Variable Sampling Interval Feature Research Articles

New weighted adaptive CUSUM charts for monitoring the generalized variance of a bivariate normal process

In modern and high volume manufacturing environments, it is vital to early detect a small sustained shift in a short period of time as it may have a serious impact on a manufacturing process. Adaptive memory-type control charts have the ability to swiftly detect a range of the shift sizes, and one such control chart is a weighted adaptive CUSUM (abbreviated as C) chart. In this paper, using a simple likelihood ratio test-statistic, we propose two types of the C charts (with and without the normalizing transformation) using three different shift estimators for monitoring the generalized variance (GV) of a bivariate normally distributed production process. In addition, the sensitivities of these control charts are also enhanced with the variable sampling interval feature. Monte Carlo simulations are used to compute the zero-state and steady-state run-length characteristics of the proposed charts. Based on detailed run-length comparisons, it is observed that the proposed charts may uniformly and substantially outperform the existing charts when detecting different kinds of shifts in the GV. A real dataset is also selected to illustrate the implementation of the newly proposed charts.

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.

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.

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.

New One-Sided EWMA t Charts without and with Variable Sampling Intervals for Monitoring the Process Mean

In statistical process control (SPC), t charts play a vital role in the monitoring of the process mean, especially when the process variance is unknown. In this paper, two separate upper-sided and lower-sided exponentially weighted moving average (EWMA) t charts are first proposed and the Monte Carlo simulation method is used to obtain their run length (RL) properties. Compared with the traditional one-sided EWMA t charts and several run rules t charts, the proposed charts are proven to have better performance than these competing charts. In addition, by adding the variable sampling interval (VSI) feature to the proposed charts, the new VSI one-sided EWMA t charts are shown to detect different shift sizes in the process more efficient than the chart without VSI feature. Finally, an example of a milk filling process illustrates the use of the charts.

Auxiliary information based maximum EWMA and DEWMA charts with variable sampling intervals for process mean and variance

The maximum EWMA (ME) and maximum double EWMA (MDE) charts are used to detect joints shifts in mean and variance of a normal process. In this paper, we propose the ME and MDE charts using the variable sampling interval (VSI) feature. In addition, the VSI feature is also attached with auxiliary information based (AIB) ME and MDE charts. Using Monte Carlo simulations, the run length characteristics of these control charts are computed. Based on detailed run length comparisons, it is found that the ME and MDE (AIB-ME and AIB-MDE) charts with VSI are more sensitive than those with fixed sampling interval. A real dataset is considered to demonstrate the implementation of proposed charts.

A new double EWMA-t chart for process mean

The EWMA-t chart is used to monitor the process mean as it is robust against errors in estimating the process standard deviation or changing standard deviation. In this paper, we propose a double EWMA-t (DEWMA-t) chart for monitoring the process mean, which is a natural extension of the EWMA-t chart. The idea is to first construct the t chart using the EWMA statistic, and then put it into another EWMA statistic to devise the DEWMA-t chart. The DEWMA-t chart encompasses the EWMA-t chart. The Monte Carlo simulation method is used to compute the run length properties of the DEWMA-t chart. Based on detailed run length comparisons, it turns out that, with some reasonable assumptions, the DEWMA-t chart performs uniformly and substantially better than the EWMA-t chart when detecting different kinds of shift in the process mean. A similar trend is observed when these control charts are compared with their counterparts based on the variable sampling interval feature. In addition, real datasets are also used to support the theory.

A novel approach to monitor simple linear profiles using individual observations

In some quality control applications, quality of a product or process may be characterized effectively by a functional relationship between one or more variables that is typically referred to as a profile. In this paper, we propose four control charts, namely the maximum CUSUM (MaxCUSUM), maximum Crosier CUSUM (MaxCCUSUM), maximum EWMA (MaxEWMA) and maximum double EWMA (MaxDEWMA) charts, for monitoring the parameters of a Phase II simple linear profile using the individual observations. In addition, 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. The run length comparisons reveal that the MaxCCUSUM (MaxDEWMA) chart is slightly more sensitive than the MaxCUSUM (MaxEWMA) chart. A real dataset is also considered to support the theory.

Adaptive CUSUM and EWMA charts with auxiliary information and variable sampling intervals for monitoring the process mean

AbstractThe variable sampling interval (VSI) feature enhances the sensitivity of a control chart that is based on fixed sampling interval (FSI). In this paper, we enhance the sensitivities of the auxiliary information‐based (AIB) adaptive Crosier cumulative sum (CUSUM) (AIB‐ACC) and adaptive exponentially weighted moving average (EWMA) (AIB‐AE) charts using the VSI feature when monitoring a mean shift which is expected to lie within a given interval. The Monte Carlo simulations are used to compute zero‐state and steady‐state run length properties of these control charts. It is found that the AIB‐ACC and AIB‐AE charts with VSI feature are uniformly more sensitive than those based on FSI feature. Real datasets are also considered to demonstrate the implementation of these control charts.

