X-bar Control Chart Research Articles

Economic Statistical Design of Integrated X-bar-S Control Chart with Preventive Maintenance and General Failure Distribution

The application of Preventive Maintenance (PM) and Statistical Process Control (SPC) are important practices to achieve high product quality, small frequency of failures, and cost reduction in a production process. However there are some points that have not been explored in depth about its joint application. First, most SPC is performed with the X-bar control chart which does not fully consider the variability of the production process. Second, many studies of design of control charts consider just the economic aspect while statistical restrictions must be considered to achieve charts with low probabilities of false detection of failures. Third, the effect of PM on processes with different failure probability distributions has not been studied. Hence, this paper covers these points, presenting the Economic Statistical Design (ESD) of joint X-bar-S control charts with a cost model that integrates PM with general failure distribution. Experiments showed statistically significant reductions in costs when PM is performed on processes with high failure rates and reductions in the sampling frequency of units for testing under SPC.

Damage alarming for bridge expansion joints using novelty detection technique based on long-term monitoring data

Damage alarming and safety evaluation using long-term monitoring data is an area of significant research activity for long-span bridges. In order to extend the research in this field, the damage alarming technique for bridge expansion joints based on long-term monitoring data was developed. The effects of environmental factors on the expansion joint displacement were analyzed. Multiple linear regression models were obtained to describe the correlation between displacements and the dominant environmental factors. The damage alarming index was defined based on the multiple regression models. At last, the X-bar control chart was utilized to detect the abnormal change of the displacements. Analysis results reveal that temperature and traffic condition are the dominant environmental factors to influence the displacement. When the confidence level of X-bar control chart is set to be 0.003, the false-positive indications of damage can be avoided. The damage sensitivity analysis shows that the proper X-bar control chart can detect 0.1 cm damage-induced change of the expansion joint displacement. It is reasonably believed that the proposed technique is robust against false-positive indication of damage and suitable to alarm the possible future damage of the expansion joints.

Using genetic algorithm and response surface methodology for statistically constrained optimization of VSI X-bar control charts under multiple assignable causes and non-normality

Variable sampling interval (VSI) control charts have been introduced with the aim of improving performance of traditional control charts. Usually, in the economic–statistical design of the VSI $$ \overline{X} $$ control charts, it is assumed that observations are normally distributed and process is subjected to only one assignable cause. However, in practice these assumptions could easily fail to hold, and results no longer could be realistic. This paper considers non-normal observations for the case of multiple assignable causes to develop a cost model for the economic design of VSI $$ \overline{X} $$ control chart. Being more applicable for all types of distributions, Burr distribution is employed for representing the distribution of non-normal process data. Since the proposed design consists of a complex nonlinear cost function that cannot be solved using a classical optimization method, genetic algorithm (GA) searching method as an efficient famous metaheuristic is employed to find the optimal values for the design parameters. Moreover, to improve the performances, response surface methodology is employed to calibrate GA parameters. The effectiveness of the proposed scheme is evaluated through a numerical example. Sensitivity analysis is also carried out to show the effects of cost and process parameters on the outputs of the model. Results show that in all cases, presented VSI model has better economical and statistical performances than its corresponding fixed sampling interval scheme.

Modeling interaction using trust and recommendation in ubiquitous computing environment

Abstract To secure computing in pervasive environment, an adaptive trust and recommendation based access control model (based on human notion of trust) is proposed. The proposed model provides support to calculate direct as well as indirect trust based on recommendations. It handles situations (by itself) both in which the requesting entity has a past experience with the service and a stranger entity requesting to access the service without any past interaction with the service. It encompasses the ability to reason human cognitive behavior and has the capability to adjust in accordance with behavioral pattern changes. X-bar control chart is used to handle malicious recommendation. The defense mechanisms incorporated by proposed model against attacks such as bad mouthing attack, oscillating behavior attack and conflicting behavior attack are also demonstrated.

Multi-objective economic statistical design of X-bar control chart considering Taguchi loss function

Shewhart charts are the most popular control charts that can be used to monitor variable quality characteristics in a production process. In this paper, a multi-objective model of the economic statistical design of the X-bar control chart is first proposed by incorporating the Taguchi loss function and the intangible external costs. The model minimizes the mean hourly loss cost while minimizing out-of-control average run length and maintaining reasonable in-control average run length. A multi-objective evolutionary algorithm, namely NSGA-II, is then developed and used to obtain the Pareto optimal solution of the model. Some sensitivity analyses are next performed to investigate the effect of parameter estimation on the chart performances. Finally, a comparison study with a traditional economic design model reveals that the proposed multiple objective design of the X-bar control chart offers a better approach and more practical outcomes for the practitioners.

