Variable Sampling Interval Research Articles

Another Look at the VSI EWMA x̄ Chart when Estimating Process Parameters

The performance of the variable sampling interval (VSI) exponentially weighted moving average (EWMA) chart is generally investigated under the assumption of known process parameters. Nevertheless, the process parameters need to be estimated from a historical Phase-I dataset because they are usually unknown in practice. When the process parameters are estimated, the chart’s performance differs among practitioners as different number of Phase-I samples is used. This leads to different parameter estimates in constructing the chart’s limits and variation in the average time to signal (ATS). This type of variation is crucial to be considered when evaluating the performance of the control chart with estimated process parameters. To consider practitioner-to-practitioner variation, this paper investigates the performance of the VSI EWMA x̄ chart with estimated process parameters by using standard deviation of the ATS. Monte Carlo simulation results show that the VSI EWMA x̄ chart requires many Phase-I samples to achieve the desired performance. The results also show that a greater number of Phase-I samples is needed for the VSI EWMA x̄ chart when the smoothing constants are large. This is because larger values of smoothing constants lead to higher variation in the run-length distribution.

Anomaly Detection for Compositional Data using VSI MEWMA control chart

In recent years, the monitoring of compositional data using control charts has been investigated in the Statistical Process Control field. In this study, we will design a Phase II Multivariate Exponentially Weighted Moving Average (MEWMA) control chart with variable sampling intervals to monitor compositional data based on isometric log-ratio transformation. The Average Time to Signal will be computed based on the Markov chain approach to investigate the performance of the proposed chart. We also propose an optimization procedure to obtain the optimal control limit, smoothing constant, and out-of-control Average Time to Signal for different shift sizes and short sampling intervals. The performance of the proposed chart in comparison with the standard MEWMA chart for monitoring compositional data is also provided. Finally, we end the paper with a conclusion and some recommendations for future research.

On Quantized Sampled–Data Event–Triggered Consensus Tracking of Nonlinear Multi–Agent Time–Delay Systems

In this paper, the quantized sampled-data leaderless consensus tracking problem of nonlinear time–delay multi–agent systems (MASs) is studied. In particular, a new methodology for the design of quantized sampled–data event–triggered controllers achieving the leaderless consensus tracking of nonlinear MASs affected by state–delays is proposed. Firstly, the notion of steepest descent consensus feedback (SDCF), induced by a general class of Lyapunov–Krasovskii functionals, is introduced. Then, it is proved that there exist a suitably accurate quantization of the input channels and a suitably small maximum inter–sampling time such that: the digital implementation of SDCFs (continuous or not), with an event-triggered mechanism to update the control law only when necessary, achieves the leaderless consensus tracking in a semi–global practical sense of the related closed–loop MAS. The stabilization in the sample-and-hold sense theory is used as a tool for proving the results. The possible non–uniform quantization of the input channels is considered. Moreover, the case of time–varying sampling intervals as well as the stability analysis of the inter–sampling system behaviour is included in the theory here developed. An example is studied in order to validate the results.

Joint Design of Control Chart, Production Cycle Length, and Maintenance Schedule for Imperfect Manufacturing Systems with Deteriorating Products under Stochastic Shift Size

This paper presents a hybrid model based on the integration of three interrelated topics including statistical process monitoring (SPM), production cycle length, and maintenance scheduling for imperfect production systems. The proposed model takes both process deterioration and product deterioration into account i.e., the process can go out of control and the products can deteriorate over time. In this regard, a nonuniform sampling strategy is presented for calculating the sampling intervals to avoid exceeding the hazard rate from a threshold value. Moreover, in order to make the model more adapted to real industrial systems, the shift magnitude in the mean parameter of quality characteristic is considered as a stochastic variable. To elaborate the model effectiveness, three comparative studies and a sensitivity analysis are presented. The first comparison shows that by using the variable sampling interval scheme instead of the fixed one, the ratio of in-control time interval to the whole production cycle increases dramatically. Moreover, the use of variable sampling interval scheme leads to a significant cost savings and decreases the out-of-control time interval. The second comparative study indicates that although using statistical considerations in establishment of the control chart imposes a higher cost on the manufacturer, both values of the in-control average run length ([Formula: see text]) and out-of-control average run length ([Formula: see text]) improve significantly. In other words, the use of economic-statistical model reduces the amount of non-conforming products as well as the risk of issuing a false alarm by the control chart. The third comparative study indicates that, in terms of total inventory holding cost, the proposed model performs better than the model without considering deterioration possibility due to the significant reduction in the deterioration cost.

