A Multi-Stage Two-Machines Replacement Strategy Using Mixture Models,Bayesian Inference,and Stochastic Dynamic Programming

If at least one out of two serial machines that produce a specific product in manufacturing environments malfunctions, there will be non conforming items produced. Determining the optimal time of the machines' maintenance is the one of major concerns. While a convenient common practice for this kind of problem is to fit a single probability distribution to the combined defect data, it does not adequately capture the fact that there are two different underlying causes of failures. A better approach is to view the defects as arising from a mixture population: one due to the first machine failures and the other due to the second one. In this article, a mixture model along with both Bayesian inference and stochastic dynamic programming approaches are used to find the multi-stage optimal replacement strategy. Using the posterior probability of the machines to be in state λ₁, λ₂ (the failure rates of defective items produced by machine 1 and 2, respectively), we first formulate the problem as a stochastic dynamic programming model. Then, we derive some properties for the optimal value of the objective function and propose a solution algorithm. At the end, the application of the proposed methodology is demonstrated by a numerical example and an error analysis is performed to evaluate the performances of the proposed procedure. The results of this analysis show that the proposed method performs satisfactorily when a different number of observations on the times between productions of defective products is available.     

An adaptive rule for stopping a discovery process under time censorship

Consider a finite population of large but unknown size of hidden objects. Consider searching for these objects for a period of time, at a certain cost, and receiving a reward depending on the sizes of the objects found. Suppose that the size and discovery time of the objects both have unknown distributions, but the conditional distribution of time given size is exponential with an unknown non-negative and non-decreasing function of the size as failure rate. The goal is to find an optimal way to stop the discovery process. Assuming that the above parameters are known, an optimal stopping time is derived and its asymptotic properties are studied. Then, an adaptive rule based on order restricted estimates of the distributions from truncated data is presented. This adaptive rule is shown to perform nearly as well as the optimal stopping time for large population size.     

Determination of Single Sampling Plans by Attributes Under the Conditions of Zero-Inflated Poisson Distribution

When the manufacturing process is well monitored, occurrence of nondefects would be a frequent event in sampling inspection. The appropriate probability distribution of the number of defects is a zero-inflated Poisson (ZIP) distribution. In this article, determination of single sampling plans (SSPs) by attributes using unity values is considered, when the number of defects follows a ZIP distribution. The operating characteristic (OC) function of the sampling plan is derived. Plan parameters are obtained for some sets of values of (p₁, α, p₂, β). Numerical illustrations are given to describe the determination of SSP under ZIP distribution and to study its performance in comparison with Poisson SSP.     

The Bayesian approach for highly reliable electro-explosive devices using one-shot device testing

Due to the high reliability and high testing cost of electro-explosive devices, even though an accelerated test is performed, one may observe very few failures or even no failures at all due to censoring. In this paper, we consider modelling the reliability of such devices by an exponential lifetime distribution in which the failure rate is assumed to be a function of some covariates and that the observed data are binary. The Bayesian approach, with three different prior settings, is used to develop inference on the failure rate, lifetime and the reliability under some settings. A Monte Carlo simulation study is carried out to show that this approach is quite useful and suitable for analysing data of the considered form, especially when the failure rates are very small. Finally, illustrative data are analysed using this approach.     

Bayesian acceptance sampling plans following economic criteria: An application to paper pulp manufacturing

The main purposes of this paper are to derive Bayesian acceptance sampling plans regarding the number of defects per unit of product, and to illustrate how to apply the methodology to the paper pulp industry. The sampling plans are obtained following an economic criterion: minimize the expected total cost of quality. It has been assumed that the number of defects per unit of product follows a Poisson distribution with process average 5 , whose prior information is described either for a gamma or for a non- informative distribution. The expected total cost of quality is composed of three independent components: inspection, acceptance and rejection. Both quadratic and step-loss functions have been used to quantify the cost incurred for the acceptance of a lot containing units with defects. Combining the prior information on 5 with the loss functions, four different sampling plans are obtained. When the quadratic-loss function is used, an analytical relation between the optimum settings of the sample size and the acceptance number is derived. The robustness analysis indicates that the sampling plans obtained are robust with respect to the prior distribution of the process average as well as to the misspecification of its mean and variance.     

