Average Run Length Performance Research Articles

The Difficulty in Designing Shewhart X̄ and X Control Charts with Estimated Parameters

The performance of the Shewhart X̄ control chart with estimated in-control parameters has been discussed a number of times in the literature. Previous studies showed that at least 400/(n – 1) phase I samples, where n > 1 is the sample size, are required so that the chart performs on average as if the in-control process parameter values were known. This recommendation was based on the in-control expected average run length (ARL) performance. The reliance on the expected ARL metric, however, averages across the practitioner-to-practitioner variability. This variability occurs due to the different historical data sets practitioners use, which results in varying parameter estimates, control limits, and in-control ARL values. In our article, we show that taking this type of variability into consideration leads to far larger amounts of phase I data than what was previously recommended. This is to ensure low levels of variation in the in-control ARL values among practitioners. The standard deviation of the ARL (SDARL) metric is used to evaluate performance for various amounts of phase I data. We show that no realistic phase I sample size is sufficient to have confidence that the attained in-control ARL is close to the desired value. We additionally investigate the effect of different process standard deviation estimators on the X̄-chart performance, showing that it is best to use a biased estimator. We also study the design of the X-chart for the case n = 1, drawing similar conclusions regarding the amount of phase I data. An alternative approach to designing control charts is recommended.

Run length quantiles of EWMA control charts monitoring normal mean or/and variance

Exponentially weighted moving average (EWMA) control charts are well-established devices for monitoring process stability. Typically, control charts are evaluated by considering their Average Run Length (ARL), that is the expected number of observations or samples until the chart signals. Because of the limitations of an average, various papers also dealt with the run length distribution and quantiles. Going beyond these papers, we develop algorithms for and evaluate the quantile performance of EWMA control charts with variance adjusted control limits and with fast initial response features, of EWMA charts based on the sample variance, and of EWMA charts simultaneously monitoring mean and variance. Additionally, for the mean charts we consider medium, late and very late process changes and their impact on appropriately conditioned run length quantiles. It is demonstrated that considering run length quantiles can protect from constructing distorted EWMA designs while optimising their zero-state ARL performance. The implementation of all the considered measures in the R package ‘spc’ allows any control chart user to consider EWMA schemes from the run length quantile prospective in an easy way.

Synthetic Double Sampling X̄ Chart with Estimated Process Parameters

The synthetic double sampling (SDS) X̄ chart comprises the double sampling (DS) X̄ and conforming run length (CRL) sub-charts. The SDS X̄ chart has been studied under the assumption of known process parameters in the literature. Nevertheless, in practice, process parameters are usually unknown and are estimated from an in-control Phase I dataset. This paper investigates the performance of the SDS X̄ chart with estimated process parameters, in terms of the average run length (ARL), average number of observations to signal (ANOS) and standard deviation of the run length (SDRL). The performance of the SDS X̄ chart with estimated process parameters by minimizing the out-of-control ARL and the out-of-control ANOS is compared with the corresponding chart’s performance with known process parameters. In addition, the minimum number of Phase I samples required by the SDS X̄ chart with estimated process parameters so that it has approximately the same in-control ARL and ANOS performances as the chart with known process parameters is studied. The ARL, ANOS and SDRL properties of the SDS X̄ chart with estimated process parameters differ significantly from that of the chart with known process parameters. Therefore, suitable optimal charting parameters are introduced so that the SDS X̄ chart with estimated process parameters has an adequate performance as its known process parameters counterpart without having to use large number of Phase I samples and sample size.

Statistical Monitoring of Nominal Logistic Profiles in Phase II

Statistical analysis of profile monitoring, a relatively new sub-area of statistical process control due to its applications in different industries, have urged researchers and practitioners to contribute to the developments of new monitoring methods. A statistical profile is a relationship between a quality characteristic (a response) and one or more independent variables to characterize quality of a process or a product. In this article, statistical profiles based on nominal responses are studied, where logistic regression is used to model the responses. Three approaches including likelihood ratio test (LRT), multivariate exponentially weighted moving average (MEWMA), and support vector machines (SVM) approaches are proposed to monitor quality of a process or product in Phase II. Performances of the proposed approaches are evaluated and compared using a case study. Moreover, the effect of two important factors on average run length (ARL) performance, number of levels and number of covariates, has been considered. Results indicate that performance of all approaches depends on the number of covariates and levels. As the number of these factors increases, SVM performance improves while performance of the other approaches deteriorates.

