A Visual Interactive Method for Solving the Multiple-Criteria Problem

In this paper we propose an interactive method for solving multiple criteria decision problems with convex constraints and a pseudoconcave and differentiable utility function. The general framework of our method is similar to that of the so-called GDF method (Geoffrion, Dyer and Feinberg 1972). However, the Frank-Wolfe algorithm used by Geoffrion et al. does not operate solely with efficient solutions. Since comparisons between inefficient solutions may not seem relevant from the decision maker’s point of view, we use a modified gradient projection method instead of the Frank-Wolfe algorithm. However, instead of the gradient vector we use reference directions that reflect the decision maker’s preferences, as suggested by Andrzej Wierzbicki (1980), instead of trying to estimate the gradient. The reference directions are projected on the efficient surface and an interactive line search is performed. The values of the objectives on the efficient surface are displayed for the decision maker’s evaluation both numerically and graphically.