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 Max A s.t. B с Department A Department B Department C P1, P₂, P3 20 1.50 2.00 (P1, P2, P3) = 0.25 3.00 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 $27 for product 1, $29 for product 2, and $32 for product 3. (a) Formulate a linear programming model for maximizing total profit contribution. (Let P; = units of product i produced, for i = 1, 2, 3.) 1.00 0.25 2.00 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)? - (C 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 Max A s.t. B с Department A Department B Department C P1, P₂, P3 20 1.50 2.00 (P1, P2, P3) = 0.25 3.00 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 $27 for product 1, $29 for product 2, and $32 for product 3. (a) Formulate a linear programming model for maximizing total profit contribution. (Let P; = units of product i produced, for i = 1, 2, 3.) 1.00 0.25 2.00 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)? - (C with profit $
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(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, $550 for product 2, and $640 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 155 units of product 1, 170 units of product 2, or 175 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 [ ]
\[P_1, P_2, P_3 \geq 0; \; y_1, y_2, y_3 = 0, 1\]
(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?
\((P_1, P_2, P_3, y_1, y_2, y_3) = ([ ])\) with profit $ [ ] .](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2fdea76-24b5-4e92-ab93-0d275ab88ec3%2F9ed8ed99-fc30-4552-bbc2-ede367cb2893%2F87l70jg_processed.png&w=3840&q=75)
Transcribed Image Text:(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, $550 for product 2, and $640 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 155 units of product 1, 170 units of product 2, or 175 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 [ ]
\[P_1, P_2, P_3 \geq 0; \; y_1, y_2, y_3 = 0, 1\]
(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?
\((P_1, P_2, P_3, y_1, y_2, y_3) = ([ ])\) with profit $ [ ] .
![### Hart Manufacturing Production Problem
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:
#### Labor-Hour Requirements Table
| 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 available labor-hours 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 $32 for product 3.
#### Tasks:
(a) **Formulate a Linear Programming Model**
The objective is to maximize the total profit contribution. Let \( P_i \) be the units of product \( i \) produced (for \( i = 1, 2, 3 \)).
**Objective Function:**
Maximize
\[ 27P_1 + 29P_2 + 32P_3 \]
**Subject to Constraints:**
- Department A:
\[ 1.5P_1 + 3P_2 + 2P_3 \leq 450 \]
- Department B:
\[ 2P_1 + P_2 + 2.5P_3 \leq 350 \]
- Department C:
\[ 0.25P_1 + 0.25P_2 + 0.25P_3 \leq 50 \]
Non-negativity constraints:
\[ P_1, P_2, P_3 \geq 0 \]
(b) **Solve the Linear Program**
Determine the quantity of each product that should be produced and the projected total profit contribution in dollars:
\[ (P_1, P_2, P_3) = (\_, \_, \_) \]
Profit = \$\_\_\_\_
(Note: Calculations need to be done to fill in the blank spaces for the optimal values and profit.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2fdea76-24b5-4e92-ab93-0d275ab88ec3%2F9ed8ed99-fc30-4552-bbc2-ede367cb2893%2F0k53oh_processed.png&w=3840&q=75)
Transcribed Image Text:### Hart Manufacturing Production Problem
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:
#### Labor-Hour Requirements Table
| 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 available labor-hours 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 $32 for product 3.
#### Tasks:
(a) **Formulate a Linear Programming Model**
The objective is to maximize the total profit contribution. Let \( P_i \) be the units of product \( i \) produced (for \( i = 1, 2, 3 \)).
**Objective Function:**
Maximize
\[ 27P_1 + 29P_2 + 32P_3 \]
**Subject to Constraints:**
- Department A:
\[ 1.5P_1 + 3P_2 + 2P_3 \leq 450 \]
- Department B:
\[ 2P_1 + P_2 + 2.5P_3 \leq 350 \]
- Department C:
\[ 0.25P_1 + 0.25P_2 + 0.25P_3 \leq 50 \]
Non-negativity constraints:
\[ P_1, P_2, P_3 \geq 0 \]
(b) **Solve the Linear Program**
Determine the quantity of each product that should be produced and the projected total profit contribution in dollars:
\[ (P_1, P_2, P_3) = (\_, \_, \_) \]
Profit = \$\_\_\_\_
(Note: Calculations need to be done to fill in the blank spaces for the optimal values and profit.)
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