Relative Average Run Length Research Articles

A weighted dual cumulative sum chart for monitoring the process mean

AbstractThe cumulative sum (CUSUM) charts are well‐structured and widely used for detecting small‐to‐moderate changes in the process parameters. However, their performance can be limited when the shift size is unknown or varies over time. Various methods have been developed to address this issue, including dual CUSUM (DCUSUM), adaptive CUSUM, adaptive EWMA, and weighted CUSUM charts, all of which offer better sensitivity compared to traditional CUSUM charts. In this study, we propose a new weighted DCUSUM chart designed to effectively detect the mean shifts in a normally distributed process. Monte Carlo simulations are used to estimate the zero‐state and steady‐state average run‐length (ARL) profiles of the control charts. The ARL performances of the control charts are evaluated in terms of expected relative ARL (ERARL) and expected weighted run‐length (EWRL). Our results show that the proposed chart may outperform existing counterparts in terms of the ARL, ERARL, and EWRL measures, indicating superior performance in detecting a range of the mean shift sizes. Finally, we apply the existing and proposed charts to a real dataset, showcasing their practical utility in process monitoring.

Ameliorating the diagnostic power of combined shewhart-memory-type control chart strategies for mean

A great deal of research in statistical process control (SPC) involves the integration of different ideas to achieve the optimal detection of anomalies in the location parameter of a process. Likewise, the prime rationale of the combined (Shewhart-Memory-type) strategies is to promptly detect the shift of any size: small, moderate or large. An effort, along the same lines, is made to detect shifts in the mean of a normally distributed process. The usual difference estimator of the process mean instead of the sample mean is employed in the Mixed EWMA-Crosier CUSUM Combined Shewhart Mixed EWMA-CUSUM and Combined Shewhart Mixed EWMA-Crosier CUSUM charts to make their upgraded versions. This enhances the sensitivity of these charts against any shift size. The performance of the charts is appraised and compared with some notable charts of the same ilk, with reference to the average run length (ARL), relative average run length, extra quadratic loss and performance comparison index. An illustrative example is also provided.

Novel combined Shewhart-CUmulative EWMA-SUM mean charts without- and with measurement error

The precision of process monitoring often encounters challenges in determining the exact shift size. Therefore, combined control charts have gained considerable attention because of their excellent speed to detect simultaneously small-to-moderate and large-size shifts. The effectiveness of the applied quality control methods strongly depends on the performance of the measurement system. Measurement error presence contributes significantly negatively toward the performance of the usual control charting schemes. This article proposes novel two-sided combined Shewhart-Cumulative EWMA-sum (Shewhart-CUESUM) control charts designed to efficiently monitor the mean of normally distributed processes. In addition, to address measurement errors, the M-Shewhart-CUESUM chart is proposed, incorporating an additive measurement error model. Evaluation of the charts through Monte-Carlo simulations, considering metrics such as average run length (ARL), extra quadratic loss, relative ARL, and performance comparison index. It is found that the combined Shewhart-CUESUM outperforms than CUESUM chart. The results show that the presence of measurement errors can significantly diminish the charts’ performance, which can be mitigated by utilizing a multiple measurements scheme. Among the different well-established combined charts examined, the M-Shewhart-CUESUM chart shows considerably more sensitive to detecting simultaneously detect small and large size shifts. To employ simulated datasets to illustrate the impact of measurement errors and demonstrate the implications of the proposed charts on process mean shifts.

