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The purpose of this paper is to compare four life data models, namely the exponential and the Weibull models, and their corresponding generalized versions, q-exponential and q-Weibull models, by means of one practical application.

Application of the models to a practical example (a welding station), with estimation of parameters by the use of the least squares method, and the Akaike Information Criterion (AIC).

The data of the example considered in this paper is divided into three regimes, decreasing, constant and increasing failure rate, and the q-Weibull model describes the bathtub curve displayed by the data with a single set of parameters.

The simplicity and flexibility of the q-Weibull model may be very useful for practitioners of reliability analysis, and its benefits surpasses the inconvenience of the additional parameter, as AIC shows.

The q-Weibull model is compared in detail with other three models, through the analysis of one example that clearly exhibits a bathtub curve, and it is shown that it can describe the whole time range with a single set of parameters.

This paper deals with the estimation of the stress–strength reliability R = P(X < Y), when X and Y follow (1) independent generalized gamma (GG) distributions with only a common shape parameter and (2) independent Weibull random variables with arbitrary scale and shape parameters and generalize the proposal from Kundu and Gupta (2006), Kundu and Raqab (2009) and Ali et al. (2012).

First, a closed form expression for R is derived under the conditions (1) and (2). Next, sufficient conditions are given for the convergence of the infinite series expansions used to calculate the value of R in case (2). The models GG and Weibull are fitted by maximum likelihood using Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton method. Confidence intervals and standard errors are calculated using bootstrap. For illustration purpose, two real data sets are analyzed and the results are compared with the existing recent results available in the literature.

The proposed approaches improve the estimation of the R by not using transformations in the data and flexibilize the modeling with Weibull distributions with arbitrary scale and shape parameters.

The proposals of the paper eliminate the misestimation of R caused by subtracting a constant value from the data (Kundu and Raqab, 2009) and treat the estimation of R in a more adequate way by using the Weibull distributions without restrictions in the parameters. The two cases covered generalize a number of distributions and unify a number of stress–strength probability P(X < Y) results available in the literature.

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.

In order to reduce avoidably lengthy duration required to test highly reliable products under usage stress, accelerated life test sampling plans (ALTSPs) are employed. This paper aims to build a decision model for obtaining optimal sampling plan under accelerated life test setting using Type-I hybrid censoring scheme for products covered under warranty.

The primary decision model proposed in this paper determines ALTSP by minimizing the relevant costs involved. To arrive at the decision model, the Fisher information matrix for Type-I hybrid censoring scheme under accelerated life test setting is derived. The optimal solution is attained by utilizing appropriate techniques following a nonlinear constrained optimization approach. As a special case, ALTSP for Type-I censoring is obtained using the same approach. ALTSP under Type-I hybrid censoring using the variance minimization approach is also derived.

On comparing the optimal results obtained using the above mentioned approaches, it is found that the cost minimization approach does better in reducing the total cost incurred. Results also show that the proposed ALTSP model under cost function setting has considerably lower expected testing time. Interesting findings from the sensitivity analysis conducted using a newly introduced failure dataset pertaining to locomotive controls are highlighted.

The research introduces a model to design optimum ALTSP for Type-I hybrid censoring scheme. The practical viability of the model makes it valuable for real-life situations. The practical application of the proposed model is exemplified using a real-life case.

The objective of the paper is to consider the problem of the strength of a manufactured item against stress, when the component follows Weibull failure law. Different cases of stress and strength with varying parameters are discussed for the Weibull‐Weibull stress‐strength model considered in this paper. The application of the proposed technique will help in understanding the design methodology of the system and addressing the risks involved in perceived quality and reliability levels by eliminating or at least reducing the risk impact at the design phase.

Generalised Weibull‐Weibull stress‐strength models have been analysed for different cases of shape parameters for stress and strength to estimate the reliability of the system. The model is generalized using semi‐regenerative stochastic processes with the help of a state space approach to include a repair facility.