The variable sampling interval S2 chart with known or unknown in-control variance

The chart has been known as a powerful tool to monitor the variability of the normal process. This paper considers the variable sampling interval (VSI) chart to improve the performance of the original chart. A VSI chart based on the average run length (ARL) unbiased and average time to signal (ATS) unbiased criteria is developed when the in-control variance is known. The performance of the proposed chart is investigated in terms of the mean and standard deviation of the RL and total time to signal. The impact of parameter estimation on the proposed VSI chart is also evaluated. The results show that the performance of the control chart is seriously different from the known parameter case when the parameter is estimated. Then, a new VSI chart with the desired in-control ARL and ATS is considered when the in-control variance is estimated. In evaluating the usefulness of the VSI feature, the VSI charts are compared with the fixed sampling interval (FSI) versions in terms of their statistical properties. The comparison results show that the VSI chart can detect the shifts in the process variance quicker than the corresponding FSI chart, regardless of whether the in-control variance is known or unknown. Finally, two examples are used to illustrate the proposed control chart.

Variable Sampling Interval Cumulative Count of Conforming Chart with Runs Rules

The cumulative count of conforming (CCC) chart is effective in detecting very low fraction of nonconforming items for high yield manufacturing processes. In this study, a combination of runs rules and variable sampling interval feature is proposed to a lower sided CCC chart by inspecting the items one by one. The performance measures of the control chart are derived by using the Markov chain approach. The numerical comparisons show that the performance of the CCC chart can be improved by adding the runs rules and varying the sampling interval.

Multivariate Synthetic |S| Control Chart with Variable Sampling Interval

A variable sampling interval (VSI) feature is introduced to the multivariate synthetic generalized sample variance |S| control chart. This multivariate synthetic control chart is a combination of the |S| sub-chart and the conforming run length sub-chart. The VSI feature enhances the performance of the multivariate synthetic control chart. The comparative results show that the VSI multivariate synthetic control chart performs better than other types of multivariate control charts for detecting shifts in the covariance matrix of a multivariate normally distributed process. An example is given to illustrate the operation of the VSI multivariate synthetic chart.

The Design of the Variable Sampling Interval Generalized Likelihood Ratio Chart for Monitoring the Process Mean

This paper considers the problem of monitoring a normally distributed process variable in which some sustained shift in the mean may occur. A generalized likelihood ratio (GLR) control chart with a variable sampling interval (VSI) scheme is proposed to quickly detect a wide range of such shifts. The performance of the VSI GLR chart is evaluated, and the results show that using the VSI feature can greatly reduce the expected detection time. Design methodology for the VSI GLR chart is provided so that practitioners can easily use this chart in applications. An illustrative example is given to explain how to apply the VSI GLR chart. Copyright © 2013 John Wiley & Sons, Ltd.

Cumulative conformance count charts with variable sampling intervals for correlated samples

The cumulative conformance count (CCC) control chart is often employed to monitor the fraction nonconforming of high-yield processes. Traditional CCC chart is used when the items from a process are inspected one-at-a-time following the production order. In recent years, the CCC chart has been generalized to accommodate some industrial practices where items from a process are inspected sample by sample and not according to the production order. In order to increase the sensitivity of the generalized CCC (GCCC) chart to changes in fraction nonconforming, the variable sampling interval (VSI) scheme is used in this study. The output characteristic within each sample is assumed with correlation. The statistical properties of the GCCC chart with the VSI scheme are deduced using the Markov chain method. In evaluating the usefulness of the VSI feature, GCCC charts with VSI and fixed sampling interval (FSI) schemes are compared in terms of their statistical properties. The comparison results show that using the VSI scheme can improve the speed of GCCC chart in detecting changes in fraction nonconforming. Finally, according to the comparison results, a design procedure is applied to an industrial example to validate its practicability.

Cumulative Sum Control Charts for Simultaneously Monitoring Means and Variances of Multiple Quality Variables

Multivariate cumulative sum (CUSUM) control charts for simultaneously monitoring both means and variances under multivariate normal process are investigated. Performances of multivariate CUSUM schemes are evaluated for matched fixed sampling interval (FSI) and variable sampling interval (VSI) features in terms of average time to signal (ATS), average number of samples to signal (ANSS). Multivariate Shewhart charts are also considered to compare the properties of multivariate CUSUM charts. Numerical results show that presented CUSUM charts are more efficient than the corresponding Shewhart chart for small or moderate shifts and VSI feature with two sampling intervals is more efficient than FSI feature. When small changes in the production process have occurred, CUSUM chart with small reference values will be recommended in terms of the time to signal.

