A survey of credibility theory. (English) Zbl 1133.90426

Liu, Baoding

Fuzzy Optim. Decis. Mak. 5, No. 4, 387-408 (2006).

Summary: This paper provides a survey of credibility theory that is a new branch of mathematics for studying the behavior of fuzzy phenomena. Some basic concepts and fundamental theorems are introduced, including credibility measure, fuzzy variable, membership function, credibility distribution, expected value, variance, critical value, entropy, distance, credibility subadditivity theorem, credibility extension theorem, credibility semicontinuity law, product credibility theorem, and credibility inversion theorem. Recent developments and applications of credibility theory are summarized. A new idea on chance space and hybrid variable is also documented.

MSC:

90C70	Fuzzy and other nonstochastic uncertainty mathematical programming
03E72	Theory of fuzzy sets, etc.
60A05	Axioms; other general questions in probability
62A01	Foundations and philosophical topics in statistics
68T37	Reasoning under uncertainty in the context of artificial intelligence

Keywords:

Fuzzy variable; Credibility measure; Credibility theory; Fuzzy random variable; Random fuzzy variable; Conditional credibility

References:

[9]	Feng, Y., Yang, L. (2006). A two-objective fuzzy k-cardinality assignment problem. Journal of Computational and Applied Mathematics, to be published. · Zbl 1098.90064
[15]	Gao, X. (2005). Fuzzy multi-criteria minimum spanning tree problem. Proceedings of the fourth international conference on information and management sciences. China: Yunnan, (pp. 498–504).
[23]	Ji, X. Y., Iwamura, K., Shao, Z. (2006). New models for shortest path problem with fuzzy arc lengths. Applied Mathematical Modelling, to be published. · Zbl 1152.90674
[25]	Ke, H., Liu, B. (2004). Project scheduling problem with fuzzy activity duration times. Proceedings of 2004 IEEE international conference on fuzzy systems, pp. 819–823.
[26]	Ke, H., Liu, B. (2005). Project scheduling problem with mixed uncertainty of randomness and fuzziness. Proceedings of the first international conference on complex medical engineering, (pp. 957–962) Japan: Takamatsu.
[32]	Li, P., Liu, B. (2003). Critical path problem with random fuzzy activities. Proceedings of the Fourth International Conference on Intelligent Technologies, (pp. 451–458). Chiang Mai: Thailand. · Zbl 1028.35153
[33]	Li, P., Liu, B. (2005). Entropy of credibility distributions for fuzzy variables. Technical Report.
[34]	Li, X., Liu, B. (2006a). A sufficient and necessary condition of credibility measure. International Journal of Uncertainty & Knowledge-Based System, 14(5): 2006. · Zbl 1113.28014
[35]	Li, X., Liu, B. (2006b). Maximum variance theorems for various types of uncertain variable. Technical Report.
[36]	Li, X., Liu, B. (2006c). On distance of fuzzy variables. Technical Report.
[48]	Liu, L. (2006a). The maximum fuzzy weighted matching models and hybrid genetic algorithm. Applied Mathematics and Computation, to be pubished. · Zbl 1152.05358
[59]	Liu, Y. K., Gao, J. (2005a). The independence of fuzzy variables in credibility theory and its applications. Technical Report.
[62]	Lu, M., & Gao, J. (2001). Fuzzy expected value integer programming models for capital budgeting problem. Proceedings of the Fourth National Youth Conference on Operations Research and Management (pp. 242–249).
[66]	Peng, J., & Liu, B. (2004b). Some properties of optimistic and pessimistic values of fuzzy variables. Proceedings of the thirteenth IEEE international conference on fuzzy systems (Vol. 2, pp. 745–750). Hungary: Budapest.
[68]	Peng J., Mok H.M.K., Tse W.M. (2005b). Credibility programming approach to fuzzy portfolio selection problems. Proceedings of 2005 International Conference on Machine Learning and Cybernetics. Guangzhou, China 4:2523–2528
[70]	Shao, Z., & Ji, X. Y. (2006). Fuzzy multi-product constraint newsboy problem. Applied Mathematics and Computation, to be published. · Zbl 1139.90436
[71]	Wang, G., & Liu, B. (2003). New theorems for fuzzy sequence convergence. Proceedings of the second international conference on information and management sciences (pp. 100–105). China: Chengdu.
[72]	Wen, M., & Iwamura, K. (2004). Fuzzy facility location-allocation problem under Hurwicz criterion. Proceedings of the third international conference on information and management sciences (pp. 310–315) China: Dunhuang. · Zbl 1149.90366
[73]	Wu, R., Tang, W., & Zhao, R. (2004). An efficient algorithm for fuzzy web-mining. IEEE Conference on Information Reuse and Integration, 576–581.
[82]	Yang, L., & Sun, Y. (2004). Expected value model for a fuzzy random warehouse layout problem. IEEE International Conference on Fuzzy Systems – Proceedings (pp. 751–756). Hungary: Budapest.
[84]	Yang, L., & Liu, B. (2005b). A multi-objective fuzzy assignment problem: new model and algorithm. Proceedings of the 2005 Conference on Fuzzy Systems (pp. 551–556). Reno.
[92]	Zhao, R., Tang, W., & Yun, H. (2004). Fuzzy renewal process, fuzzy renewal reward process and their applications. Proceedings of the thirteenth IEEE international conference on fuzzy systems (pp. 739–743). Budapest, Hungary.
[98]	Zhou, J., & Liu, B. (2003). Bifuzzy programming and hybrid intelligent algorithms. Proceedings of the second international conference on information and management sciences (pp. 440–445). Chengdu, China.
[100]	Zhou, J., & Liu, B. (2006). Modeling capacitated location-allocation problem with fuzzy demands. Computers and Industrial Engineering, to be published.
[103]	Zhu, Y., Ji, X. Y. (2006). Expected values of functions of fuzzy variables. Journal of Intelligent and Fuzzy Systems, to be published. · Zbl 1122.68138