Tutorial8sol - Tutorial 8 Outline of Solutions Q1 Consider...

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Unformatted text preview: Tutorial 8: Outline of Solutions Q1. Consider the following primal LP problem: min 2 x 1 + x 2 + x 3 s.t. 3 x 1 + x 2 ≥ 1 x 1 + 2 x 2 + x 3 ≥ 4 x 1 ,x 2 ,x 3 ≥ (a) Determine the dual problem and solve it graphically. (b) Use the Complementary Slackness Optimality Conditions to find the primal optimal solution. Solution : (a) Dual: max p 1 + 4 p 2 s.t. 3 p 1 + p 2 ≤ 2 p 1 + 2 p 2 ≤ 1 p 2 ≤ 1 p 1 ,p 2 ≥ Solving graphically: p 1 = 0 ,p 2 = 0 . 5, with dual objective value 2. (b) p 1 (3 x 1 + x 2- 1) = 0 p 2 ( x 1 + 2 x 2 + x 3- 4) = 0 x 1 (3 p 1 + p 2- 2) = 0 x 2 ( p 1 + 2 p 2- 1) = 0 x 3 ( p 2- 1) = 0 At p 1 = 0 ,p 2 = 0 . 5. Since p 2 6 = 0, x 1 + 2 x 2 + x 3- 4 = 0. 3 p 1 + p 2- 2 =- 1 . 5 6 = 0 implies x 1 = 0. p 2- 1 =- . 5 6 = 0 implies x 3 = 0. Thus, x 2 = 2. Note that the primal constraint 3 x 1 + x 2 = 2 ≥ 1 is satisfied. Thus ( x 1 ,x 2 ,x 3 ) = (0 , 2 , 0) is primal optimal. 1 Q2. Solve the following linear programming problem by the dual simplex method....
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This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.

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Tutorial8sol - Tutorial 8 Outline of Solutions Q1 Consider...

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