Generalized multivariate EWMA control chart supplemented with bivariate ranked set techniques

ABSTRACT Existing univariate exponential weighted moving average (EWMA) control charts with ranked set schemes are used for monitoring related characteristics independently. Also, existing multivariate EWMA control charts are generally applicable for simple random sampling (SRS) based processes. In this work, a generalized multivariate EWMA control chart is proposed with bivariate ranked set techniques (BVRST) for simultaneously monitoring. Furthermore, an extensive comparative analysis between proposed generalized multivariate EWMA and classical control chart is conducted. In this regard, we have used various performance measures, which cover average run length, median of run length, relative average run length, performance comparison index, and extra quadratic loss. The performance measures are obtained using the Monte Carlo simulation (MCS) approach. Results revealed that the proposed generalized multivariate EWMA control chart has an outstanding detection ability relative to the existing multivariate control charts. Additionally, existing multivariate control charts are found special cases of the proposed control chart. A real-life example from the agriculture field is included in which chemical properties (residual sodium and calcium-magnesium contents) of irrigation water are monitored through the control charts. The results revealed that the proposed control chart is proficient in diagnosing small variations in the chemical characteristics relative to the existing control charts.

An adaptive EWMA chart for Poisson process

ABSTRACT This paper introduces an adaptive EWMA (AEWMA) chart designed for monitoring Poisson processes. This chart offers a streamlined implementation process, eliminating the need for additional parameters. Notably, it demonstrates the ability to effectively detect a range of the mean shift sizes. Using Monte Carlo simulations, this study provides estimates for various performance metrics, including the zero-state and steady-state average run-length (ARL), expected weighted run-length, and expected relative ARL profiles for existing and proposed charts. The results indicate that the AEWMA chart surpasses the performances of existing EWMA, CUSUM, and two other AEWMA charts in detecting multiple shifts in the process mean. Additionally, an illustrative example is presented to showcase the practical implementation of the considered charts.

A new adaptive CUSUM control chart based on Hampel function to monitor location parameter

AbstractTo identify small‐to‐moderate shifts (also known as outliers and anomalies) in the process parameters (location and/or dispersion), the classical cumulative sum (CUSUM) and exponential weighted moving average (EWMA) control charts are famous. The traditional CUSUM control chart is less capable to detect out‐of‐control signals of 1.0 standard deviation shifts. The advanced versions of the traditional CUSUM and EWMA control charts are ACUSUM (adaptive CUSUM) and AEWMA (adaptive EWMA) control charts, respectively. The aforementioned issue of the traditional CUSUM control chart can be addressed through the ACSUUM control chart. It also identifies various sizes of shifts in the process parameters. In order to efficiently identify and remove outliers and anomalies (i.e., shifts) from datasets, the score functions are typically used to establish ACUSUM control charts. The goal of the study is to propose a new ACUSUM control chart based on the Hampel (score) function and it is symbolized as control chart. Different sizes of shift in the process location parameter can be distinguished using the control chart. The primary novelty is the substitution of a proposed time‐varying parameter for the typical CUSUM control chart's prioritization of the process characteristic by appropriate weights. The proposed control chart's run length feature is obtained through the Monte Carlo simulation technique for zero‐state and steady‐states. For evaluating a single shift, performance measures like average run length (ARL), standard deviation of run length (RL), and different percentiles of run length are preferred. However, for evaluating a range of shifts, extra quadratic loss, relative average run length, and performance comparison index are used. Comparison based on performance measures shows the superiority of the proposed control chart against the classical CUSUM and EWMA, ACUSUM, AEWMA, mixed EWMA‐CUSUM, and mixed CUSUM‐EWMA control charts. Furthermore, the proposed control chart is also implemented with real‐life data which is taken from cost of medical insurance to demonstrate its competency in statistical process control field for quality engineers, practitioners, experts, and researchers.

