Yves De Smet

Towards multicriteria clustering: An extension of the k-means algorithm

Abstract The research within the multicriteria classification field is mainly focused on the assignment of actions to pre-defined classes. Nevertheless the building of multicriteria categories remains a theoretical question still not studied in detail. To tackle this problem, we propose an extension of the well-known k-means algorithm to the multicriteria framework. This extension relies on the definition of a multicriteria distance based on the preference structure defined by the decision maker. Thus, two alternatives will be similar if they are preferred, indifferent and incomparable to more or less the same actions. Armed with this multicriteria distance, we will be able to partition the set of alternatives into classes that are meaningful from a multicriteria perspective. Finally, the examples of the country risk problem and the diagnosis of firms will be treated to illustrate the applicability of this method.

D-Sight: A New Decision Making Software to Address Multi-Criteria Problems

PROMETHEE and GAIA belong to the family of multi-criteria outranking methods. A key aspect of their successful application to real problems relies on the existence of user-friendly software implementing these approaches. Following PROMCALC and DECISION LAB 2000, D-Sight is the third generation of PROMETHEE and GAIA based applications. It offers multiple interactive and visual tools that help the decision maker to better understand and manage his multi-criteria problem. The aim of this paper is to provide a description of D-Sight by presenting its main characteristics. An illustrative case study about the outsourcing of IT infrastructure and application development is detailed.

A ranking model in uncertain, imprecise and multi-experts contexts: The application of evidence theory

We consider ranking problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be imperfect and is provided by one or several experts. We model each imperfect evaluation as a basic belief assignment (BBA). In order to rank the BBAs characterizing the performances of the actions according to each criterion, a new concept called RBBD and based on the comparison of these BBAs to ideal or nadir BBAs is proposed. This is performed using belief distances that measure the dissimilarity of each BBA to the ideal or nadir BBAs. A model inspired by Xu et al.s method is also proposed and illustrated by a pedagogical example.

Building a binary outranking relation in uncertain, imprecise and multi-experts contexts: The application of evidence theory

We consider multicriteria decision problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be uncertain and/or imprecise and is provided by one or several experts. We model this evaluation as a basic belief assignment (BBA). In order to compare the different pairs of alternatives according to each criterion, the concept of first belief dominance is proposed. Additionally, criteria weights are also expressed by means of a BBA. A model inspired by ELECTRE I is developed and illustrated by a pedagogical example.

Multi-criteria auctions without full comparability of bids

The emergence of auction mechanisms that support bids characterized by several attributes is one of the most recent evolutions within auction theory. These mechanisms, referred to as multi-attribute, multiple issue or multi-dimensional auctions, are at the intersection between multi-criteria decision and auction theories. The purpose of this paper is to introduce multi-criteria auctions the originality of which is not to require full comparability between bids. We claim that this distinctive feature is of great interest, especially in procurement situations. Furthermore, the existence of potential incomparability between multi-dimensional offers will permit us to manage different bidding niches coexisting within the same bidding space. A theoretical framework based on a general preference structure will be introduced and then referenced to existing approaches such as multi-attribute auctions or new ones such as dominance based multi-criteria auctions or butterfly auctions.

An experimental study of preference model integration into multi-objective optimization heuristics

The usage of preference models in algorithms for multi-objective optimization has recently received an increasing attention by the research community. Motivated by this trend, we experimentally study the impact that the integration of preference models into evolutionary multi-objective search algorithms has on performance. In this article, we consider three preference models, ranging from rather simple to more complex ones; these are (i) reference point, (ii) guided dominance, and (iii) Promethee II. As a benchmark problem we consider multi-objective traveling salesman problem instances of various sizes and with a varying number of objectives.

A bi-objective optimization model to eliciting decision maker's preferences for the PROMETHEE II Method

Eliciting the preferences of a decision maker is a crucial step when applying multi-criteria decision aid methods on real applications. Yet it remains an open research question, especially in the context of the Promethee methods. In this paper, we propose a bi-objective optimization model to tackle the preference elicitation problem. Its main advantage over the widely spread linear programming methods (traditionally proposed to address this question) is the simultaneous optimization of (1) the number of inconsistencies and (2) the robustness of the parameter values. We experimentally study our method for inferring the Promethee II preference parameters using the NSGA-II evolutionary multi-objective optimization algorithm. Results obtained on artificial datasets suggest that our method offers promising new perspectives in that field of research.

Some results about rank reversal instances in the PROMETHEE methods

The multicriteria methods based on pairwise comparisons suffer from possible rank reversal occurrences when the set of alternatives is modified. We study this distinctive feature in the scope of the PROMETHEE I and II methods. First, empirical tests are conducted on the basis of artificial datasets in order to quantify the likelihood of rank reversal instances. Then conditions to avoid this phenomenon are provided. Finally, a comparison with a procedure based on a distillation process is performed.

GAIA Map: A Tool for Visual Ranking Analysis in Spatial Multicriteria Problems

Spatial multicriteria decision problems are decision problems where one needs to take multiple conflicting criteria as well as geographical knowledge into account. In such a context, exploratory spatial analysis is known to provide tools to visualize as much data as possible on maps but does not integrate multicriteria aspects. Also, none of the tools provided by multicriteria analysis were initially destined to be used in a geographical context.In this paper, we propose an application of the PROMETHEE and GAIA ranking methods to Geographical Information Systems (GIS). The aim is to help decision makers obtain rankings of geographical entities and understand why such rankings have been obtained. To do that, we make use of the visual approach of the GAIA method and adapt it to display the results on geographical maps. This approach is then extended to cover several weaknesses of the adaptation. Finally, it is applied to a study of the region of Brussels as well as an evaluation of the Human Development Index (HDI) in Europe.

New GAIA Visualization Methods

In this paper, we consider multicriteria decision aid (MCDA) problems. GAIA is a descriptive extension of the PROMETHEE methods. It provides the decision maker with a two dimensional graphical representation of the multicriteria problem. A limit of the GAIA method is the loss of information resulting from the underlying principal components analysis that can result in inconsistencies with the PROMETHEE rankings. The aim of this paper is to address this limit by introducing new complementary GAIA-type visual representations. At first we introduce the context and briefly recall the principles of the PROMETHEE methods. We then introduce the GAIA method and propose two new extensions: GAIA-Stick gives a better view of the PROMETHEE ranking, while GAIA-Criterion enables the decision maker to focus on one specific criterion. Finally, a numerical example is used to illustrate the completed GAIA approach and to show how it can improve the decision process.