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

3. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu 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

4. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Why I do not possibility measure?

5. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Five Axioms

6. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Credibility Subadditivity Theorem

7. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Credibility Semicontinuity Laws

8. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Credibility Extension Theorem

9. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Fuzzy Variable

10. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Credibility Measure by Membership Function Liu and Liu (IEEE TFS, 2002)

11. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Independent Fuzzy Variables

12. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Theorem: Extension Principle of Zadeh

13. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Expected Value

14. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Why the Definition Reasonable?

15. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Credibility Distribution

16. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu A Sufficient and Necessary Condition

17. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Entropy (Li and Liu, 2005)

18. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu

19. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Essential of Uncertainty Theory

20. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu What Mathematics Made?

21. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Fuzzy Programming

22. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu The Simplest The Most Fundamental Problem

23. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Expected Value Criterion

24. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Liu and Liu (IEEE TFS, 2002) Find the decision with maximum expected return subject to some expected constraints. Fuzzy Expected Value Model

25. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Optimistic Value Criterion

26. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu (Maximax) Chance-Constrained Programming Liu and Iwamura (FSS, 1998) Maximize the optimistic value subject to chance constraints.

27. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Pessimistic Value Criterion

28. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu (Minimax) Chance-Constrained Programming Liu (IS, 1998) Maximize the pessimistic value subject to chance constraints.

29. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Credibility Criterion

30. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Fuzzy Dependent-Chance Programming Liu (IEEE TFS, 1999) Find the decision with maximum chance to meet the event in an uncertain environment.

31. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu Classify Uncertain Programming via Graph Information Philosophy Single-Objective P MOP GP DP MLP Structure EVMCCPDCP Random Fuzzy Fuzzy random Fuzzy Stochastic

32. Uncertainty Theory & Uncertain Programming U T L A B Baoding Liu Tsinghua University http://orsc.edu.cn/~liu 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