Design of Experiments for Reliability Achievement E-Book

Steven E. Rigdon

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Probability and Statistics
• Sprache: Englisch

Seitenzahl: 582

Veröffentlichungsjahr: 2022

Beschreibung

ENABLES READERS TO UNDERSTAND THE METHODS OF EXPERIMENTAL DESIGN TO SUCCESSFULLY CONDUCT LIFE TESTING TO IMPROVE PRODUCT RELIABILITY

This book illustrates how experimental design and life testing can be used to understand product reliability in order to enable reliability improvements. The book is divided into four sections. The first section focuses on statistical distributions and methods for modeling reliability data. The second section provides an overview of design of experiments including response surface methodology and optimal designs. The third section describes regression models for reliability analysis focused on lifetime data. This section provides the methods for how data collected in a designed experiment can be properly analyzed. The final section of the book pulls together all of the prior sections with customized experiments that are uniquely suited for reliability testing. Throughout the text, there is a focus on reliability applications and methods. It addresses both optimal and robust design with censored data.

To aid in reader comprehension, examples and case studies are included throughout the text to illustrate the key factors in designing experiments and emphasize how experiments involving life testing are inherently different. The book provides numerous state-of-the-art exercises and solutions to help readers better understand the real-world applications of experimental design and reliability. The authors utilize R and JMP® software throughout as appropriate, and a supplemental website contains the related data sets.

Written by internationally known experts in the fields of experimental design methodology and reliability data analysis, sample topics covered in the book include:

• An introduction to reliability, lifetime distributions, censoring, and inference for parameter of lifetime distributions
• Design of experiments, optimal design, and robust design
• Lifetime regression, parametric regression models, and the Cox Proportional Hazard Model
• Design strategies for reliability achievement
• Accelerated testing, models for acceleration, and design of experiments for accelerated testing

The text features an accessible approach to reliability for readers with various levels of technical expertise. This book is a key reference for statistical researchers, reliability engineers, quality engineers, and professionals in applied statistics and engineering. It is a comprehensive textbook for upper-undergraduate and graduate-level courses in statistics and engineering.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

About the Companion Website

Part I: Reliability

1 Reliability Concepts

1.1 Definitions of Reliability

1.2 Concepts for Lifetimes

1.3 Censoring

Problems

2 Lifetime Distributions

2.1 The Exponential Distribution

2.2 The Weibull Distribution

2.3 The Gamma Distribution

2.4 The Lognormal Distribution

2.5 Log Location and Scale Distributions

Problems

3 Inference for Parameters of Life Distributions

3.1 Nonparametric Estimation of the Survival Function

3.2 Maximum Likelihood Estimation

3.3 Inference for the Exponential Distribution

3.4 Inference for the Weibull

3.5 The SEV Distribution

3.6 Inference for Other Models

3.7 Bayesian Inference

3.A Kaplan–Meier Estimate of the Survival Function

Problems

Notes

Part II: Design of Experiments

4 Fundamentals of Experimental Design

4.1 Introduction to Experimental Design

4.2 A Brief History of Experimental Design

4.3 Guidelines for Designing Experiments

4.4 Introduction to Factorial Experiments

4.5 The

Factorial Design

4.6 Fractional Factorial Designs

Problems

5 Further Principles of Experimental Design

5.1 Introduction

5.2 Response Surface Methods and Designs

5.3 Optimization Techniques in Response Surface Methodology

5.4 Designs for Fitting Response Surfaces

Problems

Part III: Regression Models for Reliability Studies

6 Parametric Regression Models

6.1 Introduction to Failure‐Time Regression

6.2 Regression Models with Transformations

6.3 Generalized Linear Models

6.4 Incorporating Censoring in Regression Models

6.5 Weibull Regression

6.6 Nonconstant Shape Parameter

6.7 Exponential Regression

6.8 The Scale‐Accelerated Failure‐Time Model

6.9 Checking Model Assumptions

Problems

7 Semi‐parametric Regression Models

7.1 The Proportional Hazards Model

7.2 The Cox Proportional Hazards Model

7.3 Inference for the Cox Proportional Hazards Model

7.4 Checking Assumptions for the Cox PH Model

Problems

Part IV: Experimental Design for Reliability Studies

8 Design of Single‐Testing‐Condition Reliability Experiments

8.1 Life Testing

8.2 Accelerated Life Testing

Problems

9 Design of Multi‐Factor and Multi‐Level Reliability Experiments

9.1 Implications of Design for Reliability

9.2 Statistical Acceleration Models

9.3 Planning ALTs with Multiple Stress Factors at Multiple Stress Levels

9.4 Bayesian Design for GLM

9.5 Reliability Experiments with Design and Manufacturing Process Variables

Problems

A The Survival Package in R

Notes

B Design of Experiments using JMP

C The Expected Fisher Information Matrix

C.1 Lognormal Distribution

C.2 Weibull Distribution

C.3 Lognormal Distribution

C.4 Weibull Distribution

D Data Sets

E Distributions Used in Life Testing

Bibliography

Index

Wiley End User License Agreement

List of Tables

Chapter 1

Table 1.1 General relationships between PDF, CDF, hazard function, and cumu...

