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# a6 - CO350 Assignment 6 Wing Man Fok 20300650 Instructor...

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CO350 Assignment 6 Wing Man Fok 20300650 , June 25 2010 : - Instructor Mathieu Guay Paquet 1a Suppose there is a tie for the choice of leaving variable in an iteration of the . simplex method , = = , , ∉ ≠ . That is t biaik bjajk for some k N i j B with i j Assume WLOG that the new = ∪ ∖ . , , basis is B' B k i When pivoting k i we subtract ajkaik times the xi row from . = - = . , the xj row So bj' bj ajkaikbi 0 Hence the new basic variable has value zero in , . - the new solution and B' is degenerate Since every perturbed LP is non , degenerate there is never a tie for the choice of leaving variable in the . , , perturbation method Otherwise the new basis is degenerate which is . impossible 2a Suppose there is an iteration of the simplex method which starts with basis B . > = - . =- and ends with basis B' We can find t 0 and Ak such that xB' xB tAk Let d . , = ( )= - = ( - )= . Ak Since B and B' are both feasible Ad 1tA td 1tAxB' AxB 1t b b 0 Also note that = = - > cTd 1tcTtd 1tcTxB' cTxB 0 = - = - > dk 1txB'k xBk 1txB'k 0 0 = - = - < dr 1txB'r xBr 1t0 xBr 0 =- = di aik 0 for i B k 2b

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a6 - CO350 Assignment 6 Wing Man Fok 20300650 Instructor...

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