Question 1.2 of Test 1

Ace Widget must determine how many regular and deluxe widgets to produce. Each regular widget requires 5 hours of labour, while 8 hours are needed to complete a deluxe widget; only 80 hours are available. Each unit requires exactly 1 frame and there are just 12 frames available. The following information applies:

 Regular Deluxe Selling Price \$30 \$60 Unit Cost \$20 \$45 Unit Profit \$10 \$15

The following goals must be met:

Goal 1: Sales Revenue ≥ \$500

Goal 2: Production Cost ≤ \$400

Goal 3: Profit ≥ \$140

Goal 4: At least as many regular widget as deluxes must be made

The cost violating Goals 1-3 is \$1 per dollar while there is a \$2 cost per unit violation of Goal 4.

a)      Formulate the decision as a goal program

b)      Determine the value of each goal deviation variables under a production plan to make 4 regular and 7 deluxe widgets. What is the omnibus objective values under the plan

Solution:

Let XR = no. of regular widgets produced

XD = no. of deluxe widgets produced

Yi+ = amount that goal i is surpassed

Yi- = amount that goal i is fallen short by

(Note for Goal 4: XR ≥ XD)

Min Y1- + Y2+ + Y3- + 2Y4-

s.t.

30 XR + 60 XD – Y1+ + Y1- = 500                                (Goal 1 – Sales revenue)

30 XR + 45 XD – Y2+ + Y2- = 400                                (Goal 2 – Production Cost)

10 XR + 15 XD – Y3+ + Y3- = 140                                (Goal 3 - Profit)

XR  –      XD – Y4+ + Y4- = 0                                    (Goal 4 – Regular and Deluxe Prod.)

5    XR +  8  XD                    ≤ 80                                 (Labour)

XR +      XD                             ≤ 12                               (Frames)

XR, XD, Yi-, Yi+ ≥ 0

b) XR = 4 and XD = 7 (production plan given)

Deviation for each goal

Goal 1: 30 (4) + 60(7) = 540

Goal 1 is over by 40 i.e. Y1+ = 40

Goal 2: 30(4) + 45 (7) = 395

Goal 2 has fallen short of the goal by 5 i.e. Y2- = 5

Goal 3: 10(4) + 15(7) = 145

Goal 3 is over by 5 i.e. Y3+ = 5

Goal 4: (4) – (7) = - 3

Goal 4 has fallen short by 3 i.e. Y4- = 3

The omnibus objective value is

Y1- + Y2+ + Y3- + 2Y4- = (0) + (0) + (0) + 2(3) = \$6