Weighted adaptive multivariate CUSUM control charts

An adaptive multivariate cumulative sum (AMCUSUM) control chart has received considerable attention because of its ability to dynamically adjust the reference parameter whereby achieving a better performance over a range of mean shifts than the conventional multivariate cumulative sum (CUSUM) charts. In this paper, we introduce a progressive mean–based estimator of the process mean shift and then use it to devise new weighted AMCUSUM control charts for efficiently monitoring the process mean. These control charts are easy to design and implement in a computerized environment compared with their existing counterparts. Monte Carlo simulations are used to estimate the run‐length characteristics of the proposed control charts. The run‐length comparison results show that the weighted AMCUSUM charts perform substantially and uniformly better than the classical multivariate CUSUM and AMCUSUM charts in detecting a range of mean shifts. An example is used to illustrate the working of existing and proposed multivariate CUSUM control charts.     

Robustness properties of multivariate EWMA control charts

In this paper, the robustness of the multivariate exponentially weighted moving average (MEWMA) control chart to non‐normal data is examined. Two non‐normal distributions of interest are the multivariate distribution and the multivariate gamma distribution. Recommendations for constructing MEWMA control charts when the normality assumption may be violated are provided. Copyright © 2002 John Wiley & Sons, Ltd.     

Percentile‐based control chart design with an application to Shewhart X̅ and S2 control charts

We present a method to design control charts such that in‐control and out‐of‐control run lengths are guaranteed with prespecified probabilities. We call this method the percentile‐based approach to control chart design. This method is an improvement over the classical and popular statistical design approach employing constraints on in‐control and out‐of‐control average run lengths since we can ensure with prespecified probability that the actual in‐control run length exceeds a desired magnitude. Similarly, we can ensure that the out‐of‐control run length is less than a desired magnitude with prespecified probability. Some numerical examples illustrate the efficacy of this design method.     

The effects of constructed bivariate copulas on multivariate control charts effectiveness

The average control chart monitors the shifts in the process. The familiar multivariate control charts are used to detect the mean vector of the process such as multivariate cumulative sum (MCUSUM) and Hotelling's T² control charts. In this paper, the effects of constructing bivariate copulas on multivariate control charts, that is, MCUSUM and Hotelling's T² control charts are intensively investigated when observations are drawn from the exponential distribution. Moreover, the dependence levels of observations are classified to be weak, moderate, and strong in both positive and negative values by Kendall's tau. The numerical results were obtained by Monte Carlo simulation to explore the average run length (ARL). The simulation results show that the performance of Hotelling's T² control chart is superior to the MCUSUM control chart for all shifts in the mean vector of process. Furthermore, from applying the presented control chart to two sets of real data, data set of the strength of 1.5 cm glass fibers measured at the National Physical Laboratory, England and data set of the strength of glass of the aircraft window, it was found that for a small shift ($\delta \le 0.1$), the MCUSUM control chart is better than Hotelling's T² control chart.     

Memory‐type multivariate control charts with auxiliary information for process mean

In statistical process control, it is a common practice to increase the sensitivity of a control chart with the help of an efficient estimator of the underlying process parameter. In this paper, we consider an efficient estimator that requires information on several study variables along with one or more auxiliary variables when estimating the mean of a multivariate normally distributed process. Using this auxiliary‐information‐based (AIB) process mean estimator, we propose new multivariate EWMA (MEWMA), double MEWMA (DMEWMA), and multivariate CUSUM (MCUSUM) charts for monitoring the process mean, denoted by the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts, respectively. The run length characteristics of the proposed multivariate charts are computed using Monte Carlo simulations. The proposed charts are compared with their existing counterparts in terms of the run length characteristics. It turns out that the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts are uniformly and substantially better than the MEWMA, DMEWMA, and MCUSUM charts, respectively, when detecting different shifts in the process mean. A real dataset is considered to explain the implementation of the proposed and existing multivariate control charts.     

Efficient shift detection using multivariate exponentially-weighted moving average control charts and principal components

This paper demonstrates the use of principal components in conjunction with the multivariate exponentially-weighted moving average (MEWMA) control procedure for process monitoring. It is demonstrated that the number of variables to be monitored is reduced through this approach, and that the average run length to detect process shifts or upsets is substantially reduced as well. The performance of the MEWMA applied to all the variables may be related to the MEWMA control chart that uses principal components through the non-centrality parameter. An average run length table demonstrates the advantages of the principal components MEWMA over the procedure that uses all of the variables. An illustrative example is provided.     

About Shewhart control charts to monitor the Weibull mean

In this paper, a new reparametrization expressed in terms of the process mean for Weibull distribution is studied; thus, the monitoring of the process mean can be made directly. Additionally, we call attention that the asymptotic control limits for control chart by central limit theorem (CLT) may lead to a serious erroneous decision. Definitively, they can only be used to signal small/medium shifts in the process mean but with a very very large sample size. We present guidelines for practitioners about the minimum sample size needed to match out‐of‐control average run length (ARL₁) with the exact and asymptotic control limits in function of the shape parameter after an extensive simulation study. The proposed schemes are applied to monitoring the Weibull mean parameter of the strength distribution of a carbon fibber used in composite materials.     

