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The purpose of this paper is to find the proper statistical distribution function, which can cover the failure time of a single machine or a group of machines. To this end, an innovative program is written in an Excel software, capable of assessing at least six statistical distribution functions. This research study intends to show the advantages of applying statistical distribution functions in an integrated model format to create or increase productive reliability machines. Productive reliability is a simultaneous combination of efficiency and effectiveness in reliability.

The method of theoretical research methodology comprises data collection tools, reference books and articles in addition to exploiting written reports of the Iranian Center for Defence’s Standards. The practical research method includes deploying and assessing the proposed model for a selected machine (in this case a computerized numerical control machine).

A comprehensive program in an Excel software having the capability of assessing at least six statistical distribution functions was developed to find the most efficient option for covering the failure times of each machine in the shortest time with the highest precision. This is regarded as the most important achievement of the present study. Furthermore, the advantages of applying the developed model are discussed and a large group of which have direct influences on the productivity of equipment reliability.

The originality of the research was ascertained by managers and experts working in maintenance issues at the different levels of the Defense Industries Organization.

The purpose of this paper is to analyze a parallel system consisting of n dependent components with lifetimes following Weibull distribution. FGM Copula in multivariate case is used to generate the reliability function of the original system. A reduction method is introduced to improve system reliability. Other methods of hot, cold and warm duplication are established to improve system reliability. An application is introduced to show the results and compare between different improvement methods.

In this paper, a study of a parallel system consisting of n dependent and non-identical components is introduced. Reliability function of the original system is derived by using the concepts of copula, subject to Weibull distribution. Reliability function of the original system is improved according to reduction, hot duplication, warm and cold duplication methods. Reliability equivalence factors are introduced to compare between different system designs. Numerical illustration and real-time data application are discussed to show the results obtained in this paper.

Copulas can be used to model the reliability of systems with dependent units.

This paper is original. Improvement of the reliability of dependent systems is not discussed in literature. Copula is a useful tool to analyze the reliability of dependent systems. The introduced model is considered as a generalization of the models discussed in literature.

The purpose of this paper is to evaluate the reliability and signature reliability of solar panel k-out-of-n-multiplex system with the help of universal generating function.

Energy scarcity and global warming issues have become important concerns for humanity in recent decades. To solve these problems, various nations work for renewable energy sources (RESs), including sun, breeze, geothermal, wave, radioactive and biofuels. Solar energy is absorbed by solar panels, referred to as photovoltaic panels, which then transform it into electricity that can be used to power buildings or residences. Remote places can be supplied with electricity using these panels. Solar energy is often generated using a solar panel that is connected to an inverter for power supply. As a result, a converter reliability evaluation is frequently required. This paper presents a study on the reliability analysis of k-out-of-n systems with heterogeneous components. In this research, the universal generating function methodology is used to identify the reliability function and signature reliability of the solar array components. This method is commonly used to assess the tail signature and Barlow-Proschan index with independent and identically distributed components.

The Barlow-Proschan index, tail signature, signature, expected lifetime, expected cost and minimal signature of independent identically distributed are all computed.

This is the first study of solar panel k-out-of-n-multiplex systems to examine the signature reliability with the help of universal generating function techniques with various measures.

The purpose of this paper is to provide a new method to estimate the reliability of series system by using copula functions. This problem is of great interest in industrial and engineering applications.

The authors introduce copula functions and consider a Bayesian analysis for the proposed models with application to the simulated data.

The use of copula functions for modeling the bivariate distribution could be a good alternative to estimate the reliability of a two components series system. From the results of this study, the authors observe that they get accurate Bayesian inferences for the reliability function considering large samples sizes. The Bayesian parametric models proposed also allow the assessment of system reliability for multicomponent systems simultaneously.

Usually, the studies of systems reliability engineering assume independence among the component lifetimes. In the approach the authors consider a dependence structure. Using standard classical inference methods based on asymptotical normality of the maximum likelihood estimators for the parameters the authors could have great computational difficulties and possibly, not accurate inference results, which there is not found in the approach.

The purpose of this paper is to introduce a novel approach of designing, specifying, and describing the behavior of software systems in a way that helps to predict their reliability from the reliability of the components and their interactions.

Design imperatives and relevant mathematical documentation techniques for improved reliability predictability of software systems are identified.

The design approach, which is named design for reliability predictability (DRP), integrates design for change, precise behavioral documentation and structure based reliability prediction to achieve improved reliability predictability of software systems. The specification and documentation approach builds upon precise behavioral specification of interfaces using the trace function method (TFM) and introduces a number of structure functions or connection documents. These functions capture both the static and dynamic behavior of component‐based software systems and are used as a basis for a novel document driven structure based reliability predication model.