An Extension of Economic Design of x-Bar Control Charts for Non Normally Distributed Data Under Weibull Shock Models

Three parameters—sample size, sampling intervals, and the control limits—must be determined when the x bar chart to monitor a manufacturing process. The constant sampling intervals were widely employed because of its administrative simplicity. However, the variable sampling interval (VSI) has recently been shown to give substantially faster detection of most process shifts than fixed-sampling-interval (FSI) for x-bar charts. In addition, these measurements in the subgroup are assumed to be normally distributed. That assumption may not be tenable. This investigation compares the economic design of x-bar control charts for non normal data under Weibull shock models with various sampling avenues.

Economic statistical design for x-bar control charts under non-normal distributed data with Weibull in-control time

This paper proposes an approach which simultaneously considers the properties of cost and quality based on the Burr distribution to determine three parameters (including sample size, sampling interval between successive samples, and the control limits) when an x-bar chart monitors a manufacturing process with Weibull failure characteristic and non-normal data. Also, the cost model of Banerjee and Rahim (1988) is used as the objective function, and the probability density function of the Burr distribution is applied to derive the statistical constraints of economic statistical design of the x-bar control charts for non-normal data. The example of Banerjee and Rahim (1988) is adopted to indicate the solution procedure and sensitivity analyses. Meanwhile, the design parameters of the x-bar control charts can be obtained through the grid search method. The results show that an increase of skewness coefficient (α3) results in a slight decrease for sample size (n), but is robust to the control limit width (L). Also, an increase of kurtosis coefficient (α4) leads to a wider control limit width.

Monitoring The Level Of Students GPA's Over Time

A nonparametric (or distribution-free) statistical quality control chart is used to monitor the cumulative grade point averages (GPAs) of students over time. The chart is designed to detect any statistically significant positive or negative shifts in student GPAs from a desired target level. This nonparametric control chart is based on the signed-ranks of the GPAs of the sampled students. The exact false alarm rate and the in-control average run length of the proposed chart can be computed exactly and are independent of the underlying probability distribution of GPAs. The traditional Shewhart X-bar control chart for monitoring the mean of a process is based on the assumption that data follows a normal distribution. However, student GPAs may differ significantly from the normal distribution. As a result, using a traditional control chart to monitor the GPAs of students may lead to incorrectly specifying the control limits and the average run length and/or the false alarm rate of the chart. A test study was conducted at the College of Business Administration at Alabama State University. The study monitored the median cumulative GPAs of management majors during the period Spring 2005 through Spring 2009. The study revealed that the GPAs of students were stable at a median level of 2.6 over the period of the study.

Economic Design of VSI x-Bar Control Charts for Non Normally Distributed Data under Gamma (λ, 2) Failure Models

This investigation develops the economic design of x-bar control charts with variable sampling interval (VSI) for non normal data under Gamma (λ, 2) shock models. In the past, most of the economic design of control charts follows the Poisson process. However, pragmatically speaking, this process is usually not appropriate. Banerjee and Rahim (1988) presented a cost model which uses variable sampling intervals. This can be contrasted with sampling intervals of fixed length under a process-failure mechanism, which follows a Gamma (λ, 2) model with an increased hazard rate. In a separate study by Al-Oraini and Rahim (2003), an economic statistical design of x-bar control charts with the assumption of the Gamma (λ, 2) failure mechanism was proposed. When the x bar chart is designed to monitor a manufacturing process, three parameters need to be determined via engineers or participators. These parameters are sample size, sampling interval between successive samples, and control limits. However, measurements in the subgroup are assumed to be normally distributed when designing control charts. Said assumption may be untenable. This article employs a numerical example to indicate the solution procedure and to implement sensitivity analysis. Also, the situation of non normal and normal in the Gamma (λ, 2) model was compared. Results in non normal assumption revealed the following: smaller sample size (n) is needed; the initial sampling interval (h 1) can become longer; the sampling interval (h 2) can become shorter; the control limit width (L) can become narrower; and the expected cost per hour (ECT) can likewise be reduced.