Integration of hybrid score functions to design new adaptive EWMA control chart to monitor the process mean shift

Small-to-moderate shifts in the process parameters can be detected using the traditional Exponentially Weighted Moving Average (EWMA) control chart. The classical EWMA control chart's parameter helps only to identify the specific size of the shift. However, various shifts occur in real-world processes, so the conventional EWMA control chart performs inefficiently for off-target shifts. This problem has been confronted by applying an adaptive EWMA (AEWMA) control chart, an advanced version of the classical EWMA control chart. The score functions are utilized to design the AEWMA control chart. Generally, the Huber and Bi-square (score) functions have unique characteristics, such as monotone and redescended, respectively.There has not been any investigation into the simultaneous and/or joint deployment of score functions. So, by taking this point as motivation, this study aims to propose a new score function by blending the Huber and Bi-square characteristics into one hybrid formulation and then utilize it to design a new AEWMA control chart to monitor different sizes of the shift in the process location parameter. To evaluate the proposed control chart performance against other control charts, Monte Carlo simulation is applied to obtain the run length properties under zero-state and steady-state environments. Performance measures, such as average run length, standard deviation of run length, and median of run length, are considered to evaluate the performance at a single shift. In contrast, extra quadratic loss, relative average run length, and performance comparison index are utilized to evaluate effectiveness for a specific range of shifts. The comparison based on performance measures and visual presentation reveals the superiority of the proposed control chart. Besides, the proposed control chart is employed in the real-world chemical engineering process industry problem, demonstrating excellent detection ability of anomalies in air pollutant data. The comparison based on performance measures and visual presentation reveals the good performance of the proposed control chart.

A maximum dual CUSUM chart for joint monitoring of process mean and variance

ABSTRACT A dual chart provides more sensitivity than the conventional chart when it is known that a shift size varies within a given interval. In this paper, we propose a maximum dual CUSUM (MDC) chart for monitoring the joint shifts (that lie in different intervals) in the mean and variance of a normally distributed process. The Monte Carlo simulation method is used to estimate the zero-state and steady-state run-length properties of the MDC chart, which include the average run-length (ARL), expected weighted run-length (EWRL) and expected relative ARL (ERARL). Based on detailed run-length comparisons in terms of the EWRL and ERARL, it is found that the MDC chart outperforms the maximum adaptive EWMA (MAE) and maximum weighted adaptive CUSUM (MWAC) charts when detecting a range of the joint shift sizes. Moreover, the diagnostic abilities of the MDC chart are also studied. Real and simulated datasets are considered to demonstrate the implementation of the MAE, MWAC and MDC charts.

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.

On Designing of Bayesian Shewhart-Type Control Charts for Maxwell Distributed Processes with Application of Boring Machine

The quality characteristic(s) are assumed to follow the normal distribution in many control chart constructions, although this assumption may not hold in some instances. This study proposes the Bayesian-I and Bayesian-II Shewhart-type control charts for monitoring the Maxwell scale parameter in the phase II study. The posterior and predictive distributions are used to construct the control limits for the proposed Bayesian-I and Bayesian-II Shewhart-type control charts, respectively. Various performance indicators, including average run length, quadratic loss, relative average run length, and performance comparison index, are utilized to evaluate the performance of the proposed control charts. The Bayesian-I and Bayesian-II Shewhart-type control charts are compared to their competitive CUSUMV, EWMAV and V control charts. Sensitivity analysis is also performed to study the effect of hyperparameter values on the performance behavior of the proposed control charts. Finally, real-life data is analyzed for the implementation of the proposed control charts.

An adaptive EWMA control chart based on Hampel function to monitor the process location parameter

AbstractThe classical memory Exponential Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) control charts (CCs) are important to detect small‐to‐moderate sizes of shift in the process location and/or dispersion parameters. To make the memory control chart (CC) more efficient for broad range of shift, the score functions in particular Huber and Bi‐square functions are incorporated in the classical memory CCs. These score functions help to formulate the parameters of memory CCs and are called Adaptive EWMA (AEWMA) and Adaptive CUSUM (ACUSUM). The AEWMA and ACUSUM are capable to handle various sizes (i.e., small, moderate, and large) of shift. The aim of this study is to utilize Hampel function as the alternative of the mentioned score functions, and propose a new AEWMA CC, symbolized as AEWMAHample. To develop the proposed AEWMAHampleCC, the standard EWMA statistic and Hampel function are incorporated in the classical EWMA CC structure. The proposed AEWMAHampleCC is also efficient to handle certain sizes of shift in the process location parameter. Performance evaluation procedures, like average run length is considered for a particular shift. Similarly, the performance comparison index, relative average run length, and extra quadratic loss function are estimated over a range of shift. All these performance measures are calculated from run length (RL) characteristic of the proposed control chart. An algorithm in MATLAB is developed based on Monte Carlo simulation procedure to generate the RLs. Some existing control charts including EWMA, CUSUM, MEC, MCE, , , IACCUSUM, , and are taken for comparison reason. For specific sizes of shift, the proposed AEWMAHampleCC reveals the superiority. Besides, to address the practical significance for practitioners and quality engineers, the real‐life and simulated data examples are considered.

