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hour constraints, the most valuable hour is for resource B, followed by C, then D.
Of course, it is important to consider the cost of each extra unit of resource. For
example, if I can obtain an additional hour of resource D for \$155, it would not be
wise to make the investment given that the return will only lead to \$152.54 in beneﬁt
the objective function. The results would be a net loss of \$2.46.
In the last two columns of the Constraints area, additional valuable information is
provided regarding the Allowable Increase and Allowable Decrease for the change
in the RHS. As the titles suggest, these are the allowable changes in the RHS of each
constraint for the shadow price to remain at the same value. So, we can state that if
our increase of resource B had been for a 30 hour increase (the Allowable Increase
is 31), our return would be 110,593.20 (30 3686.44). Beyond 31 units, there may
still be a beneﬁt to the objective function, but it will not have the same shadow price.
In fact, the shadow price will be lower.
In the Adjustable Cells section of the Sensitivity Report , we also have information
regarding the estimated opportunity cost for variables. These opportunity costs are
found in the Reduced Cost column, and they apply to variables in our solution that
are currently “0”; they relate to the damage done to the objective function if we force
a variable to enter the solution. We have one such variable, X 3 . If we force the type
3 project variable to equal 1, that is, we require our solution to accept exactly one of
these contracts, the reduction in the objective function value will be approximately
\$12,923.90. Why do we expect a reduction in the objective function value Z ?The
value of Z will be reduced because forcing the entry of a variable that was heretofore
set to “0” by the optimization algorithm will obviously damage, rather than help,
our solution. The reduced costs for the X 5 , X 6 , and X 7 , although appearing to have
a value, cannot be interpreted as the positive effect on the objective function if we
increase the RHS by one unit, e.g. X 5 =
17.
One last, but important, use of the Sensitivity Report is related to the Allowable
increase and Allowable decrease in the coefﬁcients of the objective function vari-
ables (revenues for projects). These increases and decreases represent changes that
can be made, one at a time, without changing the solution values for the decision
variables. The objective function value will change, but not the selected decision
variable values. As often is the case with complex problems, uncertainty will play a
part in our analysis. The objective function coefﬁcients of our decision variables are
likely to be estimates that are dependent on many factors. This form of analysis will
therefore be very useful in understanding the range of possible coefﬁcient change,
without affecting the current decision variable values.
9.3.6 Some Questions for YRA
Now, let us put our understanding of LP to the test. Elizabeth has received an urgent
message from Nick. He wants her to meet him at a local coffee shop, the Daily
Grind, to discuss some matters related to the quarterly budgeting process they have
been working on. The following discussion occurs at the Daily Grind:
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