The authors use fuzzy set theory to improve classical decision-making problems by incorporating the inherent vagueness in decision-makers’ preferences into the model. They specifically study two representative models: the p-median problem and the portfolio selection problem. The first one is a location problem, which on the one hand fits many real world management situations and on the other hand is suitable for a theoretical analysis of the techniques. The version of the portfolio selection problem presented here is a harder problem, which allows the authors to show the scope of their methods. Some numerical examples are provided to illustrate how fuzzy optimal solutions improve classical ones. Finally, the authors present some results about how fuzzy solutions depend on the membership functions of fuzzy parameters.
TopIntroduction
The classical way of handling a real-world problem is to build a mathematical model, taking into account all its relevant aspects and trying to evaluate all its parameters in the most accurate way. However, this approach is clearly naive in most cases, for two main reasons: On the one hand, many real-world characteristics that must be incorporated into a model are subject to a certain degree of uncertainty which must be handled by statistical or fuzzy techniques. On the other hand, some parameters of a decision model are not intended to reflect real-world data, but a decision-maker's preferences, which are not subject to any kind of stochastic uncertainty but rather to inherent vagueness. For example, there is not much sense in saying that an investor wants a portfolio with an expected return not less than 3% with a probability of 0.95. All we can say is that he wants a 3% expected return, more or less.
Whereas there are many appropriate techniques for dealing with stochastic uncertainty when modelling real-world problems, subjective vagueness becomes more problematic and the quality of solutions provided by mathematical models is easily jeopardized by inadequate handling.
The natural way of handling subjective vagueness in those decision-maker related parameters is by means of soft constraints. In fact, since this means extending the preferences of the decision-maker from a single objective function to every vague aspect of the problem, we are led to a multi-objective environment in which even the “constraint” of optimizing the objective function becomes a soft constraint, since a slightly worse value for the objective function can be thought of as more satisfactory if it has any kind of significant counterpart.
The purpose of this chapter is to study the consequences of incorporating in this way the vagueness of the decision-maker’s criteria into the mathematical formulation of a decision problem, both from a practical and theoretical point of view. With regard to the practical aspects, we will see that fuzzy solutions, i.e. the solutions of the fuzzy version of the problem, often turn out to be substantially more satisfactory than the crisp ones, but we will also check the theoretical suitability of the fuzzy models. Namely, we will see that whereas optimal solutions are very sensitive to crisp parameters expressing subjective criteria, they show very low sensitivity to reasonably chosen fuzzy tolerances.
We will analyze two classical decision problems: the p-median location problem and a generalized version of the Markowitz portfolio selection problem. Both of them are mixed-integer problems of unquestionable practical interest, but the first one, being a linear-integer program, provides an opportunity to make an in-depth theoretical sensitivity study. On the other hand, the portfolio selection problem is non-linear and, by extending it with some computationally complex additional constraints, it becomes quite representative of the difficulties we can expect to find when dealing with fuzzy versions of general decision problems.