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(a) Formulate a linear programming model for maximizing total profit contribution. (Let P,= units of product i produced, for i=1, 2, 3.) Max 25P₁ +27P2+29P3 s.t. ✓ Department A 1.5P₁ +3P₂+2P3 ≤450 Department B2P₁+1P₂ +2.5P3 ≤ 350 ✓ ✓ Department c 0.25P₁ +0.25P₂ +0.25P3 ≤ 50 P₁ P₂ P3 20 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, $600 for product 2, and $630 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? $ 3890 x Max 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, 155 units of product 2, or 190 units of product 3. (Let P, = units of product i produced and y, be the 0-1 variable that is one if any quantity of product / is produced and zero otherwise, for i= 1, 2, 3.) What is the objective function of the mixed-integer linear program?

(a) Formulate a linear programming model for maximizing total profit contribution. (Let P,= units of product i produced, for i=1, 2, 3.) Max 25P₁ +27P2+29P3 s.t. ✓ Department A 1.5P₁ +3P₂+2P3 ≤450 Department B2P₁+1P₂ +2.5P3 ≤ 350 ✓ ✓ Department c 0.25P₁ +0.25P₂ +0.25P3 ≤ 50 P₁ P₂ P3 20 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, $600 for product 2, and $630 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? $ 3890 x Max 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, 155 units of product 2, or 190 units of product 3. (Let P, = units of product i produced and y, be the 0-1 variable that is one if any quantity of product / is produced and zero otherwise, for i= 1, 2, 3.) What is the objective function of the mixed-integer linear program?

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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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. s.t. Department Product 1 A B с 1.50 Max 2.00 0.25 Product 2 Product 3 3.00 1.00 0.25 Department A 1.5P₁+3P₂ +2P3 ≤ 450 Department B2P₁ + 1P₂ +2.5P3 ≤ 350 Department c 0.25P₁ +0.25P₂ +0.25P3 ≤ 50 P1 P₂ P3 20 2.00 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; = units of product produced, for i = 1, 2, 3.) Max 25P₁ +27P2+29P3 2.50 0.25 (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, $600 for product 2, and $630 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? $3890 X (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, 155 units of product 2, or 190 units of product 3. (Let P, = units of product i produced and y, 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?
Transcribed Image Text: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. s.t. Department Product 1 A B с 1.50 Max 2.00 0.25 Product 2 Product 3 3.00 1.00 0.25 Department A 1.5P₁+3P₂ +2P3 ≤ 450 Department B2P₁ + 1P₂ +2.5P3 ≤ 350 Department c 0.25P₁ +0.25P₂ +0.25P3 ≤ 50 P1 P₂ P3 20 2.00 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; = units of product produced, for i = 1, 2, 3.) Max 25P₁ +27P2+29P3 2.50 0.25 (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, $600 for product 2, and $630 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? $3890 X (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, 155 units of product 2, or 190 units of product 3. (Let P, = units of product i produced and y, 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?
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 2 0; Y₁ Y₂Y3 = 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? = (C 1) (P₁ P2 P3 Y₁1 Y2r Y 3) = with profit $
Transcribed Image Text: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 2 0; Y₁ Y₂Y3 = 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? = (C 1) (P₁ P2 P3 Y₁1 Y2r Y 3) = with profit $
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