Median Run Length Research Articles

Effects of measurement errors on semicircle control chart

In recent years, the impacts of measurement errors on the performance of various control charts have been well evaluated. The existing semicircle (SC) control chart has been proposed based on the assumption that there are no measurement errors. In this paper, we investigated the effect of measurement errors using the linear covariate error model with constant variance and linearly increasing variance on the performance of the SC chart. We used simulation to evaluate the performance of the SC chart, in terms of average run length (ARL), standard deviation of the run length (SDRL), median run length (MDRL), and expected average run length (EARL) metrics. The results obtained indicate that measurement errors negatively affect the performance of the SC chart and taking multiple measurements on each item or/and increasing the slope coefficient of the covariate error model can effectively reduce the adverse impact of errors on the chart’s performance. Finally, an illustrative real life example is provided to demonstrate the undesired impact of measurement errors on the detection capability of the SC chart.

An efficient MEWMA chart for Gumbel's bivariate Pareto distribution

Control charts play a vital role in process monitoring to ensure the product's desired standards. Due to rapid improvements in data collection methods, multivariate charts are preferred over univariate charts. This paper proposes a bivariate exponentially weighted moving average chart for the simultaneous monitoring of the mean vector of Gumbel's bivariate Pareto type II (also known as the Lomax distribution) time-between-events data. The performance of the proposed chart is assessed through average run length, median run length, and the standard deviation of the run length criteria. To show the implementation of the chart in the real world, illustrative examples are also presented.

A distribution-free Max-EWMA scheme for multi-aspect process monitoring with industrial applications

In Statistical Process Monitoring, Distribution-free charting schemes play a vital role in Industry 4.0 for their superior effectiveness compared to parametric methods, especially in cases where the underlying process distribution is not well-defined or poses challenges in estimation. Traditional distribution-free charting schemes focus on surveilling only one process aspect: the location or scale parameter. The past two decades have also witnessed a rapid expansion of research on joint monitoring of the two aspects, location and scale, using combined bi-aspect charting schemes. However, there is a notable scarcity of methods that addressing complex industrial scenarios where multiple parameters (like location, scale, and shape) need simultaneous monitoring with high sensitivity and specificity. Our research is motivated by this challenge, and this paper describes a distribution-free tri-aspect monitoring procedure called the Maximum Exponentially Weighted Moving Average (Max-EWMA) scheme This scheme can effectively monitor shifts in three process characteristics, encompassing aspects related to process location, scale, and shape parameters. We utilize weighted versions of well-known statistical measures such as the Wilcoxon, Ansari-Bradley, and Savage-type statistics to construct the plotting statistics. An essential feature of our proposed Max-EWMA chart is its robustness under various continuous distributions, ensuring reliable performance in in-control situations. Competing schemes are compared via intensive computing techniques based on Monte Carlo using the Median Run Length metric, which evaluates the effectiveness of the proposed scheme. Finally, our proposed schemes are illustrated with two examples. Some concluding remarks and limitations of the study are noted.

The Development and Evaluation of Homogenously Weighted Moving Average Control Chart based on an Autoregressive Process

This research aims to investigate a Homogenously Weighted Moving Average (HWMA) control chart for detecting minor and moderate shifts in the process mean. A mathematical model for the explicit formulae of the average run length (ARL) of the HWMA control chart based on the autoregressive (AR) process is presented. The efficacy of the HWMA control chart is evaluated based on the average run length, the standard deviation of run length (SDRL), and the median run length (MRL). As illustrations of the design and implementation of the HWMA control chart, numerical examples are provided. In numerous instances, a comparative analysis of the HWMA control chart relative to the Extended Exponentially Weighted Moving Average (Extended EWMA) and cumulative sum (CUSUM) control charts with mean process shifts is performed in detail. Additionally, the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI) are utilized to evaluate the performance of control charts. For various shift sizes, the HWMA control chart is superior to the Extended EWMA and CUSUM control charts. This study applies empirical data from the area of economics to validate the explicit formula of ARL values for the HWMA control chart. Doi: 10.28991/HIJ-2024-05-01-02 Full Text: PDF

Performance Evaluation of HWMA Control Chart based on AR(p) with Trend Model to Detect Shift Process Mean

The main goal of this study is to establish explicit solutions for the average run length (ARL) of the Homogenously Weighted Moving Average control chart when subjected to autoregressive with trend process. The accuracy of the explicit formula for the ARL is evaluated in comparison to the numerical integral equation method. To evaluate the two approaches, the accuracy percentage was employed. A determination is carried out of the HWMA control chart’s effectiveness using the median run length (MRL), the standard deviation of run length (SDRL), and the average run length (ARL). A comprehensive comparison is performed between the HWMA control chart, the Extended Exponentially Weighted Moving Average (EEWMA), and the cumulative sum (CUSUM) control charts with mean process shifts to illustrate the design and implementation of the HWMA control chart. As criteria for various values of design parameters, the performance of these control charts can also be evaluated using the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI). To evaluate the effectiveness of our explicit formula approach, we employ this formula on copper price data.

