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assign2.sol

# assign2.sol - CO 350 Assignment 2 Winter 2010 Solutions...

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CO 350 Assignment 2 – Winter 2010 Solutions Question # Max. marks Part marks 1 8 3 2 2 1 2 6 3 3 3 6 2 2 2 4 6 3 3 5 4 Total 30 1. Consider the following linear programming problem ( P ): min 2 x 1 + x 2 + 3 x 3 subj. to 3 x 1 + 2 x 2 = 10 x 2 + 2 x 3 8 2 x 1 + x 2 - x 3 8 x 1 0 x 2 0 (a) Convert ( P ) into standard inequality form. Denote the LP problem after the transfor- mation by ( P 0 ). (b) Convert ( P ) into standard equality form. Denote the LP problem after the transformation by ( P 00 ). (c) Let x = ( x 1 , x 2 , x 3 ) T be a feasible solution of ( P ). Determine feasible solutions x 0 of ( P 0 ) and x 00 of ( P 00 ) that correspond to x . (d) Apply your results from (c) to the feasible solution x = (4 , - 1 , 5) T of ( P ) and determine corresponding feasible solutions of ( P 0 ) and ( P 00 ). Solution: (a) Write x 2 = - x 0 2 and x 3 = u 3 - v 3 . The problem in SIF is max - 2 x 1 + x 0 2 - 3 u 3 + 3 v 3 subj. to 3 x 1 - 2 x 0 2 10 - 3 x 1 + 2 x 0 2 - 10 x 0 2 - 2 u 3 + 2 v 3 - 8 2 x 1 - x 0 2 - u 3 + v 3 8 x 1 , x 0 2 , u 3 , v 3 0 . (b) As in (a), write x 2 = - x 0 2 and x 3 = u 3 - v 3 . The problem in SEF is max - 2 x 1 + x 0 2 - 3 u 3 + 3 v 3 subj. to 3 x 1 - 2 x 0 2 = 10 x 0 2 - 2 u 3 + 2 v 3 + x 4 = - 8 2 x 1 - x 0 2 - u 3 + v 3 + x 5 = 8 x 1 , x 0 2 , u 3 , v 3 , x 4 , x 5 0 where x 4 and x 5 are slack variables. 1

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(c) If x 3 0, we let u 3 = x 3 and v 3 = 0. If x 3 < 0, we let u 3 = 0 and v 3 = - x 3 . We obtain x 0 = ( x 1 , - x 2 , x 3 , 0) T if x 3 0 x 0 = ( x 1 , - x 2 , 0 , - x 3 ) T if x 3 < 0 and x 00 = ( x 1 , - x 2 , x 3 , 0 , x 2 + 2 x 3 - 8 , - 2 x 1 - x 2 + x 3 + 8) T if x 3 0 x 00 = ( x 1 , - x 2 , 0 , - x 3 , x 2 + 2 x 3 - 8 , - 2 x 1 - x 2 + x 3 + 8) T if x 3 0 (d) x 0 = (4 , 1 , 5 , 0) T x 00 = (4 , 1 , 5 , 0 , 1 , 6) T 2. Consider the following linear programming problem (
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