Chapter 4.3, Problem 22E

In most examples of this section, the original problem is a minimization problem and the dual is a maximization problem whose solution gives shadow costs. The reverse is true in Exercises 21 and 22. The dual here is a minimization problem whose solution can be interpreted as shadow profits.

22. Toy Manufacturing A small toy manufacturing firm has 200 squares of felt, 600 oz of stuffing, and 90 ft of trim available to make two types of toys, a small bear and a monkey. The bear requires 1 square of felt and 4 oz of stuffing. The monkey requires 2 squares of felt, 3 oz of stuffing, and 1 ft of trim. The firm makes $1 profit on each bear and $1.50 profit on each monkey.

1. (a) Set up the linear programming problem to maximize profit.
2. (b) Solve the linear programming problem in part (a).
3. (c) What is the corresponding dual problem?
4. (d) What is the optimal solution to the dual problem?
5. (e) Use the shadow profits to calculate the profit the firm will make if its supply of felt increases to 210 squares.
6. (f) How much profit will the firm make if its supply of stuffing is cut to 590 oz and its supply of trim is cut to 80 ft?
7. (g) Explain why it makes sense that the shadow profit for trim is 0.