Question

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

Department	Product 1	Product 2	Product 3
A	1.95	3.45	2.45
B	2.90	1.90	3.40
C	0.70	0.70	0.70

During the next production period, the labor-hours available are 540 in department A, 440 in department B, and 140 in department C. The profit contributions per unit are $34 for product 1, $37 for product 2, and $39 for product 3. Use a software package LINGO.

1. Formulate a 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 constant is “1”, it must be entered in the box.
   
   Let P_i = units of product i produced

   Max	P1	+	P2	+	P3
   s.t.
   	P1	+	P2	+	P3	≤
   	P1	+	P2	+	P3	≤
   	P1	+	P2	+	P3	≤
   P1, P2, P3 ≥ 0

2. Solve the linear program formulated in part (a). How much of each product should be produced, and what is the projected total profit contribution?
   
   P₁ =  
   P₂ =  
   P₃ =  
   
   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 $490 for product 1, $640 for product 2, and $690 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 220 units of product 1, 195 units of product 2, or 185 units of product 3. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300) If constant is “1”, it must be entered in the box. Enter “0” if your answer 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

   P1

   +

   P2

   +

   P3

   +

   y1

   +

   y2

   +

   y3

   s.t.

   P1

   +

   P2

   +

   P3

   ≤

   P1

   +

   P2

   +

   P3

   ≤

   P1

   +

   P2

   +

   P3

   ≤

   P1

   +

   y1

   ≤

   P2

   +

   y2

   ≤

   P3

   +

   y3

   ≤

   P1, P2, P3 ≥ 0; y1, y2, y3 = 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.
   
   P₁ =  
   P₂ =  
   P₃ =  
   
   Profit = $   
   
   The profit is increased  by $  .

Answer 1

Max	34 P1	+	37 P2	+	39 P3
s.t.
	1.95 P1	+	3.45 P2	+	2.45 P3	≤	540
	2.90 P1	+	1.90 P2	+	3.40 P3	≤	440
	0.70 P1	+	0.70 P2	+	0.70 P3	≤	140
P1, P2, P3 ≥ 0

Lingo Code

Max = 34*P1 + 37*P2 + 39*P3;

1.95*P1 + 3.45*P2 + 2.45*P3 < 540;

2.90*P1 + 1.90*P2 + 3.40*P3 < 440;

0.70*P1 + 0.70*P2 + 0.70*P3 < 140;

end

Solution

P₁ = 78.10
P₂ = 112.38
P₃ = 0

Profit = $6,813.33

----------------------------------------------------

Profit with the earlier solution including the setup cost = 6,813.33 - (490 + 640 + 0) = $5,683.33

----------------------------------------------------

Max

34 P1

+

37 P2

+

39 P3

+

(-490) y1

+

(-640) y2

+

(-690) y3

s.t.

1.95 P1

+

3.45 P2

+

2.45 P3

≤

540

2.90 P1

+

1.90 P2

+

3.40 P3

≤

440

0.70 P1

+

0.70 P2

+

0.70 P3

≤

140

P1

+

(-220) y1

≤

0

P2

+

(-195) y2

≤

0

P3

+

(-185) y3

≤

0

P1, P2, P3 ≥ 0; y1, y2, y3 = 0, 1

Solution

P₁ = 0
P₂ = 107.14
P₃ = 69.54

Profit = $5,986.18

The profit is increased by 5,986.18 - 5,683.33 = $302.85