hw06solutions

hw06solutions - 3 x 1 + x 2 + x 3 ≤ 60 x 1-x 2 + 2 x 3...

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IEOR 162 Linear Programming Spring 2007 Homework 6 Solutions 4.5.2 (5 points) Standard Form of the LP: max 2 x 1 + 3 x 2 s.t. x 1 + 2 x 2 + x 3 = 6 2 x 1 + x 2 + x 4 = 8 x i 0 Simplex Algorithm Iteration 1: (note z is the objective value) B = { 3 , 4 } , N = { 1 , 2 } , z = 0 ¯ c N = c N - c B A - 1 B A N = ( 2 3 ) - ( 0 0 ) ± 1 0 0 1 ² - 1 ± 1 2 2 1 ² = ( 2 3 ) We could pick either x 1 or x 2 to enter the basis since they both have positive reduced costs. Let’s choose x 2 . x B = A - 1 B b - A - 1 B a 2 x 2 = ± 6 8 ² - ± 2 1 ² x 2 ± 0 0 ² x 2 = 3 z = 0 + ¯ c 2 x 2 = 9 x 3 = 6 - 2(3) = 0 x 4 = 8 - 1(3) = 5 So x 3 leaves the basis. Simplex Algorithm Iteration 2: B = { 2 , 4 } , N = { 1 , 3 } , z = 9 ¯ c N = ( 2 0 ) - ( 3 0 ) ± 2 0 1 1 ² - 1 ± 1 1 2 0 ² = ( 0 . 5 - 1 . 5 ) Only x 1 has a positive reduced cost, so it must enter the basis. x B = ± 3 5 ² - ± . 5 1 . 5 ² x 1 ± 0 0 ² x 1 = 10 3 z = 9 + ¯ c 1 x 1 = 9 + . 5 10 3 = 32 3 x 2 = 3 - . 5 10 3 = 4 3 x 4 = 5 - 1 . 5 10 3 = 0 So x 4 leaves the basis. Simplex Algorithm Iteration 3: B = { 1 , 2 } , N = { 3 , 4 } , z = 32 3 ¯ c N = ( 0 0 ) - ( 2 3 ) ± 1 2 2 1 ² - 1 ± 1 0 0 1 ² = ( - 4 3 - 1 3 ) All reduced costs are negative, which implies that the current basis is optimal. So the optimal solution is: x = ( 10 3 4 3 0 0 ) , z * = 32 3 1
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4.5.3 (5 points) Formulation: max 2 x 1 - x 2 + x 3 s.t.
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Unformatted text preview: 3 x 1 + x 2 + x 3 ≤ 60 x 1-x 2 + 2 x 3 ≤ 10 x 1 + x 2-x 3 ≤ 20 x i ≥ Simplex Algorithm using tableaus: ↓ z x 1 x 2 x 3 x 4 x 5 x 6 RHS Ratio 1-2 1-1 3 1 1 1 60 20 1-1 2 1 10 10 * 1 1-1 1 20 20 ↓ z x 1 x 2 x 3 x 4 x 5 x 6 RHS Ratio 1-1 3 2 20 4-5 1-3 30 7 . 5 1-1 2 1 10-2-3-1 1 10 5 * z x 1 x 2 x 3 x 4 x 5 x 6 RHS Ratio 1 3 2 3 2 1 2 25 1 1-1-2 10 1 1 2 1 2 1 2 15 1-3 2-1 2 1 2 5 All entries in first row are non-negative, which implies that we have an optimal solution. x * = ( 15 5 10 ) , z * = 25. 2...
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This note was uploaded on 09/23/2009 for the course IEOR 162 taught by Professor Zhang during the Fall '07 term at Berkeley.

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hw06solutions - 3 x 1 + x 2 + x 3 ≤ 60 x 1-x 2 + 2 x 3...

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