reading

3.2 Mechanism to Link Users’ Operational Measures with Machine Reliability and Maintenance Parameters¹

Chapter 4: Expert Judgement-Based Parameter Estimation Method for Machine Tool Reliability Analysis

5.4 Preventive Maintenance Optimization Models for Different Maintenance Scenarios

7.5 Joint Optimization of Maintenance Planning and Quality Control Policy Using

-Control Chart

7.6 Joint Optimization of Preventive Maintenance and Quality Policy Incorporating Taguchi Quadratic Loss Function

7.7 Joint Optimization of Preventive Maintenance and Quality Policy Based on Taguchi Quadratic Loss Function Using CUSUM Control Chart

7.8 Extension of the Joint Optimization of Maintenance Planning and Quality Control Policy for Multi-component System

Chapter 8: Joint Optimization of Production Scheduling with Integrated Maintenance Scheduling and Quality Control Policy

8.5 Integration of Production Scheduling with Jointly Optimized Preventive Maintenance and Quality Control Policy

Performability Engineering Series
Series Editors: Krishna B. Misra (kbmisra@gmail.com)
and John Andrews (John.Andrews@nottingham.ac.uk)

Scope: A true performance of a product, or system, or service must be judged over the entire life cycle activities connected with design, manufacture, use and disposal in relation to the economics of maximization of dependability, and minimizing its impact on the environment. The concept of performability allows us to take a holistic assessment of performance and provides an aggregate attribute that reflects an entire engineering effort of a product, system, or service designer in achieving dependability and sustainability. Performance should not just be indicative of achieving quality, reliability, maintainability and safety for a product, system, or service, but achieving sustainability as well. The conventional perspective of dependability ignores the environmental impact considerations that accompany the development of products, systems, and services. However, any industrial activity in creating a product, system, or service is always associated with certain environmental impacts that follow at each phase of development. These considerations have become all the more necessary in the 21st century as the world resources continue to become scarce and the cost of materials and energy keep rising. It is not difficult to visualize that by employing the strategy of dematerialization, minimum energy and minimum waste, while maximizing the yield and developing economically viable and safe processes (clean production and clean technologies), we will create minimal adverse effect on the environment during production and disposal at the end of the life. This is basically the goal of performability engineering.

It may be observed that the above-mentioned performance attributes are interrelated and should not be considered in isolation for optimization of performance. Each book in the series should endeavor to include most, if not all, of the attributes of this web of interrelationship and have the objective to help create optimal and sustainable products, systems, and services.

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Reliability engineering as a subject matter is developed vastly in last few decades. Numerous books have been published on the subject, discussing basic principles, theories, models, tools and techniques, in general. However, every system is unique and some of them may require specific treatment while applying various tools and techniques of reliability engineering. This book explores the domain of reliability engineering for one such very important industrial system, called machine tools.

Machine tools are at the heart of the manufacturing systems. Manufacturing industries rely on machine tools to fulfil their customers’ demand. Failure of machine tool hampers their production efficiency and creates uncertainties in managing the shop floor operations resulting into significant economic losses. Moreover, the users of such systems are now sharing the risk of failures with the machine tool manufacturers by engaging into long term maintenance or availability contracts. This has created new business avenue for machine tool manufacturers for “Servicizing” their traditionally product focused business. Machine tool manufacturers have the opportunity to package effective life cycle maintenance services with the hardware products, i.e. machine tools. It is therefore important for machine tool manufactures as well as users to focus on core of reliability engineering to model machine tool’s failure/repair and its interaction with other measures of system performances.

This advanced text on machine tool reliability modelling aims to provide a consolidated volume on various dimensions of machine tool reliability and its implications from manufacturers and users point of view. From manufacturers point of view novel methodologies for reliability and maintenance based design of machine tools are covered. From users point of view novel methodologies are presented to integrate reliability and maintenance of machine tools with production scheduling and quality control. Application area, i.e. machine tools is very important and it covers entire manufacturing sector.

