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 $28 for product 1, $30 for product 2, and $32 for product 3. (a) Formulate a linear programming model for maximizing total profit contribution (in $). (Let Pi = units of product i produced, for i = 1, 2, 3.) Max s.t.Department A Department B Department C 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) = with profit $ . (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. The supervisor noted that setup costs are $360 for product 1, $600 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 in part (c) into account. Management also stated that we should not consider making more than 145 units of product 1, 150 units of product 2, or 160 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 contribution (in dollars)? (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 $28 for product 1, $30 for product 2, and $32 for product 3. (a) Formulate a linear programming model for maximizing total profit contribution (in $). (Let Pi = units of product i produced, for i = 1, 2, 3.) Max s.t.Department A Department B Department C 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) = with profit $ . (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. The supervisor noted that setup costs are $360 for product 1, $600 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 in part (c) into account. Management also stated that we should not consider making more than 145 units of product 1, 150 units of product 2, or 160 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 contribution (in dollars)? (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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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 $28 for product 1, $30 for product 2, and $32 for product 3.

(a)

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

P_i =

units of product i produced, for

i = 1, 2, 3.)

Max

s.t.Department A

Department B

Department C

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) =

with profit $ .

(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. The supervisor noted that setup costs are $360 for product 1, $600 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 in part (c) into account. Management also stated that we should not consider making more than 145 units of product 1, 150 units of product 2, or 160 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 contribution (in dollars)?

(P₁, P₂, P₃, y₁, y₂, y₃) =

with profit $ .

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For step 8, I'm trying to understand why the Constraints for P1, P2, and P3 are all 1 in the table in cells B9, C10, D11

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