Hart Manufacturing makes three products. Each product requiresmanufacturing operations in three departments: A, B, and C. Thelabor-hour requirements, by department, are as follows:
Department Product 1 Product 2 Product 3 A 2 3 2 B 2.5 1 3 C 0.25 0.25 0.5
During the next production period, the labor-hours available are400 in department A, 325 in department B, and 50 in department C.The profit contributions per unit are $24 for product 1, $30 forproduct 2, and $31 for product 3.
Formulate a linear programming model for maximizing total profitcontribution. If required, round your answers to two decimalplaces. For those boxes in which you must enter subtractive ornegative numbers use a minus sign. (Example: -300)
Let Pi = units of product iproduced
Max P1 + P2 + P3 s.t. P1 + P2 + 2P3 ⤠2.5P1 + P2 + P3 ⤠P1 + 0.25P2 + P3 ⤠P1, P2,P3 ⥠0
Solve the linear program formulated in part (a). How much ofeach product should be produced, and what is the projected totalprofit contribution?
P1 =
P2 =
P3 =
Profit = $
After evaluating the solution obtained in part (b), one of theproduction supervisors noted that production setup costs had notbeen taken into account. She noted that setup costs are $650 forproduct 1, $750 for product 2, and $550 for product 3. If thesolution developed in part (b) is to be used, what is the totalprofit contribution after taking into account the setupcosts?
Profit = $
Management realized that the optimal product mix, taking setupcosts into account, might be different from the one recommended inpart (b). Formulate a mixed-integer linear program that takes setupcosts into account. Management also stated that we should notconsider making more than 160 units of product 1, 125 units ofproduct 2, or 130 units of product 3. Here introduce a 0-1 variableyi that is one if any quantity ofproduct i is produced and zero otherwise. If required,round your answers to two decimal places. For those boxes in whichyou must enter subtractive or negative numbers use a minus sign.(Example: -300)
Max P1 + P2 + P3 + y1 + y2 + y3 s.t. P1 + P2 + 2P3 ⤠2.5P1 + P2 + P3 ⤠P1 + 0.25P2 + P3 ⤠P1 + y1 ⤠P2 + y2 ⤠P3 + y3 ⤠P1, P2,P3 ⥠0; y1,y2, y3 = 0, 1
Solve the mixed-integer linear program formulated in part (d).How much of each product should be produced, and what is theprojected total profit contribution? Compare this profitcontribution to that obtained in part (c).
P1 =
P2 =
P3 =
Profit = $
The profit is increased by $ .
Hart Manufacturing makes three products. Each product requiresmanufacturing operations in three departments: A, B, and C. Thelabor-hour requirements, by department, are as follows:
| Department | Product 1 | Product 2 | Product 3 |
| A | 2 | 3 | 2 |
| B | 2.5 | 1 | 3 |
| C | 0.25 | 0.25 | 0.5 |
During the next production period, the labor-hours available are400 in department A, 325 in department B, and 50 in department C.The profit contributions per unit are $24 for product 1, $30 forproduct 2, and $31 for product 3.
Formulate a linear programming model for maximizing total profitcontribution. If required, round your answers to two decimalplaces. For those boxes in which you must enter subtractive ornegative numbers use a minus sign. (Example: -300)
Let Pi = units of product iproduced
| Max | P1 | + | P2 | + | P3 | ||
| s.t. | |||||||
| P1 | + | P2 | + | 2P3 | ⤠| ||
| 2.5P1 | + | P2 | + | P3 | ⤠| ||
| P1 | + | 0.25P2 | + | P3 | ⤠| ||
| P1, P2,P3 ⥠0 | |||||||
Solve the linear program formulated in part (a). How much ofeach product should be produced, and what is the projected totalprofit contribution?
P1 =
P2 =
P3 =
Profit = $
After evaluating the solution obtained in part (b), one of theproduction supervisors noted that production setup costs had notbeen taken into account. She noted that setup costs are $650 forproduct 1, $750 for product 2, and $550 for product 3. If thesolution developed in part (b) is to be used, what is the totalprofit contribution after taking into account the setupcosts?
Profit = $
Management realized that the optimal product mix, taking setupcosts into account, might be different from the one recommended inpart (b). Formulate a mixed-integer linear program that takes setupcosts into account. Management also stated that we should notconsider making more than 160 units of product 1, 125 units ofproduct 2, or 130 units of product 3. Here introduce a 0-1 variableyi that is one if any quantity ofproduct i is produced and zero otherwise. If required,round your answers to two decimal places. For those boxes in whichyou must enter subtractive or negative numbers use a minus sign.(Example: -300)
| Max | P1 | + | P2 | + | P3 | + | y1 | + | y2 | + | y3 | ||
| s.t. | |||||||||||||
| P1 | + | P2 | + | 2P3 | ⤠| ||||||||
| 2.5P1 | + | P2 | + | P3 | ⤠| ||||||||
| P1 | + | 0.25P2 | + | P3 | ⤠| ||||||||
| P1 | + | y1 | ⤠| ||||||||||
| P2 | + | y2 | ⤠| ||||||||||
| P3 | + | y3 | ⤠| ||||||||||
| P1, P2,P3 ⥠0; y1,y2, y3 = 0, 1 | |||||||||||||
Solve the mixed-integer linear program formulated in part (d).How much of each product should be produced, and what is theprojected total profit contribution? Compare this profitcontribution to that obtained in part (c).
P1 =
P2 =
P3 =
Profit = $
The profit is increased by $ .
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