Interactive relaxation method for a broad class of integer and continuous nonlinear multiple criteria problems

Abstract

An interactive decomposition algorithm for solving a broad class of multiple criteria (MC) problems is presented in this paper. On the one hand, the method is designed to solve a broad class of MC problems, including complex ones; i.e., problems with nonlinear objective functions and nonlinear constraints , with continuous and/or integer-valued decision variables , and with an unknown underlying nonlinear preference function . On the other hand, the method is intended to reduce the assessment burden on the decision maker (DM) by simplifying and facilitating the task of making preference assessments and tradeoffs. The method is efficient from a computational standpoint, since the only program that needs to be solved at each iteration is linear in the decision variables. Also the algorithm is capable of producing lower and upper bounds (LB and UB) on the optimal objective value of the MC problem at each iteration, although the DM's preference function is not assumed to be known explicitly. LB and UB can be used to terminate the search prematurely at a satisfactory (“good enough”) solution and, consequently, to reduce the computational time as well as the number of preference assessments that the DM must make. Finally, all the tradeoffs that the DM is required to make are of the form of ordinal paired comparisons , most of which are particularly simple since they involve changes in only two criteria at a time. Also, all the preference assessments are made in the context of feasible solutions only.