The target audience of the book are researchers and practicing engineers in the field of reliability engineering and operations management. The book can also be helpful to undergraduate students in the area of reliability to get an application flavour of the subject. It opens up various research dimensions for researchers. All the approaches are illustrated with the help of numerical examples. This makes the approaches easy to understand.

This book does not intend to provide coverage to basic of reliability engineering. It is expected here that the readers have some basic knowledge of the reliability engineering, probability and statistics. However, Chapter 2 is provided for the reader to refresh their basic of probability and statistics required to follow the text.

Authors would like to acknowledge the help received from Dr. Avinash Samvedi and Mr. Vikas Sankhla in writing some of the codes used in this book. We also acknowledge the help of Mr. Sandeep Kumar who helped in editing the references.

Chapter 1

Introduction

Reduced cost of production, timely delivery and high quality of products are the prime objectives for manufacturing industries. Breakdowns of production machinery or machine tools affect the manufacturer’s ability to meet the goals of Cost, Time and Quality (CTQ). One of the studies suggests that the economic loss due to an unexpected stoppage in industry can be as high as US $70,000 to US $420,000 per day [1]. Application of reliability engineering tools and techniques to machine tools for improving the manufacturing system performance is therefore a vital area of study.

The machine tool industry is one of the supporting pillars for the competitiveness of the entire manufacturing sector since it produces capital goods which in turn may produce manufactured goods. Customers of machine tool manufacturers (termed as “users” in this book) are, in many cases, vendors to other customers and have commitments to meet. Breakdowns of machine tools may jeopardize their ability to meet these commitments and also cost a lot of money to the users in terms of poor quality, slower production, downtime, etc. Since poor reliability and improper maintenance of a machine tool greatly increase the life cycle cost to the users, many machine tool users have changed their purchase criteria for a machine tool from initial acquisition cost to Life Cycle Cost (LCC) or Total Cost of Ownership (TCO).

As reliability engineering plays an important role in reducing the LCC of machine tools, this book will be equally appealing to machine tool manufacturers and users.

The book covers both the manufacturer’s and user’s viewpoint of machine tool reliability. Decisions made during the design phase of a product have the largest impact on the life cycle cost of a system. The inherent failure and repair characteristics of components and assemblies are frozen with the selection of the machine tool configuration at the design stage. Therefore, the maintenance requirements of the machine tools are also fixed at the design stage itself. For example, a higher reliability component may require a lower replacement frequency for the same operating profile compared to a lower reliability component. Therefore, machine tool manufacturers need to consider the reliability and maintenance aspects at the design stage itself. On the other hand, the cost effectiveness of machine tools at the user’s end also depends on the shop-floor level operations planning decisions, i.e., scheduling, inventory, quality control, etc. These shop-floor level operations planning decisions have interaction effect with machine tool reliability and maintenance. Therefore, machine tool users need to consider the reliability and maintenance aspects during operations planning. The goal of this book is to provide a consolidated volume on various dimensions of machine tool reliability and its implications from the manufacturer’s and user’s point of view.

The introductory chapter of the book describes basic reliability terms and defines machine tool failures. The importance of machine tool reliability from the manufacturers’ and users’ point of view is also discussed.

1.1 Basic Reliability Terms and Concepts

This section introduces important reliability terms and concepts which will help the reader in following the rest of the sections of the book.

Reliability: This is the probability that an item can perform its intended function for a specified interval under stated conditions [2].

In other words, it is the probability of survival over time. To determine the reliability of a particular component or system, an unambiguous and observable description of failure is essential. The machine tool failures are defined in the next section.

If T is a random variable, representing time to failure of the system or component, then reliability can be expressed as:

It is contextual here to clearly differentiate the term “quality” and “reliability.” If quality is the conformance to the specifications at t = 0, then reliability can be considered as conformance to the specifications at t > 0. However, in this book, “reliability” is used in the context of the machine tools, while “quality” is used in the context of the products produced using machine tools.

Failure Rate (Hazard Rate): Failure rate or hazard rate is the instantaneous (at time t) rate of failure [3]. It is the instantaneous failure rate. This index is normally used for non-repairable components. A component of the system may have increasing, decreasing, or constant failure rate. It is further discussed in Chapter 2.

