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One of the common approaches to improve systems reliability is using standby redundancy. Although many works are available in the literature on the applications of standby redundancy, the system components are assumed to be independent of each other. But, in reality, the system components can be dependent on one another, causing the failure of each component to affect the failure rate of the remaining active components. In this paper, a standby two-unit system is considered, assuming a dependency between the switch and its associated active component.

This paper assumes that the failures between the switch and its associated active component follow the Marshall–Olkin exponential bivariate exponential distribution. Then, the reliability analysis of the system is done using the continuous-time Markov chain method.

The derived equations application to determine the system steady-state availability, system reliability and sensitivity analysis on the mean time to failure is demonstrated using a numerical illustration.

All previous models assumed independency between the switch and the associated active unit in the standby redundancy approach. In this paper, the switch and its associated component are assumed to be dependent on each other.

The purpose of this paper is the determination of reliability sampling plans in the Bayesian approach assuming that the lifetime distribution is exponential.

Sampling plans are used in manufacturing companies as a tool for carrying out sampling inspections, in order to make decisions about the disposition of many finished products. If the quality characteristic is considered as the lifetime of the products, the plan is known as a reliability sampling plan. In life testing, censoring schemes are adopted in order to save time and cost of life test. The inverted gamma distribution is employed as the natural conjugate prior to the average lifetime of the products. The sampling plans are developed assuming various probability distributions to the lifetime of the products.

The optimum plans n and c are obtained for some sets of values of (p1, a, p2, ß). The selection of sampling plans is illustrated through numerical examples.

Results obtained in this paper are original and the study has been done for the first time in this regard. Reliability sampling plans are essential for making decisions either to accept or reject based on the inspection of the sample.

Acceptance sampling plans are a decision-making process on the basis of a randomly selected sampling from a party, where it is not possible to completely scan the products for reasons such as time and cost being limited or the formation of damaged products during the inspection. For some products, the life span (time from beginning to failure) may be an important quality characteristic. In this case, the quality control adequacy of the products can be checked with an acceptance sampling plan based on the truncated life test with a censored scheme for the lifetime of the products. In this study, group acceptance sampling plans (GASPs) based on life tests are studied under the Type-I censored scheme for the compound Weibull-exponential (CWE) distribution.

GASPs based on life tests under the Type-I censored scheme for the CWE distribution are developed by using both the producer's risk and the consumer's risk.

In this study, optimum sample size, optimum number of groups and acceptance number are obtained under the Type-I censored scheme for the CWE distribution. Real data set illustration is given to show GASPs how to be used for the industry applications.

Different from acceptance sampling plans with just considering the producer's risk, GASPs are constructed by using two-point approach included both the producer's risk and the consumer's risk for CWE distribution.

This paper introduces group chain acceptance sampling plans (GChSP) for a truncated life test at preassumed time by using the minimum angle method. The proposed method is an approach, where both risks associated with acceptance sampling, namely consumers’ and producer’s risks, are considered. Currently, the GChSP only considers the consumer's risk (CR), which means the current plan only protects the consumer not the producer since it does not take into account the producer's risk (PR) at all.

There are six phases involved when designing the GChSP, which are (1) identifying the design parameters, (2) implementing the operating procedures, (3) deriving the probability of lot acceptance, (4) deriving the probability of zero or one defective, (5) deriving the proportion defective and (6) measuring the performance.

The findings show that the optimal number of groups obtained satisfies both parties, i.e. consumer and producer, compared to the established GChSP, where the number of group calculated only satisfies the consumer not the producer.

There are three limitations identified for this paper. The first limitation is the distribution, in which this paper only proposes the GChSP for generalized exponential distribution. It can be extended to different distribution available in the literature. The second limitation is that the paper uses binomial distribution when deriving the probability of lot acceptance. Also, it can be derived by using different distributions such as weighted binomial distribution, Poisson distribution and weighted Poisson distribution. The final limitation is that the paper adopts the mean as a quality parameter. For the quality parameter, researchers have other options such as the median and the percentile.

The proposed GChSP should provide an alternative for the industrial practitioners and for the inspection activity, as they have more options of the sampling plans before they finally decide to select one.

This is the first paper to propose the minimum angle method for the GChSP, where both risks, CR and PR, are considered. The GChSP has been developed since 2015, but all the researchers only considered the CR in their papers.

The purpose of this paper is to develop a group acceptance sampling plan (GASP) for a resubmitted lot when the lifetime of a product follows odds exponential log logistic distribution introduced by Rao and Rao (2014). The parameters of the proposed plan such as minimum group size and acceptance number are determined for a pre-specified consumer’s risk, number of testers and the test termination time. The authors compare the proposed plan with the ordinary GASP, and the results are illustrated with live data example.

The parameters of the proposed plan such as minimum group size and acceptance number are determined for a pre-specified consumer’s risk, number of testers and the test termination time.

The authors determined the group size and acceptance number.

No specific limitations.

This methodology can be applicable in industry to study quality control.

This methodology can be applicable in health study.