Chapter 2

Table 2.1 Failure times in days for equipment given in Bartholomew (1957)....

Chapter 3

Table 3.1 Calculations for Kaplan–Meier estimate for the failure data

.

Table 3.2 Failure times two designs of a snubber.

Table 3.3 Failure times in millions of cycles for deep groove ball bearings...

Table 3.4 Summary of point estimates, log‐likelihood function evaluated at ...

Table 3.5 Lifetimes of engine starters, measured in thousands of starts.

Table 3.6 Data for Problem 3.4.

Table 3.7 Data for Problem 3.5.

Table 3.8 Data for Problem 3.6.

Chapter 4

Table 4.1 Guidelines for designing and experiment.

Table 4.2 Data for the battery life experiment.

Table 4.3 General arrangement of responses for a two‐factor factorial exper...

Table 4.4 Summary of dot notation.

Table 4.5 ANOVA table for two‐factor experiment with interaction.

Table 4.6 ANOVA table for battery lifetime experiment.

Table 4.7 Data from three replicates in the vibration experiment.

Table 4.8 The alloy experiment is a

design.

Table 4.9 Custom 28‐run design.

Table 4.10 Design evaluation and estimation efficiency.

Table 4.11 Correlation matrix for estimates of main effects and two‐factor ...

Table 4.12 The one‐half fraction of the

design (denoted

) defined by

....

Table 4.13 The

design defined by

and

.

Table 4.14 Regular Resolution III or higher designs in 8 and 16 runs.

Table 4.15 The

fractional factorial design.

Table 4.16 Some regular Resolution IV or higher fractional factorial design...

Table 4.17 Alias matrix for the

design with

.

Table 4.18 Augmented design, including original eight runs and the four add...

Table 4.19 Alias matrix for augmented design.

Table 4.20 Power analysis assuming

and an RMSE equal to 1.

Chapter 5

Table 5.1

design and results for the worsted yarn example.

Table 5.2 A three‐variable Box–Behnken design with three runs at the center...

Table 5.3 Box–Behnken design in

variables.

Table 5.4 Definitive screening design for

factors.

Table 5.5 Definitive screening design for

factors with categorical variab...

Table 5.6 Definitive screening design in two blocks.

Table 5.7 Data for Problem 5.1.

Table 5.8 Data for Problem 5.2.

Table 5.9 Data for Problem 5.3.

Table 5.10 Data for Problem 5.7.

Table 5.11 The ranitidine separation experiment.

Table 5.12 Starbucks' packaging of one‐pound coffee.

Table 5.13 Definitive screening design from Jones and Nachtsheim (2011b).

Table 5.14 Experimental data for Problem 5.15.

Chapter 6

Table 6.1 Twenty eight failure times (min) from the LOGN(2, 0.5) distributi...

Table 6.2 Regression parameter estimates and standard errors.

Table 6.3 Canonical links for generalized linear models.

Table 6.4 Number of coupons redeemed for each of eight treatments.

Table 6.5 JMP® output for full model in the coupon redemption experiment.

Table 6.6 JMP® parameter estimates for the full model.

Table 6.7 JMP® output for reduced model in the coupon redemption experiment...

Table 6.8 Comparing worsted yarn reliability results.

Table 6.9 Lifetimes of rolling ball bearings in a

experiment.

Table 6.10 Transmission experimental design and responses.

Table 6.11 Correlation between estimators of main effects and two‐factor in...

Table 6.12 The 11 control factors and 5 noise factors and their levels for ...

Table 6.13 Lifetimes of nickel superalloy specimens measured in kilocycles ...

Table 6.14 AIC (Akaike information criterion) for the four models for lifet...

Table 6.15 Data for Problem 6.10.

Chapter 7

Table 7.1 Units still operating just before each observed failure.

Table 7.2 Data set where units 1–4 were assigned to treatment group A, and ...

Table 7.3 Lifetimes of epoxy in minutes as a function of voltage.

Table 7.4 Lifetimes in cycles solder joints on a printed circuit board (PCB...

Table 7.5 Rolling contact fatigue life times in millions of cycles for stee...

Chapter 9

Table 9.1 The interval censored solder joint failure time data.

Table 9.2 The data format of solder joint failure times for fitting a GLM m...

Table 9.3 The D‐optimal design for the adhesive bond ALT.

Table 9.4 The D‐optimal design based on the incorrect linear predictor mode...

