2. Región de Tarapacá, Mapa de Riesgo, Variable de Riesgo Tsunami-Volcánica
Sessión 2-1 Riesgo
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Transcript of Sessión 2-1 Riesgo
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Introduction Part 1
1-1
1
©A.K.S. Jardine
Risk in MaintenanceDecisions
What is risk? (2.1)&
How can we estimate risk (2.2)
Andrew K.S.Jardine [email protected]
October, 2002
2
©A.K.S. Jardine
Basic Statistics
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Introduction Part 1
1-2
3
©A.K.S. Jardine
Maintenance Management
Optimizing Equipment Maintenance & Replacement Decisions
1.Best PreventiveReplacement Time
a) Replace only onfailure
b) Constant Interval
c) Age-Based
d) Deterministic
Performance
Deterioration
2.Glasser’s Graphs
3.Spare PartsProvisioning
4.RepairableSystems
5.Software RELCODE
1.Economic Life
a) Constant AnnualUtilization
b) Varying AnnualUtilization
c) Technological
2.Tracking IndividualUnits
3.Repair vs Replace
4.Software PERDEC &AGE / CON
1.InspectionFrequency for aSystem
a) ProfitMaximization
b) AvailabilityMaximization
2.A, B, C, D ClassInspection Intervals
3.FFI’s for ProtectiveDevices
4.CBM (Oil & Vib.Analysis)
5.Blended HealthMonitoring & AgeReplacement
6.Software EXAKT
1.WorkshopMachines /CrewSizes.
2.Right SizingEquipment
a) Own Equipment
b) Contracting OutPeaks inDemand
3.Lease / Buy
ComponentReplacement
Capital EquipmentReplacement
InspectionProcedures
ResourceRequirements
Probability & Statistics
(Weibull Analysis)
Time Value of Money
(Discounted Cash Flow)DynamicProgramming Queueing Theory
Simulation
DATA BASE (CMMS/EAM/ERP)
4
©A.K.S. Jardine
Normal Distribution
2
2
2
)(
2
1)( ?
?
??
??
?
t
et f
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1f(t)f(t)
tt
f(t)f(t)
MTTFMTTF
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Introduction Part 1
1-3
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©A.K.S. Jardine
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Risk ofRisk of
failurefailure
timetimetptp tptp
CCpp : total cost of a preventive replacement.: total cost of a preventive replacement.
CCf f : total cost of a failure replacement.: total cost of a failure replacement.
PR
6
©A.K.S. Jardine
Exponential Distribution
t et f ?? ??)(
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f(t)f(t)
MTTFMTTF
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Introduction Part 1
1-4
7
©A.K.S. Jardine
What Happens Now?
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
timetime
Risk of Risk of
failurefailure
Replace Only On FailureReplace Only On Failure
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©A.K.S. Jardine
Normal Distribution
µ (MTTF)
0
5
10
15
20
25
30
35
40
45
f(t)-probability density function (p.d.f).
f(t)
t
?
?
??
? 0.1)( t f
2
2
2
)(
2
1)( ?
?
??
??
?
t
et f
µ-mean
s -standard deviation
50% 50%
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Introduction Part 1
1-5
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©A.K.S. Jardine
Exponential Distribution
0
10
20
30
40
50
60
70
?-mean arrival rate of failure
1/?- mean
1/ ?=MTTF
63.2%
f(t)
t
t et f ?? ??)(
10
©A.K.S. Jardine
Weibull Distribution
? : shape parameter
? : characteristic life
?? ???
?? ???
?? ???
0
10
20
30
40
50
60
?=1/2 (Hyperexponential)
?=1 (Exponential)
?=2 (Rayleigh)
?=3.5 (Normal)
f(t)
t
?
?
?
??
? ???
?
??
?
??
?
??
?
?
??
?
??
t
et
t f
1
)(
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Introduction Part 1
1-6
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©A.K.S. Jardine
?
N.B. when ?=1 then ?=MTTF
MTTF: Mean Time To Failure
N.B. ? : time at which cumulative probability =63.2%
f(t)
t0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
63.2%
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©A.K.S. Jardine
FAILURE RATE [r(t)]
For normal distribution:
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time
r(t)
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Introduction Part 1
1-7
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©A.K.S. Jardine
FAILURE RATE [r(t)]
time
For Exponential distribution:
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r(t)
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©A.K.S. Jardine
FAILURE RATE [r(t)]
time
r(t)
For Weibull distribution:
ß>1
ß
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Introduction Part 1
1-8
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©A.K.S. Jardine
Cumulative distribution function[F(t)]
F(t): Probability of failure before time t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time
f(t)
t0
F(t)
f(t)
??t
dt t f t F 0
)()(
16
©A.K.S. Jardine
For all p.d.f ’s we have:
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time
F(t)
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Introduction Part 1
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©A.K.S. Jardine
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
RELIABILITY FUNCTION[R(t)]
R(t): Probability of survival at least to
time t.
time
f(t)
t
R(t)
f(t)
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©A.K.S. Jardine
For all density functions we have:
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time
R(t)
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Introduction Part 1
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©A.K.S. Jardine
SUMMARY
time
F(t) + R(t) = 1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f(t)
t
R(t)F(t)
f(t)
20
©A.K.S. Jardine
Failure Rate (Hazard rate)
[r(t)]
f(t)
r(t) =
1 – F(t)
This is a conditional probability, with r(t)dt being the
probability that an item fails during the interval dt,
given that it has survived to time t.
r(t) = f(t) / R(t)
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Introduction Part 1
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©A.K.S. Jardine
Failure Rate, r(t)
• Also known as HAZARD rate.
• It is the conditional probability that an item
fails during the interval dt, given that it has
survived to time t.
)()(
)(1)()(
t Rt f
t F t f t r ?
??
timetime
f(t)f(t)
t t+?t0
1
22
©A.K.S. Jardine
System Hazard Function
Stress Related
Failures
Quality
Failures
?
Infant
Mortality
? ?
Useful Life
? ? ?
Wearout
Overall Life
Characteristic Curve
Wearout
Failures
Equipment Life Periods
Time
Failure
Rate,
Risk, or
Hazardfunction
Source: Department of National Defence
