Fundamentals of the Dempster -Shafer Theory and its Applications … · 2017. 11. 30. · Methods introduced (FTA, ETA, RCM, FMECA) Fundamentals of the Dempster-Shafer Theory Intro

Fundamentals of theFundamentals of the

DempsterDempster--Shafer Theory and its Shafer Theory and its

Applications to SystemApplications to System

Safety and Reliability ModellingSafety and Reliability Modelling

Uwe Kay RakowskyUwe Kay Rakowsky

University of Wuppertal, GermanyUniversity of Wuppertal, Germany

Department D Department D –– Safety EngineeringSafety Engineering

Uwe Kay Rakowsky Dempster-Shafer Theory & its Applications to System Safety & Reliability Modelling

2

DempsterDempster--Shafer ApplicationsShafer ApplicationsIntroductionIntroduction

Objective

Modelling & expressing uncertainties in safety & reliability analyses

Evidence measures offer a different kind of flavour to RAMS engineers

What’s new? Evidence measures belief and plausibility are applied

instead of →→→→ probabilities

instead of →→→→ membership function (fuzzy set theory)

What’s not new?

Methods introduced (FTA, ETA, RCM, FMECA)

Fundamentals of the Dempster-Shafer Theory

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DempsterDempster--Shafer ApplicationsShafer ApplicationsIntroductionIntroduction

Note

ESREL 2007 →→→→ special DS approach tailored to RCM

SSARS 2007 →→→→ general DS approach to Safety & Reliability Modelling

→→→→ more details

Disclaimer

Nobody is forced to apply evidence measures

Not faster, bigger, better, higher →→→→ just different

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OutlineOutlineIntroductionIntroduction

Part 1 – Fundamentals

History

Scenario

Interpretations

Part 2 – Illustration

The DS calculus in eight steps

Part 3 – Applications to System Safety & Reliability Modelling

FTA – Fault Tree Analysis

ETA – Event Tree Analysis

RCM – Reliability-centred Maintenance

Further Analyses

Part 4 – Outroduction

Pros & Cons

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HistoryHistoryIntroductionIntroduction

1966 – Arthur P. Dempster

Developed Theory

“Upper & lower probabilities”

Suitable to express uncertain expert judgements

1976 – Glenn Shafer

Extended, refined, recast

“Upper probabilities & degrees of belief”

“DS Theory of Evidence”, “DS Evidential Theory” →→→→ DST

1988 – George J. Klir & Tina A. Folger

Introduce →→→→ “Degrees of belief & plausibility” Evidence measures depart from being probabilities

G. Shafer

A. P. Dempster

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The ScenarioThe ScenarioFundamentalsFundamentals

Scenario

System

Hypotheses

Frame of discernment

Pieces of evidence

Data sources

System Borders

In- and outputs

Elements (e.g. components or modules)

Links between the elements

Interactions of the elements

Task of the system

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Scenario

System

Hypotheses


Pieces of evidence

Data sources

Hypotheses

Single hypothesis →→→→ e.g. represents one state, one answer

Example →→→→ “functioning”, “marginal”, “faulty”

Example →→→→ “yes”, “uncertain”, “no”

Properties

Unique and

not overlapping and

mutually exclusive

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Scenario

System

Hypotheses


Pieces of evidence

Data sources


Representation →→→→ universal set Ω Hypotheses →→→→ elements of frame of discernment

Ω = “functioning”, “marginal”, “faulty”

Power set 2Ω →→→→ set of all subsets

Power set 2Ω →→→→ single and conjunctions of hypotheses

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Scenario

System

Hypotheses


Pieces of evidence

Data sources

Pieces of Evidence

Symptoms or events →→→→ e.g. failures Assignment

evidence →→→→ hypothesis(es) corresponds to cause →→→→ consequence(s) Assignment

1 p-of-e assigned to 1 hypothesis or 1 set of hypotheses

>1 p-of-e may not be assigned* to same hypothesis, same set

*) by the same data source

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Scenario

System

Hypotheses


Pieces of evidence

Data sources

Data Sources

Information provider →→→→ experts, empirical studies, data

Task →→→→ quantifying strength p-of-e →→→→ hypothesis assignments →→→→ m(A)

