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L19 - CO350 Linear Programming Chapter 7 The Two-Phase...

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CO350 Linear Programming Chapter 7: The Two-Phase Method 16th June 2005

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Chapter 7: The Two-Phase Method 1 Recap To construct auxiliary problem ( A ) for ( P ) max c T x s.t. Ax = b x 0 we make sure that b 0 , introduce one artificial variable u i for each constraint, and change the objective to - m i =1 u i . After solving ( A ) , if ( A ) has optimal value < 0 , ( P ) is infeasible; if ( A ) has optimal value = 0 , we can get a feasible basis for ( P ) from the optimal tableau for ( A ) .
Chapter 7: The Two-Phase Method 2 Infeasibility ( § 7.2) Example of infeasibility Given the LP problem ( P ) max ( z =) x 1 s.t. 3 x 1 + 5 x 2 + 2 x 3 - x 4 = 7 2 x 1 + 5 x 2 + 3 x 3 + x 4 = 3 x 1 , x 2 , x 3 , x 4 0 The auxiliary problem is ( A ) max ( w =) - x 5 - x 6 s.t. 3 x 1 + 5 x 2 + 2 x 3 - x 4 + x 5 = 7 2 x 1 + 5 x 2 + 3 x 3 + x 4 + x 6 = 3 x 1 , x 2 , x 3 , x 4 , x 5 , x 6 0 Tableau corresponding to B = { 5 , 6 } is w - 5 x 1 - 10 x 2 - 5 x 3 = - 10 3 x 1 + 5 x 2 + 2 x 3 - x 4 + x 5 = 7 2 x 1 + 5 x 2 + 3 x 3 + x 4 + x 6 = 3 Note: the w -row is obtained by subtracting x 5 -row and x 6 -row from w = - x 5 - x 6 .

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Chapter 7: The Two-Phase Method 3
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