One of the products of the G. A. Tanner Company is a special kind of toy that provides an estimated unit profit of $3. Because of a large demand for this toy, management would like to increase its production rate from the current level of 1,000 per day.

However, a limited supply of two subassemblies (A and B) from vendors makes this difficult. Each toy requires two subassemblies of type A, but the vendor providing these subassemblies would only be able to increase its supply rate from the current 2,000 per day to a maximum of 3,000 per day. Each toy requires only one subassembly of type B, but the vendor providing these subassemblies would be unable to increase its supply rate above the current level of 1,000 per day.
Because no other vendors currently are available to provide these subassemblies, management is considering initiating a new production process internally that would simultaneously produce an equal number of subassemblies of the two types to supplement the supply from the two vendors. It is estimated that the company’s cost for producing one subassembly of each type would be $2.50 more than the cost of purchasing these subassemblies from the two vendors. Management wants to determine both the production rate of the toy and the production rate of each pair of subassemblies (one A and one B) that would maximize the total profit.
The following table summarizes the data for the problem.

(a) Formulate a linear programming model for this problem and use the graphical method to obtain its optimal solution.
C

(b) Use a software package based on the simplex method to solve for an optimal solution.
C

(c) Since the stated unit profits for the two activities are only estimates, management wants to know how much each of these estimates can be off before the optimal solution would change. Begin exploring this question for the first activity (producing toys) by using the same software package to resolve for an optimal solution and total profit as the unit profit for this activity increases in 50-cent increments from $2.00 to $4.00. What conclusion can be drawn about how much the estimate of this unit profit can differ in each direction from its original value of $3.00 before the optimal solution would change?
C

(d) Repeat part

(c) for the second activity (producing subassemblies) by re-solving as the unit profit for this activity increases in 50-cent increments from $3.50 to $1.50 (with the unit profit for the first activity fixed at $3).
C

(e) Use the same software package to generate the usual output (as in Table 6.23) for sensitivity analysis of the unit profits.

Use this output to obtain the allowable range to stay optimal for each unit profit.

(f) Use graphical analysis to verify the allowable ranges obtained in part (e).
(g) For each of the 16 combinations of unit profits considered in parts

(c) and

(d) where both unit profits differ from their original estimates, use the 100 percent rule for simultaneous changes in objective function coefficients to determine if the original optimal solution must still be optimal.
(h) For each of the combinations of unit profits considered in part (g) where it was found that the original optimal solution is not guaranteed to still be optimal, use graphical analysis to determine whether this solution is still optimal.