A nonparametric mixed exponentially weighted moving average-moving average control chart with an application to gas turbines.

This study aims to develop a nonparametric mixed exponentially weighted moving average-moving average (NPEWMA-MA) sign control chart for monitoring shifts in process location, particularly when the distribution of a critical quality characteristic is either unknown or non-normal. In literature, the variance expression of the mixed exponentially weighted moving average-moving average (EWMA-MA) statistic is calculated by allowing sequential moving averages to be independent, and thus the exclusion of covariance terms results in an inaccurate variance expression. Furthermore, the effectiveness of the EWMA-MA control chart deteriorates when the distribution of a critical quality characteristic deviates from normality. The proposed NPEWMA-MA sign control chart addresses these by utilizing the corrected variance of the EWMA-MA statistic and incorporating the nonparametric sign test into the EWMA-MA charting structure. The chart integrates the moving average (MA) statistic into the exponentially weighted moving average (EWMA) statistic. The EWMA-MA charting statistic assigns more weight to recent w samples, with weights for previous observations decling exponentially. Monte Carlo simulations assess the chart's performance using various run length (RL) characteristics such as average run length (ARL), standard deviation of run length (SDRL), and median run length (MRL). Additional measures for overall performance include the average extra quadratic loss (AEQL) and relative mean index (RMI). The proposed NPEWMA-MA sign control chart demonstrates superior performance compared to existing nonparametric control charts across different symmetrical and asymmetric distributions. It efficiently detects process shifts, as validated through both a simulated study and a real-life example from a combined cycle power plant.

Variable Sample Size Control Charts for Monitoring the Multivariate Coefficient of Variation Based on Median Run Length and Expected Median Run Length

The monitoring of a well-functioning process system has always held significant importance. In recent times, there has been notable attention towards employing control charts to oversee both univariate and multivariate coefficients of variation (MCV). This shift is in response to the concern of erroneous outcomes that can arise when traditional control charts are applied under the condition of dependent mean and standard deviation, as highlighted by prior research. To address this, the remedy lies in adopting the coefficient of variation. Furthermore, this study underscores the application of MCV in scenarios where multiple quality attributes are simultaneously under surveillance within an industrial process. This aspect has demonstrated considerable enhancement in chart performance, especially when incorporating the variable sample size (VSS) feature into the MCV chart. Adaptive VSS, evaluated through metrics like median run length (MRL) and expected median run length (EMRL), is also integrated for MCV monitoring. In contrast to earlier studies that predominantly focused on average run length (ARL), this research acknowledges the potential inaccuracies in ARL measurement. In this study, two optimal designs for VSS MCV charts are formulated by minimizing two criteria: firstly, MRL; and secondly, EMRL, both accounting for deterministic and unknown shift sizes. Additionally, to assess the distribution's variability in run lengths, the study provides the 5th and 95th percentiles. The research delves into two VSS schemes: one with a defined small sample size (nS), and another with a predetermined large sample size (nL) for the initial subgroup (n(1)). The approach taken involves the development of a Markov chain method for designing and deriving performance measures of the proposed chart. These measures include MRL and EMRL. Moreover, a comparative analysis between the proposed chart's performance and the standard MCV chart (STD) is presented in terms of MRL and EMRL criteria. The outcomes illustrate the superiority of the proposed chart over the STD MCV chart for all shift sizes, whether they are upward or downward, and when n(1) equals nS or nL.

