Question

2. Consider the following LP: Min z = -4x1 - 5x2 + 3x3 Subject to X1 + x2 + x3 = 10 X1 X2 > 1 X1 + 3x2 + x3 = 20 X1, X2, X3 20 (a) Solve the problem by Big M method. (b) Solve the problem by two-phase method.

Answer 1

2 S consider the following LA. Min z = - 434 - 5 Hz + 3x3 subject to 24+ xy +z = 10 24-H > 1 4.430, +43 520 4, lg, Kz zo. solve the problem by BigM method. the proble Min z = - 484-5 *2 +313 May = -(Minz) Max Z = 42, + 5x2-3xz - M2-M17 using slack variable ao surplus variable NA and artificial variable as and at subject to i.e = 10 24 +2 +13 + x6 34 - 12 -nythy = 1 84 +3m + xy + 25 = 20 24, Hq , A3, ly, as, Ho, 1770 where M is a very big positive quan tity. solve the Tableau.

ci 4 5 5 -3 o -M-M XBO You CB B b e -M de x6 10 - o 1 O 1 -ML7 x7 1 -1 -I 0 251 25 20 - 3 O 11 0 H -2M-41 - 5 -M43 M 0 о 2-g' -M 06 x L5 26 9 O 12 1 0 1 4 1 -1 0 -1 -> 45 19 O 4 1 1. 소 -2M-9-M+3 -M-4 0 7-99 5. 22 시 X2 1 4 지 اور =ع کر 1 t O -In-Hy i on 1 O zj - ei 9/ / 0 all Since Zj ej > o and no artificial is present in Basic feasible Basic feasible solution ( 3.6.s) the solution 4 3g Hz=0 ef ३ Zmin -434 - 502 +303 at a = 1/4 = 9 .. Zmin +3.0 2 4x4 5x9 2 2 - 22 45 २ zmin - 44545 (Ans). - 89 2

b. solve the problem by two-phase method -) the proble minz =-40, -542 +343 . Max = -(mine) as max 2 = un t. 512-383 use surplus variable slack variable as and artificial artificial variable 16 and xz. xy Max i. om +0.1 +0. Uz tony toong -26 -17 subject to 24thy thy the = 10 24-12- ny th7 ²1 2 + 302 + Uz +15=20 Now the successive simplex tableau are as follows TPhasel

Phase I O 0 g CB KB b 21 22 dz du ha d & 46 a6 10 - 1 0 - - | at 27 1 1 1 -> -1 0 aras 20 1 3 1 1 -21 1 0 O Zjg - de 26 9. [2 110 1 -1 → 4 ny 1 -1 0 o X5 45 19 4 1 1 1 O -] o O2 ZG 22 X₂ g 4 x 1 as as 1 o ER-1 놀 1 2 0 0 놀 o Ž - 1/2 Ę O 1 -1 1 - 2 - 1 0 ZG o 0 0 O 1 1 so Sinee All Z - Cj 70 and optimal solution has been reached for phase-I problem Also =0 and no artificial rector is appeared in the basis. so pass for phase I For phase II The objective function is 2- 414 + 502-303 *