Pan-integral is an important kind of nonlinear integral, and credibility measure is a monotone measure with self-duality. In this paper, we define the pan-integral on credibility space, a new type of integral construction, whose basic properties are discussed in detail. The relationship between the pan-integral and H-fuzzy integral on a special pan-credibility space is established as well. Moreover, some monotone convergence theorems for a sequence of nonnegative measurable functions are investigated.
X.Deng, J.F.Zhao and Z.F.Li, Sensitivity analysis of the fuzzy mean-entropy portfolio model with transaction costs based on credibility theory, International Journal of Fuzzy Systems20(1) (2018), 209–218.
2.
S.Doria, B.Dutta and R.Mesiar, Integral representation of coherent upper conditional prevision with respect to its associated Hausdorff outer measure: A comparison among the Choquet integral, the pan-integral and the concave integral, International Journal of General Systems47(6) (2018), 569–592.
3.
B.Q.Hu, W.C.Ip and H.Wong, Fuzzy integral on credibility measure, Fuzzy Information and Engineering2(4) (2010), 389–397.
4.
H.Jalota, M.Thakur and G.Mittal, Modelling and constructing membership function for uncertain portfolio parameters: A credibilistic framework, Expert Systems With Applications71 (2017), 40–56.
5.
Z.Q.Jiang, P.Sun, C.M.Ji and J.Z.Zhou, Credibility theory based dynamic control bound optimization for reservoir flood limited water level, Journal of Hydrology529(3) (2015), 928–939.
6.
G.Li, The relationships between Choquet integral, panintegral, upper integral and lower integral, International Journal of Intelligent Systems26(5) (2011), 464–473.
7.
X.Li, Credibilistic Programming: An Introduction to Models and Applications, Springer-Verlag, Berlin, 2013.
8.
B.Liu and Y.K.Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems10(4) (2002), 445–450.
9.
B.Liu, Uncertainty Theory: An Introduction to its Axiomatic Foundations, Springer-Verlag, Berlin, 2004.
10.
B.Liu, A survey of credibility theory, Fuzzy Optimization and Decision Making5(4) (2006), 387–408.
B.Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.
13.
W.L.Liu, X.Q.Song, Q.Z.Zhang and S.B.Zhang, (T) fuzzy integral of multi-dimensional function with respect to multivalued measure, Iranian Journal of Fuzzy Systems9(3) (2012), 111–126.
14.
W.L.Liu, X.Q.Song, J.B.Liu and Q.Z.Zhang, A new kind of triangular integrals based on T-norms and T-conorms, Fuzzy Information and Engineering4(1) (2012), 13–27.
15.
Y.Ouyang, J.Li and R.Mesiar, Relationship between the concave integrals and the pan-integrals on finite spaces, Journal of Mathematical Analysis and Applications424(2) (2015), 975–987.
16.
Y.Ouyang, J.Li and R.Mesiar, On the equivalence of the Choquet, pan-and concave integrals on finite spaces, Journal of Mathematical Analysis and Applications456(1) (2017), 151–162.
17.
Y.Ouyang, J.Li and R.Mesiar, A sufficient condition of equivalence of the Choquet and the pan-integral, Fuzzy Sets and Systems355 (2019), 100–105.
18.
D.Ralescu and G.Adams, The fuzzy integral, Journal of Mathematical Analysis and Applications75(2) (1980), 562–570.
19.
F.Suárez García and P.Gil Álvarez, Two families of fuzzy integrals, Fuzzy Sets and Systems18(1) (1986), 67–81.
20.
M.Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Dissertation, Tokyo Institute of Technology, 1974.
21.
C.Tan, Z.Feng, C.Li and W.Yi, Optimal Bayesian equilibrium for n-person credibilistic non-cooperative game with risk aversion, Journal of Intelligent and Fuzzy Systems33(2) (2017), 741–751.
22.
Z.Wang, W.Wang and G.J.Klir, Pan-integrals with respect to imprecise probabilities, International Journal of General Systems25(3) (1996), 229–243.
23.
Z.Wang and G.J.Klir, PFB-integrals and PFA-integrals with respect to monotone set functions, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems5(2) (1997), 163–175.
24.
Z.Wang and G.J.Klir, Generalized Measure Theory, Springer, New York, 2009.
25.
C.Wu, S.Wang and S.Song, 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.
26.
Q.Yang, The pan-integral on the fuzzy measure space, Fuzzy Mathematics5(3) (1985), 107–114.
27.
Q.Yang and R.Song, Further discussion of the pan-integral on fuzzy measure spaces, Fuzzy Mathematics5(4) (1985), 27–36.
28.
L.A.Zadeh, Fuzzy sets, Information and Control8(3) (1965), 338–353.
29.
L.A.Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems1(1) (1978), 3–28.
30.
L.A.ZadehA theory of approximate reasoning, in: J.Hayes, D.Michie and L.I.Mikulich (Eds.), Machine Intelligence, Vol. 9, Halstead Press, New York, 1979, pp. 149–194.
31.
Y.Zhao, T.Yan and Y.Ouyang, Several inequalities for the pan-integral, Information Sciences372 (2016), 625–633.
32.
W.Zhang and R.Zhao, Generalized fuzzy measures and fuzzy integrals, Fuzzy Mathematics3(3) (1983), 1–8.
