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The purpose of this paper is to investigate the stochastic comparisons of the parallel system with independent heterogeneous Gumbel components and series and parallel systems with independent heterogeneous truncated Gumbel components in terms of various stochastic orderings.

The obtained results in this paper are obtained by using the vector majorization methods and results. First, the components of series and parallel systems are heterogeneous and having Gumbel or truncated Gumbel distributions. Second, multiple-outlier truncated Gumbel models are discussed for these systems. Then, the relationship between the systems having Gumbel components and Weibull components are considered. Finally, Monte Carlo simulations are performed to illustrate some obtained results.

The reversed hazard rate and likelihood ratio orderings are obtained for the parallel system of Gumbel components. Using these results, similar new results are derived for the series system of Weibull components. Stochastic comparisons for the series and parallel systems having truncated Gumbel components are established in terms of hazard rate, likelihood ratio and reversed hazard rate orderings. Some new results are also derived for the series and parallel systems of upper-truncated Weibull components.

To the best of our knowledge thus far, stochastic comparisons of series and parallel systems with Gumbel or truncated Gumble components have not been considered in the literature. Moreover, new results for Weibull and upper-truncated Weibull components are presented based on Gumbel case results.

In reliability studies, interests in discrete failure data came relatively late in comparison to its continuous analogue. Also, discrete failure data arise in several common situations. So, in this paper the authors try to study some reliability concepts such as reversed variance and reversed mean residual life functions based on discrete lifetime random variable.

Supposed T be a non‐negative discrete random variable, then based on reversed residual random variable T_k^*=(k−T|T≤k), some useful and applicable relations and bounds are achieved.

In this paper, the authors study the reversed variance residual life in discrete lifetime distributions, the results of which are not similar to the continuous case. Its relationship with reversed mean residual life and reversed residual coefficient of variation are obtained. Also, its monotonicity and the associated ageing classes of distributions are discussed. Some characterization results of the class of increasing reversed variance residual life, which is denoted by IRVR, are presented and the upper bound for reversed variance residual life under some conditions is obtained.

There are many situations where a continuous time is inappropriate for describing the lifetime of devices and other systems. For example, the lifetime of many devices in industry, such as switches and mechanical tools, depends essentially on the number of times that they are turned on and off or the number of shocks they receive. In such cases, the time to failure is often more appropriately represented by the number of times they are used before they fail, which is a discrete random variable.

All the results based on discrete reversed residual lifetime, such as the relationships among reversed mean, variance and coefficient of variation residual lifetime and also their monotonicity ageing classes, are new.

The purpose of this paper is to study likelihood ratio order for mixture and its components via their Glaser’s functions for weighted distributions. So, some theoretical examples using exponential family and their mixtures are presented.

First, Glaser’s functions of mixture and its components for weighted distributions in different scenarios are computed. Then by them the likelihood ratio order is investigated between mixture and its components.

The authors find conditions for weight functions under which the mixture random variable is between of its components in likelihood ratio order.

Results are obtained for weight function in general. It is well known that the some special weights are order statistics, up and down records, hazard rate, reversed hazard rate, moment generating function, etc. So, the results are valid for all of them.

The purpose of this paper is to introduce a new probability density function having both unbounded and bounded support with a wider applicability. While the distribution with bounded support on [0, 1] has applications in insurance and inventory management with ability to fit risk management data on proportions better than existing bounded distributions, the same with unbounded support is used as a lifetime model and is considered as an attractive alternative to some existing models in the reliability literature.

The new density function, called modified exponential-geometric distribution is derived from the exponential-geometric distribution introduced by Adamidis and Loukas (1998). The support of the density function is shown to be both unbounded and bounded depending on the values of one of the shape parameters. Various properties of the density function are studied in detail and the parameters are estimated through maximum likelihood method of estimation. A number of applications related to reliability, insurance and inventory management are exhibited along with some useful data analysis.

A single probability distribution with both unbounded and bounded support, which does not seem to exist in the reliability literature, is introduced in this paper. The proposed density function exhibits varying shapes including U-shape, and the failure rate also shows increasing, decreasing and bathtub shapes. The Monte Carlo simulation shows that the estimates of the parameters are quite stable with low standard errors. The distribution with unbounded support is shown to have competitive features for lifetime modeling through analysis of two data sets. The distribution with bounded support on [0, 1] is shown to have application in insurance and inventory management and is found to t data on proportions related to risk management better than some existing bounded distributions.

The authors introduce an innovative probability distribution which contributes significantly in insurance and inventory management besides its remarkable statistical and reliability properties.

The purpose of this paper is to estimate the multicomponent reliability by assuming the unit-Gompertz (UG) distribution. Both stress and strength are assumed to have an UG distribution with common scale parameter.

The reliability of a multicomponent stress–strength system is obtained by the maximum likelihood (MLE) and Bayesian method of estimation. Bayes estimates of system reliability are obtained by using Lindley’s approximation and Metropolis–Hastings (M–H) algorithm methods when all the parameters are unknown. The highest posterior density credible interval is obtained by using M–H algorithm method. Besides, uniformly minimum variance unbiased estimator and exact Bayes estimates of system reliability have been obtained when the common scale parameter is known and the results are compared for both small and large samples.

Based on the simulation results, the authors observe that Bayes method provides better estimation results as compared to MLE. Proposed asymptotic and HPD intervals show satisfactory coverage probabilities. However, average length of HPD intervals tends to remain shorter than the corresponding asymptotic interval. Overall the authors have observed that better estimates of the reliability may be achieved when the common scale parameter is known.

