System Reliability and Availability Estimation Under Uncertainty Tongdan Jin, Ph.D. Ingram School of Engineering Texas State University, San Marcos, TX [email protected] 4/11/2012 1
Mar 29, 2015
System Reliability and Availability Estimation Under Uncertainty
Tongdan Jin, Ph.D.
Ingram School of EngineeringTexas State University, San Marcos, TX
4/11/2012 1
Contents
2
System Reliability Estimation* Variance of reliability estimate* Series, and parallel systems
Operational Availability* Performance based maintenance/logistics/contracting* Reliability growth or spare parts stocking ?
* A unified availability model
Conclusion
3
Topic One
Modeling System ReliabilityWith
Uncertain Estimates
4
Two Components having Same Reliability?
Component
Test Plan 1Testing 100 hoursSample n=10, survivals=9
9.010
91 r
Test Plan 2Testing 100 hoursSample n=20, survivals=18
9.020
182 r
Which component is more reliable?
5
Risk-Averse vs. Risk-Neutral Design
• = probability density function for reliability estimate • risk-neutral design would always choose system 1• risk-adverse design might choose system 2
system 1
system 2
R
)ˆ(Rf
)ˆ(Rf R
)ˆ()ˆ(
]ˆ[]ˆ[
21
21
RVarRVar
RERE
6
Variance of Reliability Estimate
Test Plan 1Testing 100 hoursSample n=10, survivals=9
01.0110
)9.01(9.0)ˆr(av
9.010
91
r
r
Test Plan 2Testing 100 hoursSample n=20, survivals=18
Which component is more reliable?
0047.0120
)9.01(9.0)ˆr(av
9.020
181
r
r
1
)ˆ1(ˆ)ˆr(av
n
rrr
7
Variance vs. Sample Size
1
)ˆ1(ˆ)ˆr(av
1
n
rrr
n
xr n=sample size
x=survivals
Variance of Component Reliability Estimate
0.000
0.050
0.100
0.150
0.200
0 10 20 30 40
Sample Size
Va
rian
ce
r=0.8
r=0.9
8
Reliability Variance of Series Systems
Component 1 Component 2
k
iii
k
iis
k
iis
rrrR
rR
1
2
1
2
1
))ˆr(avˆ(ˆ)ˆr(av
ˆˆ
)ˆr(avˆ)ˆr(avˆˆˆ)ˆr(av
ˆˆˆ
22
212
12
22
1
21
rrrrrrR
rrR
s
s
k Components in Series
9
Numerical Example
Component 1 Component 2
Test Plan 1Testing 100 hoursn1=10, x1=9n2=20, x2=17
0067.0)ˆr(av
01.0)ˆr(av
85.0ˆ
9.0ˆ
2
1
2
1
r
r
r
r
0126.0
)0067.085.0()01.09.0(
)85.09.0()ˆr(av
765.0)85.0)(9.0(ˆ
22
22
s
s
R
R
k
iii
k
iis
k
iis
rrrR
rR
1
2
1
2
1
))ˆr(avˆ(ˆ)ˆr(av
ˆˆ
10
Reliability Confidence Estimate
)ˆr(av]ˆ[ˆsss RZRER
sRAssuming is normally distributed, the lower bound
0126.0)ˆr(av
765.0]ˆ[
s
s
R
RE
58.00126.0)64.1(765.0ˆ
62.00126.0)28.1(765.0ˆ
s
s
R
R With 90% confidence
With 95% confidence
11
Reliability Variance of Parallel System
Component 1
Component 2
2121 ˆˆ1)ˆ1)(ˆ1(1ˆ qqrrRp
Where
ii rq ˆ1ˆ for i=1, and 2
12
Estimates for Reliability and Unreliability
1
)ˆ1(ˆ)ˆr(av
