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lect7 - ISE 536Fall03 Linear Programming and Extensions...

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ISE 536–Fall03: Linear Programming and Extensions September 22, 2003 Lecture 7: Simplex Method, The Tableau Lecturer: Fernando Ord´ o˜nez 1 Outline of Algorithm The makings of the Simplex algorithm, and formulas to keep in mind: Find a BFS Ax = b x 0 A = [ BN ] x = ( x B , x N ) c = ( c B , c N ) x B = B - 1 b 0 x N = 0 . Find a direction d that is feasible and improves the cost ( d N ) j = 1 j N ( d N ) i = 0 i N, i = j d B = - B - 1 A j b = Bx B + θ ( Bd B + Nd N ) d B = - B - 1 A j and c t ( x + θd ) < c t x c t d < 0 which means that the reduced cost for variable j ¯ c j := ( c N ) j - c t B B - 1 A j < 0 Move as much as possible in that direction, i.e. need to enforce that x + θd 0 2 The Tableau Recall the example min c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 + c 5 x 5 s . t . x 1 + x 2 + x 3 + x 4 = 5 x 2 - x 3 + x 5 = 3 2 x 1 + x 2 = 4 1
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2.1 Graphically: 2.2 Obtain a BFS 2.3 Find Basic Directions and Reduced Costs 2.4 Tableau The tableau is a simple way of being organized, we represent all coefficients of interest in a table, with the goal of keeping I as the matrix for the basic variables, with operations Multiply a row by a scalar Multiply a row by a scalar and add to another row For example 0 c t B c t N b B N simple operations - c t B B - 1 b 0 c t N - c
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