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 13 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:
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Let X_{R} = no. of regular widgets produced
X_{D} = no. of deluxe widgets produced
Y_{i}^{+} = amount that goal i is surpassed
Y_{i}^{} = amount that goal i is fallen short by
(Note for Goal 4: X_{R} ≥ X_{D})
Min Y_{1}^{} + Y_{2}^{+} + Y_{3}^{} + 2Y_{4}^{}
s.t.
30 X_{R} + 60 X_{D} – Y_{1}^{+} + Y_{1}^{} = 500 (Goal 1 – Sales revenue)
30 X_{R} + 45 X_{D} – Y_{2}^{+} + Y_{2}^{} = 400 (Goal 2 – Production Cost)
10 X_{R} + 15 X_{D} – Y_{3}^{+} + Y_{3}^{} = 140 (Goal 3  Profit)
X_{R }– X_{D} – Y_{4}^{+} + Y_{4}^{} = 0 (Goal 4 – Regular and Deluxe Prod.)
5 X_{R }+ 8 X_{D} ≤ 80 (Labour)
X_{R }+ X_{D } ≤ 12 (Frames)
X_{R}, X_{D}, Y_{i}^{}, Y_{i}^{+} ≥ 0
b) X_{R} = 4 and X_{D} = 7 (production plan given)
Deviation for each goal
Goal 1: 30 (4) + 60(7) = 540
Goal 1 is over by 40 i.e. Y_{1}^{+} = 40
Goal 2: 30(4) + 45 (7) = 395
Goal 2 has fallen short of the goal by 5 i.e. Y_{2}^{} = 5
Goal 3: 10(4) + 15(7) = 145
Goal 3 is over by 5 i.e. Y_{3}^{+} = 5
Goal 4: (4) – (7) =  3
Goal 4 has fallen short by 3 i.e. Y_{4}^{} = 3
The omnibus objective value is
Y_{1}^{} + Y_{2}^{+} + Y_{3}^{} + 2Y_{4}^{} = (0) + (0) + (0) + 2(3) = $6