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Elizabeth: Nick, your phone call sounded like something related to the budget
needs urgent attention.
Nick: It does. I have been talking to a couple of our big clients and they are
upset with the idea that we may not choose to accept their projects. They
are threatening to take their business elsewhere.
Elizabeth: That is a problem. What project types are they talking about?
Nick: The problem is that they want us to take on 2 projects of type 3, and
as you know, the solution that you determined suggested that we should
not take on any type 3 projects. They are terrible projects from the stand-
point of the low revenue they return, and the high resource hours they
Elizabeth: Yes I know, they are a disaster for YRA, but if we want to keep these
clients I guess we have to bend a bit and accept a couple of type 3
projects. It’s the price of being customer focused.
Nick: There is also another issue we have to talk about. We need to reconsider
the revenue we assumed for project type 2 and 4. I believe these are
coefﬁcients in the objective function of the formulation. Well, I think
that we may have been inaccurate for both. In the case of project type
2, the revenue is more accurately between $60,500 and $68,000. For
project type 4, the range is likely to be between $19,000 and $32,000.
Elizabeth: Nick, this could be a serious issue. I’ll have to take a look at the sensitiv-
ity analysis for the solution. We may be ﬁne with the current selection of
projects, but with the requirement that we must select 2 projects of type
3, we will clearly have to make changes.
Nick: Also, while you are looking, can you also consider an increase in
resource C of 12 hours. You know that many of the numbers we used
in the formulation were point estimates of some range of values, and I
have a feeling I was incorrect about that one.
Elizabeth: Nick, I want to use the current Sensitivity Report wherever I can, but I
may have to reformulate and solve the problem again. Don’t worry; these
things happen all the time. Solver can answer many of the questions.
Nick: Thanks for stopping by; we really need to answer these questions
Elizabeth: You are right. I should have answers for you by tomorrow.
Elizabeth considered the tasks ahead of her. She decided to start with the issue of
accepting 2 of the unproﬁtable type 3 projects. Depending on the results with this
analysis, she then will consider the issues related to the revenues (coefﬁcients of the
objective function), and ﬁnally, the change in the resource hours.
She begins by returning to Exhibit 9.9 where she ﬁnds that the Reduced Cost
of the Project Type 3 is approximately
12,924 (see cell E11). As you recall, this
number represents the penalty associated with forcing a unit of a decision variable
into the solution that is currently set to 0. Therefore an estimate of the reduction
of the objective function by including 2 units of type 3 projects is approximately
$25,848. Exhibit 9.10 veriﬁes that this is approximately the change to the objective
function if we re-solve the LP with a change in the constraint for resource type 3.