Solving the Discrete Multiple Criteria Problem using Convex Cones

Abstract

An interactive method employing pairwise comparisons of attainable solutions is developed for solving the discrete, deterministic multiple criteria problem assuming a single decision maker who has an implicit quasi-concave increasing utility (or value) function. The method chooses an arbitrary set of positive multipliers to generate a proxy composite linear objective function which is then maximized over the set of solutions. The maximizing solution is compared with several solutions using pairwise judgments asked of the decision maker. Responses are used to eliminate alternatives using convex cones based on expressed preferences, and then a new set of weights is found that satisfies the indicated preferences. The requisite theory and proofs as well as a detailed numerical example are included. In addition, the results of some computational experiments to test the effectiveness of the method are described.decision analysis, utility/preference: multiattribute, programming: multiple criteria, convex cones

Equitable decision making approaches over allocations of multiple benefits to multiple entities

“…[25] provides an interactive method to find the best alternative, where the DM is asked pairwise comparison questions and the underlying value function of the DMs is assumed to be a non-decreasing and quasi concave value function. They use the pairwise comparison information to generate the corresponding cones and eliminate the inferior alternatives which are inferior to any of these cones.…”

“…The authors discuss the ways of obtaining appropriate artificial alternatives. They managed to ask smaller number of pairwise comparison questions to find the best alternative compared to [25].…”

“…In this chapter, we discuss the convex cone based approach, which is widely used in the MCDM literature ( [53], [25], [27], [54], [28]). Convex cones are used in MCDM problems to incorporate preference information in the model.…”

“…Convex cones are used in MCDM problems to incorporate preference information in the model. This method assumes that the underlying value function of the DM is quasi-concave and is based on eliminating the alternatives that are inferior with respect to the cones generated based on preference information that she provides [25].…”

CUT: A Multicriteria Approach for Concavifiable Preferences

“…Hazen (1983) provides a rigorous analysis of this concept. Korhonen, Wallenius and Zionts (1984), Ramesh, Karwan and Zionts (1988) and Prasad, Karwan and Zionts (1997) provide interactive procedures utilizing convex cones. This idea has proved influential in the literature -Karsu, Morton and Argyris (2012) cite 15 references using cones, a success we attribute to the modest and natural appeal of the quasi-concavity assumption.…”