Process monitoring research with various estimator-based MEWMA control charts

Multivariate exponentially weighted moving average (MEWMA) control chart with five different estimators as population covariance matrix is rarely applied to monitor small fluctuations in the statistical process control. In this article, mathematical models of the five estimators (S₁, S₂, S₃, S₄, S₅) are established, with which the relevant MEWMA control charts are obtained, respectively. Thereafter, the process monitoring performance of the five control charts is simulated. And the simulation results show that the S₄ estimator-based MEWMA control chart is of the best performance both in step offset failure mode and ramp offset failure mode. Since the inline process monitoring of photovoltaic manufacturing is intended to be a problem of multivariate statistics process analysis, the feasibility and effectiveness of the proposed model are elaborated in the case study during the cell testing and sorting process control for the fabrication of multicrystalline silicon solar cells.     

Combined Application of Shewhart and Cumulative Sum R Chart for Monitoring Process Dispersion

This study analyzes the performance of combined applications of the Shewhart and cumulative sum (CUSUM) range R chart and proposes modifications based on well‐structured sampling techniques, the extreme variations of ranked set sampling, for efficient monitoring of changes in the process dispersion. In this combined scheme, the Shewhart feature enables quick detection of large shifts from the target standard deviation while the CUSUM feature takes care of small to moderate shifts from the target value. We evaluate the numerical performance of the proposed scheme in terms of the average run length, standard deviation of run length, the average ratio average run length, and average extra quadratic loss. The results show that the combined scheme can detect changes in the process that were small or large enough to escape detection by the lone Shewhart R chart or CUSUM R chart, respectively. We present a comparison of the proposed schemes with several dispersion charts for monitoring changes in process variability. The practical application of the proposed scheme is demonstrated using real industrial data. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

The average run length (ARL) is usually used as a sole measure of performance of a multivariate control chart. The Hotelling's T², multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts are commonly optimally designed based on the ARL. Similar to the case of univariate quality control, in multivariate quality control, the shape of the run length distribution changes in accordance to the magnitude of the shift in the mean vector, from highly skewed when the process is in‐control to nearly symmetric for large shifts. Because the shape of the run length distribution changes with the magnitude of the shift in the mean vector, the median run length (MRL) provides additional and more meaningful information about the in‐control and out‐of‐control performances of multivariate charts, not given by the ARL. This paper provides a procedure for optimal designs of the multivariate synthetic T² chart for the process mean, based on MRL, for both the zero and steady‐state modes. Two Mathematica programs, each for the zero state and steady‐state modes are given for a quick computation of the optimal parameters of the synthetic T² chart, designed based on MRL. These optimal parameters are provided in the paper, for the bivariate case with sample sizes, nin{4, 7, 10}. The MRL performances of the synthetic T², MEWMA and Hotelling's T² charts are also compared. Copyright © 2011 John Wiley & Sons, Ltd.     

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

In 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.     

An economic comparison of CUSUM and Shewhart charts

This paper compares the economic performance of CUSUM and Shewhart schemes for monitoring the process mean. We develop new simple models for the economic design of Shewhart schemes and more accurate ways to evaluate the economic performance of CUSUM schemes. The results of the comparative analysis show that the economic advantage of using a CUSUM scheme rather than the simpler Shewhart chart is substantial only when a single measurement is available at each sampling instance, i.e., only when the sample size is always n = 1, or when the sample size is constrained to low values.     

Memory-type multivariate charts with fixed and variable sampling intervals for process mean when covariance matrix is unknown

Memory-type multivariate charts have been widely recognized as a potentially powerful process monitoring tool because of their excellent speed in detecting small-to-moderate shifts in the mean vector of a multivariate normally distributed process, namely, the multivariate EWMA (MEWMA), double MEWMA, Crosier multivariate CUSUM (MCUSUM), and Pignatiello and Runger MCUSUM charts. These multivariate charts are based on the assumption that the covariance matrix is known in advance; but, it may not be known in practice. It is thus not possible to use these multivariate charts unless a large Phase I dataset is available from an in-control process. In this paper, we propose multivariate charts with fixed and variable sampling intervals for the process mean vector when the covariance matrix is estimated from sample. Using the Monte Carlo simulation method, the run length characteristics of the multivariate charts are computed. It is shown that the in-control and out-of-control run length performances of the proposed multivariate charts are robust to the changes in the process covariance matrix, while the existing multivariate charts are not. A real dataset is taken to explain the implementation of the proposed multivariate charts.     

