Credibility Theory

1. Credibility Theory It is a new branch of mathematics that studies the behavior of fuzzy phenomena. Baoding Liu Uncertainty Theory Laboratory Department of Mathematical Sciences Tsinghua University

2. Fashion of Mathematics • 2300 Years Ago: Euclid: “Elements”, First Axiomatic System • 1899: Hilbert: Independence, Consistency, Completeness • 1931: K. Godel: Incompleteness Theorem • 1933: Kolmogoroff: Probability Theory • 2004: B. Liu: Credibility Theory Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

3. Why I do not possibility measure? Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

4. Five Axioms Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

5. Credibility Subadditivity Theorem Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

6. Credibility Semicontinuity Laws Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

7. Credibility Extension Theorem Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

8. Fuzzy Variable Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

9. Credibility Measure by Membership Function Liu and Liu (IEEE TFS, 2002) Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

10. Independent Fuzzy Variables Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

11. Theorem: Extension Principle of Zadeh Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

12. Expected Value Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

13. Why the Definition Reasonable? Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

14. Credibility Distribution Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

15. A Sufficient and Necessary Condition Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

16. Entropy (Li and Liu, 2005) Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

17. Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

18. Essential of Uncertainty Theory Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

19. What Mathematics Made? Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

20. Fuzzy Programming Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

21. The Simplest Problem The Most Fundamental Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

22. Expected Value Criterion Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

23. Fuzzy Expected Value Model Liu and Liu (IEEE TFS, 2002) Find the decision with maximum expected return subject to some expected constraints. Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

24. Optimistic Value Criterion Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

25. (Maximax) Chance-Constrained Programming Liu and Iwamura (FSS, 1998) Maximize the optimistic value subject to chance constraints. Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

26. Pessimistic Value Criterion Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

27. (Minimax) Chance-Constrained Programming Liu (IS, 1998) Maximize the pessimistic value subject to chance constraints. Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

28. Credibility Criterion Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

29. Fuzzy Dependent-Chance Programming Liu (IEEE TFS, 1999) Find the decision with maximum chance to meet the event in an uncertain environment. Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

30. Classify Uncertain Programming via Graph Information Random Fuzzy Fuzzy random Fuzzy Stochastic Philosophy Single-Objective P EVM CCP DCP MOP GP DP MLP Structure Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

31. Last Words • [1] Liu B., Foundation of Uncertainty Theory. • [2] Liu B., Introduction to Uncertain Programming. • If you want an electronic copy of my book, • or source files of hybrid intelligent algorithms, • please download them from • http://orsc.edu.cn/~liu Baoding Liu Tsinghua University http://orsc.edu.cn/~liu