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124 CHAPTER 3 LINEAR PROGRAMMING: SENSITIVITY ANALYSIS AND INTERPRETATION OF SOLUTION

leads to the formulation:
Max 40F þ 30S
s:t
2 1
/ F þ / S 20 Material 1
5 2
1
/ S 5 Material 2
2
3 3
/ F þ / S 21 Material 3
10
5
F; S 0
Use the graphical sensitivity analysis approach to determine the range of optimality for
the objective function coefficients.
2 For Problem 1 use the graphical sensitivity approach to determine what happens if an
additional three kilos of material 3 become available. What is the corresponding dual price
for the constraint?
3 Consider the following linear programme:
Max 2x 1 þ 3x 2
s:t
x 1 þ x 2 10
2x 1 þ x 2 4
x 1 þ 3x 2 24
2x 1 þ x 2 16
x 1 ; x 2 0

a. Solve this problem using the graphical solution procedure.
b. Calculate the range of optimality for the objective function coefficient of x 1 .
c. Calculate the range of optimality for the objective function coefficient of x 2 .
d. Suppose the objective function coefficient of x 1 is increased from 2 to 2.5. What is the
new optimal solution?
e. Suppose the objective function coefficient of x 2 is decreased from 3 to 1. What is the
new optimal solution?

4 Refer to Problem 3. Calculate the dual prices for constraints 1 and 2 and interpret them.
5 Consider the following linear programme:
Min x 1 þ x 2
s:t
x 1 þ 2x 2 7
2x 1 þ x 2 5
x 1 þ 6x 2 11
x 1 ; x 2 0

a. Solve this problem using the graphical solution procedure.
b. Calculate the range of optimality for the objective function coefficient of x 1 .
c. Calculate the range of optimality for the objective function coefficient of x 2 .
d. Suppose the objective function coefficient of x 1 is increased to 1.5. Find the new optimal
solution.
e. Suppose the objective function coefficient of x 2 is decreased to one-third. Find the new
optimal solution.

6 Refer to Problem 5. Calculate and interpret the dual prices for the constraints.

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