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SENSITIVITY ANALYSIS: COMPUTER SOLUTION 99

is the same as obtained by performing graphical sensitivity analysis for C S in
Section 3.2.
Using the objective function coefficient range information for deluxe bags, we see
the following range of optimality:

6:6667 C D 14:8572

This result tells us that as long as the profit contribution associated with the deluxe
bag is between $6.67 and $14.29, the production of S ¼ 540 standard bags and D ¼
252 deluxe bags will remain the optimal solution.
Try Problem 11 to test The final section of the computer output (RIGHT HAND SIDE RANGES)
your ability to use provides the limits within which the dual prices are applicable. As long as the
computer output to
determine the ranges of constraint right-hand side is between the lower and upper limit values, the associated
optimality and the ranges dual price gives the improvement in the value of the optimal solution per unit increase
of feasibility. in the right-hand side. For example, let us consider the cutting and dyeing constraint
with a current right-hand-side value of 630. Because the dual price for this constraint
is $4.37, we can conclude that additional hours will increase the objective function by
$4.37 per hour. It is also true that a reduction in the hours available will reduce the
value of the objective function by $4.37 per hour. From the range information given,
we see that the dual price of $4.37 is valid for increases up to 682.36316 and decreases
down to 495.59998. A similar interpretation for the finishing constraint’s right-hand
side (constraint 3) shows that the dual price of $6.94 is applicable for increases up to
900 hours and decreases down to 580.00146 hours.
As mentioned, the right-hand side ranges provide limits within which the dual
prices are applicable. For changes outside the range, the problem must be resolved
to find the new optimal solution and the new dual price. We shall call the range over
which the dual price is applicable the range of feasibility. The ranges of feasibility
for the GulfGolf problem are summarized here:

Constraint Min RHS Max RHS
Cutting and dyeing 495.6 682.4
Sewing 480.0 No upper limit
Finishing 580.0 900.0
Inspection and packaging 117.0 No upper limit

As long as the values of the right-hand sides are within these ranges, the dual prices
shown on the computer output will not change. Right-hand side values outside these
limits will result in changes in the dual price information.

Simultaneous Changes
The sensitivity analysis information in computer output is based on the assumption
that only one objective function coefficient changes; it is assumed that all other
coefficients will remain as stated in the original problem. In many cases, however,
we may be interested in what would happen if two or more coefficients are changed
simultaneously. As we will demonstrate, some analysis of simultaneous changes is
1
possible with the help of the 100 per cent rule. We begin by showing how the 100 per
cent rule applies to simultaneous changes in the objective function coefficients.

1
See S. P. Bradley, A. C. Hax, and T. L. Magnanti, Applied Mathematical Programming (Reading, MA: Addison-
Wesley, 1977).

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