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 $27 for product 1, $29 for product 2, and $31 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.) (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)? (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 $370 for product 1, $520 for product 2, and $570 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 170 units of product 1, 180 units of product 2, or 190 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 ______________________ 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 = ___________ units of Product 2 produced = ___________ units of Product 3 produced = ___________ 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 $27 for product 1, $29 for product 2, and $31 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.) (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)? (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 $370 for product 1, $520 for product 2, and $570 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 170 units of product 1, 180 units of product 2, or 190 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 ________________ 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 = _ units of Product 2 produced = _ units of Product 3 produced = _ 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.

Chapter12: Queueing Models

Section: Chapter Questions

Problem 59P

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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 $27 for product 1, $29 for product 2, and $31 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.)

(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)?

(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 $370 for product 1, $520 for product 2, and $570 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 170 units of product 1, 180 units of product 2, or 190 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 ______________________

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 = ___________

units of Product 2 produced = ___________

units of Product 3 produced = ___________

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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