Most of the lifetime distributions used in reliability analysis, such as exponential, Lindley, gamma, lognormal, Weibull and Chen, only exhibit constant, monotonically increasing, decreasing and bathtub-shaped hazard rates. However, in many applications in reliability and survival analysis, the most realistic hazard rates are upside-down bathtub and bathtub-shaped, which are found in the unit-Gompertz distribution. Furthermore, when reliability is measured as percentage or ratio, it is important to have models defined on the unit interval in order to have plausible results. Therefore, the authors have studied the multicomponent stress–strength reliability under the unit-Gompertz distribution by comparing the MLEs, Bayes estimators and UMVUEs.

We show why investors may prefer not to be a firm’s unique lender, even if they are in a strong bargaining position. Some firms need additional funds after a first investment: providing additional funds is rational after the first investment is sunk, but together the two investments are unprofitable. A unique lender will always provide additional funds and make losses. Two creditors can commit not always to provide funds: inefficient negotiations over debt forgiveness may end with a project’s liquidation, which is harmful ex post, but helpful ex ante, if it keeps entrepreneurs with nonpromising projects from initially requesting funds.

The purpose of this article is to derive inference for multicomponent reliability where stress-strength variables follow unit generalized Rayleigh (GR) distributions with common scale parameter.

The authors derive inference for the unknown parametric function using classical and Bayesian approaches. In sequel, (weighted) least square (LS) and maximum product of spacing methods are used to estimate the reliability. Bootstrapping is also considered for this purpose. Bayesian inference is derived under gamma prior distributions. In consequence credible intervals are constructed. For the known common scale, unbiased estimator is obtained and is compared with the corresponding exact Bayes estimate.

Different point and interval estimators of the reliability are examined using Monte Carlo simulations for different sample sizes. In summary, the authors observe that Bayes estimators obtained using gamma prior distributions perform well compared to the other studied estimators. The average length (AL) of highest posterior density (HPD) interval remains shorter than other proposed intervals. Further coverage probabilities of all the intervals are reasonably satisfactory. A data analysis is also presented in support of studied estimation methods. It is noted that proposed methods work good for the considered estimation problem.

In the literature various probability distributions which are often analyzed in life test studies are mostly unbounded in nature, that is, their support of positive probabilities lie in infinite interval. This class of distributions includes generalized exponential, Burr family, gamma, lognormal and Weibull models, among others. In many situations the authors need to analyze data which lie in bounded interval like average height of individual, survival time from a disease, income per-capita etc. Thus use of probability models with support on finite intervals becomes inevitable. The authors have investigated stress-strength reliability based on unit GR distribution. Useful comments are obtained based on the numerical study.

In order to succeed in an action under the Equal Pay Act 1970, should the woman and the man be employed by the same employer on like work at the same time or would the woman still be covered by the Act if she were employed on like work in succession to the man? This is the question which had to be solved in Macarthys Ltd v. Smith. Unfortunately it was not. Their Lordships interpreted the relevant section in different ways and since Article 119 of the Treaty of Rome was also subject to different interpretations, the case has been referred to the European Court of Justice.

The paper explores the pattern of early closure risks for worker cooperatives, whether this pattern involves a “liability of newness” or a “liability of adolescence” and whether it applies identically to coops created from scratch, to rescues of failing firms and to conversions from sound conventional businesses to the cooperative form. Non-parametric hazard curves are estimated using data on the 2,740 worker cooperatives created in France in 1977–1993, 1,660 of which exited during the period. Comparisons are drawn with conventional French firms whenever data for the same cohorts are available.

Contemporary literature reveals that, to date, the poultry livestock sector has not received sufficient research attention. This particular industry suffers from unstructured supply chain practices, lack of awareness of the implications of the sustainability concept and failure to recycle poultry wastes. The current research thus attempts to develop an integrated supply chain model in the context of poultry industry in Bangladesh. The study considers both sustainability and supply chain issues in order to incorporate them in the poultry supply chain. By placing the forward and reverse supply chains in a single framework, existing problems can be resolved to gain economic, social and environmental benefits, which will be more sustainable than the present practices.

The theoretical underpinning of this research is ‘sustainability’ and the ‘supply chain processes’ in order to examine possible improvements in the poultry production process along with waste management. The research adopts the positivist paradigm and ‘design science’ methods with the support of system dynamics (SD) and the case study methods. Initially, a mental model is developed followed by the causal loop diagram based on in-depth interviews, focus group discussions and observation techniques. The causal model helps to understand the linkages between the associated variables for each issue. Finally, the causal loop diagram is transformed into a stock and flow (quantitative) model, which is a prerequisite for SD-based simulation modelling. A decision support system (DSS) is then developed to analyse the complex decision-making process along the supply chains.

The findings reveal that integration of the supply chain can bring economic, social and environmental sustainability along with a structured production process. It is also observed that the poultry industry can apply the model outcomes in the real-life practices with minor adjustments. This present research has both theoretical and practical implications. The proposed model’s unique characteristics in mitigating the existing problems are supported by the sustainability and supply chain theories. As for practical implications, the poultry industry in Bangladesh can follow the proposed supply chain structure (as par the research model) and test various policies via simulation prior to its application. Positive outcomes of the simulation study may provide enough confidence to implement the desired changes within the industry and their supply chain networks.