Rate of Occurrence of Failure (ROCOF): This index is often used in place of hazard rate for repairable system. Failures occur as a given system ages and the system is repaired to a state that may be the same as new, or better, or worse. Let N(t) be a counting function that keeps track of the cumulative number of failures a given system has had from time zero to time t. N(t) is a step function that jumps up one every time a failure occurs and stays at the new level until the next failure. The ROCOF is the total number of failures within an item population, divided by the total number of life units expended by that population during a particular measurement period under stated conditions [2].

Every system will have its own observed N(t) function over time. If we observed the N(t) curves for a large number of similar systems and “averaged” these curves, we would have an estimate of M(t) = the expected number (average number) of cumulative failures by time t for these systems.

Maintenance: All actions necessary for retaining an item in or restoring it to a specified condition [2].

Corrective Maintenance (CM): All actions performed as a result of failure, to restore an item to a specified condition [2]. Corrective maintenance can include any or all of the following steps: localization, isolation, disassembly, interchange, reassembly, alignment and checkout.

Preventive Maintenance (PM): All actions performed to retain an item in a specified condition by providing systematic inspection, detection, and prevention of incipient failures [2].

Predictive Maintenance: Predictive maintenance or Condition Based Maintenance (CBM) is carried out only after collecting and evaluating enough physical data on performance or condition of equipment, such as temperature, vibration, particulate matter in oil, etc., by performing periodic or continuous (online) equipment monitoring [4].

Maintainability: It is the relative ease and economy of time and resources with which maintenance can be performed. More precisely, it is the probability that an item can be retained in, or restored to, a specified condition within a specified time when maintenance is performed by personnel having specified skill levels, using prescribed procedures and resources, at each prescribed level of maintenance and repair [2].

Availability: Depending on the purpose of analysis, a number of different definitions are used in the literature, some of which are given below [3]:

Instantaneous or Point Availability, A(t): It is the probability that a system will be operational at any random time t. Unlike reliability, the instantaneous availability measure incorporates maintainability information.

Average Availability: It is the proportion of time a system is available for use during a mission. Mathematically, it is calculated as the mean value of the instantaneous availability function over the period (0, T).

(1.2) equation

Steady State Availability: The steady state availability of the system is the limit of the instantaneous availability function as the time approaches infinity.

(1.3) equation

Inherent Availability: It is the steady state availability when considering only the corrective maintenance downtime of the system. It does not include delays due to unavailability of maintenance personnel, unavailability of spare parts, administrative procedures, etc. The inherent availability of a system is a function of the reliability of its components and maintainability, which more or less get defined at the design stage of the equipment.

(1.4) equation

where MTBF is the mean time between failures and MTTR is the mean time to repair.

Operational Availability: It is a measure of the average availability over a period of time, including all the delays due to unavailability of maintenance personnel, spare parts, administrative procedures, etc. Operational availability is the availability that the customer actually experiences.

(1.5) equation

where MTBM is the mean time between maintenance, SDT and MDT are the supply and maintenance delays respectively.

Inherent availability and operational availability are used in this book and are discussed further in Chapter 3.

Life Cycle Cost (LCC): It is the sum of acquisition, logistics support, operating, and retirement and phase-out expenses [2].

1.2 Machine Tool Failure

The first step in applying any reliability engineering technique to any system is to clearly define the failures of that particular system. The Society of Automotive Engineering (SAE) defines the failure of production machinery/equipment as: “any event due to which the machinery/equipment is not available to produce parts at specified conditions when scheduled, or is not capable of producing parts or performing scheduled operations to specification” [5].

However, care should be taken in expressing the failure criteria as different users may have different expectations in terms of the product performance. There may also be a diversity of opinion between machine tool users and manufacturers as to what exactly constitutes a degraded performance or failure. Therefore, while the SAE definition of failure can serve as a guideline, it is necessary that the failure criteria are clearly and quantitatively (wherever possible) defined by the designer, keeping in mind the user’s viewpoint. In this book, failures of machine tools are defined in terms of failure consequences. These consequences express the user’s view of failure under the mutually agreed upon operating conditions between the user and the manufacturer. Whenever failure occurs, it leads to one of the following Failure Consequences (FCs).