An exact method for designing Shewhart and S2 control charts to guarantee in-control performance

The in-control performance of Shewhart and S² control charts with estimated in-control parameters has been evaluated by a number of authors. Results indicate that an unrealistically large amount of Phase I data is needed to have the desired in-control average run length (ARL) value in Phase II. To overcome this problem, it has been recommended that the control limits be adjusted based on a bootstrap method to guarantee that the in-control ARL is at least a specified value with a certain specified probability. In this article we present simple formulas using the assumption of normality to compute the control limits and therefore, users do not have to use the bootstrap method. The advantage of our proposed method is in its simplicity for users; additionally, the control chart constants do not depend on the Phase I sample data.     

Monitoring multivariate coefficient of variation with upward Shewhart and EWMA charts in the presence of measurement errors using the linear covariate error model

In practice, measurement errors exist and ignoring their presence may lead to erroneous conclusions in the actual performance of control charts. The implementation of the existing multivariate coefficient of variation (MCV) charts ignores the presence of measurement errors. To address this concern, the performances of the upward Shewhart-MCV and exponentially weighted moving average MCV charts for detecting increasing MCV shifts, using a linear covariate error model, are investigated. Explicit mathematical expressions are derived to compute the limits and average run lengths of the charts in the presence of measurement errors. Finally, an illustrative example using a real-life dataset is presented to demonstrate the charts’ implementation.     

Advanced multivariate cumulative sum control charts based on principal component method with application

Existing multivariate cumulative sum (MCUSUM) control charts involve entire associated variables of a process to monitor variations in the mean vector. In this study, we have offered MCUSUM control charts with principal component method (PCM). The proposed MCUSUM control charts with PCM capture the whole process variations using fewer latent variables (principal components) while preserving as much data variability as possible. To show the significance of proposed MCUSUM control charts with PCM, various performance measures are considered including average run length, extra quadratic loss, relative average run length, and performance comparison index. Furthermore, performance measures are calculated through advanced Monte Carlo simulation method to explore the behavior of proposed MCUSUM control charts and to conduct comparative analysis with existing models. Results revealed that proposed MCUSUM control charts with PCM are efficient to detect variations timely by involving smaller number of principal components instead of considering entire associated variables. Also, proposed MCUSUM control charts have the ability to accommodate the features of existing control charts, which are illustrated as the special cases. Besides, to highlight the implementation mechanism and advantages of proposed MCUSUM control charts with PCM, a real-life example from wind turbine process is included.     

Correction factors for Shewhart and control charts to achieve desired unconditional ARL

In this paper we derive correction factors for Shewhart control charts that monitor individual observations as well as subgroup averages. In practice, the distribution parameters of the process characteristic of interest are unknown and, therefore, have to be estimated. A well-known performance measure within Statistical Process Monitoring is the expectation of the average run length (ARL), defined as the unconditional ARL. A practitioner may want to design a control chart such that, in the in-control situation, it has a certain expected ARL. However, accurate correction factors that lead to such an unconditional ARL are not yet available. We derive correction factors that guarantee a certain unconditional in-control ARL. We use approximations to derive the factors and show their accuracy and the performance of the control charts – based on the new factors – in out-of-control situations. We also evaluate the variation between the ARLs of the individually estimated control charts.     

A head-to-head comparative study of the conditional performance of control charts based on estimated parameters

Implementation of the Shewhart, CUSUM, and EWMA charts requires estimates of the in-control process parameters. Many researchers have shown that estimation error strongly influences the performance of these charts. However, a given amount of estimation error may differ in effect across charts. Therefore, we perform a pairwise comparison of the effect of estimation error across these charts. We conclude that the Shewhart chart is more strongly affected by estimation error than the CUSUM and EWMA charts. Furthermore, we show that the general belief that the CUSUM and EWMA charts have similar performance no longer holds under estimated parameters.     

The economic performance of the Shewhart t chart

In the production of small batches of customized parts, high flexibility and frequent switching of production from one product variant to another could not allow for the implementation of a control chart to monitor the process. In fact, when a short‐run production should be started, the distribution parameters of the quality characteristics to be monitored are frequentlytextitunknown and the production run is too short to get sufficient Phase I samples. To overcome this problem, the statistical properties of Shewhart t charts monitoring a short production run have been recently discussed in literature. In this paper, we investigate their economic performance: the SPC inspection cost optimization is constrained by the manufacturing and the inspection activities configuration. The decision variables of the problem include the chart design parameters and the size of batches of parts to be worked and released to the local inspection area. A numerical analysis aimed at evaluating the economic performance of the Shewhart t chart vs the Shewhart chart with known parameters has been performed. The expected economic loss associated with the implementation of the Shewhart t chart is acceptable with respect to the ‘ideal’ condition of the control chart with known parameters when the cost optimization is achieved without a statistical constraint limiting the number of expected false alarms. Finally, the effect of an erroneous initial set‐up on the correctness of the inspection cost estimation has been investigated. Copyright © 2011 John Wiley & Sons, Ltd.