Requirements

→→→→ free from bias (esp. experts)

→→→→ representative (esp. studies)

→→→→ no source is more important than another one

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Scenario

System

Hypotheses


Pieces of evidence

Data sources

Data Sources

Information provider →→→→ experts Expert group

Safety →→→→ system eng., software eng., reliability eng., service eng.

RCM →→→→ service eng., maintenance personnel, reliability eng.

Task →→→→ give subjective quantifiable statements

Basis →→→→ data, experience, intuition … ←←←← biased?

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Scenario

System

Hypotheses


Pieces of evidence

Data sources

Pieces of Evidence

Expert group →→→→ piece of evidence?

→→→→ expert judgement (experience, intuition)

→→→→ experts’ subjectivity Critical issue?

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Objectivity versus SubjectivityObjectivity versus SubjectivityFundamentalsFundamentals

Objectivity

Exactly one single hypothesis is objectively true

Subjectivity Uncertain which hypothesis fits subjectively best to reality

Dempster-Shafer Theory Calculus describes & quantifies the subjective viewpoint

as an assessment for an unknown objective fact

Safety & Reliability Engineering

PSAM/ESREL 2004 →→→→ hypotheses

→→→→ “component i is functioning” and “component i is faulty”

→→→→ … same set?!

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Sets

Ω universal set

A, B, Z ⊆⊆⊆⊆ Ω sets, containing a single hypothesis or a set of hypotheses

Basic Assignment

m, m(A) quantifies if the element belongs exactly to the set A

m: 2Ω→→→→[0, 1] mapping (prob. Ω →→→→ [0, 1])

ΣA⊆⊆⊆⊆Ωm(A)=1 all statements of an expert are normalised

m(A) > 0 focal element, only substantial statements

m(∅∅∅∅) = 0 simplicity (not required)

Differences in Properties to Probabilities

m(Ω) = 1 not required

m(A) vs. m(¬A) no relationship required

If A ⊂ B ⊆⊆⊆⊆ Ω, then m(A) ≤ m(B) not required

Assignments & SetsAssignments & SetsFundamentalsFundamentals

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The Basic AssignmentThe Basic AssignmentFundamentalsFundamentals

Interpretation of m

Task of m →→→→ assign evidential weight to hypothesis(es) →→→→ A ⊆ Ω Mathematical interpretation of m →→→→ “evidential weight”

Probability →→→→ no concept, no interpretation ( ESREL‘05 Proceedings)

Denotations of m

“Basic probability assignment” ←←←← no probability

“Basic belief assignment” ←←←← conflicts belief measure

“Basic structure” ←←←← conflicts Boolean structure function

“Mass assignment function” ←←←← mass confuses in engg. applications

“Basic assignment” ←←←←

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Evidential FunctionsEvidential FunctionsFundamentalsFundamentals

Belief Measure bel(A)

Belief is the degree of evidence

that the element in question belongs to the set A

as well as to the various special subsets of A.

Plausibility Measure pl(A)

Plausibility is the degree of evidence

that the element in question belongs to the set A

or to any of its subsets or to any set that overlaps with A.