Comparison of Novelty Score-Based Multivariate Control Charts

Control charts are widely used in various industries to improve product quality. One recent trend in developing control charts is based on novelty score algorithms that can effectively describe reality and reflect the unique characteristics of the data being monitored. In this study, we compared eight novelty score algorithms—the T2, Local T2, Dmax, Dmean, K2, the k-nearest neighbor data description, the local density outlier factor, and the hybrid novelty score (HNS)—in terms of their average run length performance. A rigorous simulation was conducted to compare the novelty score-based multivariate control charts under both normal and non-normal scenarios. The simulation showed that in both normal and lognormal scenarios, Dmax-based control charts produced the most promising results. In skewed distribution with high kurtosis non-normal scenarios, HNS- and K2-based control charts performed best. In symmetric with kurtosis non-normal scenarios, local T2-based control charts outperformed the others.

A new exponentially weighted moving average control chart for monitoring the coefficient of variation

Monitoring coefficient of variation is one of the successful approaches to Statistical Process Control (SPC) when the process mean and standard deviation are not constants. This paper presents a modified Exponentially Weighted Moving Average (EWMA) chart in order to further enhance the sensitivity of the EWMA control chart proposed by Castagliola et al. (2011). Tables are provided for the statistical properties of the new chart. Some numerical results and comparisons are given and show that the new chart has an average run length performance that is superior to some other competing procedures. A real data example from manufacturing shows that it performs quite well in applications.

CUSUM Charts for Monitoring the Characteristic Life of Censored Weibull Lifetimes

Companies routinely perform life tests on their products. Each of these life tests typically involves testing several units simultaneously with interest in the times to failure. Two aspects often associated with lifetime data that make the development of a control-charting procedure more demanding are that the data tend to be nonnormally distributed and censored. In this paper, one-sided lower and upper likelihood-ratio—based cumulative sum (CUSUM) control charting procedures are developed for Type I right-censored Weibull lifetime data to monitor changes in the scale parameter, also known as the characteristic life, for a fixed value of the Weibull shape parameter. Because a decrease in the characteristic life indicates a decrease in the mean lifetime of a product, a one-sided lower CUSUM chart is the main focus. We illustrate the development and implementation of the chart and evaluate its properties through a simulation study. The proposed CUSUM chart is compared with an exponentially weighted moving-average (EWMA) chart using the steady-state average run length (ARL) performance. The CUSUM chart is shown to perform better than the EWMA chart in detecting shifts for which it is designed.

A Reevaluation of the Adaptive Exponentially Weighted Moving Average Control Chart When Parameters are Estimated

The performance of control charts can be adversely affected when based on parameter estimates instead of known in‐control parameters. Several studies have shown that a large number of phase I observations may be needed to achieve the desired in‐control statistical performance. However, practitioners use different phase I samples and thus different parameter estimates to construct their control limits. As a consequence, there would be in‐control average run length (ARL) variation between different practitioners. This kind of variation is important to consider when studying the performance of control charts with estimated parameters. Most of the previous literature has relied primarily on the expected value of the ARL (AARL) metric in studying the performance of control charts with estimated parameters. Some recent studies, however, considered the standard deviation of the ARL metric to study the performance of control charts. In this paper, the standard deviation of the ARL metric is used to study the in‐control and out‐of‐control performance of the adaptive exponentially weighted moving average (AEWMA) control chart. The performance of the AEWMA chart is then compared with that of the Shewhart and EWMA control charts. The simulation results show that the AEWMA chart might represent a good solution for practitioners to achieve a reasonable amount of ARL variation from the desired in‐control ARL performance. In addition, we apply a bootstrap‐based design approach that provides protection against frequent false alarms without deteriorating too much the out‐of‐control performance. Copyright © 2014 John Wiley & Sons, Ltd.

A Study on ARL Performance of Ewma Control Chart

Shewhart chart has been widely used in determining shift in a process, however the main disadvantage of any Shewhart charts is that they use only the last information of the combined process and ignore any other process given by the sequence of all points. This characteristics makes Shewhart control charts relatively insensitive to a small changes in a process, because the cumulative or weight of the previous observation are disregarded. Small changes could be detected through Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) control charts. Crowder (1987, 1989) and Lucas & Saccucci (1990) presented the exponentially weighted moving average control chart, as good choice to detect small change in process average. A number of authors have studied the design of EWMA control scheme based on Average Run length (ARL) computation. Ideally, the ARL should be short when a shift occurs and should be long when there is no shift. In this paper, an attempt has been made to evaluate the run-length properties of EWMA control chart by using the Markov Chain approach. This study provides ARL tables for a range of values of λ and L .

