Diversity Maximization Approach for Multiobjective Optimization

One of the most common approaches for multiobjective optimization is to generate the whole or partial efficient frontier and then decide about the preferred solution in a higher-level decision-making process. In this paper, a new method for generating the efficient frontier for multiobjective problems is developed, called the diversity maximization approach (DMA). This approach is capable of solving mixed-integer and combinatorial problems. The DMA finds Pareto optimal solutions by maximizing a proposed diversity measure and guarantees generating the complete set of efficient points. Given a subset of the efficient frontier, DMA finds the next Pareto optimal solution which, combined with the existing ones, yields the most diversified subset of efficient points. This solution is defined as the most diverse solution. In fact, it aims to maximize the distance between the new efficient point and the closest point in the given subset of the efficient frontier. The proposed approach can be applied to any problem that can be solved for the single-objective case. We can use the DMA by solving directly a modified version of the mixed-integer programming (MIP) formulation of the multiobjective problem. In this case, the Pareto optimal solutions are found sequentially in an iterative way. Consequently, as we terminate the procedure before completion, a partial efficient frontier is available. The diversity measure assures that in every stage of the procedure, the partial efficient frontier is well diversified. This partial efficient frontier can be perceived as a filtered set of the complete efficient frontier and can be used by the decision maker in case the complete efficient frontier contains too many points. An additional way of using DMA is by incorporating it in a problem oriented branch-and-bound algorithm. Detailed examples of both approaches are given.