0

Plausibility pl(A)

1

Belief bel(A)

Uncertainty

Doubt 1 – bel(A)

Disbelief 1 – pl(A)

∑ ≠⊆= φBAB mbel ; )()( BA

∑ ≠∩= φAB mpl )()( BA

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ComplementsComplementsFundamentalsFundamentals

Direct Complements

Belief versus doubt

Plausibility versus disbelief

Contextual Complements

Certainty →→→→ belief versus disbelief

Uncertainty included →→→→ plausibility versus doubt

0

Plausibility pl(A)

1

Belief bel(A)

Uncertainty

Doubt 1 – bel(A)


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PhenomenaPhenomenaFundamentalsFundamentals

Difference in Concepts – Existence of Phenomena

Evidential measures

No causal relationship between

belief in existence bel(A)

and belief in non-existence bel(¬A) = 1 – pl(A)

Probabilities

The belief in existence pr(xi = 1)

implies belief in non-existence pr(xi = 0) = 1 – pr(xi = 1)

0

Plausibility pl(A)

1

Belief bel(A)

Uncertainty

Doubt 1 – bel(A)


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OutlineOutlinePart 2Part 2


History

Scenario

Interpretations







Further Analyses


Pros & Cons

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ContextContextlllustrationlllustration

Typical Situation in a Power Plant

Operators @control panel →→→→ detect serious changes of system properties

Causes →→→→ failures detectable

Consequence →→→→ system fault

→→→→ neither be determined exactly nor interpreted certainly Widely discussed ( ATHEANA Report, Eric Hollnagel, etc.)

DST Approach Collects pieces of evidence

Postulates hypotheses

Proposes conclusions

Dempster-Shafer approach →→→→ supports operators in reasoning

Objective →→→→ to avoid an error forcing context

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ProcedureProcedurelllustrationlllustration

Eight Steps

Step – Creating the Scenario

Step – Quantification of Statements

Step – Combining Hypotheses

Step – Reducing the Combination Table

Step – Calculating Products & Sums of Combined Basic Assignments

Step – Combining Basic Assignments

Step – Evidence Measures of Combined Hypotheses

Step – Interpretation

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Creating the ScenarioCreating the Scenariolllustrationlllustration –– Step Step

Scenario

System →→→→ power plant

Data sources →→→→ operators →→→→ 2 persons

Pieces of evidence →→→→ failures detected →→→→ 4 considered

Hypotheses →→→→ system fault states →→→→ 3 considered

Frame of discernment →→→→ Ω = h1, h2, h3

Qualitative Failure-fault(s) Assignments

1st operator →→→→ h1, h2 consequences

2nd operator →→→→ h1, h3 consequences Same p-o-e, different hypotheses

DST Restrictions

No more than one failure

lead to the same fault (hypothesis)*

*) each data source

h1

h3

h1, h

3

h1, h

2, h

3

ev1

ev2

ev3

ev4

2nd

h1

h2

h1, h

2

h1, h

2, h

3

ev1

ev2

ev3

ev4

1st

Fault(s)FailureOp.

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Quantification of StatementsQuantification of Statementslllustrationlllustration –– Step Step

Quantification

Operators quantify statements, basis →→→→ data, intuition & experience

m(Ak) = 0 →→→→ no focal element

Belief

Example: 1st Operator, set h1 ∪ h2

Set and all its subsets

h1, h2, h1 ∪ h2 ⊆ h1 ∪ h2

bel(A4) = m(A1) + m(A2) + m(A4) = 0.9

Plausibility

At least 1 hypothesis in common

h1, h2, h1 ∪ h2, h1 ∪ h3,

h2 ∪ h3, h1 ∪ h2 ∪ h3

∩ h1 ∪ h2 ≠ ∅ pl(A4) = m(A1) + m(A2) + m(A4)

+ m(A5) + m(A6) + m(A7) = 1

m(B1) = 0.2

m(B2) = 0

m(B3) = 0.2

m(B4) = 0

m(B5) = 0.4

m(B6) = 0

m(B7) = 0.2

h1

h2

h3

h1

∪∪∪∪ h2

h1

∪∪∪∪ h3

h2

∪∪∪∪ h3

h1

∪∪∪∪ h2

∪∪∪∪ h3

m(A1) = 0.2

m(A2) = 0.1

m(A3) = 0

m(A4) = 0.6

m(A5) = 0

m(A6) = 0

m(A7) = 0.1

2nd operator2ΩΩΩΩ1st operator

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Quantification of StatementsQuantification of Statementslllustrationlllustration –– Step Step

