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Performability of a product, a system or a service has been defined by this author (Misra in Inaugural Editorial of International Journal of Performability Engineering 1:1–3, 2005 [1] and Misra in Handbook of performability engineering, Springer, London, 2008 [2]) as an attribute of the holistic performance reckoned over its entire life cycle ensuring not only high dependability (quality, reliability, maintainability and safety) but also sustainability. Sustainability is a characteristic specific to a product, system or service. At the same time, a dependable product, system or service may not be sustainable. It may also be necessary here to point out that without dependability, sustainability is meaningless. Therefore, both dependability as well as sustainability attributes should be considered and must be evaluated in order to evaluate performability of a product. All other attributes of the definition of performability have been defined and can be computed except sustainability. In order to evaluate performability, it is therefore essential to define and compute sustainability. For developing sustainable products, systems and services in the twenty-first century, it is essential that we should be able to define precisely and quantify sustainability since one cannot improve what cannot be measured or assessed. The objective of the present chapter is to understand the implications of sustainability in order to facilitate computation of sustainability and thereby the performability. The purpose of 13 chapters in the Handbook (Misra in Handbook of performability engineering, Springer, London, 2008 [2]) by the author was to provide detailed introduction to each constituent elements of the definition of performability, namely quality, reliability, maintainability, safety and sustainability, and these chapters were received very well by the international academic community as is evident from Table 1.1. This was done with the intent to evoke interest among researchers across the world in the concept of performability leading to a way to compute or assess performability. But this did not happen in the past 12 years after the publication of the Handbook in 2008. The main impediment in this effort is the procedure to evaluate sustainability.

Weibull distribution is one of the most widely used model for failure data in reliability studies. In this paper a sequential estimation procedure for estimating the parameters of Weibull distribution is proposed, which is, in principle similar to Kalman filtering. The main advantage of this approach is that it shows the variation of parameters over a time as new failure data becomes available to the analyst for estimation. Also once an available data has been used, the method does not require that data for further processing as and when the new data becomes available for updating the estimates of parameters. Its use in Quality control asa control chart has been indicated and the procedure is illustrated with the help of examples.

A method of obtaining a minimized system reliability or unreliability expression using Demorgan's theorem is presented. Some formulae based on this theorem are provided, from which the minimized expression of system unreliability can be obtained by knowledge of pathsets. At the same time, this expression can also provide minimum cutsets of the system easily. On the other hand, one can obtain the minimum pathsets of a system without much difficulty, if the minimum cutsets are known.

It is well known that conventional reliability analysis using probabilities has been found to be inadequate to handle uncertainty of failure data and modelling. To overcome this problem, the concept of “fuzzy probability” has been used in the evaluation of reliability of a general non-series parallel network. Basically, only two operations, i.e. multiplication and complementation, need to be performed in assessing system reliability of a non-series parallel system. A procedure is developed to carry out these operations efficiently.

In many reliability design problems, the decision variables can only have integer values. The redundancy allocation is an example of one such problem. Besides the redundancy optimization problem, there are many other problems, such as spare pans allocation, repairmen allocation which all necessitate an integer programming formulation. In other words, integer programming plays an important role in system reliability optimization. An algorithm is presented which provides an exact, simple and economical solution to any general class of integer programming problems and thereby offers reliability designers an efficient tool for system design. The algorithm can effectively be used to solve a wide variety of reliability design problems. The scope of use of this algorithm is also indicated and the procedure is adequately illustrated through an example.

Several applications are proposed for an efficient, fast and exact technique for solving integer programming problems that normally arise in optimal reliability design problems. The computational experience of solving these problems is described and the algorithm is found to be superior to any of the earlier methods available to date in solving a wide variety of reliability design problems. The paper aims to demonstrate this capability of the algorithm; as the algorithm is based on functional evaluations only, it solves even very large system problems economically.

In many reliability design problems, the decision variables can only have integer values. The redundancy allocation is an example of one such problem; others include spare parts allocation, or repairmen allocation, which necessitate an integer programming formulation. In other words, integer programming plays an important role in system reliability optimization. In this paper, an algorithm is presented which provides an exact, simple and economical solution to any general class of integer programming problems and thereby offers reliability designers an efficient tool for system design. The algorithm can be used effectively to solve a wide variety of reliability design problems. The scope of use of this algorithm is also indicated and the procedure is illustrated by an example.

