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Unformatted text preview: IE 495 Lecture 25 November 30, 2000 Reading for This Lecture Primary Â¡ Bazaraa, Sherali, and Sheti, Chapter 2. Â¡ Chvatal, Chapters 6 and 7. Linear Programming Introduction Consider again the system Ax = b, A âˆˆ R m Ã— n , b âˆˆ R m . In this problem, there are either Â¡ no solutions Â¡ one solution Â¡ infinitely many solutions (if n > m) The problem of linear programming is min c T x s.t. Ax = b x â‰¥ 0 The Simplex Algorithm Note that x B = B1 b  B1 Nx N Hence, c T x = c B T x B + c N T x N = c B T B1 b + ( c N T c B T B1 N)x N So if c N T c B T B1 N â‰¥ , we have found the optimal solution (why?). Otherwise, suppose some component of c N T c B T B1 N is negative. Then we raise the value of the corresponding variable as much as possible while maintaining feasibility. Summary of the Simplex Algorithm Simplex algorithm Â¡ Compute yB = c B T Â¡ Choose a column of a j of N such ya j < c j Â¡ Compute Bd = a j Â¡ Find the largest t such that x B *  td â‰¥ 0 Â¡ Set the value of x j to t and the values of the basic variables to x B *  td ....
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 Fall '08
 Linderoth
 Operations Research, Linear Programming, Optimization, Simplex algorithm

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