Average Run Length Performance Research Articles

A study of attribute control charts for censored lifetime data

In many recent articles authors have advocated the use of attribute charts based on the number of censored values for monitoring censored lifetime data. In this article we compare the average run-length performance of one-sided and two-sided Shewhart charts based on the conditional expected value (CEV) with the corresponding attribute charts. The overall findings indicate that using an attribute chart to monitor the mean of a Weibull lifetime probability distribution with right censoring does not perform as well as the CEV-based chart.

On the performance and comparison of various memory-type control charts

Several versions of the exponentially weighted moving average (EWMA) control chart, such as the generally weighted moving average (GWMA), the double, triple, and quadruple EWMA (DEWMA, TEWMA, and QEWMA), and the homogeneously weighted moving average (HWMA) charts, have been developed to enhance its detection ability, especially for small and moderate shifts. These charts have been criticized as it is considered that an appropriately designed EWMA chart provides as good or better zero-state and steady-state average run-length (ARL) performance. In this article, we perform a careful comparison study of the above control charts under similar in-control (IC) run-length properties. We show that the “new” memory-type control charts have better out-of-control (OOC) run-length characteristics, especially for small to moderate shifts in the process mean for both the zero-state and steady-state cases, except for the HWMA chart where its steady-state OOC performance is very poor. We also present a comparison study based on the worst-case ARL measure. Finally, the extensions of the EWMA chart have better IC robustness ability than the EWMA and HWMA charts.

The impact of imputation methods on the performance of Phase I Hotelling’s T 2 control chart

The objective of this study was to evaluate the impact of three different methods of handling missing data on the performance of Phase I Hotelling’s T 2 multivariate control chart. Using a Monte Carlo simulation, we studied the average, median, and standard deviation of the run length performance of multivariate data imputed using mean substitution, regression imputation, and predictive mean matching at three different levels of missingness ( 1 % , 10 % , and 25 % ) and three levels of variable correlation coefficients (0.2, 0.4, and 0.8). We found that predictive mean matching has average run length performance results comparable to that of the complete in-control data set at all levels of missingness and variable correlation, while the performance of mean substitution was adversely affected by high levels of missingness and by strong variable correlation. Based on the simulation (multivariate normal data), we concluded that predictive mean matching is superior to both regression imputation and mean substitution as a method for imputing missing values for the analysis of Phase I Hotelling’s T 2 control chart. Two applications were presented using the Altenrhein wastewater treatment plant and Olive oil datasets.

A CUSUM control chart for gamma distribution with guaranteed performance

AbstractEven though most control chart developments have revolved around the normal distribution, productive operations, in fact, often use non‐normal data, and some even use skewed distributions. This occurs with cycle times and many processes with a single specification limit. Under these conditions, gamma distributions often offer a better characterization where CUSUM schemes provide optimal performance to detect changes of a known in‐control (IC) parameter. In practice, gamma parameters have to be estimated, and limited phase I samples have undesirable effects on the phase II average run length (ARL) that has not been examined. To address this situation, this study uses Markovian models to assess the ARL of a CUSUM developed to monitor scale changes when IC parameters are estimated. This paper discusses the conditional ARL performance and proposes an adaptation of a stochastic estimation algorithm to guarantee a minimum performance. IC and out‐of‐control performance is compared with the conditional performance of a traditional approach where no corrections are made to guarantee a specific behavior. To demonstrate the practical application of this new approach, an implementation was made in a packaging process using multihead weighing machines. Practitioners interested in using reliable monitoring schemes will find, in this proposal, several implementation guidelines such as algorithms and look up tables to facilitate their implementations.

