Notes 6 SimplexMethod corrected

# Notes 6 - Simplex Method and the Tableau Procedure The solution procedure presented in these notes is the simplex method for solving linear

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Simplex Method and the Tableau Procedure The solution procedure presented in these notes is the simplex method for solving linear programming problems. A key observation about the linear programming solution method (using Basic Feasible Solutions) is that when we select a basis, the set B x , and develop the coefficient matrix of ( ) , T B zx , call it M , then 1 M times the original data coefficients always yields an identity matrix. Consider the data 10 0 T zc x zA x b −= += ( 1 ) partitioned by ( ) ,, TT BN zx x then 0 BB NN zcx cx zB x N x b −− = ++ = ( 2 ) Now, 1 0 T B c M B ⎡⎤ = ⎢⎥ ⎣⎦ , 1 1 1 0 T B cB M B = and 1 1 1 0 T B B 0 x c x x N x b = ++= yields () 11 0 TTT T N N B zxc B N c x c B b zI x B N x B b −+ = = (3) The key observation here is that the mm + × + identity matrix comprises the ( ) , T B coefficients. Thus, by setting 0 N x = , we have the basic solution 1 1 T B B zcBb x Bb = = . The matrix multiplication method for creating (3) from (2) is not the only method for obtaining this solution.

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A computationally convenient method is to use Gaussian reduction to diagonalize the ( ) , T B zx coefficients. This procedure is particularly useful for moving from one BFS to an adjacent BFS. To illustrate, consider the three tableaux used to solve the example problem: Max z = 1 x + 5 2 x s.t. 2 1 x + 3 2 x 12 -
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## This note was uploaded on 10/26/2011 for the course ISEN 620 taught by Professor Curry during the Fall '10 term at Texas A&M.

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Notes 6 - Simplex Method and the Tableau Procedure The solution procedure presented in these notes is the simplex method for solving linear

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