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L25 - CO350 Linear Programming Chapter 9 The Revised...

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CO350 Linear Programming Chapter 9: The Revised Simplex Method 29th June 2005
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Chapter 9: Revised Simplex Method 1 Motivation The most computationally intensive part of the simplex method is pivoting ; i.e., the construction of the new tableau. Tableau: z - j N ¯ c j x j = ¯ v x i + j N ¯ a ij x j = ¯ b i ( i B ) In a tableau, there are n - m nonbasic ¯ c j ’s one ¯ v , ( n - m ) × m ¯ a ij ’s, m ¯ b i ’s. In all, there are ( m + 1)( n - m + 1) numbers in a tableau. However, we only need n - m nonbasic ¯ c j ’s to decide entering variable x k , and m ¯ a ik ’s and m ¯ b i ’s to decide leaving variable x r . In all, we only need n + m numbers. Revised Simplex Method An implementation of the simplex method that computes only the necessary coefficients instead of the whole tableau.
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Chapter 9: Revised Simplex Method 2 Entering variable We need to compute nonbasic ¯ c j . By definition, a tableau is derived from z - c T x = 0 Ax = b using elementary row operations. Thus there exists y = [ y 1 , y 2 , . . . , y m ] T such that the z -row z - j N ¯ c j x j = ¯ v is equivalent to the linear combination z - c T x + y T Ax = y T b ; i.e. z - ( c - A T y )
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