This document presents a Dempster-Shafer approach to modelling uncertainties in Reliability Centred Maintenance (RCM). It discusses limitations of qualitative and probabilistic RCM methods and introduces the Dempster-Shafer theory as an alternative. The key aspects covered include defining the frame of discernment and hypotheses, collecting evidence from expert sources, determining basic assignments, and calculating belief and plausibility measures to provide weighted recommendations on maintenance strategies. The approach aims to better represent uncertainties and allow non-definite expert judgements compared to traditional RCM methods.

1. Modelling of Uncertainties inModelling of Uncertainties in
Reliability Centred MaintenanceReliability Centred Maintenance ––
a Dempstera Dempster--Shafer ApproachShafer Approach
Uwe Kay Rakowsky, Ulrike GochtUwe Kay Rakowsky, Ulrike Gocht
University of Wuppertal, GermanyUniversity of Wuppertal, Germany
University of Applied SciencesUniversity of Applied Sciences Zittau/GZittau/Göörlitzrlitz, Germany, Germany

2. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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Reliability Centred MaintenanceReliability Centred Maintenance
IntroductionIntroduction
Objective
RCM →→→→ find a suitable maintenance strategy
DS-RCM →→→→ express uncertainties of experts in reasoning
DS-RCM →→→→ give weighted recommendations during the RCM process
What’s new?
Evidence measures belief and plausibility are applied instead of
→→→→ either “yes” or “no” decisions
→→→→ probabilities ( ESREL 98)
→→→→ fuzzy membership functions ( ESREL 98)
What’s not new?
RCM process conducted
RCM diagram applied
Fundamentals of the Dempster-Shafer Theory ( Proceedings)
Intro OutroRCM Application
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Tailoring a DS Approach
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3. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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Reliability Centred MaintenanceReliability Centred Maintenance
Brief IntroductionBrief Introduction
Detailed Introduction
IEC 60300-3-11
Proceedings →→→→ references
Five Steps of the RCM Process
Step – Establishing an expert group
Step – Functional breakdown of the system
Step – Collecting of data
Step – Tailoring & applying the RCM decision diagram
Step – Documenting results
Intro OutroRCM Application
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Tailoring a DS Approach
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The RCM Decision DiagramThe RCM Decision Diagram
Brief IntroductionBrief Introduction
RCM Decision Diagram – Objective
Find a suitable strategy →→→→ component, module, system
Framework of eight questions, six strategies
Testability
of failure
Detectability
of a failure
Scheduled
maintenance
Periodical
tests
Cond based
maintenance
yes
First line
maintenance
Corrective
maintenance
First line
maint
First line
maint, alone?
Significant
consequences
Other reasons
for prev maint
nonoyesyesno
yes no no yes
yes
yes
no
yes
nono Find a
better design
Cond based
maint effective
Increasing
failure rate
Intro OutroRCM Application
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Tailoring a DS Approach
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5. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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The RCM Decision DiagramThe RCM Decision Diagram
Brief IntroductionBrief Introduction
Note
Example →→→→ mix of Det Norske Veritas & Marintek [ ]
Tailor the diagram to your needs!
Testability
of failure
Detectability
of a failure
Scheduled
maintenance
Periodical
tests
Cond based
maintenance
yes
First line
maintenance
Corrective
maintenance
First line
maint
First line
maint, alone?
Significant
consequences
Other reasons
for prev maint
nonoyesyesno
yes no no yes
yes
yes
no
yes
nono Find a
better design
Cond based
maint effective
Increasing
failure rate
Intro OutroRCM Application
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Tailoring a DS Approach
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6. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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Testability
of failure
Detectability
of a failure
Scheduled
maintenance
Periodical
tests
Cond based
maintenance
yes
First line
maintenance
Corrective
maintenance
First line
maint
First line
maint, alone?
Significant
consequences
Other reasons
for prev maint
nonoyesyesno
yes no no yes
yes
yes
no
yes
nono Find a
better design
Cond based
maint effective
Increasing
failure rate
The Qualitative ApproachThe Qualitative Approach
Brief IntroductionBrief Introduction
Drawbacks of Qualitative RCM
Expert group →→→→ consensus required
“Yes” or “no” →→→→ no percentage answer
Any doubt? →→→→ no quantification of doubt
No weighted recommendations →→→→ no ranking of strategies
Intro OutroRCM Application
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Tailoring a DS Approach
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The Probabilistic ApproachThe Probabilistic Approach
Brief Introduction toBrief Introduction to pp--RCMRCM
Introducing Probabilities
Decision variable xi, i = 1, …, 8 (eight questions/decisions)
“yes” →→→→ xi = 1, “no” →→→→ xi = 0
Expected value pi
Some Interpretations of pi
Probability that an expert answers “yes” to question i
Degree of an expert's belief that “yes” is the right decision
Mean value for decision i resulting from questioning many experts
Expected value of the binomial distributed decision variable xi
Results ri
Decision variable ri, i = 1, …, 6 (six maintenance strategies)
Apply the Event Tree method
r1 = (p1 + q1⋅p2)(q3 + p3⋅q4) q5⋅p6 , …, r6 = q1⋅q2 ←←←← DS-RCM