An Extension of Banerjee and Rahim's Model for an Economic X-Bar Control Chart for Non-Normally Distributed Data Under Gamma Failure Models

This study compares an x-bar control chart economic design with a variable sampling interval (VSI) and a fixed-length sampling (FSI) for non-normal data under gamma (λ, 2) shock models. In earlier investigations, most control chart economic designs assumed the failure mechanism, which belongs to the Poisson distribution. Furthermore, these subgroup measurements were assumed to be normally distributed. However, these assumptions may not be tenable. Hence, this study employs Rahim and Banerjee (1993) model and a numerical example to indicate the solution procedure and to implement the sensitivity analysis while comparing the results of using various sampling avenues in non-normally data.

Optimal operating policies of X-bar control chart for non-normal variables with generalised Pareto in-control times

In designing the control charts, the choice of control chart parameters like sample size and sampling interval are most important. To determine the optimal design parameters, it is required to consider the model for statistical economic design of control charts with suitable probability distribution for quality characteristic and process in-control times, which are random in nature. In this paper, we consider the economic statistical design of X-bar control chart for non-normal quality characteristic. Here, it is assumed that the sample average of the quality characteristic X-bar has a Johnson distribution and the process in-control times follow generalised Pareto distribution. With suitable cost considerations and modifying the Lorenzen and Vance (1986) cost model, the expected cost per unit time is derived. The optimal sample size and the time interval between samples are obtained by minimising the expected cost per unit time. The sensitivity of the design with respect to the parameters and costs is also studied. This model is much useful for reducing the production cost and to improve product quality in places like, glassware industries, chemicals and films industries, etc.

Computation of the probability of a process being in control at the third stage for the TS X chart

He et al., in their paper ‘Design of double- and triple-sampling X-bar control charts using genetic algorithm’ [2002, International Journal of Production Research, 40 (6), 1387–1404], used a genetic algorithm to optimize the obtained model of double- and triple-sampling control charts. In order to obtain the type I and II error probabilities, they calculated the probabilities that the TS chart indicates that the process is in control at the first, second and third stages of sampling. In this article, we use a different approach to calculate the probability that the TS chart indicates that the process is in control at the third stage (P a3). The advantage of this new computational approach is that it requires only single integration rather than the double integration required by He et al. The new approach reduces the computational complexity and increases the accuracy of the probability estimation, since such integration normally involves an approximation.

Economic design of Xbar control charts with continuously variable sampling intervals

AbstractIn the context of economic design, a two‐stage model is proposed that utilizes continuously variable sampling intervals. Specifically, each successive sampling interval is determined by the extremity of the latest sample. Modeling the situation as a Markov chain, the hourly cost is developed for any arbitrary set of design parameters. This proposed approach is found to be more economically desirable, possessing a smaller average out‐of‐control production time, when compared with a standard two‐stage approach where the sampling interval alternates between two fixed values. Copyright © 2009 John Wiley & Sons, Ltd.

Gráficos de controle de EWMA e de X-barra para monitoramento de processos autocorrelacionados

Este artigo trata dos gráficos de controle de EWMA e de X-barra utilizados no monitoramento de processos cujas observações podem ser descritas por um modelo auto-regressivo de primeira ordem. Os gráficos são planejados levando em conta a correlação em série e utilizando-se o conceito de subgrupos racionais como estratégia de amostragem. As propriedades das cartas de controle são obtidas e comparadas. Os resultados numéricos mostram que a correlação positiva dentro dos subgrupos afeta o desempenho dos gráficos. O gráfico de EWMA é substancialmente mais ágil do que o gráfico de X-barra na detecção de perturbações no processo, especialmente quando tais perturbações geram pequenos desajustes na média.

Monitoring capability study and genetic economic design of X-bar control chart

Employing automatic gauges is one of the fundamental steps in advanced manufacturing system not only for assuring product quality, but also monitoring process stability. Statistical process control (SPC), therefore, is developed to detect the occurrence of assignable causes so that unnecessary quality costs can be avoided. However, it is necessary to provide a practical way to evaluate the monitoring capability of an automatic gauge for its application. That is, the monitoring capability study should be conducted before applying any automatic gauge on SPC application. The economic SPC chart then can be properly designed. Rather than conventionally considering the relationship between both manufacturing cappability and control limits, gauge capability is further and concurrently considered to analyze the monitoring error of an automatic gauge in this paper. After interpreting the effect of manufacturing capability, gauge capability, and control limits to construct the monitoring error model, furthermore, the engineer can then distinctly evaluate monitoring capability by the expected cost incurred in the SPC process. Also, genetic algorithm is successfully applied to determine the economic design of X-bar control chart with the realistic monitoring error model embedded.

