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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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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
For convenience: ......(1)
First pivoting(i) Suppose is chosen as pivot Eliminate from (1) by substitution: (*)
First pivoting(ii) Eliminate from other rows by substitution: g ≠ i
First pivoting(iii) (1)= Sum of each rows = sum of those rows not containing pivot Proof:
First pivoting(iv) Similarly,
Pivoting invariant(i) The last row = -1* (sum of those rows not containing any pivot) Proof:
Terminal condition and infeasibility When simplex method terminates, the last row (equation) must be of the form: where It’s a contradiction!!
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