Design Under Uncertainty

Design under uncertainty

E. Nikolaidis Aerospace and Ocean Engineering Department Virginia Tech Acknowledgments

Sophie Chen (VT) Harley Cudney (VT) Raphael Haftka (UF) George Hazelrigg (NSF) Raluca Rosca (UF) Outline

• Decision making problem • Why we should consider uncertainty in design • Available methods • Objectives and scope • Comparison of probabilistic and fuzzy set methods • Concluding remarks 1. Decision making problem Noise Design 1 level (db)

Initial target

Cost ($) Which design is better ? Taxonomy of decision problems (Keney and Raiffa, 1994)

Certainty about Uncertainty about outcomes of actions outcomes of actions ONE Type I problems Type II problems ATTRIBUTE IS Approach: Approaches: Utility SUFFICIENT Deterministic theory, fuzzy set FOR optimization theory DESCRIBING AN OUTCOME MULTIPLE Type III problems Type IV problems ATTRIBUTES Approaches: Utility Approaches: Utility ARE NEEDED theory, fuzzy set theory, fuzzy set FOR theory theory DESCRIBING AN OUTCOME Types of uncertainty

Irreducible: due to inherent Reducible: due to use Statistical: randomness of imperfect models due to lack in physical to predict of data for phenomena and outcomes of an action processes modeling uncertainty Preferences • An outcome is usually described with one or more attributes • Preferences are defined imprecisely: no clear sharp boundary between success and failure • Need a rational approach to quantify value of an outcome to decision maker – Utility theory – Fuzzy sets 2. Why we should consider uncertainty in design • Design parameters are uncertain -- there is no way to make a perfectly safe design • Ignoring uncertainty and using safety factors usually leads to designs with inconsistent reliability levels 3. Available methods

• Safety factor • Worst case scenario-convex models • Taguchi methods • Fuzzy set methods • Probabilistic methods Probabilistic methods

• Approach – Model uncertainties using PDF’s – Estimate failure probability – Minimize probability of failure and/or cost • Advantage: account explicitly for probability of failure • Limitations: – Insufficient data – Sensitive to modeling errors (Ben Haim et al., 1990) Fuzzy set based methods • Possibility distributions • Possibility of event = 1-degree of surprise (Shackle, 1969) • Relation to fuzzy sets (Zadeh, 1978): X is about 10:

Possibility distribution 1

0.25 Probability distribution

8 10 12 Fuzzy sets in structural design

• Uncertainty in mechanical vibration: Chiang et al., 1987, Hasselman et al., 1994 • Vagueness in definition of failure of reinforced plates (Ayyub and Lai, 1992) • Uncertainty and imprecision in preferences in machine design (Wood and Antonsson, 1990) • Relative merits of probabilistic methods and fuzzy sets may depend on: – Amount and type of available information about uncertainty – Type of failure (crisp or vague) – Accuracy of predictive models Important issues

• Are fuzzy sets better than probabilities in modeling random uncertainty when little information is available? • How much information is little enough to switch from probabilities to fuzzy sets? • Compare experimentally fuzzy set and probabilistic designs 4. Objectives and scope • Objectives: – Compare theoretical foundations of probabilistic and fuzzy set methods – Demonstrate differences on example problems – Issue guidelines -- amount of information • Scope: – Problems involving uncertainty – Problems involving catastrophic failure Æ clear, sharp boundary between success and failure 5. Comparison of probabilistic and fuzzy set methods • Comparison of theoretical foundations – Axiomatic definitions – Probabilistic and possibility-based models of uncertainty – Risk assessment – Design for maximum safety • Comparison using a design problem Axiomatic definitions

Probability measure, P(⋅) Possibility measure, Π(⋅) 1) P(A) ≥ 0 ∀ A∈S 1) Boundary requirements: Π(∅)=0, Π(Ω)=1 2) Boundary requirement: 2) Monotonicity: P(Ω)=1 ∀A, B ∈ S, if A ⊆ B, then Π(A) ≤ Π(B) 3) Probability of union of 3) Possibility of union of a events finite number of events ∀ ∈ Ai , i I, Ai are disjoint ∀Ai , i∈ I, Ai disjoint I I Π = Π P( U Ai ) = ∑ P(Ai ) ( U Ai ) maxi∈I ( (Ai )) i=1 i∈I i=1 Differences in axioms

• Probability measure can be assigned to the members of a s-algebra. Possibility can be assigned to any class of sets. • Probability measure is additive with respect to the union of sets. Possibility is subadditive. P(A) + P(AC ) =1 Π(A) + Π(AC ) ≥1 Probability density and possibility distribution functions fX(x) ≠ 1 Area=1

x0 x

P(X=x0)=0

ΠX(x) Area≥1 1

x x0

Π(X=x0)≠0 Modeling an uncertain variable when very little information is available Maximum uncertainty principle: use model that maximizes uncertainty and is consistent with data

