1. Mechanical Engineering Department
Non-Probabilistic Design Optimization with Insufficient Data using Possibility and Evidence Theories
Zissimos P. Mourelatos
Jun Zhou
Mechanical Engineering Department
Oakland University
Rochester, MI 48309, USA

2. Overview Introduction Design under uncertainty Uncertainty theories
Possibility – Based Design Optimization (PBDO)
Uncertainty quantification and propagation
Design algorithms
Evidence – Based Design Optimization (EBDO)
Examples
Summary and conclusions

3. Design Under Uncertainty
Analysis /
Simulation
Input
Output
Uncertainty (Quantified)
Uncertainty (Calculated)
Propagation
Design
Quantification
2. Propagation
3. Design

4. Uncertainty Types Aleatory Uncertainty (Irreducible, Stochastic)
Probabilistic distributions
Bayesian updating
Epistemic Uncertainty (Reducible, Subjective,
Ignorance, Lack of Information)
Fuzzy Sets; Possibility methods (non-conflicting information)
Evidence theory (conflicting information)

5. Uncertainty Theories Evidence Theory Probability Theory
Possibility Theory
Probability Theory

6. Non-Probabilistic Design Optimization:
Set Notation
Universe
(X)
Power Set (All sets)
Element A B C
Evidence Theory

7. Possibility-Based Design Optimization (PBDO)

8. Possibility-Based Design Optimization (PBDO)
Evidence Theory
No Conflicting Evidence
(Possibility Theory)

9. Quantification of a Fuzzy Variable:
Membership Function
- cut provides confidence level
At each confidence level, or cut, a set is defined as
convex normal set

10. Propagation of Epistemic Uncertainty
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

11. Optimization Method Global where : s.t. and s.t. 1.0 1.0 a a 0.0 0.0

12. Possibility-Based Design Optimization (PBDO)
(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

13. Possibility-Based Design Optimization (PBDO)
Considering that
, we have

14. Possibility-Based Design Optimization (PBDO)
s.t. ; ;
s.t. OR
Double Loop

15. PBDO with both Random and Possibilistic Variables,
with
Triple Loop ; ,
s.t.

16. Evidence-Based Design Optimization (EBDO)

17. Evidence-Based Design Optimization (EBDO)
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)

18. Evidence-Based Design Optimization (EBDO) Assuming independence,
For
where
define:
Assuming independence,

19. Evidence-Based Design Optimization (EBDO)
BPA structure for a two-input problem

20. Evidence-Based Design Optimization (EBDO) Uncertainty Propagation
If we define,
then
where
and

21. Evidence-Based Design Optimization (EBDO)
Position of a focal element w.r.t. limit state
Contributes to Belief
Contributes to Plausibility

22. Evidence-Based Design Optimization (EBDO) If non-negative null form
Design Principle
If non-negative null form
is used for feasibility,
feasible
infeasible
failure
Therefore,
is satisfied if OR

23. Evidence-Based Design Optimization (EBDO)
Formulation ,

24. Evidence-Based Design Optimization (EBDO)
Calculation of

25. Geometric Interpretation of PBDO and EBDO

26. Possibility-Based Design Optimization (PBDO)
Implementation x1
initial design point
Feasible Region
g1(x1,x2)=0
frame of discernment
g2(x1,x2)=0
Objective Reduces
PBDO optimum
deterministic optimum x2

27. Evidence-Based Design Optimization (EBDO) deterministic optimum
Implementation
Feasible Region x2 x1
g1(x1,x2)=0
g2(x1,x2)=0
Objective Reduces
initial design point
frame of discernment
EBDO optimum
deterministic optimum

28. Evidence-Based Design Optimization (EBDO) deterministic optimum
Implementation
Feasible Region x2 x1
g1(x1,x2)=0
g2(x1,x2)=0
Objective Reduces
hyper-ellipse
initial design point
frame of discernment B
MPP for g1=0
deterministic optimum

29. Evidence-Based Design Optimization (EBDO) deterministic optimum
Implementation
Feasible Region x2 x1
g1(x1,x2)=0
g2(x1,x2)=0
Objective Reduces
hyper-ellipse
initial design point
frame of discernment
EBDO optimum B
MPP for g1=0
deterministic optimum

30. Cantilever Beam Example: RBDO Formulation
s.t.
where :

31. Cantilever Beam Example: PBDO Formulation
s.t.
RBDO
s.t.
PBDO

32. Cantilever Beam Example: EBDO Formulation
s.t.
EBDO
BPA Structure

33. Cantilever Beam Example: EBDO Formulation
BPA structure for y, Y, Z, E

34. Cantilever Beam Example: Comparison of Results

35. Thin-walled Pressure Vessel Example
yielding
s.t.

36. Thin-walled Pressure Vessel Example BPA structure for R, L, t, P and Y

37. Thin-walled Pressure Vessel Example

38. More Conservative Design
Summary and Conclusions
Possibility and evidence theories were used to quantify and propagate uncertainty.
PBDO and EBDO algorithms were presented for design with incomplete information.
EBDO design is more conservative than the RBDO design but less conservative than PBDO design.
Deterministic
RBDO
EBDO
PBDO
Less Information
More Conservative Design

39. Q & A