Results

Input by operators →→→→ m(Ak)

Output by calculus →→→→ bel(Ak), pl(Ak)

0.8

0.2

0.8

0.8

1

0.8

1

0.2

0

0.2

0.2

0.8

0.2

1

0.2

0

0.2

0

0.4

0

0.2

h1

h2

h3

h1

∪∪∪∪ h2

h1

∪∪∪∪ h3

h2

∪∪∪∪ h3

ΩΩΩΩ

0.9

0.8

0.1

1

0.9

0.8

1

0.2

0.1

0

0.9

0.2

0.1

1

0.2

0.1

0

0.6

0

0

0.1

pl(Bk)bel(B

k)m(B

k)2ΩΩΩΩpl(A

k)bel(A

k)m(A

k)

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Combining HypothesesCombining Hypotheseslllustrationlllustration –– Step Step

Combination

Combining each set of hypotheses of both operators

Building cut sets ∩∩∩∩ of both

h1

h2

h3

h1∪h

2

h1∪h

3

h2∪h

3

Ω

∅h

2

h3

h2

h3

h2∪h

3

h2∪h

3

h1

∅h

3

h1

h1∪h

3

h3

h1∪h

3

h1

h2

∅h

1∪h

2

h1

h2

h1∪h

2

∅∅h

3

∅h

3

h3

h3

∅h

2

∅h

2

∅h

2

h2

h1

∅∅h

1

h1

∅h

1

B1

B2

B3

B4

B5

B6

B7

A7

A6

A5

A4

A3

A2

A1∩

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Reducing the Combination TableReducing the Combination Tablelllustrationlllustration –– Step Step

Combination

Objective →→→→ avoid mathematical effort

Drop rows & columns →→→→ non-focal elementsm(Ak) = 0, m(Bk) = 0

h1

h2

h3

h1∪h

2

h1∪h

3

h2∪h

3

Ω

∅h

2

h3

h2

h3

h2∪h

3

h2∪h

3

h1

∅h

3

h1

h1∪h

3

h3

h1∪h

3

h1

h2

∅h

1∪h

2

h1

h2

h1∪h

2

∅∅h

3

∅h

3

h3

h3

∅h

2

∅h

2

∅h

2

h2

h1

∅∅h

1

h1

∅h

1

B1

B2

B3

B4

B5

B6

B7

A7

A6

A5

A4

A3

A2

A1∩

h1

h3

h1∪h

3

Ω

h1

∅h

1

h1∪h

2

∅∅∅h

2

h1

∅h

1

h1

B1

B3

B5

B7

A7

A4

A2

A1∩

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Calculating Products & SumsCalculating Products & Sumslllustrationlllustration –– Step Step

Calculating Products

h1 ⇒ m(Z1) = m(A1) ⋅ m(B1) = 0.04

h1 ⇒ m(Z2) = m(A1) ⋅ m(B5) = 0.08

…

h1 ∪ h2 ∪ h3 ⇒ m(Z11) = m(A7) ⋅ m(B7) = 0.02

Calculating Sum(s)

Just h1 ⇒

0.02

0.02

0.04

0.02

0.12

0.24

0.12

0.02

0.04

0.08

0.04

B1

B3

B5

B7

A7

A4

A2

A1•

h1

h3

h1∪h

3

Ω

h1

∅h

1

h1∪h

2

∅∅∅h

2

h1

∅h

1

h1

B1

B3

B5

B7

A7

A4

A2

A1∩

54.0)(6

1

=∑=k

km Z

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Combining Basic AssignmentsCombining Basic Assignmentslllustrationlllustration –– Step Step

