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This presentation describes the test method, sample size computation, degradation models, and cost function for the lognormal bogey testing. Then the presentation discusses the optimum test plans, which choose the optimal sample size and the expected test time by minimizing the total test cost and simultaneously satisfying the constraints on the type II error and the available sample size. An example is given to illustrate the test method.

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- 1. Efficient Reliability Demonstration Test (快速可靠性验证试验) Guangbin Yang (杨广斌), Ph.D. ©2012 ASQ & Presentation Yang Presented live on Feb 19th, 2012http://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_liability Calendar/Webinars ‐_Chinese/Webinars_‐_Chinese.html
- 2. ASQ Reliability Division ASQ Reliability Division Chinese Webinar Series Chinese Webinar Series One of the monthly webinars One of the monthly webinars on topics of interest to reliability engineers. To view recorded webinar (available to ASQ Reliability Division members only) visit asq.org/reliability ) / To sign up for the free and available to anyone live webinars visit reliabilitycalendar.org and select English Webinars to find links to register for upcoming eventshttp://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_liability Calendar/Webinars ‐_Chinese/Webinars_‐_Chinese.html
- 3. Efficient Reliability Demonstration Tests 快速可靠性验证试验 Guangbin Yang (杨广斌), Ph.D. Ford Motor Company, Dearborn, Michigan, U.S.A. Email: gbyang@ieee.org
- 4. Overview1. Introduction2. Sample sizes for bogey tests (zero-failure tests)3. Principles of test time reduction4. Test cost modeling5. Risk of early termination of the test6. Optimal test plans7. Procedures of test time reduction8. Application example9. Summary and conclusions 2
- 5. Bogey Testing (Zero-Failure Test) Bogey test is widely used in industry to demonstrate, at a high confidence, that a product achieves a specified reliability. This test method requires a sample of predetermined size to be tested for a specified length of time. The required reliability is demonstrated if no failures occur in the testing. So a bogey test is sometimes called the zero- failure test. 3
- 6. Motivation A bogey test requires a large sample size and excessive test time. For example, to demonstrate that a product has 95% reliability at 1 million cycles with 95% confidence, a bogey test requires 59 samples, each tested for 1 million cycles. In the current competitive business environment, the sample size and test time must be reduced. 4
- 7. Sample Size for Conventional BinomialBogey Testing In some applications, life distribution is unknown. To demonstrate at a 100(1–)% confidence that a product achieves the reliability R0 at time t0, a sample of size n1 is drawn from a population, where ln( ) n1 ln( R0 ) Each of the n1 units is tested for t0. If zero failures occur during testing, the reliability is demonstrated. 5
- 8. Sample Size for Conventional LognormalBogey Testing In some situations, the life of products can be reasonably modeled by lognormal distribution. The minimum sample size to demonstrate the reliability requirement is ln( ) n2 ln{[ ln( ) / 1 (1 R0 )]} where is called the bogey ratio, which is the ratio of actual test time to t0. The equation indicates that the sample size can be reduced by increasing the test time. 6
- 9. Sample Sizes for Different Values ofRequired Reliability and Bogey Ratio 90 80 bogey ratio=1.5 bogey ratio=2 70 bogey ratio=2.5 60 n2 50 bogey ratio=3 40 30 20 10 0 0.8 0.85 0.9 0.95 1 Reliability 7
- 10. Principles of Test Time Reduction For some products, a failure is said to have occurred when a performance characteristic exceeds its threshold. For these products, it is possible to measure the performance characteristic during testing. The degradation measurements can be used to reduce the test time. 8
- 11. Principles of Test Time Reduction (Conted) y G tm t0 t 9
- 12. Principles of Test Time Reduction (Conted) y G tm t0 t 10
- 13. Principles of Test Time Reduction (Conted) y G tm t0 t 11
