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# Slides18 - Lecture 18 The 2-Phase Simplex Method Example We...

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1 Lecture 18 The 2-Phase Simplex Method Example We will discuss the 2-phase simplex method by looking at an example (Winston, page 179-184). Let’s solve the following LP by the 2-phase simplex method. s.t. 1/2 x 1 + 1/4 x 2 4 x 1 + 3x 2 20 x 1 + x 2 = 10 x 1 , x 2 0. min 2x 1 + 3x 2

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2 Step 0: Phase I In standard form we do not have an obvious starting basis (for constraint 2 and 3 – see )! s.t. 1/2 x 1 + 1/4 x 2 4 x 1 + 3x 2 20 x 1 + x 2 = 10 x 1 , x 2 0. min 2x 1 + 3x 2 s.t. z-2x 1 -3x 2 = 0 1/2 x 1 + 1/4 x 2 + s 1 = 4 x 1 + 3x 2 -s 2 = 20 x 1 + x 2 = 10 x 1 , x 2 , s 1 , s 2 0. min 2x 1 + 3x 2 We simply insert artificial basic variables for these rows: standard form * * * Step 0: Phase I s.t. z-2x 1 -3x 2 = 0 1/2 x 1 + 1/4 x 2 + s 1 = 4 x 1 + 3x 2 2 = 20 x 1 + x 2 = 10 x 1 , x 2 , s 1 , s 2 0. min 2x 1 + 3x 2 s.t. z-2x 1 -3x 2 = 0 1/2 x 1 + 1/4 x 2 + s 1 = 4 x 1 + 3x 2 -s 2 +a 1 = 20 x 1 + x 2 +a 2 = 10 x 1 , x 2 , s 1 , s 2 , a 1 , a 2 0. min 2x 1 + 3x 2 any feasible solution feasible solution to to the old problem new system with a 1 =a 2 =0 Note: (( ))
3

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Slides18 - Lecture 18 The 2-Phase Simplex Method Example We...

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