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Set_5_dual_simplex

# Set_5_dual_simplex - Dr Maddah ENMG 500 Engineering...

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1 Dr. Maddah ENMG 500 Engineering Management I 11/28/06 The Dual Simplex Method Underlying Theory ¾ Consider the LP (P) and its dual (D) (P) min (D) max s.t. s.t. Z w = = cx by Ax b y yA c x 0 y 0 ¾ The primal (P) optimality conditions are 0, for all variables . j j j z c x -1 B c B A c 0 ¾ Letting y = c B B 1 , implies that (P) optimality conditions are . yA c 0 yA c ¾ I.e., (P) optimality conditions are (D) feasibility conditions. Fact At optimality (and only then) both (P) and (D) are feasible . ¾ Suppose that at the origin O, where x = 0, the optimality conditions for (P) (i.e. (D) feasibility conditions) are satisfied but the feasibility conditions of (P) (i.e. (D) optimality conditions) are not satisfied. ¾ This could be the case if c 0 , and one of the b i ’s is positive. ¾ The dual simplex method maintains the optimality of (P) (i.e. feasibility of (D)) and iterates, similar to the usual “primal” simplex, until (P) feasibility (i.e. optimality of (D)) is reached.

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Set_5_dual_simplex - Dr Maddah ENMG 500 Engineering...

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