Enhanced fault detection models with real-life applications

<abstract> <p>Nonconforming events are rare in high-quality processes, and the time between events (TBE) may follow a skewed distribution, such as the gamma distribution. This study proposes one- and two-sided triple homogeneously weighted moving average charts for monitoring TBE data modeled by the gamma distribution. These charts are labeled as the THWMA TBE charts. Monte Carlo simulations are performed to approximate the run length distribution of the one- and two-sided THWMA TBE charts. The THWMA TBE charts are compared to competing charts like the DHWMA TBE, HWMA TBE, THWMA TBE, DEWMA TBE, and EWMA TBE charts at a single shift and over a range of shifts. For the single shift comparison, the average run length (ARL) and standard deviation run length (SDRL) measures are used, whereas the extra quadratic loss (EQL), relative average run length (RARL) and performance comparison index (PCI) measures are employed for a range of shifts comparison. The comparison reveals that the THWMA TBE charts outperform the competing charts at a single shift as well as at a certain range of shifts. Finally, two real-life data applications are presented to evaluate the applicability of the THWMA TBE charts in practical situations, one with boring machine failure data and the other with hospital stay time for traumatic brain injury patients.</p> </abstract>

Effect of measurement uncertainty on combined quality control charts

The accuracy of the measurement system is vital for reliable process monitoring using statistical process control charts. The applied chart’s effectiveness depends on the measurement system's performance. Measurement uncertainty can lead to incorrect decisions like unnecessary stops or failure to intervene. In this paper, we investigated the effect of measurement errors on the performance of four well-established combined charts for monitoring the mean of normally distributed processes: Shewhart-CUSUM, Shewhart-Crosier’s CUSUM, Shewhart-EWMA and Shewhart-GWMA charts. To deal with measurement errors we considered the additive measurement error model. Detailed run length profiles of these charts are studied in terms of average run length (ARL), extra quadratic loss, relative ARL, and performance comparison index through Monte Carlo simulations under different sizes of measurement errors. It was found that measurement errors significantly reduce the power of the combined charts. Thus, multiple measurements scheme is incorporated as a remedy to this effect. The Shewhart-Crosier’s CUSUM performed best of four charts, while the Shewhart-EWMA chart did worst. To demonstrate the effect of measurement uncertainty and highlight implications further, a simulated dataset with a shift in the process mean is considered.

Improved adaptive EWMA control chart for process location with applications in groundwater physicochemical parameters and glass manufacturing industry.

The adaptive exponentially weighted moving average (AEWMA) control charts are the advanced form of classical memory control charts used for efficiently monitoring small-to-large shifts in the process parameters (location and/or dispersion). These AEWMA control charts estimate the unknown shifts using exponentially weighted moving average (EWMA) or cumulative sum (CUSUM) control charts statistics. The hybrid EWMA (HEWMA) control chart is preferred over classical memory control charts to detect early shifts in process parameters. So, this study presents a new auxiliary information-based (AIB) AEWMA (IAEWMAAIB) control chart for process location that estimates the unknown location shift using HEWMA statistic. The objective is to develop an unbiased location shift estimator using HEWMA statistic and then adaptively update the smoothing constant. The shift estimation using HEWMA statistic instead of EWMA or CUSUM statistics boosts the performance of the proposed IAEWMAAIB control chart. The Monte Carlo simulation technique is used to get the numerical results. Famous performance evaluation measures like average run length, extra quadratic loss, relative average run length, and performance comparison index are used to evaluate the performance of the proposed chart with existing counterparts. The comparison reveals the superiority of the proposed control chart. Finally, two real-life applications from the glass manufacturing industry and physicochemical parameters of groundwater are considered to show the proposed control chart’s implementation procedure and dominance.

