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Unformatted text preview: NAME: PAGE 2 OF 8 Question 1. [10 marks] Consider the linear program: (P) max 2 x 2 +2 x 4 subject to x 1 2 x 2 + x 4 = 2 4 x 2 + x 3 +3 x 4 = 10 x 2 x 4 + x 5 = 1 x 1 , x 2 , x 3 , x 4 , x 5 ≥ . Use the simplex method to solve ( P ) starting with the feasible basis { 1 , 3 , 5 } . Do not do more than three iterations. If ( P ) has an optimal solution, give an optimal basic solution and an optimal basic dual solution. If ( P ) is unbounded, provide a parametric set of feasible solutions x ( t ) = x + td , (for t ≥ ) such that c T x ( t ) → + ∞ as t → + ∞ . Solution: The initial tableau is: z + 2 x 2 2 x 4 = x 1 2 x 2 + x 4 = 2 4 x 2 + x 3 + 3 x 4 = 10 x 2 x 4 + x 5 = 1 Entering variable: x 4 (since ¯ c 4 = 2 > ). Leaving variable: x 1 (since min( 2 1 , 10 3 ) = 2 1 ). z + 2 x 1 2 x 2 = 4 x 1 2 x 2 + x 4 = 2 3 x 1 + 2 x 2 + x 3 = 4 x 1 x 2 + x 5 = 3 Entering variable: x 2 (since ¯ c 2 = 2 > ). Leaving variable: x 3 (since min( 4 2 ) = 4 2 ). z x 1 + x 3 = 8 2 x 1 + x 3 + x 4 = 6 3 2 x 1 + x 2 + 1 2 x 3 = 2 1 2 x 1 + 1 2 x 3 + x 5 = 5 Entering variable: x 1 (since ¯ c 1 = 1 > ). ¯ A 1 < so the problem is unbounded. For each real t ≥ , define x ( t ) := 2 6 5 + t 1 3 2 2 1 2 . Then x ( t ) is feasible for every t ≥ and c T x ( t ) → ∞ as t → ∞ . NAME: PAGE 3 OF 8 Question 2. [10 marks] Consider the linear program ( P ) max { c T x : Ax = b,x ≥ } , where A = 1 1 1 0 0 2 1 2 0 1 1 0 3 1 2 0 1 2 , b = 4 9 14 , and c = [2 , 2 , , 1 , 1 , 2] T . Solve this linear program using the revised simplex method, starting with the feasible basic solu tion x * = (4 , , , 3 , 2 , 0) T . If the problem has an optimal solution, provide one. If ( P ) has an optimal solution, give an optimal basic solution and an optimal basic dual solution. If ( P ) is unbounded, provide a parametric set of feasible solutions x ( t ) = x + td , (for t ≥ ) such that c T x ( t ) → + ∞ as t → + ∞ ....
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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 Fall '07 term at Waterloo.
 Fall '07
 S.Furino,B.Guenin

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