System Reliability and Availability Estimation Under Uncertainty Tongdan

System Reliability and Availability Estimation Under Uncertainty Tongdan Jin, Ph. D. Ingram School of Engineering Texas State University, San Marcos, TX tj 17@txstate. edu 4/11/2012 1

Contents § 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 2

3 Topic One Modeling System Reliability With Uncertain Estimates

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? 4

Risk-Averse vs. Risk-Neutral Design system 2 system 1 • = probability density function for reliability estimate • risk-neutral design would always choose system 1 • risk-adverse design might choose system 2 5

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? 6

Variance vs. Sample Size n=sample size x=survivals r=0. 8 r=0. 9 7

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

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

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

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

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

Variance of Parallel System k components in parallel Where 13

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

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

Series-Parallel Systems 5 4 1 2 6 3 Variance Estimation 1 7 4 1’ 5’ 2’ 7 4’ 1’’ 7’ 16

Compute r and var(r) over Time time (hours) Sample Size Failures Cum Failures 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 Reliability Variance 17

18 Topic Two Operational Availability under Performance Based Contract (PBC)

19 Service Parts Logistics Business • Representing 8 -10% of GDP in the US. • US airline industry is $45 B on MRO in 2008. • US auto industry is $190 B and $73 B for parts in 2010. • US Do. D maintenance budget $125 B and $70 B inventory with 6, 000 suppliers. • Joint Strike Fighter (F-35): $350 B for R/D/P, and $600 B for after-production O/M for 30 years. • EU Wind turbine service revenue € 3 B in 2011 • IBM computing/network servers, etc.

20 Total Ownership Cost Distribution ct l u d o r ver p ife sts o o c e tiv umula Cost ($) C 30 -40% 50 -60% 10 -20% Research Manufacturing Development Operation and Support 5% Retirement PBC aims to lower the cost of ownership while ensuring system performance goals Reference Do. D 5000, University of Tennessee

21 Reliability Allocation and Spare Parts Logistics Reliability Allocation Spare Parts Logistics r 5 (t) r 1(t) r 2(t) r 6(t) r 4(t) r 8(t) s r 7(t) s 21 s 22 s 32 Fleet 1 s 32 Fleet 2 s 3, n-1 Fleet n-1 s 3, n • • • 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. . . • • • 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. . . Fleet n

A 4 -Step Performance-Based Contracting Step 1 Performance Outcome Step 2 Performance Measures Step 3 Performance Criteria 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 22

Five Performance Measures by US Do. D • Operational availability (OA) • Inherent reliability or mission reliability (MR) • Logistics response time (e. g. MTTR, LDT) • Cost per unit usage (CUU) • Logistics footprint 23

Interactions of Five Performance Measures MTBF=Mean Time Between Failures MTTR=Mean Time to Repair MLDT=Mean Logistics Delay Time Mission Reliability (MR) Operational Availability(OA) Logistics Footprint (LF) Cost Per Unit Usage (CUU) Logistics Response Time (LRT) 24

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

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

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

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

29 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 tr =repair turn-around time ts =time for repair-by-replacement Ref: Jin & Wang (2011)

Trading Reliability with Spares Stocking (II) =0. 5, n=50, tr=60 days Ao=0. 8 Ao=0. 95 Note: here lambda=alpha in previous slide 30

Trading Reliability with Spares Stocking (I) =0. 5, n=50, tr=30 days Ao=0. 8 Ao=0. 95 31

Trading Reliability and Spares Stocking (III) =0. 8, n=50, tr=30 days Ao=0. 8 Ao=0. 95 32

Key Terminologies 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 33

Conclusion 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/3 PL to maximize the profit by optimizing the development, production and logistics delivery. 34

References Reliability Estimation 1. 2. 3. 4. 5. 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. 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. E. Borgonov, “A new uncertainty measure”, Reliability Engineering and System Safety, vo; . 92, pp. 771784, 2007. T. Jin, D. Coit, "Unbiased variance estimates for system reliability estimate using block decompositions, " IEEE Transactions on Reliability , vol. 57, 2008, pp. 458 -464. 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 Availability Estimation 1. 2. 3. 4. 5. 6. 7. 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. Kang, K. , M. Mc. Donald. 2010. Impact of logistics on readiness and life cycle cost: a design of experiments approach, Proceedings of Winter Simulation Conference. pp. 1336 -1346. Kim, S. H. , M. A. Cohen, S. Netessine. 2007. Performance contracting in after-sales service supply chains. Management Science, vol. 53, pp. 1843 -1858. 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. 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. T. Jin, P. Wang, “Planning performance based contracts considering reliability and uncertaint system usage, ” Journal of the Operational Research Society , 2012 (forthcoming) 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 35

For Questions E-mail to tj 17@txstate. edu 36