An improved variable parameter mean square error control chart

In order to improve the efficiency of the mean square error control chart in detecting small to moderate shifts, an improved variable parameter mean square error (VPt MSE) control chart is proposed, which uses three different sampling intervals, two different sample sizes and two different control limits. Markov chain method is used to calculate the average time to signal and average number of observations to signal. The performance of the adaptive chart was compared with the variable sample size and sampling interval (VSSI), the improved variable sample size and sampling interval (VSSIt), the variable parameter (VP), the exponentially weighted moving average (EWMA), and the variable sampling interval exponentially weighted moving average (VSI EWMA) mean square error control chart. Numerical results show that the VPt MSE chart surpasses the other charts in detecting small to moderate shifts. At the same time, we also improve the VPt MSE chart, which greatly improves the statistical performance. An example is provided to illustrate the implementation of the VPt MSE chart.

A variable sampling interval EWMA t chart with auxiliary information – A robustness study in the presence of estimation error

This article proposes the variable sampling interval (VSI) exponentially weighted moving average (EWMA) t chart for the process mean using auxiliary information (AI) (denoted as the VSI EWMA t-AI chart). The proposed chart allows different sampling intervals to be adopted based on prior process quality. The average time to signal (ATS) and expected average time to signal (EATS) criteria are used to evaluate the performance of the proposed chart. Numerical analyses are conducted and the results show that the proposed VSI EWMA t-AI chart surpasses the EWMA t-AI chart in the detection of all sizes of shifts in the process mean when there are no changes (or estimation error) in the process standard deviation. Additionally, the proposed VSI EWMA t-AI chart is found to be more robust than the VSI EWMA X¯ -AI chart when there are changes in the process standard deviation.

Time to signal distribution of multivariate bayesian control chart with dual sampling scheme

The ability of a control chart to detect a shift in the process can be determined by the average run length (ARL), a widely used performance indicator for single sampling interval (SSI) control charts. When it comes to variable sampling interval (VSI) schemes, average time to signal (ATS) is utilized instead of ARL. Although analysis of ARL has been extensively studied for SSI schemes, calculation of the ATS for VSI control charts is still in its infancy due to several theoretical challenges. In this context, the paper derives closed-form expressions for time to signal distribution and the ATS for a spatial design of VSI multivariate Bayesian control chart referred to as the dual sampling scheme (DSS). In a DSS strategy, the process is being monitored initially with a longer sampling interval, which is changed to the shorter sampling interval at a designed switching point. Unlike previous approximate developments where the switching point is ignored, the paper derives exact expressions for computation of time to signal distribution and ATS within such a Bayesian framework. Derivations of the ATS are based on an intuitively pleasing definition of an artificial absorbing state and coded values of the posterior probability using Markov chain theory.

