s4 - CO350 Assignment 4 Solutions Exercise 1: Consider the...

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Assignment 4 Solutions Exercise 1: Consider the linear program: (P) max 2 x 1 +2 x 2 - 3 x 3 +3 x 5 subject to x 1 - x 2 +2 x 3 - 2 x 5 = 0 x 2 - x 3 +2 x 5 = 1 x 1 + x 2 + x 4 +4 x 5 = 5 x 1 , x 2 , x 3 , x 4 , x 5 0 . (a) Construct the tableau associated with the basis B = { 1 , 2 , 4 } . Solution. Consider the linear system: z - 2 x 1 - 2 x 2 +3 x 3 - 3 x 5 = 0 x 1 - x 2 +2 x 3 - 2 x 5 = 0 x 2 - x 3 +2 x 5 = 1 x 1 + x 2 + x 4 +4 x 5 = 5 Apply column elimination for the variable x 1 . z - 4 x 2 +7 x 3 - 7 x 5 = 0 x 1 - x 2 +2 x 3 - 2 x 5 = 0 x 2 - x 3 +2 x 5 = 1 2 x 2 - 2 x 3 + x 4 +6 x 5 = 5 Apply column elimination for the variable x 2 . z -row z +3 x 3 + x 5 = 4 x 1 -row x 1 + x 3 = 1 x 2 -row x 2 - x 3 +2 x 5 = 1 x 4 -row + x 4 +2 x 5 = 3 We do not need to apply column elimination for x 4 , since this is already a tableau. (b) Find the basic solution x determined by B = { 1 , 2 , 4 } and its objective value. Is x feasible? Solution. The basic solution is x = (1 , 1 , 0 , 3 , 0) T , which is feasible, and has objective value 4. (c) Find the basic dual solution y determined by B = { 1 , 2 , 4 } and its objective value. Is y feasible? Solution. The basic dual solution satisfies: 1 0 1 - 1 1 1 0 0 1 y = 2 2 0 . So
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This note was uploaded on 05/28/2011 for the course CO 350 taught by Professor S.furino,b.guenin during the Winter '07 term at Waterloo.

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s4 - CO350 Assignment 4 Solutions Exercise 1: Consider the...

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