Multivariate multiple sampling charts

A new multivariate statistical process control scheme, the Multivariate Multiple Sampling (MMS) control chart scheme, is proposed in this paper. A MMS chart is a multivariate extension of a double sampling X-bar control chart with at least two sampling stages. In the paper, a statistical design optimization procedure to design the MMS chart is presented and the performance of the MMS chart is investigated. The statistical efficiency in terms of average run length of the MMS chart is compared with that of the Hotelling chart both with and without variable sampling schemes, a multivariate CUMulative SUM (CUSUM) chart, and a multivariate Exponentially Weighted Moving Average (EWMA) chart. The ability of the MMS chart to handle the worst-case scenario is also investigated and compared with that of the multivariate EWMA and CUSUM charts. The results of the investigation show that even with only two sampling stages, the MMS chart provides an improvement in efficiency in detecting small shifts over the Hotelling chart without variable sampling schemes. When the number of sampling stages is equal to two, the MMS chart is better in detecting large shifts and the multivariate EWMA and CUSUM charts are better in detecting relatively small shifts. As the number of sampling stages is increased beyond two, the improvement in sensitivity of the MMS chart in detecting the small shifts increases. When the number of sampling stages ≥3, the MMS chart begins to give a better performance than a Hotelling chart with a variable sampling scheme for small shifts and is also better than a multivariate EWMA chart for both small and large shifts. As the number of sampling stages ≥4, the MMS chart begins to give a better performance than a multivariate CUSUM chart for both small and large shifts. The results of the investigation also show that the MMS chart outperforms the multivariate EWMA and CUSUM charts in the worst-case scenario situation.

Identificação de variáveis fora de controle em processos produtivos multivariados

A maioria dos processos produtivos, envolvendo múltiplas variáveis, geram grande quantidade de dados, nos quais o monitoramento simultâneo se faz necessário. Assim, o objetivo principal deste trabalho é mostrar, por meio de um exemplo, um procedimento de identificação de variáveis responsáveis pela instabilidade do processo. Ele considera variáveis fraca e fortemente correlacionadas. O comportamento destas variáveis é analisado, inicialmente, por meio do gráfico de controle T² de Hotelling, para a verificação da estabilidade do processo. Quando o processo é considerado fora de controle, uma investigação adicional é realizada. Para as variáveis fracamente correlacionadas o gráfico X-barra, com limites de Bonferroni, é utilizado. Para as variáveis com forte correlação, os dados são decompostos em componentes principais e estas são analisadas pelo gráfico de controle X-barra. Em ambos os casos, as técnicas utilizadas identificaram corretamente a variável responsável pela instabilidade do processo, bem como o período do distúrbio.

Note on ‘Design of double- and triple-sampling X-bar control charts using genetic algorithms’

He et al. formulated the designs of double-sampling (DS) X-bar control charts as optimization problems and solved these problems with a genetic algorithm. Based on the results in solving the DS chart design problems, triple-sampling (TS) X-bar control charts were developed. The efficiency of the TS charts was compared with that of the DS charts. They concluded that the TS charts are more efficient in terms of minimizing the average sample size. We explain that, since they only considered the average sample size when the process is in control, their conclusion is questionable. In fact, the question of which control chart (i.e. the standard Shewart X-bar control chart, DS chart or TS chart) is more efficient depends on both the probability of the process shifting from an in-control to an out-of-control state and the time the control chart will need to detect such a shift.

Design of double- and triple-sampling X-bar control charts using genetic algorithms

As today's manufacturing firms are moving towards agile manufacturing, quick and economic on-line statistical process control solutions are in high demand. Multiple sampling X-bar control charts are such an alternative. They can be designed to allow quick detection of a small shift in process mean and provide a quick response in an agile manufacturing environment. In this paper, the designs of double-sampling (DS) X-bar control charts are formulated and solved with a genetic algorithm. Based on the results in solving the DS chart design problems, triple sampling (TS) X-bar control charts are developed. The efficiency of the TS charts is compared with that of the DS charts. The results of the comparison show that TS charts are more efficient in terms of minimizing the average sample size.

An attempt to model distributions of machined component dimensions in production

In this study, normal, log-normal, triangular, uniform, Weibull, Erlang and unit beta probability density functions are tried to represent the behaviour of frequency distributions of workpiece dimensions collected from various manufacturing firms. Among the distribution functions, the unit beta distribution function is found to be the best fit using the chi-square test of fit. An attempt is made for the adoption of the unit beta model to x-bar charts of quality control in manufacturing. In this direction, upper and lower control limits (UCL and LCL) of x-bar control charts of dimension measurements are estimated for the beta model, and the observed differences between the beta and normal model control limits are discussed for the measurement sets.