Convex cone-based partial order for multiple criteria alternatives ☆ , ☆☆ Akram Dehnokhalaji a,b , Pekka J. Korhonen a, ⁎, Murat Köksalan a,c , Nasim Nasrabadi a,b , Jyrki Wallenius a a Aalto University, School of Economics, Department of Business Technology, P.O. Box 21220, 00079 Aalto, Helsinki, Finland b Tarbiat Moallem University, Department of Mathematics, Tehran, Iran c Middle East Technical University, Department of Industrial Engineering, Ankara, Turkey abstract article info Available online 25 November 2010 Keywords: Strict partial order Discrete alternative Convex cone Evaluation Multiple criteria In this paper, we consider the problem of finding a preference-based strict partial order for a finite set of multiple criteria alternatives. We develop an approach based on information provided by the decision maker in the form of pairwise comparisons. We assume that the decision maker's value function is not explicitly known, but it has a quasi-concave form. Based on this assumption, we construct convex cones providing additional preference information to partially order the set of alternatives. We also extend the information obtained from the quasi-concavity of the value function to derive heuristic information that enriches the strict partial order. This approach can as such be used to partially rank multiple criteria alternatives and as a supplementary method to incorporate preference information in, e.g. Data Envelopment Analysis and Evolutionary Multi-Objective Optimization. © 2010 Elsevier B.V. All rights reserved. 1. Introduction The purpose in Multiple Criteria Decision Making (MCDM) is to find the most preferred solution among a set of implicitly or explicitly defined alternatives characterized by several criteria, or to rank such alternatives. The problems where alternatives are implicitly defined using constraints are called multiple criteria design problems and the problems where alternatives are explicitly given are called multiple criteria evaluation problems. In this paper, we consider multiple criteria evaluation problems where a Decision Maker (DM) evaluates the explicitly given alternatives. Which kind of approach is most suitable to solving evaluation problems is heavily dependent on the characteristics of the problem. The outranking approach [21], the multi-attribute value function approach [6], the analytic hierarchy process [22], the regime method [4], the hierarchical interactive approach [7], the visual reference direction approach [8], the aspiration-level interactive method (AIM) [18,19], and the hybrid method [17] are typical examples of approaches developed to solve evaluation problems. A class of methods is based on implicitly known value functions. No attempt is made to construct the value function, but assumptions of its functional form are used to structure the search process. Typical assumptions are linearity, Chebyshev-type min–max function, quasi- concavity, pseudo-concavity, etc. of the value function. Examples of such methods are presented in Refs. [10–13,15,26]. There are also approaches to find which form of value function the DM's preferences are consistent with [14,24]. Various interaction styles have been proposed for interactive approaches in general. Examples include requiring pairwise compar- ison of alternatives [27], local tradeoff ratios [2], interval local tradeoff ratios [23], comparative tradeoff ratios [5], reference points [25], and reference directions [9]. A good interactive approach does not waste the DM's time, and its communication language is easy. Furthermore, it is a good idea to increase the intelligence of the system, but it is important to remember that the DM wants to keep the control of the system in his/her own hands. There are several ways to implement a dialogue between an interactive approach and the DM. In this paper we require pairwise comparison information as we think it is easy and relevant for a DM to compare pairs of alternatives. Our aim in this paper is to produce a preference-based strict partial order for a finite set of multiple criteria alternatives. We try to create the strict partial order by making maximum use of the available preference information in the form of pairwise comparisons. We assume that the DM's value function is unknown to us, but it has a quasi-concave form. Based on the available preference information and exploiting the implications of a quasi-concave value function, we construct convex cones [10] and polyhedrons to provide additional preference relations that enrich the strict partial order of alternatives. We also introduce heuristics to extract further approximate prefer- ence relations that can be used in the partial order. This paper unfolds as follows. Section 2 provides preliminary considerations. Section 3 develops the main idea and formulations. Decision Support Systems 51 (2011) 256–261 ☆ The research was supported by the Academy of Finland (Grant number 121980). ☆☆ All rights reserved. This study may not be reproduced in whole or in part without the authors' permission. ⁎ Corresponding author. Aalto University, School of Economics, Department of Business Technology, P.O. Box 21220, 00079 Aalto, Helsinki, Finland. Tel.: +358 9 47001; fax: +358 9 431 38535. E-mail address: Pekka.Korhonen@aalto.fi (P.J. Korhonen). 0167-9236/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.dss.2010.11.019 Contents lists available at ScienceDirect Decision Support Systems journal homepage: www.elsevier.com/locate/dss