two phase simplex algr - STOR 415 HANDOUT Two-phase Simplex...

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STOR 415 HANDOUT Two-phase Simplex algorithm Given any LP in general form, we can always convert it into standard form. If the standard-form LP turns out be in canonical form, then one can apply the simplex method right away. Otherwise, one needs to go through extra steps. The entire procedure is what we call the two-phase simplex algorithm . Step 0. Start with an LP in standard form. max z = c T x s.t. Ax = b , x 0 Step 1. Make the rhs nonnegative, by multiplying equations with negative right- hand-side by - 1. After this step, we have b 0 . Step 2. Add one nonnegative artificial variable to each equation. Ignore the original LP’s objective function, and set the objective function to be maxi- mizing the sum of the negatives of all artificial variables. The resulted LP is called the Phase I LP . max t = - y 1 - y 2 -···- y m s.t. Ax + y = b , x 0 y 0 Then express the objective function of the phase I LP as a function of the non-artificial variables. The phase I LP is now in canonical form. The original LP is feasible if and only if the optimal value of the Phase I LP is zero.
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