Lecture 20110325

# Lecture 20110325 - IMSE2008 Operational Research Techniques...

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IMSE2008 Operational Research Techniques ecture 03/25 Lecture 03/25 Duality Miao Song Dept of Industrial & Manufacturing Systems Engineering

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genda Agenda Duality Pricing Dual Problem Weak Duality and Strong Duality yg y Complementary Slackness 3/25/2011 2
Quick Review of Shadow Prices A Quick Review of Shadow Prices min z = 3x 1 – 2x 2 s.t. –3x 1 + 3x 2 + x 3 = 6 x 2x x 2 Shadow Prices –1/3 /2 –4x 1 + 2x 2 + x 4 = 2 x 1 ,x 2 ,x 3 ,x 4 0 –1/2 Suppose that the 6 on the RHS decreases from 6 to 5.6. What is the impact on the optimal obj value? uppose that the 6 on the RHS increases from 6 to 7 Suppose that the 6 on the RHS increases from 6 to 7. What is the impact on the optimal obj value? Suppose that the 2 on the RHS decreases from 2 to 1.7. What is the impact on the optimal obj value? Do we need any assumption in the calculations? tay in the allowable range of the shadow price 3/25/2011 3 Stay in the allowable range of the shadow price

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Quick Review of Shadow Prices A Quick Review of Shadow Prices min z = 3x 1 – 2x 2 s.t. –3x 1 + 3x 2 + x 3 = 6 x 2x x 2 Shadow Prices –1/3 /2 –4x 1 + 2x 2 + x 4 = 2 x 1 ,x 2 ,x 3 ,x 4 0 –1/2 What are the optimal reduced cost for each decision variable? n i i j i j j s a c c 1 , What are the basic variables? 3/25/2011 4
rices for a Linear Program Prices for a Linear Program min z = 3x 1 – 2x 2 s.t. –3x 1 + 3x 2 + x 3 = 6 x 2x x 2 Prices 0.33 7 2 –4x 1 + 2x 2 + x 4 = 2 x 1 ,x 2 ,x 3 ,x 4 0 –17.2 A set of prices for a linear program is a collection of real numbers associated with each constraint, other than the onnegativity constraints nonnegativity constraints Shadow prices are some special prices for the LP 3/25/2011 5

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ricing Out to Get Reduced Costs Pricing Out to Get Reduced Costs min z = 3x 1 – 2x 2 s.t. –3x 1 + 3x 2 + x 3 = 6 x 2x x 2 Prices –1 –4x 1 + 2x 2 + x 4 = 2 x 1 ,x 2 ,x 3 ,x 4 0 –2 Optimal reduced cost = original costs minus column coefficients time shadow prices Reduced cost = original costs minus column coefficients time prices n i i j i j p a c 1 , 3/25/2011 6
ricing Out to Get Reduced Costs Pricing Out to Get Reduced Costs min z = 3x 1 – 2x 2 s.t. –3x 1 + 3x 2 + x 3 = 6 x 2x x 2 min –8x 1 + 5x 2 + x 3 + 2x 4 – 10 s.t. –3x 1 + 3x 2 + x 3 = 6 x x 2 –4x 1 + 2x 2 + x 4 = 2 x 1 ,x 2 ,x 3 ,x 4 0 –4x 1 + 2x 2 + x 4 = 2 x 1 ,x 2 ,x 3 ,x 4 0 Are these two problems equivalent? Do they have the same optimal solution & optimal value? For a problem with equality constraints, optimizing wrt to the reduced costs is the same as optimizing wrt the riginal costs original costs 3/25/2011 7

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genda Agenda Duality Pricing rices: real numbers assigned to constraints Prices: real numbers assigned to constraints Reduced costs can be obtained from prices Optimizing wrt reduced costs is equivalent to optimizing wrt original costs
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