U.S. citizens’ complicity in and accountability for their government’s and businesses’ policies and actions” but notes, tellingly, that in teach- ing about “comfort women,” students became more “appreciative” because “when I presented the ‘comfort women’ issue in a way that was very critical of Japan, the students knew that I had not been ‘bash- ing’ the United States but had been applying my critical analysis to all countries and issues” (quoted in Stetz 2003). As Hase’s experi- ence illuminates, it may be all too easy for students to read “comfort women”/military sex slaves/halmoni according to the dynamics of response Volpp articulates (i.e., condemnatory versus laudatory, “that is me” versus “that is not me”), especially when issues of cultural dif- ference are operative and moral judgments are differentiated on the basis of “us” and “them” distinctions.

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In response, Stetz (2003) suggests that while the importance of bringing this interdisciplinary, multiracial, transnational material into women’s studies courses is crucial for a host of obvious reasons (among them the need to express solidarity with women who deserve support, helping to recognize the trauma and legacy of survivors, and to demonstrate that feminist concerns are not primarily white, middle class, or exclusively American), perhaps the most indispensable role that the “comfort women” issue has to play has to do with the (re)mili- tarization of US society in the wake of 9/11. Specifically, she advo- cates for the importance of teaching about “comfort women”/military sex slaves/halmoni in an effort to address the conflation of US patriot- ism with an uncritical embrace of the military establishment.4 With this insight, she gestures to how crucial it is to perform ethical alien- ation as a critical as well as a pedagogical stance, to clarify the ways in which “around the issue of sexual violence, victimization as a tech- nology of subjectification can proceed by installing and deploying ‘cultural difference’ such that the culture of the other is seen as patri- archal and oppressive while the United Stated appears in contrast as liberatory” (Chuh 2003a, 7).

The Pedagogical Paradoxes of Ethical Alienation

In the two most well-known novels about comfort women to date, Nora Okja Keller’s Comfort Woman (1997) and Chang-rae Lee’s A Ges- ture Life (1999), the comfort woman issue is addressed within a US context that can make the narratives more approachable to a US audi- ence as well as obscure the hermeneutic process by which “com- fort women”/military sex slaves/halmoni emerge as “object[s] of a distinctly American regime of knowledge” (Kang 2003, 42). Keller’s novel centers its attention on the filial (dis)connection between a mother and daughter: Akiko, whose horrific experiences in the “rec- reation camps” of the Japanese army dislocate her from her home and former identity as Soon Hyo, and Beccah, who fears her mother is crazy and who does not fully understand the nature of her painful rela- tionship with Akiko. Comfort Woman enacts the kind of ethical alien- ation effect of which I write through Keller’s careful deployment of Korean shamanism as a critical, performative, and even pedagogical practice. Through shamanism, which allows Akiko the possibilities of a double reinscription into the world as both victim and agent,

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traumatized and healer, readers come to see what the “not-me/not- not-me” both looks and feels like.5 In teaching Comfort Woman, it may be all too easy to emphasize Japanese atrocity and responsibility in ways that encourage students to engage in Volpp’s “condemnatory reaction,” but critically Keller also encourages her readers to recog- nize how the United States’s neocolonial presence in Korea creates a gendered dynamic in US-Korean relations and consequentially impacts Korean American subject formation—particularly via her delineation of the similarities between Akiko’s relationship to her min- ister husband, Richard Bradley, and her objectification and repression in the comfort camps. Lee makes the connection between Japanese imperialism and US

neocolonialism even more explicit in A Gesture Life by emphasizing both the failure of institutional structures in fostering ethical behavior and the transcultural dynamics by which women are subordinated to the demands of militarized prostitution. Lee’s novel explores the conti- nuities and disjunctions between protagonist Franklin Hata’s present and past, his current identity as an institution of his bucolic suburban neighborhood, a veritable town father known popularly as “good Doc Hata,” and his previous life as the young medical officer Jiro Kurohata who is “uncertain of himself [and whether he] could find meaning amidst the camaraderie of his fellows working in such shared pur- pose” (1999, 224). Through Hata’s “gesture life,” Lee foregrounds a crisis of faith in institutional structures that the war—and particularly the institutionalized system of comfort women as part of the Japanese military effort of which Hata is a part—sets in motion. Critical to Lee’s critique and how the novel attempts to clarify

the ways in which “victimization as a technology of subjectification can proceed by installing and deploying ‘cultural difference’” (Chuh 2003a, 7) is the figure of Sunny, Hata’s adopted daughter. Half-black and half-Korean, Sunny is estranged from Hata, who adopts her in an effort to rewrite his earlier ethical failure in saving “K.” Although Kang (2003, 32) has argued that Sunny represents “an especially forced attempt to conjoin the histories of Japanese military sexual slavery and U.S. military-related prostitution in Korea,” such a con- joining is nonetheless significant in light of the student responses Hase and others have encountered in the classroom. More concretely, it becomes increasingly difficult to hold on to the us-versus-them, oppressor-versus-liberator identity positions identified by Volpp and

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Chuh when “comfort women”/military sex slaves/halmoni are intro- duced in relation to the context of US militarized prostitution in Korea (and other parts of Asia)—especially when we acknowledge the ways in which camptowns and the system of militarized prostitution they embody operate in ways not entirely different from the Japanese mili- tarization of sex slaves during World War II (see Moon 1997; and Yuh 2004). Although these systems are clearly not the same, there are striking similarities—such as the coercion of women through third- party brokers or placement services, the euphemistic terminology ascribed to the women (“comfort women” in the case of the Japanese military and yang gongju [Western princess] and yang saeksi [Western bride] in the case of US-style militarized prostitution), the military hierarchies’ demand for and orchestration of campaigns to regulate the testing and treatment of venereal disease, and the requiring of women to wear numbered tags for identification purposes. In this case, we need to be willing to be alienated from the process that histo- rian Lisa Yoneyama (2003, 59) has termed the “Americanization of redress and historical justice,” a process that depends on “an imperial- ist myth of ‘liberation and rehabilitation’ in which violence and recov- ery are enunciated simultaneously [such that] the enemy population’s liberation from the barbaric and the backward and its successful reha- bilitation into an assimilated ally are both anticipated and explained as an outcome of the U.S. military interventions.” Ji-Yeon Yuh (2004, 36–37) concurs, arguing, “To see the camptown women as victims of militarized prostitution and as modern-day comfort women would be to shatter [a] vision of America” as a South Korean ally and friend whose freedoms and material abundance are ideals to be envied and emulated.

At this juncture, I focus on a poem by Ishle Park titled “House of Sharing Comfort Women” (2001, 51–53) in order to render more concrete how ethical alienation helps me, as a critic and a teacher, embrace a problematic Asian American agency. Park’s poem features an unnamed speaker and a halmoni whose daily life is initially charac- terized by activities—playing cards “worn smooth,” cooking “white corn over a portable stove,” smoking cigarettes, and dancing with Jap- anese students—that do not necessarily differentiate her from the other older women of her generation (51–52). The poem’s central con- ceit is the tension between the unnamed speaker, who observes the halmoni and whose descriptions are stubbornly descriptive, making

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clear her exteriority, and the halmoni herself, whose thoughts come across sporadically but never really give either the speaker or the poem’s reader a point of identificatory access. The alienation between the poem’s speaker and the halmoni,

between its reader/critic and its subjects, is precisely what allows for the consideration of the ethical dimensions of the ways in which the figures of “comfort women”/military sex slaves/halmoni have been represented and come to assume meaning as part of a (in this case) particularly American regime of knowledge. Although the poem always names the woman as halmoni, its title also names her as a very specific kind of former “comfort woman”—someone affiliated with the House of Sharing, a communal living center established in 1992 by a Buddhist charity that houses a small group of former “com- fort women”/military sex slaves/halmoni who have since become the public faces for the redress campaign.6 Significantly, the halmoni’s past in the comfort camps is gestured to only elliptically in the first half of the poem in her internal thoughts:

3 pine trees. my parents thought they sent me to a good place. my hands like rubber gloves. my heart bleeding. I was 14. (Park 2001, 51)

Given the overt signaling by the poem’s title and its reference to the House of Sharing, the poem’s refusal to provide and affirm immedi- ately the details of victimization rather than the halmoni’s successful daily functioning produces a sense of expectation in the reader that is exposed as something created not so much by an ethical orienta- tion toward the subject matter as part of a reobjectivizing and exploit- ative gaze that has been trained to interpellate the “comfort women”/ military sex slaves/halmoni in specific ways. As documentary film- maker Dai Sil Kim-Gibson (1997, 259) notes, the narrative formula by which “comfort women”/military sex slaves/halmoni have been pro- duced by activists, academics, and artists is quite rigid:

Each of the grandmas I interviewed preferred to start out citing her age at the time of capture and then move directly to the initial period of bondage. Each wanted to tell me immediately about “how many soldiers a day,” the medical examination, the venereal

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diseases, her menstruation, fees for services, and more than any- thing else, about the particular sadism of the soldiers, recruiters, and managers. . . . If indeed the old women had disclosed material beyond “that period,” it rarely made its way into the stories. It was as if their existence was justified by the horrendous years they suf- fered; nothing before or after that seemed to matter. Saddest of all, the grandmas themselves were convinced of that. They have become issues, numbers, and objects of studies, not full blooded human beings.

As soon as the reader comes to an awareness of the ethical conse- quences of expecting the halmoni’s story to conform to a narrative for- mula that focuses almost exclusively on the period of sexual slavery, Park (2001, 52) seemingly undercuts herself by obliging us with the kind of graphic image we initially expected: that of the young girl being cut open with rusted scissors because “she was too small.” As the poem continues to make clear, though, the halmoni is herself not that girl imprisoned and tortured in the comfort camps; this alienation between the girl of the past and the halmoni of the present is then what lays bare the reader’s affiliation with a victimizing position and allows the speaker to address the ethical problems of resolving such disjunction by relying on identification’s “wishful trajectory” (Kang 2003, 27). The alienation between the poem’s speaker and the hal- moni, between its reader/critic and its subjects, and between the hal- moni’s past and present identities thus emerges, at poem’s end, as the poem’s central preoccupation:

I cannot reconcile this halmoni with a girl 50 years ago, lips like pressed heart, neck long as reed, who never learned to write her own name, this halmoni, bundled thick in 2 wool coats, bus ticket clenched tightly in gloved fist to attend her hundredth rally, pushing the glass-covered police young enough to be her grandsons, to be in spitting distance of the Japanese embassy.

She draws a painting larger than herself of a soldier in mustard green, strapped to a cherry blossom tree with black barbed wire,

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guns pointed at his chest from 3 directions, white doves taking flight from its branches.

