Sin Yin Teh

Monitoring Process Mean and Variability with One Double EWMA Chart

In this article, we extend the single Max-EWMA chart to a single double EWMA chart, called the Max-DEWMA chart. The statistics of the Max-DEWMA chart are based on the maximum of the absolute values of the two DEWMA statistics, one controlling the mean while the other the variance. We show that the Max-DEWMA chart performs better than the Max-EWMA chart in detecting small and moderate shifts in the mean and/or variance.

Optimal Designs of the Median Run Length Based Double Sampling X̄ Chart for Minimizing the Average Sample Size

Designs of the double sampling (DS) chart are traditionally based on the average run length (ARL) criterion. However, the shape of the run length distribution changes with the process mean shifts, ranging from highly skewed when the process is in-control to almost symmetric when the mean shift is large. Therefore, we show that the ARL is a complicated performance measure and that the median run length (MRL) is a more meaningful measure to depend on. This is because the MRL provides an intuitive and a fair representation of the central tendency, especially for the rightly skewed run length distribution. Since the DS chart can effectively reduce the sample size without reducing the statistical efficiency, this paper proposes two optimal designs of the MRL-based DS chart, for minimizing (i) the in-control average sample size (ASS) and (ii) both the in-control and out-of-control ASSs. Comparisons with the optimal MRL-based EWMA and Shewhart charts demonstrate the superiority of the proposed optimal MRL-based DS chart, as the latter requires a smaller sample size on the average while maintaining the same detection speed as the two former charts. An example involving the added potassium sorbate in a yoghurt manufacturing process is used to illustrate the effectiveness of the proposed MRL-based DS chart in reducing the sample size needed.

The variable sampling interval run sum X control chart

Sampling intervals for VSI run sum X ? chart are varied according to process quality.Optimization programs to minimize ATS 1 ( ? opt ) and AATS 1 ( ? opt ) for the VSI run sum X ? chart are developed.The zero state and steady state VSI run sum X ? charts performances are evaluated.The VSI run sum X ? chart performs well compared with other competing charts.An example of application explains the construction of the VSI run sum X ? chart. Traditional control charts for process monitoring are based on taking samples from the process at fixed length sampling intervals. More recently, research works focused on the use of variable sampling intervals (VSIs), where the lengths of the sampling intervals are varied according to the process quality. A short sampling interval is considered when the process quality indicates a possible out-of-control situation while a long sampling interval is considered, otherwise. In this paper, the VSI run sum (RS) X ? chart is proposed with its optimal scores and parameters determined using an optimization technique to minimize the out-of-control average time to signal (ATS) or the adjusted average time to signal (AATS). A Markov-chain method is used to evaluate both the ATS and AATS of the proposed chart, for the zero and steady state cases, respectively. Results show that the VSI RS X ? chart is considerably more efficient than the basic RS X ? chart. The VSI RS X ? chart performs generally well compared with other competing charts, such as the standard X ? , synthetic X ? , exponentially weighted moving average (EWMA) X ? , VSI X ? and VSI EWMA X ? charts. The sensitivity of the VSI RS X ? chart can be enhanced further by adding more scoring regions or a head-start feature. An illustrative example is presented to explain the implementation of the proposed VSI RS X ? chart.

Run-sum control charts for monitoring the coefficient of variation

The coefficient of variation (CV) is a unit-free and effective normalized measure of dispersion. Monitoring the CV is a crucial approach in Statistical Process Control when the quality characteristic has a distinct mean value and its variance is a function of the mean. This setting is common in many scientific areas, such as in the fields of engineering, medicine and various societal applications. Therefore, this paper develops a simple yet efficient procedure to monitor the CV using run-sum control charts. The run-length properties of the run-sum CV (RS-γ) charts are characterized by the Markov chain approach. This paper proposes two optimization algorithms for the RS-γ charts, i.e. by minimizing (i) the average run length (ARL) for a deterministic shift size and (ii) the expected ARL over a process shift domain. Performance comparisons under both the zero- and steady-state modes are made with the Shewhart-γ, Run-rules-γ and EWMA-γ charts. The results show that the proposed RS-γ charts outperform their existing counterparts for all or certain ranges of shifts in the CV. The application of the optimal RS-γ charts is illustrated with real data collected from a casting process.

A variable sampling interval synthetic Xbar chart for the process mean

The usual practice of using a control chart to monitor a process is to take samples from the process with fixed sampling interval (FSI). In this paper, a synthetic X¯ control chart with the variable sampling interval (VSI) feature is proposed for monitoring changes in the process mean. The VSI synthetic X¯ chart integrates the VSI X¯ chart and the VSI conforming run length (CRL) chart. The proposed VSI synthetic X¯ chart is evaluated using the average time to signal (ATS) criterion. The optimal charting parameters of the proposed chart are obtained by minimizing the out-of-control ATS for a desired shift. Comparisons between the VSI synthetic X¯ chart and the existing X¯, synthetic X¯, VSI X¯ and EWMA X¯ charts, in terms of ATS, are made. The ATS results show that the VSI synthetic X¯ chart outperforms the other X¯ type charts for detecting moderate and large shifts. An illustrative example is also presented to explain the application of the VSI synthetic X¯ chart.