Possibility distribution 1

Probability distribution 0.25

8 10 12

8.5 • Increase range of variation from [8,12] to [7,13]: – Failure probability: 0.13Æ0.08 – Failure possibility: 0.50 Æ0.67 • Design modification that shifts failure zone from [8,8.5] to [7.5,8] – failure probability: 0.13 Æ0 (if range of variation is [8,12]) – failure probability remains 0.08 (if range of variations is [7,13]) • Easy to determine most conservative possibility based model consistent with data • Do not know what modeling assumptions will make a probabilistic model more conservative • Probabilistic models may fail to predict effect of design modifications on safety • The above differences are due to the difference in the axioms about union of events Risk assessment: Independence of uncertain variables • Assuming that uncertain parameters are independent always makes a possibility model more conservative. This is not the case with probabilistic models

2 P, P PFS=P if independent PFS=P if perfectly correlated

P, P PFS= P in both cases where components are independent or correlated A paradox Probability-possibility consistency: The possibility of any event should always be greater or equal to its probability

P, P ...

PFS=1-(1-P)n PFS=P 1 System failure probability P System failure possibility P

1 Number of components To ensure that failure possibility remains equal or greater than failure possibility need to impose the condition:

∀A: P(A) f 0, Π(A) =1 Design for maximum safety

• Probabilistic design : • Possibility-based

– find d1,…, dn design:

– to minimize PFS – find d1,…, dn – so that g0 – to minimize PFS – so that g0

Two failure modes: PFS=PF1+PF2-PF12 PFS=max(PF1, PF2) Optimality conditions

Assume PF12 small

P PF1 P PF1

PF2 PF2 d 0 d d0 d ∂ ∂ PF1 = − PF2 PF1=PF2 ∂ d ∂ d Comparison using a design problem

• No imprecision in defining failure • Only random uncertainties • Only numerical data is available about uncertainties

How to evaluate methods: Average probability of failure General approach for comparison

Information about uncertainties

Incomplete Budget information

Optimization: Maximize Safety

Fuzzy Design Probabilistic Design

Probabilistic Analysis Probabilistic Analysis

Compare relative frequencies of failure m, wn2 absorber

Normalized M, wn1 original system system amplitude y F=cos(wet)

Figure 2. Tuned damper system Original SDOF system Failure modes 1. Excessive vibration 2. Cost > budget (cost proportional to m)

48 b normalized natural 36 frequencies System amplitude 24 (assumed equal) 12

0 0.8 0.84 0.88 0.92 0.96 1 1.04 1.08 1.12 1.16 1.2 β :R=0.05; : R=0.01

Figure 4. Amplitude of system vs. β, ζ=0.01 Uncertainties 1 Normalized frequencies 2 Budget • Know true probability distribution of budget • Know type of probability distribution of normalized frequencies and their mean values, but not their scatter • Samples of values of normalized frequencies are available Design problem

• Find m • to minimize PF (PF)

• PF=P(excessive vibration¯cost overrun) • ½F=P(excessive vibration ¯cost overrun)

• heavy absorber, low vibration but high cost Estimation of variance of b

Concept of maximum uncertainty: if little information is available, assume model with largest uncertainty that is consistent with the data Comparison of ten probabilistic and ten possibility-based designs. Three sample values were used to construct models of uncertainties. Blue bars indicate possibility-based designs. Red bars indicate probabilistic designs.

- Inflation factor method, unbiased estimation

0.35

0.3

0.25

0.2

0.15

0.1 Actual probability of failure 0.05

0 12345678910 Data group sample size equal to 3,000

0.25

0.2

0.15

0.1 Actual probability of failure of probability Actual 0.05

0 12345678910 Data group Probabilistic approach cannot predict design trends

0.3

0.2

0.1 to excessive vibration excessive to Failureprobability due

0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 R

distribution of frequency - U(1,0.05) distribution of frequency - U(1,0.075) distribution of frequency - U(1,0.1)

Figure 5. Effect of standard deviations of b1 and b2 on the probability of failure vs. R , b1 and b2 are equal Comparison in terms of average failure probability as a function of amount of information Sample size Best design 3 5 10 20 100 1000 Blue bullet: on average possibility is better, red bullet: on average probability is better Concluding remarks

• Overview of problems and methods for design under uncertainty • Probabilistic and fuzzy set methods -- comparison of theoretical foundations • Probabilistic and fuzzy set methods -- comparison using design problem Concluding remarks • Important to consider uncertainties • There is no method that is best for all problems involving uncertainties • Probabilistic design better if sufficient data is available • Possibility can be better if little information is available – easier to construct most conservative model consistent with data – probabilistic methods may fail to predict effect of design modifications • Major difference in axioms about union of events