Kalyan Singhal Corp makes three products and it has three ma

Kalyan Singhal Corp. makes three products, and it has three machines available as resources as given in the following LP problem:

Maximize contribution = 5X1 + 4X2 + 3X3

Subject to: 1X1 + 7X2 + 4X3 <= 90 (hours on machine 1)

2X1 + 1X2 + 7X3 <= 96 (hours on machine 2)

8X1 + 4X2 + 1X3 <= 90 (hours on machine 3)

X1, X2, X3 >=0

(a) Determine the optimal solution using LP software. the optimal achieved is

X1=

X2-

X3=

contribution =

(b) machine 1 has __ hours of unused time available at the optimal solution

machine 2 has __ hours of unused time available at the optimal solution

machine 3 has __ hours of unused time available at the optimal solution

(d) an additional 8 hours of time available for th esecond machine, at no cost to the firm, are going to increase the objective value by __ dollars

Given:

Maximize p = 5X1 + 4X2 + 3X3 subject to
1X1 + 7X2 + 4X3 <= 90

2X1 + 1X2 + 7X3 <= 96

8X1 + 4X2 + 1X3 <= 90

X1 >=0
X2 >=0
X3 >=0

Optimal Solution: Maximum contribution = 90.547; x1 = 7.07692, x2 = 5.62393, x3 = 10.8889

hours unused for machine 1: X1 + 7X2 + 4X3 -90= 90-90 = 0

hours unused for machine 2: 2X1 + 1X2 + 7X3 - 96 = 96-96 =0

hours unused for machine 3: 8X1 + 4X2 + 1X3 -90 = 90 - 90 =0

if additional hour is available on machine 3 , the last constraint becomes :

8X1 + 4X2 + 1X3 =< 91

and corresponding contribution is = 91.0883,

thus increase in contribution is 91.0883 - 90.547 = 0.5413

an additional 8 hours of time available for the second machine, causes second constraint to become:

2X1 + 1X2 + 7X3 <= 104 , and corresponding optimal solution is = 92.302

thus increase in objective function is 92.302 - 90.547 = 1.755

Hope steps were clear!!