Wei Lin Teoh

Optimal design of the double sampling X chart with estimated parameters based on median run length

The double sampling (DS) X@? chart when the process parameters are unknown and have to be estimated from a reference Phase-I dataset is studied. An expression for the run length distribution of the DS X@? chart is derived, by conditioning and taking parameter estimation into account. Since the shape and the skewness of the run length distribution change with the magnitude of the mean shift, the number of Phase-I samples and sample sizes, it is shown that the traditional charts performance measure, i.e. the average run length, is confusing and not a good representation of a typical charts performance. To this end, because the run length distribution is highly right-skewed, especially when the shift is small, it is argued that the median run length (MRL) provides a more intuitive and credible interpretation. From this point of view, a new optimal design procedure for the DS X@? chart with known and estimated parameters is developed to compute the charts optimal parameters for minimizing the out-of-control MRL, given that the values of the in-control MRL and average sample size are fixed. The optimal chart which provides the quickest out-of-control detection speed for a specified shift of interest is designed according to the number of Phase-I samples commonly used in practice. Tables are provided for the optimal chart parameters along with some empirical guidelines for practitioners to construct the optimal DS X@? charts with estimated parameters. The optimal charts with estimated parameters are illustrated with a real application from a manufacturing company.

The revised m-of-k runs rule based on median run length.

Runs rules are used to increase the sensitivity of the Shewhart control chart in detecting small and moderate process mean shifts. Most of the charts incorporating runs rules are designed based on the average run length (ARL). It is known that the shape of the run length distribution changes according to the magnitude of the shift in the process mean, ranging from highly skewed when the process is in-control to approximately symmetric when the shift is large. Since the shape of the run length distribution changes with the magnitude of the shift in the mean, the median run length (MRL) provides a more meaningful explanation about the in-control and out-of-control performances of a control chart. In this article, we propose the design of the revised m-of-k runs rule based on MRL. In addition, the standard deviation of the run length (SDRL) of the revised m-of-k rule will also be studied. The revised m-of-k runs rule, suggested by Antzoulakos and Rakitzis (2008), was originally designed based on ARL. The Markov chain technique is employed to obtain the MRLs. Compared with the standard chart, the MRL results show that the revised rules give better performances for small and moderate mean shifts, while maintaining the same sensitivity towards large mean shifts. The MRL results are in accordance with the results obtained by Antzoulakos and Rakitzis (2008), where the rules are designed based on ARL.

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 Exact Run Length Distribution and Design of the Shewhart Chart with Estimated Parameters Based on Median Run Length

This paper investigates the Shewhart chart with estimated parameters. When parameters are unknown, we show that the average run length is a confusing performance measure and that the median run length, which is a better representation of the central tendency associated with different shapes of the run length distribution, is more intuitive. A new design model for the Shewhart chart with estimated parameters is developed. With this proposed design model, the Shewhart chart for both known and estimated parameters, even with a small number of Phase-I samples, will have an almost similar sensitivity for a specified shift.

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.

Optimal Designs of the Variable Sample Size X¯Chart Based on Median Run Length and Expected Median Run Length

The variable sample size (VSS) X chart, devoted to the detection of moderate mean shifts, has been widely investigated under the context of the average run-length criterion. Because the shape of the run-length distribution alters with the magnitude of the mean shifts, the average run length is a confusing measure, and the use of percentiles of the run-length distribution is considered as more intuitive. This paper develops two optimal designs of the VSS X chart, by minimizing (i) the median run length and (ii) the expected median run length for both deterministic and unknown shift sizes, respectively. The 5th and 95th percentiles are also provided in order to measure the variation in the run-length distribution. Two VSS schemes are considered in this paper, that is, when the (i) small sample size (nS) or (ii) large sample size (nL) is predefined for the first subgroup (n1). The Markov chain approach is adopted to evaluate the performance of these two VSS schemes. The comparative study reveals that improvements in the detection speed are found for these two VSS schemes without increasing the in-control average sample size. For moderate to large mean shifts, the optimal VSS X chart with n1 =nL significantly outperforms the optimal EWMA X chart, while the former is comparable to the latter when n1 = nS. Copyright

Monitoring the Coefficient of Variation Using a Variable Sampling Interval EWMA Chart

In recent years, the coefficient of variation (CV) chart is receiving increasing attention in quality control. A number of studies demonstrated that adaptive charts could detect process shifts faster than traditional charts. This paper proposes an EWMA chart with variable sampling interval (VSI) to monitor the CV. Formulas for computing the performance measures of the VSI EWMA-γ2 chart are derived using Markov chain, where γ2 denotes the CV squared. Comparative studies show that the VSI EWMA-γ2 chart significantly outperforms other competing charts. An example using real manufacturing data shows that the VSI EWMA-γ2 chart performs well in applications.

Run sum chart for monitoring multivariate coefficient of variation

Abstract Coefficient of variation (CV) is an important quality characteristic to take into account when the process mean and standard deviation are not constants. A setback of the existing chart for monitoring the multivariate CV is that the chart is slow in detecting a multivariate CV shift in the Phase-II process. To overcome this problem, this paper proposes a run sum chart for monitoring the multivariate CV in the Phase-II process. The average run length (ARL), standard deviation of the run length (SDRL) and expected average run length (EARL), under the zero state and steady state cases, are used to compare the performance of the proposed chart with the existing multivariate CV chart. The proposed chart’s optimal parameters are computed using the Mathematica programs, based on the Markov chain model. Two one-sided run sum charts for monitoring the multivariate CV are considered, where they can be used simultaneously to detect increasing and decreasing multivariate CV shifts. The effects of different in-control CV values, number of regions, shift and sample sizes, and number of variables being monitored are studied. The implementation of the proposed chart is illustrated with an example using the data dealing with steel sleeve inside diameters.

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.

Optimal Designs of the Synthetic t Chart with Estimated Process Mean

Abstract The synthetic t chart, which integrates a t chart and a conforming run length chart, is robust against changes in the process standard deviation. Traditionally, the synthetic t chart is studied by assuming that the in-control process mean is known. Practically, this is not always the case. The process mean is rarely known and it has to be estimated from a Phase-I dataset. Therefore, this paper presents the Markov chain approach for studying the run-length properties of the synthetic t chart with estimated process mean for both zero- and steady-state cases. The impact of the mean estimation on the synthetic t chart is evaluated and compared with its known-process-mean counterpart and the synthetic X ¯ chart. For optimum implementation, this paper develops two optimal design strategies for the synthetic t chart with estimated process mean, by minimizing (i) the average run length (ARL) and (ii) the expected ARL, for deterministic and unknown shift sizes, respectively. By taking the number of Phase-I samples and sample sizes adopted in practice into consideration, tables listing the new optimal charting parameters of the proposed chart are provided in this paper. Comparative studies show that there are some potential benefits, especially the desirable robustness property, of the synthetic t chart with estimated process mean over the synthetic X ¯ , Shewhart X ¯ and t charts with estimated process parameters or mean. The application of the synthetic t chart with estimated process mean is illustrated with real industrial data gathered from a silicon epitaxy process.