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Basic Concepts of Reliability Engineering

This chapter reviews basic concepts and common reliability engineering practices in the manufacturing industry. In addition, we briefly introduce the history of Bayesian statistics and how it relates to advances in the field of reliability engineering.

Experienced reliability engineers who are very familiar with reliability basics and would like to start learning Bayesian statistics right away, may skip this chapter and start with Chapter 2. Bayesian statistics has unique advantages for reliability estimations and predictive analytics in complex systems. In other cases, Bayesian methods may provide flexible solutions to aggregate various sources of information to potentially reduce necessary sample sizes and therefore achieve cost effectiveness. The following chapters provide more specific discussions and case study examples to expand on these topics.

1.1 Introduction

High product quality and reliability are critical to any industry in today's competitive business environment. In addition, predictable development time, efficient manufacturing with high yields, and exemplary field reliability are all hallmarks of a successful product development process.

Some of the popular best practices in industry include Design for Reliability and Design for Six Sigma programs to improve product robustness during the design phase. One core competency in these programs is to adopt advanced predictive analytics early in the product development to ensure first-pass success, instead of over-reliance on physical testing at the end of the development phase or on field performance data after product release.

The International Organization for Standardization () defines reliability as the "ability of a structure or structural member to fulfil the specified requirements, during the working life, for which it has been designed" (ISO 2394:2015 General principles on reliability for structures, Section 2.1.8). Typically, reliability is stated in terms of probability and associated confidence level. As an example, the reliability of a light bulb can be stated as the probability that the light bulb will last 5000 hours under normal operating conditions is 0.95 with 95% confidence.

Accurate and timely reliability prediction during the product development phase provides inputs for the design strategy and boosts understanding and confidence in product reliability before products are released to the market. It is also desirable to utilize and aggregate information from different sources in an effective way for reliability predictions.

Textbooks on reliability engineering nowadays are dominated by frequentist statistics approaches for reliability modeling and predictions. In a frequentist/classical framework, it is often difficult or impossible to propagate individual component level classical confidence intervals to a complex system comprising many components or subsystems. In a Bayesian framework, on the other hand, posterior distributions are true probability statements about unknown parameters, so they may be easily propagated through these system reliability models. Besides, it is often more flexible to use Bayesian models to integrate different sources of information, and update inferences when new data becomes available.

Given the benefits mentioned above, potential applications of Bayesian methods on reliability prediction are quite extensive. Historically, Bayesian methods for reliability engineering were applied on component reliability assessment where conjugate prior (will be discussed in Chapter 2) distributions were widely used due to mathematical tractability. Recent breakthroughs in computational algorithms have made it feasible to solve more complex Bayesian models, which have greatly boosted advancement and applications of Bayesian modeling. One popular algorithm is Markov chain Monte Carlo () sampling, a method of simulating from a probability distribution based on constructing a Markov chain. MCMC methods along with rapid advancement in high-speed computing have made it possible for building and solving complex Bayesian models for system reliability.

Over the past one or two decades, Bayesian statistics books have appeared in different scientific fields. However, most existing Bayesian statistics books do not focus on reliability analysis/predictions, thus real-life practical examples on reliability modeling are often absent. This challenge prevents reliability engineers from adopting the Bayesian approach to solve real-life problems. The goal of our book is to address this gap.

A few general topics covered in this book are:

• Design for reliability
• Basic concepts of Bayesian statistics and models
• Bayesian models for component reliability estimation
• Bayesian models for system reliability estimation
• Bayesian networks
• Advanced Bayesian reliability models.

Specifically, the topics covered are:

• Design for reliability

  This topic includes reliability definition, basic probability theory and computations, statistical models, basics of component reliability prediction, basics of system reliability prediction, critical feature capability prediction, Monte Carlo simulations, and accelerated life testing (), etc.

• Basic concepts of Bayesian statistics and models

  This topic includes Bayes' theorem and history, Bayesian inference vs. frequentist inference, basic statistical concepts: point estimate, confidence interval, discrete and continuous probability distributions, censored data, and selection of prior distributions (conjugate priors, non-informative priors, and informative priors), likelihood function, model selection criteria, introduction of MCMC algorithms and sampling methods, and Bayesian computation software (WinBUGS, OpenBUGS, Just Another Gibbs Sampler (), R, etc.).

• Bayesian models for component reliability estimation

  This topic includes component level reliability prediction from reliability life testing, binomial distribution, Poisson distribution, exponential distribution, Weibull distribution, normal distribution, log-normal distribution, and reliability prediction from ALT (Arrhenius model, inverse power law model, etc.).

• Bayesian models for system reliability estimation

  This topic includes reliability block diagram, series system, parallel system, mixed series and parallel system, fault tree analysis with uncertainty, process capability or design capability analysis with uncertainty, Monte Carlo simulation, and two-level nested Monte Carlo simulation and examples (strength-stress interference, tolerance stack up, etc.).

• Bayesian networks

  This topic includes basics of conditional probability, joint probability distributions, marginal probability distributions, structures of a Bayesian network, examples, and basic steps to construct a Bayesian network model.

• Advanced Bayesian reliability models

  This topic includes using hierarchical Bayesian models to predict reliability during iterative product development, to predict reliability of specific failure mechanisms, to aggregate different sources of imperfect data, to aggregate component level and system level data for system reliability prediction, and to borrow partial strength from historical product reliability information.

The first three chapters introduce commonly used reliability engineering methods and basics of Bayesian concepts and computations. The following chapters focus more on applications related to the individual topics introduced above. Readers are free to tailor their reading to specific chapters according to their interests and objectives.

1.1.1 Reliability Definition

In reliability engineering, product reliability is defined as the probability that a component or a system performs a required function under specified use conditions for a stated period of time. Note that the three key elements in the reliability definition are probability, use condition, and duration. Probability measures the likelihood of something happening. For example, when tossing a fair coin there is a 50% probability of the coin landing heads. When throwing a six-faced fair dice, the probability of observing each of the six outcomes (1, 2, 3, 4, 5, 6) is 1/6. Use conditions describe the conditions a product is operated under, e.g. temperature, humidity, pressure, voltage. Duration is usually related to the lifetime of a product. Reliability is usually estimated based on time to failure data from bench tests, accelerated life tests, or field service.

In engineering practices, it is common to define design requirements and use different types of tests, such as design verification tests or qualification tests, to ensure the product or the incoming parts meet these requirements. Here quality is measured by the probability of meeting a certain requirement, which can be thought of as reliability at time zero. Though these are quality assurance practices, the term "reliability" is sometimes used to refer to the probability of meeting a certain requirement.

Often in design verification tests, the samples are preconditioned through an equivalent lifecycle under specified stress conditions (to ensure reliable products, the stress conditions applied in the tests are usually as aggressive as or more aggressive than the actual use conditions in the field) before being tested against a requirement. In such cases, the...