Integration of Two Multiobjective Optimization Methods for Nonlinear Problems

In this paper, we bring together two existing methods for solving multiobjective optimization problems described by nonlinear mathematical models and create methods that benefit from both heir strengths. We use the Feasible Goals Method and the NIMBUS method to form new hybrid approaches. The Feasible Goals Method (FGM) is a graphic decision support tool that combines ideas of goal programming and multiobjective methods. It is based on the transformation of numerical information given by mathematical models into a variety of feasible criterion vectors (that is, feasible goals). Visual interactive display of this variety provides information about the problem that helps the decision maker to detect the limits of what is possible. Then, the decision maker can identify a preferred feasible criterion vector on the graphic display. NIMBUS is an interactive multiobjective optimization method capable of solving nonlinear and even nondifferentiable and nonconvex problems. The decision maker can iteratively evaluate the problem to be solved and express personal preferences in a simple form: the method is based on the classification of the criteria, where the decision maker can indicate what kind of changes to the current solution are desirable. We describe two possible hybrids of the FGM and the NIMBUS method for helping in finding the most preferable decision (using simple questions posed to the decision maker). First, feasible criterion values are explored, and the decision maker's preferences are expressed roughly in the form of a preferable feasible goal (FGM stage). Then, the identified goal is refined using the classification of the criteria (NIMBUS stage). Alternatively, the two methods can be used interactively. Both the hybrid approaches are here illustrated with an example.