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GRAPHICAL SENSITIVITY ANALYSIS 93

If C S is changed to 13 and simultaneously C D is changed to 8, the new objective
function slope will be given by:
C S 13
¼ ¼ 1:625
C D 8
Because this value is less than the lower limit of 1.5, the current solution of
S ¼ 540 and D ¼ 252 will no longer be optimal. By re-solving the problem with
C S ¼ 13 and C D ¼ 8 we will find that extreme point › is the new optimal
solution.
Looking at the ranges of optimality, we concluded that changing either C S to $13
or C D to $8 (but not both) would not cause a change in the optimal solution. But in
recalculating the slope of the objective function with simultaneous changes for both
C S and C D , we saw that the optimal solution did change. This result emphasizes the
fact that a range of optimality, by itself, can only be used to draw a conclusion about
changes made to one objective function coefficient at a time.

Right-Hand Sides
Let us now consider how a change in the right-hand side of a constraint may affect
the feasible region and perhaps cause a change in the optimal solution to the
problem. To illustrate this aspect of sensitivity analysis, let us consider what happens
if an additional ten hours of production time become available in the cutting and
dyeing department of GulfGolf. The right-hand side of the cutting and dyeing
constraint is changed from 630 to 640, and the constraint is rewritten as:
0:7S þ 1D 640
By obtaining an additional ten hours of cutting and dyeing time, we expand the
feasible region for the problem, as shown in Figure 3.3. With an enlarged
feasible region, we now want to determine whether one of the new feasible
solutions provides an improvement in the value of the objective function. Appli-
cation of the graphical solution procedure to the problem with the enlarged
feasible region shows that the extreme point with S ¼ 527.5 and D ¼ 270.75 now
provides the optimal solution. The new value for the objective function is
10(527.5) + 9(270.75) ¼ $7711.75, with an increase in profit of $7711.75
$7668.00 ¼ $43.75. Thus, the increased profit occurs at a rate of $43.75/10
hours ¼ $4.375 per hour added.
Although we used a graphical approach here to undertake sensitivity analysis on
the cutting and dyeing constraint, it is actually easier to use the relevant equations.
We know that for the GulfGolf problem, the optimal solution is at the intersection
of the cutting and dyeing constraint line and the finishing constraint line. So, we
have (as in Chapter 2):

0:7S þ 1D ¼ 630 ð3:5Þ
1S þ 0:6667D ¼ 708 (3:6)

at the current optimal solution. If we now want to do a sensitivity analysis on the
cutting and dyeing constraint (3.5) we can re-write the equation as:

0:7S þ 1D ¼ 631 (3:7)

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