The parameters of the proposed plan such as minimum group size and acceptance number are determined for a pre-specified consumer’s risk, number of testers and the test termination time.

The main aim of this paper is to improve reliability characteristics namely availability, mean time to failure (MTTF), and expected profit of a complex system.

The paper discusses the availability of a complex system, which consists of two independent repairable subsystems A and B in (1‐out‐of‐2: F) and (1‐out‐of‐n: F) arrangement respectively. Subsystem A has two identical units arranged in parallel redundancy (1‐out‐of‐2: G), subsystem B has n units in series (1‐out‐of‐n: F) with two types of failure, namely, partial and catastrophic. Except at two transitions where there are two types of repair namely exponential and general possible. The failure and repair time for both subsystems follow exponential and general distributions respectively. The model is analysed under “preemptive‐repeat repair discipline” where A is a priority and B is non‐priority.

By employing supplementary variable technique, Laplace transformation and Gumbel‐Hougaard family copula various transition state probabilities, availability, MTTF and cost analysis (expected profit) are obtained along with steady‐state behaviour of the system. Inversions have also been carried out so as to obtain time dependent probabilities, which determine availability of the system at any instant.

This paper, through a systematic view, presents a mathematical model of a complex system from which the reliability characteristics namely availability, MTTF, and expected profit of a complex system can be improved.

The purpose of this paper is to consider the estimation of multicomponent stress-strength reliability. The system is regarded as alive only if at least s out of k (s<k) strengths exceed the stress. The reliability of such a system is obtained when strength, stress variates are from Erlang-truncated exponential (ETE) distribution with different shape parameters. The reliability is estimated using the maximum likelihood (ML) method of estimation when samples are drawn from strength and stress distributions. The reliability estimators are compared asymptotically. The small sample comparison of the reliability estimates is made through Monte Carlo simulation. Using real data sets the authors illustrate the procedure.

The authors have developed multicomponent stress-strength reliability based on ETE distribution. To estimate reliability, the parameters are estimated by using ML method.

The simulation results indicate that the average bias and average mean square error decreases as sample size increases for both methods of estimation in reliability. The length of the confidence interval also decreases as the sample size increases and simulated actual coverage probability is close to the nominal value in all sets of parameters considered here. Using real data, the authors illustrate the estimation process.

This research work has conducted independently and the results of the author’s research work are very useful for fresh researchers.

Lifetime data are used in many different applied sciences, like biomedicine, engineering, insurance and finance and others. The purpose of this paper is to develop a new acceptance sampling plans for Rama distribution when the mean lifetime test is truncated at a pre-determined time. The minimum sample sizes required to assert the specified life mean is obtained for a given customer’s risk. The operating characteristic function values of the sampling plans and producer’s risk are calculated.

The results are illustrated using numerical examples and a real data set is considered to illustrate the performance of the suggested acceptance sampling plans and how it can be used for the industry applications.

This paper shows a new acceptance sampling plans based on Rama distribution in the particular case when the mean life time test is truncated.

The results calculated in this paper demonstrate the differences between OC values for different distributions taken into account. In particular, OC values of Rama distribution are found to be less than the proposed distribution counterparts.

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.

This paper introduces a Bayesian two-sided group chain sampling plan (BT-SGChSP) by using binomial distribution to estimate the average proportion of defectives. In this Bayesian approach, beta distribution is used as a suitable prior of binomial distribution. The proposed plan considers both consumer's and producer's risks. Currently, group chain sampling plans only consider the consumer's risk and do not account for the producer's risk. All existing plans are used to estimate only a single point, but this plan gives a quality region for the pre-specified values of different design parameters. In other words, instead of point wise description for the designing of sampling plan based on a range of quality by involving a novel approach called quality region.

The methodology is based on five phases, which are (1) operating procedure, (2) derivation of the probability of lot acceptance, (3) constructing plans for given acceptable quality level (AQL) and limiting quality level (LQL), (4) construction of quality intervals for BT-SGChSP and (5) selection of the sampling plans.

The findings show that the operating characteristic (OC) curve of BT-SGChSP is more ideal than the existing Bayesian group chain sampling plan because the quality regions for BT-SGChSP give less proportion of defectives for same consumer's and producer's risks.

There are four limitations in this study: first is the use of binomial distribution when deriving the probability of lot acceptance. Alternatively, it can be derived by using distributions such as Poisson, weighted Poisson and weighted binomial. The second is that beta distribution is used as prior distribution. Otherwise, different prior distributions can be used like: Rayleigh, exponential and generalized exponential. The third is that we adopt mean as a quality parameter, whereas median and other quintiles can be used. Forth, this paper considers probabilistic quality region (PQR) and indifference quality region (IQR).

The proposed plan is an alternative of traditional group chain sampling plans that are based on only current lot information. This plan considers current lot information with preceding and succeeding lot and also considers prior information of the product.

This paper first time uses a tight (three acceptance criteria) and introduces a BT-SGChSP to find quality regions for both producer's and consumer's risk.