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:

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:

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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 Y₁^- + Y₂⁺ + Y₃^- + 2Y₄^-

 s.t.

30 X_R + 60 X_D – Y₁⁺ + Y₁^- = 500                                (Goal 1 – Sales revenue)

30 X_R + 45 X_D – Y₂⁺ + Y₂^- = 400                                (Goal 2 – Production Cost)

10 X_R + 15 X_D – Y₃⁺ + Y₃^- = 140                                (Goal 3 - Profit)

     X_R  –      X_D – Y₄⁺ + Y₄^- = 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. 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