Table 9.5 The

‐optimal design for the glass capacitor ALT.

Table 9.6 The

‐optimal design for the glass capacitor ALT with a shorter t...

Table 9.7 The

‐optimal design for the glass capacitor ALT.

Table 9.8 Bayesian D optimal design for the logistic regression in Example ...

Table 9.9 Bayesian D optimal design for the first‐order logistic regression...

Table 9.10 The

‐optimal design for the drill bit ALT.

Appendix D

Table D.1 Electrical appliance lifetime data.

Table D.2 Glass capacitor accelerated life testing data.

Table D.3 Solder joint survival times in cycles for 3 PCB types and 3 level...

Table D.4 Clutch spring durability testing data.

Table D.5 Drill bit life testing data.

Table D.6 Low‐cycle fatigue test of nickel super alloy.

Table D.7 Failure and censoring times for diesel generator fans.

List of Illustrations

Chapter 1

Figure 1.1 Properties of the PDF for a lifetime distribution.

Figure 1.2 Approximation

.

Figure 1.3 Hazard and probability mass function for the case of constant haz...

Figure 1.4 Hazard and probability mass function for the case of constant pro...

Figure 1.5 Times to failure for six items. The X indicates a failure time an...

Figure 1.6 Times to failure for seven items. The X indicates a failure event...

Figure 1.7 Intervals containing the failures for items that are inspected on...

Figure 1.8 Failure times for two repairable systems. For the first system (b...

Chapter 2

Figure 2.1 Probability density functions, survival functions, and hazard fun...

Figure 2.2 Example of a bathtub‐shaped hazard function, which cannot be mode...

Figure 2.3 Empirical CDF (the step function) and the CDF estimated assuming ...

Figure 2.4 PDFs, survival functions, and hazard function s for three Weibull...

Figure 2.5 PDFs, survival functions, and hazard function s for three gamma d...

Figure 2.6 The generalized gamma distribution and special cases.

Figure 2.7 PDFs, survival functions, and hazard functions for three lognorma...

Figure 2.8 Plots of the PDFs for SEV distributions. (a) The standard SEV dis...

Figure 2.9 Plots of the PDFs and CDFs for Logistic distributions for varying...

Figure 2.10 Plots of the PDFs and CDFs for Logistic distributions for varyin...

Chapter 3

Figure 3.1 The empirical survival function when there is no censoring.

Figure 3.2 The Kaplan–Meier estimate of the survival function for the data i...

Figure 3.3 Kaplan–Meier estimates of the survival functions for the NCOG tri...

Figure 3.4 Kaplan–Meier estimates of the survival functions for the old and ...

Figure 3.5 Contributions to the likelihood for left, interval, and right cen...

Figure 3.6 Q‐Q plots for the deep groove ball bearings example for the five ...

Figure 3.7 Estimates of the hazard function for the five lifetime distributi...

Figure 3.8 Prior distributions for the parameters

and

of the Weibull dis...

Figure 3.9 Trace plots (on left side) for parameters

and

. The posterior ...

Figure 3.10 Joint posterior distribution for

and

based on 10,000 MCMC si...

Figure 3.11 Posterior mean lifetime for starter measured in 1000s.

Figure 3.12 Predictive distribution for starter lifetime measured in 1000s o...

Figure 3.13 Posterior distribution of reliability for a mission of 58,440 st...

Chapter 4

Figure 4.1 A two‐factor factorial experiment without interaction.

Figure 4.2 A two‐factor factorial experiment with interaction.

Figure 4.3 A factorial experiment with no significant interaction.

Figure 4.4 A factorial experiment with an interaction.

Figure 4.5 JMP fit model analysis for the battery life experiment.

Figure 4.6 JMP fit model analysis for the battery life experiment (continued...

Figure 4.7 Scatter plots created in R for the battery life data. Plot on lef...

Figure 4.8 JMP fit model analysis for the vibration experiment.

Figure 4.9 JMP fit model output for the router experiment in Table 4.7.

Figure 4.10 JMP power calculations.

Figure 4.11 Illustration of the

factorial design. (a) Geometric view and (...

Figure 4.12 Contrasts associated with main effects and interactions. (a) Mai...

Figure 4.13 Additional JMP fit model output.

Figure 4.14 JMP fit model output for the bottle filling experiment.

Figure 4.15 JMP fit model output for the bottle filling experiment continued...

Figure 4.16 The reduced model for the bottle filling experiment.

Figure 4.17 Crack length contrasts for screening design.

Figure 4.18 Half normal plot for crack length experiment.

Figure 4.19 JMP output for crack length experiment.

Figure 4.20 Additional JMP output for crack length experiment.

Chapter 5

Figure 5.1 A three‐dimensional response surface showing the expected yield a...