Failure Consequence 1 (FC1): failure is detected immediately and the machine has to be stopped.
Failure Consequence 2 (FC2): machine continues to operate, but at a lower production rate than designed (i.e., with increased cycle time).
Failure Consequence 3 (FC3): machine continues to run, but produces more rejections than the normal rejection rate.

In many cases, failure consequences 2 and 3 are detected by the users after a time lag, during which the machine tool runs at a reduced performance level. The last two failure consequences can be considered as the result of partial failures and can be defined as degradation in performance without complete failure [6–8]. Figure 1.1 depicts these failure consequences on a time-performance curve. It clearly indicates the relation of a machine tool failure with the user’s shop-floor performance measures like Availability (A), Performance Rate (PR), Quality Rate (QR) and failure costs.

It was observed during one of the research projects carried out by the authors with a machine tool industry in India that many failure events of machine tools lead to failure consequences 2 and 3 and finally to consequence 1 when detected. Thus, such failure consequences must be considered explicitly by machine tool manufacturers, as well as by users, to reduce the life cycle cost of the machine tools. Table 1.1 provides examples of all three types of failure consequences for a CNC grinding machine.

Table 1.1 Failure consequences and affected performances (Reprinted with permission from [8]; Inderscience Publishers).

1.3 Machine Tool Reliability: Manufacturer’s View Point

Historically, machine tool designers have done a good job of evaluating the functions and form of products at the design phase. Once the functional design of the machine tool is done, designers generally have multiple alternatives for many of the components/subassemblies that can satisfy the functional requirements of the system. Such alternatives, apart from their cost, also differ in their inherent failure and repair characteristics, like time-to-failure distribution, time-to-repair distribution, failure consequences, etc. For example, a designer may have two alternatives for spindle. viz., motorized and belted spindle. Even though both these alternatives may satisfy the functional requirements of the machine, they will have different failure and repair characteristics. Therefore, each of these alternatives will contribute differently to the reliability performance of the system. Further, Preventive Maintenance (PM) can also be used to improve the reliability performance of the system. However, preventive maintenance again consumes resources and time which could otherwise be used for production, thereby affecting profit. Therefore, from the view point of a machine tool designer, the problem of reliability and maintenance-based design of machine tool finally boils down to selecting the optimal machine tool configuration from the available alternatives for different components/subassemblies by simultaneously considering reliability and maintenance parameters such that the final configuration meets the user’s performance requirements and budget constraints.

However, optimization of reliability and maintenance schedule poses a challenge when users are unable to explicitly express their reliability requirements quantitatively. It was observed during a survey done by the authors that only a few corporate customers express their reliability requirements explicitly in terms of Mean Time Between Failures (MTBF). But, even these users are more concerned about their shop-floor level performance and they judge the reliability of a machine tool based on how well it performs in terms of performance measures like Overall Equipment Effectiveness (OEE), Life Cycle Cost (LCC), Cost Per Piece (CPP), etc. These performance measures are closer to the heart of the users and are affected by the inherent failure and repair characteristics of the machine tool components/subassemblies and the maintenance plans. However, the extent to which the inherent failure as well as repair characteristics and preventive maintenance of machine components/subassemblies affect a user’s performance measures also depends on the user’s cost structure and shop-floor level policies. For example, if a user has alternative machines available to bear the load of a failed machine, then the downtime cost of that machine may not be as significant as in the case where there is no alternative machine available to bear the load of the failed machine [10]. Similarly, if a machine is being used as a stand-alone machine, its downtime cost will be different than that in the case when the same machine is being used in a production line. As can be seen from the examples, the cost structures will be different for each of the cases. Similarly, a tighter quality control policy at the user’s end will detect process shifts due to failure of machine components/subassemblies much earlier, thereby reducing the rejection rate. Thus, the effect of machine failures and maintenances on LCC, OEE, and other performance measures, may be different for different users.

