Multiple Criteria Hierarchy Process in Robust Ordinal Regression Salvatore Corrente a , Salvatore Greco a , Roman Słowiński b, ⁎ a Department of Economics and Business, University of Catania, Corso Italia, 55, 95129 Catania, Italy b Institute of Computing Science, Poznań University of Technology, 60–965 Poznań, and Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw, Poland abstract article info Article history: Received 3 August 2011 Received in revised form 5 January 2012 Accepted 13 March 2012 Available online 28 March 2012 Keywords: Multiple Criteria Decision Aiding Hierarchy of criteria Multiple Criteria Hierarchy Process Robust Ordinal Regression Preference modeling A great majority of methods designed for Multiple Criteria Decision Aiding (MCDA) assume that all evaluation criteria are considered at the same level, however, it is often the case that a practical application is imposing a hierarchical structure of criteria. The hierarchy helps decomposing complex decision making problems into smaller and manageable subtasks, and thus, it is very attractive for users. To handle the hierarchy of criteria in MCDA, we propose a methodology called Multiple Criteria Hierarchy Process (MCHP) which permits consideration of preference relations with respect to a subset of criteria at any level of the hierarchy. MCHP can be applied to any MCDA method. In this paper, we apply MCHP to Robust Ordinal Regression (ROR) being a family of MCDA methods that takes into account all sets of parameters of an assumed preference model, which are compatible with preference information elicited by a Decision Maker (DM). As a result of ROR, one gets necessary and possible preference relations in the set of alternatives, which hold for all compatible sets of parameters or for at least one compatible set of parameters, respectively. Applying MCHP to ROR one gets to know not only necessary and possible preference relations with respect to the whole set of criteria, but also necessary and possible preference relations related to subsets of criteria at different levels of the hierarchy. We also show how MCHP can be extended to handle group decision and interactions among criteria. © 2012 Elsevier B.V. All rights reserved. 1. Introduction It is well known that the dominance relation established in the set of alternatives evaluated on multiple criteria is the only objective informa- tion that comes out from a formulation of a multiple criteria decision problem (including sorting, ranking and choice). While dominance relation permits to eliminate many irrelevant (i.e. dominated) alterna- tives, it does not compare completely all of them, resulting in a situation where many alternatives remain incomparable. This situation may be addressed by taking into account preferences of a Decision Maker (DM). Therefore, all Multiple Criteria Decision Aiding (MCDA) methods (for state-of-the-art surveys on MCDA see [7]) require some preference information elicited by a DM. Information provided by a DM is used within a MCDA process to build a preference model which is then applied on a non-dominated (Pareto-optimal) set of alternatives to arrive at a recommendation. A great majority of methods designed for MCDA, assume that all evaluation criteria are considered at the same level, however, it is often the case that a practical application is imposing a hierarchical structure of criteria. For example, in economic ranking, alternatives may be evaluated on indicators which aggregate evaluations on several sub- indicators, and these sub-indicators may aggregate another set of sub- indicators, etc. In this case, the marginal value functions may refer to all levels of the hierarchy, representing values of particular scores of the alternatives on indicators, sub-indicators, sub-sub-indicators, etc. Considering hierarchical, instead of flat, structure of criteria, permits decomposition of a complex decision problem into smaller problems involving less criteria. To handle the hierarchy of criteria, we introduce in this paper a Multiple Criteria Hierarchy Process (MCHP). The basic idea of MCHP relies on consideration of preference relations at each node of the hierarchy tree of criteria. These preference relations concern both the phase of eliciting preference information, and the phase of analyzing a final recommendation by the DM. Let us consider a very simple and well known preference model, the linear value function, which assigns to each alternative a ∈ A the value U(a)=w 1 g 1 (a)+…+ w n g n (a), w i ≥ 0, i = 1, …n, where g i (a) is an evaluation of alternative a on criterion g i , i = 1, …, n. If in the phase of eliciting preference information, the DM declares that alternative a is preferred to alternative b with respect to a criterion which, in a node of the hierarchy tree, groups a set of sub-criteria G r , this can be modeled as ∑ i∈G r w i g i a ðÞ > ∑ i∈G r w i g i b ðÞ; which puts some constraints on the values of admissible weights w i . In the phase of analyzing a final recommendation, even more important, MCHP shows preference relations c r on A with respect to the set of subcriteria G r , such that, for all a, b ∈ A, a c r b⇔ ∑ i∈G r w i g i a ðÞ≥∑ i∈G r w i g i b ðÞ; Decision Support Systems 53 (2012) 660–674 ⁎ Corresponding author. E-mail addresses: salvatore.corrente@unict.it (S. Corrente), salgreco@unict.it (S. Greco), roman.slowinski@cs.put.poznan.pl (R. Słowiński). 0167-9236/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.dss.2012.03.004 Contents lists available at SciVerse ScienceDirect Decision Support Systems journal homepage: www.elsevier.com/locate/dss