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The simplex method for infeasible solution

The simplex method for infeasible solution. Example. -x1-2x2+3x3<=3; -2x1-6x2+x5>=9; 3x1-x2-7x5>=2; x1+5x2+4x4>=5; x2-3x3>=6;. slack variables artificial variables. x1 x2 x3 x4 x5 s1 s2 s3 s4 s5 y1 y2 y3 y4 y5 b. Min Σ y i ≠ 0 → infeasible. For convenience:.

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The simplex method for infeasible solution

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  1. The simplex method for infeasible solution

  2. Example -x1-2x2+3x3<=3; -2x1-6x2+x5>=9; 3x1-x2-7x5>=2; x1+5x2+4x4>=5; x2-3x3>=6; slack variablesartificial variables x1 x2 x3 x4 x5s1 s2 s3 s4 s5y1 y2 y3 y4 y5b Min Σ yi ≠ 0 → infeasible

  3. For convenience: ......(1)

  4. First pivoting(i) Suppose is chosen as pivot Eliminate from (1) by substitution: (*)

  5. First pivoting(ii) Eliminate from other rows by substitution: g ≠ i

  6. First pivoting(iii) (1)= Sum of each rows = sum of those rows not containing pivot Proof:

  7. First pivoting(iv) Similarly,

  8. Pivoting invariant(i) The last row = -1* (sum of those rows not containing any pivot) Proof:

  9. Pivoting invariant(ii) Ex:

  10. Terminal condition and infeasibility When simplex method terminates, the last row (equation) must be of the form: where It’s a contradiction!!

  11. UNSAT proof Ex: 768*(-2x1-6x2+x5>=9) 512*(3x1-x2-7x5>=2) -5120x2-2816x5>=7936 768*(x2-3x3>=6) -4352x2-2304x3-2816x5>=12544 4352*(X2>=0) -2304x3-2816x5>=12544 2304*(X3>=0) 2816*(X5>=0) -2816x5>=12544 0>=12544

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