An Expected Average Run Length (EARL) Performance Comparison of the SSGR and EWMA Control Charts

The acceleration use of control charts in industrial processes has led to the effectiveness in their evaluation by quality practitioners. This is crucial, as it influences their decisions on the choice of which control charts to employ. This study aims to explore and compare the performance of the side sensitive group runs (SSGR) and exponentially weighted moving average (EWMA) control charts. In general, the average run length (ARL) characteristics were used to evaluate the performance of these control charts. The ARL, which considers the exact shift size in the process, is restricted in the case when the practitioner cannot identify the process shift size (unknown shift size). In this situation, the expected average run length (EARL) is an alternative performance criterion. Upon comparison of the findings obtained, the EWMA chart has superior performance when (δmin, δmax) = (0.1, 0.4). In contrast, the SSGR chart overtakes the EWMA chart when (δmin, δmax) = (0.5, 0.8) and (δmin, δmax) = (0.9, 1.2), except when the sample size n = 3 for (δmin, δmax) = (0.5, 0.8). For this particular combination, the EWMA chart performs slightly better than the SSGR chart. The outcome of this study is expected to contribute to practitioners in identifying suitable control charts in process monitoring and implementation.

Enhancing Control: Unveiling the Performance of Poisson EWMA Charts through Simulation with Poisson Mixture DATA

Poisson-Exponentially Weighted Moving Average (PEWMA) charts are one of the most frequently used control charts for monitoring count data. But as real-world data often shows overdispersion—prevalent in manufacturing, health care, economics, and marketing—the standard Poisson distribution falls short. One of the ways to tackle overdispersion is to use Poisson mixture distributions. Our study examines Average Run Length (ARL) performance in the presence of Poisson mixture distribution in the PEWMA control charts. Through meticulously designed experiments, we explore different control parameter combinations and employ simulation to evaluate the process. Our graphs illustrate the performance of the PEWMA control chart, offering desired in-control ARL across parameter combinations. Finally, the performance of the PEWMA control chart is presented for the real process data of fastener production.

A Study on the Laney p′ Control Chart with Parameters Estimated from Phase I Data: Performance Evaluation and Applications

The Laney p′ control chart is a new type of attribute control chart that can be applied in situations where the process exhibits either overdispersion or underdispersion. While it has gained acceptance in the industry, there is still limited knowledge about its effectiveness in detecting process variation. It is well known that before applying a control chart, understanding its performance is crucial, especially when the parameters of the control chart need to be estimated from historical data. In this study, we used simulations to investigate the ability of the Laney p′ control chart to detect process variations when the parameters are estimated. We designed appropriate experiments to assess the impact of overdispersion on the average run length (ARL) performance. In this study, we assumed that the overdispersion comes from the variation in the mean fraction nonconforming of each sample. The mean value varies according to a uniform distribution. This study evaluated the performance of the Laney p′ control chart using the average of the ARL (AARL) and the standard deviation of the ARL (SDARL). Additionally, real-world data were utilized to illustrate the practical applications of the Laney p′ control chart in the PCB and IC substrate industries. The research findings can serve as valuable guidance for practical implementation.

Optimal design of S2‐EWMA control chart for short production runs

AbstractIn recent fast pace smart manufacturing environment, the short production runs (SPR) control charts offer a higher degree of flexibility compared to the traditional control charts. This is due to Phase I data collection not needed for SPR control charts and this eases practitioners in estimating the parameters and setting up the control charts. This paper proposes the methods to optimally design the SPR S‐chart and SPR S2‐EWMA control chart for process variance monitoring in terms of truncated average run length (TARL), truncated standard deviation of the run length (TSDRL), and expected truncated average run length (ETARL) performance measures. The optimal parameters in minimizing the TARL and ETARL values for SPR S2‐EWMA control chart are presented. The control chart performances are compared between both the SPR S‐chart and SPR S2‐EWMA control charts. Results show that SPR S2‐EWMA control chart has a higher chart sensitivity and it triggers the out‐of‐control signal faster than the SPR S‐chart. An illustration using the real electronic manufacturing industry data is given in this paper to explain the implementation of the SPR S‐chart and SPR S2‐EWMA control chart.

