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Unformatted text preview: 8 Phase 1 of the simplex method Let us summarize the method we sketched in the previous section, for finding a feasible basis for the constraint system n X j =1 a ij x j = b i ( i = 1 , 2 , . . . , m ) x j ≥ ( j = 1 , 2 , . . . , n ) . (8.1) We can assume each righthand side b i is nonnegative: otherwise we multiply the corresponding equation by 1. The idea of “Phase 1” of the simplex method is, corresponding to each constraint, to introduce an “artificial variable” u i ≥ 0 (for i = 1 , 2 , . . . , m ), which we think of as the error in that constraint. We therefore consider the system ( S ) X j a ij x j + u i = b i ( i = 1 , 2 , . . . , m ) . We then try to force the values of these artificial variables u i to zero, by using the simplex method to minimize their sum: min n m X i =1 u i : ( S ) holds , x ≥ , u ≥ o (8.2) We transform this Phase 1 problem to standard equality form by multiplying the objective function by 1. An initial feasible tableau is readily available for this problem: we choose u 1 , u 2 , . . . , u m as the initial list of basic variables, and eliminate them from the top equation defining the Phase 1 objective function using row operations. We then begin the simplex method. If, after some iteration of the Phase 1 simplex method, we arrive at a tableau ( T ) where one of the variables, u r say, has become nonbasic, then before proceeding to the next iteration we delete all the appearances of the artificial variable...
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 Fall '07
 BLAND
 Linear Programming, Optimization, feasible tableau

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