A painting of a hand picking ripe plum, Inside each one the face of a young Korean girl, Hair plaited in two thick braids, inside each plum, one girl. (Park

2001, 53)

The poem’s final stanzas can be read as paradoxically problematic: even as the speaker claims that she “cannot reconcile this halmoni with a girl 50 years ago,” she arguably tries to resolve this inability through her recounting of the halmoni’s paintings. The details of the paintings as well as the fact of the halmoni’s painting might be read as part of an effort to reconcile artistically through a visual register what cannot be spoken by either the speaker or the halmoni. However, I suggest that this concluding moment might also be read as one that embraces irreconcilability and the lack of identificatory correspon- dence on which a position of ethical alienation depends. In order to clarify how each of these critical readings operates, I

direct our attention to paintings produced by Kang Deuk-kyung, a now-deceased former “comfort woman”/military sex slave/halmoni who spent the last years of her life living at the House of Sharing and participating in weekly Wednesday protests at the Japanese con- sulate.7 Clearly, Kang Deuk-kyung’s paintings, which were created as part of an art therapy program, are the ones being referenced in Park’s poem. There is the “soldier in mustard green,” the “white doves taking flight,” and the “face of a young Korean girl” (Park 2001, 53). The actuality of these paintings and the fact that they were cre- ated by a former “comfort woman”/military sex slave/halmoni would seem to produce the reconciliation that the speaker professes not to be able to perform at poem’s end. And yet, a closer inspection of the paintings reveals that the poem’s visual images, however vivid, detailed, and referential, are also deliberately rendered imperfectly. Although the soldier in Kang’s painting is strapped to a tree, the tree is not a cherry tree in bloom but rather a tree without leaves or flow- ers, denuded and acting as a stark background to the blindfolded sol- dier strapped to its trunk. The image of the cherry blossom tree is actually taken from another picture—“Purity Lost Forever”—which features a soldier who seems to be part of the tree itself. He is half- man, half-tree, and he looms over a naked woman who lies at the roots

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of the cherry tree, her hands over her eyes and her body curled in a fetal position. At the foreground of the picture is a field of skulls, which might represent other female victims of the camps or might be meant to invoke the death of “purity” that is referenced in the pic- ture’s title. The last painting, which indeed shows hands plucking fruit on which are figured the images of young Korean girls, is also “mis- translated” by the poem; rather than plums and a girl with “hair plaited in two thick braids,” the painting features pears and a girl with a single braid.

Beyond Identification’s “Wishful Trajectory”

What do such mistranslations tell us about the ethical dilemma of how to study and teach about comfort women and our own disciplin- ary investments as Asian Americanists and American studies practi- tioners? In using the work of Kang Deuk-kyung, I believe Park is clearly signaling the importance of remembering that even as “com- fort women”/military sex slaves/halmoni operate as a “term of analy- sis and history rather than personhood” (Chuh 2003a, 9) they also gesture, however inadequately, to the “complex personhood” (Gordon 1997, 4–5) of the women who survived this brutal history. Park’s poetic use of the paintings both pays homage to and refigures Kang’s work. In her transposition of the cherry tree from “Purity Lost For- ever,” a tree that represents both the young girl’s innocence and her traumatic sexualization, to the scene of “Punish Those Responsible,” Park brings together the separate images of violation and retribution from Kang’s oeuvre to fashion a statement about the possibility of rec- onciling the halmoni with the young girl in order to make sense of the political will revealed in the weekly demonstrations at poem’s end.

This is a critical statement, one that is crucial to the redress move- ment and represents a “coming to voice” that we want to celebrate. And yet, being ethically alienated from this possibility means recog- nizing that there can be no easy reconciliation—the transfiguration of the image is poetic license that can bring both halmoni and young girl together artistically only by acknowledging the disjunction that Kang’s painting insists on—and understanding that the significance of making such a statement locates not so much to the women con- scripted to sexual slavery but to us, critics and artists and teachers, to

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satisfy our will to knowledge. In this effort, Park’s resistance to “the core values of aesthetic realism—correspondence, mimesis, and equivalence—and [her] approach to these notions as contradictions” (Lowe 1996, 130), her commitment to what Lisa Lowe might call “an aesthetic of infidelity,” insists on an ethical alienation that recognizes critical distance as crucial to the embrace of a problematic Asian American agency that both acknowledges a desire to practice and works to counter what Rey Chow has termed “‘self-subalternization,’ a process by which the critic identifies with a position of powerlessness in order, paradoxically, to claim a certain kind of academic power” (quoted in Chuh 2003a, 8). Academic power comes in manifest forms. As a field that came into

being by challenging the various disciplinary mechanisms by which the university produces and maintains knowledge, Asian American lit- erary studies has always viewed its own agency and successes in somewhat suspect terms. From the early denunciations of mainstream success by the Aiiieeee! editors to a contemporary tendency toward “recursive feedback loops”—what Rachel Lee (2014, 4) has character- ized as a “fractal incorporation” of “critiques of its self-critiques of its self-critiques of its self-critiques, and so on”—Asian American literary studies would do well to attend to the possibilities of being ethically alienated from itself. Although it is easy to understand the dissemina- tion of knowledge hierarchically, a dynamic seemingly reinforced by the publication of four compendiums within a single year to mark the institutional successes of the field, this article has argued for the possibilities of reassessing that hierarchy by considering the ways in which pedagogical practice is not simply an application of scholarly work but can itself serve to respond to, refine, and complicate the intellectual and critical assumptions of the discipline. In some ways, this represents a logical next move for Asian American studies, so long used to protesting others but now, in the vein of recognizing how “adaptive hegemony” works, perhaps ready to move its self-critiques from a stance of theoretical open-endedness to wrestling with what Roderick Ferguson (2012) has named “the reorder of things,” wherein the successful institutionalization of new curricula might be what we need to problematize—not just celebrate. In other words, what does it mean in both pedagogical and institutional terms to recognize our alignments with dominance and repression, to engage with those positions as a meaningful part of the field’s professional responsibility?

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The trauma suffered by the women subjected to work in comfort stations has been compounded in multiple ways: in addition to surviv- ing their sexual slavery by the Japanese military, these women have had to struggle against a US Cold War hegemony that supported the suppression of their reparation efforts against Japan, the refutation of their testimony concerning the brutalities they suffered, and the Japa- nese government’s refusal to offer an official apology.8 Much of the public emphasis regarding the comfort women has been on restitu- tion and reparation, with both “comfort women”/military sex slaves/ halmoni organizations and war crimes commissions demanding an offi- cial apology as well as remuneration from the Japanese government. As the human rights and activist activity surrounding the redress move- ment reveals, the ethical dimensions to “comfort women”/military sex slaves/halmoni issues are multiple—related to historical memory and amnesia, the debate over what really happened, and how public witness and personal testimony can be used to produce “bodies” of evidence in support of what has been historically disavowed or repressed. And yet, the artistic representation of comfort women in Asian American litera- ture and culture, which in many cases does not explicitly address the issue of reparations and government accountability that constitutes the ethical center of the redress movement, foregrounds for Asian Ameri- canists different kinds of questions and problems about what consti- tutes ethical knowledge production and practice in Asian American studies as a site of both academic inquiry and pedagogy.

Even as this article has employed the figures of “comfort women”/ military sex slaves/halmoni as a means of critiquing the consolidation of Asian Americanist academic knowledge production, the practices it espouses—a pedagogy of active interference and ethical alienation as a productive response to the institutionalization of specific episte- mological frameworks—can also usefully be turned on themselves. At the end of 2015, two significant events occurred illustrating both the drive toward consolidating knowledge practices and the impor- tance of ongoing interference through alienating practices. First, the state of California released a draft of guidelines for the teaching of history recommending that tenth graders be taught in world history about “comfort women”/military sex slaves/halmoni “as an example of institutionalized sexual slavery, and one of the largest cases of human trafficking in the 20th century.” This recommendation has generated ongoing debates and even an online petition on Change.org

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in response to a conservative Japanese newspaper’s reporting (Kim 2016). Second, the Japanese and South Korean governments announced that they had come to an agreement about the “comfort women”/military sex slaves/halmoni issue (Ministry of Foreign Affairs of Japan 2015). Heralded as a diplomatic success, the agree- ment included Prime Minister Shinzo Abe’s “sincere apologies and remorse” and the establishment of “a foundation for the purpose of providing support for the former comfort women” funded by an $8.3 million contribution as well as the South Korean government’s pub- licly affirmed efforts to address the installation of a “comfort woman”/ military sex slave/halmoni statue in front of the Embassy of Japan in Seoul. Japan’s apologies and monetary contributions were not made contingent on the removal of the statue, and critically, the ongoing debates about the statue of a barefoot girl and an empty chair (see figure 1) placed in front of the embassy in 2011 to commemorate the unresolved “comfort women”/military sex slaves/halmoni issue serve as an example of how important ethical alienation can be in response to what one might argue are the consolidating impera- tives articulated through both the curricular changes proposed by

Figure 1. The Statue of Peace, located in front of the Embassy of Japan in Seoul. Image courtesy of YunHo Lee

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California and the “comfort women”/military sex slaves/halmoni agreement between Japan and Korea. The statue reminds us that there are many other statues commemorating “comfort women”/mili- tary sex slaves/halmoni established around the world—most notably in Manila, Nanjing, and the United States—and that the agreement between Japan and Korea continues to be vigorously protested by fac- tions in both countries as well as Filipina comfort women, who are angry about their exclusion from this official response. As such, it seems important to remember that the very worthwhile accomplish- ments of such consolidating moves can and must be subject to contin- ued critique, one made visible through a willingness to be alienated ethically from prevailing ideas about disciplinary, political, aesthetic, pedagogical, and epistemological priorities.

Tina Chen is associate professor of English and Asian American studies at the Penn- sylvania State University. Her book, Double Agency: Acts of Impersonation in Asian American Literature and Culture, was named a CHOICE Outstanding Academic Title in 2005. She is also the founding editor of Verge: Studies in Global Asias.

Notes

1 For an overview of the redress movement, see Hicks 1999. Several oral histories and testimonies have also been published, including Kim- Gibson 1997; Schellstede 2000; Howard 1996; and Henson 1999. Addition- ally, a number of scholarly studies in English have also appeared. See, for instance, Choi 1997; Hicks 1994; Soh 2009; Tanaka 1998, 2002; and Yoshiaki 2000.

2 Although Korean women made up approximately 80 percent of the com- fort women population during the war, data reveals that a wide variety of women—including Japanese, Taiwanese, Chinese, Indonesian, Dutch, Burmese, Malay, White Russian, Filipina, Vietnamese, and Burmese— were used by the Japanese military. Historian George Hicks estimates that if we assume a ratio of comfort women to soldiers as 50:1, then there were about 139,000 comfort women in service during the war. Yoshimi Yoshiaki (2000, 93–94) acknowledges that we have no documents from which to derive accurate estimates; however, the number of women is between 50,000 and 200,000.

3 For a fuller discussion of how teaching minority writing can disrupt or problematize traditional Western pedagogical goals and assumptions in the classroom, see Chen 2005b, 2006.

4 This is due to the difference in attitude between the publication of Stetz’s essay and now, including changing notions of the appropriateness of war.

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5 For a more detailed discussion of how shamanism operates in Keller’s novel, see chapter 5, “Shamanism and the Subject(s) of History in Nora Okja Keller’s Comfort Woman,” in Chen 2005a, 113–51.

