Crisp dimension theory and valued preference relations

Abstract

Representation of binary preference relations in a real space where each coordinate suggests the existence of underlying criteria is a standard and indeed suggestive approach. Classical dimension theory addresses this problem, showing that whenever crisp preferences define a partial order set, it can be represented in a real space, and then we can search for a minimal representation. Valued preference relation being a much more complex structure, there is an absolute need for meaningful representations, being manageable by decision-makers. In this paper, we continue analyzing the concept of a generalized dimension function of valued preference relations, i.e. a mapping assigning a generalized dimension value to every α-cut of any given valued preference relation, as introduced in a previous paper. We should of course be expecting deep computational problems within this generalized dimension context, since they are already present in crisp dimension theory. In this paper, we present some properties of such a generalized dimension function, pointing out that our approach allows alternative representations depending on some underlying rationality core the decision-maker may change.

Keywords:

• Valued preference relations
• dimension theory
• preference representation
• generalized dimension function

Acknowledgements

This research has been partially supported by the Government of Spain, grants PB98-0825 and BFM2002-00281.

Notes

Daniel Gómez is Assistant Professor in the Department of Statistics and Operational Research III at the Faculty of Statistics, Complutense University of Madrid, Spain. He holds a Ph.D. in Mathematics from Complutense University since 2003. He is author of more than five research papers in refereed journals and more than five papers as book chapters. His research interests are in multicriteria decision-making, preference representation, aggregation, classification problems, fuzzy sets, graph theory and game theory.

Javier Montero is Associate Professor in the Department of Statistics and Operational Research I at the Faculty of Mathematics, Complutense University of Madrid, Spain. He holds a Ph.D. in Mathematics from Complutense University since 1982. He is author of more than 30 research papers in refereed journals such as the European Journal of Operational Research, Fuzzy Sets and Systems, Approximate Reasoning, Intelligent Systems, General Systems, Kybernetes, Mathware and Soft Computing, IEEE Transactions on Systems, Man and Cybernetics, and Uncertainty, Fuzziness and Knowledge-Based Systems, and more than 30 papers as book chapters. His research interests are in preference representation, multicriteria decision-making, group decision-making, system reliability theory and classification problems, mainly viewed as application of fuzzy sets theory.

Javier Yáñez is Associate Professor in the Department of Statistics and Operational Research I at the Faculty of Mathematics, Complutense University of Madrid, Spain. He holds a Ph.D. in Mathematics from Complutense University since 1983. His research papers have been published in refereed journals such as the European Journal of Operational Research, Fuzzy Sets and Systems, Neural Networks, Journal of Algorithms, etc. His research interests are in preference representation, network optimization and neural network.

Jacinto González-Pachón is Associate Professor in the School of Computer Science at the Technical University of Madrid. He holds a B.S. in Mathematics from the University of Extremadura and a Ph. D. in Computer Science from the Technical University of Madrid. His teaching and research specialities are in the field of preference modelling and rational choice, multi-criteria decision-making problems, group decision-making problems and uncertainty in decision analysis. He is the author of papers which have appeared in international refereed journals and monographs such as Computers and Operation Research, European Journal of Operational Research, Journal of the Operational Research Society, Lecture Note in Economics and Mathematical Systems, OMEGA, Theory and Decision, and others.

Vincenzo Cutello is Professor of Computer Science in the Department of Mathematics and Computer Science, University of Catania, Italy. He holds a Ph.D. in Mathematics from the University of Catania, and a Ph.D. in Computer Science from New York University. He is author of many research papers in refereed journals such as the European Journal of Operational Research, Fuzzy Sets and Systems, Approximate Reasoning, Intelligent Systems, IEEE Transactions on Fuzzy Systems, Uncertainty, Fuzziness and Knowledge-Based Systems, Automated Reasoning, Communication on Pure and Applied Mathematics, and many other papers as book chapters and international conference proceedings. His research interests are in evolutionary computation applied to combinatorial optimization, preference representation, multi-criteria decision-making, group decision-making, as application of fuzzy sets theory.