Conforming Run Length Research Articles

Evaluating the Performance of Synthetic Double Sampling np Chart Based on Expected Median Run Length

To keep an eye on the status of high-quality processes for fraction nonconforming, the synthetic double sampling (SDS) np chart is a helpful tool. The SDS np chart is a hybrid between the double sampling (DS) np chart and the conforming run length (CRL) chart. The performance of a control chart is typically judged solely using the average run length (ARL). However, as the shape of the run length (RL) distribution varies with the magnitude of the shift in the process fraction nonconforming, the ARL no longer provides clear interpretation of a chart’s performance. Subsequently, enhanced DS np charts that use median run length (MRL) and expected median run length (EMRL) measures, including SDS np with MRL have recently been proposed for addressing this setback. To broaden the functionality of SDS np, in this work, the unexplored use of EMRL as alternative performance measure is developed by means of Markov chain model. Additionally, in both the zero-state (ZS) and steady-state (SS) modes, the novel optimal designs algorithms are described for computing the optimal charting parameters of the SDS np chart, for both MRL1 and EMRL1 minimizations, without any unfavourable feature of bilateral sensitivity. Both the MRL and EMRL performances of SDS np, synthetic np, and DS np charts are compared. Optimal designs charting parameters and sensitivity analyses are provided to aid the practical application of SDS np chart.

Statistical design of an adaptive synthetic X¯$\bar{X}$ control chart for autocorrelated processes

AbstractThis paper aims to design a variable parameters synthetic control chart for first‐order AR(1) autocorrelated data following a Gaussian process. To improve the statistical performance in detecting small changes and maintaining a low false alarm rate, the variable parameters control chart was combined with the Conforming Run Length (CRL) sub‐chart to determine when to implement tight or relaxed control and intervene to identify an assignable cause. The statistical design of the proposed chart was performed under a discrete‐time Markov chain approach and a non‐linear programming mathematical model to obtain, using a genetic algorithm (GA), the values of the design parameters that minimise the average time to signal the out‐of‐control state (). A sensitivity analysis was performed to examine the behaviour of the in the face on changes in the parameters. Also, a performance evaluation was conducted to compare the proposed chart with other adaptive synthetic charts, the synthetic control chart and the chart developed by Costa and Machado. Our results indicate that this proposal is faster to detect small changes in the process mean considering the autocorrelation of the data.

A synthetic exponentially weighted moving average control chart to monitor process median based on ranked set sampling

Exponentially Weighted Moving Average (EWMA) control charts yield insights into data in a way more comprehensible to the practitioners and researchers because of its capability in discovering small to moderate process mean shifts. EWMA control chart is incorporated with conforming run length (CRL) chart, named synthetic EWMA chart, to enhance the performance of the chart in detecting the out-of-control signal. Synthetic EWMA chart based on ranked set sampling (RSS) for monitoring process mean has been proposed as it advanced the detection of chart over a series of mean shifts. With the situation that normality assumption is scarcely attain in practice, we proposed synthetic EWMA median chart based on RSS. Rather than select average run length (ARL) as sole performance evaluating tool, the median and percentiles of run-length distribution are used to examine the performance of the proposed chart as it provides more information on the entire run-length distribution. Near-optimal parameters of the proposed chart will be acquired by setting the incontrol ARL at a designated value. The run length performances of the proposed chart are then compared with the existing charts such as EWMA median chart based on RSS.

Improved CUSUM monitoring of Markov counting process with frequent zeros

AbstractThere has been a growing interest in monitoring processes featuring serial dependence and zero inflation. The phenomenon of excessive zeros often occurs in count time series because of the advancement of quality in manufacturing process. In this study, we propose three control charts, such as the cumulative sum chart with delay rule (CUSUM‐DR), conforming run length (CRL)‐CUSUM chart, and combined Shewhart CRL‐CUSUM chart, to enhance the performance of monitoring Markov counting processes with excessive zeros. Numerical experiments are conducted based on integer‐valued autoregressive time series models, for example, zero‐inflated Poisson INAR and INARCH, to evaluate the performance of the proposed charts designed for the detection of mean increase. A real example is also illustrated to demonstrate the usability of our proposed charts.

