Interactive Multiobjective Mixed-Integer Optimization Using Dominance-Based Rough Set Approach

We present a new methodology for dealing with interactive multiobjective optimization in case of mixed-integer variables. The preference information elicited by the Decision Maker (DM) in course of the interaction is processed using the Dominance-based Rough Set Approach (DRSA). This permits to ask the DM simple questions and obtain in return a decision model expressed in terms of easily understandable “if..., then...” decision rules. In each iteration, the current set of decision rules is presented to the DM with the proposal of selecting one of them considered the most representative. The selected decision rule specifies some minimal requirements that the DM desires to be achieved by the objective functions. This information is translated into a set of constraints which are added to the original problem restricting the space of feasible solutions. Moreover, we introduce one simple but effective algorithm, called bound-and-cut, that efficiently reduces the set of feasible values of the integer variables. This process continues iteratively until the part of the Pareto front that is interesting for the DM can be exhaustively explored with respect to the integer variables. The bound-and-cut algorithm can be embedded in an Evolutionary Multiobjective Optimization (EMO) method, which permits to compute a reasonable approximation of the considered part of the Pareto front. A subset of representative solutions can be selected from this approximation and presented to the DM in the dialogue phase of each iteration.