Reliability Growth in JMP 10

Transcription

1 Reliability Growth in JMP 10 Presented at Discovery Summit 2012 September 13, 2012 Marie Gaudard and Leo Wright

2 Purpose of Talk The goal of this talk is to provide a brief introduction to: The area of reliability growth The Reliability Growth platform in JMP Reliability Growth platform future directions 2012 North Haven Group 2

3 What is Reliability Growth? Reliability growth addresses the techniques used to understand and improve a product s reliability over time. Detect failure sources Monitor redesign efforts based on root cause identification Continue testing until performance goals are met 2012 North Haven Group 3

4 What is Reliability Growth? Reliability Growth Testing Process From Army Material Systems Analysis Activity Design for Reliability Handbook, p North Haven Group 4

5 What is Reliability Growth? Given a failure mode, a decision must be made as to whether to address that failure mode during testing or to delay corrective action until the end of test phase. There are three ways to respond to a failure mode during a test phase: Implement a corrective action during the test phase (Test-Fix-Test) Implement a corrective action at the end of the test phase (Test-Find-Test) Implement no corrective action 2012 North Haven Group 5

6 What is Reliability Growth? Test-Fix-Test: All fixes implemented during test phase MTBF Test Duration 2012 North Haven Group 6

7 What is Reliability Growth? Test-Find-Test: All fixes implemented at the end of the test phase MTBF Jump due to implementation of delayed fixes Test Duration 2012 North Haven Group 7

8 What is Reliability Growth? Test-Fix-Test with Delayed Fixes: Some fixes implemented during test phase, some at the end of the test phase Jump due to implementation of delayed fixes MTBF Test Duration 2012 North Haven Group 8

9 What is Reliability Growth? If goals are not met, testing is then continued into another phase. MTBF Jump due to implementation of delayed fixes Phase 1 Phase North Haven Group 9

10 What is Reliability Growth? Often, the management strategy is Test-Fix-Test with Delayed Fixes over several test phases North Haven Group 10

11 Why Reliability Growth? Keep consumers: Safe Satisfied Loyal Build healthy and competitive organizations: Improve quality Reduce expenses returns, rework, warranty repairs Remain profitable in a tough global marketplace 2012 North Haven Group 11

12 Models Available in JMP s Reliability Growth Platform Depending on the choices made in the launch window, the RG platform fits various non-homogeneous Poisson Process (NHPP) models and performs automatic change-point detection: Crow AMSAA Fixed Parameter Crow AMSAA Piecewise Weibull NHPP Reinitialized Weibull NHPP Piecewise Weibull NHPP Change Point Detection 2012 North Haven Group 12

13 Crowe AMSAA Model A common form for the intensity function of an NHPP is ν () t = λβt β 1 This special case of an NHPP is called the power law process or the Weibull process. Scale parameter: λ Shape parameter: β Reliability growth parameter: 1 - β 2012 North Haven Group 13

14 Crowe AMSAA Model Intensity function: β ν = λβ 1 () t t Mean time between failures (MTBF): MTBF = 1/ ( t) = 1/ t β ν λβ 1 Cumulative failures: Nt () = λt β 2012 North Haven Group 14

15 Intensity for Various Betas, Lambda = North Haven Group 15

16 MTBF for Various Betas, Lambda = North Haven Group 16

17 Launch Window 2012 North Haven Group 17

18 Launch Window: Data Formats The launch window includes a tab for each of two data formats: Time to Event Format Assumes that time is recorded as the number of time units, for example, days or hours, since initial start-up of the system. The test start time is assumed to be time zero. Dates Format Assumes that time is recorded in a date/time format, indicating an absolute date or time North Haven Group 18

19 Event Count Phase Launch Window: Event Count and Phase This is the number of events, usually failures addressed by corrective actions (fixes), occurring at the specified time or within the specified time interval. Reliability growth programs often involve several periods, or phases, of active testing. These testing phases can be specified in the optional Phase column North Haven Group 19

20 Data Table: Exact Failure Times versus Interval Censoring Exact Failure Times A single column is entered. It is assumed that the column gives the exact elapsed times at which events occurred. Two columns are entered, but the interval start and end times are identical. Interval Censoring (Grouped Data) Two columns are entered, giving interval start and end times. If an interval s start and end times differ, it is assumed that the corresponding events occurred at some unknown time within that interval North Haven Group 20

