Lecture4

# If an optimal tableau contains an artificial variable

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/31/2014 for the course MS 602 at NJIT.

Ask a homework question - tutors are online