6 In addition to housing former “comfort women”/military sex slaves/ halmoni, the House of Sharing is also a human rights museum dedicated to the theme of sexual slavery. Its website can be accessed at www.nanum .org/eng/. The women who live there have been featured in many inter- views, in films such as Dai Sil Kim-Gibson’s Silence Broken and The House of Sharing, and as the subjects of various art exhibits featuring their paint- ings (produced as part of an ongoing art therapy program). Additionally, several of the House of Sharing halmoni participated in the weekly protests at the Japanese Embassy sponsored by the Korean Council. The weekly Wednesday demonstrations have been ongoing since January 8, 1992.

7 Kang Deuk-kyung is also spelled Kang Duk-Kyung and Kang Deok- keong.

8 For nuanced articles on the Japanese response to comfort women’s demands for apology and reparations, see Field 1997; and Park 1997. For an argument about how US hegemony is complicit in the sexual enslave- ment of Asian women, see Thiesmeyer 1996.

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51–53. Park, Won Soon. 1997. “Japanese Reparations Policies and the ‘Comfort

Women Question.’” positions: East Asia Cultures Critique 5, no. 1: 107– 34.

Schellstede, Sangmie Choi. 2000. Comfort Women Speak: Testimony by Sex Slaves of the Japanese Military. New York: Holmes and Meier.

Schlund-Vials, Cathy J., Linda Trinh Vo, and K. Scott Wong, eds. 2015. Key- words for Asian American Studies. New York: New York Univ. Press.

Soh, C. Sarah. 2009. The Comfort Women: Sexual Violence and Postcolonial Memory in Korea and Japan. Chicago: Univ. of Chicago Press.

Sommer, Doris. 1999. Proceed with Caution, when Engaged by Minority Writing in the Americas. Cambridge, MA: Harvard Univ. Press.

Srikanth, Rajini, and Min Hyoung Song, eds. 2015. The Cambridge History of Asian American Literature. New York: Cambridge Univ. Press.

Stetz, Margaret. 2003. “Teaching ‘Comfort Women’ Issues in Women’s Studies Courses—Military Sex Slaves.” Radical Teacher, no. 66.

Stetz, Margaret, and Bonnie B. C. Oh. 2001. Legacies of the Comfort Women of World War II. Armonk, NY: M. E. Sharpe.

Tanaka, Yuki. 1998. Hidden Horrors: Japanese War Crimes in World War II. Boulder, CO: Westview.

. 2002. Japan’s Comfort Women: Sexual Slavery and Prostitution during World War II and the U.S. Occupation. London: Routledge.

Thiesmeyer, Lynn. 1996. “U.S. Comfort Women and the Silence of the Ameri- can Other.” Hitting Critical Mass: A Journal of Asian American Cultural Criticism 3, no. 2: 47–67.

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Watanabe, Kazuko. 1994. “Militarism, Colonialism, and Trafficking of Women: Military ‘Comfort Women’ Forced into Sexual Labor for Japanese Sol- diers.” Bulletin of Concerned Asian Scholars 26, no. 4: 3–16.

Yoneyama, Lisa. 2003. “Traveling Memories, Contagious Justice: American- ization of Japanese War Crimes at the End of the Post-Cold War.” Journal of Asian American Studies 6, no. 1: 57–93.

Yoshiaki, Yoshimi. 2000. Comfort Women: Sexual Slavery in the Japanese Mili- tary during World War II. New York: Columbia Univ. Press.

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Pedagogies of Dissent Kandice Chuh

American Quarterly, Volume 70, Number 2, June 2018, pp. 155-172 (Article)

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| 155Pedagogies of Dissent

2018 The American Studies Association

Pedagogies of Dissent

Kandice Chuh

T hat academic freedom is under attack is something of a commonplace observation, and understandably so. There is ample evidence to be drawn from across the world and more locally in support of such a

claim. The Scholars at Risk Network’s recent publication, Free to Think 2017, covers 257 reported attacks on higher education communities in thirty-five countries in just the past year.1 These include the violent suppression of orga- nized student protests by state authorities in Venezuela, South Africa, Niger, Cameroon, Turkey, and India, as well as repression through travel restrictions in such countries as China, Turkey, Israel, Uganda, Thailand, and the United States. The report also includes legislative activities that threaten the autonomy and viability of higher education institutions, a phenomenon clearly notable in the United States. The consequences of all this, Scholars at Risk finds, run the gamut from killings and disappearances to restrictions on the content of teaching and research, and to fines, dismissal, and imprisonment. Other organizations and studies report similar repressive and suppressive activities, and importantly take note of the growing influence of private interests—cor- porations and overtly ideological organizations—on restrictions to academic freedom.2 In brief, there is no shortage of evidence that academic freedom is insecure, fragile, vulnerable.

In the United States, we know such vulnerability across all levels of educa- tion is produced through a wide range of state and state-authorized activities including the Arizona anti-ethnic studies movement and legislation; the enactment in ten states and the pending status of legislation in several others of “campus carry” laws providing for the possession of concealed weapons in public universities and colleges;3 the intensive defunding of public schools again at all levels;4 “bathroom bills” that regulate access to public facilities; the promised defunding of science and especially climate research; and the repeated attempts at both state and federal levels to enact laws that would refuse public funding or otherwise penalize individuals and institutions who criticize Israel in any way. Indeed, people who have mounted criticisms of Israel, or who have spoken out against racism or sexualized violence, have found themselves

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targets of administrative censure and have faced aggressive harassment through online campaigns.5 The autonomy of both educational institutions and educa- tors on which the principle of academic freedom is grounded, arguably always fictional, is these days aggressively challenged both by state activities and by unfettered incursions of the interests of capital, as well as by private, ideologi- cally driven groups.

All this contributes to the sense that we are under siege, that public educa- tion in particular and especially higher education is in crisis, and that it and both its current and aspirational communities need defense. The question that has long been and currently resurges before us is the matter of how to respond—how to protect and defend colleges and universities from unwanted and unwarranted intrusion by nonacademic forces. It is this question I linger with in my remarks today. An enormous amount of literature already exists that details the conditions we currently face, including some that provide strategies for mobilizing against them.6 My remarks are somewhat differently skewed though similarly oriented. In this address, I take what I think of as a pedagogical approach to the question of defense, to raise some questions I believe necessary to engage in the formulation of responses to the current scene. For, it seems to me, there is work to do in sinking into the question of defense, or more precisely, to question defensiveness itself and the defensive posture that current conditions (try to) impose so strongly on us. This work is related to what we do as people committed to thinking hard together in our capacities as teachers, students, and scholars, oftentimes all at once. To my mind, while there is no question of the need to stand by and with people living the consequences of expressing dissent, there are many that arise in relation to the defense of academic freedom, a principle, or the university, an institution.7 I want to bring to bear how we fight against defensiveness—our own as well as our students’—in classroom settings, in perhaps tacit recogni- tion of how it is categorically antithetical to learning, of how defensiveness, as a self-justificatory stance, serves as a form of an ego-driven protection of one’s rightness and results in the unwillingness to entertain error in one’s own position. Such defensiveness inhibits learning, prohibits collaboration, and makes collegiality, groupness, or in other words, robust support of each other difficult if not impossible. I’m asking us to bring this awareness of the effects of defensiveness to our thinking about the vulnerability of academic freedom and what that vulnerability tells us about contemporary conditions, and to do so by working in a pedagogical mode. I’m asking us, that is, to do so by sinking into the time stretched out by the intentionally pedagogical to allow for the thickening and potential transformation of understanding. More specifically,

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by doing so, my aim is to think with and through the streams of thought that, in often overlapping forms and under the administrative rubrics of queer of color critique, women of color feminism, Native and Indigenous studies, Black studies, ethnic studies, and decolonial discourses among others, enjoin us to disidentify from the pedagogies of dissent that, however inadvertently, feed rather than defunction the systems producing the conditions that induce dissent in the first place.

Primary among such pedagogies is the nationalism that carries forward liberalism and neoliberalism in the United States, the values of which argu- ably have become dominant intellectual values through not only the histori- cal foundations upon which education has been built but also the structured embrace of academic freedom in a liberal key.8 The declaration that academic freedom is under attack accordingly acts here as an invitation to identify the conditions that give such statements traction. Who and what are being attacked and by whom, how, and why? But also, what histories are brought forward and which are occluded, what ideologies are affirmed and which discounted, by the assumed value of academic freedom accompanying declarations of its besieged state?

I want to remember how the pedagogical refuses the reactive temporalities of the language and production of crisis, temporalities that are fundamentally anti-intellectual in their inhibition of thoughtful consideration. Certainly, the rapidity with which highly objectionable, dissent-inducing events and actions are these days pressed on us, in an important sense, demands reactive imme- diacy. It is breathtaking—literally for some—to experience and apprehend the manifold ways the US state in its partnership with capitalism innovates to diminish life. The need for immediate redress is clear in the face of the state’s aggressive abandonment of people, its refusal to provide basic relief to the people of Puerto Rico in the aftermath of hurricanes and in disavowal of the responsibility the United States bears for the crumbling infrastructure of the island, for example, and by its moves to eliminate the already very minimal affordance that DACA offers to undocumented students, as well as by its ef- forts to restrict travel and entry, its plans to defund health care, to divert public monies to corporations and away from student loan programs—the list goes on.

This cascading cruelty has given rise to what I have elsewhere called “protest nationalism” to refer to expressions of dissent aligned with and framed by the rationalities of the US nation-state.9 Protest nationalism remains faithful to the idea of the fundamental goodness of US liberal democracy and identifies its retrievability from the hands of a spastic executive anomalous to the nation’s otherwise laudable if imperfect past as source of political hope and objective. It

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is in this way a form of consent in the cloak of dissent. Of protest nationalism we might say, simply, metonymically and empirically speaking, we are not all immigrants, nor do we aspire to be, for a variety of reasons including the settler colonialism that originates the possibility of immigration. The availability and traction of “We Are All Immigrants” as a response to US nationalism attest to the continuing potency of the liberal nation-state as the framework that governs dominant understandings and imaginations of both US history and its possible futures.

Stretch time around academic freedom and in ways mindful of these nor- mative, nationalist pedagogies of dissent, and it becomes sharply clear that contestations over academic freedom bespeak what Edward Said describes as the “nationalization of intellectual activity” through formal education. Like Said, here, “I want to raise the question of how the central importance and authority given [to] national identity impinges on and greatly influences, surreptitiously and often unquestioningly . . . what transpires in the name of academic freedom.”10 The project that emerges, and in some important respects has long been ongoing, is not one of “decentering” the nation or “transnational- izing” our paradigms; rather, it may be understood as the delegitimation of the authority of the US nation-state and the bourgeois liberalism that secures it, by the elaboration and activation of frameworks and sensibilities fundamentally incommensurate to it.

All this is to remember that dissent is in itself not an end but is instead a point of departure. The question is, where will we go, and how will we get there?

