Five male characters (Billie, John, Fish, Glen, and Larry) and five female characters (Ally, Georgia, Jane, Rene, and Nell) from Ally McBeal are marooned on a desert island. The problem is to determine what percentage of time each woman on the island should spend with each man. For example, Ally could spend 100% of her time with John or she could “play the field” by spending 20% of her time with each man. Table 55 shows a “happiness index” for each potential pairing of a man and woman. For example, if Larry and Rene spend all their time together, they earn 8 units of happiness for the island. a Play matchmaker and determine an allocation of each man and woman’s time that earns the maximum total happiness for the island. Assume that happiness earned by a couple is proportional to the amount of time they spend together. b Explain why the optimal solution to this problem will, for any matrix of “happiness indices,” always involve each woman spending all her time with one man. Ally Georgia Jane Rene Nell Billie 8 6 4 7 5 John 5 7 6 4 9 Fish 10 6 5 2 10 Glen 1 0 0 0 0 Larry 5 7 9 8 6 a) Formulate a linear programming (LP) model for this problem. b) Use solver to find the optimal assignment rul
Five male characters (Billie, John, Fish, Glen, and Larry) and five female characters (Ally, Georgia, Jane, Rene, and Nell) from Ally McBeal are marooned on a desert island. The problem is to determine what percentage of time each woman on the island should spend with each man. For example, Ally could spend 100% of her time with John or she could “play the field” by spending 20% of her time with each man. Table 55 shows a “happiness index” for each potential pairing of a man and woman. For example, if Larry and Rene spend all their time together, they earn 8 units of happiness for the island. a Play matchmaker and determine an allocation of each man and woman’s time that earns the maximum total happiness for the island. Assume that happiness earned by a couple is proportional to the amount of time they spend together.
b Explain why the optimal solution to this problem will, for any matrix of “happiness indices,” always involve each woman spending all her time with one man.
| Ally | Georgia | Jane | Rene | Nell | |
| Billie | 8 | 6 | 4 | 7 | 5 |
| John | 5 | 7 | 6 | 4 | 9 |
| Fish | 10 | 6 | 5 | 2 | 10 |
| Glen | 1 | 0 | 0 | 0 | 0 |
| Larry | 5 | 7 | 9 | 8 | 6 |
a) Formulate a linear programming (LP) model for this problem.
b) Use solver to find the optimal assignment rule
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