System Reliability and Availability Estimation Under Uncertainty Tongdan Jin, Ph.D. Ingram School of Engineering Texas State University, San Marcos, TX.

System Reliability and Availability Estimation Under Uncertainty

Tongdan Jin, Ph.D.

Ingram School of EngineeringTexas State University, San Marcos, TX

tj17@txstate.edu

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

36

For QuestionsE-mail to

tj17@txstate.edu