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

Suppose that in the GulfGolf problem the accounting department concluded
that the original profit contributions of $10 and $9 for the standard and deluxe
bags, respectively, were incorrectly calculated; the correct values should have been
$11.50 and $8.25. To determine what effect, if any, these simultaneous changes
have on the optimal solution, we need to first define the terms allowable increase
and allowable decrease. For an objective function coefficient, the allowable
increase is the maximum amount the coefficient may increase without exceeding
the upper limit of the range of optimality; the allowable decrease is the maximum
amount the coefficient may decrease without dropping below the lower limit of the
range of optimality.
From Figure 3.4 we see that the upper limit for the objective function coefficient
of S is 13.49993; so, the allowable increase is 3.49993 (13.49993 10). In terms of
percentage change, the increase of $1.50 in the objective function coefficient (from
10 to 11.50) for the standard bags is (1.50/3.49993)(100) ¼ 42.86 per cent of the
allowable increase. Given the lower limit of 6.66670 for D, the allowable decrease for
D is 2.33330 (9 6.66670). In terms of percentage change, the decrease of $0.75 in
the objective function coefficient (from 9 to 8.25) for the deluxe bags is (0.75/
2.33330)(100) ¼ 32.14 per cent of the allowable decrease. The sum of the percentage
change of the allowable increase (42.86 per cent) and the percentage change of the
allowable decrease (32.14 per cent) is 75.00 per cent.
Let us now state the 100 per cent rule as it applies to simultaneous changes in the
objective function coefficients.

100 Per cent Rule for Objective Function Coefficients
For all objective function coefficients that are changed, sum the percentages
of the allowable increases and the allowable decreases represented by the
changes. If the sum of the percentage changes does not exceed 100 per cent
the optimal solution will not change.
So, because the sum of the two percentage changes in the objective function
coefficients for the problem is 75 per cent, these simultaneous changes will not
affect the optimal solution. Note, however, that although the optimal solution is still
S ¼ 539.99841 and D ¼ 252.00113, the value of the optimal solution will change
because the profit contribution for the standard bags has increased to $11.50 and the
profit contribution of the deluxe bags has decreased to $8.25.
The 100 per cent rule does not, however, say that the optimal solution will
change if the sum of the percentage changes exceeds 100 per cent. It is possible
that the optimal solution will not change even though the sum of the percentage
changes exceeds 100 per cent. When the 100 per cent rule is not satisfied, we
must re-solve the problem to determine what affect such changes will have on
the optimal solution.
A similar version of the 100 per cent rule also applies to simultaneous changes in
the constraint right-hand sides.

100 Per cent Rule for Constraint Right-Hand Sides
For all right-hand sides that are changed, sum the percentages of allowable
increases and allowable decreases. If the sum of percentages does not
exceed 100 per cent, then the dual values will not change.
Let us illustrate the 100 per cent rule for constraint right-hand sides by consid-
ering simultaneous changes in the right-hand sides for the GulfGolf problem.
Suppose, for instance, that in this problem we could obtain 20 additional hours of
cutting and dyeing time and 100 additional hours of finishing time. The allowable
increase for cutting and dyeing time is 682.36316 630.0 ¼ 52.36316, and the
allowable increase for finishing time is 900.0 708.0 ¼ 192.0 (see Figure 3.4).

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