Computing min-max regret solutions in possibilistic combinatorial optimization problems Adam Kasperski and Pawel Zieli´ nski Abstract In this chapter we discuss a wide class of combinatorial optimization prob- lems 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. 1 Introduction In many optimization problems we seek an object composed of elements of a given set to achieve some goal. For instance, in a wide class of network problems the ele- ment set consists of all edges of a given graph and we seek an optimal path, spanning tree, cut, matching etc. in this graph. A comprehensive review of various problems of this type can be found in [1, 30, 35]. While describing a particular system we often meet some parameters associated with the elements whose values are not pre- cisely known. For instance, in a traffic network the traveling times between distinct Adam Kasperski Wroclaw University of Technology, Institute of Industrial Engineering and Management, Wybrze˙ ze Wyspia´ nskiego 27, 50-370, Wroclaw, Poland, e-mail: adam.kasperski@pwr.wroc.pl Pawel Zieli´ nski Wroclaw University of Technology, Institute of Mathematics and Computer Science, Wybrze˙ ze Wyspia´ nskiego 27, 50-370, Wroclaw, Poland, e-mail: pawel.zielinski@pwr.wroc.pl 1