Figure 5.2 A three‐dimensional response surface showing the expected yield a...

Figure 5.3 The sequential application of RSM.

Figure 5.4 (a) Response surface and (b) contour plot illustrating a surface ...

Figure 5.5 (a) Response surface and (b) contour plot illustrating a surface ...

Figure 5.6 (a) Response surface and (b) contour plot illustrating a surface ...

Figure 5.7 A stationary ridge system.

Figure 5.8 A rising ridge system.

Figure 5.9 The Worsted yarn experiment.

Figure 5.10 JMP output for the first‐order model for the log of the cycles t...

Figure 5.11 The central composite design for

and

factors.

Figure 5.12 Face centered designs for

and

.

Figure 5.13 Box–Behnken design for

.

Figure 5.14 Fraction of design space plot comparing a face centered cube and...

Chapter 6

Figure 6.1 Simulated failure time data.

Figure 6.2 Parametric lifetime distribution fit to the failure time data.

Figure 6.3 Quantile profiler showing the median estimate and 95% confidence ...

Figure 6.4 Quantile profiler from JMP® survival fit showing factor settings ...

Figure 6.5 Prediction profiler for the percent coupon redemption.

Figure 6.6 Illustration of the proportional hazards property for Weibull reg...

Figure 6.7 Display of lifetimes of rolling ball bearings from

experiment....

Figure 6.8 The window in JMP® for lifetime regression model specification. S...

Figure 6.9 The parameter estimation table provided by JMP's® Weibull regress...

Figure 6.10 Rolling ball bearing data with cage design ignored.

Figure 6.11 Scatterplot of transmission shaft lifetimes versus factors

thr...

Figure 6.12 Contour plot for characteristic life

as a function of factors

Figure 6.13 Contour plot for characteristic life

as a function of factors

Figure 6.14 Output from JMP® for the first‐order model applied to the transm...

Figure 6.15 Output from JMP® for the second‐order model applied to the trans...

Figure 6.16 Scatter plots and estimated mean functions for four models using...

Figure 6.17 Trace plots and histograms for parameters

,

,

,

, and

.

Figure 6.18 The effect on the nonconstant shape parameter of the lifetime di...

Figure 6.19 The shape of the estimated survival densities. Top, a constant

Figure 6.20 CDF for three SAFT models with Weibull baseline. The bottom curv...

Figure 6.21 Battery life against material type and temperature.

Figure 6.22 Cox–Snell residuals for the battery life data. The top and middl...

Figure 6.23 Cox–Snell residuals for a first‐order model applied to the trans...

Figure 6.24 Cox–Snell residuals for a first‐order model applied to the trans...

Figure 6.25 Cox–Snell residuals for a second‐order model applied to the tran...

Figure 6.26 Cox–Snell residuals for a second‐order model applied to the tran...

Figure 6.27 Histogram of the 28 failure times compared with lognormal fit (s...

Figure 6.28 (a) Lognormal probability plot, shown on lognormal quantile scal...

Chapter 7

Figure 7.1 Illustration of likelihood contribution for Cox proportional haza...

Figure 7.2 Partial log‐likelihood function for data in Table 7.2.

Figure 7.3 Scatter plot of Voltage versus Lifetime. Gray lines indicate poss...

Figure 7.4 Survival functions for each level of Voltage using the Cox propor...

Figure 7.5 The JMP PH model parameter estimation outputs.

Figure 7.6 Survival functions for each level of Voltage using the Cox propor...

Figure 7.7 Log survival time versus log cumulative hazard function for the e...

Chapter 8

Figure 8.1 The event plot of 36 electrical appliance lifetimes.

Figure 8.2 Comparing the Weibull and exponential distributions fitted the el...

Figure 8.3 Determine the sample size for the electrical appliance demonstrat...

Figure 8.4 Determine the sample size for the electrical appliance reliabilit...

Figure 8.5 Determine the sample size for the lognormal distribution specifie...

Figure 8.6 Determine the sample size for the lognormal distribution by JMP R...

Figure 8.7 The probability of acceptance of two life testing plans: (100, 10...

Chapter 9

Figure 9.1 The three cases of optimal ALT plans for exponential lifetime dis...

Figure 9.2 The JMP's optimal ALT plans depend on the specification of the in...

Figure 9.3 The two

‐optimal ALT plans with different testing duration.

Figure 9.4 The

‐optimal ALT plan with expected prediction variance.

Figure 9.5 Bayesian D‐optimal design for the first‐order logistic regression...

Figure 9.6 Bayesian D‐optimal design for the logistic regression model in Ex...

Figure 9.7 The optimal ALT plans for worst yarn experiment with

and

.

Figure 9.8 The optimal ALT plans for worst yarn experiment with adjusted str...

Figure 9.9 The Weibull regression parameter estimations for the clutch sprin...

