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# 06 - Spring 2010 CS530 Analysis of Algorithms Homework 6...

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Spring 2010 CS530 – Analysis of Algorithms Homework 6 H OMEWORK 6, DUE M ARCH 17 You must prove your answer to every question. Problems with a ( * ) in place of a score may be a little too advanced, or too challenging to most students, so I do not assign a score to them. But I will still note if you solve them. Problem 1 (29.5-6) . (25pts) Solve the following linear programming prob- lem, by the simplex method. (This is the politician’s problem we started with.) minimize x 1 + x 2 + x 3 + x 4 subject to - 2 x 1 + 8 x 2 + 10 x 4 50, 5 x 1 + 2 x 2 100, 3 x 1 - 5 x 2 + 10 x 3 - 2 x 4 25, x 1 , x 2 , x 3 , x 4 0. Solution. The auxiliary variable for the first phase (to find a feasible solu- tion) is called x 5 . We muliply by - 1 all the inequalities and the objective function, in order to get a standard form. I present the solution as a list of tableaux. The variable numbers appear at the head of their columns for non- basic variables (following the number of the column) and at the head of their rows for basic variables (folowing the number of the row). The last two lines are the value (with reversed sign) and the coefficients of the two objective functions: the original one, and the function - x 5 . 3 times 5 1 1 2 2 3 3 4 4 5 5 1 6 -50 2 -8 0 -10 -1 2 7 -100 -5 -2 0 0 -1 3 8 -25 -3 5 -10 2 -1 0 -1 -1 -1 -1 0 0 0 0 0 0 -1 Bringing x 5 into the basis: 3 times 5 1 1 2 2 3 3 4 4 5 7 1 6 50.000 7.000 -6.000 0.000 -10.000 -1.000 2 5 100.000 5.000 2.000 -0.000 -0.000 -1.000 3 8 75.000 2.000 7.000 -10.000 2.000 -1.000 0.000 -1.000 -1.000 -1.000 -1.000 0.000 100.000 5.000 2.000 0.000 0.000 -1.000

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Spring 2010 CS530 – Analysis of Algorithms Homework 6 Optimizing the objective function found in the last line. 3 times 5 1 6 2 2 3 3 4 4 5 7 1 1 7.143 0.143 -0.857 0.000 -1.429 -0.143 2 5 64.286 -0.714 6.286 -0.000 7.143 -0.286 3 8 60.714 -0.286 8.714 -10.000 4.857 -0.714 7.143 0.143 -1.857 -1.000 -2.429 -0.143 64.286 -0.714 6.286 0.000 7.143 -0.286 3 times 5 1 6 2 8 3 3 4 4 5 7 1 1 13.114 0.115 0.098 -0.983 -0.951 -0.213 2 5 20.489 -0.508 -0.721 7.214 3.639 0.229 3 2 6.967 -0.033 0.115 -1.148 0.557 -0.082 20.081 0.082 0.213 -3.131 -1.394 -0.295 20.489 -0.508 -0.721 7.214 3.639 0.229 3 times 5 1 6 2 8 3 5 4 4 5 7 1 1 15.906 0.046 -0.000 0.136 -0.455 -0.182 2 3 2.840 -0.070 -0.100 0.139 0.504 0.032 3 2 10.228 -0.114 0.000 0.159 1.136 -0.046 28.974 -0.138 -0.100 0.434 0.185 -0.196
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