On importance indices in multicriteria decision making. (English) Zbl 1431.91106
Summary: We address the problem of how to define an importance index in multicriteria decision problems, when a numerical representation of preferences is given. We make no restrictive assumption on the model, which could have discrete or continuous attributes, and in particular, it is not assumed that the model is monotonically increasing or decreasing with respect to the attributes. Our analysis first considers discrete models, which are seen to be equivalent to multichoice games. We propose essentially two importance indices, namely the signed importance index and the absolute importance index, both based on the average variation of the value of the model induced by a given attribute. We provide several axiomatizations for these importance indices, extend them to the continuous case, and finally illustrate them with examples: classical simple models and an example of discomfort evaluation based on real data.
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| [1] | Banzhaf, J., Weighted voting doesn’t work: A mathematical analysis, Rutgers Law Review, 19, 317-343 (1965) |
| [2] | Choquet, G., Theory of capacities, Annales de l’institut Fourier, 5, 131-295 (1953) · Zbl 0064.35101 |
| [3] | Crama, Y.; Hammer, P., Boolean functions. Number 142 in encyclopedia of mathematics and its applications (2011), Cambridge University Press |
| [4] | Datta, A.; Sen, S.; Zick, Y., Algorithmic transparency via quantitative input influence: Theory and experiments with learning systems, Proceedings of the IEEE symposium on security and privacy (SP), 598-617 (2016) |
| [5] | Denguir, A. (2014). Modèle de performance agrégée et raisonnement approché pour l’optimisation de la consommation énergétique et du confort dans les bâtiments. Montpellier 2. PhD thesis; Denguir, A. (2014). Modèle de performance agrégée et raisonnement approché pour l’optimisation de la consommation énergétique et du confort dans les bâtiments. Montpellier 2. PhD thesis |
| [6] | Fanger, P., Thermal comfort: Analysis and applications in environmental engineering (1970), Danish Technical Press: Danish Technical Press Copenhagen |
| [7] | Fishburn, P., Interdependence and additivity in multivariate, unidimensional expected utility theory, International Economic Review, 8, 335-342 (1967) · Zbl 0153.49302 |
| [8] | Grabisch, M., The application of fuzzy integrals in multicriteria decision making, European Journal of Operational Research, 89, 445-456 (1996) · Zbl 0916.90164 |
| [9] | Grabisch, M.; Labreuche, C., Capacities on lattices and \(k\)-ary capacities, Proceedings of the International conference of the euro society for fuzzy logic and technology (EUSFLAT), 304-307 (2003), Zittau, Germany |
| [10] | Grabisch, M.; Labreuche, C., Bipolarization of posets and natural interpolation, Journal of Mathematical Analysis and Application, 343, 1080-1097 (2008) · Zbl 1137.91316 |
| [11] | Grabisch, M.; Labreuche, C., A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid, Annals of Operations Research, 175, 247-286 (2010) · Zbl 1185.90118 |
| [12] | Grabisch, M.; Labreuche, C., Fuzzy measures and integrals in MCDA, (Greco, S.; Ehrgott, M.; Figueira, J. R., Multiple criteria decision analysis (2016), Springer), 553-603 |
| [13] | Grabisch, M.; Labreuche, C., Monotone decomposition of 2-additive generalized additive independence models, Mathematical Social Sciences, 92, 64-73 (2018) · Zbl 1397.91132 |
| [14] | Grabisch, M.; Lange, F., Games on lattices, multichoice games and the Shapley value: a new approach, Mathematical Methods of Operations Research, 65, 1, 153-167 (2007) · Zbl 1154.91319 |
| [15] | Grabisch, M.; Marichal, J. L.; Mesiar, R.; Pap, E., Aggregation functions, Encyclopedia of mathematics and its applications: Vol. 127 (2009), Cambridge University Press · Zbl 1196.00002 |
