Lecture28aa - Lecture 1 More Examples of Big M Method 2 LP...

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Lecture 4/02/08 1. More Examples of Big M Method 2. LP Anomalies a. Infeasibility b. Degeneracy c. Variables unrestricted in sign.
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Infeasible LP (Prob #4 on P.178) LP Problem Min z= 3x 1 LP in Standard Form Create an artificial problem by adding artificial variables a 1 and a 2 . s. t. 2x 1 + x 2 6 3x 1 + 2x 2 = 4 x 1 , x 2 0 z- 3x 1 =0 2x 1 + x 2 - e 1 =6 3x 1 + 2x 2 = 4 x 1 , x 2 ,e 1 0 Min z= 3x 1 +Ma 1 +Ma 2 ==> z- 3x 1 -Ma 1 -Ma 2 =0 2x 1 + x 2 - e 1 + a 1 =6 3x 1 + 2x 2 + a 2 = 4 x 1 , x 2 ,e 1 , a 1 , a 2 0
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Infeasible LP (Prob #4 on P.178) Initial Simplex Tableau z x 1 x 2 e 1 a 1 a 2 RHS 1 -3 0 0 -M -M 0 0 2 1 -1 1 0 6 0 3 2 0 0 1 4 To get tableau in conical form: New Row 0 = Old Row 0 + M*Row1+M*Row2 z x 1 x 2 e 1 a 1 a 2 RHS 1 -3+5M 3M -M 0 0 10M 0 2 1 -1 1 0 6 0 3 2 0 0 1 4 Ratio 6/2=3 4/3
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Infeasible LP (Prob #4 on P.178) x 1 enters basis replacing a 2 New Row 0= Old Row 0 +(3-5M)*Row2 Divide Pivot Row by 3: z x 1 x 2 e 1 a 1 RHS 1 -3+5M 3M -M 0 10M 0 2 1 -1 1 6 0 1 2/3 0 0 4/3 New Row 1 = Old Row 1 -2*Row 2 z x 1 x 2 e 1 a 1 RHS 1 0 2-M/3 -M 0 4 +(10/3)M 0 0 -1/3 -1 1 10/3 0 1 2/3 0 0 4/3
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Infeasible LP (Prob #4 on P.178) Tableau appears optimal but artificial variable a 1 is still positive.
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