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only 2,775 3 hours available, some projects will not be accepted. The LP will choose
the best combination of projects that meets the constraints imposed by the resources,
while maximizing the revenue returned. It is not obvious which projects should
be accepted. To simply assume that the highest revenue projects will be selected
until resources run out is not advisable. There may be lower revenue projects that
consume fewer resources that are more attractive from a cost/beneﬁt perspective.
Fortunately for us, the LP will make this determination and guarantee an “optimal
solution”, or indicate that no solution is possible ( infeasibility ).
One of the features of an LP solution is the continuous nature of the resulting
decision variables. Thus, it is likely that our solution will suggest fractional val-
ues of contracts; for example, it is possible that a solution could suggest 12.345
units of Project Type 1. Obviously, accepting fractional contracts is a problem;
but if the numbers of projects that can be accepted are relatively large, rounding
these fractional numbers up or down, while making sure that we do not exceed
important constraints, should lead to a near optimal solution. Later we will see
that by imposing a constraint on a variable to be either binary (0 or 1) or inte-
ger (1, 2, 3, etc.) we convert our LP into a non-linear program. As we suggested
earlier, these are problems that are far more complex to solve, and will require
careful use of the Excel Solver to guarantee that an optimal solution has been
achieved.
9.3.1 Formulation
Now, let us state the problem in mathematical detail. First, the decision variables
from which we can construct our objective function and constraints are:
X 1 , X 2 ,..., X 7 =
the number of each of the seven types of projects selected for the quarter
For example, X 4 is the number of projects of type 4 selected. The decision vari-
ables must all be non-negative values; selecting a negative number of projects does
not make any practical sense.
Next, consider the objective function , Z , for the YRA in terms of the decision
variables:
Z
=
45,000 X 1 +
63,000 X 2 +
27,500 X 3 +
19,500 X 4 +
71,000 X 5 +
56,000 X 6 +
48,500 X 7
The objective function sums the revenue contribution of the projects that are
selected. If X 1 is selected to be 10, then the contribution of X 1 projects to the
objective function, Z , is \$450,000 (10 45,000
=
450,000).
3 Total resource hours available in the quarter are 2775 (800 + 900 + 700 + 375
=
2775) for Res-A
through Res-D, respectively.
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