A Brief History of Professionalization, or The Ruse of Academic Freedom

Many of you will recognize that I’m riffing Lisa Lowe in using this phrase, “the ruse of academic freedom.” In Intimacies of Four Continents, Lowe resoundingly details how freedom, as given to us by liberalism, is a ruse that covers over the ongoingness of unfreedoms organized by the entangled procedures and effects of colonialism and settler colonialism, racial capitalism, and empire.11 Freedom as narrated under and by liberalism denies slavery and its afterlife, and disavows the dispossession of the land and lives of indigenous peoples. Liberalism in this way develops freedom for a narrow portion of humanity under the sign of Western Man while subjugating all others as not-yet and perhaps never to be capable of its possession. Liberal freedom in this regard must be understood to belie its materiality, its uses against people and the worlds we inhabit in the name of humanity, and its evocation must cause pause.

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It is in that pause that I situate academic freedom. In and from that space, it is unmistakable how, as developed in the US context, academic freedom delimits its affordances to a select few, as it not only occludes the labor condi- tions that characterize the academy but also affirms the onto-epistemologies of liberalism that continue to rationalize and thereby legitimate the ravages of capitalism and colonialism.

As may be familiar to you, in the United States, academic freedom is gener- ally understood in the ways elaborated by the 1940 Statement of Principles on Academic Freedom and Tenure from the American Association of University Professors, to which in 1970, the AAUP offered “interpretive comments.”12 Most US universities and colleges have adopted its principles in the formula- tion of their employment procedures and institutional aims. They are in this sense embedded in the very architecture of contemporary higher education.

The AAUP’s introduction of the statement reads as follows:

The purpose of this statement is to promote public understanding and support of academic freedom and tenure and agreement upon procedures to ensure them in colleges and uni- versities. Institutions of higher education are conducted for the common good and not to further the interest of either the individual teacher or the institution as a whole. The common good depends upon the free search for truth and its free exposition. Academic freedom is essential to these purposes and applies to both teaching and research. Freedom in research is fundamental to the advancement of truth. Academic freedom in its teaching aspect is fundamental for the protection of the rights of the teacher in teaching and of the student to freedom in learning. It carries with it duties correlative with rights.

Tenure is a means to certain ends; specifically: (1) freedom of teaching and research and of extramural activities, and (2), a sufficient degree of economic security to make the profes- sion attractive to men and women of ability. Freedom and economic security, hence, tenure, are indispensable to the success of an institution in fulfilling its obligations to its students and to society.

Academic freedom, the statement continues, is constituted by three features:

1. Teachers are entitled to full freedom in research and in the publication of the results, subject to the adequate performance of their other academic duties; but research for pecuniary return should be based upon an understanding with the authorities of the institution.

2. Teachers are entitled to freedom in the classroom in discussing their subject, but they should be careful not to introduce into their teaching controversial matter which has no relation to their subject. Limitations of academic freedom because of religious or other aims of the institution should be clearly stated in writing at the time of the appointment.

3. College and university teachers are citizens, members of a learned profession, and officers of an educational institution. When they speak or write as citizens, they should be free

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from institutional censorship or discipline, but their special position in the community imposes special obligations. As scholars and educational officers, they should remember that the public may judge their profession and their institution by their utterances. Hence they should at all times be accurate, should exercise appropriate restraint, should show respect for the opinions of others, and should make every effort to indicate that they are not speaking for the institution.

I want to draw our attention to how the statement links academic freedom and tenure—that is, by means of constellating special privileges (that come with being a professional), with obligations (to behave appropriately), and job security. This three-part constellation, the AAUP makes clear, is necessary to avoid embarrassing the profession, which is itself justified as necessary for the common good. Academic freedom, in other words, comes to us as a facet of and bound by professionalization.

Part of the significance of the professional boundedness of academic freedom lies in helping us understand how the structures that give academic freedom meaning bolster the neoliberalization of the academy we are experiencing now—for example, the exponential growth of contingent labor, and here I return to the AAUP’s history to explain.

The association was founded in 1915 by a group of professors from such private, exclusive research universities as Johns Hopkins and Columbia, who were concerned with limiting the power of governing boards in curricular and thus personnel matters. College and university governing boards were at that time formed by businesspeople and others with vested interests in the kinds of education offered, and exercised substantial control over basic operations. In the nineteenth century, US tertiary education had undergone considerable challenge in the face of the establishment of Darwinism, which undermined the explanatory power of religious orthodoxies that had defined the curricula of colleges to date, and the establishment of public research universities in the Humboldtian model drawn from Germany. The social status and authority of faculty as well as universities and colleges were in flux. By 1915, the founders of AAUP were also responding to the dismissal or refusal to hire faculty on the basis of economic viewpoints (specifically socialist) at odds with those on gov- erning boards.13 In short, AAUP’s founders had many reasons to be concerned about the prospects of teaching and research without external constraints on content given the governance structures of higher education of the era.

That they decided on professionalization as the route necessary to giving faculty the kind of authority to self-government enjoyed by professions like medicine and law is suggestive of the strength and contours of the bourgeois imaginary in play at the time. And it is in this context and with these aims that

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the principles in the 1915 articulation were formulated and expressed in 1940. As Timothy Cain persuasively documents, the elaboration of academic freedom in the 1940 statement was consequent to sometimes contestatory interactions among several different organizations in the interwar years, including the AAUP, the Association of American Colleges (now the AACU), the American Civil Liberties Union, and the American Federation of Teachers.14 Cain notes among these interactions the different orientations of the AFT and the AAUP, differences that can be understood as between the interests of labor and the aims of establishing an elite professional class, as the AAUP sought to do.15 In other words, the stratification of education between K-12 educators, at the core of the AFT’s mission, and college and university professors, underwrites the crystallization of academic freedom.

The elitism of the AAUP of these early days was expressed in part by its insistence on the expertise of faculty and their unique ability to judge the merits of each other’s work. At the time, what was understandably construed as the overreach of college and university governing boards in making hiring and firing decisions underwrote this insistence on expert knowledge. In short, this early iteration of the AAUP understood and rationalized professionaliza- tion by means of insistence on the distinctive judgments of disciplinary peers as necessary to securing freedom from external forces.

It is in the context of McCarthyism that academic freedom as elaborated in the 1940 statement is legalized, specifically by the US Supreme Court’s recognition of a constitutional interest in it, and in ways that sediment an understanding of expert or disciplinary knowledge as necessary to the national interest. In providing universities with cover of academic freedom, the Court concretized disciplinarity insofar as the test of academic freedom defers to professional scholarly standards.16

In Sweezy v. New Hampshire, a 1957 case involving University of New Hampshire professor Paul Sweezy, the Court found that the state’s attorney general had unconstitutionally invaded Sweezy’s guaranteed rights of expres- sion and due process.17 Sweezy had refused to answer questions about his suspected affiliations with the Communist Party. In its decision, the Court found Sweezy’s academic freedom as it pertains to the “intellectual life of a university” more compelling than it did the government’s interest. The Court in 1967, in Keyishian v. Board of Regents, more specifically identified academic freedom “as a special concern of the First Amendment, which does not tolerate laws that cast a pall of orthodoxy over the classroom. . . . Our Nation is deeply committed to safeguarding academic freedom, which is of transcendent value to all of us and not just the teachers involved.”18

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The Court’s interest in academic freedom may be understood, as Robert Post suggests, in terms of the furtherance of “democratic competence,” the “cognitive empowerment of persons within public discourse, which in part depends on their access to disciplinary knowledge.”19 This is knowledge, Post explains, that is understood to be necessary for self-governance and democratic legitimation. That is, the Supreme Court affirmed the importance of disinterested expert knowledge by positing society’s reliance on it as it invested universities with distinct constitutional—which is to say, national—value.20

It takes no special insight to recognize that the legacies of the professional elitism through which academic freedom crystallizes are part of the very terrain against which we contemporarily struggle.21 While the AAUP certainly was not solely responsible for the concretization of disciplinarity and elitist forms of expert knowledge, that these aims and logics grounded the articulation of the principles that would come to be elaborated in the 1940 statement deepens our understanding of the difficulty of delegitimizing disciplinarity, the purview of expert knowledge, and the competencies necessary to gauge each other’s work. The absence of the language of labor in the articulation of academic freedom also helps explain why unionization has come generally late and haltingly if at all to the professoriat.22

This history of professionalization also makes obvious that because academic freedom is reserved for permanent faculty members, the split between tenure track and contract instructors in higher education acts as a built-in leverage against the ability to teach and research freely in the contemporary context, when the vast and growing majority of teachers at the college level are contin- gent workers. Simply, the fewer tenured faculty there are, the less academic freedom there is or can be. In other words, even staying within the given parameters of academic freedom and its relationship to the common good as instated by the 1940 statement, we can recognize its limitations as a defense against the kinds of incursions by external forces—specifically, here, incursions by the forces constituting the reigning political economy—that it was designed to forestall. Indeed, a perhaps paranoid but nonetheless plausible take on the phenomenal growth of contingent labor is its advantage to those who benefit from the minimization of academic freedom, that is, those who benefit from the anti-intellectualism that so robustly accompanies the convergence of the interests of the state and of capital characterizing the current conjuncture.

But neither can we be satisfied with this framing of academic freedom, or by the invocation of the version of the common good it posits. For, we know that the prevailing conception of the common good—prevailing today as it was in 1915 and 1940, if in different iterations—is that which obscures its

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own historicity, including especially how it secures itself in modern nationalist terms through rationalization of the dispossession of native peoples and through logics that buttress the dichotomies between good and bad subjects—that is, the public versus the criminal, the immigrant versus the alien, the citizen versus the terrorist, and so on—in brief, the dichotomies that are a key technology of racialization.

Understanding that professionalization, academic freedom, and US na- tionalism are intimately linked projects vis-à-vis this posited “common good” also reminds us how academic freedom “here” is implicated in the US nation’s contribution to producing the social and political unrest characterizing those places identified in the Scholars at Risk Network’s report—Venezuela, South Africa, Niger, Cameroon, Turkey, and India—where violence on campuses has erupted, whether through its economic policies or military incursion, as recently exemplified in Niger. US academic freedom may be understood in this way to be linked to the suppression of academic freedom elsewhere.

We can also recall how academic freedom was used to criticize the curricular transformations wrought by the social movements and students and teachers of the 1960s and 1970s—how “political correctness” emerged as a term of the culture wars to criticize what was cast as the politicization of the academy against its true nature.23 Such criticisms and deployments of academic freedom eluci- date the specifically nationalist character of education. That academic freedom can be used by both those arguing for inclusion and exclusion is symptomatic of its cathection to nationalism and liberalism in the United States.