Sum of Product

Example: hypothesis h1 again →→→→

Calculating the Focal Sum

Sum of all basic assignment products →→→→

Basic Assignment of the Comb. Hypothesis

Example: hypothesis h1 →→→→

76.0)(11

1

=∑=k

km Z

54.0)(6

1

=∑=k

km Z

7105.0

)(

)(

)(11

1

6

11 ≈=

∑

∑

=

=

kk

kk

m

m

hm

Z

Z

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Measures of Combined HypothesesMeasures of Combined Hypotheseslllustrationlllustration –– Step Step

Evidence Measures

Input by Step →→→→ m(Zk)

Output by calculus of Step →→→→ bel(Zk), pl(Zk)

Ranking according to pl(Zk) & certainty

1

0.9737

0.9737

0.9471

0.2105

0.1053

1

0.8947

0.7895

0.7105

0.0263

0.0263

0.0263

0.1579

0.0526

0.7105

0.0263

0.0263

ΩΩΩΩh

1∪∪∪∪ h

2

h1

∪∪∪∪ h3

h1

h2

h3

plbelm2ΩΩΩΩ

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InterpretationInterpretationlllustrationlllustration –– Step Step

Interpretation

Which fault may be responsible for the serious changes of system

properties?

Probabilistic approach →→→→ blames h1 alone

Dempster-Shafer approach →→→→ points h1 and gives a hint to h2

Different mappings Ω →→→→ [0, 1] versus 2Ω →→→→ [0, 1]

1

0.9737

0.9737

0.9471

0.2105

0.1053

1

0.8947

0.7895

0.7105

0.0263

0.0263

0.0263

0.1579

0.0526

0.7105

0.0263

0.0263

ΩΩΩΩh

1∪∪∪∪ h

2

h1

∪∪∪∪ h3

h1

h2

h3

plbelm2ΩΩΩΩ

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History

Scenario

Interpretations







Further Analyses


Pros & Cons

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74

≥1

2 3 8 14 9

&

12 15 1 10 13 16 11 17 5 6

≥1

&

≥1≥1

&18 20 & 2119

≥1 ≥1

&

Fault Tree AnalysisFault Tree AnalysisBrief IntroductionBrief Introduction

Detailed Introduction

IEC 61025

Proceedings →→→→ references

Four Steps of the FTA Step – Define the top event of interest

Step – Define the analytical boundaries

Step – Define the tree-top structure

Step – Develop the path of faults

for each branch to the basic event

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DempsterDempster--Shafer FTA ApproachShafer FTA ApproachFault Tree AnalysisFault Tree Analysis

Scenario

Ω = h1, h2, h3 →→→→ “state occurs”, “uncertain”, “state does not occur”

Gates →→→→ And or Or

Inputs →→→→ e.g. two states, m(A) and m(B)

Output →→→→ state m(Z)

The Guth Approach to DS-FTA

m(A1) ≡ bel(A)

m(A2) ≡ pl(A) – bel(A)

m(A3) ≡ 1 – pl(A)

m(A1) + m(A2) + m(A3) = 1

Same for B0

Plausibility pl(A)

1

Belief bel(A)

Uncertainty

Doubt 1 – bel(A)


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AND and OR Combination

Similar to min/max operations

AND Gate According to Step

m(Z1) = m(A1) m(B1)

m(Z2) = m(A1) m(B2) + m(A2) m(B1) + m(A2) m(B2)

m(Z3) = … = m(A1) m(B3) + m(A2) m(B3) + m(A3)

OR Gate According to Step

m(Z1) = … = m(A1) + m(A2) m(B1) + m(A3) m(B1)

m(Z2) = m(A2) m(B2) + m(A2) m(B3) + m(A3) m(B2)

m(Z3) = m(A3) m(B3)

h1

h2

h3

h1

h2

h2

h1

h1

h1

B1

B2

B3

h3

h3

h3

h2

h2

h3

h1

h2

h3

B1

B2

B3

A3

A2

A1

OrA3

A2

A1

And

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Develop the path

Output of the lower gate →→→→ m(Z1), m(Z2), m(Z3)

