Fuzzy Combinatorial Optimization Problem

Contents

Modeling the weights by means of closed intervals is perhaps the simplest form of uncertainty representation. For every weight we must only specify a range of possible values. In this chapter we discuss a more sophisticated uncertainty evaluation. The key idea is to extend the notion of the classical closed interval to the fuzzy one. A fuzzy interval can be seen as a family of closed intervals parametrized by the value of λ ∈ [0,1]. It is reacher in information than a classical one and allows a representation that is at once both pessimistic and optimistic. A fuzzy interval has an interpretation in the setting of possibility theory, which is described for instance in a book by Dubois and Prade [44]. In Section 10.1 we describe the concept of a fuzzy interval and its possibilistic interpretation. We will consider then a combinatorial optimization problem in which the uncertain weights are modeled by means of fuzzy intervals.

In Chapter 2 we have introduced the concepts of possibly and necessarily optimal solutions and elements. The optimality evaluation has a clear interpretation in the setting of possibility theory. Intuitively, a solution X is possibly optimal if the event that X will be optimal may happen and it is necessarily optimal if the event that it will be optimal is sure. The same holds for elements. In the fuzzy case, the Boolean notions of possible and necessary optimality are generalized to degrees of possible and necessary optimality, which are numbers from interval [0,1]. Both degrees can also be deriven from a fuzzy deviation, which generalizes the interval deviation discussed in Chapter 2 and provides a full possibilistic characterization of optimality. The optimality evaluation and some methods of computing the fuzzy deviation are discussed in Section 10.3.

In Section 10.4 we consider the problem of choosing a solution under fuzzy weights. We discuss two solution concepts that are closely related to the minmax regret approach. The first concept consists in computing a solution that has the greatest necessary optimality degree. This solution can be obtained in polynomial time if the underlying deterministic problem is polynomially solvable. The second concept consists in determining so called necessarily soft optimal solution, which can be determined in polynomial time if the underlying minmax regret problem is polynomially solvable. Similar solution concepts were applied to the linear programming problem with a fuzzy objective function [67,68].