Figure 9.10 Quantile profiler of the Weibull regression model for the clutch...

Figure 9.11 Sample size determination for the demonstration test of clutch s...

Figure 9.12 The exponential regression parameter estimations for the drill b...

Figure 9.13 Quantile profiler of the exponential regression model for the dr...

Appendix A

Figure A.1 Kaplan–Meier estimate of the old snubber design.

Figure A.2 Comparison of fitted exponential distribution fit with the Kaplan...

Figure A.3 Comparison of fitted Weibull distribution fit with the Kaplan–Mei...

Figure A.4 Scatter plot of lifetimes in kilocycles versus pseudostress.

Figure A.5 Scatter plot of lifetimes in kilocycles versus pseudostress with ...

Figure A.6 Scatter plot of lifetimes in kilocycles versus pseudostress with ...

Appendix B

Figure B.1 The custom design interface in JMP for generating optimal designs. ...

Figure B.2 Evaluation of the 24 run D‐optimal design using prediction as a funct...

Figure B.3 Map of correlations among estimates of main effects and two‐facto...

Guide

Cover

Table of Contents

Title Page

Copyright

Dedication

Preface

About the Companion Website

Begin Reading

A The Survival Package in R

B Design of Experiments using JMP

C The Expected Fisher Information Matrix

D Data Sets

E Distributions Used in Life Testing

Bibliography

Index

Wiley End User License Agreement

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WILEY SERIES IN PROBABILITY AND STATISTICS

Established by Walter A. Shewhart and Samuel S. Wilks

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Design of Experiments for Reliability Achievement

Steven E. RigdonSaint Louis UniversityEpidemiology and BiostatisticsSaint Louis, MO

Rong PanArizona State UniversitySchool of Computing and Augmented IntelligenceTempe, AZ

Douglas C. MontgomeryArizona State UniversitySchool of Computing and Augmented IntelligencePhoenix, AZ

Laura J. FreemanStatisticsVirginia TechBlacksburg, VA

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Preface

Techniques of design of experiments (DOE) have for decades been used in industry to achieve quality products and processes. These methods often involve analyzing models that assume, at least approximately, that the outcome is normally distributed. When the outcome is a lifetime, similar techniques can be applied, although the methods require more complicated models. Reliability experiments are special in two respects: (i) there is almost always censoring, i.e. the termination of the experiment before all units have failed, and (ii) lifetime distributions are usually not well approximated by the normal. This book is about designing experiments and analyzing data when the outcome is a lifetime.

Condra (2001b) suggests three aspects of reliability methods:

Methods for measuring and predicting failures

Methods for accommodating failures

Methods for preventing failures

The first, measuring and predicting failures, usually involves fitting models to lifetime data in order to assess the reliability of a system. The second, accommodating failures, involves concepts like parallel redundancy (where failure of a single component does not cause failure of the system), repairability (the ability to quickly fix a problem and return the system to working condition), maintainability (the ability to keep a system in working condition), and others. The last, methods for preventing failures, is potentially the most useful. DOE methods can be used to find characteristics of the product, or maybe the process used to make the product, that lead to the highest possible reliability. Of course, this involves methods for measuring and predicting failures (the first item earlier) and it could involve the second (accommodating failures), but the idea of designing experiments to improve reliability is a powerful one. DOE has been used successfully in a number of areas where a normally distributed response is reasonable, but applications in reliability are rather sparse.

We have divided the book into four parts:

Reliability

 Here we cover the basic concepts and definitions of reliability. We present models for lifetimes, including the exponential, Weibull, gamma, and log‐normal. In addition, we discuss log‐location‐scale distributions, such as the

smallest extreme value

(

SEV

) distribution, which is a general class of distributions that can be used to model the logarithm of lifetimes. Inference for lifetime distributions, or log lifetime distributions, is the topic of

Chapter 3

. There, we develop point and interval estimate of model parameters and ways we could test hypotheses regarding those parameters.

Design of Experiments

 In the second part we present the basic ideas of experimental design and analysis. We cover the DOE for linear and generalized linear models.

Chapter 4

covers factorial designs in general, the

design, and fractional two‐level designs.

Chapter 5

covers designs for response surfaces.

Regression Models for Reliability Studies

 This part consists of two chapters.

Chapter 6

covers parametric regression models. This includes models on transformed data, exponential regression, and Weibull regression.

Chapter 7

covers semi‐parametric regression models including the Cox proportional hazards model.

Experimental Design for Reliability Studies

 The final part addresses experimental designs for reliability studies.

Chapter 8

covers tests done under a single test condition.

Chapter 9

covers multiple‐factor experiments, including accelerated life tests.

