Average Run Length Research Articles

Monitoring of zero-inflated COM-Poisson processes

This paper investigates control charts for count data with a significant number of zeros. The proposed models are generalized to handle data exhibiting different types of dispersion, i.e. equidispersion, overdispersion, and underdispersion. To this effect, the zero-inflated Conway–Maxwell Poisson (ZICMP) distribution is employed. A Shewhart and an exponentially weighted moving average (EWMA) control chart are developed, referred to as ZICMP-Shewhart and ZICMP-EWMA charts, respectively. The ZICMP distribution incorporates various distributions, including the Conway–Maxwell–Poisson (COM-Poisson) and zero-inflated Poisson (ZIP), Geometric, and Bernoulli distributions, as special cases. Consequently, the flexibility and versatility of the ZICMP distribution enhance the applicability of the proposed control charts, thereby providing practitioners with an adaptable tool suitable for various scenarios.The control charts are examined for their ability to detect upward shifts in the process mean level, and their statistical performance is evaluated in terms of the average run-length (ARL). Through a simulation study, we demonstrate that the ZICMP-Shewhart chart is more effective for monitoring over-dispersed data, while the ZICMP-EWMA chart is more suitable for under-dispersed data. Finally, we provide two real-life examples to illustrate the applicability of the proposed charts.

Hotelling T2 control chart based on minimum vector variance for monitoring high‐dimensional correlated multivariate process

AbstractMultivariate control charts are practical tools that simultaneously monitor several correlated quality characteristics in a process. Monitoring high‐dimensional data structures is challenging because, in most cases, the process sample size for monitoring parameters is greater than the number of process characteristics. Many researchers have used the multivariate Hotelling's T2 chart to monitor high‐dimensional data using the maximum‐likelihood methods (MLM) to estimate the covariance matrices. However, the multivariate Hotelling's T2chart based on MLM suffers from low statistical performance. In this paper, we proposed a multivariate Hotelling's T2 chart based on the minimum vector variance (MVV) and some regularized methods for monitoring high‐dimensional data structures. The performance of the proposed chart is evaluated in terms of the average run length (ARL). The results reveal the superiority of the proposed MVV Hotelling's T2 chart over the existing Hotelling's T2 charts for high‐dimensional correlated processes.

Flexible Control Limits for Interval-Valued Data Using Mixture Probability Distributions

Availability of data in different forms is a boon or bane as it may require different approaches to deal with. In fact, while the construction of control charts is straightforward if the data are real-valued, in many practical situations data are essentially interval-valued with observations often represented by minimum and maximum values. While there are only fewer attempts to develop control charts for interval-valued data, no simple-to-use approaches have been proposed to deal with such type of data. In this paper, considering the idea of the development of control limits for multi-stream processes, we propose to make use of the mixture probability distributions approach to determine control limits for the interval-valued data. The resulting control limits are flexible in nature as they are linked to the control limits of the overall distribution that is based on the average of the means of two upper and lower populations along with their respective standard deviations. We have studied and numerically evaluated the average run length (ARL) properties of the proposed control chart to demonstrate the ways of determining the control chart coefficients. The proposed control chart is then illustrated using a dataset and conclusions are drawn based on the results.

Investigating the average kiloelectron-volt emission of partially observed events in nuclear physics through distance weighted mean based censored control chart.

The average lifespan of particles, a crucial parameter in nuclear physics, is essential for identification purposes. Modern particle detectors excel at recognizing individual radioactive nuclei arrivals and their subsequent decay events. However, challenges arise when matching arrivals with departures, especially when departures are only partially observed. One inefficient approach involves conducting experiments with very low arrival rates to facilitate matching. The kiloelectron-volt E(keV) emission is obtained during this radio active process. This study focuses on the meticulous surveillance of the average keV emission from partially observed events within the domain of nuclear physics. To accomplish this, the methodology employs the statistical approach known as Distance Weighted Mean (DWM), integrated with the application of censored control charts. The utilization of censored control charts allows for the effective management of incomplete data, enabling researchers to make informed decisions despite potential limitations in observation. We propose a DWM based exponentially weighted moving average-cumulative sum (DWM-EC) control chart for monitoring kiloelectron-volt E(keV) data. The proposed charts is developed for Weibull lifetimes under type-I censoring. For the construction of an efficient control charting structure, we employed the conditional median (CM) methods. The goal is to find changes in the mean of Weibull lifetimes with censored data with known and estimated parameter conditions. The performance of the proposed DWM-EC chart is evaluated by the average run length (ARL). Besides a simulation study, a real-life data set on E(keV) related to the alpha decays of 177 Lutetium isotope is also discussed.