Reevaluating the performance of control charts based on ranked-set sampling

Nearly one hundred types of control charts have been proposed that incorporate ranked-set sampling (RSS) methods. The performance of these charts has been evaluated with comparisons to existing charts based on simple random sampling. The reduction of the standard error in estimating the parameter being monitored with RSS leads to uniformly better average run length performance. We show, however, that these performance comparisons can be very misleading once the sampling strategy over time is considered more carefully with the benefits of RSS being considerably overstated. We consider the most basic RSS method when monitoring the mean of the process, but the approach can be applied to evaluate other RSS monitoring methods.

Multivariate auto‐correlated process control by a residual‐based mixed CUSUM‐EWMA model

AbstractMultivariate auto‐correlated process control issues in industrial systems are a concern for statistical process monitoring (SPM). Traditional control charts produce large false alarms and/or miss timely detections of quality deterioration because they are unable to recognize the signals from multivariate auto‐correlated response variables. To track multivariate auto‐correlated processes, this paper presents a new residual‐based mixed multivariate control chart using cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) approaches. Using in‐control data, the multi‐output least square support vector regression model's optimal hyper‐parameters are determined, and a bootstrap method is used to estimate the upper control limit of the proposed control chart. The suggested control chart has strong detection performance for a small magnitude mean vector shift based on the average run length (ARL) performance for a particular range of shifts. Experimental result elaborates that the proposed control chart is more sensitive to detecting the mean vector shift compared with the existing commonly used models, such as multivariate CUSUM and multivariate EWMA control charts. The proposed control chart model and corresponding computational algorithm are successfully applied to SPM in an electronic conductor production line with multivariate auto‐correlated attributes.

The Performance of S Control Charts for the Lognormal Distribution with Estimated Parameters

Control charts, one of the powerful tools in statistical process control (SPC), are widely used to monitor and detect out-of-control processes in the manufacturing industry. Many researchers have pointed out the effects of using estimated parameters on the average run length (ARL) performance metric. Most of the previous literature has studied the expected value of the average run length (AARL) and the standard deviation of the average run length (SDARL) to evaluate the performance of control charts. In this article, we study the performance of three S control charts, the Shewhart S-chart, the median absolute deviation (MAD) control chart, and the lognormal S control chart, for a lognormal distribution in terms of the AARL and SDARL. Simulation results indicate the sample size to reach the specified in-control ARL value is very large. The lognormal S control chart has a smaller SDARL value than the other two S-charts.

Control Charts for Monitoring the Mean of Skew-Normal Samples

The presence of asymmetric data in production processes or service operations has prompted the development of new monitoring schemes. In this article, an adapted version of the exponentially weighted moving averages (EWMA) control chart with dynamic limits is proposed to monitor the mean of samples from the skew-normal distribution. The detection ability of the proposed control chart in online monitoring was investigated by simulating the average run length (ARL) performance for different out-of-control scenarios. The results of the simulation study suggest that the proposed scheme overcomes the main drawback of the recently developed Shewhart-type control scheme. As shown in this article, the existing Shewhart-type procedure exhibits the undesirable property of taking longer to detect changes in the mean value of skewed normal observations due to increases in the shape parameter of the basic distribution than in stable conditions. The proposed control chart was shown to work fairly acceptably in all considered out-of-control scenarios.

Monitoring count time series: Robustness to nonlinearity when linear models are utilized

AbstractLinear models are typically utilized for time series analysis as these are often simple to implement and interpret, as well as being useful in modeling many practical phenomena. Hence, most of the literature on control charts for monitoring time series also consider linearity of the data generating processes (DGP). In practice, however, a nonlinear DGP can be modeled as if it is linear, either due to overlook or illusion, when it is approximately linear. This study quantifies the effects of nonlinear DGPs, misspecified and modeled as linear, on the performance of Shewhart‐type and cumulative sum (CUSUM) control charts for count time series data. Time series models for bounded and unbounded counts with several parametrizations are considered for studying the sensitivity of linear approximations to nonlinear DGPs. The Markov chain (MC) approach and simulations are used to compute the average run length (ARL) performance of the control charts. Robustness of the performance is evaluated with respect to the extent that the DGP violates linearity, with and without errors in parameter estimation. It is shown that the chart designs are, in general, robust to model misspecification when the parameters are specified. However, the CUSUM performance can be affected significantly if both the model is misspecified and the parameters are estimated. Real‐world data implementations are provided to illustrate the sensitivity of the control charts' performances to the type of model and to the estimation approach.