As Frances Julia Riemer explains of the “discursive packaging” of Arizona legislation regarding Mexican American studies and guns on campus, “the stars and stripes of liberalism” are used to justify both advocacy for and opposition to them: “Liberalism . . . affords the vocabulary to both establish and contest political authority.”24 According to her convincing account, state officials took the position that Mexican American studies advances “ethnic chauvinism” and is thus antithetical to the sacrosanct liberal value of equality, while teachers and students countered by invoking civil rights history and idioms, to argue that Mexican American studies advanced the liberal democratic project. Meanwhile, advocates for campus carry relied on individual rights arguments while “even those who advocate for increased gun control pay their respects [to the Second Amendment] before they offer their rebuttals.” As Riemer rightly notes, “This rights-based discourse obscures the economic interests behind gun initiatives,” that is, the interests of the $4 billion gun industry. Riemer concludes: “In 21st century Arizona, liberalism is our system of representation. . . . Both sides of the political continuum, assimilationists and pluralists, pro-gun and anti-gun

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activists, we all wrap ourselves in the flag.”25 Until and unless we can defunc- tion this epistemology, academic freedom will remain precisely that, that is, a construct the usefulness of which is delimited to the defense of liberal ideology as it rationalizes US nationalism.

Extramural Speech, or The Problem of the Individual

In 1960 Leo Koch, an assistant professor of biology at the University of Illinois, Urbana-Champaign, wrote a letter in response to an article published in the Daily Illini, the student newspaper, which he signed with his institutional title. As John Wilson reports, Koch was responding to a multiauthored forum titled “Sex Ritualized,” which, in Wilson’s words, “lamented how unfortunate it was that men were obliged to be ‘smooching’ with women in a sorority ‘until the one o’clock ‘dong’ relieves [them] from their chivalrous duty.”26 The authors complained in this forum that “‘male-female relations on campus’ have ‘stul- tified into a predetermined ritual.’”27 In essence, they were complaining that they were unable to do anything more than “smooch.” Koch’s response, which was published a few days later, chided the paper for failing to acknowledge the moral and social norms that sustained everything from outdated Victorian codes of behavior to reigning double standards for the sexual lives of men and women. Koch concluded: “‘With modern contraceptives and medical advice readily available at the nearest drugstore, or at least a family physician, there is no valid reason why sexual intercourse should not be condoned among those sufficiently mature to engage in it without social consequences and without violating their own codes of morality and ethics.’”28

Driven in large part by a figure described as “‘a right-wing anti-communist former missionary to China, whose daughter was a student at the university,” a campaign to fire Koch was organized in the state legislature and among par- ents.29 A pamphlet denouncing Koch was circulated, which described his letter as “‘an audacious attempt to subvert the religious and moral foundations of America.’”30 University of Illinois administrators, a mere ten days after Koch’s response was published, and without consulting his faculty peers as the prin- ciples of academic freedom warranted, determined his letter to be irresponsible and voted to fire him. The university president at that time, David Dodds Henry, “declared that Koch’s views were ‘offensive, repugnant and contrary to commonly accepted standards of morality and [that] his espousal of these views could be interpreted as an encouragement of immoral behavior . . . for these reasons he should be relieved of his University duties.’”31

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This case turned out to be formative in the shaping of the AAUP’s stance on extramural utterances. Until then, as established in the 1940 statement, extramural utterances had to meet professional standards insofar as faculty were enjoined to “exercise appropriate restraint, . . . show respect for the opinions of others, and . . . make every effort to indicate that they are not speaking for the institution.” At issue in 1960 for the AAUP was whether a standard of “responsibility,” conceptualized in terms of a “gentleman scientist model,” should be retained. While the organization censured the university for failing to follow due process by not consulting with faculty before dismissal, as well as for what it judged to be the disproportionate punishment for Koch’s behavior, the politics of professional respectability loomed large over its consideration of this case. It is bitingly ironic given the University of Illinois’s recent flagrant demonstration of its unwillingness to heed its expert faculty in matters of personnel and curriculum when those faculty are in American Indian studies, that the school responded to censure following its treatment of Koch by trans- forming its processes and articulating strongly worded policies on academic freedom and tenure. Such irony, of course, attests to the ongoingness of settler colonialism and its multifaceted minimization of indigeneity.

The problem the AAUP faced was how to isolate speech as a citizen from that of speech as a professional. The 1915 principles and the 1940 statement had made the distinction based on the relationship of such speech to areas of expertise: if a professor spoke on a topic within his realm of expertise, that speech was to be held up to the normative standards of the profession; if that speech was unrelated to his expertise and removed from the spaces where professional obligations hold—the classroom, for one—then it should be protected under the umbrella of academic freedom as a matter of the rights of a citizen to engage freely in public discourse. Moreover, the manner of speech matters insofar as it ought to take into account the unique social position of professional scholars and thus be presented in an appropriately dignified way. While the AAUP undertook renovation to its understanding of extramural speech following the Koch case, and in part as a result of both McCarthyism and the social and intellectual movements of the 1950s and 1960s—renova- tion resulting in a statement that more actively affirms the professor’s right to extramural speech—its current position retains a relationship between grounds for dismissal and extramural speech as an indicator of professional competence. More important, beyond the AAUP’s debates on these matters, we can observe across the landscape of education at all levels an energetic and proliferating reliance on codes insisting on civility.32

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We can readily acknowledge the historic and ongoing usefulness of the mandate to civility in the service of dispossession, subjugation, and devasta- tion of people and planet. The civilizing and Christianizing mission of early higher education in the United States resonates strongly today. Perhaps espe- cially in the aftermath of 9/11, with the unapologetic resurgence of the kinds of efforts to subordinate and censor intellectual activity to the interests of the US nation-state associated with McCarthyism—remember the PATRIOT Act—and the corollary ways that the state has redoubled its efforts to secure Christianity as the US nation’s religious and moral foundation by regulating the nation’s borders, that both the content and manner of expression continues to be at issue is patently clear. As the dismissal of and/or refusal to hire people expressing support for Palestinians or criticism of Israel in ways that challenge the gentlemanly model of speech attests, as well as the suspension of others for public remarks made in response to anti-Black racism and white supremacy indicates, the judgment of incivility still operates as part of the machinery of suppression of illiberal dissent.33

We may refuse the liberal framework of academic freedom and the dictates of professionalism that ask us to consider the civility of speech, to ask instead, who gets to express themselves with incivility—through state-sanctioned violence and/or racist, sexually violent, and white supremacist discourse and activities, say—and for whom is incivil speech impermissible?34 In fact, we know that it is speech emergent outside the powerfully structured domain of civility that is a practice of freedom—of the freedom that is before and beyond the academy, civic society, or broadly, the liberal order. It is the freedom that is not given or even taken but is already and has always been ours, the freedom that inheres in existence. This freedom punctures perhaps the biggest ruse of liberal freedom, namely, the idea that freedom is something to be given or earned.

Stretch time around the case of Koch’s academic freedom and the centrality of the regulation of desires and behaviors to which we give the name gender and sexuality, and the political uses of “sex panics” come to the fore as key facets of the world liberal freedom attempts to secure;35 stretch time and the histories of orientalism, US imperialism, and the rationalities of racial capitalism that give rise to the figure of the “right-wing anti-communist party former missionary to China” may be brought to bear. Stretch time and the voracious logics and effects of settler colonialism come to the fore, that is, the dispossession of the indigenous people who lived on the lands on which the University of Illinois is built emerges with great clarity—through the Morrill Act of 1862 that gave rise to land grant universities by territorial acquisition advanced by such US legislation as the 1887 Dawes Act, the 1862 Homestead Act, the Indian

| 167Pedagogies of Dissent

Removal Act of 1830, and, by the establishment of Illinois as a state in 1818, each of which is a marker in a long history of compulsory cession and warfare, and involving ongoing incursions in the territory referred to as Illinois, and resulting in the university’s location on lands seized from the peoples of the Kaskaskia, Miami, Shawnee, Sauk, Fox, Chippewa, Ottawa, Chickasaw, and Potawatomi tribes. Those gender and social codes to which Koch’s case refers us are, we remember, facets of the settler colonial regime realized in the university.

The given paradigms by which we are to understand and use academic free- dom isolate utterances and individuals to insist that the contexts that matter are professional and institutional. But if we stretch time, the potent context of modern nationalism/settler colonialism becomes strongly palpable. In other words, the pedagogical in an illiberal mode allows us to remember that the problem isn’t an individual’s decision to speak on a given subject and in a given way; the problem is the interested material power behind the very conception of the autonomous rights-bearing citizen iterated by academic freedom and that marks its liberal-nationalist grounds. This is not simply a matter of com- plicating the distinctions between professional and citizenly speech but one of understanding that the distinction itself relies on the assumption of individual autonomy in decision-making without regard for either the historicity of that understanding of autonomy or the social nature of choice.

Speech deemed uncivil is, I believe, better (illiberally) understood as speech expressed in the absence of a choice to do so: when conditions are such that speech is not willed but eruptive, when one cannot choose not to speak—to wit, in confronting racism and settler colonialism and cis-heteropatriarchy as the expressions provoking recent academic freedom cases have done. Such speech, uncivil speech, is an expression of the freedom that was/is ours before and beyond.

By Way of Closing . . . the Work of Associating

I am inspired by how this association, its constituency and our collective com- mitments to speaking truth to power, has come to take its current shape. I think of the planning and labor of people—of women of color and like-minded people—inside and outside the ASA who refused the exclusionary nature of the academy and organized in ways to remake it, who afforded the shifts to the world and this association that allow this annual meeting to be, for so many of us, a place where defensiveness is neither necessary nor useful. In one sense, what I’ve been doing in and with this address is asking us collectively, given all the conditions to which the declaration of academic freedom’s vulnerability

| 168 American Quarterly

refers, what will we have wanted the ASA to have been and done some decades hence? What difference can our work of associating make? Through associating, the AAUP professionalized the activities of teaching and research. Through associating, what will we do?

Here are some of the things I’m thinking. It will be necessary to continue to proliferate the unlearning of inherited

ways of knowing and being, to take flight from dissent in ways that reckon with the contradictions and entanglements of history that necessitate thinking nation, race, gender, sexuality, coloniality, indigeneity, class, and ableism all at once, which is to say, to be ever attentive to the complex ways in which power operates to dis/organize life. This kind of reckoning is difficult: it requires not merely acknowledging but living and thinking inside the cacophony, to bor- row Jodi Byrd’s term, to allow un/learning to remake us and elucidate other worlds.36 In this, what I am referring to as illiberal intellectual traditions are vital to undoing powerful ignorance and the power of ignorance, that is, of anti-intellectualism. One of the things we learn from these traditions over and over again is that it will take all of us to make sense out of cacophony, to elabo- rate the rationalities and sensibilities and structures that allow life to flourish.

Stretch time around the work of associating and part of what you will find, I believe, is, instead of defensiveness, the kind of collegiality that helps us make and hold up our worlds—of a form of relationality only possible when uninvested in professional advancement or in the defense of institutions or principles but rather inclined always toward people and the planet in their inseparability. Collegiality gives rise to the deep joy that inheres, as much as risk, in the fact of being, and being together: it lives in what Sylvia Wynter described as the unimaginability of writing an autobiography because of the imperative of the collective; it resounds in the ethos of interrelationality or- ganizing the Combahee River Collective so vital to the critique of capital and empire they advance; it is a form of the mutuality Sandy Grande emphasizes as necessary to undoing settler colonial education; it pervades the collective creative imaginaries of This Bridge Called My Back; it is an iteration of Jose Muñoz’s “being with”; it is what brings us back to the table, to this table, to this annual meeting, even when there is conflict, antagonism, difficulty, risk. The collegiality of all these people, of all of us, of all their and our work, makes us possible, continues in and through us.