The most important attribute of performance is reliability and this is defined as the probability of failure free operation over a specified duration of time under a given set of conditions, which depend on location and the kind of application the item is put to. In designing for reliability, the objective must be to maximize inherent and operational reliability such that occurrences of failures are considerably reduced. Component or system reliability goals can be achieved through design and testing. Failures and consequently accidents can be avoided if their causes of can be traced and taken care of during the design process.

Maintenance is another important aspect of system performance after reliability. There are several facets of maintenace management, and in this introductory chapter we would like to have these surveyed. Broadly speaking, maintenance is the process of maintaining equipment in its operational state either by preventing its transition to a failed state or by restoring it to an operational state following a failure. This leads to various types of maintenace activities that can be planned to realize the objective of maitenance, such as preventive, predictive, or corrective. Recent developments in maintenance engineering and management are also discussed in this chapter.

For better system performance, attributes such as quality, reliability, maintainability, safety, and risk that are closely related and govern the dependability of the system, must be considered. It is also necessary to understand the inter-relationship between these factors so that one not only minimizes the chances of occurrence of any untoward incident at the design and fabrication stage but also minimizes the chances of occurrences and the consequences of such an event during the system operation and use phase.

Reliability is one of the most important attributes of performance in arriving at the optimal design of a system since it directly and significantly influences the system’s performance and its life cycle costs. Poor reliability would greatly increase life-cycle costs of the system, and reliability based design must be carried out if the system is to achieve its desired performance. An optimal reliability design is one in which all possible means available to a designer have been explored to enhance the reliability of the system with minimum cost under the constraints imposed on the development of a system.

The 15 papers in this special section present new tools, techniques, applications, approaches, and surveys related to various aspects of reliability, and risk analysis and management.

In this chapter, we discuss an entirely new development in the area of system reliability evaluation. This development promises to provide an analyst a handy tool for carrying out various reliability analyses and make system reliability assessment a routine job, which need not be confined to highly qualified analysts only. This desk-top facility is considered preferable and economical over the sophisticated softwares on a PC, or on a mini or microcomputer keeping in view the several iterative trials that are required during the reliability design of a system. Besides, using this gadget, a reliability analyst or designer can observe the consequences of his ramifications of the designed system configuration, interactively. This gadget can also help in the learning process of system reliability designers.

There was a serious discussion on Reliability Engineering for the Future during the European Conference on Safety and Reliability held at Copenhagen, in June 1992. The panellists discussed the current state-of-art and critically examined the shortcomings of present models, practices, methodologies and approaches in the context of dependability engineering. The editor's motivation to put these thoughts into a detailed logical and meaningful presentation of involved problems in the form of a book came as a consequence of this event. This stocktaking will guide and channelize the on-going efforts and the future developments in the area of system reliability evaluation.

Fuzzy sets being relatively a new concepts, a brief summary of important definitions and relevant mathematical background are essential for understanding the fuzzy set theory (FST) and its applications. Our objective, in this chapter, is to restrict our discussion to such topics in FST which will have direct bearing on their applications to system reliability or performance evaluation and other related areas like fault tree analysis and risk assessment. The applications of FST are being presented in Part-II of this chapter, separately.

A large number of research articles have appeared in the literature during the last two decades on the subject of system reliability optimisation, each with a view to providing simple, exact and efficient techniques. Here, an efficient, fast and exact technique is proposed for solving integer‐programming problems that normally arise in optimal reliability design problems. The algorithm presented is superior to any of the earlier methods available so far, being based on functional evaluations and a limited systematic search close to the boundary of resources. Thus it can quickly solve even very large system problems. It can also be effectively used with other operations research problems involving integer‐programming formulations.

Reliability evaluation of complex systems is of interest to engineers of all disciplines. The techniques for reliability evaluation depend upon the logic diagram of the system. System reliability evaluation is straight forward in case of series-parallel systems; while this is not so in general nonseries parallel systems. In this paper, many different techniques for reliability evaluation of general systems have been presented. Merits and demerits of every method are discussed. An example is solved by all the methods to have a comparison of computational labour involved and the size of final derived reliability expression.

Very few techniques exist for reliability evaluation of communication systems where links as well as nodes have certain probability of failure. This correspondence describes a technique by which the reliability expression for such a system can be conveniently derived. It is also shown that using the concept of this correspondence, it is possible to extend all the existing reliability-evaluation algorithms to communication systems with little effort.

