1. Punk and Pawn manufactures hockey sticks and chess sets. Each hockey stick yi
ID: 453012 • Letter: 1
Question
1. Punk and Pawn manufactures hockey sticks and chess sets. Each hockey stick yields a profit of #2, and each chess set yields $4 profit. A hockey stick requires 4 hours of processing at Machine Center A and 2 hours at Machine Center B. Each chess set requires 6 hours at machine center A and 6 hour at machine center B, and 1 hour at Machine Center C. Machine center A has a maximum of 120 hours of available capacity per day; Machine Center B has 72 hour capacity per day; and Machine Center C has 10 hours capacity per day. The manufacturer seeks to maximize daily profits.
Formulation - Decision variables: Let XH = # of hockey sticks produced per day, and let XC = # of Chase Sets produced per day. The linear program is as below:
MAX P = 2XH + 4XC
S.T. 4XH + 6XC £ 120 (Machine Center A Capacity)
2XH + 6XC £ 72 (Machine Center B Capacity)
1XC £ 10 (Machine Center C Capacity)
XH, XC£ 0 (Non-negativity)
This Linear Program was run using computer soft ware and results were as shown below:
PUNK & PAWN COMPANY
OPTIMAL SOLUTION - DETAILED REPORT
Variable Value Cost Red. cost Status
1 XH 24.0000 2.0000 0.0000 Basic
2 XC 4.0000 4.0000 0.0000 Basic
Slack Variables
3 CONSTR 1 0.0000 0.0000 -0.3333 Lower bound
4 CONSTR 2 0.0000 0.0000 -0.3333 Lower bound
5 CONSTR 3 6.0000 0.0000 0.0000 Basic
Objective Function Value = 64
PUNK & PAWN COMPANY
OPTIMAL SOLUTION - DETAILED REPORT
Constraint Type RHS Slack Shadow price
1 CONSTR 1 <= 120.0000 0.0000 0.3333
2 CONSTR 2 <= 72.0000 0.0000 0.3333
3 CONSTR 3 <= 10.0000 6.0000 0.0000
Objective Function Value = 64
(a) From the above computer output, how many hockey sticks and how many chess sets are to be produced per day and how much profit does the company make per day.
(b) How much capacity is left unused in machine center A? Machine Center B, and Machine Center C?
(c) Explain the meaning and logic behind the shadow prices.
Explanation / Answer
a). From the above computer output, 24 hockey sticks and 4 chess sets re to be produced per day and company make $64 profit per day.
b) The two constraint (Machine center A capacity and Machine center B capacity) is utiizing full capacity whereas third constraint (Machine center C capacity) is left 6 units unused.
c) The meaning and logic of shadow price is if the units of the constraints increased or decreased by 1, it will have an effect on optimal function. For example if the constraints 1 and 2 have been increased by 1 it would give the profit of 0.33 respectively.
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