The multi-objective economic statistical design of the p-chart: NSGA II approach

A control chart is a crucially important tool in statistical process control that is essentially used to differentiate between assignable and chance causes of variation. With the advent of Six Sigma in the parlance of quality management, the control chart assumes more significance to hold the gain in the control phase after attaining process improvement in the improve phase. The sample size, sampling interval, and control limits’ multiplier are three important parameters required to design an effective as well as efficient control chart. Many approaches like economic design, statistical design, and economic statistical design of control charts have been studied by several researchers. All these approaches consider a single objective function. In this paper, we have proposed a multi-objective design of the p-chart, where we are minimizing both out-of-control Average Run Length ( ARL δ ) and the expected cost per cycle ( C E ). Non-dominated Sorting Genetic Algorithm II (NSGA II) is used to solve the proposed multi-objective model. The proposed approach is demonstrated with the help of two numerical examples and the corresponding results are found to be quite encouraging for minimizing both objectives in comparison to another pertinent model given in the literature. Sensitivity analysis has been carried out to give credence to the worth of the proposed approach.

A comparative analysis of mean charts assuming Weibull and generalized exponential distributions

This study discusses Shewhart control charts to monitor the mean of non-normally distributed data using extensive Monte Carlo simulations. In particular, Weibull, exponential, and generalized exponential distributions are used to conduct a systematic study. Different parameters settings are considered to evaluate the effectiveness of the Shewhart charts based on the normal approximation. The results show significant differences between the charts using generalized exponential and Weibull distributions. In particular, the generalized exponential distribution based chart achieves the desired average run length of 370 with a single threshold value across a wide range of shape parameter combinations. On the other hand, the Weibull distribution requires different threshold values depending on its shape parameter. These findings highlight the significance of the two most commonly used distributions in reliability studies and their impact on control chart performance.

A new omnibus SPRT chart for monitoring process mean and variability based on the average number of observations to signal

The recent development of the omnibus sequential probability ratio test (OSPRT) chart marks a significant contribution to the advancement of joint monitoring schemes. As the OSPRT chart is a variable-sample-size control chart, practitioners often wish to understand its inspection efficiency, i.e. the number of observations it samples before producing a signal. In this article, we propose two enhanced optimization designs for the OSPRT chart based on the average number of observations of signal (ANOS) and expected value of the ANOS (EANOS) metrics under deterministic and unknown shift sizes, respectively. The ANOS metric is central to our design as it perfectly combines both the average run length (ARL) and the average sample number. A comparative analysis reveals that the OSPRT chart outperforms four benchmarking control charts in terms of the ANOS and EANOS metrics. Finally, an implementation of the OSPRT chart is presented with a ball shear test dataset.

Pengendalian Kualitas Produksi Air Minum Dalam Kemasan Menggunakan Peta Kendali

The most popular drinking water product produced by PT.X is the 220 ml glass packaging product. This product experienced the most production defects at 0.086% of total production. Based on observations that have been made, there are several problems that cause defective products, such as damaged packaging and poor water quality. So PT.X in maintaining production quality must improve the process of maintaining quality control. The aim of this research is in line with the problems faced by PT.X, namely controlling the quality of bottled drinking water using a decision on belief control chart. Control charts are used to monitor whether product defect data is statistically controlled or not. One of the control charts used to monitor whether product defect data is statistically controlled or not is the Decision on Belief control chart, because the Decision on Belief control chart is more sensitive to data shifts so that faster in detecting data that goes outside control limits or is out of control. Based on the graph of the results of the decision on belief control chart, of the 25 data there are 24 data that are out of the upper control limit and the lower control limit, meaning that the decision on belief control chart is sensitive to data shifts in detecting out of control data. Based on the results of the average run length calculation, it is concluded that the decision on belief control chart is weak in detecting out of control data because the shift value obtained is getting bigger.