Assessing risk for rare events

This paper develops a method for assessing the risk for rare events based on the following scenario. There exists a large population with an unknown percentage p of defects. A sample of size N is drawn from the population and, in the sample, 0 defects are drawn. Given these data, we want to determine the probability that no more than n defects will be found in another random sample of N drawn from the population. Estimates on the range of p and n are calculated from a derived joint distribution which depends on p, n and N. Asymptotic risk results based on an infinite sample are then developed. It is shown that these results are applicable even with relatively small sample spaces.     

Finding the maximum efficiency for multistage ranked-set sampling

Multistage ranked-set sampling (MRSS) is a generalization of ranked-set sampling in which multiple stages of ranking are used. It is known that for a fixed distribution under perfect rankings, each additional stage provides a gain in efficiency when estimating the population mean. However, the maximum possible efficiency for the MRSS sample mean relative to the simple random sampling sample mean has not previously been determined. In this paper, we provide a method for computing this maximum possible efficiency under perfect rankings for any choice of the set size and the number of stages. The maximum efficiency tends to infinity as the number of stages increases, and, for large numbers of stages, the efficiency-maximizing distributions are symmetric multi-modal distributions where the number of modes matches the set size. The results in this paper correct earlier assertions in the literature that the maximum efficiency is bounded and that it is achieved when the distribution is uniform.     

On Non-parametric Estimation of the Survival Function with Competing Risks

The non-parametric maximum likelihood estimators (MLEs) are derived for survival functions associated with individual risks or system components in a reliability framework. Lifetimes are observed for systems that contain one or more of those components. Analogous to a competing risks model, the system is assumed to fail upon the first instance of any component failure; i.e. the system is configured in series. For any given risk or component type, the asymptotic distribution is shown to depend explicitly on the unknown survival function of the other risks, as well as the censoring distribution. Survival functions with increasing failure rate are investigated as a special case. The order restricted MLE is shown to be consistent under mild assumptions of the underlying component lifetime distributions.     

Bayesian Estimation of the Size of a Closed Population Using Photo-ID Data with Part of the Population Uncatchable

Abstract

We develop a Bayesian statistical model for estimating bowhead whale population size from photo-identification data when most of the population is uncatchable. The proposed conditional likelihood function is a product of Darroch's model, formulated as a function of the number of good photos, and a binomial distribution of captured whales given the total number of good photos at each occasion. The full Bayesian model is implemented via adaptive rejection sampling for log concave densities. We apply the model to data from 1985 and 1986 bowhead whale photographic studies and the results compare favorably with the ones obtained in the literature. Also, a comparison with the maximum likelihood procedure with bootstrap simulation is considered using different vague priors for the capture probabilities.     

Degradation data analysis for samples under unequal operating conditions: a case study on train wheels

Traditionally, reliability assessment of devices has been based on life tests (LTs) or accelerated life tests (ALTs). However, these approaches are not practical for high-reliability devices which are not likely to fail in experiments of reasonable length. For these devices, LTs or ALTs will end up with a high censoring rate compromising the traditional estimation methods. An alternative approach is to monitor the devices for a period of time and assess their reliability from the changes in performance (degradation) observed during the experiment. In this paper, we present a model to evaluate the problem of train wheel degradation, which is related to the failure modes of train derailments. We first identify the most significant working conditions affecting the wheel wear using a nonlinear mixed-effects (NLME) model where the log-rate of wear is a linear function of some working conditions such as side, truck and axle positions. Next, we estimate the failure time distribution by working condition analytically. Point and interval estimates of reliability figures by working condition are also obtained. We compare the results of the analysis via an NLME to the ones obtained by an approximate degradation analysis.     