ˆ
n
rrr
n
xr n=sample size
x=survivals
)ˆr(av1
)ˆ1(ˆ
1
ˆ)ˆ1()ˆr(av
ˆ
rn
rr
n
qqq
n
xnq
Ser
ies
Sys
tem
Par
alle
l S
yste
m
13
Variance of Parallel System
k
iii
k
iip
k
ii
k
iip
qqqR
qrR
1
2
1
2
11
))ˆr(avˆ(ˆ)ˆr(av
ˆ1)ˆ1(1ˆ
Where ii rq ˆ1ˆ
k components in parallel
1q
2q
kq
14
Numerical Example
Test Plan 1
Testing 100 hoursn1=10, x1=9n2=20, x2=17
0067.0)ˆr(av
01.0)ˆr(av
15.0ˆ1ˆ;85.0ˆ
1.0ˆ1ˆ;9.0ˆ
2
1
222
111
q
q
rqr
rqr
000225.0
)0067.015.0()01.01.0(
)15.01.0()ˆr(av
985.0)15.0)(1.0(1ˆ
22
22
p
p
R
R
Component 1
Component 2
k
iii
k
iip
k
ii
k
iip
qqqR
qrR
1
2
1
2
11
))ˆr(avˆ(ˆ)ˆr(av
ˆ1)ˆ1(1ˆ
15
Reliability Confidence Estimate
)ˆr(av]ˆ[ˆppp RZRER
pRAssuming is normally distributed, then
000225.0)ˆr(av
985.0]ˆ[
p
p
R
RE
960.0000225.0)64.1(985.0ˆ
966.0000225.0)28.1(985.0ˆ
p
p
R
R With 90% confidence
With 95% confidence
16
Series-Parallel Systems
1
1
2
4
7
3
5
6
1’
4
7
2’
5 ’
1’’
4 ’
7
1’’ 7 ’
Var
ianc
e E
stim
atio
n
17
Compute r and var(r) over Time
time (hours)Sample
Size FailuresCum
Failures Reliability Variance
1 20 0 0 1 0
2 20 0 0 1 0
3 20 0 0 1 0
4 20 1 1 0.95 0.0025
5 20 0 1 0.95 0.0025
6 20 0 1 0.95 0.0025
7 20 1 2 0.9 0.0047
8 20 1 3 0.85 0.0067
9 20 2 5 0.75 0.0099
10 20 1 6 0.7 0.0111
18
Topic Two
Operational Availabilityunder
Performance Based Contract (PBC)
Service Parts Logistics Business
• Representing 8-10% of GDP in the US.
• US airline industry is $45B on MRO in 2008.
• US auto industry is $190B and $73B for parts in 2010.
• US DoD maintenance budget $125B and $70B inventory with 6,000 suppliers.
• Joint Strike Fighter (F-35): $350B for R/D/P, and $600B for after-production O/M for 30 years.
• EU Wind turbine service revenue €3B in 2011
• IBM computing/network servers, etc.
19
20C
ost
($)
10-20%
30-40% 50-60%
ResearchDevelopment Manufacturing Operation and Support Retirement
Cumulative costs over product life
Total Ownership Cost Distribution
5%
PBC aims to lower the cost of ownership while ensuring system performance goals
Reference DoD 5000, University of Tennessee
• Scherbrooke (1968, 1992)• Muckstadt (1973)• Graves (1985)• Lee (1987)• Cohen et al. (1990)• Diaz & Fu (1996)• Alfredsson (1997)• Zamperini & Freimer (2005)• Lau & Song (2008)• Kutanoglu et al. (2009)• More .....
Spare Parts Logistics
Reliability Allocation and Spare Parts Logistics
• Tillman et al. (1977)• Kuo et al. (1987)• Chen (1992)• Jin & Coit (2001)• Levitin & Lisnianski (2001)• Coit et al. (2004)• Ramirez-Marquez et al. (2004)• Marseguerra, Zio (2005)• Jin & Ozalp (2009) • Ramirez-Marquez & Rocco (2010)• More .....