Improving Sensitivity of the DEWMA Chart with Exact ARL Solution under the Trend AR(p) Model and Its Applications

The double exponentially weighted moving average (DEWMA) chart is a control chart that is a vital analytical tool for keeping track of the quality of a process, and the sensitivity of the control chart to the process is evaluated using the average run length (ARL). Herein, the aim of this study is to derive the explicit formula of the ARL on the DEWMA chart with the autoregressive with trend model and its residual, which is exponential white noise. This study shows that this proposed method was compared to the ARL derived using the numerical integral equation (NIE) approach, and the explicit ARL formula decreased the computing time. By changing exponential parameters that were relevant to evaluating in various circumstances, the sensitivity of AR(p) with the trend model with the DEWMA chart was investigated. These were compared with the EWMA and CUSUM charts in terms of the ARL, standard deviation run length (SDRL), and median run length (MRL). The results indicate that the DEWMA chart has the highest performance, and when it was small, the DEWMA chart had high sensitivity for detecting processes. Digital currencies are utilized to demonstrate the efficacy of the proposed method; the results are consistent with the simulated data. Doi: 10.28991/ESJ-2023-07-06-03 Full Text: PDF

Analytical Explicit Formulas of Average Run Length of Homogenously Weighted Moving Average Control Chart Based on a MAX Process

Statistical process control (SPC) is used for monitoring and detecting anomalies in processes in the areas of manufacturing, environmental studies, economics, and healthcare, among others. Herein, we introduce an innovative SPC approach via mathematical modeling and report on its application via simulation studies to examine its suitability for monitoring processes involving correlated data running on advanced control charts. Specifically, an approach for detecting small to moderate shifts in the mean of a process running on a homogenously weighted moving average (HWMA) control chart, which is symmetric about the center line with upper and lower control limits, is of particular interest. A mathematical model for the average run length (ARL) of a moving average process with exogenous variables (MAX) focused only on the zero-state performance of the HWMA control chart is derived based on explicit formulas. The performance of our approach was investigated in terms of the ARL, the standard deviation of the run length (SDRL), and the median run length (MRL). Numerical examples are given to illustrate the efficacy of the proposed method. A detailed comparative analysis of our method for processes on HWMA and cumulative sum (CUSUM) control charts was conducted for process mean shifts in many situations. For several values of the design parameters, the performances of these two control charts are also compared in terms of the expected ARL (EARL), expected SDRL (ESDRL), and expected MRL (EMRL). It was found that the performance of the HWMA control chart was superior to that of the CUSUM control chart for several process mean shift sizes. Finally, the applicability of our method on a HWMA control chart is provided based on a real-world economic process.

Efficient signed-rank based EWMA and HWMA repetitive control charts for monitoring process mean with and without auxiliary information

Control charts are powerful tools to observe the presentation of the manufacturing process. Mostly, when the data in industries come from the process may follow non-normal or unknown distributions. So, the distribution-free control charts are useful in practice when the possibility model of the process productivity is unknown. In such situations, the correct selection of the sampling mechanism is beneficial for process examination. This paper proposes a nonparametric exponentially weighted moving average signed-rank (EWMA-SR) and also proposed a homogeneously weighted moving average Signed-Rank (HWMA-SR) control charts for examining the small shift in process with the help of an auxiliary variable (in the form of a regression estimator) by using an efficient plan, namely, a repetitive sampling plan. The proposal’s presentation is evaluated and matched with its complements for different symmetric distributions by using some famous run length properties including average run length, median run length, and standard deviation of run length.

The design of two-sided S 2-chart with estimated in-control process variance based on conditional median run length

ABSTRACT Monitoring both decrease and increase in the process variance has a significant value in effectively implementing any quality improvement program. This paper considers the Phase II two-sided S 2 -chart with estimated in-control (IC) process variance and evaluates its performance under the conditional perspective. The conditional perspective focuses on the conditional (given parameter estimates) IC run length distribution and its associated properties to evaluate chart’s performance with estimated parameters. Thus, it accounts for the practitioner-to-practitioner variability and, hence, provides a comprehensive understanding of the chart’s performance. To guard against unnecessary false alarms, we design the chart using the exceedance probability criterion under the conditional perspective, which ensures the desired IC performance with a high probability. Furthermore, because the median run length (MRL) is a more suitable representative of run length (RL) distribution, we examine the chart’s performance in terms of the median of the conditional RL distribution. We also discuss the biasedness property of the two-sided S 2 -chart, which is an outcome of the skewed nature of the charting statistic. It is shown that the proposed MRL-unbiased chart requires about within 500 Phase I observations to achieve a reasonably good IC and OOC performance.

First to signal criterion for comparing control chart performance

Control chart performance is often measured using average run length or median run length, which gives the expected or median number of samples to signal. It is often argued that on average, one chart will signal a process change quicker than another, and is therefore a better choice. Average and median run length do not, however, answer the question of which method will be more likely to signal first. We introduce the idea of “first to signal” and compare charts based on this criterion.