21 Data Table: Failure and Time Termination Termination method is inferred from the entries in the last row in a phase. Single Test Phase If the last row contains an exact failure time with a nonzero event count, the test is considered failure terminated. If the last row contains an exact failure time with a zero event count, the test is considered time terminated. If the last row contains an interval with nonzero width, the test is considered time terminated with termination time equal to the right endpoint of that interval North Haven Group 21

22 Data Table: Failure and Time Termination Multiple Test Phases: Phase start times must be entered as rows in the data table. This may required showing event counts of zero. Suppose that Phase A ends and that Phase B begins at time t B. If the failure time for the last failure in Phase A is exact and if that time differs from t B, then Phase A is considered to be time terminated. The termination time is equal to t B. If the last failure time in Phase A is exact and is equal to t B, then Phase A is assumed failure terminated. If the last failure in Phase A is interval-censored, Phase A is considered to be time terminated with termination time equal to t B North Haven Group 22

23 Report Options: Models and Estimation Crow AMSAA model Fixed Parameter Crow AMSAA - When you fix one parameter, the other is re-estimated and plots update Piecewise Weibull NHPP Fits Crow AMSAA models to two or more phases under the constraint that the cumulative number of events at the start of a phase equals the number at the end of the previous phase Reinitialized Weibull NHPP Fits an independent growth model to each test phase Piecewise Weibull NHPP Change Point Detection Estimates a time point where the reliability model changes 2012 North Haven Group 23

24 Report Options: Achieved MTBF The observed data represent only one of infinitely many possible failure-time sequences from an NHPP. Suppose that the test is failure terminated at the nth failure. The confidence interval computed in the Achieved MTBF report takes into account the fact that the n failure times are random. If the test is time terminated, then the number of failures as well as their failure times are random. This is why the confidence interval for the Achieved MTBF differs from the confidence interval provided by the MTBF Profiler at the last observed failure time North Haven Group 24

25 Report Options: Goodness of Fit Tests Do the data follow an NHPP with Weibull intensity? One time column: Cramér-von Mises Intended for exact failure times Compares theoretical to empirical distribution function Large values lead to rejection Uses unbiased estimator for β 2012 North Haven Group 25

26 Report Options: Goodness of Fit Tests Do the data follow an NHPP with Weibull intensity? Two time columns: Chi-squared test Intended for interval-censored failure times Compares observed to expected numbers of failures in the time intervals defined. Large values lead to rejection. Intended for interval-censored data where the time intervals specified in the data table cover the entire time period of the test North Haven Group 26

27 Recurrence Platform The Recurrence platform is meant for repairable systems. The goal is to estimate the mean cumulative function (MCF). The MCF is the total cost per unit as a function of time (can be simply the number of repairs). The Recurrence platform models NHPP, HPP, Proportional Intensity Poisson Process, and Loglinear NHPP intensities. The parameters of the intensity functions can be constant or linear functions of effects North Haven Group 27

28 Recurrence Platform Note that the NHPP parametrizations in Reliability Growth and Recurrence differ (due to conventions in the literature for both areas) Recurrence: Reliability Growth: ν () t β t = θ θ β 1 () t = t ν λβ β North Haven Group 28

29 Supporting Material 2012 North Haven Group 29

30 Brief History 1936 T.P. Wright described manufacturing improvements mathematically. He showed that as the quantity of airplanes were produced in sequence, the direct labor input per plane decreased in a mathematical pattern, forming a straight line when plotted on log-log paper J.T. Duane (GE Motors), found that cumulative estimates of MTBF plotted against cumulative operating time on log-log paper follow an approximately straight line. Mid 1970 s L. Crow, while working the Army Materials Systems Activity (AMSAA), converted Duane s postulate into rigorous statistics and developed the more general Crow- AMSAA model North Haven Group 30

