Question

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

(c) an additional hour of time available for third machine s worth __ dollars to the firm

(d) an additional 12 hours of time available for the second machine, at no cost to the firm, are going to increase the objective value by __ dollars

Answer 1

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

a]

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

b]

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

c]

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

d]

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