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Multi-state System Reliability Analysis and Optimization for Engineers and Industrial Managers presents a comprehensive, up-to-date description of multi-state system (MSS) reliability as a natural extension of classical binary-state reliability. It presents all essential theoretical achievements in the field, but is also practically oriented. New theoretical issues are described, including: • combined Markov and semi-Markov processes methods, and universal generating function techniques; • statistical data processing for MSSs; • reliability analysis of aging MSSs; • methods for cost-reliability and cost-availability analysis of MSSs; and • main definitions and concepts of fuzzy MSS. Multi-state System Reliability Analysis and Optimization for Engineers and Industrial Managers also discusses life cycle cost analysis and practical optimal decision making for real world MSSs. Numerous examples are included in each section in order to illustrate mathematical tools. Besides these examples, real world MSSs (such as power generating and transmission systems, air-conditioning systems, production systems, etc.) are considered as case studies. Multi-state System Reliability Analysis and Optimization for Engineers and Industrial Managers also describes basic concepts of MSS, MSS reliability measures and tools for MSS reliability assessment and optimization. It is a self-contained study resource and does not require prior knowledge from its readers, making the book attractive for researchers as well as for practical engineers and industrial managers.



1. Multi-state Systems in Nature and in Engineering

All systems are designed to perform their intended tasks in a given environment. Some systems can perform their tasks with various distinctive levels of efficiency usually referred to as performance rates. A system that can have a finite number of performance rates is called a multi-state system (MSS). Usually a MSS is composed of elements that in their turn can be multi-state. Actually, a binary system is the simplest case of a MSS having two distinctive states (perfect functioning and complete failure).
The basic concepts of MSS reliability were primarily introduced in the mid of the 1970's by Murchland (1975), El-Neveihi et al. (1978), Barlow and Wu (1978), and Ross (1979). Natvig (1982), Block and Savits (1982), and Hudson and Kapur (1982) extended the results obtained in these works. Since that time MSS reliability began intensive development. Essential achievements that were attained up to the mid 1980's were reflected in Natvig (1985) and in El-Neveihi and Prochan (1984) where can be found the state of the art in the field of MSS reliability at this stage. Readers that are interested in the history of ideas in MSS reliability theory at next stages can find the corresponding overview in Lisnianski and Levitin (2003) and Natvig (2007).

2. Modern Stochastic Process Methods for Multi-state System Reliability Assessment

The purpose of this chapter is to describe basic concepts of applying a random process theory to MSS reliability assessment. Here, we do not present the basics of the measure-theoretic framework that are necessary to pure mathematicians. Readers who need this fundamental framework and a more detailed presentation on stochastic processes can find it in Kallenberg (2002), Karlin and Taylor (1981) and Ross (1995). For reliability engineers and analysts, the books of Trivedi (2002), Epstein and Weissman (2008), Aven and Jensen (1999), and Lisnianski and Levitin (2003) are especially recommended. A great impact to stochastic processes application to MSS reliability evaluation was done by Natvig (1985) and Natvig et al. (1985).
In this chapter, the MSS system reliability models will be consequently studied based on Markov processes; Markov rewards processes, and semi-Markov processes. The Markov processes are widely used for reliability analysis because the number of failures in arbitrary time intervals in many practical cases can be described as a Poisson process and the time up to the failure and repair time are often exponentially distributed. This chapter presents a detailed description of a discrete- time Markov chain as well as a continuous-time Markov chain in order to provide for readers a basic understanding of the theory and its engineering applications. It will be shown how by using the Markov process theory MSS reliability measures can be determined. It will also be shown how such MSS reliability measures as the mean time to failure, mean number of failures in a time interval, and mean sojourn time in a set of unacceptable states can be found using the Markov reward models. These models are also the basis for reliability-associated cost assessment and life-cycle cost analysis. In practice, basic assumptions about exponential distributions of times between failures and repair times sometimes do not hold. In this case, more complicated mathematical techniques such as semi-Markov processes and embedded Markov chains may be applied. Corresponding issues are also considered in this chapter.

3. Statistical Analysis of Reliability Data for Multi-state Systems

The purpose of this chapter is to describe basic concepts of applying statistical methods to MSSs reliability assessment. Here we will stay in the Markov model framework and consider modern methods for estimation of transition intensity rates. But first basic concepts of statistical estimation theory will be briefly presented. Readers who need more fundamental and detailed development of estimation theory may wish to consult such texts as Bickel and Doksum (2007) or Lehmann and Casella (2003). Engineering applications can be found in Hines and Montgomery (1997), Ayyub and McCuen (2003), etc.

4. Universal Generating Function Method

In recent years a specific approach called the universal generating function (UGF) technique has been widely applied to MSS reliability analysis. The UGF technique allows one to find the entire MSS performance distribution based on the performance distributions of its elements using algebraic procedures. This technique (sometimes also called the method of generalized generating sequences) (Gnedenko and Ushakov 1996) generalizes the technique that is based on a well-known ordinary generating function. The basic ideas of the method were primarily introduced by I. Ushakov in the mid 1980s (Ushakov 1986, 1987). Then the method was described in a book by Reinshke and Ushakov (1988), where one chapter was devoted to UGF. (Unfortunately, this book was published only in German and Russian and so remained unknown for English speakers.) Wide application of the method to MSS reliability analysis began in the mid-1990s, when the first application was reported (Lisnianski et al. 1994) and two corresponding papers (Lisnianski et al. 1996; Levitin et al. 1998) were published. Since then, the method has been considerably expanded in numerous research papers and in the books by Lisnianski and Levitin (2003), and Levitin (2005).
Here we present the mathematical fundamentals of the method and illustrate the theory by corresponding examples in order to provide readers with a basic knowledge that is necessary for understanding the next chapters.

