In order to contain the risk of the investment to an acceptable level, the amount of money allocated to the aggressive growth plus the corporate bond funds cannot exceed 60% of the portfolio, and the average risk level of the portfolio cannot exceed 2. The total amount invested should be $10 million. What is the optimal portfolio allocation for achieving the maximum expected return at the end of the year, if no short selling is allowed?

Problem 2. (Capital Budgeting) The operating manager at a pharmaceutical company is considering funding proposals for eight different research and development (R&D) projects, but she has limited funds C(C = $1,000.00). The cost of investing in Project i is ci, and bi is the estimated present value of benefit of Project i, as shown in the following table.

Each project is on a “take it or leave it” basis, that is, it is not possible to fund a project partially. Which projects should the operating manager fund in order to maximize the present value of the expected total benefit?

Problem 3. (Cash Flow Matching) Consider an asset manager who is managing funds for the corporate sponsor of a defined benefit pension plan that needs to ensure a particular stream of semiannual cash payments over the next four years for retiring plan participants. For example, the pension plan may have semiannual obligations representing annuity payments. Let the cash obligations for the eight payments dates of the next four years be represented by a vector m = (m1,...,m8). The asset manager is considering investing in five different high investment-grade quality bonds. Over the next eight payment dates (i.e., semiannually), bond i pays out coupons ci = (ci1,...,ci8). If the bond matures at date t, the corresponding cit equals the coupon plus the principal. The bonds currently trade at ask prices p = (p1,...,p5). The relevant data are presented in the following table.

The asset manager would like to ensure that the coupon payments from the bonds over the pension plan’s obligations. The objective is to minimize the cost of acquiring the bonds today while still meeting all expected future obligations.

Problem 4. An oil refinery can buy two types of oil: light crude oil and heavy crude oil. The cost per barrel of these types is respectively $20 and $15. The following quantities of gasoline, kerosene and jet fuel are produced per barrel of each type of oil.

Note that 5 percent and 8 percent, respectively, of the crude are lost during the refining process. The refinery has constructed to deliver 1 million barrels of gasoline, 500,000 barrels of kerosine, and 300,000 barrels of jet fuel. Formulate the problem of finding the number of barrels of each crude oil that satisfies the demand and minimizes the total cost as a linear program.

Problem 5. A company makes three products: kettlebell, barbell and dumbbell. Each kettlebell requires 1/4 hours of production labor, 1/8 hours of testing, and $1.2 worth of raw materials. Each barbell requires 1 hours of production labor, 1/2 hours of testing, and $1.5 worth of raw materials. Each dumbbell requires 1/3 hours of production labor, 1/3 hours of testing, and $0.9 worth of raw materials. Given the current personnel of the company, there can be at most 90 hours of labor and 80 hours of testing, each day. Products of the first, second and third types have a market value of $9, $10 and $8, respectively. Formulate the problem of maximization of profit as a linear program.

Problem 6. (Currency Conversion) Current cash reserves and their desired position in six different currencies (in millions) are as follows.

We want to convert the initial position amounts so that we have reserves at least equal to the amounts in the desired position. The conversion rates are presented in the second table.

The numbers in this table should be understood this way: buying 1 U.S. dollar for Japanese Yen requires 76.925 U, 1 U.S. dollar costs 0.7091¤, etc. In order to buy 1 Japanese Yen for U.S. dollars we need to pay $0.013, etc.

Find the optimal conversion plan to satisfiy the goals and to maximize the value of U.S $ of the new position.

Problem 7. Solve the following LP problems geometrically (graphically).

(a)

minimize − 2x1− 3x2 subject to

x1 + 2x2 ≤ 2 x1,x2 ≥ 0

(b)

maximize x1 + 2x2 subject to

−x1 + x2 ≤ 1 x1 ≤ 2 x2 ≤ 2

x1,x2 ≥ 0