In the case of machine tools, users provide their functional requirements, like cycle time, process capability, material to be machined, etc., to the manufacturers. Based on these requirements, the manufacturer designs the machine tool. In general, the design is for one of the following:

A general purpose machine tool is one which can be used for a wide variety of operations, on a wide range of size of work pieces [11]. Thus, they are designed for a wide range of users. A special purpose machine tool is one which is designed for some specific operations on a limited range of work piece sizes and shapes [11]. These are generally engineered to meet the requirements of a specific user. In a customized machine tool, some of the components/subassemblies, especially structural elements, are standard components, while others are designed based on the specific requirements of a customer. Thus, the machine tool designer has three different functional design scenarios (also referred to as manufacturer’s business scenarios in this book).

While considering reliability and maintenance at the design stage of a machine tool, each of the above three functional design scenarios offers different opportunities and challenges to the designer. For example, a general purpose machine tool design must be able to meet the reliability requirements of a wide range of users. On the other hand, the designer of a special purpose machine tool must be able to capture reliability requirements of a specific user. Therefore, a reliability- and maintenance-based design approach for machine tools must also be able to address the design needs in each of these functional design scenarios. Figure 1.2 depicts the entire concept of reliability- and maintenance-based design of machine tools. Many times, the existing alternatives available to the designer for some of the components/subassemblies may not be able to give satisfactory performance under the specified operating environment of the users. The designer may then need to improve the existing design of such components/subassemblies. For example, one of the main causes of failure of workhead spindle is seal failure, thereby allowing the coolant and chips to enter into the spindle bearing and causing it to fail early. In this case, the designer may have to change the design of the spindle to accommodate a different sealing technology such that it restricts coolant entry or provides better chip separation arrangement.

Similarly, users may also be interested in improving the reliability performance of their existing machines. Users can do this in two ways:

1. By improving the design of components/subassemblies in collaboration with machine tool manufacturer. The designer can search for a better alternative design for the critical components/subassemblies of the machine.

2. By changing the operating environment, shop-floor policies and cost structure. For example, users may introduce more stringent Statistical Process Control (SPC) procedures to reduce the cost of failures through early detection, thereby reducing the criticality of the failures.

However, any improvement effort needs investment. The designer and users need to make a trade-off between the cost of improvements and benefits from the improvements. Thus, an approach for considering reliability and maintenance at the design stage will also help in making such decisions.

1.4 Machine Tool Reliability: User’s View Point

Users of machine tools are other manufacturing industries, which use them for producing consumer or capital goods and are under continuous pressure to meet their customers’ requirements of high quality, low cost and timely delivery of products. Failures of production equipment affect the shop-floor level performance of the users. Thus, the users evaluate the reliability of a machine tool based on how well it performs in their production environment to meet their customers’ requirements. Moreover, shop-floor level performance also depends on the operations policy pertaining to scheduling, maintenance and quality. Furthermore, these three aspects of operations planning are affected by machine tool failures and also have some interaction effect, and hence joint consideration of various policy options pertaining to quality, maintenance and scheduling, along with their effect on the performance of manufacturing systems, are important areas of investigation.

Only in recent years have researchers started to develop approaches that try to simultaneously optimize their parameters [12–16]. For example, older approaches for production scheduling do not consider the effect of machine unavailability due to failures or preventive maintenance activities. Similarly, the older maintenance planning models did not consider the impact of maintenance on due dates to meet customer requirements. However, maintenance effectiveness cannot be measured in a meaningful way without taking into account whether the maintenance addresses the production requirements [17]. Delaying the maintenance actions to meet the production requirement may increase the process variability and risk of machine failure, which in turn may cause higher rejections or downtime losses. Ollila and Malmipuro [18] observed that maintenance has a major impact on efficiency and quality along with equipment availability. In a case study carried out in five Finnish industries, it was shown that well-functioning machinery is a prerequisite to quality products. It was also shown that a lack of proper maintenance is usually among the three most important causes of quality deficiencies.

Due to the operational complexity and the presence of deterministic and stochastic events, obtaining optimal policies for manufacturing systems is both theoretically and computationally difficult. The literature as well as input from industries has clearly indicated the need to explore the problem of joint consideration of these shop-floor operational aspects. Thus, from the point of view of users, reliability of machine tools can be used to model the interaction of various shop-floor level operations policies. Figure 1.3 depicts the concept of interaction of reliability of machine tool and various shop-floor operations planning like scheduling, maintenance and quality control.