Once a system has been designed to meet its operational requirements, an analysis is usually carried out to redesign the system such that reliability goals are satisfied. The objective of this analysis is to work out a tradeoff between the design parameters based on system reliability effectiveness, and the availability of resources. Efforts have been made to systematize reliability design procedures, however much remains to be done in this area. Although each system is unique, a general design philosophy and methodology could be described that may be found useful for designing reliable systems. The paper describes state of art on optimum design of systems based on reliability effectiveness.

The present paper develops mathematical models for evaluating the exact reliability and mean time to failure of k-out-of-m: G systems with different unit failure probabilities. The ith unit is assumed to be characterized by a general hazard rate hi(t) = λitb. The models are based on the concepts of tie sets; they are fairly simple and can be used for any values of m and k. The algorithm developed for these models is suitable for adoption for a computer code.

Reliability of a network containing only link failures is well attempted in the reliability literature. In a communication network, node failure probability is sometimes as significant as link failure probability. The problem has been given little attention. The procedure evolved is tedious and the resulting reliability expression is quite lengthy. In this paper, the flow graph method of reliability evaluation has been extended to consider simultaneously the node as well as link failure. The method suggested is simple and leads to a simpler reliability expression. An example has been solved doing all the steps of the procedure clearly.

Availability and other related parameters, such as failure frequency, mean down Lima, otc, aro important reliability parameters to be considered at the design stage of a system; and this needs appropriate mathematical models for their assessment. This paper presents algorithms to assess quantitatively the above parameters of a repairable m-ordor system comprising units with unequal failuro and repair rates. The concepts of minimal cut sets and minimal tie sets have been employed. The modols developed here can be applied to series and parallol systems as well.

This paper presents a method of maximizing reliability of a series system subject to multiple constraints. The use of both active series parallel and standby redundancy is considered to increase the reliability of the system. A flexible tree search method has been used in this paper which is found to be an efficient method of solution, as it requires only simple mathematical computations. A three stage system is used for illustration.

A laboratory model of a reliability analyzer is described. The idea behind the development of such an analyzer is to provide an analogue model for determining all pathsets or cutsets of a system and thereby determining the system reliability when the system configuration is known along with the reliability of its constituent elements. Any system with unreliable branches and nodes can be simulated on the reliability analyzer using the logic gates. The analyzer provides tremendous flexibility in changing of the system configuration for reliability studies. The users require manually plugging in or out of a particular link to simulate addition or removal of a branch. The reliability parameters such as terminal reliability and global reliability in the case of a communication network can also be studied. Block diagram or fault tree approaches can be used to analyze a system on the analyzer described in this paper.

A system design engineer often resorts to redundancies while maximizing the probability of success of a system designed to perform a certain job over a period, This will have to be done within the resources available. The problem of maximizing system reliability, subject to some linear constraints such as cost, weight or volume, is considered in this paper- The usual problem is transformed to a suitable form and the least square concept is used to develop a solution technique. This type of approach is found to be very simple and faster than earlier methods although the solution is an approximate one.

An algorithm is presented for evaluation of reliability of any redundant network. It uses the properties of digraphs and is especially suitable for the computer analysis of large complex networks. A method for deriving the reliability expression for any type of network is also described along with a Fortran computer program for the reliability evaluation of series-parallel networks.

A new formulation for the problem of system reliability optimization when constrained by some linear constraints is presented in this paper. The formulation provided is easily adaptable to geometric programming form. The problem is further reduced to that of an optimization of an unconstrained objective function with variables one less than the number of constraints, when its dual is defined. It is amply demonstrated through this paper that the formulation and approach of this paper are simpler than earlier attempts described elsewhere. An example is also given.

An algorithm is developed to obtain a simplified reliability expression for a general network. All the success paths of the network are determined; then they are modified to be mutually disjoint. The reliability expression follows directly from the disjoint paths. The algorithm is easy and computationally economical. The reliability expression involves fewer terms and arithmetic operations than any of the existing methods. This is an advantage, considering the size of the system a reliability engineer normally handles and the frequency with which the expression is used for reliability studies.

Reliability expression of a system derived by various methods of reliability evaluation varies considerably. Therefore, computational time needed for the numerical evaluation of reliability of a particular system shall depend upon the method applied. Also, the absolute error in the final result for reliability shall depend upon the complexity of reliability expression, and hence on the method applied. A quantitative study of these two important aspects is given in this paper.