A Study on ARL Performance of Ewma Control Chart

Shewhart chart has been widely used in determining shift in a process, however the main disadvantage of any Shewhart charts is that they use only the last information of the combined process and ignore any other process given by the sequence of all points. This characteristics makes Shewhart control charts relatively insensitive to a small changes in a process, because the cumulative or weight of the previous observation are disregarded. Small changes could be detected through Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) control charts. Crowder (1987, 1989) and Lucas & Saccucci (1990) presented the exponentially weighted moving average control chart, as good choice to detect small change in process average. A number of authors have studied the design of EWMA control scheme based on Average Run length (ARL) computation. Ideally, the ARL should be short when a shift occurs and should be long when there is no shift. In this paper, an attempt has been made to evaluate the run-length properties of EWMA control chart by using the Markov Chain approach. This study provides ARL tables for a range of values of λ and L .

The Average Run Length Performance of a Multivariate Exponentially Weighted Moving Average Control Chart Procedure with Application

The drawbacks to multivariate charting schemes is their inability to identify which variable was the source of the signal. The multivariate exponentially weighted moving average (MEWMA) developed by Lowry, et al (1992) is an example of a multivariate charting scheme whose monitoring statistic is unable to determine which variable caused the signal. In this paper, the run length performance of multivariate exponentially weighted moving average (MEWMA) chart with application is studied. Industry fertilizers is important one of the chemical industries in Egypt, so that this work concerns the fertilizers industries quality control, especially urea fertilizer with application on Delta fertilizer and chemical industries which is considered on of the leading companies the field of fertilizer production in Middle east with application of multivariate quality control procedures to achieve best one procedure for multivariate quality control.

Designing of Gini-chart for Exponential, t, Logistic and Laplace Distributions

This article extended the work of Saghir and Lin (2012) to develop the probability limits of Gini chart, a process dispersion chart based on gini's mean difference proposed by Riaz and Saghirr (2007), for the exponential, t(5), Logistic and Laplace distributions. The asymmetrical control limits of the Gini chart are proposed for the distributions under study. The estimated factors and quantile points used in the construction of the Gini chart are provided for the exponential, t(5), Logistic and Laplace distributions. The effect of improper use of constants and quantile points in the construction of Gini chart is studied in terms of the associated false alarm rates. The performance of the asymmetrical control limits of the given chart is evaluated in terms of average run length (ARL) for the exponential, t(5), Logistic and Laplace distributions and compared with the 3σ limits proposed by Saghir and Lin (2012). The ARL performance of the proposed probability limits is compared with the existing limits of R and S charts for the under study parent distributions. Finally, the design scheme of chart associated with Gini chart is developed.

Moving range EWMA control charts for monitoring the Weibull shape parameter

In this article, we propose an exponentially weighted moving average (EWMA) control chart for the shape parameter β of Weibull processes. The chart is based on a moving range when a single measurement is taken per sampling period. We consider both one-sided (lower-sided and upper-sided) and two-sided control charts. We perform simulations to estimate control limits that achieve a specified average run length (ARL) when the process is in control. The control limits we derive are ARL unbiased in that they result in ARL that is shorter than the stable-process ARL when β has shifted. We also perform simulations to determine Phase I sample size requirements if control limits are based on an estimate of β. We compare the ARL performance of the proposed chart to that of the moving range chart proposed in the literature.

Monitoring the Weibull Shape Parameter with Type II Censored Data

In this article, we introduce a method for monitoring the Weibull shape parameter β with type II (failure) censored data. The control limits depend on the sample size, the number of censored observations, the target average run length, and the stable value of β. The method assumes that the scale parameter α is constant during each sampling period, which is true under rational subgrouping. The proposed method utilizes the relationship between Weibull and smallest extreme value distribution. We propose an unbiased estimator of σ = 1/β as the monitoring statistic. We derive the control limits for one-sided and two-sided charts for several stable process average run lengths. We discuss two schemes, namely, the control-limits-only scheme and the control-limits-with-warning-lines scheme. The stable process average run length performance of the proposed charts is studied and compared with those of other charts for monitoring β under similar assumptions. Copyright © 2014 John Wiley & Sons, Ltd.

An accurate evaluation of adaptive exponentially weighted moving average schemes

As a natural generalization of the conventional Exponentially Weighted Moving Average (EWMA) monitoring scheme, the Adaptive EWMA (AEWMA) scheme has received a great deal of attention. The Markov chain method was originally used to approximate the average run length performance of the AEWMA chart; however, this method may suffer from the issue of slow convergence and unstable approximation due to kernel discontinuity. In order to overcome this issue, this article extends the piecewise collocation method and the Clenshaw–Curtis (CC) quadrature (method) to the evaluation of AEWMA chart performance. It is shown that both the collocation and CC quadrature methods are very competitive and can provide more accurate and fast approximation to the run length performance of AEWMA charts than the conventional Markov chain approach.

