Lecture4

If an optimal tableau contains an artificial variable

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Unformatted text preview: ial Variable has a Non-zero value then the problem has no solution If it has Zero value, then the problem had 1 or more redundant constraints . l l l l Numerical value used for M can affect results Numerical evaluation of (M-3) should be c (M-2) Computer algorithms should be able to correctly distinguish between them Truncation I Rounding Error I L OSS of Precision Numerical instability Two-Phase Technique Preventive Measures introduction Phase I Create a new (but related) LP with the objective of minimizing the Artiftcial Variables Solve the Phase I Problem and (if possible) ftnd a Basic Feasible Solution in which all the Artificials have zero value Phase II Starting with the (hopefully successful) results from the Phase I Problem, set up and solve the Phase II LP Treat the value M Symbolically throughout the algorithm wherever it appears Select an appropriate numerical value for M which does not introduce instability (A rule of thumb is 3 to 10 times the largest value found elsewhere in the LP) Remove all artificial variables using the Two-Phase Method l l l .. EM-602 I QM-710 (NJ) Lecture 4 Page 4-5 Two-Phase (example) Two-phase (example) Phase 1A - Substitution . Substitute out all of the Artiftcial Variables in the Objective Function Consider the problem previously solved using M-Technique r=R,+R? R( = 3-3.5 -xl R? = 6 --Ix, - 3.5 + .r) r=(3-3x, -.5)+(6-4x, -3x,+x,) Minimize r = R, +R2 subject to 3.5 +x2 + R, = 3 4x, +3x2 - x, + RT = 6 I, +1x2 +x1 = 4 r = -7.q -4x? +x, +9 x,.x? ..r) .R, .R> ..(I, 2 0 Two-phase (example) Two-Phase (example) Phase ZA - Formulation Phase 16. - Std Form 6 Se&&ion l Put Phase I Problem into Standard Form r+7x,+4Xl-.r,=9 l subject to 3x, +x2 +R, = 3 - -iI, + From Optimal Phase I Tableau obtain the constraint equations for the Phase II LP Minimize z = 4x, +x1 1 3 subject to x, +-I, = 3 5 3.5 - x, + R_ = 6 x, +2x, + .r, = -I 3 6 .I: - 7 X) = Y _ .r,..r,..r,.R,.R..r, 20 l Solve the Phase I problem normally using Simplex Method 3 3 .x...
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