Geoscience Reference
In-Depth Information
We call “scenario” a complete realization of all the uncertain parameters. This
notion is independent of whether or not probabilistic information is available.
Nevertheless, if uncertain parameters can be represented by random variables, some
probability can be associated with each scenario. Depending on the problem, we
may have a finite or infinite number of scenarios. As it will be discussed later, this
fact has impact on the models and techniques that can be used.
One important feature that influences the type of model to be considered,
regards the attitude of the decision maker towards risk. Two attitudes are typically
considered: risk neutral and risk averse. In the first case, the decision maker does
not take risk into account when making a decision and a linear function is a
correct representation of the utility associated with the decision maker. When a
probability can be associated with each scenario, a risk neutral decision maker looks
for the decision which minimizes the expected cost (or maximizes the expected
return or utility). A risk averse decision maker can be associated with a concave
utility function (when utility is measured on the vertical axis and monetary value
is measured on the horizontal axis). In this case, the decision maker wants to
avoid unnecessary risk and the expected value of the future assets is no longer an
appropriate objective. Such decision maker may look, for instance, for the solution
minimizing the maximum cost across all scenarios.
Finally, in some classes of problems, there is another aspect that influences the
mathematical model to be considered: the identification of the ex ante and ex post
decisions. In the first case, we have the here-and-now decisions, i.e., the decisions
that must be implemented before uncertainty is revealed; in the second case, we
have the decisions to be implemented after uncertainty is disclosed. The latter set
of decisions is often used as a reaction to the values observed for the uncertain
parameters. In a facility location problem, the location of the facilities is often an
ex ante decision. This is a consequence of the strategic nature of such decisions
in many problems, which imposes their fully implementation before uncertainty is
revealed. Regarding the allocation or distribution decisions, they will depend on the
specific problem addressed whether they will be ex ante or ex post decisions. In the
following sections we address both situations.
8.3
Robust Facility Location Problems
We start by assuming that uncertainty is appropriately captured by a finite set of
scenarios. As mentioned above, each scenario fully determines the value of all the
uncertain parameters. If no probabilistic information is available, one possibility for
measuring the performance of a system is to use a robustness measure. Two classical
objectives are often considered: minmax cost and minmax regret.
For illustrative purposes, we consider a well-known facility location problem:
the p-median problem. In this problem, we have a set of demand nodes, J, each
of which to be served by one out of p new facilities to be located. The potential
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