Lecture4

Program the constraints in standard form become

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , + .r, subject to 3x, +x, = 3 -Ix, + 3x, 2 6 I, +2x, 5 -I X, ..K. > 0 Minimize z = 4.x, + x, subject to 3x, + x1 = 3 (no changes made) 4.q + 3.5 - .q = 6 (surplus is subtracted) .Y, + 2x, + .x, = 4 (slack is added) .x, .SI ,.x1, x, 2 0 M-Technique M-Technique (Step 2 add Artiicialr) (Step 3 - Modify Objective Function) Augment all constraints lacking a Slack variable by adding an Artiftcial Variable Augment the Objective Function to include the two Artificial Variables Minimize z = -Ix, + .x, subpxt to 3.~, + rZ + R, = 3 (Artlticlal added) 4r, + 7~: - Y, + R1 = 6 (Artificial added) v, + 2.~. +,x1 =-I (no further changes) .x, .x.. .s, _.I., 2 0 Minmiize z = -I; + .v, 1s aug111enteLl to Mulmnze z = 1.s, +.x2 + R, + Rz \\lnch when penalized by + M becomes Minunize z = 1.x, +x1 + .\IR, + .\fR_ EM-602 I QM-710 (NJ) Lecture 4 Page 4-3 M-Technique M-Technique (Summarizing) (Step 4 - Solving for R’s) The Artificial variables are solved for in terms of the Veal” variables The complete problem becomes: Mitknize z=4x,+x,+MR,+MR, 3.5 + xl + R, = 3 a R, = 3 - 3x, - .r: 4.u,+3.~2-.~,+R,=6=>R=6-4x,-3x,+x, M-Technique M-Technique (Step (Step 5 -Conditioning Objective Function) The Basic Feasible Solutiof~ for the problem with 6 variables and 3 equations can now be wrttten: The Artificials are removed from the Objective Function by substitution and the Objective Function is put into Standard Form: z = 4x, +x2 + .\fR, + l/R2 Z = -bX, + XI + .\ f( 3 - 3X, - .KI 6 - Sating Solution) R, = 3 R,=6 x4 = 4 )+ .\/(6-4.5 -3-r: +.r,) 2=(4-7J/).r, +(I--l.\f).r, +.\l.r,+9.\1 z-(-I-7.\f).r, -(l--l.\f).rz - .\f.r, = 9.\f M-Techruque lterabon 0 Basic -- x. x, x, M-Technique lterabon 1 R. R. x, Soln Basic x. x, x, EM-602 I QM-710 (NJ) Lecture 4‘ Page 4-4 R. R, x, Soln M-Technque ItcrattOn 3 H-Technqx lteratm 2 Basic x. x1 x, R, R, X, Soln Basic x. x_ x, R, R, x, Soln M-Technique l l l M-Technique Drawbacks Observations Numerical instability Artiftcial Variables should be ftrst to leave basis since their elimination is essential Once an Artificial Variable has left the basis, it can never return - Eliminate it! If an optimal tableau contains an Artificial variable in the basic column, two conclusions are possible If the Artific...
View Full Document

Ask a homework question - tutors are online