Construction of double sampling s‐control charts for agile manufacturing

Double sampling (DS) ‐control charts are designed to allow quick detection of a small shift of process mean and provides a quick response in an agile manufacturing environment. However, the DS ‐control charts assume that the process standard deviation remains unchanged throughout the entire course of the statistical process control. Therefore, a complementary DS chart that can be used to monitor the process variation caused by changes in process standard deviation should be developed. In this paper, the development of the DS s‐charts for quickly detecting small shift in process standard deviation for agile manufacturing is presented. The construction of the DS s‐charts is based on the same concepts in constructing the DS ‐charts and is formulated as an optimization problem and solved with a genetic algorithm. The efficiency of the DS s‐control chart is compared with that of the traditional s‐control chart. The results show that the DS s‐control charts can be a more economically preferable alternative in detecting small shifts than traditional s‐control charts. Copyright © 2002 John Wiley & Sons, Ltd.     

Note on ‘Design of double- and triple-sampling X-bar control charts using genetic algorithms’

He et al. formulated the designs of double-sampling (DS) X-bar control charts as optimization problems and solved these problems with a genetic algorithm. Based on the results in solving the DS chart design problems, triple-sampling (TS) X-bar control charts were developed. The efficiency of the TS charts was compared with that of the DS charts. They concluded that the TS charts are more efficient in terms of minimizing the average sample size. We explain that, since they only considered the average sample size when the process is in control, their conclusion is questionable. In fact, the question of which control chart (i.e. the standard Shewart X-bar control chart, DS chart or TS chart) is more efficient depends on both the probability of the process shifting from an in-control to an out-of-control state and the time the control chart will need to detect such a shift.     

Design of double- and triple-sampling X-bar control charts using genetic algorithms

As today's manufacturing firms are moving towards agile manufacturing, quick and economic on-line statistical process control solutions are in high demand. Multiple sampling X-bar control charts are such an alternative. They can be designed to allow quick detection of a small shift in process mean and provide a quick response in an agile manufacturing environment. In this paper, the designs of double-sampling (DS) X-bar control charts are formulated and solved with a genetic algorithm. Based on the results in solving the DS chart design problems, triple sampling (TS) X-bar control charts are developed. The efficiency of the TS charts is compared with that of the DS charts. The results of the comparison show that TS charts are more efficient in terms of minimizing the average sample size.     

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

A 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.     

An improved double sampling s chart

Double sampling (DS) s charts are designed to allow quick detection of a small shift in process standard deviations and to provide a quick response in an agile manufacturing environment. However, current developed DS s charts assume that the sample standard deviations follow a normal distribution. Although valid for relatively large sample sizes, this assumption has limited the application of DS s charts to the monitoring of manufacturing processes with relatively small sample sizes. An improved DS s chart is developed without the normality assumption of the sample standard deviations. The design of the improved DS s chart is formulated as a statistical design optimization problem and solved with a genetic algorithm. The efficiency of the improved DS s charts is compared with that of the DS s charts developed in previous research and with that of the traditional s charts.     

Optimal T2 Control Chart with a Double Sampling Scheme – An Alternative to the MEWMA Chart

Recent studies have shown that a double sampling (DS) scheme yields improvements in detection times of process shifts over variable ratio sampling (VRS) methods that have been extensively studied in the literature. Additionally, a DS scheme is more practical than some of the VRS methods since the sampling interval is fixed. In this paper, we investigate the effect of double sampling on cost, a criterion as important as detection rate. We study economic statistical design of the DS T² chart (ESD DS T²) so that designs are found that are economically optimal but yet meet desired statistical properties such as having low probabilities of false searches and high probabilities of rapid detection of process shifts. Through an illustrative example, we show that relatively large benefits can be achieved in a comparison with the classical T² chart and the statistical DS T² charts with our ESD DS T² approach. Furthermore, the economic performance of the ESD DS T² charts is favorably compared to the MEWMA and other VRS T² control charts in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Memory‐Type Control Charts for Monitoring the Process Dispersion

Control charts have been broadly used for monitoring the process mean and dispersion. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are memory control charts as they utilize the past information in setting up the control structure. This makes CUSUM and EWMA‐type charts good at detecting small disturbances in the process. This article proposes two new memory control charts for monitoring process dispersion, named as floating T ? S² and floating U ? S² control charts, respectively. The average run length (ARL) performance of the proposed charts is evaluated through a simulation study and is also compared with the CUSUM and EWMA charts for process dispersion. It is found that the proposed charts are better in detecting both positive as well as negative shifts. An additional comparison shows that the floating U ? S² chart has slightly smaller ARLs for larger shifts, while for smaller shifts, the floating T ? S² chart has better performance. An example is also provided which shows the application of the proposed charts on simulated datasets. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive multivariate double sampling and variable sampling interval Hotelling's T2 charts