- 14. Sample Size for Reduced Test Time When the test time is reduced, the type II error is increased by c. The minimum sample size for the lognormal distribution is ln( ) ln(1 ) n3 ln{[ ln( ) / 1 (1 R0 )]} where = c /. 12
- 15. Test Cost Modeling The cost of a bogey test consists of the cost of conducting the test, the cost of samples, and the cost of measurements. Cost model ln( ) ln(1 ) TC ( , ) c1t0 (c2 c3 m) ln{[ ln( ) / (1 R0 )]} 1 13
- 16. Consumer’s Risk Due to Early Termination y F(t0) G tm t0 t For a test unit that has y0<G, terminating test earlier increases the consumer’s risk. 14
- 17. Producer’s Risk Due to Early Termination y F(t0) G tm t0 t For a test unit that has y0>G, terminating test earlier increases the producer’s risk. 15
- 18. Risk Formulation For a linear or transformed linear degradation model, the risk can be formulated as G y0 ˆ F (t 0 ) Pr T 1 3(m 2m) 2 1 ˆ m m 2 (m 2 1) where T has the student-t distribution with m-2 degrees of freedom. 16
- 19. Optimal Test Plans The test plans are characterized by and . The values of and are optimized by minimizing the total cost TC(, ), while the following constraints are satisfied: (1) The risk associated with early termination of a test must not exceed c/n3. (2) The sample size must not be greater than the number of available test units. The optimization model can be calculated using Excel Solver. 17
- 20. Procedures of Test Time Reduction During testing, each unit is inspected periodically to measure y. When there are three measurements, a degradation model is fitted to the data, and the estimates of the model parameters and the risk F(t0) are calculated. Then we make one of the following decisions based on the estimates. 18
- 21. Test Termination Rules ˆ (1) If F (t 0 ) / n3 , then terminate the test of the unit. This test unit passes the bogey test. (2) If F (t 0 ) 1 / n3 , where is the specified type I error, ˆ then terminate the test of the unit. This test unit fails to pass the bogey test. ˆ (3) If / n3 F (t 0 ) 1 / n3 , continue the test until decision rule (1) or (2) is met, or until t0 is reached, whichever occurs sooner. 19
- 22. Application Example Problem statement A part is required to have a reliability of 95% at a design life of 1.5105 cycles under the 95th percentile of the customer usage profile. The part fails due to its stiffness degradation; a failure is said to have occurred when the stiffness degrades to 20% of the initial value. We want to demonstrate the reliability at a 95% confidence level. 20
- 23. Test Plans The calculation of optimization model for the test plan gives = 0.3147, and = 0.631. Then the test plan is to test 39 samples and the expected test time is 1.325105 cycles. In contrast, the conventional bogey test requires testing 59 samples for each 1.5105 cycles, or 39 units each for 2.1105 cycles. 21
- 24. Decision Rules The decision rules for terminating the test of a part are as follows. (1) If F (t 0 ) 0.403 10 3 , then terminate the test of the ˆ unit. This test unit passes the bogey test. ˆ (2) If F (t 0 ) 0.9987 , then terminate the test of the unit. This test unit fails to pass the bogey test. (3) If 0.403 10 3 F (t0 ) 0.9987 , continue the test of the ˆ unit until decision rule (1) or (2) is met, or until 2.1105 cycles is reached, whichever occurs first. 22
- 25. Summary and Conclusions The conventional binomial bogey test requires a large sample size and excessive test time. If the life is known to be lognormal, the bogey test sample size can be reduced by extending the test time. For products subject to degradation failure, the test time can be reduced substantially by using the degradation measurements. 23
- 26. Additional Readings G. Yang, “Reliability Demonstration Through Degradation Bogey Testing,” IEEE Transactions on Reliability, vol. 58, no. 4, December 2009. G. Yang, “Optimum Degradation Tests for Comparison of Products,” IEEE Transactions on Reliability, vol. 61, no. 1, March 2012. G. Yang, Life Cycle Reliability Engineering, Wiley, 2007. (Chapter 9) 24

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