5. Combined Universal Generating Function and Stochastic Process Method

As was described in Chapter 2, stochastic process methods are very effective tools for MSS reliability evaluation. According to these methods a state-space diagram of a MSS should be built and transitions between all the states defined. Then a system evolution should be represented by a continuous-time discrete-state stochastic process. Based on this process all MSS reliability measures can be evaluated.
The main disadvantage of stochastic process models for MSS reliability evaluation is that they are very difficult for application to real-world MSSs consisting of many elements with different performance levels. This is so-called the “dimension curse”. First, state-space diagram building or model construction for such complex MSSs is not a simple job. It is a difficult nonformalized process that may cause numerous mistakes even for relatively small MSSs. The problem of identifying all the states and transitions correctly is a very difficult task. Second, solving models with hundreds of states can challenge the available computer resources. For MSSs consisting of n different repairable elements where every element j has k j different performance levels one will have a model with \( K = \prod^{n}_{j=1} k_{j} \) states. This number can be very large even for relatively small MSSs.

6. Reliability-associated Cost Assessment and Management Decisions for Multi-state Systems

Reliability is an important factor in the management, planning, and design of any engineering product. Today, in the global economy and due to various market pressures, the procurement decisions for many products are not based only on initial purchasing costs, but on their total life cycle costs (LCCs) (Dhillon 2000). Any important decision such as reliability allocation, spare parts storage, operation modes, etc. is based on total life cycle cost. Total LCC analysis should include all types of costs associated with a system’s life cycle. The main part of these costs for repairable systems is operation and maintenance costs. In order to repair a system we must buy corresponding spare parts, so we must pay money for spare parts purchasing. We also must allocate for spare parts storage and pay the repair team to repair the system. In addition, there are financial losses when a system interrupts its work because of failure and so on. All these costs together are usually significantly greater than the cost of purchasing a system. Below these costs will be called reliability associated costs (RACs). In order to perform effective life cycle cost analysis RACs should be accurately assessed. The reliability engineer performing such an assessment should have a basic knowledge and cooperate with specialists in many areas (including engineering design, finance and accounting, statistical analysis, reliability and maintainability engineering, logistics, and contracting). Creating the methods for correct evaluation of RACs is one of the main problems of practical reliability engineering. This problem is partially solved only for binary-state systems. Unfortunately, for MSSs almost every individual practical case requires carrying on special research and very few research works have been devoted to this problem till now. As a result, managers often do not even recognize the problem’s existence. Therefore, there is a significant contradiction between the great theoretical achievements of reliability theory and their relatively rare successful applications in practice for MSSs.
In this chapter we present the history of LCC analysis, its principles, and corresponding standards. It will be shown that RAC is really a main part of LCC for the majority of repairable systems. The methods described in the previous chapters will be applied below in order to assess RAC for MSSs for some methodologically important cases. Based on this, the corresponding optimal management practices are established. It is shown that significant amount of money may be saved as a result of correct reliability management of MSS’s.

7. Aging Multi-state Systems

Many technical systems are subjected during their lifetime to aging and degradation. After any failure, maintenance is performed by a repair team. This chapter considers an aging MSS, where the system failure rate increases with time.
Maintenance and repair problems for binary-state systems have been widely investigated in the literature. Barlow and Proshan (1975), Gertsbakh (2000), Valdez-Flores and Feldman (1989), and Wang (2002) survey and summarize theoretical developments and practical applications of maintenance models. Aging is usually considered as a process that results in an age-related increase of the failure rate. The most common shapes of failure rates have been observed by Gertsbakh and Kordonsky (1969), Meeker and Escobar (1998), Bagdonavicius and Nikulin (2002), and Wendt and Kahle (2006), Finkelstein (2003). An interesting approach was introduced by Finkelstein (2005, 2008), where it was shown that aging is not always manifested by an increasing failure rate. For example, it can be an upside-down bathtub shape of the failure rate that corresponds to a decreasing mean remaining lifetime function.

8. Fuzzy Multi-state System: General Definition and Reliability Assessment

In conventional multi-state theory, it is assumed that the exact probability and performance level of each component state are given. However, with the progress in modern industrial technologies, the product development cycle has become shorter and shorter while the lifetime of products has become longer and longer (Huang et al. 2006). In many highly reliable applications, there may be only a few available observations. Therefore, it is difficult to obtain sufficient data to estimate the precise values of these probabilities and performance levels in these systems. Moreover, inaccuracy in system models that is caused by human error is difficult to deal with solely by means of conventional reliability theory (Huang et al. 2004). In some cases, in order to reduce the computational burden, a simplified model is used to represent a complex system and a MSS model is used to characterize a continuous-state system, which can reduce the computational accuracy. New techniques and theories are needed to solve these fundamental problems.


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