Reliability Allocation
r4(t)
r5 (t)
r6(t)
r7(t)
r8(t)
r2(t)
r1(t)
n)r(
n)r(
n)r(
,min
),(varmin
)],([max
tCost
tR
tRE
sys
sys
s32
s32
s3,n
s3,n-1
ss21
s22
Fleet 1
Fleet 2
Fleet n-1
Fleet n
),(),(min
)(max
xsxs,
xs,
EBOCost
Ap
21
22
A 4-Step Performance-Based Contracting
Step 1Performance
Outcome
Step 1Performance
Outcome
Step 2Performance
Measures
Step 2Performance
Measures
Step 3Performance
Criteria
Step 3Performance
Criteria
Step 4Performance
Compensation
Step 4Performance
Compensation
System readiness,
operational reliability,
assurance of spare parts
supply
System readiness,
operational reliability,
assurance of spare parts
supply
System availability,
MTBF, MTTR, Mean
downtime, logistics
response time
System availability,
MTBF, MTTR, Mean
downtime, logistics
response time
Mini availability, max failure
rate, max repair waiting time, max cost per
unit time
Mini availability, max failure
rate, max repair waiting time, max cost per
unit time
Cost plus incentive fee,
cost plus award fee, linear reward,
exponential reward
Cost plus incentive fee,
cost plus award fee, linear reward,
exponential reward
23
Five Performance Measures by US DoD
• Operational availability (OA)
• Inherent reliability or mission reliability (MR)
• Logistics response time (e.g. MTTR, LDT)
• Cost per unit usage (CUU)
• Logistics footprint
24
Interactions of Five Performance Measures
OperationalAvailability(OA)
MissionReliability (MR)
Logistics Response Time (LRT)
LogisticsFootprint (LF)
Cost Per Unit Usage (CUU)
MLDTMTTRMTBF
MTBFAo
MTBF=Mean Time Between FailuresMTTR=Mean Time to RepairMLDT=Mean Logistics Delay Time
Evolution of Sustainment/Maintenane Solution
25
CMPM
Tot
al O
wne
rshi
p C
ost
CM=>{Warranty, MBC}PM=>{MBC}CBM=>{Warranty, MBC}PBM/PBL=>{PBC}
PBMCBM
PBC aims to lower the cost of ownership while ensuring system performance (e.g. reliability and availability).
Note: PBM=performance-based maintenance
Repair CenterRepair Center
Local spares
stocking
Local spares
stocking
SystemfleetN(t)
Supplier or OEM
OEM for design and
manufacturing
OEM for design and
manufacturing
Customer
Emergency Repair
Integrating Manufacturing with Service
26
Repair CenterRepair Center
Local spares
stocking
Local spares
stocking
SystemfleetN(t)
Supplier or OEM
OEM for design and
manufacturing
OEM for design and
manufacturing
Customer
Emergency Repair
Availability and Variable Fleet Size
Variable Fleet Size
27
0
200
400
600
800
1000
0
400
800
1,200
1,600
2,000
1
15
30
45
60
75
90
10
5
12
0
13
5
Cu
mu
lativ
e F
lee
t Siz
e
MT
BF
(ho
urs
)
Weeks
Figure 1: System Reliability and Fleet Size
MTBF
System Population
0
20,000
40,000
60,000
80,000
100,000
120,000
140,000
160,000
98-9
9
02-0
3
2006
2008
2010
2012
2014
2016
2018
2020
2022
2024
2026
2028
2030
Inst
alle
d W
T Po
pula
tion
Cumulative Installed WT (1998 to 2030)
1.5 MW (2010-2014 )2.0 MW (2015-2020)2.5 MW (2020-2025)3.0 MW (2025-2030)
150,000
22,500
Se
mic
on
du
cto
r In
du
str
y
Win
d P
ow
er
Ind
us
try
• Availability
MDTMTBF
MTBFA
• MTBF=100 hours, MDT=5 hours
95.05100
100
A
• MTBF=200 hours, MDT=10 hours
95.010200
200
A
Performance Measures and Drivers
28
Operational Availability (Ao)
Logistics Support(s, ts, tr)
Inherent Reliability ()
Maintenance Schedule ()
System Fleet(n, )
MTBF
MTTR
MLDTCustomerControlled
OEMControlled
A Unified Operational Availability Model
s
x
tnxr
rs
sro
xetn
tt
ttnsAr
0 !)(
11
1),,,,,(
=system or subsystem inherent failure rate
s =base stock level
β =usage rate, and 0β1
n =installed base size
tr=repair turn-around time
ts=time for repair-by-replacement
Ref: Jin & Wang (2011)
29
30
Trading Reliability with Spares Stocking (II)
