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 1.95 3.45 2.45 b 2.90 1.90 3.40 c 0.70 0.70 0.70 during the next production period, the labor-hours available are 540 in department a, 440 in department b, and 140 in department c. the profit contributions per unit are $34 for product 1, $37 for product 2, and $39 for product 3. formulate a linear programming model for maximizing total profit contribution. for those boxes in which you must enter subtractive or negative numbers use a minus sign. (example: -300) let pi = units of product i produced
Max : $34X + $37y + $39Z
1.95X + 3.45Y + 2.45Z ≤ 540
2.90X + 1.90Y +3.40Z ≤ 440
0.70X + 0.70Y + 0.70Z ≤ 140
X ≥ 0 , Y ≥ 0 , Z ≥ 0
Represent product 1 by X, product 2 by Y and product 3 by Z
Missing Part of the Question
The management also stated that we should not consider making more than 175 units for product 1, 150units of product 2, or 140 units of product 3.
Answer and Explanation:
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
The profit contributions per unit are
Product 1 = $25
Product 2 = $28
product 3 = $30
The set up costs are
Product 1 = $400
Product 2 = $550
Product 3 =$600
The labor-hours available are
Department A = 450
Department B = 350
Department C = 50
Formulating a linear programming model, as have
Let X be the number of units produced
Max Z = 25X1 + 28X2 + 30X3 - 400Y1 - 550Y2 - 600Y3
Subjected to (the constraints are)
1.5X1 + 3X2 + 2X3 ≤ 450
2X1 + X2 + 2.5X3 ≤ 350
0.25X1 + 0.25X2 + 0.25X3 ≤ 50
0 ≤ X1 ≤ 175Y1
0 ≤ X2 ≤ 150Y2
0 ≤ X3 ≤ 140Y3
The first three constraints represents the labour hours in departments A, B and C.
While the last three constraints represents the maximum production of product 1,2 and 3.
The solution is given in the pictures attached
(a)Formulate a linear programming model for maximizing total profit contribution.
If the constant is "1" it must be entered in the box. If required, round your answers to two decimal places.
Let Pi = units of product i produced
Solve the linear program formulated in part (a). How much of each product should be produced, and what is the projected total profit contribution
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 $400 for product 1, $550 for product 2, and $600 for product 3. If the solution developed in part (b) is to be used, what is the total profit contribution 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 provided in part (c) into account. Management also stated that we should not consider making more than 175 units of product 1, 150 units of product 2, or 140 units of product 3. What are the new objective function and additional equation constraints?
If the constant is "1" it must be entered in the box.
Let Yi is one if any quantity of product i is produced and zero otherwise.
(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? Compare this profit contribution to that obtained in part (c).
If required, round your answers to nearest whole number. If your answer is zero enter “0”.