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.
- (a) Set up the linear programming problem to maximize profit.
- (b) Solve the linear programming problem in part (a).
- (c) What is the corresponding dual problem?
- (d) What is the optimal solution to the dual problem?
- (e) Use the shadow profits to calculate the profit the firm will make if its supply of felt increases to 210 squares.
- (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?
- (g) Explain why it makes sense that the shadow profit for trim is 0.
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