The material in this book requires a one‐ or two‐semester course in probability and statistics that uses some calculus. Readers with a background in reliability but not DOE can skip Part I and proceed to Part II on experimental design, and then to Parts III and IV. Readers with a background in experimental design but not reliability can begin with Part I, skip Part II, and proceed to Parts III and IV. Those who are well versed in both reliability and DOE can proceed directly to Parts III and IV.

The book's companion web site contains the data sets used in the book, along with the R and JMP code used to obtain the analyses. The web site also contains lists of known errors in the book.

Steven E. RigdonSaint Louis26 November 2021

Rong Pan, Douglas C. MontgomeryTempe26 November 2021

Laura J. FreemanArlington26 November 2021

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/rigdon/designexperiments

The website includes data sets and computer code.

Part IReliability

1Reliability Concepts

1.1 Definitions of Reliability

It is difficult to define reliability precisely because this term evokes many different meanings in different contexts. In the field of reliability engineering, we primarily deal with engineered devices and systems. Single‐word descriptions may depict one or two aspects of reliability in an engineering application context, but they are inadequate for a technical definition of engineering reliability. So, how do engineers and technical experts define reliability?

Radio Electronics Television Manufacturers Association (

1955

) – “Reliability is the probability of a device performing its purpose adequately for the period of time intended under the operating conditions encountered.”

ASQ (

2020

) – “Reliability is defined as the probability that a product, system, or service will perform its intended function adequately for a specified period of time, or will operate in a defined environment without failure.”

Meeker and Escobar (

1998a

) – “Reliability is often defined as the probability that a system, vehicle, machine, device, and so on will perform its intended function under operating conditions, for a specified period of time.”

Condra (

2001a

) – “Reliability is quality over time.”

Yang (

2007

) ‐ “Reliability is defined as the probability that a product performs its intended function without failure under specified conditions for a specified period of time.”

There are some variations in the aforementioned definitions, but they all either explicitly or implicitly state the following characteristics of reliability:

Reliability is a probabilistic measure – the probability of a functioning product, service, or system.

Reliability is a function of time – the probability function of successfully performing tasks, as designed, over time.

Reliability is defined under specified or intended operating conditions.

We define a function, , to be the survival, or reliability, function, which is the probability of the product, service, or system being successfully operated under its normal operating condition at time ; in other words, the unit survived past time .

1.2 Concepts for Lifetimes

When an item fails, the “fix” sometimes involves making a repair to bring it back to a working condition. Another possibility is to discard the item and replace it with a working item. In general, the more complex a system is, the more likely we are to repair it, and the simpler it is the more likely we are to scrap it and replace it with a new item. For example, if the starter on our automobile fails, we would probably take out the old starter and replace it with a new one. In a case like this, the automobile is a repairable system, but the starter is nonrepairable since our fix has been to replace it entirely.

Since complex systems, which are usually repairable, are made up of component parts that are nonrepairable, we will focus in this book on nonrepairable items. If these nonrepairable items are designed and built to have high reliability, then the system should be reliable as well. For nonrepairable systems we are interested in studying the distribution of the time to the first (and only) failure, or more generally, the effect of predictor variables on this lifetime. This lifetime need not be measured in calendar time; it could be measured in operating time (for an item that is switched on and off periodically), miles driven (for a motor vehicle like a car or truck), copies made (for a copier or printer), or cycles (for an industrial machine). For nonrepairable systems, we study the occurrence of events in time, such as failures (and subsequent repairs) or recurrence of a disease or its symptoms. See Rigdon and Basu (2000) for a treatment of repairable systems.

The lifetime of a unit is a random variable that necessarily takes on nonnegative values. Usually, but not always, we think of as a continuous random variable taking on values in the interval . There are various forms that the distribution may take, many of which, including the exponential, Weibull and gamma, are presented in detail Chapter 2. Here we present the fundamental ideas and terms for continuous random variables.

Definition 1.1The probability density function (PDF) of a continuous random variable is a function with the property that

Thus, probabilities for a continuous random variable are found as areas under the PDF. (See Figure 1.1a.) Note that and can be or . Since the probability that , that is, the probability that equals a particular value is equal to zero. This also implies that

See Figure 1.1b.

Figure 1.1 Properties of the PDF for a lifetime distribution.

Since the probability is 1 that is between and we have the property that

(1.1)

See Figure 1.1c. Also, since all probabilities must be nonnegative, the PDF must satisfy

(1.2)

The results in (1.1) and (1.2) are the fundamental properties for a PDF.

The development earlier makes no assumption about the possible values that the random variable can take on. For lifetimes, which must be nonnegative, we have for Thus, for lifetimes, the PDF must satisfy

The set of values for which the PDF of the random variable is positive is called the support of The support for a lifetime distribution is although for some distributions we exclude the possibility of .

