1. DETC06: Uncertainty Workshop; Evidence & Possibility Theories Evidence and Possibility Theories in Engineering Design Zissimos P. Mourelatos Mechanical Engineering Department Oakland University Rochester, MI 48309, USA mourelat@oakland.edu

2. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 2 Workshop Objective  Compare and evaluate uncertainty representations for design  Promote understanding and discussion on uncertainty representations

3. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 3 Design in the Context of  Decision-making; selection of alternatives  Optimization Uncertainty is involved in both cases

4. DETC06: Uncertainty Workshop; Evidence & Possibility Theories  Probabilistic (Irreducible)  Sufficient data  Non - Probabilistic (Reducible)  Insufficient (scarce) data Uncertainty Quantification Fuzzy Sets Interval Analysis Possibility & Evidence Theories

5. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 5 Evidence Theory Possibility Theory Probability Theory Uncertainty Theories

6. DETC06: Uncertainty Workshop; Evidence & Possibility Theories Basics of Evidence Theory & Application to Design Optimization

7. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 7 Set Notation and Basic Relations Universe (X) Power Set (All sets) Element A BC A B Evidence Theory

8. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 8 Universe A BC Basic Probability Assignment (BPA) Complementary Measures Set Notation and Basic Relations (Cont.)

9. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 9 Uncertainty Quantification; BPA Basic Probability Assignment (BPA): m(A) If m(A)>0 for then A is a focal element 0.2 0.1 0.4 0.3 Y 5 6.2 8 9.5 11 “Expert” A 5 0.3 0.4 Y 7 8.7 11 “Expert” B Y 0.x1 7 8.7 11 9.5 8 6.2 0.x2 0.x3 0.x6 0.x5 0.x4 5 Combining Rule (Dempster – Shafer; among others)

10. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 10 Example: Residual Strength of a Wooden Bridge Estimate Residual Strength (lb/in 2 ) BPA 1[3000, 4000]0.3 2[2000, 4000]0.4 3[2000, 5000]0.2 4[1000, 5000]0.1

11. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 11 Wooden Bridge Example (Cont.) 1000 2000 3000 4000 5000 m(A 1 )=0.3 m(A 2 )=0.4 m(A 3 )=0.2 m(A 4 )=0.1

12. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 12 1000 2000 3000 4000 5000 m(A 1 )=0.3 m(A 2 )=0.4 m(A 3 )=0.2 m(A 4 )=0.1 Bel(1000, 2000)=0Pl(1000, 2000)=0.1 Bel(2000, 3000)=0Pl(2000, 3000)=0.7 Bel(3000, 4000)=0.3Pl(3000, 4000)=1.0 Bel(4000, 5000)=0Pl(4000, 5000)=0.3 Bel(2000, 4000)=0.7 Pl(2000, 4000)=1-Bel(1000, 2000) – Bel(4000, 5000) = 1 – 0 – 0 = 1 Wooden Bridge Example (Cont.)

13. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 13 Uncertainty Propagation For where define: Assuming independence: where,

14. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 14 BPA structure for a two-input problem Uncertainty Propagation (Cont.)

15. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 15 If we define, then where and Uncertainty Propagation (Cont.)

16. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 16 Position of a focal element w.r.t. limit state Contributes to Belief Contributes to Plausibility Uncertainty Propagation (Cont.)

17. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 17 Evidence-Based Design Optimization (EBDO) Design Principle Therefore, is satisfied if OR If non-negative null form is used for feasibility, feasible infeasible failure

18. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 18, Evidence-Based Design Optimization (EBDO) Formulation

19. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 19 Evidence-Based Design Optimization (EBDO) Calculation of

20. DETC06: Uncertainty Workshop; Evidence & Possibility Theories Basics of Possibility Theory & Application to Design Optimization

21. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 21 Set Notation and Basic Relations Complementary Measures A1A1 A2A2 A3A3 Nested Sets: (Consonant Evidence)

22. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 22 Some General Remarks  Possibility theory is a special case of evidence theory  It can be applied only for non-conflicting (i.e. consonant) evidence (nested focal elements)  A fuzzy set approach is usually followed

23. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 23 Uncertainty Quantification : Membership Function - cut provides confidence level At each confidence level, or -cut, a set is defined as convex normal set

24. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 24 1000 2000 3000 4000 5000 0.2 0.4 0.8 0.6 1.0 Possibility Theory (nested sets): 1000 2000 3000 4000 5000 m(A 1 )=0.3 m(A 2 )=0.4 m(A 3 )=0.2 m(A 4 )=0.1 Wooden Bridge Example

25. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 25 Uncertainty Quantification (Cont.) Focal Element Singleton

26. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 26 Uncertainty Propagation Extension Principle The “extension principle” calculates the membership function (possibility distribution) of the fuzzy response from the membership functions of the fuzzy input variables. If where then Practical Approximations of Extension Principle Vertex Method Discretization Method Hybrid (Global-Local) Optimization Method

27. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 27 0.0 1.0 a 0.0 1.0 a where : 0.0 1.0 a 0.0 1.0 a 0.0 1.0 a 0.0 1.0 a s.t. and Global Uncertainty Propagation: Optimization Method

28. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 28 Evidence Theory No Conflicting Evidence (Possibility Theory) Possibility-Based Design Optimization (PBDO)

29. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 29 (Possibility Theory)  What is possible may not be probable  What is impossible is also improbable If feasibility is expressed with positive null form then, constraint g is ALWAYS satisfied if for or Possibility-Based Design Optimization (PBDO)

30. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 30 Considering that, we have Possibility-Based Design Optimization (PBDO)

31. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 31 Possibility-Based Design Optimization (PBDO) ; s.t. ; OR

32. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 32 Summary and Conclusions  Evidence and Possibility theories are based on well established axiomatic foundation  Uncertainty quantification and propagation are well established  Possibility theory is a subset of Evidence theory  Evidence and Possibility theories can be formally integrated with Probability theory  Evidence theory supports the P-box approach to imprecise probabilities  Evidence and Possibility theories result in conservative designs due to limited information

33. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 33 Q&A

34. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 34

35. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 35 Normative Decision Analysis is a logical process for ranking design alternatives based on :  outcomes from each alternative with assigned probabilities  value assessment  decision maker’s preferences

36. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 36 Normative Decision Analysis Decision Utility; D_M preference outcome Alternative Outcome PDF Outcome U(Out)

37. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 37 Decision Making and Optimization* s.t. The objective function f : is a scalar (multi-attribute problems??) represents a design space to performance space translator; a model (…validated model??) represents decision maker’s preferences value assessment outcome design alternatives feasibility *Hazelrigg, G.A., “The Cheshire Cat on Engineering Design,” draft to ASME J. of Mech. Design. Max

38. DETC06: Uncertainty Workshop; Evidence & Possibility Theories Design Optimization Under Uncertainty Analysis / Simulation Input Output Uncertainty (Quantified) Uncertainty (Calculated) Propagation Design 1. Quantification 2. Propagation 3. Design

39. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 39 Example: Six-Hump Camel Function

40. DETC06: Uncertainty Workshop; Evidence & Possibility Theories 40 Example: Six-Hump Camel Function (Cont.)