An overview of synthetic‐type control charts: Techniques and methodology

AbstractIn this paper, we provide an overview of a class of control charts called the synthetic charts. Synthetic charts are a combination of a traditional chart (such as a Shewhart, CUSUM, or EWMA chart) and a conforming run‐length (CRL) chart. These charts have been considered in order to maintain the simplicity and improve the performance of small and medium‐sized shift detection of the traditional Shewhart charts. We distinguish between different types of synthetic‐type charts currently available in the literature and highlight how each is designed and implemented in practice. More than 100 publications on univariate and multivariate synthetic‐type charts are reviewed here. We end with some concluding remarks and a list of some future research ideas.

On designing new optimal synthetic Tukey’s control charts

ABSTRACTTukey’s control chart is generally used for monitoring the processes where the measurement process physically damages the product. It is based on single observation and robust to outliers. In this paper, two optimal synthetic Tukey’s control charts are proposed by integrating the conforming run length chart with the Tukey’s control chart and its modification. The performance comparison of the proposed charts with the existing Tukey’s control charts is made by using out-of-control average run length and extra quadratic loss as performance metrics. The proposed charts offer better protection against the process shifts as compare to the existing Tukey’s control charts when the underlying process distribution is symmetric or asymmetric. Simulation studies also establish the supremacy of the proposed control charts over the existing Tukey’s control charts. In the end, an illustrative example based on a real data set of the combined cycle power plant is provided for practical implementation.

A synthetic double sampling control chart for process mean using auxiliary information

AbstractA control chart is a simple yet powerful tool that is extensively adopted to monitor shifts in the process mean. In recent years, auxiliary‐information–based (AIB) control charts have received considerable attention as these control charts outperform their counterparts in monitoring changes in the process parameter(s). In this article, we integrate the conforming run length chart with the existing AIB double sampling (AIB DS) chart to propose an AIB synthetic DS chart for the process mean. The AIB synthetic DS chart also encompasses the existing synthetic DS chart. A detailed discussion on the construction, optimization, and evaluation of the run length profiles is provided for the proposed control chart. It is found that the optimal AIB synthetic DS chart significantly outperforms the existing AIB Shewhart, optimal AIB synthetic, and AIB DS charts in detecting various shifts in the process mean. An illustrative example is given to demonstrate the implementation of the existing and proposed AIB control charts.

New sampling strategies to reduce the effect of autocorrelation on the synthetic T2 chart to monitor bivariate process

AbstractIn this paper, synthetic T2 chart is developed to monitor bivariate process with correlated variables and autocorrelated observations. The proposed chart is a combination of the Hotelling's T2 chart and the conforming run length chart. The operation and design of the chart are described when observations are autocorrelated and cross correlated. The first‐order vector autoregressive process VAR (1) is used to model the bivariate data from an autocorrelated process of interest. Using an average run length as performance measure criterion in the VAR (1) model, it is observed that autocorrelation seriously impact the performance of the synthetic T2 chart. To reduce the effect of autocorrelation on the performance of the synthetic T2 chart, the skip and mixed sampling strategies are implemented to form rational subgroups in the construction of synthetic T2 chart. The average run length performance of the synthetic T2 chart implementing these strategies is compared with that of the standard strategy of formation of rational subgroups. It is observed that implementing skip and mixed sampling strategies within rational subgroup improves the performance of the synthetic T2 chart.