1.5 Organization of the Book

The rest of the book is organized as follows: The second chapter presents a brief overview of the basic reliability mathematics. It presents a discussion on some of the most common lifetime distributions, viz., exponential, Weibull, Normal, etc. In the third chapter, various performance measures for machine tool reliability are discussed and detailed models are developed for each of these measures. These models relate a machine tool’s reliability and maintenance parameters with the user’s cost structure and shop-floor level operational parameters. To be more specific, models for availability, performance rate, quality rate, overall equipment effectiveness, life cycle costs and cost per piece are developed. The chapter also provides a discussion on the use of such models from the user’s and manufacturer’s point of view. Models developed in Chapter 3 rely on time-to-failure distribution parameters for estimating the number of failures for a component in a given time interval. If sufficient filed failure data are available, the designer can use the conventional methods mentioned in Chapter 2 to estimate the time-to-failure distribution parameters. However, in many of the real life situations, designers do not have sufficient field failure data. For such cases, Chapter 4 explores the possibility of utilizing expert knowledge for obtaining time-to-failure distribution parameters. Chapter 5 discusses various maintenance scenarios for machine tools. The following three maintenance scenarios are identified for a machine tool based on the types of preventive maintenance actions and the degree of restoration after a repair:

For each scenario, a maintenance optimization problem is formulated. For the case of imperfect maintenance, complexities in obtaining the optimal preventive maintenance schedule are reduced by developing some approximate models for estimating the number of failures. For the case of minimal corrective repair, a conditional number of failures model is discussed. This model, apart from regular preventive repair and replacement, also helps the designer in considering the effect of major overhauls on the optimal maintenance schedule decisions. It is demonstrated in the chapter that the optimal maintenance schedule decision also depends on the user’s cost structure and shop-floor policy parameters. In Chapter 6, two methods for reliability-based design of machine tools are provided. The first design methodology allows selection of optimal machine tool configuration based on Life Cycle Cost (LCC) and other performance requirements of the user. The optimal solution is obtained by simultaneously considering reliability and maintenance at the design stage under three different functional design scenarios, viz., general purpose machine tool design, special purpose machine tool design and customized machine tool design. The second design methodology helps the designer in improving the design of the existing system by identifying the critical components/subassemblies. A cost-based Failure Consequence Analysis (FCA) is proposed for this purpose. The proposed methodology can help a machine tool manufacturer in making effective cost-driven decisions while improving the reliability performance of the machine tool. It also provides guidance to the machine tool users in identifying the areas where they can focus to obtain better performance from the machine.

Chapter 7 and Chapter 8 present the user’s perspectives on machine tool reliability by highlighting the interaction of machine failures and maintenance with shop-floor level operations policies.

Chapter 7 presents various approaches for joint optimization of maintenance scheduling and quality control policy. In the first approach, a

control chart (goal post approach) and an imperfect time-based maintenance policy for the simultaneous economic design of preventive maintenance and quality control policy are discussed. In the second approach, Taguchi’s loss function is incorporated in the simultaneous economic design of preventive maintenance and quality control policy. A major limitation of a

control chart is that it is relatively insensitive to small shifts in the process mean. CUSUM (cumulative-sum) control charts can be used as an effective alternative to the

control chart to detect small shifts in the process mean. In the third approach, an integrated maintenance planning and quality control model is developed considering the CUSUM control chart for detecting small shifts in the process. To make the integrated preventive maintenance and quality control model more generic, approach 2 is extended from the component level to system level, where a system is assumed to be comprised of a set of independent multiple components. In Chapter 8, a methodology is developed which integrates the production schedule with the jointly optimized maintenance schedule and quality control policy for a single machine problem. Total enumeration as well as the Backward-Forward heuristic and Genetic Algorithm for solving the optimization problem are discussed. Finally, Chapter 9 discusses, in brief, various possible directions for future research in the area of machine tool reliability.

