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SamplePrelim1Solutions

# SamplePrelim1Solutions - Sample Prelim Solutions...

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Sample Prelim Solutions: Problem #1: (a): x 3 , x 4 and x 5 are slack variables that were added to put the original problem into canonical form so that the Simplex method can be used. (b): The basis is x 1 , x 2 , x 3 , so the basis matrix is: B = 3 1 1 3 2 0 1 1 0 (c): B - 1 can be determined by looking at the coefficients of the slack variables in the final tableau. In this instance: B - 1 = 0 1 - 2 0 - 1 3 1 - 2 3 (d): Recall ¯ c = c - c B B - 1 A . Hence, letting c 2 = 3: ¯ c = [43000] - [430] 0 1 - 2 0 - 1 3 1 - 2 3 3 1 1 0 0 3 2 0 1 0 1 1 0 0 1 = [000 - 1 - 1] which matches the reduced costs from the final tableau. Similarly, recalling ¯ b = B - 1 b , you can verify that 0 1 - 2 0 - 1 3 1 - 2 3 9 10 4 = 2 2 1 (e): The basic variables are x 1 , x 2 , x 3 and the nonbasic variables are x 4 , x 5 . The solution given in the final representation is: [ x 1 , x 2 , x 3 , x 4 , x 5 ] = [2 , 2 , 1 , 0 , 0] , using the corrected data from part (d). (f): Recall from above that ¯ b = B - 1 b . Hence, using the new value of b , you can verify that ¯ b is now [0 , 6 , 2] which is non-negative. Since changing the right-hand-side value does not affect the reduced costs (which are still all non-positive) and this new basic solution is feasible (since it’s

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