Monitoring kiloelectron-volt emission variability in partially observed nuclear events through distance weighted censored control charts

The variation in the lifespan of particles is a crucial parameter in nuclear physics and is essential for identification purposes. Modern particle detectors excel at recognizing individual radioactive nuclei arrivals and their subsequent decay events. However, challenges arise when matching arrivals with departures, especially when departures are only partially observed. One inefficient approach involves conducting experiments with very low arrival rates to facilitate matching. The kiloelectron-volt E(keV) emission is obtained during this radio active process. This study focuses on the meticulous surveillance of the standard deviation in keV emission from partially observed events within the domain of nuclear physics.The utilization of censored control charts allows for the effective management of incomplete data, enabling researchers to make informed decisions despite potential limitations in observation. To accomplish this, the methodology employs the statistical approach known as Distance Weighted Mean based Standard Deviation (DWMS) integrated with the application of censored control charts. We propose a DWMS based exponentially weighted moving (DWMS-E) control chart for monitoring kiloelectron-volt E(keV) data. The proposed charts is developed for Weibull lifetimes with type-I censored data. The goal is to find changes in the mean of Weibull lifetimes with known and estimated parameter conditions. The performance of the proposed DWMS-E chart is evaluated by the average run length (ARL). Besides a simulation study, a real-life data set on E(keV) related to the alpha decays of 177 Lutetium isotope is also discussed.

On control charts based on generalized multiple dependent state sampling

ABSTRACT We clarify the use of the widely used multiple dependent state sampling (MDSS) technique and its generalization (GMDSS) with control charts. We show that these rules are simple modifications of well-known supplementary runs rules. The MDSS and GMDSS rules are inefficient; however, they do not distinguish between the upper and lower warning regions. We show that the application of the more conventional supplementary runs rules, i.e. those side-sensitive with respect to the warning regions, provides significantly better performance than the MDSS and GMDSS techniques. In addition, we show that a commonly used expression for determining the average run length of the GMDSS method is incorrect and provide a Markov chain alternative approach.

Control charts for monitoring the capability of unstable process

Stability is an important criterion to be investigated at the early stage of any industrial process and stable processes are usually preferred over unstable ones. But a process can be unstable and nevertheless capable and, therefore, there is a clear need to monitor such situations to ensure an acceptable capability. In this study we introduce, in a first step, a new Shewhart-type control chart for normal processes based on the yield index S pk to monitor the capability of potentially unstable processes and then, in a second step, we extend it using an Exponentially Weighted Moving Average (EWMA) scheme. Moreover, the optimal values of the control parameters are obtained using a nonlinear optimization problem for which the out-of-control average run length is the objective function. The introduced charts’ sensitivity is examined for deterministic shift sizes. Numerical and graphical analyses are given to display the features of the proposed charts. A simulation study shows, how in such a situation, the introduced tool is successful to recognize changes in the process capability.

Deep Learning-Adjusted Monitoring of In-Hospital Mortality after Liver Transplantation.

Background: Surgeries represent a mainstay of medical care globally. Patterns of complications are frequently recognized late and place a considerable burden on health care systems. The aim was to develop and test the first deep learning-adjusted CUSUM program (DL-CUSUM) to predict and monitor in-hospital mortality in real time after liver transplantation. Methods: Data from 1066 individuals with 66,092 preoperatively available data point variables from 2004 to 2019 were included. DL-CUSUM is an application to predict in-hospital mortality. The area under the curve for risk adjustment with Model of End-stage Liver Disease (D-MELD), Balance of Risk (BAR) score, and deep learning (DL), as well as the ARL (average run length) and control limit (CL) for an in-control process over 5 years, were calculated. Results: D-MELD AUC was 0.618, BAR AUC was 0.648 and DL model AUC was 0.857. CL with BAR adjustment was 2.3 with an ARL of 326.31. D-MELD reached an ARL of 303.29 with a CL of 2.4. DL prediction resulted in a CL of 1.8 to reach an ARL of 332.67. Conclusions: This work introduces the first use of an automated DL-CUSUM system to monitor postoperative in-hospital mortality after liver transplantation. It allows for the real-time risk-adjusted monitoring of process quality.

Design of a new attribute chart based on inspection run length

The conventional np chart detects shifts in fraction nonconforming of the process depending on the total number of nonconforming units, which is obtained until all of the units within a sample are completely inspected. To improve the effectiveness of inspection capacity, this paper presents a new attribute control chart, namely the Inspection Run Length (IRL) control chart, which detects shifts in fraction nonconforming of the process depending on a novel inspection strategy. This strategy employs information about the number of conforming and nonconforming units and inspection run length during the inspection process to curtail the number of units that need to be inspected. The optimal design parameters of the proposed control chart are derived based on optimising average run length (ARL) properties. In addition, the performance of the proposed IRL chart is evaluated and compared with the np chart and CRL-type charts. Comparison studies show that the proposed chart achieves a significant reduction in inspection time and cost and an improvement in performance (a lower out-of-control ARL) while holding the false alarm rate ( in-control ARL) at a specified level. Finally, two industrial examples are given to illustrate how to simply apply the proposed chart to the manufacturing process.

