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SENSITIVITY ANALYSIS WITH THE SIMPLEX TABLEAU 265

valuable management information and should be included as part of the manage-
ment report on any linear programming project. The range of feasibility is typically
made available as part of the computer solution to the problem.

Simultaneous Changes
In reviewing the procedures for developing the range of optimality and the range
of feasibility, we note that only one coefficient at a time was permitted to vary. Our
statements concerning changes within these ranges were made with the under-
standing that no other coefficients are permitted to change. However, sometimes
we can make the same statements when either two or more objective function
coefficients or two or more right-hand sides are varied simultaneously. When the
simultaneous changes satisfy the 100 per cent rule, the same statements are
applicable. The 100 per cent rule was explained in Chapter 3, but we will briefly
review it here.
Let us define allowable increase as the amount a coefficient can be increased
before reaching the upper limit of its range, and allowable decrease as the amount a
coefficient can be decreased before reaching the lower limit of its range. Now
suppose simultaneous changes are made in two or more objective function coeffi-
cients. For each coefficient changed, we compute the percentage of the allowable
increase, or allowable decrease, represented by the change. If the sum of the
percentages for all changes does not exceed 100 per cent, we say that the 100 per
cent rule is satisfied and that the simultaneous changes will not cause a change in the
optimal solution. However, just as with a single objective function coefficient
change, the value of the solution will change because of the change in the coeffi-
cients.
Similarly, if two or more changes in constraint right-hand side values are made,
we again compute the percentage of allowable increase or allowable decrease
represented by each change. If the sum of the percentages for all changes does
not exceed 100 per cent, we say that the 100 per cent rule is satisfied. The dual prices
are then valid for determining the change in value of the objective function asso-
ciated with the right-hand side changes.

NOTES AND COMMENTS

1 Sometimes, interpreting dual prices and 2 The Notes and Comments in Chapter 3
choosing the appropriate sign can be confusing. concerning sensitivity analysis are also applicable
It often helps to think of this process as here. In particular, recall that the 100 per cent rule
follows. Relaxing a constraint means cannot be applied to simultaneous changes in the
decreasing its right-hand side, and relaxing objective function and the right-hand sides; it
a constraint means increasing its right-hand applies only to simultaneous changes in one or
side. Relaxing a constraint permits improvement the other. Also note that this rule does not mean
in value; restricting a constraint (decreasing the that simultaneous changes that do not satisfy the
right-hand side of a constraint or increasing rule will necessarily cause a change in the
the right-hand side of a constraint) has the solution. For instance, any proportional change in
opposite effect. In every case, the absolute value all the objective function coefficients will leave the
of the dual price gives the improvement in the optimal solution unchanged, and any proportional
optimal value associated with relaxing the change in all the right-hand sides will leave the
constraint. dual prices unchanged.

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