Question

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	2.00	1.50	3.00
B	2.50	2.00	1.00
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 $30 for product 1, $25 for product 2, and $28 for product 3.

(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
	Max $ P1 + $ P2 + $ P3 s.t. P1 + P2 + P3 - Select your answer -≤≥= P1 + P2 + P3 - Select your answer -≤≥= P1 + P2 + P3 - Select your answer -≤≥= 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?
	Product 1 Product 2 Product 3 Amount to Produce
	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. She noted that setup costs are $550 for product 1, $400 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.
	Max $ P1 + $ P2 + $ P3 - $ Y1 - $ Y2 - $ Y3 s.t. P1 - Select your answer -≤≥= Y1 P2 - Select your answer -≤≥= Y2 P3 - Select your answer -≤≥= Y3 P1, P2, P3 ≥ 0
(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”.
	Product 1 Product 2 Product 3 Amount to Produce
	Updated Profit $

Answer 1

a)

Let Pi = units of product i produced
Objective function Z = Max 30P1+25P2+28P3      
s.t.                              
2P1   +   1.5P2   +   3P3   <= 450 (Department A)  
2.5P1   +   2P2   +   1P3   <= 350 (Department B)  
0.25P1   +   0.25P2   +   0.25P3   <= 50 (Department C)  
P1, P2, P3 >= 0  

b) Solving in solver we get,

Amount to Produce for product 1 = 60, product 2 = 60 and product 3 = 80.
Maximized profit = 5540

Solver screenshot

Solver formula

Solver window

c)

After taking setup costs into account, profit = 5540-550-400-600 = 3990

d)

Let, Yi = 1 if any quantity of product i is produced else 0

Objective is to maximize profit = max 30P1+25P2+28P3-550y1-400y2-600y3

Let,

P1 <= 175y1
P2 <= 150y2
P3 <= 140y3

P1, P2, P3 >= 0   and yi = binary

e)

Solving in solver we get,

Amount to Produce for product 1 = 0, product 2 = 100 and product 3 = 100.
Maximized profit = 4300

Solver screenshot

Solver formula

Solver window