Monitoring process mean and variance with a single generally weighted moving average chart

Two generally weighted moving average (GWMA) charts are usually used concurrently for a simultaneous monitoring of the process mean and process variance. In this article, we propose a new GWMA chart, called the Max-GWMA chart, which uses a single statistic for a simultaneous monitoring of the process mean and variance. The statistic of the Max-GWMA chart is based on the maximum of the absolute values of two GWMA statistics, one for controlling the mean while the other the variance. We show that the Max-GWMA chart outperforms the combined GWMA chart, in terms of the average run length (ARL), standard deviation of the run length (SDRL) and diagnostic abilities performances. The combined GWMA chart consists of two GWMA charts that are run concurrently, one for monitoring the mean and the other the variance.

Monitoring of Time‐Between‐Events with a Generalized Group Runs Control Chart

Control charting methods for time between events (TBE) is important in both manufacturing and nonmanufacturing fields. With the aim to enhance the speed for detecting shifts in the mean TBE, this paper proposes a generalized group runs TBE chart to monitor the mean TBE of a homogenous Poisson failure process. The proposed chart combines a TBE subchart and a generalized group conforming run length subchart. The zero-state and steady-state performances of the proposed chart were evaluated by applying a Markov chain method. Overall, it is found that the proposed chart outperforms the existing TBE charts, such as the T, Tr, EWMA-T, Synth-Tr, and GR-Tr charts. Copyright

The Run Sum Hotelling's χ2 Control Chart with Variable Sampling Intervals

Control charts are widely used for process monitoring and quality control in manufacturing industries. Implementing variable sampling interval (VSI) control schemes on control charts rather than traditional fixed sampling interval procedure can significantly improve the control charts efficiency. In this paper, the VSI run sum (RS) Hotellings χ2 chart is proposed. The optimal scores and parameters of the proposed chart are determined using an optimization technique to minimize the following: (i) out-of-control average time to signal (ATS); (ii) adjusted ATS (AATS), when the exact shift size can be specified; (iii) expected ATS; or (iv) expected AATS, when the exact shift size cannot be specified. The Markov chain method is used to evaluate the zero-state ATS and expected ATS, and steady-state AATS and expected AATS of the proposed chart. The results show that the VSI RS Hotellings χ2 chart significantly outperforms the standard RS Hotellings χ2 chart and the former also performs well compared with other competing charts. By adding more scoring regions, the efficiency of the VSI RS Hotellings χ2 chart can be further enhanced. An illustrative example using data from a manufacturing process is presented to demonstrate the application of the VSI RS Hotellings χ2 chart. The application of the proposed chart in a quality improvement program can be extended to management and service industries. Copyright

Group Runs Double Sampling np Control Chart for Attributes

This paper proposes a group runs (GR) double sampling (DS) np chart to detect increases in the fraction of non-conforming units. It combines the charting statistics of the DS np chart and an extended version of the CRL chart. The performance of the proposed GR DS np chart is evaluated and compared with other attribute charts, namely, the np, GR np, DS np, synthetic DS np, variable sample size (VSS) np, exponentially weighted moving average (EWMA) np, and cumulative sum (CUSUM) np charts, in terms of the average run length (ARL) criterion. The ARL result showed that the optimal GR DS np chart generally performs better than the optimal version of the charts under comparison, for detecting increases in the fraction of non-conforming units, for most shift sizes. The optimal charting parameters that simplify the implementation of the GR DS np chart are provided. The implementation of the proposed chart is illustrated with an example. Based on the significant improvement in the ARL performance, the GR DS np chart is a viable substitute of existing np-type charts for the detection of increases in the fraction of non-conforming units.

Exact run length distribution of the double sampling chart with estimated process parameters

Since the run length distribution is generally highly skewed, a significant concern about focusing too much on the average run length (ARL) criterion is that we may miss some crucial information about a control chart’s performance. Thus it is important to investigate the entire run length distribution of a control chart for an in-depth understanding before implementing the chart in process monitoring. In this paper, the percentiles of the run length distribution for the double sampling (DS) X chart with estimated process parameters are computed. Knowledge of the percentiles of the run length distribution provides a more comprehensive understanding of the expected behaviour of the run length. This additional information includes the early false alarm, the skewness of the run length distribution, and the median run length (MRL). A comparison of the run length distribution between the optimal ARL-based and MRL-based DS X chart with estimated process parameters is presented in this paper. Examples of applications are given to aid practitioners to select the best design scheme of the DS X chart with estimated process parameters, based on their specific purpose.