s3 - CO350Linear Optimization Assignment 3 Solutions 1...

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CO350—Linear Optimization Assignment 3 Solutions 1. Determine whether b x = (1 , 1 , 1 , 3) T and e x = (0 , 1 , 6 , 1) T are optimal solutions of this LP. max 2 x 1 - 2 x 2 + x 3 + 2 x 4 s.t. 12 x 1 + x 2 + 2 x 3 - x 4 = 12 7 x 1 + 3 x 3 + 4 x 4 = 22 3 x 1 + x 2 + x 3 + x 4 = 8 x 1 , x 2 , x 3 , x 4 0 . Solution: Note that b x and e x are both feasible. The dual is min 12 y 1 + 22 y 2 + 8 y 3 s.t. 12 y 1 + 7 y 2 + 3 y 3 2 y 1 + y 3 ≥ - 2 2 y 1 + 3 y 2 + y 3 1 - y 1 + 4 y 2 + y 3 2 and the complementary slackness conditions are x * 1 = 0 or 12 y * 1 + 7 y * 2 + 3 y * 3 = 2 x * 2 = 0 or y * 1 + y * 3 = - 2 x * 3 = 0 or 2 y * 1 + 3 y * 2 + y * 3 = 1 x * 4 = 0 or - y * 1 + 4 y * 2 + y * 3 = 2 . For b x , this gives 12 b y 1 + 7 b y 2 + 3 b y 3 = 2 b y 1 + b y 3 = - 2 2 b y 1 + 3 b y 2 + b y 3 = 1 - b y 1 + 4 b y 2 + b y 3 = 2 , which is a system with no solution. Therefore, b x is not optimal for this LP. For e x , the complementary slackness conditions give e y 1 + e y 3 = - 2 2 e y 1 + 3 e y 2 + e y 3 = 1 - e y 1 + 4 e y 2 + e y 3 = 2 , which has e y = (0 , 1 , - 2) T as unique solution. This is not feasible for the dual, so e x is not optimal for the primal. 2. By using the complementary slackness conditions, determine for which values of
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s3 - CO350Linear Optimization Assignment 3 Solutions 1...

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