Frailty models of manufacturing effects

The median service lifetime of respirator safety devices produced by different manufacturers is determined using frailty models to account for unobserved differences in manufacturing processes and raw materials. The gamma and positive stable frailty distributions are used to obtain survival distribution estimates when the baseline hazard is assumed to be Weibull. Frailty distributions are compared using laboratory test data of the failure times for 104 respirator cartridges produced by 10 different manufacturers tested with three different challenge agents. Likelihood ratio tests indicate that both frailty models provide a significant improvement over a Weibull model assuming independence. Results are compared to fixed effects approaches for analysis of this data.     

Sample Size Requirements and Study Duration for Testing Main Effects and Interactions in Completely Randomized Factorial Designs When Time to Event is the Outcome

In this paper, we develop the methodology for designing clinical trials with any factorial arrangement when the primary outcome is time to event. We provide a matrix formulation for calculating the sample size and study duration necessary to test any effect with a prespecified type I error rate and power. Assuming that a time to event follows an exponential distribution, we describe the relationships between the effect size, the power, and the sample size. We present examples for illustration purposes. We provide a simulation study to verify the numerical calculations of the expected number of events and the duration of the trial. The change in the power produced by a reduced number of observations or by accruing no patients to certain factorial combinations is also described.     

A natural conjugate prior for the nonhomogeneous poisson process with an exponential intensity function

This article presents a natural conjugate prior for the nonhomogeneous Poisson process (NHPP) with an exponential intensity function, for modeling the failure rate of repairable systems. The behavior of the conjugate prior distribution with respect to its parameters is studied, and the use of this prior in Bayesian estimation is compared to two other estimation approaches (the use of independent prior distributions, and the bivariate normal distribution). The use of the conjugate prior proposed here facilitates Bayesian statistical analysis of aging. In particular, the proposed prior allows us to explicitly account for dependence between the initial failure rate and the aging rate. This is a significant improvement over the assumptions made in most prior work (either the assumption that the aging rate is known, or the assumption that the initial failure rate and the aging rate are independent). Monte Carlo simulation shows that Bayesian estimation using the proposed prior generally performs at least as well as Bayesian estimation using independent priors for the initial failure rate and the aging rate,except in the case where the prior distribution underestimates both the initial failure rate and the aging rate.     

Estimation of Weibull distribution parameters for irregular interval group failure data with unknown failure times

SUMMARY This paper presents three methods for estimating Weibull distribution parameters for the case of irregular interval group failure data with unknown failure times. The methods are based on the concepts of the piecewise linear distribution function (PLDF), an average interval failure rate (AIFR) and sequential updating of the distribution function (SUDF), and use an analytical approach similar to that of Ackoff and Sasieni for regular interval group data. Results from a large number of simulated case problems generated with specified values of Weibull distribution parameters have been presented, which clearly indicate that the SUDF method produces near-perfect parameter estimates for all types of failure pattern. The performances of the PLDF and AIFR methods have been evaluated by goodness-of-fit testing and statistical confidence limits on the shape parameter. It has been found that, while the PLDF method produces acceptable parameter estimates, the AIFR method may fail for low and high shape parameter values that represent the cases of random and wear-out types of failure. A real-life application of the proposed methods is also presented, by analyzing failures of hydrogen make-up compressor valves in a petroleum refinery.     

The computation of standard normal distribution integral in any required precision based on reliability method

Based on reliability theory, the value of the standard normal distribution integral can be obtained by calculating the probability of the failure domain of the linear performance function. After the sample space is divided into some sub-sample spaces, a number of sub-failure domains are obtained. In the paper, the methods of computing the probabilities of sub-failure domains are discussed. All the formulae and the steps of computing the standard normal distribution integral which meet any required precision are given in the paper. Examples show that it is easy for the method to compute the standard normal distribution integral.     

