Repairable Systems Reliability Analysis
Description
This book provides an application-oriented framework for reliability modeling and analysis of repairable systems in conjunction with the procurement process of weapon systems and throughput analysis for industries.
Most of the reliability literature is directed towards non-repairable systems, that is, systems that fail are discarded or replaced. This book is mainly dedicated towards providing coverage to the reliability modeling and analysis of repairable systems that undergo failure-repair cycles.
This unique book provides a comprehensive framework for the modeling and analysis of repairable systems considering both the non-parametric and parametric approaches to deal with their failure data. The book presents MCF based non-parametric approach with several illustrative examples and the generalized renewal process (GRP) based arithmetic reduction of age (ARA) models along with its applications to the systems failure data from the aviation industry. A complete chapter on an integrated framework for procurement process is devoted by utilizing the concepts of multi-criteria decision-making (MCDM) techniques which will of a great assistance to the readers in enhancing the potential of their respective organizations. This book also presents FMEA methods tailored for GRP based repairs.
This text has primarily emerged from the industrial experience and research work of the authors. A number of illustrations have been included to make the subject lucid and vivid even to the readers who are relatively new to this area. Besides, various examples have been provided to display the applicability of presented models and methodologies to assist the readers in applying the concepts presented in this book.
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Persons
Rajiv Nandan Rai is an assistant professor at Subir Chowdhury School of Quality and Reliability, Indian Institute of Technology Kharagpur, West Bengal, India. He obtained his PhD in Mechanical Engineering with specialization in Reliability, Maintenance and Industrial Engineering from IIT Delhi, India. He has hands on industrial experience of almost 23 years in military aviation in which he has worked at all levels of maintenance, repair and overhaul of aircraft, aero engines and their components. He has published papers in several SCI journals.
S. K. Chaturvedi is a professor and currently Head at Subir Chowdhury School of Quality and Reliability, Indian Institute of Technology Kharagpur, West Bengal, India. He has research interest in the areas of reliability modeling and analysis, network reliability, life-data analysis, maintenance and optimization. He has published papers in several international journals, is reviewer to and executed several consultancy projects of private and Govt. organizations. He's written two books He's on the editorial board of Int. J. International Journal of Reliability and Safety (IJRS), Int. J of Mathematical, Engineering and Management Sciences (IJMEMS). He's also a senior member to IEEE and has also served as Co-Editor-in-Chief to International Journal of Performability Engineering.
Nomesh Bolia received the PhD degree in operations research from the University of North Carolina at Chapel Hill, Chapel Hill, NC, USA. He is currently a professor in the Department of Mechanical Engineering, Indian Institute of Technology Delhi, India. His research interests include operations research and its applications to reliability, health, telecommunications, transportation and public systems. Dr. Bolia received the prestigious Indo-US Fellowship for Public Health Research and has published in several journals including IEEE Transactions.