All the approaches explained in this book are illustrated with the help of suitable examples. It is hoped that this book will be useful for students, researchers and practicing engineers.

End Notes

1. Printed with permission from Inderscience Publishers: Lad B.K. and Kulkarni M.S., A mechanism for linking user’s operational equirements with reliability and maintenance schedule for machine tool, Int. J. Reliability and Safety (IJRS), Vol. 4, No. 4, 2010, p. 347.

2. Printed with permission from Inderscience Publishers: Lad B.K. and Kulkarni M.S., Integrated reliability and optimal maintenance schedule design: a life cycle cost based approach, Int. J. Product Lifecycle Management (IJPLM), Vol. 3, No. 1, 2008, p. 86.

Chapter 2

Basic Reliability Mathematics

This chapter discusses in brief the reliability mathematics helpful for understanding the chapters which follow. Specifically, it discusses various functions used in reliability engineering and their interrelationships. Some of the most commonly used life distributions are discussed in brief. The chapter also presents an overview of data analysis for estimating time-to-failure distribution parameters. A note on the simulation approach for estimating numbers of failure and the Bayesian approach in reliability engineering are also presented. Extensive discussions of these topics are out of the scope of this book. For more discussion on these topics readers may refer to some of the basic text in reliability engineering, like Ebeling [3], Birolini [20], Rinne [21], etc.

2.1 Functions Describing Lifetime as a Random Variable

Time to failure of a unit is not fixed; instead it is a random variable. Generally this variable is continuous and non-negative. There are several functions which completely specify the distribution of this random variable. These functions are discussed in brief in the following paragraphs.

Let T (T ≥ 0) be a continuous random variable representing time to failures of a system (component), then Probability Density Function (PDF) of time to failures distribution f(t) has the following properties:

The Cumulative Distribution Function (CDF) of the failure distribution can be written as:

As the reliability over time t is the probability of failure free operation over t, reliability Function R(t) is the probability that the time to failure is greater than or equal to t, i.e.,

The function R(t) is normally used when reliability for any time of operation of the system (component) is being computed, and F(t) is normally used when risk of failure is of prime concern. PDF, CDF and reliability function are shown in Figures 2.1–2.3.

Another important function used in reliability studies is the Failure Rate or Hazard Rate Function λ(t). It is the instantaneous rate of failure and can be written as conditional probability of failure per unit time. It can be expressed as:

A component may have decreasing, constant or increasing failure rate based on the underlying failure mode or cause. A curve between failure rates and time is called a Bathtub Curve and is shown in Figure 2.4.

For a non-repairable component the failure rate will either be decreasing, constant or increasing. A decreasing failure rate indicates the presence of manufacturing defects. The constant failure rate represents random failures. An increasing failure rate represents the wear out failures of the component. For example, a gear wheel may face early life failures due to improper machining or heat treatment.

Generally electronic components in a machine tool show the constant failure rate behavior. Providing excessive strength or redundancy can help in improving the reliability of such components.

Most of the mechanical components exhibit increasing failures or wearout failures. Providing preventive replacement of such components may help in ensuring higher availability of the system. Optimal replacement of such components in a machine tool is discussed in Chapter 4.

2.2 Probability Distributions Used in Reliability Engineering

Various theoretical distributions are used in reliability engineering to model the failure process. These are often called reliability models. In this section the most widely used probability distributions are discussed in brief.

2.2.1 Exponential Distribution

One of the most common failure distributions in reliability engineering is the exponential or Constant Failure Rate (CFR) model. Failures due to completely random or chance events will follow this distribution. It is generally characterized by a single parameter know as failure rate (λ) in the case of time-to-failure distribution. Various functions, mentioned in the previous section, for exponential distribution are expressed as follows:

One of the important properties of exponential distribution is the memorylessness. The memorylessproperty makes the time to failure of a component independent of age or how long the component has been operating. Mathematically,

2.2.2 Weibull Distribution

Weibull distribution is a widely used distribution to model failure process. It is normally characterized by two parameters, viz., shape (β) and scale (η) parameters. The value of shape parameter or Beta (β) provides insight into the behavior of the failure process as shown in Table 2.1.