Different cases of stress and strength with varying parameters have been discussed for the Weibull‐Weibull stress‐strength models considered in this paper. The results show how the stress‐strength reliability model is affected by changes in the parameters of stress and strength. The application of the proposed technique will help in understanding the design methodology of the system, and also lead to the problem of addressing the risks involved in perceived quality and reliability levels by eliminating or at least reducing the risk impact in the design phase.

The present study is limited to a few special cases of Weibull‐Weibull stress‐strength models. The authors propose to continue to study the behaviour of general Weibull strength against exponential stress in particular and to identify the shape parameter that maximises the strength reliability.

The application of the proposed technique will help in understanding the design methodology of the system, and also lead to the problem of addressing the risks involved in perceived quality and reliability levels by eliminating or at least reducing the risk impact at the design phase. The model has been extended and generalized to include a repair facility under the assumption that all the random variables involved in the analysis are arbitrarily distributed (i.e. general).

In the Weibull‐Weibull stress‐strength model of reliability, different cases have been considered. In the first case, both parameters of stress‐strength have the same values and are independent of the distribution. In the second case, if the shape parameter of the strength is twice that of the stress, the probability will have a normal distribution with different parameter values. In the third case, if the shape parameter of the stress is twice that of the strength, then probability distribution is a parabolic cylindrical function. The study shows how to proceed in all cases. The model is generalized to include a repair facility, with all the random variables involved in the analysis being arbitrarily distributed using semi‐regenerative stochastic processes.

The purpose of this paper is to analyze mathematical aspects of the q‐Weibull model and explore the influence of the parameter q.

The paper uses analytical developments with graph illustrations and an application to a practical example.

The q‐Weibull distribution function is able to reproduce the bathtub shape curve for the failure rate function with a single set of parameters. Moments of the distribution are also presented.

The generalized q‐Weibull distribution unifies various possible descriptions for the failure rate function: monotonically decreasing, monotonically increasing, unimodal and U‐shaped (bathtub) curves. It recovers the usual Weibull distribution as a particular case. It represents a unification of models usually found in reliability analysis. Q‐Weibull model has its inspiration in nonextensive statistics, used to describe complex systems with long‐range interactions and/or long‐term memory. This theoretical background may help the understanding of the underlying mechanisms for failure events in engineering problems.

Q‐Weibull model has already been introduced in the literature, but it was not realized that it is able to reproduce a bathtub curve using a unique set of parameters. The paper brings a mapping of the parameters, showing the range of the parameters that should be used for each type of curve.

The purpose of this paper is to design a modified chain sampling plan for assuring the product percentile life where the lifetime follows Weibull or generalized exponential distributions (GEDs). In order to reduce the cost of inspection when implementing the proposed modified chain sampling plan, it is also considered the economic aspect of designing of proposed plan in this paper.

The authors have designed the proposed plan on the basis of two points on the operating characteristic (OC) curve approach. The optimization problem is used to determine the plan parameters of the proposed plan so that the specified values of producer’s risk and consumer’s risk are satisfied simultaneously.

The results we have obtained, confirm that the proposed plan will be very effective in reducing the sample size rather than other existing sampling plans. The OC curves of proposed plan, chain sampling plan and zero acceptance number single sampling plan show that the performance of proposed plan in discriminating the good and poor quality lots is better than other two plans. In this paper, it is proved that the value of number of preceding lots required for current lot disposition plays an important role.

The proposed modified chain sampling plan for assuring the percentile lifetime of the products under Weibull or GEDs is not available in the literature. The proposed plan can be used in all the manufacturing industries to assure the product percentile lifetime with minimum sample size as well as minimum cost.

The purpose of this paper is to design an optimal reliability acceptance sampling plan (RASP) using the Type-I generalized hybrid censoring scheme (GHCS) for non-repairable products sold under the general rebate warranty. A cost function approach is proposed for products having Weibull distributed lifetimes incorporating relevant costs.

For Weibull distributed product lifetimes, acceptance criterion introduced by Lieberman and Resnikoff (1955) is derived for Type-I GHCS. A cost function is formulated using expected warranty cost and other relevant cost components incorporating the acceptance criterion. The cost function is optimized following a constrained optimization approach to arrive at the optimum RASP. The constraint ensures that the producer's and the consumer's risks are maintained at agreed-upon levels.