Content
Series Editor Preface ix
Preface xi
List of Tables xv
List of Figures xix
1 Introduction to Repairable Systems 1
1.1 Introduction 1
1.2 Perfect, Minimal, and Imperfect Repairs 4
1.3 Summary 7
References 8
2 Repairable Systems Reliability Analysis: Non-Parametric 9
2.1 Introduction 9
2.2 Mean Cumulative Function 11
2.3 Construction of MCF Plot and Confidence Bounds: Exact Age Data 14
2.3.1 MCF Construction: Exact Age Data 14
2.3.2 Confidence Bounds on MCF: Exact Age Data 20
2.3.3 Construction of MCF Plot and Confidence Bounds: Grouped Data 22
2.3.4 Confidence Bounds on MCF: Grouped Data 22
2.4 Case Study: ROV System 23
2.5 Interval Age Analysis 27
2.5.1 MCF With All Types of Failure Modes Combined 27
2.5.2 MCF for Individual Failure Modes 33
2.5.3 Exact Age Analysis 33
2.6 Summary and Conclusion 40
References 41
3 Repairable Systems Reliability Analysis: Parametric 43
3.1 Introduction 43
3.2 Basic Terminologies 43
3.3 Parametric Analysis Approaches 46
3.3.1 Renewal Process 46
3.3.2 Non-Homogeneous Poisson Process (NHPP) 47
3.3.3 Generalized Renewal Process (GRP) 54
3.3.3.1 ARA Models 54
3.3.3.2 Kijima-I Model 56
3.3.3.3 Kijima-II Model 63
3.3.3.4 Virtual Age-Based Reliability Metrics 72
3.3.4 Summary 77
References 78
Further Reading 79
ARI Models 79
4 Goodness-of-Fit Tests for Repairable Systems 83
4.1 Introduction 83
4.2 Mann's Test for the Weibull Distribution 84
4.3 Laplace Trend Test 86
4.4 GOF Models for Power Law Process 87
4.4.1 Crow/AMSAA Test 87
4.4.2 Common Beta Hypothesis (CBH) Tests 88
4.4.3 CVM Test 90
4.5 GOF Model for GRP Based on Kijima-I Model 92
4.6 Summary 94
References 95
5 Maintenance Modeling in Repairable Systems 97
5.1 Introduction to Maintenance Policies Using Kijima Virtual Age Model 97
5.2 Need for HFRC Threshold 98
5.3 Reliability-Based Methodology for Optimal Maintenance Policies in MA 100
5.3.1 Reliability-Based Threshold Model for HFRC 100
5.3.1.1 Review of Present Maintenance Policy for HFRCs 102
5.4 Availability-Based HFRC Analysis 107
5.4.1 Availability-Based Criteria for HFRC (BB Approach) 107
5.4.1.1 Review of Overhaul Cycle (BB Approach) 109
5.4.2 Availability-Based HFRC Threshold Model Considering FMs 112
5.4.2.1 Maintenance Strategy for HFRCs With FM Approach 115
5.4.2.2 TBO Model Considering FMs 115
5.5 Summary 120
References 122
6 FMEA for Repairable Systems Based on Repair Effectiveness Index 125
6.1 Introduction 125
6.2 A Brief Overview on Performing FMEA 128
6.2.1 System Definition 129
6.2.2 Identification of Failure Modes 129
6.2.3 Determination of Cause 130
6.2.4 Assessment of Effect 130
6.2.5 Classification of Severity (S) 131
6.2.6 Estimation of Probability of Occurrence (O) 131
6.2.7 Detection 132
6.2.8 Computation of Conventional RPN 132
6.2.9 Determination of Corrective Action 132
6.3 Estimating RPNs Through the Modified Approach [15] 133
6.4 Corrective Actions 135
6.5 Summary 140
References 140
7 An Integrated Approach to Weapon Procurement Systems 143
7.1 Introduction 143
7.2 Analytic Network Process Model 147
7.3 AP Index and AP Value Estimation 151
7.3.1 Analytic Hierarchy Process Model 151
7.3.2 AP Index Estimation 151
7.3.3 Sample AP Index Estimation 152
7.3.4 AP Value Estimation 154
7.4 Formation of an ACU 160
7.4.1 Attack Model 160
7.4.2 Defense Model 161
7.4.3 Illustrative Example 162
7.5 Summary 164
References 166
8 Throughput Analysis of the Overhaul Line of a Repair Depot 169
8.1 Introduction 169
8.2 Basic Definitions, Parameters, and Relationships 173
8.3 Variability 174
8.3.1 Measures and Classes of Variability 174
8.3.2 Causes of Variability 175
8.3.3 Variability from Preemptive Outages (Breakdowns) 175
8.3.4 Variability in Flows 176
8.3.5 Variability Interactions Queuing 177
8.3.5.1 The M/M/1 Queue 177
8.3.5.2 The G/G/1 Queue 178
8.4 Process Batching 178
8.5 System Flow and Parameters 179
8.6 System Analysis and Discussion 181
8.6.1 Component 1: LPCR Blades 181
8.6.2 Component 2: CCOC 188
8.6.3 Component 3: LPTR Blades 190
8.7 Summary 192
References 193
Appendix A 195
The Saaty Rating Scale 195
Pairwise Comparisons and Estimation of Weights for ANP 196
Appendix B 257
Unweighted Super-Matrix (Part 1) 258
Unweighted Super-Matrix (Part 2) 265
Weighted Super-Matrix (Part 1) 273
Weighted Super-Matrix (Part 2) 280
Limit Super-Matrix (Part 1) 288
Limit Super-Matrix (Part 2) 295
Appendix C 303
