Computing Min-Max Regret Solutions in Possibilistic Combinatorial Optimization Problems

In this chapter we discuss a wide class of combinatorial optimization problems with a linear sum and a bottleneck cost function. We first investigate the case when the weights in the problem are modeled as closed intervals. We show how the notion of optimality can be extended by using a concept of a deviation interval. In order to choose a solution we adopt a robust approach. We seek a solution that minimizes the maximal regret, that is the maximal deviation from optimum over all weight realizations, called scenarios, which may occur. We then explore the case in which the weights are specified as fuzzy intervals. We show that under fuzzy weights the problem has an interpretation consistent with possibility theory. Namely, fuzzy weights induce a possibility distribution over the scenario set and the possibility and necessity measures can be used to extend the optimality evaluation and the min-max regret approach.

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