Question

Solve by solving the dual problem.

Minimize

z = 30x₁ + 15x₂ + 28x₃,

subject to

5x1	+	3x2	+	4x3	≥	45
5x1	+	6x2	+	8x3	≥	120
20x1	+	6x2	+	14x3	≥	300
x1	≥	0, x2	≥	0, x3	≥	0.

x_1=

x_2=

x_3=

z=

Answer 1

The given Linear Programming Problem is- Mimmize z= 30x4 + 15% + 2883 Subject to, 5%,+ 3x2 + 423 745 5x, + 6x₂ + 813 3 120 2021+ 6X2 + 1422> 300,- *, 20, 222,0, 132,0. corresponding dual problem is- Maximize, Zw = 45w, + 120 W, 7 300 wz subject to, 5w,+ 5W₂ + 2003 £30 3w, + 6 W2+6W3 £15 4w1+ Bw.y + 1463 = 28. Wrzo, W, 30, 6320. The

o 0 Let us solve the LPP using simplex method, Introducing slack variable wy. Ws, we one to each constrain, the converted equations are 5w, + 5W2 + 20wz + W4+0.Wz+OW= 30 3w, 6W2+6W 3 +0.04 + W5+0.we=15 4w, t 8w, +14W3 +0w40 W5+ W6 = 28 W;)0 V3 =1,2,...6. c |45|120 | 300 | o olo Basis CB b a aa a au as a min Ration Wy 0 30 5 5 200 1 100 30 =1.5* Ws o 15 3 6 6o o 18-22 22.5 o 28 4 8 14 0 0 1 - 2 Zj-ej 0 -45 -120 -360 o 0 0 W₃ 300 3/2 / 114 11 /20 o o wg o 6 3/2 The = 1.536 Z; - Cj 450 30 -45% of W3 300 7/6 1/6 o 1 7160 t o Wy 120 /3. 113 1 0 - 0 Wo o 1 too Z; - ; 510 45 o 10 27 100 As, All Z;-C; 70, so we have reached to the optimal stage, and a Min Z= - (Max Zw) = 510 at W3=7/6, W₂=4. and W6 = 1, and the corresponding stack variables are a4, as e as. 30, 2,2 27, 24 = 10, 13 20. 6* 3 1-333 * N all alla ay h an aw a kollegoi oor

Hence, the required solution is, Z=-510, ats 21-27, x2=10 23=0,