Problem # 3 (Value 50 points)

Given the following linear programming problem

Minimize

Z = 8X1 + 5X2 + 7X3

S.A.

10X1 + 8X2 + 12X3 120

2X1 + 4X2 - 3X3 16

X1, X2, X3 0

Whose final table is the following:

VB

Z

X1

X2

X3

X4

X5

X6

LD

Z

1

-11/2

0

-43/4

0

-5/4

-M + 5/4

twenty

X4

0

6

0

18

1

two

-two

88

X2

0

1

-3/4

0

-1/4

1/4

4

Determine the optimal solution of the PRIMAL problem, clearly indicating the value of each variable and of the objective function Z. Determine if the problem has a single or multiple solution, and explain why. Get the equivalent dual model. Determine the optimal solution of the DUAL problem, clearly indicating the value of each variable and of the objective function V. Determine the effect on the objective function of the PRIMAL model, of decreasing the resource of the second constraint by 2 units.

Minimizar Z = 8X1 + 5X2 + 7X3 S.A. 10X1 + 8X2 + 12x3 = 120 2X1 + 4X2 - 3X3 2 16 X1, X2, X320 Cuya tabla final es la siguiente: VB N X1 X2 X3 X4 X5 Xo LD Z 1 - 11/2 0 -43/4 0 -5/4 -M+5/4 20 X4 0 6 0 18 1 2 -2 88 X2 0 12 1 -3/4 0 -1/4 1/4 4 1. Determine la solucin ptima del problema PRIMAL, sealando claramente el valor de cada variable y de la funcin objetivo Z. 2. Determine si el problema tiene solucin nica o mltiple, y explique por qu. 3. Obtenga el modelo dual equivalente. 4. Determine la solucin ptima del problema DUAL, sealando claramente el valor de cada variable y de la funcin objetivo V. 5. Determine el efecto en la funcin objetivo del modelo PRIMAL, de decrementar el recurso de la segunda restriccin en 2 unidades