s7 - CO350 Assignment 7 Solutions. Exercise 1: For the...

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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.

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s7 - CO350 Assignment 7 Solutions. Exercise 1: For the...

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