Operation System
9. Linear programming models are used by many Wall Street firms to select a desirable bond portfolio. The following is a simplified version of such model. Solodrex is considering investing in four bonds: $1,000,000 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 Table 15. The duration of a bond is measure of the bond’s sensitivity to interest rates. Solodex wants to maximize the expected return from its bond investments, subject to three constraints.
Constraint 1. The worst-case return of the bond portfolio must be at least 8%.
Constraint 2. The average duration of the portfolio must be at most 6. For example, a portfolio that invested $ 600,000 in bond 1 and $400,000 in bond 4 would have an average duration of {(600,000) (3) + (400,000) (9)]}/ 1,000,000 = 5.4.
Constraint 3. Because of the diversification requirements, at most 40% of the total amount invested can be invested in a single bond.
Formulate an LP model to help Solodex tom achieve its objective.
Table. 15 Bond | Expected return | Worst-case return | Duration | 1 | 13% | 6% | 3 | 2 | 8% | 8% | 4 | 3 | 12% | 10% | 7 | 4 | 14% | 9% | 9 |
11. Eli Daisy produces the drug Rozac from four chemicals. Today they must produce 1,000 lb of drug. The three active ingredients in Rozac are A, B, and C. By weight, at least 8% of Rozac 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 ingredient in1 lb of each chemical are given in table below. It is necessary that at least 100 lb of chemical 2 be used. Formulate an LP whose solution would determine the cheapest way of producing today’s batch of Rozac. Chemical | Cost per lb | A | B | C | 1 | $8 | 0.03 | 0.02 | 0.01 | 2 | $10 | 0.06 | 0.04 | 0.01 | 3 | $11 | 0.10 | 0.03 | 0.04 | 4 | $14 | 0.12 | 0.09 | 0.04
Constraint 1. The worst-case return of the bond portfolio must be at least 8%.
Constraint 2. The average duration of the portfolio must be at most 6. For example, a portfolio that invested $ 600,000 in bond 1 and $400,000 in bond 4 would have an average duration of {(600,000) (3) + (400,000) (9)]}/ 1,000,000 = 5.4.
Constraint 3. Because of the diversification requirements, at most 40% of the total amount invested can be invested in a single bond.
Formulate an LP model to help Solodex tom achieve its objective.
Table. 15 Bond | Expected return | Worst-case return | Duration | 1 | 13% | 6% | 3 | 2 | 8% | 8% | 4 | 3 | 12% | 10% | 7 | 4 | 14% | 9% | 9 |
11. Eli Daisy produces the drug Rozac from four chemicals. Today they must produce 1,000 lb of drug. The three active ingredients in Rozac are A, B, and C. By weight, at least 8% of Rozac 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 ingredient in1 lb of each chemical are given in table below. It is necessary that at least 100 lb of chemical 2 be used. Formulate an LP whose solution would determine the cheapest way of producing today’s batch of Rozac. Chemical | Cost per lb | A | B | C | 1 | $8 | 0.03 | 0.02 | 0.01 | 2 | $10 | 0.06 | 0.04 | 0.01 | 3 | $11 | 0.10 | 0.03 | 0.04 | 4 | $14 | 0.12 | 0.09 | 0.04