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.

New Lepage-type control charts for joint monitoring of location and scale

This paper introduces two new Shewhart Lepage-type control charts for jointly monitoring the location and scale parameters of continuous processes. The first is based on the van der Waerden (VW) and Ansari-Bradley (AB) tests while the other is on the VW and Mood tests. Statistical performances, in terms of average run length (ARL), of these charts are evaluated numerically and compared with that of their two competitor charts in the existing literature under four distributional models for the process output, namely, the normal, lognormal, Laplace, and logistic. The numerical results show that irrespective of the process distribution, one of the proposed charts (based on VW and Mood tests) has the best performance among all the four charts in the detection of simultaneous shifts in both the parameters of all sizes as well as in the detection of shifts in the scale that have not been accompanied by shifts in the location (only-scale shifts) of all sizes. Moreover, the other proposed chart (based on VW and AB tests) performs better than one of the competitor charts in the existing literature for detecting simultaneous shifts as well as only-scale shifts of all sizes. In addition, it performs better than the other competitor chart for detecting the simultaneous shifts consisting of a large location shift and a small scale shift, however, performs worse than that for detecting the simultaneous shifts consisting of a small location shift and a scale shift of any size, as well as for detecting the only-scale shifts of all sizes. All the four charts perform almost similarly in the detection of only-location shifts of all sizes. Regression models are provided for the proposed charts, which facilitate their designs in practice. Finally, the practical implementation of the proposed charts is illustrated through a real-data example.

Gauss–Legendre Numerical Integrations for Average Run Length Running on EWMA Control Chart with Fractionally Integrated MAX Process

Gauss–Legendre Numerical Integrations for Average Run Length Running on EWMA Control Chart with Fractionally Integrated MAX Process

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.

Explicit Formulas of Average Run Length for Triple Moving Average Control Chart to Monitor Changes in Mean Parameter of Normal and Non-normal Process

In this research, a formula is presented for calculating the average run length of a triple moving average (TMA) control chart, which is used to measure control chart performance and determine control chart width. To apply appropriately to data in various processes, a TMA control chart is used to detect small changes in a process by finding the average value of the data over a specified period of time (w). However, determining measure control chart performance and control chart width is difficult because the formula for calculation is not created. The performance of the control charts is measured using the standard normal distribution probability properties and compared with the moving average (MA) control charts and the double moving average (DMA) control charts. The analysis results show that TMA is very effective at detecting small changes, when w is set to have a small value. The study determines that the data within the process are normally and non-normally distributed. Detecting changes on real data by TMA charts has the best detection performance compared to MA and DMA.

The exponentially weighted control charts for geometrically moving standard deviations

AbstractThis article develops the control limits of Exponentially Weighted Moving Standard Deviation for Monitoring Gradual Shifts in Process variability, where the sample size n ≥ 2 stays constant in all subgroups. We have developed the Geometrically Moving control limits for the simple case of process standard deviation specified at σ0, and then for the case when σ is estimated from a subgroup of size m. We then obtained the conditional average run length by approximating the power function for testing the null hypothesis H0:σ = σ0 at the common sample sizes n = 4, 5, 6, 7, 8, 9, 10, 11 and 12. The results of this article are valid only if the normality assumption of underlying distribution is not violated.

A likelihood-based monitoring method for social networks with auto-correlated labeled Poisson model

Social network monitoring aims to detect any significant change in the structure of communications from the normal condition. The normal condition can be a set of networks considered as a reference set. This set is represented in dynamic cases by a probability model. Statistical process control methods can be applied to signal changes in the reference set. These signals can be useful for decision-makers in various applications, from marketing analysis to criminal investigations. In most of the developed models, the networks are assumed independent over time. However, this assumption may be questioned in real-world applications. Accordingly, this article formulates the social network as an auto-correlated labeled Poisson model and monitors it using a control method based on likelihood statistics. A simulation study is performed to evaluate the performance of the proposed methods in terms of the average run length criterion. The obtained results revealed that the proposed methods can reliably detect significant changes.