In this article, two adaptive multivariate charts, which combine the double sampling (DS) and variable sampling interval (VSI) features, called the adaptive multivariate double sampling variable sampling interval T² (AMDSVSI T²) and the adaptive multivariate double sampling variable sampling interval combined T² (AMDSVSIC T²) charts, are proposed. The real purpose of using the proposed charts is to provide flexibility by enabling the sampling interval length of the DS T² chart to be varied so that the chart's sensitivity can be enhanced. The fundamental difference between the two proposed charts is that when a second sample is taken, the AMDSVSI T² chart uses the information of the combined sample mean vectors while the AMDSVSIC T² chart uses the information of the combined T² statistics, in deciding about the process status. This research is motivated by existing combined DS and VSI charts in the literature, which show convincing performance improvement over the standard DS chart. Consequently, it is believed that adopting this existing approach in the multivariate case will enable superior multivariate DS charts to be proposed. Numerical results show that the proposed charts outperform the existing standard T² and other adaptive multivariate charts, in detecting shifts in the mean vector, for the zero‐state and steady‐state cases. The performances of both charts when the shift sizes in the mean vector are unknown are also measured. The application of the AMDSVSI T² chart is illustrated with an example.     

Optimal Design of VSI ―X Control Charts for Monitoring Correlated Samples

This paper develops an economic design of variable sampling interval (VSI)―X control charts in which the next sample is taken sooner than usual if there is an indication that the process is off‐target. When designing VSI―X control charts, the underlying assumption is that the measurements within a sample are independent. However, there are many practical situations that violate this hypothesis. Accordingly, a cost model combining the multivariate normal distribution model given by Yang and Hancock with Bai and Lee's cost model is proposed to develop the design of VSI charts for correlated data. An evolutionary search method to find the optimal design parameters for this model is presented. Also, we compare VSI and traditional ―X charts with respect to expected cost per unit time, utilizing hypothetical cost and process parameters as well as various correlation coefficients. The results indicate that VSI control charts outperform the traditional control charts for larger mean shift when correlation is present. In addition, there is a difference between the design parameters of VSI charts when correlation is present or absent. Copyright © 2005 John Wiley & Sons, Ltd.     

Bivariate Dispersion Control Charts for Monitoring Non‐Normal Processes

Multivariate control charts are well known to be more sensitive to the occurrence of variation in processes with two or more correlated quality variables than univariate charts. The use of separate univariate control charts to monitor multivariate process can be misleading as it ignores the correlation between the quality characteristics. The application of multivariate control charts allows for the simultaneous monitoring of the quality characteristics by forming a single chart. The charts operate on the assumption that process observations are normally distributed, but in practice this is not always the case. In this study, we examine and present multivariate dispersion control charts for detecting shifts in the covariance matrix of normal and non‐normal bivariate processes. These control charts, referred to as SMAX, QMAX, MDMAX and MADMAX, rely on dispersion estimates, such as the sample standard deviation (S), interquartile range (Q), average absolute deviation from median (MD) and median absolute deviation (MAD), respectively. We compare the performances of these charts to the existing multivariate generalized variance |S| and RMAX charts for bivariate processes using normal and non‐normal parent distributions. The average run length (ARL) measure is used for the evaluation and comparison of the charts. A real life and simulated datasets are used to demonstrate the application of the charts. Copyright © 2016 John Wiley & Sons, Ltd.     

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 chart's efficiency. In this paper, the VSI run sum (RS) Hotelling's 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 Hotelling's χ² chart significantly outperforms the standard RS Hotelling's χ² chart and the former also performs well compared with other competing charts. By adding more scoring regions, the efficiency of the VSI RS Hotelling's 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 Hotelling's χ² chart. The application of the proposed chart in a quality improvement program can be extended to management and service industries. Copyright © 2016 John Wiley & Sons, Ltd.     

Alternative Hotelling's T2 Charts using Winsorized Modified One‐Step M‐estimator

Hotelling's T² chart is a popular tool for monitoring statistical process control. However, this chart is sensitive in the presence of outliers. To alleviate the problem, this paper proposed alternative Hotelling's T² charts for individual observations using robust location and scale matrix instead of the usual mean vector and the covariance matrix, respectively. The usual mean vector in the Hotelling T² chart is replaced by the winsorized modified one‐step M‐estimator (MOM) whereas the usual covariance matrix is replaced by the winsorized covariance matrix. MOM empirically trims the data based on the shape of the data distribution. This study also investigated on the different trimming criteria used in MOM. Two robust scale estimators with highest breakdown point, namely S_n and T_n were selected to suit the criteria. The upper control limits for the proposed robust charts were calculated based on simulated data. The performance of each control chart is based on the false alarm and the probability of outlier's detection. In general, the performance of an alternative robust Hotelling's T² charts is better than the performance of the traditional Hotelling's T² chart. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive ―X Control Charts with Sampling at Fixed Times

The idea of a variable sampling interval with sampling at fixed times (VSIFT) has been presented by Reynolds. This paper extends this idea to the other two adaptive ―X charts: the variable sampling rate with sampling at fixed times (VSRFT) ―X chart and the variable parameters with sampling at fixed times (VPFT) ―X chart. The VSIFT, VSRFT and VPFT ―X charts are inclusively called the adaptive with sampling at fixed times (AFT) ―X charts in this paper. The control scheme and the design issue are described and discussed for each of the AFT ―X charts. A comparative study shows that the AFT ―X charts have almost the same detection ability as the traditional adaptive ―X charts. However, from the practical viewpoint, the AFT ―X charts are considered to be more convenient to administer than the traditional adaptive ―X charts. Overall, this paper advances the application of ‘sampling at fixed times’ to the adaptive ―X control charts. Copyright © 2004 John Wiley & Sons, Ltd.     

CCC‐r charts' performance with estimated parameter for high‐quality process

CCC‐r charts are effective in detecting process shifts in the nonconforming rate especially for a high‐quality process. The implementation of the CCC‐r charts is usually under the assumption that the in‐control nonconforming rate is known. However, the nonconforming rate is never known, and accurate estimation is difficult. We investigate the effect of estimation error on the CCC‐r charts' performances through the expected value of the average number of observations to signal (EANOS) as well as the standard deviation of the average number of observations to signal (SDANOS). By comparing the in‐control performance of the CCC‐r charts, the CCC‐r chart with a larger value of r is more susceptible to the effects of parameter estimation. Meanwhile, the performance of the CCC‐r charts can converge when detecting upward shifts in p of out‐of‐control processes. We recommend the use of the CCC‐4 chart when considering its effectiveness in detecting shifts as well as its easier construction in practice. Furthermore, it is investigated that the CCC‐4 chart is less sensitive to parameter estimation while being more effective in detecting different process shifts when compared with Geometric CUSUM chart and synthetic chart.     

A New Statistical High‐Quality Process Monitoring Method: The Counted Number Between Omega‐Event Control Charts

Control chart techniques for high‐quality process have attracted great attention in modern precision manufacturing. Traditional control charts are no longer applicable because of high false alarm rate. To solve this problem, in this article a new statistical process monitoring method, the counted number between omega‐event statistical process control charts, abbreviated as CBΩ charts, is proposed. The phrase omega event denotes that one observation falls into some certain interval and the CBΩ chart is to monitor the number of consecutive parts between successive r omega events. On the basis of CBΩ charts, a dual‐CBΩ monitoring scheme is developed. This scheme sets up two CBΩ charts with symmetrical omega events, (μ + ?σ, + ∞) and (? ∞, μ ? ?σ), respectively. The performance of CBΩ charts and dual‐CBΩ monitoring is investigated. Dual‐CBΩ monitoring has shown its capability in detecting both mean and variance shift and convenience in implementation compared with other traditional charts. Dual‐CBΩ monitoring can reduce false alarm rate greatly without introducing an unacceptable loss of sensitivity in detecting out‐of‐control signals in high‐quality process control. Copyright © 2011 John Wiley & Sons, Ltd.     

A comparative study of the CRL‐type control charts

The CRL (Conforming Run Length) type control charts have attracted increasing interest recently for attribute Statistical Process Control (SPC). The two most promising charts of this type are identified as the CRL‐CUSUM chart and the SCRL (Sum of CRLs) chart. This article compares the operating characteristics of these two charts in a comprehensive manner. The general findings reveal that the CRL‐CUSUM chart excels the SCRL chart in detecting downward (decreasing) fraction nonconforming (p) shifts and large‐scale upward (increasing) p shifts. However, the SCRL chart is superior to the CRL‐CUSUM chart in detecting the small and moderate scale upward p shifts, especially when the normal p value is small. The information acquired in this study will provide Quality Assurance (QA) engineers with useful guidance for selecting and applying the CRL‐type control charts. Copyright © 2000 John Wiley & Sons, Ltd.     

