Lecture5

# Lecture5 - EM-602 I CM-710(NJ Assignments from Lecture 4...

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EM-602 I CM-710 (NJ) Management Science Lecture 5 Sensitivity Analysis Homework Problem #3 (Solve by M-Technique) Consider the following linear program: Minuiilze 2 = 8Os, +60.r1 subject to O.ZO.K, +0.32.r2 5 0.25 I, + .K, = 1 .K, . S. 2 0 Assignments from Lecture 4 l Handout Problem #3 (solve probkm by both M-Technique and Two-Phase Technique) l Problems 35, 3-22(a), 3-22(d) l Read Sections 353.6 Homework Problem #3 (Step Ml 6 M2 - Modify 6 Add Artiftcials) The constraints in standard form become: Minimize 2 = 801, +60x2 subject to .20x,+.31x, +s, =3 (slack IS added) _K, +x1 +R, = 1 (Artificial is added) I, ..‘I, 3, . R, 2 0 Homework Problem #3 Homework Problem #3 (Step M3, M4, MS - Modify Obj Function) (Step M6 - Starting Simplex Solution) Augment the Objective Function to include the Arttfkial Variable, penalize it, then sdve for and condition it to remove the artiftcial variable Minimize z = 8O.r, +6&r, IS augmented to z = 8t)s, + 6o.r, + R, and penalized to become z = ~OS, +6&K: + JR, z - ~O.K, - 60.~~ - .\I( I - .‘c, - x2 ) = 0 ~+(-80+.\/)_K~ +(-a+.\/)~: = .\f The Basic Feasible Solution for the problem with 4 variables and 2 equations can now be written: s, = 0.25 R, = I .K,_.K. = 0 ,?= .\I The problem can then be solved by the simplex method EM-602 / QM-710 (NJ) Lecture 5’ Page 5-l

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Problem p3 Two-Phase Technique Phase I - Optimal Tableau Kl 1 1 0 1 I 1 Homework Problem #3 Phase PA & 20 - Formulate & Substitute From Optimal Phase I Tableau obtain the constraint equations and Objective Function for the Phase II LP Mimmize I = 80x, +60x, subject to .12x, +x, =.OS x, +x2 = 1 Since xl is basic in Phase I. .r, = I- x2 Phase 2 Objective function is z = 8C( I - .z2) +60x, 2 +20x, = 80 Problem 1)3 Problem jf3 Two-Phase Technique Two-Phase Technique Phase 2 - Iteration 0 Phase 2 - Optimal Tableau Basic 5 4 ‘1 Z 0 a1 0 ‘(1 1 Homework Problem 3-S (a) Determine the maximum number of Extrema (b) Identify all feasible Extreme points (c) Find the Optimal Basic Feasible Solution \launilzz : = 2.x, - Jv_: + !.I-, - 6.~, \ubJect to .xL +4x. - 2, + 8.~, 5 ? 5, + 2X. + 3X, + 4.Y, 5 I v, X) \-, Y, 2 0 Homework Problem 3-5 Solution l The problem in standard form is EM-602 I QM-710 (NJ) Lecture 5 Page 5-3

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-- Homework Problem 35 (solution) 6! 4 ct=-= lj 2!4! b) ffnd all bask solutions and their feasibility by enumeration (see handout) c) Restrict to only feasible solutions and choose the extreme point with the optimal 2 (8,0,3,&Q, 0) L = 31 Homework Problem 3-22 (solution) (a) Masimize 2 = Z.r, +3x2 - 5.r, - .W?, - .W+ ct. I, +2.~~ +I, +R, = I 5, -5x, +.Q -s, +R, = IO x,.x2..r,_R,.s,.R. 20 r-(:!+3.\f)s, -(3-l.\f).r, -(-j+L\/).r, +.\h, = -1741 R, = 7:R? = Io..Y,..K:.X,.S, =0 .
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