The compromise hypersphere for multiobjective linear programming

Abstract

For a linear-programming problem with q (⩾2) objective functions (that is, a multiobjective linear-programming problem), we propose a method for ranking the full set or a subset of efficient extreme-point solutions. The idea is to enclose the given efficient solutions, as represented by q-dimensional points in objective space, within an annulus of minimum width, where the width is determined by a hypersphere that minimizes the maximum deviation of the points from the surface of the hypersphere. We argue that the hypersphere represents a surface of compromise and that the point closest to its surface should be considered as the “best” compromise efficient solution. Also, given a ranked (sub)set of efficient solutions, a procedure is given that associates to each efficient solution a set of q positive weights that causes the efficient solution to be optimal with respect to the given set of efficient solutions.