Decades of research effort have been spent in software design, mathematical/formal specification and description and reliability prediction of software systems. However, there has been little convergence among these three areas. This paper brings a new direction where the three research areas are unified to create a new design paradigm.

The purpose of this paper is to evaluate the reliability and structure function of refrigeration complex system consisted of four components in complex manner.

Although, a variety of methodologies have been used to assess the refrigeration system's reliability function that has proven to be effective, the universal generating function approach is the basis of this research study, which is used in the calculation of a domestic refrigeration system with four separate components that are related in series and parallel with a corresponding sample to form a complex machine.

In this paper, signature reliability of the refrigeration system has been evaluated with the universal generating function technique. There are four components present in the proposed system in complex (series and parallel) manner. The tail signature, signature, Barlow–Proschan index, expected lifetime and expected cost of independent identically distributed are all computed.

This is the first study of domestic refrigeration system to examine the signature reliability with the help of universal generating function techniques with various measures. Refrigeration systems are an essential process in industries and home applications as they perform cooling or the maintain temperature at the desired value. A cycle of refrigeration consists of four main components such as, heat exchange, compression and expansion with a refrigerant flowing through the units within the cycle.

This chapter of the book seeks to use famous mathematical functions (statistical distribution functions) in evaluating and analyzing supply chain network data related to supply chain management (SCM) elements in organizations. In other words, the main purpose of this chapter is to find the best-fitted statistical distribution functions for SCM data. Explaining how to best fit the statistical distribution function along with the explanation of all possible aspects of a function for selected components of SCM from this chapter will make a significant attraction for production and services experts who will lead their organization to the path of competitive excellence. The main core of the chapter is the reliability values related to the reliability function calculated by the relevant chart and extracting other information based on other aspects of statistical distribution functions such as probability density, cumulative distribution, and failure function. This chapter of the book will turn readers into professional users of statistical distribution functions in mathematics for analyzing supply chain element data.

The purpose of this paper is to evaluate various reliability measures like reliability, expected lifetime (mean time to failure), signature reliability and compare networks based on the different flows.

The reliability characteristics of complex bridge networks have been evaluated using different algorithms with the help of universal generating function (UGF). Further, the signature reliability of the considered networks has been determined using Owen’s method.

The present paper proposes an efficient algorithm to compute the reliability indices of complex bridge networks having i.i.d. lifetime components (nodes, edges) with the help of UGF and Owen’s method. This study reveals that a slight change in the complex bridge network affects the reliability significantly. Finally, by the reliability structure function, proposed algorithms are used to find the signature and MTTF. From signature, we have determined the different failure probabilities corresponding to edges of the network.

In this work, we have evaluated reliability characteristics and signature reliability of the complex bridge networks using UGF method and Owen’s method respectively unlike done in the past.

The purpose of this paper is to identify the critical components of a complex system by using survival signature. First, a complex system is abstracted with varying scales and generates a multi-levels model. Then reliability evaluations can be conducted by survival signature from rough to fine for tracing and identifying them. Finally, the feasibility of the proposed approach is demonstrated by an actual production system.

The paper mainly applies a multi-level evaluating strategy for the reliability analysis of complex systems with components of multiple types. In addition, a multi-levels model of a complex system is constructed and survival signature also used for evaluation.

The proposed approach was demonstrated to be the feasibility by an actual production system that is used in the case study.

The case study was performed on a system with simple network structure, but the proposed approach could be applied to systems with complex ones. However, the approach to generate the digraphs of abstraction levels for complex system has to be developed.

So far the approach has been used for the reliability analysis of a machining system. The approach that is proposed for the identification of critical components also can be applied to make maintenance decision.

The multi-level evaluating strategy that was proposed for reliability analysis and the identification of critical components of complex systems was a novel method, and it also can be applied as index to make maintenance planning.

The purpose of this paper is to propose new importance measures for degrading components based on Shapley value, which can provide answers about how important players are to the whole cooperative game and what payoff each player can reasonably expect.

The proposed importance measure characterizes how a specific degrading component contributes to the degradation of system reliability by using Shapley value. Degradation models are also introduced to assess the reliability of degrading components. The reliability of system consisting independent degrading components is obtained by using structure functions, while reliability of system comprising correlated degrading components is evaluated with a multivariate distribution.

The ranking of degrading components according to the newly developed importance measure depends on the degradation parameters of components, system structure and parameters characterizing the association of components.

Considering the fact that reliability degradation of engineering systems and equipment are often attributed to the degradation of a particular or set of components that are characterized by degrading features. This paper proposes new importance measures for degrading components based on Shapley value to reflect the responsibility of each degrading component for the deterioration of system reliability. The results are also able to give timely feedback of the expected contribution of each degrading component to system reliability degradation.