An exact interactive method for exploring the efficient facets of multiple objective linear programming problems with quasi-concave utility functions

Many real-world problems can be formulated as multiple-objective linear programming (MOLP) problems. In the search for the best compromise solution for conflicting and noncommensurate objectives, a quasiconcave preference structure (utility function) is used that is more flexible and general than pseudoconcave, concave, and linear utility functions. Since the complete assessment of such a utility function is very difficult or impossible, an interactive method is developed in which, with a minimum of simple questions to the decision-maker (DM), the best compromise solution can be obtained. The DM responds to either paired comparison or simple trade-off questions. Conversion of tradeoff questions to paired comparison is discussed. The method also uses linear approximations of the nonlinear utility function to improve convergence rate. A procedure is provided for identifying efficient tradeoffs on the efficient facet so that only efficient alternatives are generated. Convergence with a limited number of questions is proven for quasiconcave and pseudoconcave utility functions