sample-final-solutions - IEOR 162 Linear Programming Fall...

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Unformatted text preview: IEOR 162 Linear Programming Fall 2006 Sample Final Solutions 1. Starting from x = (1 , 1 . 5 , , 0), implies that we are starting from the basis with basic variables x B = ( x 1 , x 2 , s 1 ), since s 1 = 5- 3 x 2- x 3- x 4 = . 5. We can calculate A- 1 B = 1 3 1 2- 1 6 1- 3 2 1 2 , which we then use to create the tableau for this basis: ↓ z x 1 x 2 x 3 x 4 s 1 s 2 s 3 RHS ratio 1- 3- 1 1 2 3 1 3 1 3 1 1 . 5 1 1 6- 1 6 1 2- 1 6 1 . 5 9 .5 3 2 1- 3 2 1 2 . 5 1 * Performing one simplex iteration we get to the following tableau: z x 1 x 2 x 3 x 4 s 1 s 2 s 3 RHS 1 8 6- 9 3 3 1- 5 3- 4 3 2- 1 3 1 3 1- 2 3- 1 3 1- 1 3 4 3 1 3 2- 3 1 1 We see that this tableau is not optimal because there is a negative entry in row 0 for s 2 . 2. a. Dual: max 6 y 1 + 4 y 2 + 3 y 3 s.t. 4 y 1 + y 2 + 3 y 3 ≤ 4 3 y 1 + 2 y 2 + y 3 ≤ 1 y 1 + y 2 + y 3 ≤ y 1 ≥ , y 2 ≤ , y 3 urs b. Complementary Slackness conditions: (4 x 1 + 3 x 2 + x 3- 6) y 1 = 0 ( x 1 + 2 x 2 + x 3- 4) y 2 = 0 (3 x 1 + x 2 + x 3- 3) y 3 = 0 (4 y 1 + y 2 + 3 y 3- 4) x 1 = 0 (3 y 1 + 2 y 3 + y 3- 1) x 2 = 0 ( y 1 + y 2 + y 3 ) x 3 = 0 c. Given the solution ( x 1 , x 2 , x 3 ) = (0 . 2 , 1 . 4 , 1 . 0) and the complementary slackness conditions of part (b), we can deduce what values of ( y 1 , y 2 , y 3 ) will satisfy the complementary slackness conditions. If these values of ( y 1 , y 2 , y 3 ) also satisfy the dual constraints, then we know that both x and y are optimal for the primal and dual respectively. First note that ( x 1 , x 2 , x 3 ) = (0 . 2 , 1 . 4 , 1 . 0) satisfies all primal constraints at equality, so the first three complimentary slackness equations are already satisfied. Since all values of x are positive, we must also 1 satisfy all dual constraints at equality (because of the last 3 complimentary slackness conditions). This issatisfy all dual constraints at equality (because of the last 3 complimentary slackness conditions)....
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This homework help was uploaded on 04/02/2008 for the course IEOR 162 taught by Professor Zhang during the Spring '07 term at University of California, Berkeley.

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sample-final-solutions - IEOR 162 Linear Programming Fall...

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