Monitoring reliability under competing risks using field data

Many modern products fail due to one of multiple causes called competing risks. In this article, we propose variable features for monitoring product failure by control charts under competing risks. Failure reports arrive one at a time from a sample of population of units. Features are derived from both the reports and the assumed competing-risk statistical model. To assess the efficacy of different feature subsets in detecting shifts in the failure-time process, we consider control charts based on random forests and compare the average run length performances under different shift scenarios. We demonstrate the control charts with both simulated data sets and actual field data set from a consulting problem. We also propose graphical fault-diagnosis methods for identifying assignable causes of alarm signals. Control charts based on the proposed features will provide valuable information to manufacturers in planning for warranty, part-replacement, or repair.

Effect of Parameter Estimation on the Performance of Shewhart <img src=image/13425522_01.gif>-joint Chart Looked at in Terms of the Run Length Distribution

Using spread-charts to monitor process variation and thereafter using the <img src=image/13425522_02.gif>-chart to monitor the process mean after is a common practice. To apply these charts independently using estimated 3-sigma limits is common. Recently, some authors considered the application of <img src=image/13425522_02.gif> and R-charts together as a charting scheme, <img src=image/13425522_01.gif>-chart when the standards are known, Case KK, only the mean standard is known, Case KU and both standards unknown, Case UU. The average run length (ARL) performance criterion was used. However, because of the skewed nature of the run length (RL) distribution, many authors have frowned at the use of ARL as a sole performance measure and encouraged the percentiles of the RL distribution instead. Therefore, the cdfs of the RLs of the chart under the cases mentioned will be derived in this work, and the percentiles are used to look at the chart for Case KU and the yet to be considered case of the chart, Case UK where only the process variance is known is included for comparison. These are the contribution to the existing literature. <img src=image/13425522_01.gif>-chart performed better in Case KU than in Case UK and the unconditional in-control median run length described the behavior of the chart better than the in-control ARL.

A synthetic multivariate exponentially weighted moving average scheme for monitoring of bivariate Gamma distributed processes

AbstractSeveral scholarly works intrinsically assume that the frequency and magnitude of events are two independent variables . However, in many practical situations, it is necessary to consider the dependence between these two characteristics of events. In this paper, two Smith–Adelfang–Tubbs (SAT) models, namely, the four‐ and the five‐parameter bivariate Gamma distributions (BGD) with a particular dependency parameter, are first reviewed. Then, a synthetic multivariate exponentially weighted moving average (synthetic MEWMA, or SMEWMA) type scheme is proposed to monitor the individual BGD observations (subgroups of size one) generated from those two specific SAT models. The Monte‐Carlo simulation method is developed to evaluate the run length () properties of the recommended SMEWMA BGD scheme in both the zero‐state and the steady‐state cases. Furthermore, we discuss the average run length (ARL) of the proposed SMEWMA BGD scheme with different design parameters, and then compare both the zero‐state and the steady‐state out‐of‐control ARL performances of the recommended SMEWMA BGD scheme with those of the classic MEWMA chart, the paired individual Gamma chart, and the MCUSUM chart. Simulation results suggest that the overall performance of the recommended SMEWMA BGD scheme is superior to the other comparative schemes. Finally, the implementation of the proposed SMEWMA BGD scheme is illustrated with a natural flood event dataset .

The impracticality of homogeneously weighted moving average and progressive mean control chart approaches

AbstractThere is growing literature on new versions of “memory‐type” control charts, where deceptively good zero‐state average run‐length (ARL) performance is misleading. Using steady‐state run‐length analysis in combination with the conditional expected delay (CED) metric, we show that the increasingly discussed progressive mean (PM) and homogeneously weighted moving average (HWMA) control charts should not be used in practice. Previously reported performance of methods based on these two approaches is misleading, as we found that performance is good only when a process change occurs at the very start of monitoring. Traditional alternatives, such as exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts, not only have more consistent detection behavior over a range of different change points, they can also lead to better out‐of‐control zero‐state ARL performance when properly designed.

