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

## Presentation on theme: "System Reliability and Availability Estimation Under Uncertainty Tongdan Jin, Ph.D. Ingram School of Engineering Texas State University, San Marcos, TX."— Presentation transcript:

System Reliability and Availability Estimation Under Uncertainty Tongdan Jin, Ph.D. Ingram School of Engineering Texas 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 Reliability With Uncertain Estimates

4 Two Components having Same Reliability? Component Test Plan 1 Testing 100 hours Sample n=10, survivals=9 Test Plan 2 Testing 100 hours Sample n=20, survivals=18 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

6 Variance of Reliability Estimate Test Plan 1 Testing 100 hours Sample n=10, survivals=9 Test Plan 2 Testing 100 hours Sample n=20, survivals=18 Which component is more reliable?

7 Variance vs. Sample Size n=sample size x=survivals r=0.8 r=0.9

8 Reliability Variance of Series Systems Component 1Component 2 k Components in Series

9 Numerical Example Component 1Component 2 Test Plan 1 Testing 100 hours n 1 =10, x 1 =9 n2=20, x 2 =17

10 Reliability Confidence Estimate Assuming is normally distributed, the lower bound With 90% confidence With 95% confidence

11 Reliability Variance of Parallel System Component 1 Component 2 Where for i=1, and 2

12 Estimates for Reliability and Unreliability n=sample size x=survivals Series System Parallel System

13 Variance of Parallel System Where k components in parallel

14 Numerical Example Test Plan 1 Testing 100 hours n 1 =10, x 1 =9 n 2 =20, x 2 =17 Component 1 Component 2

15 Reliability Confidence Estimate Assuming is normally distributed, then With 90% confidence With 95% confidence

16 Series-Parallel Systems

17 Compute r and var(r) over Time time (hours) Sample SizeFailures Cum FailuresReliabilityVariance 1200010 2 0010 3 0010 4 110.950.0025 520010.950.0025 620010.950.0025 720120.90.0047 820130.850.0067 920250.750.0099 1020160.70.0111

18 Topic Two Operational Availability under 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

20 Cost (\$) 10-20% 30-40% 50-60% Research Development ManufacturingOperation 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)r4(t) r 5 (t) r6(t)r6(t) r7(t)r7(t) r8(t)r8(t) r2(t)r2(t) r1(t)r1(t) s 32 s 3,n s 3,n-1 s s 21 s 22 Fleet 1 Fleet 2 Fleet n-1 Fleet n 21

22 A 4-Step Performance-Based Contracting Step 1 Performance Outcome Step 1 Performance Outcome Step 2 Performance Measures Step 2 Performance Measures Step 3 Performance Criteria Step 3 Performance Criteria Step 4 Performance Compensation Step 4 Performance Compensation System readiness, operational reliability, assurance of spare parts supply System availability, MTBF, MTTR, Mean downtime, logistics response 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

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 Operational Availability(OA) Mission Reliability (MR) Logistics Response Time (LRT) Logistics Footprint (LF) Cost Per Unit Usage (CUU) MTBF=Mean Time Between Failures MTTR=Mean Time to Repair MLDT=Mean Logistics Delay Time

Evolution of Sustainment/Maintenane Solution 25 CM PM Total Ownership Cost CM=>{Warranty, MBC} PM=>{MBC} CBM=>{Warranty, MBC} PBM/PBL=>{PBC} PBM CBM PBC aims to lower the cost of ownership while ensuring system performance (e.g. reliability and availability). Note: PBM=performance-based maintenance

Repair Center Repair Center Local spares stocking Local spares stocking System fleet N(t) Supplier or OEM OEM for design and manufacturing Customer Emergency Repair Integrating Manufacturing with Service 26 Repair Center Repair Center Local spares stocking Local spares stocking System fleet N(t) Supplier or OEM OEM for design and manufacturing Customer Emergency Repair

Availability and Variable Fleet Size Variable Fleet Size 27 Semiconductor Industry Wind Power Industry Availability MTBF=100 hours, MDT=5 hours MTBF=200 hours, MDT=10 hours

Performance Measures and Drivers 28 Operational Availability (A o ) Logistics Support (s, t s, t r ) Inherent Reliability ( ) Maintenance Schedule ( ) System Fleet (n, ) MTBF MTTR MLDT Customer Controlled OEM Controlled

A Unified Operational Availability Model =system or subsystem inherent failure rate s=base stock level β =usage rate, and 0 β 1 n=installed base size trtr =repair turn-around time tsts =time for repair-by-replacement Ref: Jin & Wang (2011) 29

30 Trading Reliability with Spares Stocking (II) Note: here lambda=alpha in previous slide =0.5, n=50, t r =60 days A o =0.95 A o =0.8

31 Trading Reliability with Spares Stocking (I) A o =0.95 A o =0.8 =0.5, n=50, t r =30 days

32 Trading Reliability and Spares Stocking (III) A o =0.95 A o =0.8 =0.8, n=50, t r =30 days

1.Variance of reliability estimate 2.Variance propagation 3.Series/parallel reduction 4.Unbiased estimate 5.Operational availability 6.Mean downtime 7.Mean time to repair 8.Mean logistics delay time 9.Mean time between failures 10.Mean time to failure 11.Performance based logistics/contracting/maintenance 12.Performance measure 13.Performance criteria 14.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 Questions E-mail to tj17@txstate.edu

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

Similar presentations