Note that the PDF does not give probabilities directly; for example, does not give the probability that Rather, as an approximation we can write

(See Figure 1.2.) Thus, the PDF can be interpreted as

(1.3)

or equivalently

To be precise, the PDF is equal to the limit of the right side earlier as :

Figure 1.2 Approximation .

Definition 1.2The cumulative distribution function (CDF) of the random variable is defined as

(1.4)

where is the PDF for

Note that we have changed the variable of integration from to in order to avoid confusion with the upper limit on the integral. For a lifetime distribution with support we have the result

(1.5)

As on the right side earlier, the integral goes to , which equals 1. Also, since the probability of having a negative lifetime is 0, the CDF must be zero for all Finally, since the CDF “accumulates” probability up to and including increasing can only increase (or hold constant) the CDF. Thus, for a lifetime distribution, the CDF must satisfy

Equation (1.4) shows how to get the CDF given the PDF. A formula for the reverse (getting the PDF from the CDF) can be obtained by differentiating both sides of (1.4) with respect to and applying the fundamental theorem of calculus:

(1.6)

In other words, the PDF is the derivative of the CDF.

Definition 1.3The survival function, or reliability function, is defined to be

In other words, is the probability that an item survives past time while is the probability that it fails at or before time (that is, that it doesn't survive past time ). Thus, and are related by

One of the most important concepts in lifetime analysis is the hazard function.

Definition 1.4The hazard function is

(1.7)

As an approximation, we can write

(1.8)

analogous to (1.3).

The probability in the definition of the hazard is a conditional probability; it is conditioned on survival to the beginning of the interval. This is a natural quantity to consider because it makes intuitive sense to talk about the failure probability of an item that is still working. It is conceptually more difficult to talk about the probability of an item failing if the item might or might not be working. If we replace the conditional probability in the definition of the hazard function with an unconditional probability, we get

(1.9)

which is equal to the PDF . Thus, the PDF is the (limit of) the probability of failing in a small interval when viewed before testing begins. The hazard is the (limit of) the probability of failing in a small interval for a unit that is known to be working.

The hazard function can be written as

(1.10)

Indeed, many books define the hazard function in this way. We choose to define the hazard as the limit of a conditional probability because this intuitive concept is helpful for understanding the failure mechanism.

To illustrate the difference between hazard and density, consider a discrete case, say, where items are placed on test and are observed every 1000 hours. Let denote the probability that an item fails in the th interval Suppose first that so that there is a probability of that a working unit will fail in any time interval. Thus, at the end of a time interval when we inspect those units still operating, we would expect that about one‐tenth of them would fail. We could naturally ask the question “What is the probability that a unit fails in the th interval?” This is different from the question “What is the probability that a unit that is currently operating fails in the th interval?” The difference is that the latter is a conditional probability (conditioned on the unit still operating), whereas the former is an unconditional probability. The answer to the latter question is: the hazard To get at the answer to the latter question, we can observe that

and in general,

This is, of course, the geometric distribution. Plots of and for the case of a constant (discrete) hazard are shown in Figure 1.3.

Figure 1.3 Hazard and probability mass function for the case of constant hazard.

Figure 1.4 Hazard and probability mass function for the case of constant probability mass.

Suppose now that the probability mass function (rather than the hazard function) is constant, with Since we conclude that for The hazard is then

Thus, The last number, , may seem a little surprising, but if for and if an item hasn't failed up through interval then it must fail at time The hazard and probability mass for the constant probability case are shown in Figure 1.4.

The cumulative hazard is defined to be the accumulated area under the hazard function. To be precise, the cumulative hazard is defined to be

(1.11)

Any one of the PDF, CDF, survival function, hazard, or cumulative hazard function is enough to determine the lifetime distribution. In other words, knowing any one of these can get you all of the others. For example, Eq. (1.5) shows how you can get the CDF if you know the PDF. If we know the hazard function , we can use the relationship

to find . To see this, notice that this is a simple first‐order linear differential equation with initial condition , which can be solved by integrating both sides from to . This yields

(1.12)

from which we obtain the relationship

(1.13)

We leave the other relationships as an exercise. Table 1.1 shows most of the relationships.

Table 1.1 General relationships between PDF, CDF, hazard function, and cumulative hazard function.

PDF

CDF

Hazard

Cumulative hazard

—

—

—

—

1.3 Censoring

A life testing experiment, where many units are operating simultaneously, may be terminated before all of the units have failed. This is a typical scenario, because many units will be highly reliable and will not fail during a test. This leads to the concept of censoring, where we cannot observe the lifetime of an item because of the design of the life testing experiment.

Definition 1.5When we cannot observe the exact failure time of an item, but rather we can only observe an interval in which the failure was observed, we say that the observation is censored. This interval could be an unbounded interval, such as or a bounded interval such as .