A new nonparametric synthetic EWMA control chart for monitoring process mean

ABSTRACTThe exponentially weighted moving average (EWMA) control charts are widely used in chemical and process industries because of their excellent speed in catching small to moderate shifts in the process target. In usual practice, many data come from a process where the monitoring statistic is non-normally distributed or it follows an unknown probability distribution. This necessitates the use of distribution-free/nonparametric control charts for monitoring the deviations from the process target. In this paper, we integrate the existing EWMA sign chart with the conforming run length chart to propose a new synthetic EWMA (SynEWMA) sign chart for monitoring the process mean. The SynEWMA sign chart encompasses the synthetic sign and EWMA sign charts. Monte Carlo simulations are used to compute the run length profiles of the SynEWMA sign chart. Based on a comprehensive comparison, it turns out that the SynEWMA sign chart is able to perform substantially better than the existing EWMA sign chart. Both real and simulated data sets are used to explain the working and implementation of existing and proposed control charts.

Optimal designs of the synthetic t chart with estimated process mean

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.

Synthetic exponential control charts with unknown parameter

ABSTRACTThe existing synthetic exponential control charts are based on the assumption of known in-control parameter. However, the in-control parameter has to be estimated from a Phase I dataset. In this article, we use the exact probability distribution, especially the percentiles, mean, and standard deviation of the conditional average run length (ARL) to evaluate the effect of parameter estimation on the performance of the Phase II synthetic exponential charts. This approach accounts for the variability in the conditional ARL values of the synthetic chart obtained by different practitioners. Since parameter estimation results in more false alarms than expected, we develop an exact method to design the adjusted synthetic charts with desired conditional in-control performance. Results of known and unknown in-control parameter cases show that the control limit of the conforming run length sub-chart of the synthetic chart should be as small as possible.

Determining the optimum process mean in a two-stage production system based on conforming run length sampling method

We develop a profit optimization model based on Markovian approach to determine the optimum process mean in the two-stage serial production system. If quality characteristics of the items fall within the specification limits then the items are accepted. Our inspection approach in sampling plan is that when the number of conforming items between successive defective items falls below the lower control threshold, then the batch is rejected, and if it falls above the upper control threshold, then the batch is accepted and if it falls within the control thresholds then the process of inspecting the items continues. Taguchi’s symmetric quadratic quality loss function is applied for evaluating the product quality. Then a numerical example is provided to demonstrate the application of proposed model. Also a sensitivity analysis of the model is provided.

Optimal design of synthetic np control chart based on median run length

ABSTRACTThe synthetic np chart is a combination of the np sub-chart and the conforming run length sub-chart. A procedure for the optimal design of the synthetic np chart is provided by minimizing the out-of-control median run length (MRL). The comparative results show that the synthetic np chart performs better than the corresponding standard np chart for detecting process shifts in the fraction non conforming, in terms of the MRL. An example is given to illustrate the operation of the synthetic np chart.

Phase II synthetic exponential charts and effect of parameter estimation

Exponential charts for monitoring time-between-events data are widely studied recently as it has shown a wide range in different applications. To improve the sensitivity of the simple exponential chart, this paper develops a synthcontrol chart to monitor the two-sided shifts in the exponential mean. This synthetic chart consists of an exponential sub-chart and a conforming run length (CRL) sub-chart and it is designed from two new perspectives. Previous synthetic charts are constructed based on the optimization program for a specified shift size, which proves to be difficult to predict in practice. In this paper, the synthetic chart is constructed based on a new design method which takes the mean (ARL) and the standard deviation (SDRL) of the run length into consideration. The other improvement is that the exponential sub-chart is designed to ensure that the CRL sub-chart only needs a lower control limit and that the proposed synthetic chart is ARL-unbiased. The effect of parameter estimation on the proposed synthetic chart is also studied in this paper. Performance comparisons show that our proposed chart has a better performance in general.

New Synthetic CUSUM and Synthetic EWMA Control Charts for Monitoring the Process Mean using Auxiliary Information

The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are potentially powerful process monitoring tool because of their excellent speed in detecting small to moderate shifts in the process parameters. These control charts can be further improved by integrating them with the conforming run length control chart, resulting in the synthetic CUSUM (SynCUSUM) and synthetic EWMA (SynEWMA) charts. In this paper, we enhance the detection abilities of the SynCUSUM and SynEWMA charts using the auxiliary information. With suitable assumptions, the proposed control charts encompass the existing SynCUSUM, SynEWMA, CUSUM, and EWMA charts. Extensive Monte Carlo simulations are used to study the run length profiles of the proposed control charts. It turns out that the proposed near‐optimal control charts with the auxiliary information perform uniformly and substantially better than the existing near‐optimal SynCUSUM, SynEWMA, CUSUM, and EWMA charts. The proposed and existing control charts are also illustrated with the help of an example. Copyright © 2017 John Wiley & Sons, Ltd.