Input of the next upper gate →→→→ m(A1), m(A2), m(A3)

Criticism

Interval arithmetic is more concise and efficient in operation than DST

→→→→ However, fault tree structure may cause trouble withthe sub-distributivity property of subtraction operations

as known from the fuzzy FTA

Multistate modelling, upper/lower probs, Bayesian networks …

h1

h2

h3

h1

h2

h2

h1

h1

h1

B1

B2

B3

h3

h3

h3

h2

h2

h3

h1

h2

h3

B1

B2

B3

A3

A2

A1

OrA3

A2

A1

And

7412 15 1 10 13

≥1≥1

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Event Tree AnalysisEvent Tree AnalysisBrief IntroductionBrief Introduction


IEC 62502


Five Steps of the ETA Step – List all possible initiating events

Step – Identify functional responses

Step – Define failure sequences

Step – Assign probabilities to each step

Step – Calculate the total probability of occurrence for each sequence

Initialevent

r2

A2

A1

A2

A1

A2

A1 r1

r3

r4

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Initialevent

r2

A2

A1

A2

A1

A2

A1 r1

r3

r4

DempsterDempster--Shafer ETA ApproachShafer ETA ApproachEvent Tree AnalysisEvent Tree Analysis

Scenario

Ω = h1, h2, h3 →→→→ “failure”, “no failure”, “uncertain”

Inputs →→→→ data source gives 3 values m(A1), m(A2), m(A3)

Output 1 →→→→ evidence measures for “failure”

Output 2 →→→→ evidence measures for “no failure”

Evidence Measures bel(Z1) = m(A1)

pl(Z1) = m(A1) + m(A3)

bel(Z2) = m(A2)

pl(Z2) = m(A2) + m(A3)

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Initialevent

r2

A2

A1

A2

A1

A2

A1 r1

r3

r4

DempsterDempster--Shafer ETA ApproachShafer ETA ApproachEvent Tree AnalysisEvent Tree Analysis

Procedure

Calculating evidence measures of every bifurcation of the ET

Then applying interval arithmetic

More details →→→→ RCM

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Reliability Centred MaintenanceReliability Centred MaintenanceBrief IntroductionBrief Introduction


IEC 60300-3-11


Seven Steps of the RCM Process Step – Establishing an expert group

Step – Functional breakdown of the system

Step – Conducting FMECA

Step – Collecting of data

Step – Tailoring the RCM decision diagram

Step – Applying the RCM decision diagram

Step – Documenting results

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Brief IntroductionBrief IntroductionReliabilityReliability--centred Maintenancecentred Maintenance

RCM Decision Diagram – Objective

Find a suitable strategy →→→→ component, module, system Framework of eight questions, six strategies

Testabilityof failure

Detectabilityof a failure

Scheduled maintenance

Periodical tests

Cond basedmaintenance

yes

First linemaintenance

Correctivemaintenance

First linemaint

First linemaint, alone?

Significantconsequences

Other reasonsfor prev maint

nonoyesyesno

yes no no yes

yes

yes

no

yes

nono Find abetter design

Cond basedmaint effective

Increasingfailure rate

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DS-RCM Example

Condition-based maintenance effective:

Do methods exist for

effective condition monitoring

so that an item failure

can be avoided?

Two answers

Two experts (example)

→→→→ two statements

Expert AssessmentExpert AssessmentReliabilityReliability--centred Maintenancecentred Maintenance

Cond.-basedmaintenance

effective?

YesYes

NoNo

Expert1

Expert1

Expert

2

Expert

2

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Input

Statements →→→→ “yes”, “no”, or “uncertain”

Quantification →→→→ basic assignments

Input & OutputInput & OutputReliabilityReliability--centred Maintenancecentred Maintenance


effective?

