13. decision making Consider a three product firm facing a constrained linear technology.

The firm is organized into two departments, machining and assembly.

Machine hours are constraining in the first department and labor hours are constraining in the second department. The required machine and labor times for each product are listed below:
#1 #2 #3 hours of machine time in dept. 1 1 2 3 hours of direct labor in dept. 2 2 4 5 Thus, each unit of product #1 requires 1 machine hour in department #1 and 2 direct labor hours in department #2, and so on. Total capacity is 12, 000 machine hours in department #1 and 15, 000 direct labor hours in department #2. In turn, total manufacturing cost, for any feasible production plan q = [q1, q2, q3] is given by TMC = 200, 000 + 18q1 + 24q2 + 45q3. Respective selling prices are 130, 145, and 185 per unit. Finally, the only period cost is specialized shipping "foam" that protects each of the products. This "foam" is purchased from a local supplier at a cost of 100 per pound. Each unit of product #1 requires 0.3 pounds of foam, each unit of product #2 requires 0.5 pounds of foam, and each unit of product #3 requires 0.7 pounds of foam.

(a) Formulate a program to maximize the firm’s profit. Use four decision variables in your formulation, q1, q2, q3, and F (the total quantity of foam purchased). Your program should have three capacity constraints, dealing with total machine hours in department #1, total direct labor hours in department #2, and total foam consumed.

(b) Without solving the program, what is the shadow price on the foam constraint? Carefully explain your reasoning. Then solve your program and verify your conjecture.

(c) Next formulate a program to maximize the firm’s profit using but three decision variables, q1, q2, and q3. Carefully explain the relationship between your two programs.