A Bayesian condition-based maintenance and monitoring policy with variable sampling intervals

The aim of this study is to propose a Condition-Based Maintenance and Monitoring (CBM) policy which employs a Bayesian inspection scheme. A single unit system, which is subject to both operational deterioration and catastrophic failures, is considered. The equipment may operate in two different non-observable states (healthy and unhealthy). The unhealthy state is characterized by higher operational cost and higher proneness to failure. Failures are self-announced (directly observable) and thus, corrective maintenance is implemented immediately. A new double-sampling Bayesian control chart with state-dependent variable inspection frequency is proposed. The process operation is analytically modeled through a six-state Markov process, while, unlike all previous Bayesian models, there is no need for discretization of the unhealthy-state probabilities. At each inspection instance all available information regarding the equipment condition is utilized in order to schedule future inspections and preventive maintenance actions and detect possible operation in the unhealthy state. The critical parameters, namely the duration of the inspection intervals, the sample sizes and the preventive maintenance times, which minimize the expected total cost per time unit, are determined. Numerical comparisons with three other Bayesian CBM models are conducted to demonstrate the effectiveness of the proposed policy.

Monitoring Coefficient of Variation Using Variable Sampling Interval Double Exponentially Weighted Moving Average Charts

As a measure of relative variability, the coefficient of variation (CV) is a valuable charting statistic in statistical process control. Great efforts have been devoted to monitoring CV efficiently. To further improve the performance of CV charts, this paper proposes three Double Exponentially Weighted Moving Average (DEWMA) charts by incorporating Variable Sampling Interval (VSI) strategies to monitor the CV squared. The run length properties of the proposed charts are evaluated via Monte Carlo simulations. Comparative studies show that the proposed VSI DEWMA CV charts detect the process shifts faster than the existing CV charts. A real data example is presented to illustrate the VSI DEWMA CV charts.

Economic Design of EWMA Control Charts with Variable Sampling Intervals for Monitoring the Mean and Standard Deviation under Preventive Maintenance and Taguchi’s Loss Functions

The control chart and the maintenance management need to control process quality and reduce out-of-control cost. They are two key tools in the production process; however, they have usually been analyzed separately in the literature. Moreover, the existing studies integrating these two concepts suffer from three significant drawbacks as follows: (1) using control charts with fixed parameters to monitor the process, so that the small and middle shifts are detected slowly; (2) monitoring the mean and standard deviation separately, whereas, in real condition, the mean and standard deviation should be monitored simultaneously; (3) quality loss function is not usually used to design economic model, which leads to a large social quality loss in the monitoring process of control chart. To eliminate these weaknesses, the economic design of the exponential weighted moving average (EWMA) control chart with variable sampling intervals (VSI) for monitoring the mean and standard deviation under preventive maintenance and Taguchi’s loss functions is proposed. The optimal values of the parameters are determined to minimize the loss-per-item in an average cycle process. In addition, a genetic algorithm is used in a numerical example to search for the optimal values of the parameters. According to the sensitivity analysis, the effect of the model parameters on the solution of the economic model is obtained. Finally, the comparison study shows that the VSI EWMA control charts designed by the joint economic model are expected to reduce loss.

Memory‐type t charts with multiple auxiliary information for the process mean

AbstractMultiple auxiliary information‐based (MAIB) memory‐type charts are proposed with fixed and variable sampling intervals for an improved monitoring of the process mean, which include adaptive/nonadaptive cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts. These control charts are constructed based on a unique uniformly minimum variance unbiased estimator of the process mean that requires information on a study variable as well as on several correlated auxiliary variables. The Monte Carlo simulation technique is used to compute the run length characteristics of the proposed charts when sampling from a multivariate normal distribution. The run length comparisons show that the proposed MAIB‐ charts outperform their existing auxiliary information based (AIB) and non‐AIB charts, where the normalizing transformation is used for all considered charts in order to have uniformity in the comparisons. A real data application is also given to support the proposed theory.

Monitoring multivariate simple linear profiles using individual observations

In quality control applications, the quality of a product may effectively be characterized by a relationship between one or more variables that is typically referred to as a profile. In this paper, we propose two control charts, namely the maximum multivariate EWMA (MEWMA) and maximum double MEWMA charts for monitoring the parameters of a multivariate simple linear profile using the individual observations. Moreover, the sensitivities of these control charts are also enhanced with the variable sampling interval feature. The Monte Carlo simulation method is used to compute the zero-state and steady-state run-length characteristics of the proposed charts. Based on detailed run-length comparisons, it is found that the MaxDMEWMA chart is slightly more sensitive than the MaxMEWMA chart. A real dataset is also considered to support the theory.

EWMA and adaptive EWMA variable sampling intervals charts for simultaneous monitoring of Weibull parameters

ABSTRACT In this article, we first propose an exponentially weighted moving average (EWMA) chart for monitoring both the shape and scale parameters of the Weibull distribution. The proposed chart was developed by incorporating two individual EWMA charts, one for monitoring the shape parameter and the other for monitoring both the shape and scale parameters. Particularly, it is shown that these two individual EWMA charts are mutually independent. Simulation studies are presented to compare the performance of the proposed EWMA chart and the existing Shewhart schemes with the EWMA setup. Further, to reduce the time for detecting an out-of-control (OC) process, an EWMA chart with variable sampling intervals (VSIs) and an adaptive EWMA chart with VSI for monitoring both the shape and scale parameters were also developed based on the proposed EWMA chart. The EWMA chart with VSI and the adaptive EWMA chart with VSI were studied and their performances were evaluated. An example to illustrate the applicability and implementation of the proposed charts is also provided.

Monitoring logistic profiles using variable sample interval approach

Investigation the stability of the relationship between a response variable and one or more independent variables over time is termed as profile monitoring. In the area of profile monitoring, fewer researches can be found to the profiles with the non-normal response variable denoted by generalized linear models (GLM). Logistic profiles as one of common the GLM profiles, are the most common types of profiles in which we take advantage of logistic regression to model the relationship. In logistic regression, the response variable is assumed to be non-quantitative and binary. Also, to increase the efficiency of the charts and faster detection of the shifts, we can consider control charts’ parameters such as sample size, sampling intervals and etc., with an adaptive approach, which means that considering the process conditions, parameter values can be unfixed and variable. In this study, we investigate logistic profiles monitoring in phase II using three existing control charts entailing T2 control charts, multivariate exponentially weighted moving average (MEWMA), deviance control chart and two proposed control chart including generalized likelihood ratio (GLR) and support vector regression (SVR) methods by considering sampling interval as a variable, denoted with variable sample interval (VSI) approach. Simulation results indicate the much better performance of control charts by adding the VSI than the traditional fixed sampling interval in terms of average time to signal (ATS). Eventually, to clarify how to implement the VSI charts, a numerical example is provided.

A new double EWMA‐t chart with auxiliary information for the process mean

AbstractIn this paper, we propose an auxiliary‐information‐based (AIB) double EWMA‐ (AIB‐DEWMA‐) chart for monitoring the process mean. The DEWMA‐ chart encompasses the EWMA‐ and AIB‐EWMA‐ charts. The Monte Carlo simulations are used to compute the run length characteristics of the AIB‐DEWMA‐ chart. Based on detailed run length comparisons, it is found that the AIB‐DEWMA‐ chart may uniformly and substantially outperform the AIB‐EWMA‐ chart when detecting different shifts in the process mean. In addition, the AIB‐DEWMA‐ chart is uniformly more sensitive than the DEWMA‐ chart. Similar trends are observed when comparing these control charts with the variable sampling interval feature. A real dataset is also considered to demonstrate the implementation of the proposed chart.

One-sided variable sampling interval EWMA control charts for monitoring the multivariate coefficient of variation in the presence of measurement errors

Online monitoring of the multivariate coefficient of variation (MCV) can be of interest in many real situations in which the dispersion of a multivariate process is meant to remain constant with regard to its position. To this aim, several control charts have been recently proposed in the literature. In this paper, the new one-sided adaptive charts to monitor the MCV are proposed. The chart applies a variable sampling interval (VSI) strategy on an exponentially weighted moving average (EWMA) scheme, aiming at benefiting from the known advantages of both approaches. Formulas to optimally determine the parameters of the chart are derived and presented. The proposed chart is shown to outperform competing charts under most circumstances. The presence of the measurement errors is considered to understand their effects on the charts. An illustrative example is also included.

Directionally sensitive weighted adaptive multivariate CUSUM mean charts

AbstractIn many service and manufacturing industries, process monitoring involves multivariate data, instead of univariate data. In these situations, multivariate charts are employed for process monitoring. Very often when the mean vector shifts to an out‐of‐control situation, the exact shift size is unknown; hence, multivariate charts for monitoring a range of the mean shift sizes in the mean vector are adopted. In this paper, directionally sensitive weighted adaptive multivariate CUSUM charts are developed for monitoring a range of the mean shift sizes. Directionally sensitive charts are useful in situations where the aim lies in monitoring either an increasing or a decreasing shift in the mean vector of the quality characteristics of interest. The Monte Carlo simulation is used to compute the run length characteristics in comparing the sensitivities of the proposed and existing multivariate CUSUM charts. In general, the directionally sensitive and weighted adaptive features enhance the sensitivities of the proposed multivariate CUSUM charts in comparison with the existing multivariate CUSUM charts without the adaptive feature or those that are directionally invariant. It is also found that the variable sampling interval feature enhances the sensitivities of the proposed and existing charts as compared to their fixed sampling interval counterparts. The implementation of the proposed charts in detecting upward and downward shifts in the in‐control process mean vector is demonstrated using two different datasets.

Economic-statistical design of the variable sampling interval Poisson EWMA chart

An economic-statistical design is developed for the variable sampling interval (VSI) Poisson exponentially weighted moving average chart, namely the VSI PE chart, where the cost function is minimized by incorporating statistical constraints. The statistical performance measures of the VSI PE chart are derived using the Markov chain approach. A sensitivity analysis based on the design of the experiment is conducted to study the effects of the cost model parameters on the optimal solution. The results show that the VSI PE chart performs better than the fixed sampling interval PE chart, in terms of the expected cost and the statistical performances.

Enhanced adaptive multivariate EWMA and CUSUM charts for process mean

The multivariate charts are mostly used to simultaneously monitor several quality characteristics in manufacturing processes. In this study, we enhance the sensitivities of the recently proposed adaptive multivariate EWMA (AME) and weighted adaptive multivariate CUSUM (WAMC) charts with an auxiliary-information-based (AIB) estimator, namely the AIB-AME and AIB-WAMC charts, for monitoring different kinds of shifts in the mean of a multivariate normally distributed process. In addition, the variable sampling interval (VSI) feature is also incorporated into the proposed charts. The run length properties of these control charts are computed using Monte Carlo simulations. It is found that the AIB-AME and AIB-WAMC charts are uniformly and substantially more sensitive than the AME and WAMC charts, respectively. The same trend is observed when these control charts have the VSI feature incorporated into them. Real datasets are used to demonstrate the implementation of the proposed charts.

Memory‐type control charts with multiple auxiliary information for process mean

AbstractMemory‐type auxiliary‐information‐based (AIB) control charts are very effective in detecting small‐to‐moderate shifts in the process mean. In this study, we first develop a unique uniformly minimum variance unbiased estimator of the process mean that requires information on the study variable as well as on several correlated auxiliary variables. Then, based on this estimator, adaptive and nonadaptive CUSUM and EWMA charts are developed with either fixed or variable sampling interval for monitoring the process mean, namely, the multiple AIB (MAIB) charts. The proposed charts encompass existing charts with or without the auxiliary information. The run length characteristics of the proposed charts are computed with the Monte Carlo simulations when sampling from a multivariate normal distribution. Based on the run length comparisons, it is found that the MAIB charts are uniformly and substantially more sensitive than the AIB charts when monitoring the process mean. Real datasets are also considered to explain the implementation of the MAIB charts.