What I am describing is a pedagogy of dissent, an organized approach to un/learning grounded in the world and founded in generosity and compassion, understood to be essential to social transformation. This pedagogy of dissent

| 169Pedagogies of Dissent

names the doing of work even, perhaps especially, when that doing might mean an undoing of belief, of job security, social standing, sometimes of self, when that doing requires courage for the sake of the collective. (I remember here that Paulo Freire opens Pedagogy of the Oppressed by emphasizing generosity and love.) These are the grounds and affordances of collegiality as a mode of refusing subjection.

Since I heard Roderick Ferguson speak brilliantly at this annual meeting about courage as a condition of possibility for intellectual activity, about the interdisciplinarity given to us through social movements as an intellectual formation that creates and proliferates new forms of courage—specifically, of the courage to be nonaligned with the powerful—I’ve been thinking about how exactly right he is and about how courage is pressed on us as much as defensiveness.37 Courage is conditioned by the fact of vulnerability and its exacerbation; it is activated not by individual choice but because circumstances make it such that there is no choice but to be courageous, nonaligned, even insurrectionary, no choice but to take up the insurrectionary work of intel- lectual activity that refuses to be disciplined, professional, civil.

Given racism in its intersectional and state-sanctioned forms, given all the ways the world’s unevenness is indexed by the bodies with which we experi- ence it, for some, the everyday act of walking on the street is courageous; for some, academic environments require bravery simply to enter; for some, courage is in these and manifold other ways a fact of daily existence. This courage learned experientially gives rise to the understanding that courage as teachers and scholars and students elicits work in the face of uncertainty, work that elaborates the genealogies of the insurrectionary forms and formations of knowledge of nonaligned pedagogies of dissent.

This is to say that we will need to remember that risk has always accompanied dissent, and that risk asks us what we will give up, not in a sacrificial model, but through the collective expression of being, of collegiality, that remakes our selves, that reveals the self to be collective, constitutively communal, relational, plural. Indeed, we will need to remember that risk inheres in the very fact of existence, that the differential distribution of risk means there is no universal safe space, and to be ever mindful of how the desire to inoculate ourselves from this vulnerability aligns all too well with the interests of the security state. More policing, more surveillance, more laws, more bureaucracy, more defensiveness have yet to produce more life or invulnerability.

We will need to continue to mobilize to address immediate needs. And through the pedagogical—not only but distinctly through the pedagogical—

| 170 American Quarterly

we may take the longer view necessary to organize around the conditions that induce dissent as part of a collective, collaborative, collegial effort to refuse, refute, and defunction them. The work of associating may itself be pedagogi- cal in this regard. If we will have been able to continue to create time to learn and unlearn with all that that implies as to the organizing we will have to do, the structures we’ll need to put in place, simply, the work we will need to do, we will have made a world in fact worth defending.

In the end, I am simply observing that stretching time is our collective superhero power. We must use it courageously.

Notes This address reflects insights and ideas drawn from ongoing conversations with and the luminous

work of many people. With abundant gratitude, I’m very glad to acknowledge foremost among them Jodi Byrd, Lisa Duggan, Roderick Ferguson, J. Jack Halberstam, Ruth Wilson Gilmore, Gayatri Gopinath, Laura Hyun Yi Kang, Lisa Lowe, Jodi Melamed, Karen Shimakawa, Siobhan Somerville, and Alexandra T. Vazquez.

1. The period covered is September 1, 2016, to August 31, 2017. Scholars at Risk, Free to Think 2017, www.scholarsatrisk.org/wp-content/uploads/2017/09/Free-to-Think-2017.pdf.

2. See, e.g., Philip G. Altbach, “Academic Freedom: International Realities and Challenges,” Higher Education 41.1–2 (2001): 205–19; Solomon R. Benatar, “Freedom of Speech, Academic Freedom, and Challenges to Universities in South Africa,” Society 53 (2016): 383–90; Alexis Gibbs, “Academic Freedom in International Higher Education: Right or Responsibility?,” Ethics and Education 11.2 (2016): 175–85; Lis Lange, “Thinking Academic Freedom,” Arts & Humanities in Higher Education 15.2 (2016): 177–86; and Mahmood Mamdani, “Between the Public Intellectual and the Scholar: Decolonization and Some Post-independence Initiatives in African Higher Education,” Inter-Asia Cultural Studies 17.1 (2016): 68–83.

3. These states are Arkansas, Colorado, Georgia, Idaho, Kansas, Mississippi, Oregon, Texas, Utah, and Wisconsin. For a review of the legislative actions and implementation policies regarding campus carry across the country, see Andrew Morse, Lauren Sisneros, Zeke Perez, Brian A. Sponsler, Guns on Cam- pus: The Architecture and Momentum of State Policy Action (Washington, DC: NASPA and Education Commission of the States, January 2016).

4. See Christopher Newfield, The Great Mistake: How We Wrecked Public Universities and How We Can Fix Them (Baltimore, MD: Johns Hopkins University Press, 2016); Christopher Newfield, Unmaking the Public University: The Forty-Year Assault on the Middle Class (Cambridge, MA: Harvard University Press, 2008).

5. Academic freedom policies and principles have yet to attend thoroughly to social and other forms of contemporary mediation. See John K. Wilson, “The Changing Media and Academic Freedom,” Academe 102.1 (2016): 8–12; see also Kimberly W. O’Connor and Gordon B. Schmidt, “‘Facebook Fired’: Legal Standards for Social Media-Based Terminations of K-12 Public School Teachers,” Journal of Workplace Rights 5.1 (2015): 1–11; and Henry Reichman, Ashley Dawson, Martin Garnar, Chris Hoofnagle, Rana Jaleel et al., “Academic Freedom and Electronic Communications,” Academe 100.4 (2014): 18–23.

6. For a wide-ranging annotated bibliography on the literature on academic freedom, see Stephen H. Aby and James C. Kuhn IV, eds., Academic Freedom: A Guide to the Literature (Westport, CT: Greenwood).

7. In this, I am aligned with Rajini Srikanth’s reminder of the necessity of attending to the “axis of power” in discussions of academic freedom (“The Axis of Power and Academic Freedom,” Journal of Asian American Studies 19.1 [2016]: 105–10).

| 171Pedagogies of Dissent

8. On the foundations of US universities, see Craig Steven Wilder, Ebony and Ivy: Race, Slavery, and the Troubled History of America’s Universities (New York: Bloomsbury, 2013).

9. Kandice Chuh, “What Is ‘America’ in ‘The Asian Century’?,” Journal of American Studies 49.3 (2017): 299–321.

10. Edward Said, “Identity, Authority, and Freedom: The Potentate and the Traveler,” in The Future of Academic Freedom, ed. Louis Menand (Chicago: University of Chicago Press, 1996), 214–28, 221.

11. Lisa Lowe, Intimacies of Four Continents (Durham, NC: Duke University Press, 2015). 12. The AAUP provides the full text of the 1940 Statement of Principles on Academic Freedom and Tenure

on its website: www.aaup.org/report/1940-statement-principles-academic-freedom-and-tenure, along with a brief history of its formulation; the 1970 interpretive comments appear as footnotes to the statement.

13. Of particular note is the dismissal of Edward Ross from Stanford University at the behest, the AAUP reports, of Jane Lathrop Stanford, the university’s cofounder, whose husband had made his fortune in the railroad industry. Ross reportedly criticized railroad monopolies and the use of immigrant labor (www.aaup.org/about/history/timeline-first-100-years).

14. Timothy R. Cain, Establishing Academic Freedom: Politics, Principles, and the Development of Core Values (New York: Palgrave Macmillan, 2012).

15. Ibid., chap. 2. 16. Robert Post, Democracy, Expertise, Academic Freedom: A First Amendment Jurisprudence for the Modern

State (New Haven, CT: Yale University Press, 2013); on the constitutional interest in academic freedom, see J. Peter Byrne, “Academic Freedom: A ‘Special Concern of the First Amendment,’” Yale Law Journal 99.2 (1989): 251–340; David M. Rabban, “A Functional Analysis of ‘Individual’ and ‘Institutional’ Academic Freedom under the First Amendment,” in “Freedom and Tenure in the Academy: The Fiftieth Anniversary of the 1940 Statement of Principles,” special issue, Law and Contemporary Problems 53.3 (1990): 227–301.

17. Sweezy v. New Hampshire, 354 U.S. 234 (1957). 18. Keyishian v. Board of Regents, 385 U.S. 589 (1967), 603. 19. Post, Democracy, Expertise, Academic Freedom, 34. 20. Ibid., 76. 21. As Judith Butler puts it, “If disciplinary innovation becomes the price we pay in order to establish a

basis on which to legitimate an argument against unwanted political intrusions, then it would seem we establish a conservative academic culture and even suppress disciplinary innovation, as well as inter- disciplinary work, in order to preserve academic freedom. Then, of course, we have to ask, for whom is academic freedom preserved and for whom is it destroyed, and with what sense of the academic are we left?” “Critique, Dissent, Disciplinarity,” Critical Inquiry 35, 4 (Summer 2009), 773–795, 774.

22. It is telling that in a case involving a lecturer at Cal State Fullerton, the force of the California Faculty Association, the Cal State system’s union, was vital to their reinstatement. See neatoday.org/2017/07/24/ cal-state-lecturer-reinstated-victory-for-academic-freedom/. The AFT’s statement on academic freedom explicitly identifies labor conditions and contract terms as necessary to consider in relation to academic freedom. See www.aft.org/position/academic-freedom.

23. See the essays collected in Louis Menand, ed., The Future of Academic Freedom (Chicago: University of Chicago Press, 1996), which speak to these issues.

24. Frances Julia Riemer, “Wrapped in the Flag: Liberal Discourse, Mexican American Studies, and Guns on Campus,” Critical Education 5.8 (2014): 1–13, 2.

25. Ibid., 8. On constitutional issues related to campus carry laws, see Shaundra K. Lewis, “Crossfire on Compulsory Campus Carry Laws: When the First and Second Amendments Collide,” Iowa Law Review 102.2 (2017): 2109–44. For discussion of specific state laws, see, e.g., David Hsu and Jessica Weekley Truelove, “HB 859—Offenses against Public Order,” Georgia State University Law Review 33.1 (2016): 20–29; Aric K. Short, “Guns on Campus: A Look at the First Year of Concealed Carry at Texas Universities,” Texas Bar Journal 80.8 (2017): 516–17; Christopher M. Wolcott, “The Chilling Effect of Campus Carry: How the Kansas Campus Carry Statute Impermissibly Infringes upon the Academic Freedom of Individual Professors and Faculty Members,” Kansas Law Review 65 (2017): 875–911.

