Abstract
An axiomatization of the Choquet integral is proposed in the context of multiple criteria decision making without any commensurability assumption. The most essential axiom—named Commensurability Through Interaction—states that the importance of an attribute i takes only one or two values when a second attribute k varies. When the importance takes two values, the point of discontinuity is exactly the value on the attribute k that is commensurate to the fixed value on attribute i. If the weight of criterion i does not depend on criterion k, for any value of the other criteria than i and k, then criteria i and k are independent. Applying this construction to any pair i, k of criteria, one obtains a partition of the set of criteria. In each block, the criteria interact one with another, and it is thus possible to construct vectors of values on the attributes that are commensurate. There is complete independence between the criteria of any two blocks in this partition. Hence one cannot ensure commensurability between two blocks in the partition. But this is not a problem since the Choquet integral is additive between subsets of criteria that are independent.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Thermal comfort is a subjective judgment depending on temperature, humidity and air velocity, among other parameters. The Predicted Percentage Dissatisfied (PPD) has been defined, based “thermal” exchange equations between the body and the air (Fanger 1970)
Commensurability is based on the concept of scales. A scale \(L_i=\{\varphi _i(x_i) \, : \, x_i\in X_i\}\) ordered by \(\ge \) is constructed on each criterion i. The marginal utility functions \(\varphi _i\) are then said to be commensurate if all scales \(L_i\)’s can be represented by the same scale \(L_u\) (Dubois et al. 2003). This means that one shall be able to compare the values of any two scales \(L_i\) and \(L_j\), with a preference relation \(\succsim _{L_i \times L_j}\) defined on \(L_i \times L_j\). A simple way to fulfill this assumption is to ensure that all scales are identical: \(L_1=\cdots =L_n=L_u\), and \(a \sim _{L_i \times L_j} a\) (value a on scale \(L_i\) is indifferent to the same value a on scale \(L_j\)) for every \(a\in L_u\) and \(i,j\in N\). In other words, if \(\varphi _i(x_i)=\varphi _j(x_j)\), then value \(x_i\) on attribute \(X_i\) has the same satisfaction/attractiveness as value \(x_j\) on attribute \(X_j\).
As U is piecewise \(C^1\), we can extend its partial derivatives in the closure of each block in the partition. Two partial derivatives of U are defined at \(x_k = \mathrm {Co}_{i\rightarrow k}(x_i)\): the left and right derivative.
This means that for any \(x_i\), there exists \(C>0\) such that \( \left| {\mathrm {Co}_{i\rightarrow k}(y_i) - \mathrm {Co}_{i\rightarrow k}(x_i)} \right| \le C \, \left| {y_i-x_i} \right| \) for any \(y_i\) sufficiently near to \(x_i\).
References
Angilella, S., Corrente, S., & Greco, S. (2015). Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem. European Journal of Operational Research, 240, 172–182.
Angilella, S., Greco, S., Lamantia, F., & Matarazzo, B. (2004). Assessing non-additive utility for multicriteria decision aid. European Journal of Operational Research, 158, 734–744.
Bana e Costa, C. A., De Corte, J., & Vansnick, J.-C. (2012). MACBETH. International Journal of Information Technology and Decision Making, 11, 359–387.
Barbaresco, F., Deltour, J., Desodt, G., Durand, B., Guenais, T., & Labreuche, C. (2009). Intelligent M3R radar time resources management: Advanced cognition, agility & autonomy capabilities. In International radar conference, Bordeaux, France.
Benvenuti, P., Mesiar, R., & Vivona, D. (2002). Monotone set functions-based integrals. In E. Pap (Ed.), Handbook of measure theory (pp. 1329–1379). Elsevier Science.
Choquet, G. (1953). Theory of capacities. Annales de l’Institut Fourier, 5, 131–295.
Couceiro, M., & Marichal, J. (2011a). Axiomatizations of Lovász extensions of pseudo-Boolean functions. Fuzzy Sets and Systems, 181, 28–38.
Couceiro, M., & Marichal, J. (2011b). Axiomatizations of quasi-Lovász extensions of pseudo-Boolean functions. Aequationes Mathematicae, 82, 213–231.
Denguir, A. (2014). Modèle de performance agrégée et raisonnement approché pour l’optimisation de la consommation énergétique et du confort dans les bâtiments. Ph.D. thesis, Ecole des Mines d’Ales.
Diestel, R. (2005). Graph theory. New York: Springer.
