Design under uncertainty

E. Nikolaidis

Aerospace and Ocean EngineeringDepartment

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 indesign

• Available methods

• Objectives and scope

• Comparison of probabilistic and fuzzy setmethods

• Concluding remarks

1. Decision making problemNoiselevel (db)

Cost ($)

Initial target

Design 1

Design 2Design 3

Which design is better ?

Taxonomy of decision problems(Keney and Raiffa, 1994)

Certainty aboutoutcomes of actions

Uncertainty aboutoutcomes of actions

ONEATTRIBUTE ISSUFFICIENTFORDESCRIBINGAN OUTCOME

Type I problemsApproach:

Deterministicoptimization

Type II problemsApproaches: Utility

theory, fuzzy settheory

MULTIPLEATTRIBUTESARE NEEDEDFORDESCRIBINGAN OUTCOME

Type III problemsApproaches: Utility

theory, fuzzy settheory

Type IV problemsApproaches: Utility

theory, fuzzy settheory

Types of uncertainty

Irreducible:due to inherent randomnessin physical phenomena and processes

Reducible: due to use of imperfect models to predictoutcomes of an action

Statistical:due to lack of data for modelinguncertainty

Preferences• An outcome is usually described with one

or more attributes

• Preferences are defined imprecisely: noclear sharp boundary between success andfailure

• Need a rational approach to quantify valueof an outcome to decision maker– Utility theory

– Fuzzy sets

2. Why we should consideruncertainty in design

• Design parameters are uncertain -- there isno way to make a perfectly safe design

• Ignoring uncertainty and using safetyfactors usually leads to designs withinconsistent reliability levels

• Safety factor

• Worst case scenario-convex models

• Taguchi methods

• Fuzzy set methods

• Probabilistic methods

3. Available 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:

1

10 8 12

0.25

Possibility distribution

Probability distribution

Fuzzy sets in structural design

• Uncertainty in mechanical vibration:Chiang et al., 1987, Hasselman et al., 1994

• Vagueness in definition of failure ofreinforced plates (Ayyub and Lai, 1992)

• Uncertainty and imprecision in preferencesin machine design (Wood and Antonsson,1990)

• Relative merits of probabilistic methods andfuzzy 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 inmodeling random uncertainty when littleinformation is available?

• How much information is little enough toswitch from probabilities to fuzzy sets?

• Compare experimentally fuzzy set andprobabilistic 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 Æclear, sharp boundary between success andfailure

5. Comparison of probabilisticand fuzzy set methods

• Comparison of theoretical foundations– Axiomatic definitions

– Probabilistic and possibility-based models ofuncertainty

– 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)=12) Boundary requirement:

P(Ω)=12) Monotonicity:

)()(then

,,,

BA

BAifSBA

Π≤Π⊆∈∀

3) Probability of union ofevents

)()(

disjoint are ,,

1

∑=

∈∀

∈= Iii

I

ii

ii

APAP

AIiA

U

3) Possibility of union of afinite number of events

))((max)(

disjoint ,,

1iIi

I

ii

ii

AA

AIiA

Π=Π

∈∀

∈=U

• Probability measure can be assigned to themembers of a s-algebra. Possibility can beassigned to any class of sets.

• Probability measure is additive with respectto the union of sets. Possibility issubadditive.

Differences in axioms

1)()(

1)()(

≥Π+Π

=+C

C

AA

APAP

Probability density and possibility distributionfunctions

Area=1

x

fX(x)

x0

P(X=x0)=0

≠ 1

ΠX(x)

x

Area≥1

x0

Π(X=x0)≠0

1

Modeling an uncertain variable when very littleinformation is available

Maximum uncertainty principle: use model that maximizesuncertainty and is consistent with data

1

10 8 12

0.25

Possibility distribution

Probability distribution

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 ofvariation is [8,12])

– failure probability remains 0.08 (if range ofvariations is [7,13])

• Easy to determine most conservativepossibility based model consistent with data

• Do not know what modeling assumptionswill make a probabilistic model moreconservative

• Probabilistic models may fail to predicteffect of design modifications on safety

• The above differences are due to thedifference in the axioms about union ofevents

Risk assessment: Independence ofuncertain variables

• Assuming that uncertain parameters areindependent always makes a possibilitymodel more conservative. This is not thecase with probabilistic models

P, P

P, P

PFS=P2 if independentPFS=P if perfectly correlated

PFS= P in both cases where components are independent or correlated

A paradoxProbability-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

Number of components

1 System failure probability

System failurepossibility

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

1)(,0)(: =Π∀ AAPA f

P

P

1

Design for maximum safety

• Probabilistic design :– find d1,…, dn

– to minimize PFS

– so that g0

• Possibility-baseddesign:– find d1,…, dn

– to minimize PFS

– so that g0

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

Optimality conditions

dd0

d

PF

d

PF

∂∂−=

∂∂ 21

PF1

PF2

Assume PF12 small

dd0

PF1=PF2

PF1

PF2

P P

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

Optimization: Maximize Safety

Fuzzy Design Probabilistic Design

Probabilistic Analysis Probabilistic Analysis

Compare relative frequencies of failure

Informationabout uncertainties

IncompleteinformationBudget

m, wn2 absorber

originalsystem

M, wn1

F=cos(wet)

Figure 2. Tuned damper system

Normalizedsystemamplitude y

Original SDOF system

Failure modes1. Excessive vibration

2. Cost > budget (cost proportional to m)

β0.8 0.84 0.88 0.92 0.96 1 1.04 1.08 1.12 1.16 1.2

0

12

24

36

48

60

Figure 4. Amplitude of system vs. β, ζ=0.01

Syst

em a

mpl

itud

e

:R=0.05; : R=0.01

b normalizednaturalfrequencies(assumedequal)

Uncertainties1 Normalized frequencies

2 Budget

• Know true probability distribution of budget

• Know type of probability distribution ofnormalized frequencies and their meanvalues, but not their scatter

• Samples of values of normalizedfrequencies are available

Design problem

• Find m

• to minimize PF (PF)

• PF=P(excessive vibrationcost 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 modelwith 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. Redbars indicate probabilistic designs.

- Inflation factor method, unbiased estimation

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8 9 10

Data group

Act

ual

pro

bab

ility

of

failu

re

sample size equal to 3,000

0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 10

Data group

Act

ual

pro

bab

ility

of

failu

re

Probabilistic approach cannot predictdesign trends

R0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0

0.1

0.2

0.3

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 ofb1 andb2on the probability of failure vs. R ,

b1 and b2 are equal

Failu

re p

roba

bilit

y du

eto

exc

essi

ve v

ibra

tion

Comparison in terms of average failureprobability as a function of amount of

informationSample size Best design

351020100

1000Blue bullet: on average possibility is better, red bullet: on average probability is better

Concluding remarks

• Overview of problems and methods fordesign 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 problemsinvolving uncertainties

• Probabilistic design better if sufficient data isavailable

• Possibility can be better if little information isavailable– easier to construct most conservative model consistent

with data

– probabilistic methods may fail to predict effect ofdesign modifications

• Major difference in axioms about union of events