Question

Use the following Management Scientist output to answer the questions.

LINEAR PROGRAMMING PROBLEM

MAX 31X1+35X2+32X3

S.T.

	3X1+5X2+2X3>90
	6X1+7X2+8X3<150
	5X1+3X2+3X3<120

OPTIMAL SOLUTION

Objective Function Value = 763.333

Variable	Value	Reduced Cost
X1	13.333	0.000
X2	10.000	0.000
X3	0.000	10.889

Constraint	Slack/Surplus	Dual Price
1	0.000	0.778
2	0.000	5.556
3	23.333	0.000

OBJECTIVE COEFFICIENT RANGES

Variable	Lower Limit	Current Value	Upper Limit
X1	30.000	31.000	No Upper Limit
X2	No Lower Limit	35.000	36.167
X3	No Lower Limit	32.000	42.889

RIGHT HAND SIDE RANGES

Constraint	Lower Limit	Current Value	Upper Limit
1	77.647	90.000	107.143
2	126.000	150.000	163.125
3	96.667	120.000	No Upper Limit

a.	Give the solution to the problem.
b.	Which constraints are binding?
c.	What would happen if the coefficient of x1 increased by 3?
d.	What would happen if the right-hand side of constraint 1 increased by 10?

Answer 1

Ans a)

The solution to the problem is as follows

X1 = 13.333

X2 = 10

X3 = 0

The value of the objective function is 763.333

Ans b)

Constraints that have a non-zero dual price are known as binding constraints. Constraint 1 and constraint 2 have a non zero dual price. Hence they are binding constraints.

Ans c)

If the coefficient of X1 is increased by 3, there will be no change in the optimal solution. This is because the coefficient of X1 has an allowable increase of infinity. Thus any increase in the value of the coefficient of X1 would not result in the change of the optimal solution. The objective function value will increase as the coefficient increases.

Ans d)

If the right-hand side of constraint 1 increased by 10, its value will increase from 90 to 100. This lies within the upper limit for the right-hand side of constraint 1, which is 107.143

Hence there will be no change in the optimal solution as well as the objective function value.