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

infinity as the objective function line becomes vertical; in this case, the slope of the
objective function has no lower limit. Using the upper limit of 1.5, we can write:
C s
1:5
C D
Slope of the %
objective function line
Following the previous procedure of holding C D constant at its original value,
C D ¼ 9, we have:
C S C S
1:5or 1:5
9 9
Solving for C S provides the following result:

C S 13:5
In reviewing Figure 3.2 we note that extreme point › remains optimal for all values of C S
above 13.5. Thus, we obtain the following range of optimality for C S at extreme point ›:

13:5 C S < 1
That is, the current solution will remain optimal as long as C s is at least 13.5. Once
above this value, C s can increase indefinitely (to infinity, or 1) and the optimal
solution will remain unchanged.

Simultaneous Changes
The range of optimality for objective function coefficients is only applicable for
changes made to one coefficient at a time. All other coefficients are assumed to be
fixed at their initial values. If two or more objective function coefficients are changed
simultaneously, further analysis is necessary to determine whether the optimal
solution will change. However, when solving two-variable problems graphically,
expression (3.2) suggests an easy way to determine whether simultaneous changes
in both objective function coefficients will cause a change in the optimal solution. We
simply calculate the slope of the objective function ( C S /C D ) for the new coefficient
values. If this ratio is greater than or equal to the lower limit on the slope of the
objective function and less than or equal to the upper limit, then the changes made will
not cause a change in the optimal solution.
Consider changes in both of the objective function coefficients for the GulfGolf
problem. Suppose the profit contribution per standard bag is increased to $13 and the
profit contribution per deluxe bag is simultaneously reduced to $8. Recall that the
ranges of optimality for C S and C D (both calculated in a one-at-a-time manner) are:

6:3 C s 13:5 (3:3)
6:67 C D 14:29 (3:4)

For these ranges of optimality, we can conclude that changing either C S to $13 or C D
to $8 (but not both) would not cause a change in the optimal solution of S ¼ 540 and
D ¼ 252. But we cannot conclude from the ranges of optimality that changing both
coefficients simultaneously would not result in a change in the optimal solution.
In expression (3.2) we showed that extreme point fi remains optimal as long as:

C S
1:5 0:7
C D

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