Min-max and two-stage possibilistic combinatorial optimization problems

This paper deals with a class of combinatorial optimization problems with uncertain costs. The uncertainty is modeled by specifying a scenario set containing a finite number of possible realizations of the costs called scenarios. Additionally, a possibility distribution on the scenario set can be defined. Two robust models, namely the min-max and two-stage, for hedging against uncertainty of the costs in the possibilistic setting are considered. A general framework for solving the problems is proposed. For the linear sum objective a mixed integer proggraming formulation is shown. For the bottleneck objective, an algorithm is constructed which runs in polynomial time if the deterministic problem, i.e. the one with a single scenario, is polynomially solvable.