Question

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 department, are as follows:

Department	Product 1	Product 2	Product 3
A	1.60	3.10	2.10
B	2.20	1.20	2.70
C	0.35	0.35	0.35

During the next production period, the labor-hours available are 470 in department A, 370 in department B, and 70 in department C. The profit contributions per unit are $27 for product 1, $30 for product 2, and $32 for product 3.

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)
   
   Let P_i = units of product i produced
   

   Max	P1	+	P2	+	P3
   s.t.
   	P1	+	P2	+	2.1P3	≤
   	2.2P1	+	P2	+	P3	≤
   	P1	+	0.35P2	+	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 $420 for product 1, $570 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 185 units of product 1, 160 units of product 2, or 150 units of product 3. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300)
   
   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

   +

   2.1P3

   ≤

   2.2P1

   +

   P2

   +

   P3

   ≤

   P1

   +

   0.35P2

   +

   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).
   
   P₁ =  
   P₂ =  
   P₃ =  
   
   Profit = $ _____
   
   The profit is _______ by $ _____.

Answer 1

a) Linear Programming model is following:

Max 27P1+30P2+32P3

s.t.

1.6P1+3.1P2+2.1P3 <= 470

2.2P1+1.2P2+2.7P3 <= 370

.35P1+.35P2+.35P3 <= 70

P1, P2, P3 >= 0

b) Solution of the linear program using Excel Solver is following:

fx SUMPRODUCT(B3:D3,$B$9:$D$9) E3 A C E E G J K M N P3 1 P1 P2 S Solver Parameters X 2 Profit 32 3 27 30 5844 Set Objective: SES3 4 Value Of: To: Max Min 5 Dept A 1.6 3.1 2.1 470 470 <= By Changing Variable Cells: 6 Dept B 7 Dept C 2.2 1.2 2.7 370 370 SBS9:SDS9 70 0.35 0.35 0.35 70 <= Subject to the Constraints: 8 SES5:SES7 SG$5:SG$7 Result Add 88 9 76 36 10 Change 11 12 Delete 13 14 Reset All 15 Make Unconstrained Variables Non-Negative 16 17 Simplex LP Select a Solving Method: Options 18 19 Solving Method 20 Select the IPOPT Nonlinear engine for Solver Problems that are smooth nonlinear.Select the LP Simplex engine for linear Solver Problems 21 22 23 Solve Close 24 C

FORMULAS:

E3 =SUMPRODUCT(B3:D3,$B$9:$D$9) copy to E5:E7

P1 = 76

P2 = 88

P3 = 36

Profit = $ 5,844

c) Total profit contribution after taking into account the setup costs = 5844 - (420+570+620) = $ 4,234

d)

Mixed-integer programming model is following:

Max 27P1+30P2+32P3-420y1-570y2-620y3

s.t.

1.6P1+3.1P2+2.1P3 <= 470

2.2P1+1.2P2+2.7P3 <= 370

.35P1+.35P2+.35P3 <= 70

P1-185y1 <= 0

P2-160y2 <= 0

P3-150y3 <= 0

P1, P2, P3 >= 0

y1, y2, y3 = 0,1

e) Solution using Excel Solver is following:

fx =SUMPRODUCT(B3:G3, $B$12:$G$12) H3 N P A C G K M O R P1 P2 1 P3 V2 y3 S Solver Parameters X 2 Profit 27 30 32 -420 -570 3 -620 4710 Set Objective SHS3 O Max Min Value Of: To: 0 5 Dept A 1.6 3.1 2.1 470 470 By Changing Variable Cells: 340 370 6 Dept B 2,2 1,2 2.7 SB$12:SGS12 Dept C 0.35 0.35 0.35 70 70 7 <= Subject to the Constraints: SHS5:SHS10<= SJS5:SJS10 Max P1 -185 8 1 -85 0 -60 0 Add 9 Max P2 1 -160 <= SES12:SGS12 = binary -150 10 Max P3 1 0 0 <= Change 11 12 Result 1 100 100 0 1 Delete 13 14 Reset All 15 Make Unconstrained Variables Non-Negative 16 17 Simplex LP Select a Solving Method: Options 18 19 Solving Method 20 engine for nonlinear.Select the IP Simplex enaine for linear Solver Problems. 21 22 23 Solve Close 24

FORMULAS:

H3 =SUMPRODUCT(B3:G3,$B$12:$G$12) copy to H5:H10

P1 = 100

P2 = 100

P3 = 0

Profit = $ 4,710

The profit is increased by = 4710-4234 = $ 476