Yes 0.6

No 0.3

Unc 0.1

Yes 0.6

No 0.3

Unc 0.1

Yes 0.5

No 0.3

Unc 0.2

Yes 0.5

No 0.3

Unc 0.2

YesYes

NoNo

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Input

Statements →→→→ “yes”, “no”, or “uncertain”

Quantification →→→→ basic assignments

Output Values of evidential functions

Certainty

→→→→ 70% in “yes”

→→→→ 27% in “no” Uncertainty

→→→→ 3%

Input & OutputInput & OutputReliabilityReliability--centred Maintenancecentred Maintenance


effective?

Yes 0.6

No 0.3

Unc 0.1

Yes 0.6

No 0.3

Unc 0.1

Yes 0.5

No 0.3

Unc 0.2

Yes 0.5

No 0.3

Unc 0.2

Yesbel 0.70

pl 0.73

Yesbel 0.70

pl 0.73

No bel 0.27

pl 0.30

No bel 0.27

pl 0.30

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Input

Eight results of every “yes” or “no” decision

→→→→ values of evidential functions bel and pl

Calculus

Interval arithmetic →→→→ (easily)

Output Six weighted recommendations on maintenance strategies

Example, periodical testing bel = 0.51, pl = 0.62

Testabilityof failure

Detectabilityof a failure

Scheduled maintenance

Periodical tests

Cond basedmaintenance

yes

First linemaintenance

Correctivemaintenance

First linemaint

First linemaint, alone?

Significantconsequences

Other reasonsfor prev maint

nonoyesyesno

yes no no yes

yes

yes

no

yes

nono Find abetter design

Cond basedmaint effective

Increasingfailure rate

Weighted RecommendationsWeighted RecommendationsReliabilityReliability--centred Maintenancecentred Maintenance

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Further AnalysesFurther AnalysesSome HintsSome Hints

Failure Mode, Effects and Criticality Analysis

IEC 60812

Dempster-Shafer approach →→→→ Section 4.1 Proceedings

Preliminary/Potential Hazard Analysis … same holds for PHA

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History

Scenario

Interpretations







Further Analyses


Pros & Cons

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Some CommentsSome CommentsOutroductionOutroduction

Disclaimer

Nobody is forced to apply DST instead of Probability Theory

No uncertainties →→→→ no DS modelling recommended (?)

Prefer modelling uncertainties by probabilities? →→→→ apply probabilities

Prefer … interval arithmetic? →→→→ apply interval arithmetic

Prefer … fuzzy sets? →→→→ apply fuzzy sets

Applying DST is an option, not an obligation

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Disadvantages of the DSTDisadvantages of the DSTOutroductionOutroduction

Disadvantages (also valid for Probabilities)

Lack of introspection or assessment strategies

unreasonable requirement for precision →→→→ m

difficult to determine with necessary precision

Instability

estimated m may be influenced by the conditions of its estimation

Ambiguity

ambiguous or imprecise judgement could not be expressed

by the evidence measures

Disadvantages

Frame of discernment Ω →→→→ given k hypotheses →→→→ up to 2k elements

larger number of values →→→→ than after the Probability Theory DST does not offer a procedure for implementation of a diagnostic system

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Advantages of the DSTAdvantages of the DSTOutroductionOutroduction

Advantages

Calculus describes & quantifies the subjective viewpoint

as an assessment for an unknown objective fact

Applying DST is an option, not an obligation

“If the only tool you have is a hammer,

you tend to see every problem as a nail.” Abraham Maslow

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Fundamentals of theFundamentals of the

DempsterDempster--Shafer Theory and its Shafer Theory and its

Applications to SystemApplications to System

Safety and Reliability ModellingSafety and Reliability Modelling

Uwe Kay RakowskyUwe Kay Rakowsky

University of Wuppertal, GermanyUniversity of Wuppertal, Germany

Department D Department D –– Safety EngineeringSafety Engineering

Documents

Fundamentals of the Dempster -Shafer Theory and its Applications … · 2017. 11. 30. · Methods introduced (FTA, ETA, RCM, FMECA) Fundamentals of the Dempster-Shafer Theory Intro