) Consider a product mix problem, where the decision involves determining the optimal production levels for products X and Y. A unit of X requires 4 hours of labor in department 1 and 6 hours of labor in department 2. A unit of Y requires 3 hours of labor in department 1 and 8 hours of labor in department 2. Currently, 1000 hours of labor time are available in department 1, and 1200 hours of labor time are available in department 2. Furthermore, 400 additional hours of cross-trained workers are available to assign to either department (or split between both). Each unit of X sold returns a $50 profit, while each unit of Y sold returns a $60 profit. All units produced can be sold. Formulate this problem as a linear program. (Hint: Consider introducing other decision variables in addition to the production amounts for X and Y.) (17.5pts)

Let

X = the number of units of product X sold

Y = the number of units of product Y sold

C1= the number of cross-trained labor hours allocated to department 1

C2= the number of cross-trained labor hours allocated to department 2

Maximize: 50X + 60Y

Subject to: 4X + 3Y - C1 1000 (Labor Department 1)

6X + 8Y - C2 1200 (labor Department 2)

C1 + C2 400 (Cross-Trained Hours)

X, Y 0

1. What is the optimal solution for this problem?
2. How much will you be willing to pay for one additional product of X?

If cross-training was not available, how would this change the solution?