Enhanced CUSUM and dual CUSUM mean charts

The CUSUM (C) and dual C (DC) charts are well-known because of their sensitive nature against small-to-moderate shifts that occur in the process parameter(s). In this study, we propose new auxiliary information based (AIB) control charts for monitoring the process mean, which include Brownian motion-based C (BC), new BC (NBC), NBC with Crosier C (NBCC), dual NBC (DNBC) and dual NBCC (DNBCC) charts. In addition, fast initial response feature is also added to the proposed charts. We use Monte Carlo simulations to compute the run length properties of the control charts. Based on detailed run length comparisons, it is found that the proposed AIB charts are uniformly better than the existing AIB charts in terms of expected weighted and relative average run length benchmarks. An important concern all around the world is the quality and safety of drinking water. Several water quality tests are available that provide required information about the health of the waterway. Here, a water quality test dataset is taken on two variables, namely pH and total solids, and the proposed AIB charts are used to monitor the mean level of pH in the water.

Design Tukey’s Control Chart and mix with CUSUM Control Chart

Statistical process control is a collection of valuable tools for detecting alteration in a process. It has wide application in many areas field and other fields where variation is being monitored. The variation may be a natural cause variation or a particular cause variation. Statistical process control deals with the monitoring process to detect disturbances in the process. These disturbances may be from the process mean or variance. This study proposes efficient charts for detecting early shifts in dispersion parameters by applying the Fast Initial Response feature. We propose and compare the performance of different cumulative sum (CUSUM)control charts for phase II monitoring of location based on mean and median. The (CUSUM) control chart, which is a method of data analysis based on John Tukey's principles control chart (TCC), is used to compare the proposed charts with their existing counterparts is used to evaluate new charts to existing charts using performance measures such as average run length, the standard deviation of run length, additional quadratic loss, relative average run length, and performance comparison . The proposed charts detect early shifts in the process dispersion faster and have better overall. This article is a similar effort to design an improved charting structure in the form of mixed or using Tukey -CUSUM chart together, to show the process control chart., and drawing the Average Run Length ARL value.

Efficient Auxiliary Information–Based Control Charting Schemes for the Process Dispersion with Application of Glass Manufacturing Industry

The hybrid exponentially weighted moving average (HEWMA) control chart is an enhanced version of the EWMA control chart that monitors the process parameters effectively. Similarly, the auxiliary information-based (AIB) EWMA control charts are very efficient for monitoring process parameters. The purpose of this paper is to propose two new control charts for the improved monitoring of process dispersion referred to as HEWMA AIB 1 and HEWMA AIB 2 control charts. A simulation study is carried out to assess the performance of the proposed HEWMA AIB 1 and HEWMA AIB 2 control charts. Average run length, extra quadratic loss, relative average run length, and performance comparison index are used to compare the performance of the proposed control charts against the existing counterparts. The comparisons reveal the superiority of the proposed control charts against other competing control charts, particularly for small to moderate shifts in the process dispersion. Finally, a real-life data set from the glass industry is used to demonstrate the practical implementation of the proposed control charts.