Mixed memory control chart based on auxiliary information for simultaneously monitoring of process parameters: An application in glass field

Cumulative sum (CUSUM) and exponentially weighted moving (EWMA) control charts, known as memory control charts, are famous to monitor a small-to-moderate shift in the process parameters (location and/or dispersion). The EWMA control chart based on auxiliary information denoted as EWMAAIB(1) and EWMAAIB(2) are advanced forms of the classical EWMA control chart to monitor the process location and dispersion, respectively. Likewise, the combined mixed EWMA-CUSUM (CMEC) control chart is used to simultaneously monitor the process parameters. This study presents the CMEC control chart based on auxiliary information, symbolized as CMECAIB control chart for simultaneous monitoring of the process parameters. The proposed CMECAIB control chart used EWMAAIB(1) and EWMAAIB(2) control charts plotting statistics as inputs in the classical CUSUM control charts. The Monte Carlo simulation is used as a computational technique for numerical results. The proposed CMECAIB control chart based on average run length performance measure is evaluated against other control charts such as CMEC, combined double mixed EWMA-CUSUM, combined CUSUM, maximum double EWMA, maximum EWMA based on auxiliary information (MaxEWMAAIB) control and maximum Hybrid EWMA based on auxiliary information (MaxHEWMAAIB) control charts. The comparison revealed the superiority of the proposed CMECAIB control chart. Besides, the CMECAIB control chart performs better as the correlation coefficient increases. Likewise, the proposed CMECAIB control chart has a better diagnostic ability for possible directions of changes against other control charts. It is vital to mention that some existing control charts such as classical CUSUM, CMEC, and maximum CUSUM are special cases of proposed CMECAIB control chart with the specific parameter’s values. Finally, to demonstrate the vitality of the proposed study from a practical point of view, a real-life application in the glass bottle manufacturing industry is also provided for users and practitioners.

A generalized likelihood ratio test for monitoring profile data

Profile data emerges when the quality of a product or process is characterized by a functional relationship among (input and output) variables. In this paper, we focus on the case where each profile has one response variable Y, one explanatory variable x, and the functional relationship between these two variables can be rather arbitrary. The basic concept can be applied to a much wider case, however. We propose a general method based on the Generalized Likelihood Ratio Test (GLRT) for monitoring of profile data. The proposed method uses nonparametric regression to estimate the on-line profiles and thus does not require any functional form for the profiles. Both Shewhart-type and EWMA-type control charts are considered. The average run length (ARL) performance of the proposed method is studied. It is shown that the proposed GLRT-based control chart can efficiently detect both location and dispersion shifts of the on-line profiles from the baseline profile. An upper control limit (UCL) corresponding to a desired in-control ARL value is constructed.

On the conditional performance of the synthetic chart with unknown process parameters using the exceedance probability criterion.

Recent researches on the control charts with unknown process parameters have noticed the large variability in the conditional in-control average run length (ARL) performance of control charts, especially when a small number of Phase I samples is used to estimate the process parameters. Some research works have been conducted on the conditional ARL performance of different types of control charts. In this paper, by simulating the empirical distribution of the conditional ARL and especially using the exceedance probability criterion (EPC), we study the conditional ARL performance of the synthetic chart. Our results show that a large amount of Phase I samples is needed to obtain a specified EPC of the synthetic chart. For the available number of Phase I samples, the control limits of the synthetic chart are adjusted using the EPC method to improve its conditional in-control performance. It is shown that, for small mean shift sizes, a tradeoff should be made between the conditional in-control and out-of-control performances. For moderate to large shifts, the conditional performance of the synthetic chart using the adjusted control limits is generally satisfied. By comparing the results with the ones using the bootstrap approach, it can also be concluded that the conditional performances of both approaches are comparable. While the method proposed in this paper requires much less computation work than the bootstrap approach.