Ranking multiple criteria alternatives with half-space, convex, and non-convex dominating cones: Quasi-concave and quasi-convex multiple attribute utility functions

This paper introduces a quasi-concave nonlinear multiple attribute utility function (MAUF) to rank multiple criteria alternatives. It is demonstrated that the quasi-concave MAUF (QCMAUF) is more general and flexible than additive, multiplicative, and multilinear MAUFs. New definitions of weights for QCMAUFs are developed that can be used to rank alternatives. It is shown that paired comparison questions can be used to generate local partial information on the weights. Definitions and procedures for ranking alternatives with complete local unique weights (by half-space cones), partial local information or unique weights (by convex cones), and non-unique local weights (by non-convex cones) are developed. They are all based on problems that can be solved by linear programming. It is shown that if the MAUF is quasi-convex, similar results can be stated. It is possible to determine whether the decision maker’s behavior is quasi-concave or quasi-convex and use the appropriate theory to rank alternatives. Several examples are presented.