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Lecture 5

# Lecture 5 - Duality 1 The Production Planning Problem Two...

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1 Duality

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2 The Production Planning Problem Two Products: P and Q Three machines: A, B, and C Objective: maximum profit from P and Q Constraints: available machine hours 0 , 0 C) (Machine 1440 6 15 B) (Machine 1440 28 12 A) (Machine 1800 10 20 to subjec 60 45 Maximize + + + + = Q P Q P Q P Q P Q P Z
3 A Related Problem Another firm is considering to buy out all the available machine times. What is an appropriate price? Variables: X A , X B , X C : unit price for each hour of each machine A, B, C – Total amount: w =1800 X A +1440 X B +1440 X C For the buyer to minimize w, what are the constraints?

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4 Value of the Resources For the current firm, 20 hours of machine A + 12 hours of machine B + 15 hours of machine C = 1 product P which brings a profit of 45 Thus, Constraint 1: 20 X A + 12 X B +15 X C 45 Similarly, consider profit from Q, Constraint 2: 10 X A +28 X B +6 X C 60 The owner won’t sell if these constraints are not satisfied Non-negativity constraints
Primal and Dual 0 , 0 C) (Machine 1440 6 15 B) (Machine 1440 28 12 A) (Machine 1800 10 20 to subjec 60 45 Maximize + + + + = Q P Q P Q P Q P Q P Z 0 , , Q) (Product 60 6 28 10 P) (Product 45 15 12 20 to subjec 1440 1440 1800 Minimize + + + + + + = C

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Lecture 5 - Duality 1 The Production Planning Problem Two...

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