Page 159 -

THE PROCESS OF PROBLEM FORMULATION 139

2 Determine the overall objective that appears to be relevant. It will usually
be clear whether the objective relates to maximization or minimization, to
cost or profit and so on. An adequate understanding of the overall
objective can be of considerable assistance in unravelling other aspects of
the problem.
3 Determine the factors (constraints) that appear to restrict in some way
the attainment of the objective identified in the previous stage.
These three stages together will provide a detailed verbal exposition of the
complete problem under investigation. The next step is to put the
verbal description into a suitable mathematical framework.
4 Define the decision variables that are relevant to the problem and ensure that
their units of measurement are explicitly stated. Failure to do so may well lead
to difficulty in formulating appropriate constraints and in interpreting the
solution results.
5 Using these decision variables, formulate an objective function. It is clear that
this function should incorporate all of the decision variables. If it does not,
then it signifies either a lack of information or an incorrect choice of decision
variables.
6 For each of the factors identified in Stage 3, formulate a suitable mathematical
constraint. Again, each constraint must include at least some of the decision
variables and, again, the units of measurement of each constraint should be
explicit.
7 Lastly, check the entire formulation to ensure linearity of all variables and
constraints.
It should not be concluded, on the basis of this process, that problem formu-
lation will be as simple and straightforward as this. It will typically involve
considerable backtracking (the methodology structure discussed in Chapter 1 is
clearly appropriate to this process). You may consider initially that you have
identified the appropriate decision variables but are then unable to formulate a
particular constraint involving these variables. This failure suggests that a full
reconsideration of the problem is necessary. Equally you may complete the
formulation only to find that there is no apparent solution to the problem as
formulated. Typically this may imply an incorrect formulation. It is equally
important that once an optimal solution has been found, you need to ‘translate’
the solution back into the original, verbal, problem to ensure that the mathe-
matical solution is appropriate for the original problem. A frequent mistake
made by many students is to produce a formulation (often lacking some critical
constraint) to solve the problem and then simply to assume that because they
have a solution then their formulation mustbecorrect.Onlyifthe mathemat-
ical solution can be tied in with the original problem are we in a position to
assume that our problem formulation is the correct one.
To illustrate the process and to provide examples of some of the more common
areas of LP applications to business problems we shall now look at a number of
problems and their formulation in detail. These problems have been categorized
in terms of their general area of applicability but it must be stressed that the
divisions between such categories are arbitrary and serve only as a general
guide. In the real world practical applications of the technique will not fall
neatly into one particular category, although it is frequently useful to undertake
such categorization to help focus on an appropriate overall structure to the
formulation.

Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.