In the literature [40], the authors proposed the sum interaction index and discussed its basic mathematical properties. And by some comparison analyses, the authors demonstrated that the sum interaction index can be taken as an alternative of the probabilistic interaction index to measure the interaction phenomenon of the multiple decision criteria. In this paper, we further investigate the properties of the sum interaction index with respect to some particular families of monotone measures, such as the λ-monotone measures, the possibility monotone measures, the k-additive monotone measures, the p-symmetric monotone measures, and the k-tolerant and k-intolerant monotone measures. Some illustrative examples are also given to show the comparison analyses between the sum interaction index and the probabilistic interaction index.
AngilellaS., GrecoS., LamantiaF. and MatarazzoB., Assessing non-additive utility for multicriteria decision aid, European Journal of Operational Research158 (2004), 734–744.
2.
AngilellaS., GrecoS. and MatarazzoB., Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral, European Journal of Operational Research201 (2010), 277–288.
3.
BanzhafJ.F., Weighted voting doesn’t work: A mathematical analysis, Rutgers Law Review19 (1965), 317–343.
4.
BeliakovG., Construction of aggregation functions from data using linear programming, Fuzzy Sets and Systems160 (2009), 65–75.
5.
BeliakovG., JamesS. and LiG., Learning Choquet-integral-based metrics for semisupervised clustering, Fuzzy Systems, IEEE Transactions on19 (2011), 562–574.
6.
BerrahL., MaurisG. and MontmainJ., Monitoring the improvement of an overall industrial performance based on a Choquet integral aggregation, Omega36 (2008), 340–351.
7.
Bustince SolaH., GalarM., BedregalB., KolesárováA. and MesiarR., A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy sets applications, Fuzzy Systems, IEEE Transactions on22 (2013), 1150–1162.
8.
ChateauneufA. and JaffrayJ.-Y., Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion, Mathematical social sciences17 (1989), 263–283.
ChoquetG., Theory of capacities, Annales de l’institut Fourier5 (1953), 131–292.
11.
DennebergD., Non-additive measure and integral, Kluwer, Dordrecht, 1994.
12.
FujimotoK., KojadinovicI. and MarichalJ.-L., Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices, Games and Economic Behavior55 (2006), 72–99.
13.
GrabischM., The application of fuzzy integrals in multicriteria decision making, European Journal of Operational Research89 (1996), 445–456.
14.
GrabischM., Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems69 (1995), 279–298.
15.
GrabischM., A graphical interpretation of the choquet integral, Fuzzy Systems, IEEE Transactions on8 (2000), 627–631.
16.
GrabischM., K-order additive discrete fuzzy measures and their representation, Fuzzy Sets and Systems92 (1997), 167–189.
17.
GrabischM., KojadinovicI. and MeyerP., A review of methods for capacity identification in Choquet integral based multi-attribute utility theory, European Journal of Operational Research186 (2008), 766–785.
18.
GrabischM. and LabreucheC., A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid, 4or: A Quarterly Journal of Operations Research6 (2008), 1–44.
19.
GrabischM., MarichalJ.-L. and RoubensM., Equivalent representations of set functions, Mathematics of Operations Research25 (2000), 157–178.
20.
GrabischM. and RoubensM., An axiomatic approach to the concept of interaction among players in cooperative games, International Journal of Game Theory28 (1999), 547–565.
21.
GrabischM. and RoubensM., Probabilistic interactions among players of a cooperative game, in: Beliefs, Interactions and Preferences in Decision Making, Kluwer Academic, Dordrecht, 1999, pp. 205–216.
22.
GrabischM., SugenoM. and MurofushiT., Fuzzy measures and integrals: Theory and applications, Springer-Verlag, New York, 2000.
23.
GrecoS., MousseauV. and SłowińskiR., Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions, European Journal of Operational Research191 (2008), 416–436.
24.
KojadinovicI., An axiomatic approach to the measurement of the amount of interaction among criteria or players, Fuzzy Sets and Systems152 (2005), 417–435.
25.
KojadinovicI., Minimum variance capacity identification, European Journal of Operational Research177 (2007), 498–514.
26.
LiJ. and MesiarR., Lusin’s theorem on monotone measure spaces, Fuzzy Sets and Systems175 (2011), 75–86.
27.
LiginlalD. and OwT.T., On policy capturing with fuzzy measures, European Journal of Operational Research167 (2005), 461–474.
28.
MarichalJ.-L., An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria, Fuzzy Systems, IEEE Transactions on8 (2000), 800–807.
29.
MarichalJ.-L., k-intolerant capacities and Choquet integrals, European Journal of Operational Research177 (2007), 1453–1468.
30.
MarichalJ.-L., Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral, European Journal of Operational Research155 (2004), 771–791.
31.
MarichalJ.-L. and RoubensM., The chaining interaction index among players in cooperative games, in: Advances in Decision Analysis, Kluwer Academic, Dordrecht, 1999, pp. 69–85.
32.
MirandaP., GrabischM. and GilP., p-Symmetric fuzzy measures, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems10 (2002), 105–123.
33.
NätherW. and WälderK., Applying fuzzy measures for Considering interaction effects in root dispersal models, Fuzzy Sets and Systems158 (2007), 572–582.
34.
PapE., Null-additive set functions, Kluwer, Boston, 1995.
35.
ShapleyL.S., A value for n-person games, in: KuhnH.W. and TuckerA.W. (Eds.), Contributions to the Theory of Games II, Annals of Mathematics Studies, 1953, Princeton University Press, princeton, pp. 307–317.
36.
SugenoM., Theory of fuzzy integrals and its applications, in, Institute of Technology, Tokyo, Japan, 1974.
37.
WangZ. and KlirG.J., Generalized measure theory, Springer Science & Business Media, New York, 2010.
38.
WeberR.J., Probabilistic values for games in: The Shaley Value, Essays in Honor of Lloyd S. Shapley, 1988, pp. 101–119.
39.
WuJ.-Z., ChenF., NieC. and ZhangQ., Intuitionistic fuzzy-valued Choquet integral and its application in multicriteria decision making, Information Sciences222 (2013), 509–527.
40.
WuJ.-Z., YuL.-P., LiG., JinJ. and DuB., Using the monotone measure sum to enrich the measurement of the interaction of multiple decision criteria, Journal of Intelligent and fuzzy systems (2015). DOI: 10.3233/IFS-151787.
41.
WuJ.-Z. and ZhangQ., 2-order additive fuzzy measure identification method based on diamond pairwise comparison and maximum entropy principle, Fuzzy Optimization and Decision Making9 (2010), 435–453.
42.
WuJ.-Z., ZhangQ., DuQ. and DongZ., Compromise principle based methods of identifying capacities in the framework of multicriteria decision analysis, Fuzzy Sets and Systems246 (2014), 91–106.
43.
YangR., WangZ., HengP.-A. and LeungK.-S., Classification of heterogeneous fuzzy data by Choquet integral with fuzzy-valued integrand, Fuzzy Systems, IEEE Transactions on15 (2007), 931–942.
44.
ZadehL.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems1 (1978), 3–28.
