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Lecture 20110218

# Lecture 20110218 - IMSE2008 Operational Research Techniques...

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IMSE2008 Operational Research Techniques ecture 02/18 Lecture 02/18 Simplex Method Miao Song Dept of Industrial & Manufacturing Systems Engineering

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eview of Lecture 02/11 Review of Lecture 02/11 Geometry of Linear Programming Extreme Points and Optimality py Infeasibility and Unboundedness Simplex Method Idea of Simplex LP Standard Form 2/18/2011 2
implex Flow Chart Simplex Flow Chart 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 2/18/2011 3 p bounded? value goes to infinity

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genda Agenda Simplex Method Basic Feasible Solution Optimality and Unboundedness ivoting Pivoting Formalizing the approach 2/18/2011 4
implex Tableau Simplex Tableau feasible extreme point min – 2 + 2x 2 + x 5 s.t. 2x 2 + x 3 –x 5 = 4 A feasible extreme point x 1 = 3 = 0 – x 2 + x 4 + 2x 5 = 1 x 1 + 6x 2 + 3x 5 = 3 x 2 0 x 3 = 4 x 4 = 1 x 1 ,x 2 ,x 3 ,x 4 ,x 5 0 x 5 = 0 x 1 x 2 x 3 x 4 x 5 2 02001 4 0210 - 1 obj coefficients constraint - constant in obj func 1 0- 10 1 2 3 16003 constraint coefficients right hand side 2/18/2011 5

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tart with a Feasible Extreme Point Start with a Feasible Extreme Point LP has equality constraints and non- negativity constraints. There is one “basic” variable for each equality constraint. The column for the basic variable for onstraint j has a 1 in constraint j and 0’s constraint j has a 1 in constraint j and 0s elsewhere. The remaining variables are called non- basic. 2/18/2011 6
asic Feasible Solution (BFS) Basic Feasible Solution (BFS) min – 2 + 2x 2 + x 5 s.t. 2x 2 + x 3 –x 5 = 4 What are the basic variables? What are the non-basic variables? – x 2 + x 4 + 2x 5 = 1 x 1 + 6x 2 + 3x 5 = 3 Set non-basic variable to 0 hat is the basic feasible solution?

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Lecture 20110218 - IMSE2008 Operational Research Techniques...

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