26. John K. Wilson, “Academic Freedom and Extramural Utterances: The Leo Koch and Steven Salaita Cases at the University of Illinois,” AAUP Journal of Academic Freedom 6 (2015): 2.

| 172 American Quarterly

27. Ibid. 28. Quoted in Wilson, “Academic Freedom,” 3. 29. Wilson, “Academic Freedom,” 3. 30. Ibid. 31. Ibid. 32. On the challenges posed to academic freedom by civility codes, see Risa L. Lieberwitz, “Civility and

Academic Freedom,” Journal of Collective Bargaining in the Academy (April 2015), vol. 0 National Center Proceedings 2015, Article 10.

33. See Doumani for extensive discussion of academic freedom in the aftermath of 9/11. See also Evyn Lê Espiritu, “Civility, Academic Freedom, and the Project of Decolonization: A Conversation with Steven Salaita,” Qui Parle 24.1 (2015): 63–88.

34. The formulation of this question is indebted to Ruth Wilson Gilmore. 35. On the politically interested construction and uses of “sex panics,” see Lisa Duggan and Nan H.

Hunter, Sex Wars: Sexual Dissent and Political Culture, 10th anniversary ed., (New York: Routledge, 2006).

36. Jodi Byrd, Transit of Empire: Indigenous Critiques of Colonialism (Durham, NC: Duke University Press, 2011).

37. Roderick Ferguson, “The Crisis of Alignment,” paper presented at American Studies Association Annual Meeting, Chicago, November 9, 2017.

contexts.org30

being a transnational korean adoptee, becoming asian american

by wendy marie laybourn

http://crossmark.crossref.org/dialog/?doi=10.1177%2F1536504218812866&domain=pdf&date_stamp=2018-12-13

31FA L L 2 0 1 8 c o n t e x t sContexts, Vol. 17, Issue 4, pp. 30-35. ISSN 1536-5042, electronic ISSN 1537-6052. © 2018 American Sociological Association. http://contexts.sagepub.com. DOI 10.1177/1536504218812866.

Thirty years ago, Korea hosted the Summer Olympics for the first

time. While the global spotlight highlighted Korea’s miraculous

transformation from destitute to highly developed, not every head-

line was celebratory. Among critical commentary from a variety of

countries, North Korea took advantage of the political stage, char-

acterizing Korea’s sending of its children to adoptive families in

Western countries as the “ultimate form of capitalism.” In response

to this global shame, Korea’s Minister of Health and Social Welfare

announced that the country would cease international adoption.

Afterward, Korea’s international adoption slowed, but even today,

Korean children continue to be adopted to the United States.

Since the 1950s, U.S. families have adopted over 125,000

Korean children. Adoption from Korea was the first sustained

intercountry adoption program to the United States. To date,

Korean adoptees comprise about 25% of all international

adoptions to the United States and are the largest group of

transnational transracial adoptees in adulthood. It should come

as little surprise, then, that during the 2018 Winter Olympics

in Korea, Korean adoptees were again the subject of Olympic

headlines. Rather than demonstrate ire from the global com-

munity, however, these human interest stories followed U.S.

Korean adoptees who had been scouted to South Korean teams

and were returning to their birth country, often for the first time.

Over the past three years I surveyed, interviewed, and joined

hundreds of Korean adoptees across the United States and in

Korea to understand how this unique group of Asian Americans

navigates belonging. Whereas in the United States popular

press have portrayed Korean adoptees as evidence that we are

“beyond race,” in Korea, government officials herald adoptees

as global ambassadors bridging the two nations. Yet Korean

adoptees often report feeling in-between races, cultures, and

identities. To learn how Korean adoptees fit in to United States,

Korea, and Asian America, we must first go back to when Korean

adoption began.

from past to present Over honey citron tea and melon cream bread at a local

Korean bakery, Mary*, 54, told the story of the day she was

chosen by her (adoptive) mom. Mary had been at an orphanage

for the first 10 years of her life. Older children often age-out

of orphanages, as adoptive families tend to want younger chil-

dren, infants if possible. However, Mary’s mother specifically

requested an older child. “I remember they chose 10 kids, ages

ranging from 8 to 11, and that’s the age range she wanted,”

Mary reflected. “I remember going through that process. It was

almost like American Idol, being picked out.”

Although the idea of selecting children paraded on display

reduces family-building to simple consumerism, Korea’s selec-

tion of healthy children and the ease of its adoption process

established it as the “Cadillac of adoption programs.” Prior to

adoption from Korea, international adoptions were carefully

controlled family-making meant to minimize difference through

matching children and adoptive parents by physical features,

religion, and temperament. The goal was that these adoptions

appear “as if begotten.”

https://doi.org/10.1177/1536504218812866

32 contexts.org

Adoption from Korea changed these norms. White adop-

tive families were sold on adopting Korean children directly after

the Korean War. Newspapers and television showed images of

abandoned children, missionaries returned from Korea bringing

news about these children in need, and U.S. G.I.s stationed in

Korea set up some of the first Korean orphanages, often writing

home to their families in the United States asking for donations.

So Korean orphans flooded American consciousness. But,

it was after Harry and Bertha Holt’s very public 1955 adop-

tion of eight Korean children, “seeds from the East,” fulfilling

what Harry called a “mission from God,” that adoption from

Korea soared. Through media framing, first-hand accounts from

Christian missionaries to church congregations, and the Holts,

Korean adoption became linked to Christian ideals of helping

the fatherless.

On the geopolitical stage, U.S. aid to Korea secured the

country’s position as “big brother” to a fledgling Korean nation-

state, while, within the United States, White American families’

adoption of Korean children affirmed U.S. perceptions of East

Asians as “model minorities.” Adopted Korean children joined

their White adoptive families during a time of otherwise exclu-

sionary Asian immigration policies. The juxtaposition emphasized

Korean adoptees’ exceptional status.

Though Korean children were obviously racially different

from their White adoptive parents, mainstream press and adop-

tion agencies portrayed this difference as negligible. Korean

children were seen as having a racial flexibility and benign

exoticism. The assumption was that Korean children would,

and could, assimilate totally into their White families. Social

work best practices at the outset of Korean adoption were that

no attention be given to transracial adoptees’ racial difference

or heritage culture. These transnational transracial adoptees

were seen simply as family members, not racially different and

not immigrants.

Once Mary was adopted, her ties to Korea were essentially

severed. Her mother wanted her and her sister to learn English

fluently without an accent. They took ESOL (English to Speakers

of Other Languages) courses in an American school and also

had an English tutor. “She forbid me and my adopted sister to

speak Korean to each other,” Mary recalled. “It was just like

going from being a Korean to American overnight. From culture

point-of-view to language point-of-view in every way.”

Though international adoption is often seen as a firmly

middle-class phenomenon, given the timing of early adoption

from Korea (before the policies and practices of today), fami-

lies from a range of socioeconomic backgrounds were able to

adopt. Working-class, middle-class, and wealthy families from

cities, suburbs, and rural areas across the United States adopted

Korean children. The two constants across these adoptive fami-

lies were that the overwhelming majority were White and most

resided in predominantly White communities. I interviewed over

100 Korean adoptees (these survey and interview respondents

were identified through Korean adoptee organizations, adoptee

activities, Korean adoptee list servs, and snowball sampling,

and though this is a convenience sample,

the demographic data and experiences

mirror previous research on Korean adop-

tees), and virtually all had been adopted

by White adoptive parents and 92%

reported growing up in a predominately

White community. These factors, com-

bined with parents’ approaches to their

Korean children as devoid of racial differ-

ence, posed challenges to Korean adoptees’ racial and ethnic

identity development.

neither quite white nor completely korean Like the majority of Korean adoptees, Stacey, 38, grew up

in a predominantly White town. As a child, when people asked

where she was from, she would tell them Ireland. “I thought

I was Irish,” Stacey recalled. “Then, I thought I was Italian for

a little while. I really was so confused. I had no idea, but it

didn’t last very long because people would look at me and go,

‘What?’”

Stacey was the only Asian person in her otherwise Irish-

Italian community. It seemed logical to her that she was Irish or

Italian like everyone else, particularly because, like other adoptive

parents, her parents took a colorblind approach to her upbring-

ing. Still Stacey found that neither of those identities was fully

available to her.

When she was in the fifth grade, two Japanese boys moved

into Stacey’s neighborhood. Even though, by then, she “was

always reminded that [she] was Asian and adopted by everyone

else,” Stacey wanted to separate herself from other Asians,

especially these two classmates. Dozens of other Korean adop-

tees I interviewed echoed this experience. Most grew up in

predominantly White neighborhoods, attended predominantly

Social work best practices at the outset of Korean adoption were that no attention be given to transracial adoptees’ racial difference or heritage culture.

33FA L L 2 0 1 8 c o n t e x t s

White schools, and

identified as White

during childhood.

Almost none identi-

fied as immigrants

w h e n t h e y w e re

growing up. Their

experiences and ways

of thinking about

themselves lined up

with the social work

best practices of the

time, aimed at mini-

mizing transnational

transracial adoptees’

racial, ethnic, and

immigrant status dif-

ference from their

adoptive families.

T h o u g h o v e r

60% of the adoptees

I surveyed character-

ized their parents’ attitude toward their Korean heritage culture

as “not important,” they nonetheless learned that their Asian

group membership was important to how they were perceived

by others. As they interacted with people outside of their

immediate neighborhoods, even with extended family mem-

bers, respondents reported encountering the expectation that

they were knowledgeable about their heritage culture, spoke

their heritage culture language, or had ties to Asian American

communities. In my interviews, Korean adoptees also relayed

common experiences of racialization, such as being told to

“Go back to where you came from!” or being bullied because

of their racialized physical features. These experiences taught

respondents that even though they felt firmly rooted within

their White adoptive families, the expectation beyond their

homes was that they were accountable for their racial group

membership—the very identity social workers had downplayed,

diminished, and ignored.

So adoptees were reminded that they were Asian, but they

didn’t exactly know how they fit into Asian America. For some,

this came out of unfamiliarity with other Asian Americans; for

others, from internalization of negative perceptions about their

racial group.

creating community What happens when you don’t feel fully part of either of

the communities you are expected to belong to?

A critical mass

of Korean adoptees

was coming of age

as the inter net’s

mainstream expan-

sion took hold in the

mid-1990s. Korean

adoptees started to

use online message

boards to find peo-

ple like themselves.

Facilitated first by

Yahoo! Groups and

now by Facebook

G r o u p s , K o r e a n

adoptees created

spaces to find one

another, share their

experiences, and

explore their Korean

heritage culture. For

some, these online

spaces offered their first connections to other adoptees. Due

to geographic constraints, some Korean adoptees’ interactions

remain constrained to the online groups, while for many others,

in-person meet-ups extend their connections into “real-world”

spaces.

“Where did you grow up?”

“Have you been back to Korea?”

“Have you done a birth family search?”

“Any suggestions for where to take Korean [language]

classes?”