Dubois, D., Fargier, H., & Perny, P. (2003). Qualitative decision theory with preference relations and comparison uncertainty: An axiomatic approach. Artificial Intelligence, 148, 219–260.
Fanger, P. (Ed.). (1970). Thermal comfort: Analysis and applications in environnmental engineering. Copenhagen: Danish Technical Press.
Fishburn, P. (1967). Interdependence and additivity in multivariate, unidimensional expected utility theory. International Economic Review, 8, 335–342.
Goujon, B., & Labreuche, C. (2013). Holistic preference learning with the Choquet integral. In International conference of the European society for fuzzy logic and technology (EUSFLAT), Minano, Italy.
Grabisch, M. (1996). The application of fuzzy integrals in multicriteria decision making. European Journal of Operational Research, 89, 445–456.
Grabisch, M., & Labreuche, C. (2010). A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Annals of Operation Research, 175, 247–286.
Grabisch, M., & Labreuche, C. (2016). Fuzzy measures and integrals in MCDA. In J. Figueira, S. Greco, M. E., (Eds.), Multiple criteria decision analysis—State of the art surveys (pp. 553–603). Springer.
Grabisch, M., Murofushi, T., & Sugeno, M. (2000). Fuzzy measures and integrals. Theory and applications (edited volume). Studies in fuzziness. Heidelberg: Physica Verlag.
Krantz, D., Luce, R., Suppes, P., & Tversky, A. (1971). Foundations of measurement: Additive and polynomial representations (Vol. 1). Cambridge: Academic Press.
Labreuche, C. (2011). Construction of a Choquet integral and the value functions without any commensurateness assumption in multi-criteria decision making. In International conference of the European society for fuzzy logic and technology (EUSFLAT), Aix Les Bains, France.
Labreuche, C. (2012). An axiomatization of the Choquet integral and its utility functions without any commensurateness assumption. In International Conference on information processing and management of uncertainty in knowledge-based systems (IPMU), Catania, Italy.
Labreuche, C., & Grabisch, M. (2003). The Choquet integral for the aggregation of interval scales in multicriteria decision making. Fuzzy Sets & Systems, 137, 11–26.
Labreuche, C., & Grabisch, M. (2006). Generalized Choquet-like aggregation functions for handling ratio scales. European Journal of Operational Research, 172, 931–955.
Labreuche, C., & Grabisch, M. (2007). The representation of conditional relative importance between criteria. Annals of Operation Research, 154, 93–122.
Larsson, S.-O. (2001). Multi-criteria decision support in a road planning process—A Swedish case. In Euro working group on multiple criteria decision aiding meeting (pp. 167–170).
Marichal, J.-L. (1998). Aggregation operators for multicriteria decision aid. Ph.D. thesis, University of Liège.
Marichal, J.-L. (2000). An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Transactions on Fuzzy Systems, 8(6), 800–807.
Murofushi, T., & Soneda, S. (1993). Techniques for reading fuzzy measures (III): Interaction index. In 9th fuzzy system symposium (pp. 693–696). Sapporo, Japan.
Peleg, B., & Sudholter, P. (2003). Introduction to the theory of cooperative games. Boston: Kluwer Academic Publisher.
Pignon, J., Labreuche, C., & Ponthoreau, P. (2008). Combining experimentation and multi-criteria decision aid: A way to improve nec systems of systems architecting. In NATO SCI panel symposium on agility, resilience and control in NEC, Amsterdam, The Netherland.
Savage, L. J. (1972). The foundations of statistics (2nd ed.). New York: Dover.
Schmeidler, D. (1986). Integral representation without additivity. Proceedings of the American Mathematical Society, 97(2), 255–261.
Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57(3), 571–587.
Shapley, L. S. (1953). A value for $n$-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the theory of games, number 28 in annals of mathematics studies (Vol. 2, pp. 307–317). Princeton: Princeton University Press.
Tehrani, A. F., Labreuche, C., & Hüllermeier, E. (2014). Choquistic utilitaristic regression. In Decision aid to preference learning (DA2PL) workshop, Chatenay-Malabry, France.
Timonin, M. (2014). Conjoint axiomatization of the Choquet integral for two-dimensional heterogeneous product sets. In Decision aid to preference learning (DA2PL) workshop. Chatenay-Malabry, France.
Von Neuman, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University Press.
Acknowledgements
The author would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Labreuche, C. An axiomatization of the Choquet integral in the context of multiple criteria decision making without any commensurability assumption. Ann Oper Res 271, 701–735 (2018). https://doi.org/10.1007/s10479-018-3046-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-018-3046-1