Note: here lambda=alpha in previous slide
2000 4000 6000 8000 10000 12000 14000 160000
2
4
6
8
10
MTBF (1/lambda) in hours
spar
e pa
rts
num
ber
Reliability vs. Spare Parts Inventory
=0.5, n=50, tr=60 days
Ao=0.95
Ao=0.8
31
Trading Reliability with Spares Stocking (I)
1000 2000 3000 4000 5000 6000 7000 80000
2
4
6
8
10
MTBF (1/lambda) in hours
num
ber
of s
pare
par
ts
Reliability vs. Spare Parts Inventory Level
Ao=0.95
Ao=0.8
=0.5, n=50, tr=30 days
32
Trading Reliability and Spares Stocking (III)
0 2000 4000 6000 8000 10000 120000
5
10
15
20
MTBF (1/lambda) in hours
spar
e pa
rts
num
ber
Reliability vs. Spare Parts Inventory
Ao=0.95
Ao=0.8
=0.8, n=50, tr=30 days
1. Variance of reliability estimate2. Variance propagation3. Series/parallel reduction4. Unbiased estimate5. Operational availability6. Mean downtime7. Mean time to repair8. Mean logistics delay time9. Mean time between failures10. Mean time to failure11. Performance based logistics/contracting/maintenance12. Performance measure13. Performance criteria14. Material based contracting
Key Terminologies
33
1. Variance is a simple, yet accurate metric to gauge the reliability uncertainty
2. Estimating the reliability variance for series, parallel and mixed series-parallel systems
3. PBC aims to guarantee the system performance while lowering the cost of ownership
4. PBC incentivizes the OEM/3PL to maximize the profit by optimizing the development, production and logistics delivery.
Conclusion
34
1. D. W. Coit, “System reliability confidence intervals for complex systems with estimated component reliability,” IEEE Transactions on Reliability, vol. 46, no. 4, 1997, pp. 487-493.
2. J. E. Ramirez-Marquez, and W. Jiang, “An improved confidence bounds for system reliability,” IEEE Transactions on Reliability, vol. 55, no. 1, 2006, pp. 26-36.
3. E. Borgonov, “A new uncertainty measure”, Reliability Engineering and System Safety, vo;. 92, pp. 771-784, 2007.
4. T. Jin, D. Coit, "Unbiased variance estimates for system reliability estimate using block decompositions," IEEE Transactions on Reliability , vol. 57, 2008, pp.458-464.
5. H. Guo, T. Jin, A. Mettas, “Designing reliability demonstration test for one-shot systems under zero component failures," IEEE Transactions on Reliability , vol. 60, no. 1, 2011, pp. 286-294
References
35
Reliability Estimation
1. Huang, H.-Z., H.J. Liu, D.N.P. Murthy. 2007. Optimal reliability, warranty and price for new products. IIE Transactions, vol. 39, no. 8, pp. 819-827.
2. Kang, K., M. McDonald. 2010. Impact of logistics on readiness and life cycle cost: a design of experiments approach, Proceedings of Winter Simulation Conference. pp. 1336-1346.
3. Kim, S.H., M.A. Cohen, S. Netessine. 2007. Performance contracting in after-sales service supply chains. Management Science, vol. 53, pp. 1843-1858.
4. Nowicki, D., U.D. Kumar, H.J. Steudel, D. Verma. 2008. Spares provisioning under performance-based logistics contract: profit-centric approach. The Journal of the Operational Research Society. vol. 59, no. 3, 2008, pp. 342-352.
5. Öner, K.B., G.P. Kiesmüller, G.J. van Houtum. 2010. Optimization of component reliability in the design phase of capital goods. European Journal of Operational Research, vol. 205, no. 3, pp. 615-624.
6. T. Jin, P. Wang, “Planning performance based contracts considering reliability and uncertaint system usage,” Journal of the Operational Research Society , 2012 (forthcoming)
7. Jin, T., Y. Tian, “Optimizing reliability and service parts logistics for a time-varying installed base,” European Journal of Operational Research, vol. 218, no. 1, 2012, pp. 152-162
Availability Estimation