Adaptive CUSUM control charts for efficient monitoring of process dispersion

AbstractAn adaptive cumulative sum (ACUSUM) control chart is an advance form of the classical CUSUM control chart to identify different sizes of shift in the process parameters (location and/or dispersion). In the continuation of the ACUSUM control chart, this study also proposed ACUSUM control charts to enhance the performance of the process dispersion for a broad range of shift. The proposed ACUSUM control charts methodologies are based on the dispersion CUSUM statistic, score (Huber and Bi‐square) functions, generalized likelihood ratio test, and nonlinear optimization technique. The use of dispersion CUSUM statistic helps to distinguish a specific shift as the classical CUSUM statistic does, the score functions, the nonlinear optimization technique and generalized likelihood ratio test enhance the ability of the proposed ACUSUM control charts to distinguish a shift of different size. For the assessment of control charts, run length (RL) measure is used, and it is generated by developing an algorithm in MATLAB through Monte Carlo simulation method. Further, average of RL is utilized to carry out comparative analysis of control charts for a single shift, while for a certain range of shift (comprehensive analysis), extra quadratic loss (EQL), relative average run length (ARL), and performance comparison index (PCI) measures are used. Findings based on numerical results and visual presentations reveal the superiority of the proposed control charts against some existing control charts. Moreover, for real‐world point of assessment, the proposed control chart is implemented with numerical data to show the worth over other control charts.

Violating the standard assumption underlying the process monitoring: Perfect measurements

AbstractThe presence of the measurement errors is a rule rather than an exception in any process monitoring context. Since the observations are generally observed, in quick manner, by some measuring process and are time ordered. Moreover, their presence contributes negatively towards the performance of the usual control charting schemes. Therefore, it will be naïve and may lead to invalid conclusions to devise a control structure, without taking them into account, at first place, and then counteracting their adverse effect. This paper studies the foregoing impact on the performance of two eminent charts, for mean or average parameter of a process, namely: Crosier CUSUM and Dual CUSUM charts. With growing intensity of the error variation, a deteriorating performance of the said charts has been observed. The average run length (ARL) behaviors, zero‐state, and steady‐state, of both charts are studied, using Monte‐Carlo simulation algorithm, under different situations for the effect of measurement errors. Also, extra quadratic loss, relative ARL and performance comparison index (abbreviated as EQL, RARL, and PCI, respectively) are used to evaluate the performance of the charts. Repeated measurements are employed, as a remedial scheme, for the effect of measurement errors. An illustrative example is incorporated to elucidate the study.

New Hybrid EWMA Charts for Efficient Process Dispersion Monitoring with Application in Automobile Industry

The EWMA charts are the well-known memory-type charts used for monitoring the small-to-intermediate shifts in the process parameters (location and/or dispersion). The hybrid EWMA (HEWMA) charts are enhanced version of the EWMA charts, which effectively monitor the process parameters. This paper aims to develop two new uppersided HEWMA charts for monitoring shifts in process variance, i.e., HEWMA1 and HEWMA2 charts. The design structures of the proposed HEWMA1 and HEWMA2 charts are based on the concept of integrating the features of two EWMA charts. The HEWMA1 and HEWMA2 charts plotting statistics are developed using one EWMA statistic as input for the other EWMA statistic. A Monte Carlo simulations method is used as a computational technique to determine the numerical results for the performance characteristics, such as average run length (ARL), median run length, and standard deviation run length (SDRL) for assessing the performance of the proposed HEWMA1 and HEWMA2 charts. In addition, to evaluate the overall performance of the proposed HEWMA1 and HEWMA2 charts, other numerical measures consisting of the extra quadratic loss (EQL), relative average run length (RARL), and performance comparison index (PCI) are also computed. The proposed HEWMA1 and HEWMA2 charts are compared to some existing charts, such as CH, CEWMA, HEWMA, AEWMA HHW1, HHW2, AIB-EWMA-I, and AIB-EWMA-II charts, on the basis aforementioned numerical measures. The comparison reveals that the proposed HEWMA1 and HEWMA2 charts achieve better detection ability against the existing charts. In the end, a real-life data application is also provided to enhance the implementation of the proposed HEWMA1 and HEWMA2 charts practically.