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

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Unformatted text preview: CO350 Linear Programming Chapter 9: The Revised Simplex Method 29th June 2005 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- P j N c j x j = v x i + P 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. 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 = Ax = b using elementary row operations. Thus there exists y = [ y 1 , y 2 , . . . , y m ] T such that the z-row z- P 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 ) T x = y T b Comparing coefficients of basic...
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L25 - CO350 Linear Programming Chapter 9: The Revised...

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