Development of an Optimal Design for Conforming Run Length Sampling Methods in the Presence of Inspection Errors

Abstract In this research, a new sampling system was introduced based on the concept of conforming run length. The inspection error was considered in the proposed model as well. This methodology was based on Markov modeling and a negative binomial distribution. The proposed sampling system was compared with classical sampling methods such as the Dodge-Romig single sampling plan based on average outgoing quality limit (AOQL) in the presence of inspection errors. The superiority of the proposed method for larger values of lot sizes and process average was denoted using a comparison study.

Economic-statistical design of synthetic max chart

This paper studies the economic-statistical design of a synthetic max chart. The synthetic max chart consists of a combination of a max sub-chart and a conforming run length sub-chart. The design of the chart is developed by considering the minimization of the cost function, subject to constraints on the average run length. A sensitivity analysis is performed to investigate the effects of the cost and process parameters on the solution of the economic-statistical design. Furthermore, the economic-statistical designs of the synthetic max chart and the max chart are compared in this study. The results show that the synthetic max chart outperforms the max chart from an economic-statistical viewpoint. Although the economic-statistical design of the synthetic max chart has the disadvantage of a slight increase in cost over the economic design, the economic-statistical design gives better in-control statistical performance, where the desirable statistical properties are justified. This study is also extended to the multivariate case.

Performance comparison for the CRL control charts with estimated parameters for high-quality processes

The conforming run length (CRL) control charts based on Bernoulli data have been shown to be effective when monitoring the proportion nonconforming rate, especially for high-quality processes. Considering the usage simplicity in practice, the count of conforming chart, RL2 chart and the geometric cumulative sum (CUSUM) chart are subject matter of this paper. When implementing the CRL control charts, the in-control proportion nonconforming rate is seldom known and accurate estimation is needed. Thus, we investigate the effects of parameter estimation on the CRL control charts using the average number of observations to signal and the standard deviation of the average number of observations to signal with a Bayes estimator. The SDANOS values of the CRL control charts show that practitioners should rarely expect in-control performance close to that obtained under the assumption when the process parameters are known. By comparing in-control performance of the CRL control charts, the geometric CUSUM chart is most sensitive to the parameter estimation.

Control charts for the coefficient of variation

The coefficient of variation (CV) has received an extensive attention in quality control. The existing control charts for monitoring the two-sided changes in the CV do not consider the average run length (ARL) unbiased criterion. In practice, the practitioners prefer the ARL-unbiased control charts. This paper proposes an ARL-unbiased Shewhart-type control chart to monitor the two-sided shifts in the CV. Then an ARL-unbiased synthetic chart in which the conforming run length sub-chart only needs a lower control limit is proposed to improve the performance of the ARL-unbiased Shewhart-type chart. The comparison results show that the proposed synthetic chart outperforms the proposed ARL-unbiased control chart in terms of the ARL. The effects of the parameter estimation on the proposed charts are also studied. Finally, a real example is used to illustrate the proposed control charts.

Synthetic double sampling s chart

ABSTRACTThis study proposes a synthetic double sampling s chart that integrates the double sampling (DS) s chart and the conforming run length chart. An optimization procedure is proposed to compute the optimal parameters of the synthetic DS s chart. The performance of the synthetic DS s chart is compared with other existing control charts for monitoring process standard deviation. The results show that the synthetic DS s chart is more effective for detecting increases in the process standard deviation for a wide range of shifts. An example is provided to illustrate the operation procedure of the synthetic DS s chart.