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co227_a2_soln

co227_a2_soln - Assignment 2 Due Wednesday October 12 at...

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Assignment 2 Due: Wednesday October 12 at the BEGINNING of class 1. Convert the following linear programs into standard equality form: (a) minimize x 1 - 6 x 2 + 4 x 3 subject to 8 x 2 + x 3 5 x 1 - 8 x 2 0 x 2 0 Solution: maximize - x + 1 + x - 1 + 6 x 2 - 4 x + 3 + 4 x - 3 subject to 8 x 2 + x + 3 - x - 3 + x 4 = 5 x + 1 - x - 1 - 8 x 2 - x 5 = 0 x + 1 , x - 1 , x 2 , x + 3 , x - 3 , x 4 , x 5 0 (b) maximize 5 x 1 + 2 x 2 - x 3 subject to - 5 x 1 + 3 x 2 + x 3 = - 2 x 1 + x 2 - 3 x 3 7 2 x 1 + x 3 2 x 1 0 Solution: maximize 5 x 1 + 2 x + 2 - 2 x - 2 - x + 3 + x - 3 subject to - 5 x 1 + 3 x + 2 - 3 x - 2 + x + 3 - x - 3 = - 2 x 1 + x + 2 - x - 2 - 3 x + 3 + 3 x - 3 + x 4 = 7 2 x 1 + x + 3 - x - 3 - x 5 = 2 x 1 , x + 2 , x - 2 , x + 3 , x - 3 , x 4 , x 5 0 2. Write a linear program in standard equality form that satisfies the following: (a) A linear program with one variable that is unbounded. Solution: maximize x 1 subject to x 1 0 (b) A linear program with one variable that is infeasible. Solution: maximize x 1 subject to x 1 = - 1 x 1 0 (c) A linear program with one variable that has an optimal solution. Solution: maximize x 1 subject to x 1 = 1 x 1 0

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3. Given the following linear program, prove whether the linear program is infeasible,
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