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- 2017
集值函数关于模糊测度Choquet积分的表示和积分原函数性质
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Abstract:
摘要： 研究了集值函数关于模糊测度Choquet积分的分析性质: 讨论了集值函数Choquet积分的计算方法, 给出了集值函数Choquet积分的表示定理和Radon-Nikodym性质, 并且对集值函数Choquet积分的原函数进行了刻划。最后, 对集值函数关于模糊测度Choquet积分定义进行了改进, 提出了集值函数 “上方函数” 和 “下方函数” 概念, 实现了对集值函数关于模糊测度的Choquet积分的控制。
Abstract: The analytic properties of the Choquet integral of set-valued functions with respect to fuzzy measures are discussed, such as the characteristics of the primitive, representation of integral, differentiability of the primitive, and so on. Firstly, based on the previous results, the calculation of Choquet integral of set-valued function is investigated, and a representation theorem of Choquet integral for set-valued function is obtained as a Radon-Nikodym property in some sense. In addition, the characteristics of the primitive of the Choquet integral for set valued functions are given. Finally, the definition of the Choquet integral of set-valued functions with respect to fuzzy measures is improved, and the concepts of the above functions and below function of the set-valued functions are proposed, which achieved the domination of the Choquet integral of set-valued functions with respect to fuzzy measures
| [1] | SUGENO M. A note on derivatives of functions with respect to fuzzy measures[J]. Fuzzy Sets and Systems, 2013, 222(1):1-17. |
| [2] | JANG L C, KIL B M, KWON J S. Some properties of Choquet integrals of set-valued functions[J]. Fuzzy Sets and Systems, 1997, 91(1):95-98. |
| [3] | HUANG Yan, WU Congxin. Real-valued Choquet integral for set-valued mappings[J]. International Journal of Approximate Reasoning, 2014, 55(2):683-688. |
| [4] | GRABISCH M, MUROFUSHI T. Fuzzy measures and integrals[M] // Theory and Applications. Heidelberg: Physica-verlag, 2000. |
| [5] | 巩增泰, 魏朝琦. 集值函数关于非可加集值测度的Choquet积分[J]. 山东大学学报(理学版), 2015, 50(8):63-71. GONG Zengtai, WEI Zhaoqi. Choquet integral of set-valued functions with respect to multisubmeasures[J]. Journal of Shandong University(Natural Science), 2015, 50(8):63-71. |
| [6] | MUROFUSHI T, SUGENO M. An integral of fuzzy measures and the Choquet integral as integral with respect to a fuzzy measure[J]. Fuzzy Sets and Systems, 1989, 29(2):201-227. |
| [7] | MENG Fanyong, ZHANG Qiang. Induced continuous Choquet integral operators and their application to group decision making[J]. Computers and Industrial Engineering, 2014, 68(1):42-53. |
| [8] | AUMANN R J. Integrals of set-valued functions[J]. Journal of Mathematical Analysis and Applications, 1965, 12(1):1-12. |
| [9] | 吴从炘, 马明. 模糊分析学基础[M]. 北京: 国防工业出版社, 1991. WU Congxin, MA Ming. The foundament of fuzzy analysis[M]. Beijing: National Defense Industry Press, 1991. |
| [10] | LI Jun. Order continuity of monotone set function and convergence of measurable functions sequence[J]. Applied Mathematics and Computation, 2003, 135(2-3):211-218. |
| [11] | CHOQUET G. Theory of capacities[J]. Annales de linstitut Fourier, 1955, 5:131-295. |
| [12] | TORRA V, NARUKAWA Y. Numerical integration for the Choquet integral[J]. Information Fusion, 2016, 31(1):137-145. |
| [13] | JANG L C, KWON J S. On the represtentation of Choquet of set-valued functions and null set[J]. Fuzzy Sets and Systems, 2000, 112(2):233-239. |
| [14] | ZHANG Deli, WANG Zixiao. On set-valued fuzzy integrals[J]. Fuzzy Sets and Systems, 1993, 56(2):237-241. |
| [15] | LIAMAZARES B. Constructing Choquet integral-based operators that generalize weighted means and OWA operators[J]. Information Fusion, 2015, 23(1):131-138. |
| [16] | NARUKAWA Y, TORRA V, SUGENO M. Choquet integral with respect to a symmetric fuzzy measure of a function on the real line[J]. Ann Oper Res, 2016, doi: 10.1007/s10479-012-1166-6. |
| [17] | YAO Ouyang, LI Jun. Some properties of monotone set functions defined by Choquet integral[J]. Journal of Southeast University, 2003, 19(4):424-427. |