Over a family-style meal at a local restaurant, a flurry of

questions and recommendations filled the air. About a dozen

adult Korean adoptees, women and men ranging in age from

their late 20s to early 50s, were bonding. Some were new to the

Korean adoptee community and others more established, but

they were coming together over experiences such as being the

“only one”—the only Asian, the only adoptee—when they were

growing up, addressing race or avoiding race altogether with

their White family members, and visiting Korea for the first time.

Korean adoptee groups like these can be found across the

United States. Some have only a handful of members, like the

newly formed Tennessee Korean Adoptees group, while others,

like the New York City based Also-Known-As group that began

in 1996, count hundreds of members. These groups have a

sustained online presence, but also meet up for monthly dinners,

weekly workshops, and other events based on the members’

An #18MillionRising sticker pack shows the range of issues Asian American activist networks undertake, including, but also going far beyond Adam Crapser’s individual deportation case. (Available for purchase at store.alliedmedia.org/products/18-million-rising-sticker-set.)

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34 contexts.org

self-identified needs and interests. Through these groups, Korean

adoptees normalize their family formation but also carve out

space to express an Asian American identity often missing

from mainstream understandings. The impact of these groups

is demonstrated by the 32% of my survey respondents who

participated in Korean adoptee group activities and identified

distinctly as “Korean adoptees,” a reference to their Korean heri-

tage culture, American upbringing, and adoptee background.

Korean adoptees articulate feeling in-between the White-

ness of their adoptive families and the Korean-ness of their

heritage culture, yet I had heard similar feelings expressed by

second-generation Korean Americans. At a panel on “Korean

American Influencers in the Age of YouTube” at the Council for

Korean Americans’ annual summit, for example, Korean Ameri-

cans, the second-generation sons and daughters of Koreans who

immigrated to the U.S., described feeling as if they didn’t fit into

mainstream American culture because of their assumed foreign-

ness. They also felt disconnected from Koreans of their parents’

generation, because they grew up in America. As I listened to

these second-generation Korean Americans articulate their dual

exclusions, I was struck by how comparable they sounded to

Korean adoptees.

What both the Korean adoptees and the second-gen

Korean Americans were expressing were feelings of conditional

acceptance within Asian communities and a lack of visibility

in mainstream American culture. Though their conversations

seemed to miss one another, they were responding in similar

ways—through YouTube and other online platforms.

While these Korean Americans were leveraging user-gen-

erated media to create alternative Asian American content, in

mainstream news, another headline was forming.

american… without citizenship In early 2015, U.S. Immigration and Customs Enforce-

ment (ICE) served deportation paperwork to Adam Crapser, a

40-year-old Korean adoptee who had come to the United States

as a toddler. Crapser had recently applied for a green card. His

background check was flagged for a crime he had committed

and for which he had served time. By the spring of 2015, the

New York Times was covering his “bizarre deportation odyssey.”

In the article, Crapser is quoted: “I was told to be American. And

I tried to fit in. I learned every piece of slang. I studied everything

I could about American history. I was told to stop crying about

my mom, my sister, Korea. I was told to be happy because I was

an American.”

Yet Crapser was, in fact, not an American. His adoptive

parents never took the necessary steps to secure his U.S. citizen-

ship. Though currently, under the Child Citizenship Act of 2000

(CCA), international adoptees adopted by U.S. citizens receive

automatic U.S. citizenship, that was not always the case. At the

time of Crapser’s adoption (and up until the enactment of the

CCA) it was incumbent upon adoptive parents to know they

needed to naturalize their adopted children and to follow the

necessary steps to do so. Many parents

either did not know or, if they did know,

did not do so due to the high costs or out

of neglect. There are an estimated 35,000

international adoptees without citizenship.

The majority of these are Korean adoptees.

Crapser’s case sent shock waves

through the Korean adoptee community. Korean adoptees had

been told all their lives that they were American, yet here was

the most absolute refutation of that. Almost immediately after

Crapser’s deportation paperwork was served, Kevin Vollmers of

Gazillion Strong, an adoptee-created advocacy group, began

advocating for Crapser. Korean adoptee groups across the

United States joined in supporting Crapser’s case and called for

a legislative fix. Their goal was “citizenship for all adoptees.”

Crapser’s case activated a communal Korean adoptee identity,

as adoptee organizers emphasized that he “could be any of

us.” The specter of deportation emphasized Korean adoptees’

immigrant status in a way previously unimaginable.

Joining the mobilization efforts were Asian American activ-

ism networks such as 18MillionRising and the National Korean

American Service and Education Consortium (NAKASEC). In

the spring of 2015, 18MillionRising launched a campaign to

#KeepAdamHome, which included a petition against Crapser’s

deportation and fundraising for his legal defense. By the fall,

NAKASEC had taken over Crapser’s legal defense and was sched-

uling meetings on Capitol Hill to reintroduce a bill that would

retroactively grant U.S. citizenship to international adoptees

not covered under the CCA. This bill would be known as the

Adoptee Citizenship Act.

In addition to NAKASEC’s support for the Adoptee Citizen-

ship Act, within the organization, they also developed a position

to solely focus on Korean adoptee needs. Though Korean

adoptees often feel separate from other Asian Americans, the

advocacy by these groups demonstrated Korean adoptees’ inclu-

sion within Asian immigrant communities.

There are an estimated 35,000 international adoptees without citizenship. The majority of these are Korean adoptees.

35FA L L 2 0 1 8 c o n t e x t s

Despite the organizing around his case and for adoptee

citizenship rights, in October of 2016, an immigration judge

ordered Crapser deported to Korea. Unfamiliar with the lan-

guage or culture, Crapser’s outlook was bleak. Deportation

intensifies the precarious position of those who are already

vulnerable, and with little financial, social, or cultural support,

deportees face enormous hurdles to integration into their new

country. For many, integration is nearly impossible. It can lead

to fatal outcomes, such was the case for Philip Clay, another

Korean adoptee who was deported back to Korea and com-

mitted suicide in July 2017.

In the midst of continued advocacy for adoptees, NAKASEC

also began a 22-day, 24-hour vigil in front of the White House

to draw attention to other immigrant rights. From August

15-September 5, 2017, NAKASEC led “DREAM Action” or

#DreamAction17 to protest the end of the Deferred Action for

Childhood Arrivals (DACA) and the Temporary Protective Status

(TPS) programs. DREAM Action drew together wide-ranging

members of the immigrant community. Korean adoptees, who

have typically not considered themselves immigrants, joined this

around-the-clock action. In the current political climate, height-

ened immigration scrutiny and adoptees’ precarious citizenship

rights appear to have facilitated an awareness of “linked fate,”

whereby the conditions and outcomes for one are connected to

the many, among and between immigrant groups.

Support for a legislative fix for adoptee citizenship continues,

and in April 2018 a new version of the Adoptee Citizenship Act

was introduced in the House and the Senate. Ironically, although

it was Crapser’s case that reignited support for citizenship for

all adoptees, this version excludes the most vulnerable—those

adoptees who, like Crapser, have been found guilty of a violent

crime and have already been deported.

adoptees and asian america Korean adoption began at a time of exclusionary Asian

immigration policies, yet, until recently, Korean adoptees were

excluded from Asian immigration history. An appropriate cor-

rective must also incorporate an inclusion of Korean adoptees

in how we think about contemporary Asian American com-

munity and identity. Though adoption from Korea has slowed

considerably, international adoption from other Asian countries

to the United States continues. Like the critical mass of Korean

adoptees before them, other Asian adoptees will soon be coming

of age. Not only will issues of identity and belonging likely still

be key, but the new contours of this cohort of Asian adoptees,

who were adopted at older ages, often have identified medi-

cal issues, and hail from countries across Asia, will necessitate

examinations of age, disability, colorism, and adoption within

and across Asian America.

As my respondents’ experiences growing up and the fight

for adoptee citizenship rights demonstrate, adoption into White

American families does not translate into complete social or legal

U.S. citizenship. Korean adoptees still experience the world and

are treated as hyphenated Americans. By incorporating Korean

adoptees within our understandings of Asian America, another

layer of the Asian American experience is illuminated. Korean

adoptees face many of the same realities of belonging and non-

belonging as Asian Americans more broadly, but their adoptive

status provides an additional lens through which to view the

Asian American experience.

recommended resources Samantha Futerman and Ryan Miyamoto. 2015. Twinsters. Net- flix. Ignite Channel. A documentary following Korean adoptee twins who were separated at birth, adopted to families in two different countries, and reunited with the help of social media.

Eleana J. Kim. 2010. Adopted Territory: Transnational Korean Adoptees and the Politics of Belonging. Durham, NC: Duke Uni- versity Press. Based on interviews and observations with Korean adoptees and the earliest international Korean adoptee gather- ings, this book details the beginnings of a global Korean adoptee consciousness.

Jon Maxwell. 2016. AKA SEOUL. NBC Asian America. CA: Inter- national Secret Agents. The follow up to akaDAN, a documen- tary about Korean adoptee and music artist Dan Matthews, AKA SEOUL provides a candid exploration of five subjects’ experiences attending an international Korean adoptee conference in Korea.

Arissa H. Oh. 2015. To Save the Children of Korea: The Cold War Origins of International Adoption. Stanford, CA: Stanford Uni- versity Press. A robust history of U.S.-Korean adoption, situated within domestic and geopolitical contexts.

Kim Park Nelson. 2016. Invisible Asians: Korean American Adoptees, Asian American Experiences, and Racial Exceptional- ism. New Brunswick, NJ: Rutgers University Press. Drawing from Korean adoptee oral histories and archival research, Park Nelson examines the experiences, histories, and racial implications of Korean adoptees.

Wendy Marie Laybourn is in the sociology department at the University of Mem-

phis. A Korean adoptee and former child services worker, Laybourn is the author (with

Devon R. Goss) of Diversity in Black Greek-Letter Organizations.

*All names, except Adam Crapser’s, are pseudonyms.

If They Should Come for Us B Y FAT I M A H A S G H A R

these are my people & I find them on the street & shadow through any wild all wild my people my people a dance of strangers in my blood the old woman’s sari dissolving to wind bindi a new moon on her forehead I claim her my kin & sew the star of her to my breast the toddler dangling from stroller hair a fountain of dandelion seed at the bakery I claim them too the sikh uncle at the airport who apologizes for the pat down the muslim man who abandons his car at the traffic light drops to his knees at the call of the azan & the muslim man who sips good whiskey at the start of maghrib the lone khala at the park pairing her kurta with crocs my people my people I can’t be lost when I see you my compass is brown & gold & blood my compass a muslim teenager snapback & high-tops gracing

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Source: Poetry (March 2017)

the subway platform mashallah I claim them all my country is made in my people’s image if they come for you they come for me too in the dead of winter a flock of aunties step out on the sand their dupattas turn to ocean a colony of uncles grind their palms & a thousand jasmines bell the air my people I follow you like constellations we hear the glass smashing the street & the nights opening their dark our names this country’s wood for the fire my people my people the long years we’ve survived the long years yet to come I see you map my sky the light your lantern long ahead & I follow I follow

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