A simulation study on the performance of the sign test, Mann-Whitney test, Hodges-Lehmann estimator and control charts for Normal and Weibull data

Article history: Received January 6 2014 Received in Revised Format June 15 2014 Accepted July 14 2014 Available online July 14 2014 The new method to chart the Hodges-Lehmann estimator control chart is proposed in this study. The evaluation of the three nonparametric control charts the Sign test (ST), Mann-Whitney (MW), and the Hodges-Lehmann estimator (HL), for the known process distribution using normal and Weibull data represent the symmetric and asymmetric shapes of the process based on the original method through the 10000 run lengths simulation. The result illustrates that the average run length performance of the ST and MW correspond to their respective test statistics but for HL’s performance, the result indicates that the average run length is much greater than that derived from Wilcoxon signed rank statistics. The Hodges-Lehmann estimator control chart by the new approach for the known process distribution will be the alternative method for the process that needs to robust outliers’ properties from this statistics. In addition, the simulation demonstrates that the performances of the Sign test (ST) from mean and median processes are varied in the skewed distribution, and moreover, the Sign test (ST) from the median process represents more accurate performance. Meanwhile, for the control groups, MW generated within control limits or without restriction shows slightly different performance. The performance of dual scheme for the above-mentioned variable parameters control charts also produce the weighted average values that effect from the tight control scheme to the regular control scheme. © 2014 Growing Science Ltd. All rights reserved

On median control charting under double sampling scheme

Standard Shewhart control charts are often based on the assumption that the observations follow a specific parametric distribution, such as the normal, and outlier-free samples are initially selected to construct control limits for future monitoring of process parameters, e.g., location, dispersion, etc. The median is a popular measure of location which is more robust than mean for heavily skewed distributions. In ideal circumstances (where all the underlying assumptions such as normality and independence are met), the median chart is shown to be less efficient that the mean chart. To overcome the efficiency loss of the median chart, this study presents a set of auxiliary information-based median type Shewhart charts based on parent normal, t and gamma distributed process environments under double sampling scheme. The performance of these charts is evaluated in terms of run length (RL) characteristics such as: average run length (ARL), median run length (MDRL), standard deviation of the run length distribution (SDRL), extra quadratic loss (EQL) and relative ARL (RARL). Moreover, the effects of Step 1 sample size and contaminated environments are examined on the ARL performance of different median-based charting structures, under double sampling scheme. Illustrative examples are also provided to explain the working of the said charts.

A Synthetic Control Chart for Monitoring Process Variability

The control chart based on Downton's estimator (D chart) has recently been introduced in the literature for monitoring the process variability. The D chart is found to be equally efficient to the S chart in terms of detecting shifts in process variability. In this paper, salient features of D chart and the conforming run length chart are combined to produce synthetic D chart. The average run length performance of the synthetic D chart is investigated using simulation study and is compared with the originally proposed D chart and some other procedures proposed in the literature. It is found that it has an improved performance in comparison with the traditional control charts for process variability. Copyright © 2013 John Wiley & Sons, Ltd.

Using Control Chart to Detect Abnormal Changes in the Permeability

Water-vapour permeability is a critical factor of writing/ printing papers in most of practical applications; but how to monitor the manufacturing process to keep the key characteristic of the paper in control is still understudied. Therefore, in this paper, in order to monitor the water-vapour permeability of writing/ printing papers, MaxGWMA chart is first suggested due to its best effectiveness in terms of average run length performance and its high capability of detecting small shifts in the process mean and variability as well as identifying the source and the direction of an out-of-control signal. By using MaxGWMA chart, assignable causes of any out-of-control signal should be deeply examined so as to have proper corrective actions undertaken to either eliminate them from the process or reduce the variability induced by them to make the papers consistently manufactured under a stable process.

Multivariate Control Charts Based on Hybrid Novelty Scores

We propose a new nonparametric multivariate control chart that integrates a novelty score. The proposed control chart uses as its monitoring statistic a hybrid novelty score, calculated based on the distance to local observations as well as on the distance to the convex hull constructed by its neighbors. The control limits of the proposed control chart were established based on a bootstrap method. A rigorous simulation study was conducted to examine the properties of the proposed control chart under various scenarios and compare it with existing multivariate control charts in terms of average run length (ARL) performance. The simulation results showed that the proposed control chart outperformed both the parametric and nonparametric Hotelling's T 2 control charts, especially in nonnormal situations. Moreover, experimental results with real semiconductor data demonstrated the applicability and effectiveness of the proposed control chart. To increase the capability to detect small mean shift, we propose an exponentially weighted hybrid novelty score control chart. Simulation results indicated that exponentially weighted hybrid score charts outperformed the hybrid novelty score based control charts.