Abstract
In this paper, we elaborate on two important developments in the realm of multi-criteria decision aid, which have attracted increasing attention in the recent past: first, the idea of leveraging methods from preference learning for the data-driven (instead of human-centric) construction of decision models, and second, the use of hierarchical instead of “flat” decision models. In particular, we show the advantage of combining the two, that is, of learning hierarchical MCDA models from suitable training data. This approach is illustrated by means of a concrete example, namely the learning of tree-structured combinations of the Choquet integral as a versatile aggregation function.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It shall be noted that piecewise affine functions are necessarily continuous.
References
Allais M (1953) Le comportement de l’homme rationnel devant le risque, critique des postulats et axiomes de l’école américaine. Econometrica 21:503–546
Angilella S, Corrente S, Greco S, Slowinski R (2013) Multiple criteria hierarchy process for the Choquet integral. In: 7th International conference on evolutionary multi-criterion optimization, Shaffield, pp 475–489
Angilella S, Corrente S, Greco S, Slowinski R (2016) Robust ordinal regression and stochastic multiobjective acceptability analysis in multiple criteria hierarchy process for the Choquet integral preference model. Omega 63:154–169
Benabbou N, Perny P, Viappiani P (2017) Incremental elicitation of Choquet capacities for multicriteria choice, ranking and sorting problems. Artif Intell 246:152–180. https://doi.org/10.1016/j.artint.2017.02.001
Bengio Y (2009) Learning deep architectures for AI. Found Trends Mach Learn 2(1):1–127
Bresson R, Cohen J, Hüllermeier E, Labreuche C, Sebag M (2020) Neural representation and learning of hierarchical 2-additive Choquet integrals. In: Proc. 28th int. joint conf. on artificial intelligence (IJCAI), Yokohoma, pp 1984–1991
Bresson R, Cohen J, Hüllermeier E, Labreuche C, Sebag M (2021) On the identifiability of hierarchical decision models. In: Proceedings of the 18th international conference on principles of knowledge representation and reasoning (KR 2021) Hanoi, pp 151–161. https://doi.org/10.24963/kr.2021/15
Choquet G (1953) Theory of capacities. Ann Inst Fourier 5:131–295
Corrente S, Greco S, Slowiński R (2012) Multiple criteria hierarchy process in robust ordinal regression. Decis Support Syst 53:660–674
Ellsberg D (1961) Risk, ambiguity, and the Savage axioms. Q J Econom 75:643–669
Fürnkranz J, Hüllermeier E (eds) (2010) Preference learning. Springer, Berlin
Grabisch M, Labreuche C (2010) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Ann Oper Res 175:247–286
Grabisch M, Roubens M (1999) An axiomatic approach to the concept of interaction among players in cooperative games. Int J Game Theory 28:547–565
Grabisch M (1997) K-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst 92:167–189
Ioffe S, Szegedy C (2015) Batch normalization: accelerating deep network training by reducing internal covariate shift. In: Int. conf. on machine learning (ICML). http://arxiv.org/abs/1502.03167
Jacquet-Lagrèze E, Siskos Y (1982) Assessing a set of additive utility functions for multicriteria decision making: the UTA method. Eur J Oper Res 10:151–164
Keeney RL, Raiffa H (1976) Decision with multiple objectives. Wiley, New York
Krantz D, Luce R, Suppes P, Tversky A (1971) Foundations of measurement, vol 1: additive and polynomial representations. Academic, New York
Labreuche C (2018) An axiomatization of the Choquet integral and its utility functions without any commensurability assumption. Ann Oper Res 271(2), 701–735
Marichal JL (2000) An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Trans Fuzzy Syst 8(6):800–807
Murofushi T, Soneda S (1993) Techniques for reading fuzzy measures (III): interaction index. In: 9th Fuzzy system symposium Sapporo, May, pp 693–696
Ovchinnikov S (2002) Max-min representation of piecewise linear functions. Beiträge zur Algebra Geom 43(1):297–302
Perny P, Viappiani P, Boukhatem A (2016) Incremental preference elicitation for decision making under risk with the rank-dependent utility model. In: Proceedings of the conference on uncertainty in artificial intelligence (UAI-16), New York City, NY, June 2016
Roy B (1985) Méthodologie multicritère d’aide à la décision. Economica, Paris
Roy B (1999) Decision aiding today: What should we expect? In: Gal T, Stewart T, Hanne T (eds) Multicriteria decision making: advances in MCDM models, algorithms, theory and applications. Kluwer Academic, Boston, pp 1.1–1.35
Schmeidler D (1989) Subjective probability and expected utility without additivity. Econometrica 57(3):571–587
Senge R, Hüllermeier E (2011) Top-down induction of fuzzy pattern trees. IEEE Trans Fuzzy Syst 19(2):241–252
Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games, vol II, No. 28 in Annals of mathematics studies. Princeton University Press, Princeton, pp 307–317
Sobrie O, Gillis N, Mousseau V, Pirlot M (2018) UTA-poly and UTA-splines: additive value functions with polynomial marginals. Eur J Oper Res 264(2):405–418
Sugeno M (1974) Theory of fuzzy integrals and its applications. Ph.D. thesis, Tokyo Institute of Technology
Tehrani AF, Cheng W, Dembczynski K, Hüllermeier E (2012) Learning monotone nonlinear models using the Choquet integral. Mach Learn 89:183–211
Tehrani AF, Cheng W, Hüllermeier E (2012) Preference learning using the Choquet integral: the case of multipartite ranking. IEEE Trans Fuzzy Syst 20(6):1102–1113
Tehrani AF, Hüllermeier E (2013) Ordinal Choquistic regression. In: Montero J, Pasi G, Ciucci D (eds) Proc. EUSFLAT, 8th int. conf. of the European society for fuzzy logic and technology. Atlantis Press, Milano
Tehrani AF, Labreuche C, Hüllermeier E (2014) Choquistic utilitaristic regression. In: Workshop DA2PL, from decision aid to preference learning, Chatenay-Malabry, Nov 2014
Timonin M (2016) Conjoint axiomatization of the Choquet integral for heterogeneous product sets. CoRR abs/1603.08142. http://arxiv.org/abs/1603.08142
Wakker P (1991) Additive representations on rank-ordered sets I: the algebraic approach. J Math Psychol 35:501–531
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hüllermeier, E., Labreuche, C. (2022). Modeling and Learning of Hierarchical Decision Models: The Case of the Choquet Integral. In: Greco, S., Mousseau, V., Stefanowski, J., Zopounidis, C. (eds) Intelligent Decision Support Systems . Multiple Criteria Decision Making. Springer, Cham. https://doi.org/10.1007/978-3-030-96318-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-96318-7_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-96317-0
Online ISBN: 978-3-030-96318-7
eBook Packages: Business and ManagementBusiness and Management (R0)