Question

The management of Hartman Company is trying to determine the amount of each of two products to produce over the coming planning period. The following information concerns labor availability, labor utilization, and product profitability:

	Labor-Hours Required (hours/unit)
Department	Product 1		Product 2		Hours Available
A	1.00		0.35		95
B	0.30		0.20		36
C	0.20		0.50		50
Profit contribution/unit	$30.00		$15.00

(a)	Develop a linear programming model of the Hartman Company problem. Solve the model to determine the optimal production quantities of products 1 and 2.
	If required, round your answer to two decimal places.
	Product 1 Product 2 Production
(b)	In computing the profit contribution per unit, management does not deduct labor costs because they are considered fixed for the upcoming planning period. However, suppose that overtime can be scheduled in some of the departments. Which departments would you recommend scheduling for overtime?
	- Select your answer -Dept ADept BDept CItem 3
	What is the upper limit of what you would be willing to pay per hour of overtime in the department you recommended?
	If required, round your answer to two decimal places.
(c)	Suppose that 10, 6, and 8 hours of overtime may be scheduled in departments A, B, and C, respectively. The cost per hour of overtime is $18 in department A, $22.50 in department B, and $12 in department C. Formulate a linear programming model that can be used to determine the optimal production quantities if overtime is made available. What are the optimal production quantities, and what is the revised total contribution to profit?
	If required, round your answer to two decimal places.
	Product 1 Product 2 Production
	If required, round your answer to nearest whole number.
	Total Profit $
	How much overtime do you recommend using in each department?
	If required, round your answer to two decimal places. If you answer is zero, enter “0”.
	OT hours: Dept. Used A B C
	What is the increase in the total contribution to profit if overtime is used?
	If required, round your answer to nearest whole number.
	$

Answer 1

Answer:

(a) Linear programming model is following:

Let X1, X2 be the optimal production quantities of product 1 and 2 respectively

Maximize 30X1+15X2

s.t.

1X1+.35X2 <= 95

.3X1+.2X2 <= 38

.2X1+.5X2 <= 50

X1, X2 >= 0

Solution is determined using LINDO as follows:

Product 1 = 69.77

Product 2 = 72.09

Total profit = $ 3174.42

(b) Dual price of row 2 (pertaining to department A) and row 4 (pertaining to dept C) is non-zero. Which means that each additional hours in department A will yield an incremental profit of $ 27.90 and each additional hour in department C will yield an incremental profit of $ 10.47  

Therefore, overtime should be scheduled in dept A and C.

Upper limit to pay for overtime in dept A = 27.90

Upper limit to pay for overtime in dept C = 10.47

(c) The linear programming including the overtime is following:

Let A, B, C be the number of overtime hours scheduled in dept A, B, C respectively

Maximize 30X1+15X2-18A-22.5B-12C

s.t.

1X1+.35X2-A <= 95

.3X1+.2X2-B <= 38

.2X1+.5X2-C <= 50

A <= 10

B <= 6

C <= 8

X1, X2, A, B, C >= 0

Solution of this model using LINDO is following:

Product 1 = 81.40

Product 2 = 67.44

Total profit = $ 3273.49

OT hours used

Dept A = 10

Dept B = 0

Dept C = 0

Increase in total contribution to profit by using overtime = 3273.49 - 3174.42 = $ 99