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sol3 - CO350 LINEAR OPTIMIZATION SOLUTION HW3 Exercise...

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Unformatted text preview: CO350 LINEAR OPTIMIZATION - SOLUTION HW3 Exercise 3.3.1 (a) Since x 1 is free variable, we let x 1 = v 1- v 2 ,v 1 ,v 2 ≥ . The problem is changed to: min v 1- v 2- x 2 subject to 2 v 1- 2 v 2- x 2 = 2 v 1 , v 2 , x 2 ≥ We then change the equality constrain to two inequality constraints and change the objective function to maximize function to get the standard inequality form. max- v 1 + v 2 + x 2 subject to 2 v 1- 2 v 2- x 2 ≤ 2- 2 v 1 +2 v 2 + x 2 ≤ - 2 v 1 , v 2 , x 2 ≥ . (b) We add slack variables x 3 ,x 4 ≥ to the two constraints and change the objective function to maximize the function. max x 1- x 2 subject to 5 x 1- 2 x 2- x 3 = 4 2 x 1 + x 2 + x 4 = 3 x 1 , x 2 , x 3 , x 4 ≥ . Exercises 3.3.2 Since x 2 is a free variable, we use the equality constraint to eliminate, it i.e. x 2 = x 1- 1 (*) After substituting for x 2 , we get, min 3 x 1- 3 x 3- 1 subject to 2 x 1 + x 3- 1 ≥ 2 x 1- x 3 ≤ 2 x 1 , x 3 ≥ We can now add slack (and surplus) variables and remove the constant in the objective function and change the min to a max, min- 3 x 1 +3 x 3 subject to 2 x 1 + x 3- x 4 = 3 x 1- x 3 + x 5 = 2 x 1 , x 3 , x 4 , x 5 , ≥ To obtain the optimal value of the original problem, we take the optimal value of the last formulation, multiply...
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sol3 - CO350 LINEAR OPTIMIZATION SOLUTION HW3 Exercise...

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