Thus, Weibull distribution can be used to model all three types of failure behavior of the bathtub curve.

The second parameter, i.e., scale parameter or Eta (η), represents the time corresponding to the cumulative probability of failure = 0.632 regardless of the value of the shape parameter.

Various functions, mentioned in Section 2.1, for Weibull distribution can be expressed as follows:

In many cases, component may have a minimum life or shelf age. Three-parameter Weibull distribution is more appropriate for such cases. The third parameter is called location parameter (t₀). This distribution assumes that no failure will take place prior to time t₀. The failure rate function for three-parameter Weibull distribution can be written as:

2.2.3 Normal Distribution

When components fail mainly in wearout or fatigue mode, normal distribution can be used to model time to failures. However, the normal distribution allows negative values, and since the life of an item is always positive, one should be careful in using the normal distribution to represent the time-to-failure distribution. In general, if the ratio of mean and standard deviation is high (≈3 or more) one can use normal distribution to model time to failures. This would essentially ensure that only positive values occur. Normal distribution can be characterized by two parameters, viz., mean (μ) and variance (σ²). The distribution is symmetric about its mean, with the spread of the distribution determined by the standard deviation σ. Probability density function and reliability function for normal distribution with mean μ and variance (σ²) can be written as follows:

2.2.4 Lognormal Distribution

If T is the random variable representing the lognormal time to failure, the logarithm of T has a normal distribution with mean μ and standard deviation σ. The distribution is defined for only positive values of t and is therefore more appropriate than the normal as a failure distribution. The lognormal density function using mean and standard deviation of the logarithm of T as the distribution parameters can be written as follows:

2.3 Life Data Analysis

Reliability engineers use product life data to determine the probability and capability of parts, components and systems to perform their required functions for a desired period of time without failure, in specified environments. Time-to-failure data collected from field operations or tests are called life data. These life data can be measured in hours, miles, cycles-to-failure, stress cycles or any other metric with which the life of a product can be measured. Data may be complete or censored. In the case of complete data, exact time to failure of the unit is known. However, in the case of censored data, exact time to failure is not available, instead failure is known to occur before or after some known time (left or right censoring) or between time intervals (interval censoring).

These life data generally follow some probability distributions like exponential, Weibull, lognormal, etc. Weibull and exponential distributions are discussed in brief in the previous section. Once a life distribution has been selected, the parameters (i.e., the variables that govern the characteristics of the probability density function) need to be determined. Several parameter estimation methods, including least squares and maximum likelihood estimation, are available. From a statistical point of view, the method of maximum likelihood is considered to be more robust and yields estimators with good statistical properties. Parameters can also be estimated graphically using probability plotting. These methods are discussed below.

Weibull probability plotting: One method of calculating the parameters of the Weibull distribution is the use of probability plotting. The Weibull cumulative density function is:

Equation 2.19 is a linear equation of the form y_i = ax_i + b, where

and x_i = lnt_i.

Therefore, if we plot

on a simple graph paper, then the slope of the line (a) is equal to shape parameter (β) and intercept (b) is equal to –βlnη. Thus, both parameters can be easily estimated.

A Weibull probability paper is often used in place of simple graph paper to estimate these parameters. Weibull graph paper is constructed so that data generated from a Weibull distribution will plot as a straight line. The abscissa is a logarithmic scale, and the ordinate, while labeled in terms of the cumulative percentage of failures, F(t), is scaled on the basis of

Thus, the (t_i, F(t_i)) can be directly plotted on such graphs, where F(t_i) can be calculated using the median rank formula discussed in the next section.

Least Squares Estimation: As Equation 2.20 is a linear equation, a least-squares fit to the data gives a more accurate parameter estimation compared to a manual plot. Least-squares method estimates the parameters such that the square of the error between actual and estimated values of dependent variable (i.e., y_i in Equation 2.20) is minimized. For n sample observation, the parameter values of linear equations obtained from least-squares method are obtained as:

0<β<1	Decreasing failure rate behavior
β = 1	Constant failure rate behavior
β > 1	Increasing failure rate behavior