Question

The Wearever Carpet Company manufactures two brands of carpet, shag and sculptured, in 100-yard lots. It requires eight hours to produce one lot of shag carpet and six hours to produce one lot of sculptured carpet. The company has the following production goals in prioritized order.

(1) Do not underutilize production capacity, which is 480 hours.

(2) Limit production overtime to 20 hours.

(3) Achieve product demand of forty 100-yard lots for shag and fifty 100-yard lots for sculptured carpet. Meeting demand for shag is more important than meeting demand for sculptured by a ratio of 5 to 2.

(a) Formulate a goal programming model to determine the amount of shag and sculptured carpets to produce to best meet the company’s goals.

(b) Solve this problem (i.e., all the decision variables including the deviational variables should be found) using graphical method.

Answer 1

ANSWER a :

Let X1 and X2 be the number of lots of shag and sculpture carpets to be produced respectively. Also, Let Ui and Oi be the under and overachievements of the i-th goal.

Objective function: Minimize Z = 3*U1 + 2*(5/7)*U2 + 2*(2/7)*U3 + 1*O1
Subject to,
8X1 + 6X2 + U1 - O1 = 480 (min labor hours - goal 1)
X1 + U2 - O2 = 40 (demand for X1 - goal 2)
X2 + U3 - O3 = 50 (demand for X2 - goal 2)
8X1 + 6X2 + U1 - O1 = 480 (max labor hours - goal 3)
X1, X2, Ui, Oi >= 0

ANSWER b :

(b)