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Unformatted text preview: CO350 Assignment 7 Solutions. Exercise 1: For the following pairs ( A,b ) solve the feasilbility problem ( Ax = b, x ≥ 0) by first formulating the auxiliary problem ( P ) and then solving ( P ) by the simplex method. If ( Ax = b, x ≥ 0) is feasible, give a feasible solution, otherwise find a vector y satisfying ( A T y ≥ , b T y < 0). (a) A =  1 1 2 1 2 1 1 1 0 1 1 1 1 1 , and b = 1 1 4 . Solution: The auxiliary problem is: ( P ) maximize u 1 u 2 u 3 subject to x 1 + x 2 +2 x 3 + x 4 2 x 5 + u 1 = 1 x 1 + x 2 x 3 + x 5 + u 2 = 1 x 1 + x 2 x 3 + x 4 + u 3 = 4 x 1 , x 2 , x 3 , x 4 , x 5 , u 1 , u 2 , u 3 ≥ Now let z = u 1 u 2 u 3 = ( x 1 + x 2 + 2 x 3 + x 4 2 x 5 1) + ( x 1 + x 2 x 3 + x 5 1) + ( x 1 + x 2 x 3 + x 4 4) = x 1 + 3 x 2 + 2 x 4 x 5 6 . So the initial tableau is: z x 1 3 x 2 2 x 4 + x 5 = 6 x 1 + x 2 +2 x 3 + x 4 2 x 5 + u 1 = 1 x 1 + x 2 x 3 + x 5 + u 2 = 1 x 1 + x 2 x 3 + x 4 + u 3 = 4 After solving ( P ) via the simplex method we arrive at the final tableau: z +2 x 2 + x 3 + u 1 +3 u 2 = 1 x 1 +3 x 1 + x 4 + u 1 +2 u 2 = 3 x 1 + x 2 x 3 + x 5 + u 2 = 1 2 x 2 x 3 u 1 2 u 2 + u 3 = 1 Since the optimal value of ( P ) is negative, the system ( Ax = b, x ≥ 0) is infeasible....
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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.
 Winter '07
 S.Furino,B.Guenin

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