Lecture 20110325

Lecture 20110325 - IMSE2008 Operational Research Techniques...

Info icon This preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
IMSE2008 Operational Research Techniques ecture 03/25 Lecture 03/25 Duality Miao Song Dept of Industrial & Manufacturing Systems Engineering
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
genda Agenda Duality Pricing Dual Problem Weak Duality and Strong Duality yg y Complementary Slackness 3/25/2011 2
Image of page 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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern