Terms Of Average Run Length Research Articles

Control charts using half-normal and half-exponential power distributions using repetitive sampling

This manuscript presents the development of an attribute control chart (ACC) designed to monitor the number of defective items in manufacturing processes. The charts are specifically tailored using time-truncated life test (TTLT) for two lifetime data distributions: the half-normal distribution (HND) and the half-exponential power distribution (HEPD) under a repetitive sampling scheme (RSS). To assess the effectiveness of the proposed control charts, both in-control (IC) and out-of-control (OOC) scenarios are considered by deriving the average run length (ARL). Various factors, including sample sizes, control coefficients, and truncated constants for shifted phases, are taken into account to evaluate the performance of the charts in terms of ARL. The behavior of ARLs is analyzed in the shifted process by introducing shifts in its parameters. The superiority of the HEPD-based chart is highlighted by comparing it with both the HND-based ACC and the ACC based on the Exponential distribution (ED) under TTLT using RSS. The results showcase the superior performance of the proposed HEPD-based chart, indicated by smaller ARL values. Additionally, the benefits of another proposed ACC using HND are compared with the ED-based ACC under RSS, further confirming the effectiveness of the HND-based approach through smaller ARLs Finally, the proposed control charts are evaluated through simulation testing and real-life implementation, emphasizing their practical applicability in real-world manufacturing settings.

Remedial approaches to decrease the effect of measurement errors on polynomial profile monitoring

In the application of quality control charts, most of the researches in profile monitoring are based on accurate measurements. Measurement errors, however, often exist in many manufacturing and service environment. In this paper, first, we investigate the effect of neglecting measurement errors on the performance of four common control charts for Phase II monitoring of polynomial profiles. Simulation studies in terms of average run length (ARL) metric show that neglecting the measurement errors adversely affects the capability of all mentioned control charts. Then, we reconstruct the control charts based on two consistent estimators and two remedial approaches including multiple measurement and increasing sample size are utilized to decrease the effect of measurement errors. The results show that the remedial approaches compensate for the mentioned effect to a certain extent.

Control chart for half normal and half exponential power distributed process

In this manuscript, we construct an attribute control chart (ACC) for the number of defective items using time-truncated life tests (TTLT) when the lifetime of a manufacturing item follows two lifetime data distributions: the half-normal distribution (HND) and the half-exponential power distribution (HEPD). To assess the potential of the proposed charts, necessary derivations are made to obtain the value of the average run length (ARL) when the production process is in-control and out-of-control. The performance of the presented charts is evaluated for different sample sizes, control coefficients, and truncated constants for shifted phases in terms of ARL. The behavior of ARLs is studied for the shifted process by introducing shifts in its parameters. The advantages of the proposed HEPD-based chart are discussed in the form of ARLs with HND and Exponential Distribution (ED) based ACCs under TTLT, showing the excellent assessment of the proposed chart. Additionally, the advantages of another proposed ACC using HND are compared with ED-based ACC, and the findings support the HND in the form of smaller ARLs. Finally, simulation testing and real-life implementation are also discussed for functional purposes.

On efficient change point detection using a step cumulative sum control chart

Control charts are widely used to monitor the stability of processes and Shewhart is the most commonly used type of chart because of the simplicity of its structure. To improve the performance, there are several modifications on control chart suggested since its proposal like cumulative sum and exponentially weighted moving average control charts. The cumulative sum chart accumulates all the deviations from target mean higher than a specific design parameter which can be used to controls the sensitivity of this chart. The optimal choice of this design parameter is half of the target shift size. This results into an optimal performance in terms of quick detection of shift that is equal to the target shift size. For shift sizes other than target, the cumulative sum chart loses its performance and becomes inferior to other control charts in terms of average run length. Since we rarely know the process shift size that is going to occur in future, this study proposes a revision in the structure of cumulative sum chart. The adjusted cumulative sum chart is shown to be very flexible for a wide range of shifts, when designed against a target shift size. The proposed adjusted cumulative sum control chart is compared with cumulative sum, exponentially weighted, combined Shewhart-CUSUM, progressive mean, double progressive mean and mixed EWMA-CUSUM control charts in terms of zero- and steady-state average run lengths. The superiority zones are identified and the results are supported by a real-life implementation of the proposed scheme.

On updating the results of the control chart using a modified EWMA statistic

AbstractKhan et al. presented the results using the approximation for the control chart using a modified exponentially weighted moving average (EWMA) statistic. In this paper, we will correct the results using Monte Carlosimulation. We will show that modified EWMA is more efficient than traditional EWMA in terms of average run length.

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.

A generalized multiple dependent state sampling chart based on ridge penalized likelihood ratio for high-dimensional covariance matrix monitoring

Online monitoring of high-dimensional processes variability in which the number of variables is larger than the sample size is a challenging issue for quality practitioners because the sample covariance matrix is not invariable. To deal with this challenge, a generalized multiple dependent state sampling (GMDS) chart based on ridge penalized likelihood ratio (RPLR) statistic is developed for Phase II monitoring of multivariate process variability under high-dimensional setting. The developed control chart benefits from three advantages: (1) departing from the conventional covariance matrix charts, it can be efficiently employed for both spars and non-spars covariance matrices; (2) it is able to detect spars shift patterns in which only a few covariance matrix elements are deviated from their nominal values; and (3) it outperforms the detectability of the RPLR chart in terms of average run length (ARL) and standard deviation of run length (SDRL). The performance of RPLR, MDS-RPLR, and GMDS-RPLR charts are compared using extensive simulation studies by considering different diagonal and/or off-diagonal covariance matrix disturbance. Moreover, sensitivity analysis are provided to analyze how the number of process variables and GMDS parameter affect the run length properties of the developed chart.

Modified calibrated control charts for monitoring the population mean under stratified sampling

AbstractIn this research paper, we discuss the use of calibration technique to propose some new estimators under stratified sampling design and its use in reliability engineering. We first proposed the estimators using the calibration technique and then modify the already existing control charts such as Shewhart, exponentially weighted moving average and hybrid exponentially weighted moving averages using these calibration estimators. We also have studied the use of the proposed control charts in detail by showing their efficiency in terms of average run length. We find out that a hybrid control chart is more effective to detect the early shifts in the mean of the process.

Reintegration of auxiliary information based control charts

During the last decade, auxiliary information based (AIB) control charts have gained a lot of popularity for process monitoring. These charts are proved efficient for early detection of shifts in process parameters like location and dispersion. A recent study has pointed out few concerns related to AIB control charts which were already highlighted in earlier studies one way or the other. In reply to that, we reiterate the initial classification of processes by Hawkins (1993) namely non-cascading processes where the change in one variable may not affect the other variable. Under the non-cascading situations, AIB control charts are designed with the assumption that auxiliary variable remains stable even if there is a shift in study variable. This assumption is true for some processes. On the other hand, in many non-cascading processes where the auxiliary variable can experience a shift, the performance of AIB control charts is highly affected specially when the auxiliary variable(s) and study variable are strongly correlated. This study proposes a modified AIB control charting structure that safeguards the detection ability and the run length properties in case auxiliary variable experiences a shift. In addition, the usual AIB and classical Shewhart charts become a special case of the proposed chart. For several values of design parameter, the in control and out of control performance of proposed chart is evaluated in terms of average run length. It is observed that the proposed modified AIB control chart is capable of increasing its robustness against any shift in the auxiliary variable through a design parameter. An extensive comparative analysis revealed that the newly proposed modified AIB control chart has capability to absorb a shift of any size in auxiliary variable, maintaining the gain in terms of efficiency. An illustrative example is also included to support the finding of our study.

Double Moving Average Control Chart for Autocorrelated Data

The assumption of normality and independence is necessary for statistical inference of control charts. Misleading results are obtained if the traditional control chart technique is applied on the auto-correlated data. When data is correlated, a time series model is employed to produce an optimum output. The objective is to create a new control chart methodology that takes the autocorrelation of observations into account. Charts of Moving Average, Exponentially Weighted and Cumulative Sum better perform in the existence of autocorrelation data for small and moderate changes. The proposed methodology is highly skilled and receptive to minor, moderate and major changes in the process. Propsed DMA chart increases efficiency of average run length (ARL) chart for moving average (MA) to detect the small to medium magnitude shifts in the mean. The simulation also demonstrates that the DMA chart with spans of w=10 and 15 generally performs well in terms of average run length (ARL) as compared to clasical MA. This research may be extended to a multivariate autocorrelated statistical process control, but it can also be used to recognise and categorise seven categories of traditional control chart patterns, such as Downward, Upward Shift, Normal Trend, Cyclic, Systematic patterns, Increasing and Decresing Trend. In order to identify and categorize a set of subclasses of abnormal patterns, this model (multivariate autocorrelated statistical process control chart) should employ a multilayer feed forward Artificial Neural Network (ANN) architecture controlled by a back-propagation learning rule.

Attribute Control Chart for Rayleigh Distribution Using Repetitive Sampling under Truncated Life Test

A control chart is an important tool in statistical process monitoring that is useful to monitor and improve production process quality. In this article, an attribute control chart using repetitive sampling under a truncated life test is proposed for monitoring the mean life of the product where the lifetime follows the Rayleigh distribution. The repetitive sampling parameters and the control limit coefficients of the chart are determined so that the in-control average run length (ARL) is very close to the target ARL. Tables of ARL values for various shift sizes in the scale parameter were presented, and the performance of the proposed chart is compared with the existing attribute control charts using the out-of-control ARL. The proposed control chart is shown to outperform the existing control charts in terms of ARL. An illustrative example is given to demonstrate the application of the proposed chart.

Performance of the Shiryaev‐Roberts‐type scheme in comparison to the CUSUM and EWMA schemes in monitoring weibull scale parameter based on Type I censored data

AbstractThis paper introduces a process monitoring scheme based on the Shiryaev‐Roberts (SR) procedure for Type I right‐censored Weibull lifetime data for detecting shifts in the scale parameter for a fixed value of the shape parameter. The performance of the new charting scheme is evaluated under different censoring rates and sample sizes via Monte Carlo simulation. Comparison with two existing schemes, namely, the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) schemes, is performed in terms of average run length (ARL). Both zero‐state and steady‐state shifts are considered in this study for impartial comparisons. The numerical results based on simulation are highly encouraging in favour of the SR procedure. Our findings show that the new scheme outperforms the other two schemes in some cases. An illustrative example is given to illustrate the application of the proposed scheme.

An approach for improvements on the attribute npx$np_x$ control chart

AbstractThis paper proposes an attribute control chart named which is used for monitoring the process mean value of a variable (X) by employing sequential sample sizes of , with . Each item is classified as approved or not according to a discriminant limit (, ) using a GO/NO GO gauge. In the end of inspection we'll have and items classified as disapproved. Whenever or , the process is judged to be out of control and after adjustments in the process, the inspection is always restarted with a sample size of . The parameters used in the construction of the control chart were obtained through a search for values that optimize their performance, such that can compete with the standard attribute chart and the traditional control chart. The performances were compared in terms of average run length () in scenarios of shift sizes (δ) in the process mean. A numerical example illustrates its application.

The Performance of Triple Sampling x̄ Control Chart with Measurement Errors

ABSTRACT The presence of measurement errors can seriously alter the statistical performance of Phase II control charts. Up today, no research on designing the triple sampling control charts taking into account the gauge measurement errors is reported in the existing literature. In this paper, we study the adverse effect of measurement errors on detecting performance of triple sampling (TS)- control chart based on an additive covariate model. Three multiple measurement based triple sampling (MMBTS) schemes are developed to reduce the undesired impact of gauge inability on detecting performance of TS- chart. Through simulation studies in terms of average run length (ARL) and standard deviation of run length (SDRL), it is indicated that the run length characteristics of the TS- is significantly affected by the measurement errors. The results also confirm that all proposed remedial approaches can effectively reduce the undesired impact of imprecise measurements on performance of TS- chart. A sensitivity analysis is also carried out to evaluate how the covariate model parameters affect the detection performance of the TS- chart. Finally, using a real industrial data obtained from the spring production system, we demonstrate the performance of TS- chart when the measurement errors exist.

Generally weighted moving average control chart for monitoring two-parameter exponential distribution with measurement errors

Control charts for two-parameter exponential distributions are widely used in monitoring quality and lifetime processes, and more useful and flexible than the chart for one parameter exponential distributions. In order to efficiently detect small shifts for two-parameter exponential distributions, a generally weighted moving average (GWMA) control chart is developed. Since measurement errors are inevitable in practice, and they are not considered in the existing control charts, we proposed a GWMA chart with measurement errors (GWMA-M) by two strategies: one is the GWMA chart with adjusted control limits, and the other is the GWMA chart with additive measurement error model to deal with measurement errors. The performance of the proposed GWMA-M scheme is investigated in terms of average run length (ARL) and expected ARL (EARL). Comparisons of the GWMA-M chart with the EWMA-M chart and the GWMA signed-rank (GWMA-SR) chart in the presence of measurement errors are carried out, and results show the advantage and efficiency of the GWMA-M scheme. In addition, the effect of model parameters, measurement errors and sample size on the GWMA-M chart are studied through Monte-Carlo simulation. Finally, two real examples of monitoring flood data and HPM (high-voltage metal oxide semiconductor transistor) data are given to illustrate the implementation of the proposed chart.

Data Consumption Analysis by Two Ordinal Multivariate Control Charts

The process quality is described by one or more important factors called multivariate processes. Contingency tables used to demonstrate the relevance between these factors and modeled by log-linear model. There are also two types of statistical variables that are nominal and ordinal. In this paper, the variables are ordinal and two new control charts have been used to monitor the process of analyzing subscribers' consumption. These two multivariate ordinal chart are the MR chart and the multivariate ordinal categorical (MOC) used to monitor processes based on the ordinal log-linear model in Phase II. In addition, with a real numerical example, about analyzing the internet usage of mobile subscribers, two control charts are drawn and compared with each other in terms of average run length. In this case, we focus on customer behavior and in real action, by marketing department, changing in data consumption has been seen and analyzed. The study of the two proposed charts was performed using simulation based on real example in different situation, and the MOC performed relatively better.