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Tailoring a DempsterTailoring a Dempster--Shafer RCMShafer RCM
ModellingModelling
Scenario
Frame of discernment
Hypotheses
Data sources
Pieces of evidence
Frame of Discernment
Only one single question/decision is considered
Representation →→→→ universal set Ω
Intro OutroRCM Application
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Tailoring a DS Approach
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The ScenarioThe Scenario
ModellingModelling
Scenario
Frame of discernment
Hypotheses
Data sources
Pieces of evidence
Hypotheses
Single hypothesis →→→→ one answer to a question
→ “yes”, “no”, or “uncertain”
Hypotheses →→→→ unique and not overlapping and mutually exclusive
Universal set Ω represents →→→→ Ω = {“yes”, “no”, “uncertain”}
Subsets of Ω →→→→ single or conjunctions of hypotheses
Set of all subsets →→→→ power set 2Ω
Intro OutroRCM Application
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Tailoring a DS Approach
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The ScenarioThe Scenario
ModellingModelling
Scenario
Frame of discernment
Hypotheses
Data sources
Pieces of evidence
Data Sources
Expert group
→→→→ e.g. service eng., maintenance personnel, system eng., reliability eng.
Task →→→→ give subjective quantifiable statements
Basis →→→→ data, intuition & experience
Intro OutroRCM Application
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Tailoring a DS Approach
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The ScenarioThe Scenario
ModellingModelling
Scenario
Frame of discernment
Hypotheses
Data sources
Pieces of evidence
Pieces of Evidence
Piece of evidence →→→→ expert judgement
Assignment: piece evidence →→→→ hypothesis
Assignment strength →→→→ quantified by m
Intro OutroRCM Application
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Tailoring a DS Approach
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The ScenarioThe Scenario
ModellingModelling
Scenario
Frame of discernment
Hypotheses
Data sources
Pieces of evidence
Pieces of Evidence
Piece of evidence →→→→ expert judgement (data, intuition & experience)
Assignment
(evidence →→→→ hypothesis) corresponds to (cause → consequence)
Assignment
(1 p-of-e) assigned to (1 hypothesis or 1 set of hypotheses)
(different p-of-e) may not be assigned to (same hypothesis nor set)
Quantification
strength of implication →→→→ m(A)

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Intro OutroRCM Application
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Tailoring a DS Approach
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Sets
Ω universal set
A ⊆⊆⊆⊆ Ω set, contains 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
ΣA⊆⊆⊆⊆Ωm(A)=1 all statements of an expert are normalised
m(A) > 0 focal element
m(∅∅∅∅) = 0 simplicity (not required)
Differences in Properties to Probabilities
m(Ω) = 1 not required
m(A) vs. m(¬A) no relationship
If A ⊂ B ⊆⊆⊆⊆ Ω, then m(A) ≤ m(B) not required
FundamentalsFundamentals
The DempsterThe Dempster--Shafer CalculusShafer Calculus
This is noThis is no
probability mappingprobability mapping
Again & again, theseAgain & again, these
are NO probabilitiesare NO probabilities

15. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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Evidential FunctionsEvidential Functions
DempsterDempster--Shafer CalculusShafer Calculus
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
Intro OutroRCM Application
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Tailoring a DS Approach
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16. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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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 Assessment
RCM ApplicationRCM Application
Cond.-based
maintenance
effective?
YesYes
NoNo
Expert
1
Expert
1
Expert
2
Expert
2
Intro OutroRCM Application
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Tailoring a DS Approach
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17. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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Input
Statements →→→→ “yes”, “no”, or “uncertain”
Quantification →→→→ basic assignments
Input & OutputInput & Output
RCM ApplicationRCM Application
Cond.-based
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
Intro OutroRCM Application
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Tailoring a DS Approach
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18. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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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 & Output
RCM ApplicationRCM Application
Cond.-based
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
Yes
bel 0.70
pl 0.73
Yes
bel 0.70
pl 0.73
No
bel 0.27
pl 0.30
No
bel 0.27
pl 0.30
Intro OutroRCM Application
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Tailoring a DS Approach
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19. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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Complements
Belief in “yes” versus
doubt in “yes” ≡≡≡≡ plausibility of “no”
Plausibility of “yes” versus
disbelief in “yes” ≡≡≡≡ belief in “no”
ComplementsComplements
RCM ApplicationRCM Application
Yes
bel 0.70
pl 0.73
Yes
bel 0.70
pl 0.73
Cond.-based
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
No
bel 0.27
pl 0.30
No
bel 0.27
pl 0.30
Intro OutroRCM Application
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Tailoring a DS Approach
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Weighted RecommendationsWeighted Recommendations
RCM ApplicationRCM Application
Input
Eight results of every “yes” or “no” decision
→→→→ values of evidential functions bel and pl
Calculus
Interval arithmetic →→→→ proceedings (easily)
Output
Six weighted recommendations on maintenance strategies
Example, periodical testing bel = 0.51, pl = 0.62
Testability
of failure
Detectability
of a failure
Scheduled
maintenance
Periodical
tests
Cond based
maintenance
yes
First line
maintenance
Corrective
maintenance
First line
maint
First line
maint, alone?
Significant
consequences
Other reasons
for prev maint
nonoyesyesno
yes no no yes
yes
yes
no
yes
nono Find a
better design
Cond based
maint effective
Increasing
failure rate
Intro OutroRCM Application
|
Tailoring a DS Approach
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21. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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ConclusionConclusion
OutroductionOutroduction
Comments on the Methods
Probabilistic, fuzzy-, and Dempster-Shafer RCM →→→→ hardly comparable
Prob →→→→ one-dimensional value
Fuzzy →→→→ graph, peak, results depend on defuzzification method
DS →→→→ interval, no peak
Comments on the Results
Results are close to each other
Results based on
→→→→ different interpretations
Interpretations based on
→→→→ different theoretical concepts
Testability
of failure
Detectability
of a failure
Scheduled
maintenance
Periodical
tests
Cond based
maintenance
yes
First line
maintenance
Corrective
maintenance
First line
maint
First line
maint, alone?
Significant
consequences
Other reasons
for prev maint
nonoyesyesno
yes no no yes
yes
yes
no
yes
nono Find a
better design
Cond based
maint effective
Increasing
failure rate
Intro OutroRCM Application
|
Tailoring a DS Approach
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22. Rakowsky & Gocht Uncertainties in RCM – Dempster-Shafer Approach
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ConclusionConclusion
OutroductionOutroduction
Even more Comments
DS calculus →→→→ easy to apply →→→→ less than one page
Experts feel more comfortable →→→→ degrees instead “yes” or “no”
Not forced to single strategy →→→→ second best?
etc. →→→→
DS-RCM offers a different kind of flavour to the maintenance analyst
Testability
of failure
Detectability
of a failure
Scheduled
maintenance
Periodical
tests
Cond based
maintenance
yes
First line
maintenance
Corrective
maintenance
First line
maint
First line
maint, alone?
Significant
consequences
Other reasons
for prev maint
nonoyesyesno
yes no no yes
yes
yes
no
yes
nono Find a
better design
Cond based
maint effective
Increasing
failure rate
Intro OutroRCM Application
|
Tailoring a DS Approach
|

23. Modelling of Uncertainties inModelling of Uncertainties in
Reliability Centred MaintenanceReliability Centred Maintenance ––
a Dempstera Dempster--Shafer ApproachShafer Approach
Uwe Kay Rakowsky, Ulrike GochtUwe Kay Rakowsky, Ulrike Gocht
University of Wuppertal, GermanyUniversity of Wuppertal, Germany
University of Applied SciencesUniversity of Applied Sciences Zittau/GZittau/Göörlitzrlitz, Germany, Germany