Problem 7-11 (Algorithmic)

Hart Manufacturing makes three products. Each product requires manufacturing operations in three departments: A, B, and C. The labor-hour requirements, by the department, are as follows:

Department	Product 1	Product 2	Product 3
A	1.50	3.00	2.00
B	2.00	1.00	2.50
C	0.25	0.25	0.25

During the next production period, the labor-hours available are 450 in department A, 350 in department B, and 50 in department C. The profit contributions per unit are $24 for product 1, $27 for product 2, and $28 for product 3. Use a software package LINGO.

1. Formulate an integer linear programming model for maximizing total profit contribution. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300) If the constant is 1, it must be entered in the box. Let P_i = units of product i produced

   Max	P 1	+	P 2	+	P 3
   s.t.
   	P 1	+	P 2	+	P 3
   	P 1	+	P 2	+	P 3
   	P 1	+	P 2	+	P 3
   P 1, P 2, P 3 0

2. Solve the integer linear program formulated in part (a). How much of each product should be produced, and what is the projected total profit contribution? P ₁ = P ₂ = P ₃ = Profit = $
3. After evaluating the solution obtained in part (b), one of the production supervisors noted that production setup costs had not been taken into account. She noted that setup costs are $390 for product 1, $550 for product 2, and $620 for product 3. If the solution developed in part (b) is to be used, what is the total profit contribution after taking into account the setup costs? Profit = $
4. Management realized that the optimal product mix, taking setup costs into account, might be different from the one recommended in part (b). Formulate a mixed-integer linear program that takes setup costs into account. Management also stated that we should not consider making more than 160 units of product 1, 165 units of product 2, or 200 units of product 3. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300) If the coefficient of a constraint is 1, enter 1 in the answer box. Enter 0 if a coefficient or an RHS value is zero. Here introduce a 0-1 variable y _i that is one if any quantity of product i is produced and zero otherwise.

   Max

   P 1

   +

   P 2

   +

   P 3

   +

   y 1

   +

   y 2

   +

   y 3

   s.t.

   P 1

   +

   P 2

   +

   P 3

   P 1

   +

   P 2

   +

   P 3

   P 1

   +

   P 2

   +

   P 3

   P 1

   +

   y 1

   P 2

   +

   y 2

   P 3

   +

   y 3

   P 1, P 2, P 3 0; y 1, y 2, y 3 = 0, 1

5. Solve the mixed-integer linear program formulated in part (d). How much of each product should be produced, and what is the projected total profit contribution? Compare this profit contribution to that obtained in part (c). Enter 0 if your answer is zero. P ₁ = P ₂ = P ₃ = Profit = $ The profit is ______ by $ _____ .