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 1.50 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 $25 for product 1, $27 for product 2, and $29 for product 3. (a) Formulate a linear programming model for maximizing total profit contribution. (Let Pi = units of product i produced, for i = 1, 2, 3.) Max 25P1+27P2+29P3 s.t.Department A 1.5P1+3P2+2P3<=450 Department B 2P1+1P2+2.5P3<=350 Department C 0.25P1+0.25P2+0.25P3<=50 P1, P2, P3 ≥ 0 (b) Solve the linear program formulated in part (a). How much of each product should be produced, and what is the projected total profit contribution (in dollars)? (P1, P2, P3) = 60,80,60 with profit $ 5400 . (c) 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 $410 for product 1, $540 for product 2, and $650 for product 3. If the solution developed in part (b) is to be used, what is the total profit contribution (in dollars) after taking into account the setup costs? $ (d) 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 140 units of product 1, 175 units of product 2, or 180 units of product 3. (Let Pi = units of product i produced and yi be the 0-1 variable that is one if any quantity of product i is produced and zero otherwise, for i = 1, 2, 3.) What is the objective function of the mixed-integer linear program? Max 25P1+27P2+29P3−410y1−540y2−650y3 In addition to the constraints from part (a), what other constraints should be added to the mixed-integer linear program? s.t.units of Product 1 produced P1≤140y1 units of Product 2 produced P2≤175y2 units of Product 3 produced P3≤180y3 P1, P2, P3 ≥ 0; y1, y2, y3 = 0, 1 (e) 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 (in dollars) contribution? (P1, P2, P3, y1, y2, y3) = with profit $

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 1.50 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 $25 for product 1, $27 for product 2, and $29 for product 3. (a) Formulate a linear programming model for maximizing total profit contribution. (Let Pi = units of product i produced, for i = 1, 2, 3.) Max 25P1+27P2+29P3 s.t.Department A 1.5P1+3P2+2P3<=450 Department B 2P1+1P2+2.5P3<=350 Department C 0.25P1+0.25P2+0.25P3<=50 P1, P2, P3 ≥ 0 (b) Solve the linear program formulated in part (a). How much of each product should be produced, and what is the projected total profit contribution (in dollars)? (P1, P2, P3) = 60,80,60 with profit $ 5400 . (c) 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 $410 for product 1, $540 for product 2, and $650 for product 3. If the solution developed in part (b) is to be used, what is the total profit contribution (in dollars) after taking into account the setup costs? $ (d) 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 140 units of product 1, 175 units of product 2, or 180 units of product 3. (Let Pi = units of product i produced and yi be the 0-1 variable that is one if any quantity of product i is produced and zero otherwise, for i = 1, 2, 3.) What is the objective function of the mixed-integer linear program? Max 25P1+27P2+29P3−410y1−540y2−650y3 In addition to the constraints from part (a), what other constraints should be added to the mixed-integer linear program? s.t.units of Product 1 produced P1≤140y1 units of Product 2 produced P2≤175y2 units of Product 3 produced P3≤180y3 P1, P2, P3 ≥ 0; y1, y2, y3 = 0, 1 (e) 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 (in dollars) contribution? (P1, P2, P3, y1, y2, y3) = with profit $

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter4: Linear Programming Models

Section: Chapter Questions

Problem 111P

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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	1.50	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 $25 for product 1, $27 for product 2, and $29 for product 3.

(a)

Formulate a linear programming model for maximizing total profit contribution. (Let

P_i =

units of product i produced, for

i = 1, 2, 3.)

Max

25P1+27P2+29P3

s.t.Department A

1.5P1+3P2+2P3<=450

Department B

2P1+1P2+2.5P3<=350

Department C

0.25P1+0.25P2+0.25P3<=50

P₁, P₂, P₃ ≥ 0

(b)

Solve the linear program formulated in part (a). How much of each product should be produced, and what is the projected total profit contribution (in dollars)?

(P₁, P₂, P₃) =

60,80,60

with profit $

5400

(c)

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 $410 for product 1, $540 for product 2, and $650 for product 3. If the solution developed in part (b) is to be used, what is the total profit contribution (in dollars) after taking into account the setup costs?

(d)

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 140 units of product 1, 175 units of product 2, or 180 units of product 3. (Let

P_i =

units of product i produced and

y_i

be the 0-1 variable that is one if any quantity of product i is produced and zero otherwise, for

i = 1, 2, 3.)

What is the objective function of the mixed-integer linear program?

Max

25P1+27P2+29P3−410y1−540y2−650y3

In addition to the constraints from part (a), what other constraints should be added to the mixed-integer linear program?

s.t.units of Product 1 produced

P1≤140y1

units of Product 2 produced

P2≤175y2

units of Product 3 produced

P3≤180y3

P₁, P₂, P₃ ≥ 0; y₁, y₂, y₃ = 0, 1

(e)

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 (in dollars) contribution?

(P₁, P₂, P₃, y₁, y₂, y₃) =

with profit $

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