An EWMA t chart with variable sampling intervals for monitoring the process mean

This paper proposes a variable sampling interval (VSI) version of the fixed sampling interval (FSI) exponentially weighted moving average (EWMA) t chart developed by Zhang et al. for monitoring the changes in the process mean. An optimal design strategy based on the average time to signal (ATS) is presented. We determine the optimal parameters for the VSI EWMA t chart using a Markov chain approach so that the chart has the desired robustness property against errors in estimating the process standard deviation or changing standard deviation. Also, we explain how the various parameters of this VSI EWMA t chart can be computed and how the use of the VSI feature improves the statistical efficiency of the FSI EWMA t and FSI EWMA X-bar charts in terms of out-of-control ATS performances. Comparisons with the FSI EWMA t and FSI EWMA X-bar charts are performed.

Multivariate Control Charts for Monitoring the Mean Vector and Covariance Matrix with Variable Sampling Intervals

Traditional control charts for process monitoring are fixed sampling rate (FSR) control charts based on taking samples of fixed size with a fixed sampling interval between samples. Variable sampling rate (VSR) control charts vary the sampling rate as a function of the data from the process, and can detect most process changes significantly faster than traditional FSR control charts. The main focus of this article is on a type of VSR control chart called a variable sampling interval (VSI) control chart, but another type of VSR control chart based on sequential sampling is also considered. We consider the problem of detecting sustained or transient shifts in the mean or variability of a process. We investigate Shewhart control charts and multivariate exponentially weighted moving average (MEWMA) control charts based on sample means and on the sum of the squared deviations from target. We show that the combination of the standard MEWMA chart and an MEWMA-type chart based on squared deviations from target gives excellent overall performance. When the VSI feature is used with this combination, the performance is dramatically better than the performance of the standard FSR multivariate control charts that have traditionally been used.

On the Superiority of a Variable Sampling Interval Control Chart

The paper establishes the analytical grounds of the uniform superiority of a variable sampling interval (VSI) Shewhart control chart over the conventional fixed sampling interval (FSI) control chart, with respect to the zero-time performance, for a wide class of process distributions. We provide a sufficient condition on the distribution of a control chart statistic, and propose a criterion to determine the control limits and the regions in the in-control area of the VSI chart, corresponding to the different sampling intervals used by it. The condition and the criterion together ensure the uniform zero-time superiority of the VSI chart over the matched FSI chart, in detecting a process shift of any magnitude. It is shown that normal, Student's t and Laplace distributions satisfy the sufficient condition. In addition, chi-square, F and beta distributions satisfy it, provided that these are not extremely skewed. Further, it is illustrated that the superiority of the VSI feature is not trivial and cannot be assured if the sufficient condition is not satisfied or the control limits and the regions are not determined according to the proposed criterion. An application of the result to confirm the superiority of the VSI feature is demonstrated for the control chart for individual observations used to monitor a milk-pouch filling process.

EWMA control charts with variable sample sizes and variable sampling intervals

Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. Variable Sample Size (VSS) and Variable Sampling Interval (VSI) control charts vary the sampling rate from the process as a function of the data from the process. VSS and VSI control charts sample at a higher rate when there is evidence of a change in the process, and are thus able to detect process changes faster than traditional control charts. This paper considers general VSS and VSI control charts and develops integral equation and Markov chain methods for finding the statistical properties of these charts. EWMA charts with the VSS and/or the VSI features are studied in detail, and different ways of defining the EWMA control statistic are investigated. It is shown that using either the VSS or VSI feature in an EWMA control chart substantially improves the ability to detect all but very large shifts in the process mean. The VSI feature usually gives more improvement in detection ability than the VSS feature, and using both features together sometimes gives more improvement than using either one separately. Guidelines are given for choosing the possible sample sizes and the possible sampling intervals for these charts. EWMA charts with the VSS and/or VSI feature are compared to CUSUM charts and Shewhart X¯ charts with the VSS and/or VSI features.

Monitoring the Process Mean and Variance Using Individual Observations and Variable Sampling Intervals

In this paper we investigate control charts for monitoring a process to detect changes in the mean and/or variance of a normal quality variable when an individual observation is taken at each sampling point. The traditional X chart and moving range (MR) chart are evaluated. Also evaluated are the exponentially weighted moving average (EWMA) chart of the observations and the EWMA chart of the squared deviations of the observations from the target. It is shown that the combination of the X and MR charts will not detect small and moderate parameter shifts as fast as combinations involving the EWMA charts. The ability of charts to diagnose the type of parameter shift produced by a special cause is also investigated. It is shown that combinations involving the EWMA charts are just as effective at diagnosing the type of parameter shift as the traditional combination of the X and MR charts. The effect of adding the variable sampling interval (VSI) feature is also evaluated for some of the combinations of charts. The VSI feature allows the sampling interval to be varied as a function of the values of the statistics being plotted. It is shown that adding the VSI feature to the combinations of charts results in very substantial reductions in the expected time required to detect shifts in process parameters.