The Choquet integral is proven quite reasonable as an integral form with respect to monotone measures, where the credibility measure is a specific case with self-duality. The main objective of this paper is to propose the Choquet integral of measurable functions on the credibility space, which bridges the gap between the Choquet integral and credibility theory. First, the Choquet integrals for nonnegative functions with respect to the credibility measure are introduced, and their properties are investigated such as the monotonicity and translatability. Then, the symmetric Choquet integrals and translatable Choquet integrals of any measurable functions are developed through the use of the Choquet integrals of nonnegative functions. Finally, Choquet integrals on finite sets based on the credibility measure are presented to simplify the calculation procedures.
AngilellaS., GrecoS. and MatarazzoB., Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral, European Journal of Operational Research201(1) (2010), 277–288.
2.
ArcidiaconoS.G., CorrenteS. and GrecoS., As simple as possible but not simpler in Multiple Criteria Decision Aiding: the robust-stochastic level dependent Choquet integral approach, European Journal of Operational Research280(3) (2020), 988–1007.
3.
ChoquetG., Theory of capacities, Annales de l’institut Fourier5 (1954), 131–295.
4.
DongJ.Y., LinL.L., WangF. and WanS.P., Generalized Choquet integral operator of triangular Atanassov’s intu-itionistic fuzzy numbers and application to multi-attribute group decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems24(5) (2016), 647–683.
5.
DuW.S., Pan-integral on credibility space and its properties, Journal of Intelligent and Fuzzy Systems36(6) (2019), 5711–5719.
6.
GargH., A novel approach for analyzing the reliability of series-parallel system using credibility theory and different types of intuitionistic fuzzy numbers, Journal of the Brazilian Society of Mechanical Sciences and Engineering38(3) (2016), 1021–1035.
7.
GargH., AgarwalN. and TripathiA., Choquet integral-based information aggregation operators under the interval-valued intuitionistic fuzzy set and its applications to decision-making process, International Journal for Uncertainty Quantification7(3) (2017), 249–269.
8.
GargH. and Nancy, Multiple criteria decision making based on Frank Choquet Heronian mean operator for single-valued neutrosophic sets, Applied and Computational Mathematics18(2) (2019), 163–188.
9.
JalotaH., ThakurM. and MittalG., Modelling and constructing membership function for uncertain portfolio parameters: A credibilistic framework, Expert Systems with Applications71 (2017), 40–56.
10.
KlementE.P., MesiarR. and PapE., A universal integral as common frame for Choquet and Sugeno integral, IEEE Transactions on Fuzzy Systems18(1) (2010), 178–187.
11.
LiX., Credibilistic Programming: An Introduction to Models and Applications, Springer-Verlag, Berlin, 2013.
12.
LiangD., ZhangY. and CaoW., q-Rung orthopair fuzzy Choquet integral aggregation and its application in heterogeneous multicriteria two-sided matching decision making, International Journal of Intelligent Systems34(12) (2019), 3275–3301.
13.
LiuB. and LiuY.K., Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems10(4) (2002), 445–450.
14.
LiuB., Uncertainty Theory:An Introduction to its Axiomatic Foundations, Springer-Verlag, Berlin, 2004.
15.
LiuB., A survey of credibility theory, Fuzzy Optimization and Decision Making5(4) (2006), 387–408.
LiuB., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.
18.
LiuW.L., SongX.Q., LiuJ.B. and ZhangQ.Z., A new kind of triangular integrals based on-norms and-conorms, Fuzzy Information and Engineering4(1) (2012), 13–27.
19.
LiuW.L., SongX.Q., ZhangQ.Z. and ZhangS.B., (T) fuzzy integral of multi-dimensional function with respect to multivalued measure, Iranian Journal of Fuzzy Systems9(3) (2012), 111–126.
20.
LiuY.K., Credibility Measure Theory: A Modern Methodology of Handling Subjective Uncertainty, Science Press, Beijing, 2018.
21.
MaY., ChenH., SongW. and WangZ., Choquet distances and their applications in data classification, Journal of Intelligent and Fuzzy Systems33(1) (2017), 589–599.
22.
MurofushiT. and SugenoM., An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems29(2) (1989), 201–227.
23.
RalescuD. and AdamsG., The fuzzy integral, Journal of Mathematical Analysis and Applications75(2) (1980), 562–570.
24.
Suárez GarcíaF. and Gil ÁlvarezP., Two families of fuzzy integrals, Fuzzy Sets and Systems18(1) (1986), 67–81.
25.
SugenoM., Theory of fuzzy integrals and its applications, Ph.D. Dissertation, Tokyo Institute of Technology, 1974.
26.
WangZ., WangW. and KlirG.J., Pan-integrals with respect to imprecise probabilities, International Journal of General Systems25(3) (1996), 229–243.
27.
WangZ. and KlirG.J., PFB-integrals and PFA-integrals with respect to monotone set functions, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems5(2) (1997), 163–175.
28.
WangZ. and KlirG.J., Generalized Measure Theory, Springer, New York, 2009.
29.
WuC., WangS. and SongS., Generalized triangle norms and generalized fuzzy integrals, in: X. Liu (Ed.), Proceedings of Sino-Japan Symposium on Fuzzy Sets and Systems, International Academic Press, Beijing, 1990.
30.
WuX., ZhaoR. and TangW., Principal-agent problems based on credibility measure, IEEE Transactions on Fuzzy Systems23(4) (2015), 909–922.
31.
YangQ., The pan-integral on the fuzzy measure space, Fuzzy Mathematics5(3) (1985), 107–114.
32.
YangQ. and SongR., Further discussion of the pan-integral on fuzzy measure spaces, Fuzzy Mathematics5(4) (1985), 27–36.
33.
YangR. and OuyangR., Classification based on Choquet integral, Journal of Intelligent and Fuzzy Systems27(4) (2014), 1693–1702.
34.
ZadehL.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems1(1) (1978), 3–28.
35.
ZadehL.A., A theory of approximate reasoning, in: J. Hayes, D. Michie, L.I. Mikulich (Eds.), Machine Intelligence,Vol. 9, Halstead Press, NewYork, 1979, pp. 149–194.
36.
ZhangW. and ZhaoR., Generalized fuzzy measures and fuzzy integrals, Fuzzy Mathematics3(3) (1983), 1–8.
