Question

Rocket Computers is a local computer hardware company. They specialize in ordering computer components from manufacturers and assembling the components into a computer that can be sold in stores. They currently have two products on the market and the relevant information for each product is given in the table below. Rocket Computers has a goal of achieving $4800 in profits and incurs a $10 penalty for each dollar it falls short of this goal. The company has 32 hours of labor available. A $20 penalty is incurred for each hour of overtime (labor over 32 hours) used, and $10 penalty is incurred for each hour of available labor that is unused. Marketing considerations require at least 5 units of product 1 and 10 units of product 2 be produced. For each unit of either product by which production falls short of demand, a penalty of $50 is assessed. Product 1 Product 2 Labor Required 4 hours 2 hours Contribution to profit $400 $200 (a) Formulate a weighted goal programming problem that can be used to minimize the penalty incurred by Rocket Computers. (15 points) (b) Suppose that the company sets (in order of importance) the following goals: 1. Goal 1: Avoid underutilization of labor 2. Goal 2: Meet demand for product 1 3. Goal 3: Meet demand for product 2 4. Goal 4: Do not use overtime How would the model change to account for this new situation? (5 points)

Answer 1

(a)

Weighted goal programming model is as follows:

Let X1 and X2 be the number of product 1 and 2 to be produced respectively

pi and ni be the positive and negative deviation variables for goal i

Min 10n1+20p2+10n2+50n3+50n4 (total penalty $)

s.t.

Goal 1- profit) 400X1+200X2-p1+n1 = 4800

Goal 2 - overtime and undertime) 4X1+2X2-p2+n2 = 32

Goal 3 - demand of product 1) X1-p3+n3 = 5

Goal 4 - demand of product 2) X2-p4+n4 = 10

X1, X2, pi, ni >= 0

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(b)

The model would change as follows:

Let O be the overtime hours

Min 4n1+3n2+2n3+p4

s.t.

Goal 1- undertime) 4X1+2X2-O+n1 = 32

Goal 2 - demand of product 1) X1-p2+n2 = 5

Goal 3 - demand of product 2) X2-p3+n3 = 10

Goal 4 - overtime) O-p4+n4 = 0

X1, X2, O, pi, ni >= 0