Pairwise Comparisons and Estimation of Weights for AHP 304
Appendix D 339
F Distribution Table 339
Appendix E 347
Normal Distribution Table 347
Appendix F 353
Chi Square Table 353
Appendix G 363
Critical Values for Cramér-von Mises Test 363
Index 365
1
Introduction to Repairable Systems
1.1 Introduction
A system is a collection of mutually related items, assembled to perform one or more intended functions. Any system majorly consists of (i) items as the operating parts, (ii) attributes as the properties of items, and (iii) the link between items and attributes as interrelationships. A system is not only expected to perform its specified function(s) under its operating conditions and constraints but also expected to meet specified requirements, referred as performance and attributes. The system exhibits certain behavioural pattern that can never ever be exhibited by any of its constituent items or their subsets. The items of a system may themselves be systems, and every system may be part of a larger system in a hierarchy. Each system has a purpose for which items, attributes, and relationships have been organized. Everything else that remains outside the boundaries of system is considered as environment from where a system receives input (in the form of material, energy, and/or information) and makes output to the environment which might be in different form as that of the input it had received. Internally, the items communicate through input and output wherein output(s) of one items(s) becomes the input(s) to others. The inherent ability of an item/system to perform required function(s) with specified performance and attributes when it is utilized as specified is known as functionability [1]. This definition differentiates between the terms functionality and functionability where former is purely related to the function performed whereas latter also takes into considerations the level of performance achieved.
Despite the system is functionable at the beginning of its operational life, we are fully aware that even after using the perfect design, best technology available for its production or the materials from which it is made, certain irreversible changes are bound to occur due to the actions of various interacting and superimposing processes, such as corrosion, deformations, distortions, overheating, fatigue, or similar. These interacting processes are the main reason behind the change in the output characteristics of the system. The deviation of these characteristics from the specifications constitutes a failure. The failure of a system, therefore, can be defined as an event whose occurrence results in either loss of ability to perform required function(s) or loss of ability to satisfy the specified requirements (i.e., performance and/or attributes). Regardless of the reason of occurrence of this change, a failure causes system to transit from a state of functioning to a state of failure or state of unacceptable performance. For many systems, a transition to the unsatisfactory or failure state means retirement. Engineering systems of this type are known as non-maintained or non-reparable system because it is impossible to restore their functionability within reasonable time, means, and resources. For example, a missile is a non-repairable system once launched. Other examples of non-repairable systems include electric bulbs, batteries, transistors, etc. However, there are a large number of systems whose functionability can be restored by effecting certain specified tasks known as maintenance tasks. These tasks can be as complex as necessitating a complete overhaul or as simple as just cleaning, replacement, or adjustment. One can cite several examples of repairable systems one's own day-to-day interactions with such systems that include but not limited to automobiles, computers, aircrafts, industrial machineries, etc. For instance, a laptop, not connected to an electrical power supply, may fail to start if its battery is dead. In this case, replacing the battery-a non-maintained item-with a new one may solve the problem. A television set is another example of a repairable system, which upon failure can be restored to satisfactory condition by simply replacing either the failed resistor or transistor or even a circuit board if that is the cause, or by adjusting the sweep or synchronization settings.
