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artificial bases

artificial bases - Economics 146 Fall 2007 Linear...

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Economics 146 – Fall, 2007 Linear Programming – Part III Last revision: 10/22/07. Artificial Basis Vectors. If an LP problem in computational form has no obvious initial basis, one method to use to start the simplex method is to introduce artificial basis vectors. We give an example first. Consider the following LP problem in computational form (which we did not get from a max problem in standard form): minimize - x 1 - 2 x 2 - 3 x 3 + x 4 subject to x 1 +2 x 2 +3 x 3 = 15 2 x 1 + x 2 +5 x 3 = 20 x 1 + 2 x 2 + x 3 + x 4 = 10 and x 1 , . . . , x 4 0 (1) The simplex tableau (with no initial feasible basis chosen yet) is: - 1 - 2 - 3 1 a 1 a 2 a 3 a 4 c 1 2 3 0 15 2 1 5 0 20 1 2 1 1 10 (2) Note that there is no obvious choice for an initial feasible basis. (We did not get this problem by adding slack variables to a problem in standard maximum form.) There is a good choice for one basis vector, viz. a 4 . We will augment the problem by adding two new artificial variables x 5 , x 6 and two new artificial basis vectors a 5 = [1 , 0 , 0] T , a 6 = [0 , 1 , 0] T .
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