e. nikolaidis aerospace and ocean engineering department...
TRANSCRIPT
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)
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
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