Lecture 20110225

Lecture 20110225 - IMSE2008 Operational Research Techniques...

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IMSE2008 Operational Research Techniques ecture 02/25 Lecture 02/25 Simplex Method Miao Song Dept of Industrial & Manufacturing Systems Engineering
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eview of Simplex Method Review of Simplex Method tart with an Start with an feasible extreme point solution eturn the optimal ind an improved Is it optimal? Return the optimal solution Find an improved extreme point solution Yes No o Is the optimum un- Return the feasible direction along which the objective No 2/24/2011 2 p bounded? value goes to infinity
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genda Agenda Simplex Method Formalizing the approach gp p Obtaining an initial bfs 2/24/2011 3
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P Canonical Form LP Canonical Form What are the basic variables? hat are the non- asic variables? What are the non basic variables? What is the bfs? What is the objective value? anonical Form: x 1 x 2 x 3 x 4 x 5 3 10020 9 20130 Canonical Form: Zero cost coefficient for asic variables 1 110 - 1 0 4 00021 basic variables 2/24/2011 4
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The Simplex Pivot Rule and Optimality Condition Is the bfs optimal? If all cost-row coefficients are non-negative then the current basic feasible solution is optimal (basic variable have 0 cost coefficients) ivot in a non asic variable whose cost w Pivot in a non-basic variable whose cost-row coefficient is negative x 1 x 2 x 3 x 4 x 5 3 10020 9 20130 1 110 - 1 0 4 00021 2/24/2011 5
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ptimality Condition Optimality Condition Optimality conditions: the bfs is optimal if c j 0 for all non-basic variable j and c j = 0 for all basic variable j The bar indicates that it is possibly the coefficients after some pivots Reduced Cost x 1 x 2 x 3 x 4 x 5 3 c 1 00 c 4 0 9 20130 1 110 - 1 0 4 00021 2/24/2011 6
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etermining the Entering Variable Determining the Entering Variable Choose an entering variable with negative cost-row coefficient Set the value of the entering variable to Δ and make Δ as large as possible while maintaining feasibility x 1 x 2 x 3 x 4 x 5 3 100 - 2 0 9 20130 1 110 - 1 0 4 00021 2/24/2011 7
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etermining the Exiting Variable Determining the Exiting Variable hat is the exiting variable? What is the exiting variable? To determine the exiting variable, take the min ratio of the RHS coefficient divided by the coefficient of the entering ariable among all those whose variable, among all those whose coefficient is strictly positive x 1 x 2 x 3 x 4 x 5 3 100 - 2 0 9 20130 1 110 - 1 0 4 00021 2/24/2011 8
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inimum Ratio Rule Minimum Ratio Rule Entering variable is x s with negative c s < 0 Pivot out the basic variable in row r where = argmin a 0} and thus r = argmin i {b i /a is : a is > 0}, and thus b r /a rs = min {b i /a is : a is > 0} a 0 for all i then the solution is If a is 0 for all i, then the solution is unbounded x 1 x 2 x 3 x s x 5 z 0 100 c s 0 b 201 a s 0 1 1s b 2 110 a 2s 0 b 3 000 a 3s 1 2/24/2011 9
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ivoting Pivoting he objective value strictly improves so long as 0
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This note was uploaded on 12/09/2011 for the course IMSE 0301 taught by Professor Song during the Spring '11 term at HKU.

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Lecture 20110225 - IMSE2008 Operational Research Techniques...

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