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Lecture 20110311

# Lecture 20110311 - IMSE2008 Operational Research Techniques...

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IMSE2008 Operational Research Techniques Lecture 03/11 Si l M th d Simplex Method Sensitivity Analysis Miao Song D t f I d t i l & M f t i Dept of Industrial & Manufacturing Systems Engineering

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Review of Previous Lecture Formalizing Simplex Method Min Ratio Rule Degeneracy Multiple Optima 3/11/2011 2
Agenda Simplex Method Obtaining an initial bfs The two-phase simplex method The big-M method The big M method Sensitivity Analysis 3/11/2011 3

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Phase 1 LP I th Ph 1 LP f ibl ? Is the Phase 1 LP feasible? Is the Phase 1 LP unbounded? Does there exist an optimal bfs for the Phase 1 LP? If the original problem is feasible, what is the optimal bfs to the Phase 1 LP? If the original problem is infeasible, what is the optimal bfs to the Phase 1 LP? How to obtain an optimal bfs to the Phase 1 LP? min – 2x 1 – 3x 2 – x 3 t 2 min w = y 1 + y 2 t 2 s.t. x 1 + x 2 + x 3 = 2 x 1 + 2x 2 + 3x 3 = 4 x 1 x 2 x 3 0 s.t. x 1 + x 2 + x 3 + y 1 = 2 x 1 + 2x 2 + 3x 3 + y 2 = 4 x 1 x 2 x 3 0 3/11/2011 4 , x , x , x , x
Phase 1 Tableau min w = y 1 + y 2 s.t. x 1 + x 2 + x 3 + y 1 = 2 x 1 + 2x 2 + 3x 3 + y 2 = 4 x 1 , x 2 , x 3 0 x 1 x 2 x 3 y 1 y 2 0 0 0 0 1 1 2 1 1 1 1 0 4 1 2 3 0 1 x 1 x 2 x 3 y 1 y 2 -2 -1 -1 -1 0 1 R1 – R2 2 1 1 1 1 0 4 1 2 3 0 1 x 1 x 2 x 3 y 1 y 2 -6 -2 -3 -4 0 0 R1 – R3 3/11/2011 5 2 1 1 1 1 0 4 1 2 3 0 1

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Phase 1 Tableau min w = y 1 + y 2 s.t. x 1 + x 2 + x 3 + y 1 = 2 x 1 + 2x 2 + 3x 3 + y 2 = 4 x 1 , x 2 , x 3 0 x 1 x 2 x 3 y 1 y 2 -6 -2 -3 -4 0 0 2 1 1 1 1 0 4 1 2 3 0 1 x 1 x 2 x 3 y 1 y 2 -6 -2 -3 -4 0 0 R3 – R2 2 1 1 1 1 0 2 0 1 2 -1 1 x 1 x 2 x 3 y 1 y 2 -2 0 -1 -2 2 0 R1 + 2R2 3/11/2011 6 2 1 1 1 1 0 2 0 1 2 -1 1