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IEOR162_hw04 - IEOR 162 Fall 2011 Homework 04 1(MODIFIED...

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IEOR 162, Fall 2011 Homework 04 1. (MODIFIED from Problem 4.4.1; 10 points) Recall that in Problem 4 of Homework 3, we have converted the following LP max 3 x 1 + 2 x 2 s.t. 2 x 1 + x 2 100 x 1 + x 2 80 x 1 40 x 1 , x 2 0 . (1) to its standard form max 3 x 1 + 2 x 2 s.t. 2 x 1 + x 2 + x 3 = 100 x 1 + x 2 + x 4 = 80 x 1 + x 5 = 40 x j 0 j = 1 , ..., 5 , (2) where x 3 , x 4 , and x 5 are slack variables. Now, for the standard form LP in (2), show how the basic feasible solutions correspond to the extreme points of the feasible region of the original LP in (1). 2. (15 points) Consider the following LP: min - x 1 + x 2 s.t. - 2 x 1 + x 2 6 x 1 - x 2 = - 6 x 1 0 , x 2 0 . (3) (a) (5 points) Convert the following LP to its standard form: (b) (10 points) For the LP in (3), show how the basic feasible solutions to its standard form correspond to the extreme points of the feasible region of the original LP in (3). Note. Keep in mind that the original LP is the one having x 1 nonpositive . 3. (10 points) Consider the following LP: max x 1 s.t. 2 x 1 + x 2 = 2 x 1 + x 2 3 x 1 , x 2 0 .
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