Optimal solution using the above approach is obtained for Type-I GHCS. As a special case of Type-I GHCS, the proposed approach is also used to arrive at the optimal design for Type-I hybrid censoring scheme as shown in Chakrabarty et al. (2019). Observations regarding the change in optimal design and computational times between the two censoring schemes are noted. An extensive simulation study is performed to validate the model for finite sample sizes and the results obtained are found to be in strong agreement. In order to analyze the sensitivity of the optimal solution due to misspecification of parameter values and cost components, a well-designed sensitivity analysis is carried out using a real-life failure data set from Lawless (2003). Interesting observations are made regarding the change in optimal cost due to change in parameter values, the impact of warranty cost in optimal design and change in optimal design due to change in lot sizes.

The research presents an approach for designing optimal RASPs using Type-I generalized hybrid censoring. The study formulates optimum life test sampling plans by minimizing the average aggregate costs involved, which makes it valuable in dealing with real-life problems pertaining to product quality management.

The development of digital technology has provided technical support to various industries. Specifically, Internet-based freight platforms can ensure the high-quality development of the logistics industry. Online freight platforms can use cargo transportation insurance to improve their service capabilities, promote their differentiated development, create products with platform characteristics and increase their core competitiveness.

This study uses a generalised linear model to fit the claim probability and claim intensity data and analyses freight insurance pricing based on the freight insurance claim data of a freight platform in China.

Considering traditional pricing risk factors, this study adds two risk factors to fit the claim probability data, that is, the purchase behaviour of freight insurance customers and road density. The two variables can significantly influence the claim probability, and the model fitting outcomes obtained with the logit connection function are excellent. In addition, this study examines the model results under various distribution types for the fitting of the claim intensity data. The fitting outcomes under a gamma distribution are superior to those under the other distribution types, as measured by the Akaike information criterion.

With actual data from an online freight platform in China, this study empirically proves that a generalised linear model is superior to traditional pricing methods for freight insurance. This study constructs a generalised linear pricing model considering the unique features of the freight industry and determines that the transportation distance, cargo weight and road density have a significant influence on the claim probability and claim intensity.

While Six Sigma metrics have been studied by researchers in detail for normal distribution-based data, in this paper, we have attempted to study the Six Sigma metrics for two-parameter Weibull distribution that is useful in many life test data analyses.

In the theory of Six Sigma, most of the processes are assumed normal and Six Sigma metrics are determined for such a process of interest. In reliability studies non-normal distributions are more appropriate for life tests. In this paper, a theoretical procedure is developed for determining Six Sigma metrics when the underlying process follows two-parameter Weibull distribution. Numerical evaluations are also considered to study the proposed method.

In this paper, by matching the probabilities under different normal process-based sigma quality levels (SQLs), we first determined the Six Sigma specification limits (Lower and Upper Six Sigma Limits- LSSL and USSL) for the two-parameter Weibull distribution by setting different values for the shape parameter and the scaling parameter. Then, the lower SQL (LSQL) and upper SQL (USQL) values are obtained for the Weibull distribution with centered and shifted cases. We presented numerical results for Six Sigma metrics of Weibull distribution with different parameter settings. We also simulated a set of 1,000 values from this Weibull distribution for both centered and shifted cases to evaluate the Six Sigma performance metrics. It is found that the SQLs under two-parameter Weibull distribution are slightly lesser than those when the process is assumed normal.

The theoretical approach proposed for determining Six Sigma metrics for Weibull distribution is new to the Six Sigma Quality practitioners who commonly deal with normal process or normal approximation to non-normal processes. The procedure developed here is, in fact, used to first determine LSSL and USSL followed by which LSQL and USQL are obtained. This in turn has helped to compute the Six Sigma metrics such as defects per million opportunities (DPMOs) and the parts that are extremely good per million opportunities (EGPMOs) under two-parameter Weibull distribution for lower-the-better (LTB) and higher-the-better (HTB) quality characteristics. We believe that this approach is quite new to the practitioners, and it is not only useful to the practitioners but will also serve to motivate the researchers to do more work in this field of research.