On quasi-orderings and multi-objective functions

In the paper the relationship between quasi-orderings (i.e. transitive and reflexive relations) and multi-objective functions (i.e. functions being a finite vector of continuous value functions) is investigated. A characterization of quasi-orderings representable by multi-objective functions is given.     

On some sheaves of special groups

Using sheaves of special groups, we show that a general local-global principle holds for every reduced special group whose associated space of orderings only has a finite number of accumulation points. We also compute the behaviour of the Boolean hull functor applied to sheaves of special groups. The research leading to this note was carried out with the partial support of the European RTN Networks HPRN-CT-2002-00287 “Algebraic K-Theory, Linear Algebraic Groups and Related Structures”, and HPRN-CT-2001-00271 “Real Algebraic and Analytic Geometry”     

Two algorithms for the Student-Project Allocation problem

We study the Student-Project Allocation problem (SPA), a generalisation of the classical Hospitals/Residents problem (HR). An instance of SPA involves a set of students, projects and lecturers. Each project is offered by a unique lecturer, and both projects and lecturers have capacity constraints. Students have preferences over projects, whilst lecturers have preferences over students. We present two optimal linear-time algorithms for allocating students to projects, subject to the preference and capacity constraints. In particular, each algorithm finds a stable matching of students to projects. Here, the concept of stability generalises the stability definition in the HR context. The stable matching produced by the first algorithm is simultaneously best-possible for all students, whilst the one produced by the second algorithm is simultaneously best-possible for all lecturers. We also prove some structural results concerning the set of stable matchings in a given instance of SPA. The SPA problem model that we consider is very general and has applications to a range of different contexts besides student-project allocation.     

A box‐shaped cyclically reduced operator

We propose a new procedure of partial cyclic reduction, where we apply a 2^d‐color ordering (with d=2, 3 the dimension of the problem), and use different operators for different gridpoints according to their color. These operators are chosen so that the gridpoints can be readily decoupled, and we then eliminate all colors but one. This yields a smaller cartesian mesh and box‐shaped 9‐point (in 2D) or 27‐point (in 3D) operators that are easy to analyze and implement. Multi‐line and multi‐plane orderings are considered, and we perform convergence analysis and numerical experiments that demonstrate the merits of our approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Preference disaggregation and statistical learning for multicriteria decision support: A review

Disaggregation methods have become popular in multicriteria decision aiding (MCDA) for eliciting preferential information and constructing decision models from decision examples. From a statistical point of view, data mining and machine learning are also involved with similar problems, mainly with regard to identifying patterns and extracting knowledge from data. Recent research has also focused on the introduction of specific domain knowledge in machine learning algorithms. Thus, the connections between disaggregation methods in MCDA and traditional machine learning tools are becoming stronger. In this paper the relationships between the two fields are explored. The differences and similarities between the two approaches are identified, and a review is given regarding the integration of the two fields.     

The role of fuzzy sets in decision sciences: Old techniques and new directions

We try to provide a tentative assessment of the role of fuzzy sets in decision analysis. We discuss membership functions, aggregation operations, linguistic variables, fuzzy intervals and the valued preference relations they induce. The importance of the notion of bipolarity and the potential of qualitative evaluation methods are also pointed out. We take a critical standpoint on the state-of-the-art, in order to highlight the actual achievements and question what is often considered debatable by decision scientists observing the fuzzy decision analysis literature.     

Likelihood estimation of consumer preferences in choice-based conjoint analysis

In marketing research the measurement of individual preferences and assessment of utility functions have long traditions. Conjoint analysis, and particularly choice-based conjoint analysis (CBC), is frequently employed for such measurement. The world today appears increasingly customer or user oriented wherefore research intensity in conjoint analysis is rapidly increasing in various fields, OR/MS being no exception. Although several optimization based approaches have been suggested since the introduction of the Hierarchical Bayes (HB) method for estimating CBC utility functions, recent comparisons indicate that challenging HB is hard. Based on likelihood maximization we propose a method called LM and compare its performance with HB using twelve field data sets. Performance comparisons are based on holdout validation, i.e. predictive performance. Average performance of LM indicates an improvement over HB and the difference is statistically significant. We also use simulation based data sets to compare the performance for parameter recovery. In terms of both predictive performance and RMSE a smaller number of questions in CBC appears to favor LM over HB.     

Revealed preference theory: An algorithmic outlook

Revealed preference theory is a domain within economics that studies rationalizability of behavior by (certain types of) utility functions. Given observed behavior in the form of choice data, testing whether certain conditions are satisfied gives rise to a variety of computational problems that can be analyzed using operations research techniques. In this survey, we provide an overview of these problems, their theoretical complexity, and available algorithms for tackling them. We focus on consumer choice settings, in particular individual choice, collective choice and stochastic choice settings.     

Families of relations modelling preferences under incomplete information

Let us consider a preferential information of type preference–indifference–incomparability (P, I, J), with additional information about differences in attractiveness between pairs of alternatives. The present paper offers a theoretical framework for the study of the “level of constraint” of this kind of partial preferential information. It suggests a number of structures as potential models being less demanding than the classical one in which differences in utilities can be used to represent the comparison of differences in attractiveness. The models are characterized in the more general context of families of non-complete preference structures, according to two different perspectives (called “semantico-numerical” and “matrix”). Both perspectives open the door to further practical applications connected with elicitation of the preferences of a decision maker.