An optimally designed distribution‐free CUSUM procedure for tri‐aspect surveillance of continuous processes

AbstractIn the past few decades, research on nonparametric process monitoring schemes mainly dealt with the uni‐aspect or bi‐aspect schemes, focusing on monitoring process location or scale separately or jointly. Another critical process characteristic, namely, the process shape, is not explicitly dealt with in‐depth in most existing charting schemes. In classical hypothesis testing, some recent literature clearly showed that the multi‐aspect test statistics, although designed for very restrictive alternatives, often perform as well or better than many statistics for arbitrary distributional shifts. They are often better than Kolmogorov‐Smirnov or Cramér‐von Mises statistics and some empirical likelihood‐based test statistics. The current paper aims to use a tri‐aspect statistic to design a distribution‐free Phase‐II Cumulative Sum (CUSUM) charting scheme for monitoring any arbitrary process changes in the process. The proposal is nonparametric and is equivalent to an unknown standard case. It reflects, in addition, which parameters, among location, scale, or shape, are more responsible for a signal. The construction of the CUSUM scheme from an existing tri‐aspect Shewhart‐type chart is simple, so significant attention is devoted to determining a near‐optimal reference parameter of the chart in keeping an unknown shift type in mind. Comparisons of the optimal performance of various competitors are considered in terms of the median run length (MRL) metric. The proposed charting scheme designed with the tri‐aspect statistic compares highly favorably with many existing SPM schemes. The same is evident from our findings based on the Monte‐Carlo simulation. Finally, the proposed schemes are illustrated with a flow‐width measurement monitoring example.

Distribution-free Phase-II monitoring of high-dimensional industrial processes via origin and modified interpoint distance based algorithms

Improvements in measuring devices, the development of sensing technologies, automated record-keeping and cloud storage facilities have offered us a large volume of data streams involving many variables for data-driven decision-making. Consequently, monitoring high-dimensional (HD) processes has become an essential and exciting topic in statistical process monitoring. HD monitoring poses a significant challenge because considerable uncertainty is involved in estimating process parameters when an adequately large reference sample is unavailable before monitoring. Thus, designing an effective, conveniently implementable monitoring scheme for the HD processes remains a challenging task that has yet to be effectively addressed. This paper develops two distribution-free exponentially weighted moving average (EWMA) schemes to monitor multivariate and HD processes based on origin and interpoint distances. Previous works on interpoint distance-based charts are generally not distribution-free unless only one reference point is used as a benchmark to compute the interpoint distances. We offer an improved and general way to achieve conditionally distribution-free properties closely. The proposal also combines the advantage of the nonparametric approaches for jointly monitoring the process location and scale parameters with the better shift detection performance of an EWMA scheme. The new EWMA schemes can be easily implemented even when the data dimension is large with a small number of Phase-I samples. Control limits using the median run length as a metric are evaluated and tabulated for some cases to implement the proposed chart quickly. Extensive comparative studies have been conducted to verify the efficacy of the proposed monitoring schemes. Finally, the proposed schemes are illustrated with a real problem of monitoring high-dimensional quality characteristics from semiconductor manufacturing processes.

An improved adaptive EWMA control chart for monitoring time between events with application in health sector

AbstractNonconforming events rarely occur in high‐quality processes, and the time between events () is likely to be followed by a skewed distribution, such as an exponential distribution. This paper proposes the upper and lower‐sided improved adaptive exponentially weighted moving control charts for monitoring data modeled by the exponential distribution. The proposed control charts are labeled as control charts, that is, the and control chart, respectively. The control chart detects the upward shifts, while the control chart identifies the downward shifts in the process. The Monte Carlo simulations are used to approximate the run length distribution of the proposed control charts. Numerical results associated with various performance measures, such as average run length (), standard run length (), median run length (), extra quadratic loss (), relative average run length (), and performance comparison index () are computed. The proposed control charts compared to the respective existing control charts, such as , , , and control charts, at a single shift, as well as over a specified range of shifts. The comparison reveals that the proposed control charts outperform the existing control charts for both single specified shifts and over a certain range of shifts. Finally, real‐life data of hospital stay time for male patients with traumatic brain injury are analyzed for the applicability of the control chart in the health sector.