31 Duane Model Introduced in 1964 Duane, J.T., "Learning Curve Approach to Reliability Monitoring," IEEE Transactions on Aerospace, Vol. 2, No. 2, Addressed the issue of variation in reliability of complex electromechanical and mechanical systems in the early stages of development. Goals were to: monitor development progress, predict growth patterns, and plan programs for reliability improvement. Duane used date from five different products to support the idea of a common learning curve North Haven Group 31

32 Duane Model Duane s Figure 2: Plot of Cumulative Failure Rate against Operating Hours 2012 North Haven Group 32

33 Duane Model Let N(t) = Cumulative number of failures at time t. Duane plotted the log of cumulative failure rate, N(t)/t, against the log of cumulative operating hours, t. Duane concluded that the plots showed linear behavior: log( Nt ( ) / t) = ( β 1)log( t) + λ, for β > 0 It follows that the cumulative failure rate is: Nt () And that the instantaneous failure rate at time t is: β ν = λβ 1 () t t Beta is a shape parameter, lambda is a scale parameter North Haven Group 33 = λt β

34 Duane Model The cumulative failure rate is N(t)/t. This means that the MTBF is t/n(t). If we express log( Nt ( ) / t) = ( β 1)log( t) + λ, for β > 0 in terms of MTBF, we obtain: log(mtbf) = log( t/ Nt ( )) = (1 β)log( t) λ, for β > 0 The quantity 1 β is called the Reliability Growth Slope North Haven Group 34

35 Crowe AMSAA Model Developed by Dr. Larry H. Crow, working at the U.S. Army Materiel Systems Analysis Activity (AMSAA) Growth model details published in MIL-HDBK-189, "Reliability Growth Management," February 13, 1981 (pp ). Applies for fixes introduced as failures occur, as opposed to fixes that are delayed to the end of a test phase North Haven Group 35

36 Crowe AMSAA Model Crow AMSAA model assumes that reliability growth is a nonhomogeneous Poisson process. The number of failures in an interval is a random variable with a Poisson distribution, but the mean of the Poisson distribution varies with time. This model can be applied to either continuous or discrete reliability systems, single or multiple systems, and tests which are time or failure truncated North Haven Group 36

37 Crowe AMSAA Model Definition. N((a,b]) is a random variable that counts the number of failures in the interval (a,b]. N(t) = N((0,t]) is number of failures up to time t. A counting process N(t) follows a Poisson process if: 1. N(0) = 0 2. For any a < b c < d, N((a,b]) and N((c, d]) are independent 3. There is a function ν, called the intensity function of the Poisson process, such that PNtt ( (, + t] = 1) ν ( t) = lim t 0 t 4. There are no simultaneous failures 2012 North Haven Group 37

38 Crowe AMSAA Model A homogeneous Poisson process (HPP) is a Poisson process with constant intensity function ν(t) = ν. A process is HPP with intensity function ν if and only if the times between failures are iid exponential random variables with mean 1/ν North Haven Group 38

39 Crowe AMSAA Model A nonhomogeneous Poisson process (NHPP) is a Poisson process with non-constant intensity function ν(t). Example. Suppose that a repairable system is modeled by a NHPP with intensity function ν ( t) = 0.2t North Haven Group 39

40 Crowe AMSAA Model Intensity function: β ν = λβ 1 () t t Mean time between failures (MTBF): MTBF = 1/ ( t) = 1/ t β ν λβ 1 Cumulative failures: Nt () = λt β 2012 North Haven Group 40

41 Crowe AMSAA Model Test Phase Termination: Fixed number of failures Fixed period of time (number of failures is random) Failure times: Exact Censored or grouped data 2012 North Haven Group 41

42 Crowe AMSAA Model Failure-truncated study, truncated at n failures: n n β 1 β L( λ, β t1, t2,..., tn) = ( λβ ) ti exp( λtn ) i= 1 Time-truncated study, truncated at time T: n n 1 L(, nt, 1, t2,..., tn) ( ) t β β λ β = λβ i exp( λt ) i= North Haven Group 42

43 Crowe AMSAA Model Grouped data: There are k intervals, [b i, b i+1 ), i=1,,k. Let m i = number of failures in interval [b i, b i+1 ). L( λβ, m,..., m ) 1 K ( β β ) m i λ λ( β β ) K = bi+ b + 1 i exp bi b 1 i i= 1 mi! 2012 North Haven Group 43