Klein's supplementary runs rules on X¯$\overline{X}$ chart to control the mean of a Weibull random variable

AbstractMost of the papers found in the literature concerned monitoring a process mean of a quality characteristic normally distributed. However, in several practical situations (like in reliability studies), the engineering team also desires to monitor a process mean in which the quality characteristic follows other probabilistic distributions, such as the Weibull. In this paper, we monitor a process mean of a random variable following a Weibull distribution using the traditional control chart. To improve the performance, Klein's two of two supplementary run rule is included in the chart. Such adoption promotes better performance, producing faster signals of abnormal situations (in terms of average run length) than the traditional chart in a great variety of shifts. An empirical example illustrates the proposed scheme.

On designing a new VEWMA control chart for efficient process monitoring

This study proposes a new memory type control chart named V-exponentially weighted moving average (VEWMA) control chart to monitor Maxwell distributed quality characteristics. We evaluated the performance of the chart using different run-length properties, including average run length (ARL), the median of run-length (MDRL), the standard deviation of run-length (SDRL), and some useful percentiles of the run-length. A comparative analysis is conducted with the existing V -cumulative sum (VCUSUM) and V charts for Maxwell distribution. We observed that the overall performance of the proposed chart is superior as compared to other control charts for the Maxwell distribution in terms of ARL and MDRL. Moreover, the VEWMA chart outperforms the V chart for small shifts, while for larger shifts the V chart dominates. We have provided a simulation study and a real-life example based on the Maxwell distribution to monitor the lifetime of a vertical boring machine of an ongoing process.

Bootstrap beta control chart for monitoring proportion data

PurposeAs an alternative to the standard p and np charts along with their various modifications, beta control charts are used in the literature for monitoring proportion data. These charts in general use average of proportions to set up the control limits assuming in-control parameters known. The purpose of the paper is to propose a control chart for detecting shift(s) in the percentiles of a beta distributed process monitoring scheme when in-control parameters are unknown. Such situations arise when specific percentile of proportion of conforming or non-conforming units is the quality parameter of interest.Design/methodology/approachParametric bootstrap method is used to develop the control chart for monitoring percentiles of a beta distributed process when in-control parameters are unknown. Extensive Monte Carlo simulations are conducted for various combinations of percentiles, false-alarm rates and sample sizes to evaluate the in-control performance of the proposed bootstrap control charts in terms of average run lengths (ARL). The out-of-control behavior and performance of the proposed bootstrap percentile chart is thoroughly investigated for several choices of shifts in the parameters of beta distribution. The proposed chart is finally applied to two skewed data sets for illustration.FindingsThe simulated values of in-control ARL are found to be closer to the theoretical results implying that the proposed chart for percentiles performs well with both positively and negatively skewed data. Also, the out-of-control ARL values for the percentiles decrease sharply with both downward and upward small, medium and large shifts in the parameters. The phenomenon indicates that the chart is effective in detecting shifts in the parameters. However, the speed of detection of shifts varies depending on the type of shift, the parameters and the percentile being considered. The proposed chart is found to be effective in comparison to the Shewhart-type chart and bootstrap-based unit gamma chart.Originality/valueIt is worthwhile to mention that the beta control charts proposed in the literature use average of proportion to set up the control limits. However, in practice, specific percentile of proportion of conforming or non-conforming items should be more useful as the quality parameter of interest than average. To the best of our knowledge, no research addresses beta control chart for percentiles of proportion in the literature. Moreover, the proposed control chart assumes in-control parameters to be unknown, and hence captures additional variability introduced into the monitoring scheme through parameter estimation. In this sense, the proposed chart is original and unique.

The effect of gauge measurement errors on double sampling control chart

In recent years, the impact of measurement errors on the performance of various control charts have been well evaluated. However, as far as we know, the multiple sampling control charts in the literature have been proposed based on the assumption that the measurements are precise. In this paper, we investigate the effect of measurement errors with both constant and linearly increasing variance on the performance of double sampling (DS)- chart. We study the capability of multiple measurements procedure to reduce the undesired effect of imprecise measurements on detection capability of DS- chart. Using simulation experiments, we compare the performance of DS- and classical charts in terms of average run length (ARL) and standard deviation of run length (SDRL) metrics. The results show that the measurement errors negatively affect the performance of both DS- and classical charts. The comparison results also confirm that under measurement errors, the DS- control chart outperforms the classical chart in all out-of-control scenarios. Moreover, the results indicate that taking several measurements on each item can effectively reduce the undesired impact of error on chart’s performance. Finally, the undesired impact of measurement errors on detection capability of DS- control chart is illustrated using a real-life example.