Survival rates of Hungarian sand martins and their relationship with Sahel rainfall

Since 1986, I have carried out an intensive field survey of 10 000-30 000 pairs of a sand martin ( Riparia riparia ) population. Direct survey of the size and distribution of the breeding population and estimation of adult survival rates by SURGE, based on extensive ringing data sets, allow us to analyze the effects of different environmental factors with high precision. The model selection showed that the S s+t , P t model, in which the survival rates differ by sex for adults and vary in parallel by year and the capture rate varies by year, fits the data. The adult females had a lower survival rate compared to the males. The capture rate could be modelled as a quotient of the number of captured birds and the number of breeding birds along the upper part of the river Tisza. The survival rates of adults were related to the rainfall of the southern Sahel, which has an important role in the extension of the winter foraging habitat in the Sahel. Although the severe decrease in the population size, which may reach 50%, coincided with a large decline in the adult survival rate, there was not a significant relation between the adult survival rate and population size during the studied period. The population recruitment by first breeders and immigrant-emigrant adults could have a key role in the determination of population size. In the case of the studied subpopulation along the river, which is a core of the Carpathian Bend population, the immigration-emigration of adults had an important effect on the population size. The significant difference between juvenile male and female dispersal indicates the importance of separate estimation of juvenile survival for the sexes in further studies.     

The exact hypothesis test for the shape parameter of a new two-parameter distribution with the bathtub shape or increasing failure rate function under progressive censoring with random removals

This study considers the exact hypothesis test for the shape parameter of a new two-parameter distribution with the shape of a bathtub or increasing failure rate function under type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial or a uniform distribution. Several test statistics are proposed and one numerical example is provided to illustrate the proposed hypothesis test for the shape parameter. Finally, a simulation study is performed to compare the power performances of all proposed test statistics. We concluded that the test statistic w ₁ is more attractive than other methods as it has better performance than other test statistics for most cases based on the criteria of maximum power.     

Detecting multiple change points in piecewise constant hazard functions

The National Cancer Institute (NCI) suggests a sudden reduction in prostate cancer mortality rates, likely due to highly successful treatments and screening methods for early diagnosis. We are interested in understanding the impact of medical breakthroughs, treatments, or interventions, on the survival experience for a population. For this purpose, estimating the underlying hazard function, with possible time change points, would be of substantial interest, as it will provide a general picture of the survival trend and when this trend is disrupted. Increasing attention has been given to testing the assumption of a constant failure rate against a failure rate that changes at a single point in time. We expand the set of alternatives to allow for the consideration of multiple change-points, and propose a model selection algorithm using sequential testing for the piecewise constant hazard model. These methods are data driven and allow us to estimate not only the number of change points in the hazard function but where those changes occur. Such an analysis allows for better understanding of how changing medical practice affects the survival experience for a patient population. We test for change points in prostate cancer mortality rates using the NCI Surveillance, Epidemiology, and End Results dataset.     

Existence of every possible distribution for any sample order statistic

Given are any stated value for sample size and an arbitrary but specified univariate distribution. Let an arbitrary but specified order statistic for a univariate random sample of this size be considered. It is shown that a statistical population always exists, for yielding the sample, such that the distribution of the order statistic is exactly the specified distribution. Some asymptotic implications of these results are outlined.     

Factorial experiments in the analysis of assembly systems

Assembly systems are a key tool for mass production and are increasingly being implemented in the manufacturing industry. Since the performance of such systems depends on the levels of many design variables, they are not well understood. In this paper, the performance of free transfer automatic assembly systems with closed inspection and repair loops is studied via factorial experiments of a simulated system. Five factors were identified that affect the throughput of the system: buffer size, number of pallets in the system, number of repair stations, repair time of jammed assembly machines, and subcomponent defect rate. Initially, two levels of each factor were considered, so that a full 2 5 factorial design of the experiment was used to study the system. Next, to develop a deeper understanding of the linear or non-linear effect of each factor, additional levels were investigated. Finally, a predictive model is proposed. Engineers and system designers can use this predictive model to estimate the performance of the system, given a combination of levels of each of the five factors that we studied.