Control Charts For Monitoring The Weibull Shape Parameter Based On Type‐II Censored Sample

In this paper, a new statistic is proposed to monitor the Weibull shape parameter when the sample is type II censored. The one‐sided and two‐sided average run length‐unbiased control charts are derived based on the new monitoring statistic. The control limits of the proposed control charts depend on the sample size, the failure number and the false alarm rate. Using Monte Carlo simulation, the performance of the proposed control charts is studied and compared with the range‐based charts proposed by Pascual and Li (2012), which is equivalent to the proposed control charts when r = 2. The simulation results show that the proposed control charts perform better than the ones of Pascual and Li (2012). This paper also evaluates the effects of parameter estimation on the proposed control charts. Finally, an example is used to illustrate the proposed control charts. Copyright © 2012 John Wiley & Sons, Ltd.     

The variable sampling rate X̄ control charts for monitoring autocorrelated processes

Recent studies demonstrated that the adaptive X? control charts are more efficient than fixed parmeters (FP) X? control chart from statistical and economic criteria. The usual assumption for designing a control chart is that the observations from the process are independent. However, for many processes, such as chemical processes, consecutive measurements are often highly correlated, especially when the interval between samples is small. In the present paper, the observations are modeled as an AR(1) process plus a random error, and the properties of the variable sampling rate (VSR) X? charts are evaluated and studied under this model. Based on the study, the VSR X? chart is faster than the FP, variable sampling interval and variable sample size X? control charts in detecting mean shifts. However, when the level of autocorrelation is high or the process mean shift is large, the advantage of the VSR X? chart is reduced. Copyright © 2008 John Wiley & Sons, Ltd.     

Steady‐state ARL analysis of ARL‐unbiased EWMA‐RZ control chart monitoring the ratio of two normal variables

Very recently, control charts for monitoring the ratio of 2 normal variables have been investigated in statistical process control. In the two‐sided case, however, these control charts tend to be average run length (ARL) biased, in the sense that some out‐of‐control ARL values are larger than the in‐control ARL. This paper proposes an ARL‐unbiased EWMA control chart for monitoring of this kind of ratio with each subgroup consisting of n?1 sample units. Also, to study the long‐term properties of ARL‐unbiased EWMA‐RZ control chart, we investigate the steady‐state ARL. Several tables and figures are given to show the statistical properties of the proposed control charts. The comparison results show that the proposed ARL‐unbiased chart outperforms other two‐sided control charts in terms of the zero‐state and steady‐state ARL. An example illustrates the use of this chart on a real quality control problem from the food industry.     

A comparative study of phase II robust multivariate control charts for individual observations

Use of Hotelling's T² charts with high breakdown robust estimates to monitor multivariate individual observations are the recent trend in the control chart methodology. Vargas (J. Qual. Tech. 2003; 35: 367‐376) introduced Hotelling's T² charts based on the minimum volume ellipsoid (MVE) and the minimum covariance determinant (MCD) estimates to identify outliers in Phase I data. Studies carried out by Jensen et al. (Qual. Rel. Eng. Int. 2007; 23: 615‐629) indicated that the performance of these charts heavily depends on the sample size, amount of outliers and the dimensionality of the Phase I data. Chenouri et al. (J. Qual. Tech. 2009; 41: 259‐271) recently proposed robust Hotelling's T² control charts for monitoring Phase II data based on the reweighted MCD (RMCD) estimates of the mean vector and covariance matrix from Phase I. They showed that Phase II RMCD charts have better performance compared with Phase II standard Hotelling's T² charts based on outlier free Phase I data, where the outlier free Phase I data were obtained by applying MCD and MVE T² charts to historical data. Reweighted MVE (RMVE) and S‐estimators are two competitors of the RMCD estimators and it is a natural question whether the performance of Phase II Hotelling's T² charts with RMCD and RMVE estimates exhibits similar pattern observed by Jensen et al. (Qual. Rel. Eng. Int. 2007; 23: 615‐629) in the case of MCD and MVE‐based Phase I Hotelling's T² charts. In this paper, we conduct a comparative study to assess the performance of Hotelling's T² charts with RMCD, RMVE and S‐estimators using large number of Monte Carlo simulations by considering different data scenarios. Our results are generally in favor of the RMCD‐based charts irrespective of sample size, outliers and dimensionality of Phase I data. Copyright © 2010 John Wiley & Sons, Ltd.