| [16] | Grabisch, M.; Marichal, J. L.; Roubens, M., Equivalent representations of set functions, Mathematics of Operations Research, 25, 2, 157-178 (2000) · Zbl 0982.91009 |
| [17] | Haldi, F.; Robinson, D., Modelling occupants personal characteristics for thermal comfort prediction, International Journal of Biometeorology, 55, 5, 681-694 (2011) |
| [18] | Hammer, P. L.; Rudeanu, S., Boolean methods in operations research and related areas, Econometrics and operations research (1968), Springer · Zbl 0155.28001 |
| [19] | Hsiao, C. R.; Raghavan, T. E.S., Shapley value for multi-choice cooperative games, I, Games and Economic Behavior, 5, 240-256 (1993) · Zbl 0795.90092 |
| [20] | Klijn, F.; Slikker, M.; Zarzuelo, J. M., Characterizations of a multi-choice value, International Journal of Game Theory, 28, 521-532 (1999) · Zbl 0942.91007 |
| [21] | Labreuche, C.; Fossier, S., Explaining multi-criteria decision aiding models with an extended Shapley value, Proceedings of the International joint conference on artificial intelligence (IJCAI’2018), 331-339 (2018) |
| [22] | Labreuche, C.; Grabisch, M., Using multiple reference levels in multi-criteria decision aid: the generalized-additive independence model and the Choquet integral approaches, European Journal of Operational Research, 267, 2, 598-611 (2018) · Zbl 1403.91118 |
| [23] | Lundberg, S.; Lee, S. I., A unified approach to interpreting model predictions, Proceedings of the 31st conference on neural information processing systems (NIPS 2017), 4765-4774 (2017), Curran Associates, Inc: Curran Associates, Inc Long Beach, CA, USA. |
| [24] | Marichal, J. L. (1998). Aggregation operators for multicriteria decision aid. Ph.d. thesis University of Liège.; Marichal, J. L. (1998). Aggregation operators for multicriteria decision aid. Ph.d. thesis University of Liège. |
| [25] | Murofushi, T. (1992). A technique for reading fuzzy measures (I): the Shapley value with respect to a fuzzy measure. 2nd Fuzzy Workshop, (pp. 39-48). Nagaoka, Japan, October In Japanese.; Murofushi, T. (1992). A technique for reading fuzzy measures (I): the Shapley value with respect to a fuzzy measure. 2nd Fuzzy Workshop, (pp. 39-48). Nagaoka, Japan, October In Japanese. |
| [26] | van den Nouweland, A.; Tijs, S.; Potters, J.; Zarzuelo, J., Cores and related solution concepts for multi-choice games, Zeitschrift für Operations Research, 41, 289-311 (1995) · Zbl 0837.90133 |
| [27] | Owen, G., Multilinear extensions of games, Management Science, 18, 5, 64-79 (1972) · Zbl 0239.90049 |
| [28] | Peleg, B.; Sudhölter, P., Introduction to the theory of cooperative games (2003), Kluwer Academic Publisher |
| [29] | Peters, H., Game Theory: A Multilevel Approach (2008), Springer · Zbl 1147.91001 |
| [30] | Peters, H.; Zank, H., The egalitarian solution for multichoice games, Annals of Operations Research, 137, 399-409 (2005) · Zbl 1138.91328 |
| [31] | Ridaoui, M.; Grabisch, M.; Labreuche, C., An alternative view of importance indices for multichoice games, Proceedings of the 5th International Conference on Algorithmic Decision Theory 2017, Luxembourg, October 2017, 81-92 (2017) · Zbl 1398.91047 |
| [32] | Ridaoui, M.; Grabisch, M.; Labreuche, C., Axiomatization of an importance index for generalized additive independence models, Proceedings of the 14th European conference, ECSQARU 2017, Lugano, Switzerland, july 2017, 340-350 (2017) · Zbl 1493.91040 |
| [33] | Rota, G. C., On the foundations of combinatorial theory I. Theory of Möbius functions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2, 4, 340-368 (1964) · Zbl 0121.02406 |
| [34] | Shapley, L. S., A value for \(n\)-person games, (Kuhn, H. W.; Tucker, A. W., Contributions to the theory of games, vol. II, number 28 in annals of mathematics studies (1953), Princeton University Press), 307-317 · Zbl 0050.14404 |
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