The system, in fact, wavers and stays between satisfactory and unsatisfactory states during its operational life until a decision is taken to dispense with it. The proportion of the time, during which the system is functionable, depends on the interaction between the inherent characteristics of a system from the design and utilization function given by the users' specific requirements and actions. The prominent inherent characteristics could be reliability, maintainability, and supportability. Note that these characteristics are directly related to the frequency of failures, the complexity of a maintenance task, and ease to support that task. The utilization characteristics are driven by the users' operational scenarios and maintenance policy adopted, which are further supported by the logistics functions, which is related to the provisioning of operational and maintenance resources needed. In short, the pattern followed by an engineering system can be termed as funtionability profile whose specific shape is governed by the inherent characteristics of design and system's utilization. The metric Availability or its variants quantitatively summarize the functionability profile of an item/system. It is an extremely important and useful measure for reparable systems; besides, a technical aid in the cases where user is to make decisions regarding the acquisition of one item among several competing possibilities with differing values of reliability, maintainability, and supportability. Functionability and availability brought together indicates how good a system is. It is referred as system technical effectiveness representing the inherent capability of the system. Clearly, the biggest opportunity to make an impact on systems' characteristics is at the design stage to won or lost the battle when changes and modifications are possible almost at negligible efforts. Therefore, the biggest challenge for engineers, scientists, and researchers has been to assess the impact of the design on the maintenance process at the earliest stage of the design through field experiences, analysis, planning and management. And, the repairable system analysis is not just constricted on finding out the reliability metrics.
Most complex systems, such as automobiles, communication systems, aircraft, engine controllers, printers, medical diagnostics systems, helicopters, train locomotives, and so on so forth are repaired once they fail. In fact, when a system enters into utilization process, it is exposed to three different performance influencing factors, viz., operation, maintenance, and logistics, which should be strategically managed in accordance with the business plans of the owners. It is often of considerable interest to determine the reliability and other performance characteristics under these conditions. Areas of interest may include assessing the expected number of failures during the warranty period, maintaining a minimum reliability for an interval, addressing the rate of wear out, determining when to replace or overhaul a system, and minimizing its life cycle costs.
Traditional reliability life or accelerated test data analysis-nonpara-metric or parametric-is based on a truly random sample drawn from a single population and independent and identically distributed (i.i.d.) assumptions on the reliability data obtained from the testing/fielded units. This i.i.d. assumption may also be valid, intuitively, on the first failure of several identical units, coming from the same design and manufacturing process, fielded in a specified or assumed to be in an identical environment. Life data of such items usually consists of an item's single failure (or very first failure for reparable items) times with some items may be still surviving-referred as censoring or suspension. The reliability literature is in plenty to cover such aspects in reliability data analysis where the failure times are modeled by appropriate life distributions [2].
However, in repairable system, one generally has times of successive failures of a single system, often violating the i.i.d assumption. Hence, it is not surprising that statistical methods required for repairable system differ from those needed in reliability analysis of non-repairable items. In order to address the reliability characteristics of complex repairable systems, a process rather than a distribution is often used. For a repairable system, time to next failure depends on both the life distribution (the probability distribution of the time to first failure) and the impact of maintenance actions performed after the first occurrence of a failure. The most popular process model is the Power Law Process (PLP). This model is popular for several reasons. For instance, it has a very practical foundation in terms of minimal repair-a situation when the repair of a failed system is just enough to get the system operational again by repair or replacement of its constituent item(s). Second, if the time to first failure follows the Weibull distribution, then the Power Law model repair governs each succeeding failure and adequately models the minimal repair phenomenon. In other words, the Weibull distribution addresses the very first failure and the PLP addresses each succeeding failure for a repairable system. From this viewpoint, the PLP can be regarded as an extension of the Weibull distribution and a generalization of Poisson process. Besides, the PLP is generally computationally easy in providing useful and practical solutions, which have been...