Evaluating the Performance of Synthetic Double Sampling np Chart Based on Expected Median Run Length

To keep an eye on the status of high-quality processes for fraction nonconforming, the synthetic double sampling (SDS) np chart is a helpful tool. The SDS np chart is a hybrid between the double sampling (DS) np chart and the conforming run length (CRL) chart. The performance of a control chart is typically judged solely using the average run length (ARL). However, as the shape of the run length (RL) distribution varies with the magnitude of the shift in the process fraction nonconforming, the ARL no longer provides clear interpretation of a chart’s performance. Subsequently, enhanced DS np charts that use median run length (MRL) and expected median run length (EMRL) measures, including SDS np with MRL have recently been proposed for addressing this setback. To broaden the functionality of SDS np, in this work, the unexplored use of EMRL as alternative performance measure is developed by means of Markov chain model. Additionally, in both the zero-state (ZS) and steady-state (SS) modes, the novel optimal designs algorithms are described for computing the optimal charting parameters of the SDS np chart, for both MRL1 and EMRL1 minimizations, without any unfavourable feature of bilateral sensitivity. Both the MRL and EMRL performances of SDS np, synthetic np, and DS np charts are compared. Optimal designs charting parameters and sensitivity analyses are provided to aid the practical application of SDS np chart.

Exponentially weighted moving average Lepage-type schemes based on the lower-order percentile of the run-length metrics and their use in monitoring time-occupancy in Google applications

ABSTRACT Many existing monitoring schemes in the literature are based on the in-control (IC) average or median run-length. Several Phase-II schemes frequently fail to protect against the high rate of early false alarms. The problem may worsen when the average run-length metric is used, and the scheme is based on unknown and estimated parameters. Early false alarms can be avoided using monitoring schemes based on the lower-order percentiles of the IC run-length distribution. The exponentially weighted moving average (EWMA)-Lepage scheme is presented in this paper. The new design is based on a percentile-based approach that can effectively reduce and control the rate of early false alarms. The run-length properties of the EWMA scheme with the lower-order percentile-based design were investigated and compared with the double EWMA-Lepage and homogeneously weighted moving average-Lepage schemes. Detailed simulation studies show no clear winner among the three schemes for given sample sizes for the unknown shift. Instead, the size of the Phase-I and Phase-II samples heavily influences the choice of a potentially beneficial scheme. A case study on monitoring the time occupation of users on the Google application is presented to demonstrate the design and implementation of lower-percentile-based techniques. Some future research directions are offered.

Conditional design of the Shewhart <i>X̄</i> chart with unknown process parameters based on median run length

Numerous researches have been done on the Shewhart X̄ chart based on the average run length (ARL) metric. Since the shape of the run length (RL) distribution changes with the mean shift size, the median run length (MRL) is argued to be a better criterion for evaluating the performance of the Shewhart X̄ chart. Moreover, when the process parameters are unknown, the phase 2 properties of the Shewhart X̄ chart are conditioned on the parameter estimates arising from different practitioners in phase 1. This variability among the estimated process parameters is usually called as the between-practitioners variability. In order to investigate this variability in the conditional MRL values, both the average MRL (AMRL) and the standard deviation of MRL (SDMRL) will be used together in our article. The performance analyses of the MRL-based Shewhart X̄ chart are provided. To prevent too many lower in-control MRL values than the desired one, an appropriate bootstrap approach is adopted to adjust the control limits, and to further balance the in- and out-of-control MRL values of the Shewhart X̄ chart. [Submitted: 26 September 2018; Accepted: 15 January 2022]

Conditional design of the Shewhart Xbar chart with unknown process parameters based on median run length

Conditional design of the Shewhart Xbar chart with unknown process parameters based on median run length

Efficient control chart-based monitoring of scale parameter for a process with heavy-tailed non-normal distribution

<abstract> <p>Statistical process control is a procedure of quality control that is widely used in industrial processes to enable monitoring by using statistical techniques. All production processes are faced with natural and unnatural variations. To maintain the stability of the production process and reduce variation, different tools are used. Control charts are significant tools to monitor a production process. In this article, we design an extended exponentially weighted moving average (EEWMA) chart under the assumption of inverse Maxwell (IM) distribution, an IM EEWMA (IMEEWMA) control chart. We have estimated the performance of the proposed chart in terms of various run-length (RL) properties, including the average RL, standard deviation of the RL and median RL. We have also carried out a comparative analysis of the proposed chart with the existing Shewhart-type chart for IM distribution (VIM chart) and IM exponential weighted moving average (IMEWMA) chart. We observed that the proposed IMEEWMA chart performed better than the VIM chart and IMEWMA chart in terms of the ability to detect small and moderate shifts. To demonstrate its practical application, we have applied the IMEEWMA chart, along with existing control charts, to monitor the lifetime of car brake pad data. This real-world example illustrates the superiority of the IMEEWMA chart over its counterparts in industrial scenarios.</p> </abstract>