The most common type of censoring occurs when the life test is stopped before all items have failed. Let denote the censoring time, that is, the time of termination of the test. In this case, we would know the exact failure times of all items that failed before time , but for those still operating at the end of the experiment, we know only that the failure would occur past time . For those item still operating, we would only know that the failure time was in the interval . Actually, the censoring time need not be the same for all items. For example, if the items were placed into service at different times, then the censoring times would be different even if the test was terminated at the same (calendar) time. This type of lifetime censoring, where we observe a survival event (where we know that the failure must occur in the interval ), is called right censoring.

Figure 1.5 shows an illustration of right censoring. In this situation six items were placed on test at the same time. Items 1, 2, 3, and 5 failed during the test at times 2.1, 3.7, 8.1, and 8.6, respectively. The other two units were still operating at times 9.4 and 10.0. The failure times are denoted by an X and the censoring times are denoted by an O. The solid lines cover the times for which it is known that the items were operating.

Figure 1.5 Times to failure for six items. The X indicates a failure time and the O indicates a censored time.

It sometimes happens that observation of an item may begin well after the it was placed into service. This can occur when items are observed in the field. In some instances, it may be the case that the item is observed to have already failed when it is first inspected. For example, suppose an item is placed into service and then not inspected until an age of 100 days. If it was observed to be in a failed condition at that time, then we know only that it failed before time . In other words, the failure must have occurred sometime in the interval , but the exact failure time is unknown. This is called left censoring, because the failure is known to have occurred before time 100. Figure 1.6 illustrates the concept of left (and right) censoring. Here, items 1, 2, and 4 were placed on test and the failure times were observed to be 24, 37, and 77, respectively. Items 3 and 6 were observed at times 10 and 28, respectively, to be in a failed condition. A dashed line is used to indicate the possible times of actual failure. In general, we use a solid line to indicate times when the units were known to be operating. Finally, items 5 and 7 were still operating at time 99 when the test was terminated. Thus, items 3 and 6 were left censored and items 5 and 7 were right censored.

Figure 1.6 Times to failure for seven items. The X indicates a failure event and the O indicates a right censoring event, whereby the failure occurred to the right of the of the O

2Lifetime Distributions

While the normal distribution is a useful model for a number of circumstances, it is generally a poor model for lifetimes. Usually lifetimes have a distribution that is skewed to the right, that is, there are more observations far in the right tail of the distribution than in the left tail. Also, lifetimes have a natural lower boundary of zero (since lifetimes cannot ever be negative).

In this chapter, we present a number of lifetime distributions, including the exponential, Weibull, gamma, lognormal, and log‐logistic. We also consider some distributions that are related to these, such as the extreme value distribution. The choice of a lifetime distribution is important, because these distributions have different properties, especially in the left and right tails. In reliability applications, making an inference about the behavior in the tails is often important.

2.1 The Exponential Distribution

The simplest model for lifetimes is the exponential distribution, which has the probability density function (PDF)

(2.1)

The survival function is

(2.2)

and the hazard function is

(2.3)

The cumulative distribution function (CDF) is related to the survival function by

(2.4)

The exponential distribution is characterized by a single parameter, which is equal to the mean (and also to the standard deviation) of the distribution. If has an exponential distribution with mean , then we will write . Thus, if , then

(2.5)

and

(2.6)

Figure 2.1 shows the PDFs, survival functions, and hazard functions for the exponential distributions with means , , and .

Figure 2.1 Probability density functions, survival functions, and hazard functions for several exponential distributions.

The simplicity of having just one parameter is also a drawback because the one parameter limits the flexibility of the model. The hazard function for the exponential distribution, given in (2.3), is independent of . This implies that the hazard function, which is the (limit of) the probability of failure in a small interval divided by the length of the interval, is constant regardless of the age of the unit. Most mechanical systems have an increasing hazard function; that is, as the unit ages, the probability of an imminent failure increases. This is because the moving parts wear and often the connections among parts become less reliable. For example, a car engine that has run for 100,000 miles is in a weaker condition (i.e. is less reliable) than one that has run for 10,000 miles. Electrical systems with no moving parts are a different matter. Often, the wiring does not degrade, or degrades very little, so that a two‐year‐old part is essentially equivalent to a new part. Thus, the exponential distribution can sometimes be used as a model for electrical parts. Another aspect of reliability that cannot be modeled by the exponential distribution is that of infant mortality, followed by a gradually increasing hazard. Computer hard drives (which are both electrical and mechanical) provide a good example of this. Some drives have one or more manufacturing flaws that will cause failure at, or soon after, the time that they are first put into service. Most drives, however, do not have this type of flaw and will work without failure for a long time. Only after the mechanical parts begin to wear or there is a sudden electrical failure will the drive fail. This phenomenon leads to a “bathtub” shaped hazard function, as shown in Figure 2.2