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.     

Improved design of robust exponentially weighted moving average control charts for autocorrelated processes

Residual‐based control charts for autocorrelated processes are known to be sensitive to time series modeling errors, which can seriously inflate the false alarm rate. This paper presents a design approach for a residual‐based exponentially weighted moving average (EWMA) chart that mitigates this problem by modifying the control limits based on the level of model uncertainty. Using a Bayesian analysis, we derive the approximate expected variance of the EWMA statistic, where the expectation is with respect to the posterior distribution of the unknown model parameters. The result is a relatively clean expression for the expected variance as a function of the estimated parameters and their covariance matrix. We use control limits proportional to the square root of the expected variance. We compare our approach to two other approaches for designing robust residual‐based EWMA charts and argue that our approach generally results in a more appropriate widening of the control limits. Copyright © 2010 John Wiley & Sons, Ltd.     

An Empirical Indicator of Product Appearance for Process Control

An innovative approach based on multiattribute utility theory and group processes was used to directly monitor and control cosmetic characteristics affecting product appearance to improve the quality of a product. The main advantage of this approach is its ability to provide a direct and reliable measure of visual appearance for products for which this is the critical quality characteristic. The paper is divided into two main parts. The first part briefly presents a systematic approach to develop an empirical indicator of product appearance, called the index of visual condition (IOVC). The second part presents a study of the applicability of the IOVC during process control to directly monitor and improve product appearance. This study was performed during industrial production of a ceramic product for a period of seven months. Several Shewhart charts were developed, including the first exploratory X-bar and R charts for product appearance (IOVC charts) and c-charts for number of defects. The performance of these charts was evaluated by their ability to detect special causes of variation and improve the product. The case study indicates that these two approaches complement each other. While the c-chart allows one to monitor the number of defects that impair product functionality, the IOVC chart brings a new level of capability, providing the ability to directly monitor product appearance considering defects that do not affect functionality. Both approaches were useful for process control and improvement.     

Robust economic control chart design

Designing a control chart involves making fundamental decisions about the control chart parameters. Traditionally, practitioners select design parameters using ad hoc procedures. Academic research has been challenging this tradition by introducing a more rigorous criterion for selecting the design parameters based on economic criteria. However, there has been limited success in implementing economic designs. This research develops the concept of Robust Economic Design of control charts where multiple economic and process scenarios are considered in control chart design. By developing a robust optimization technique for economic control chart design, we hope to promote a better understanding of industrial implementation of economic designs of control charts. The effectiveness of this technique is illustrated through examples and a sensitivity analysis.     

An Economic Approach to the Management of High‐Quality Processes

Modern manufacturing developments have forced researchers to investigate alternative quality control techniques for high‐quality processes. The cumulative count of conforming (CCC) control chart is a powerful alternative approach for monitoring high‐quality processes for which traditional control charts are inadequate. This study develops a mathematical model for the economic design of the CCC control chart and presents an application of the proposed model. On the basis of the results of the application, the economic and classical CCC control chart designs of the CCC control chart are compared. The optimal design parameters for different defective fractions are tabulated, and a sensitivity analysis of the model is presented for the CCC control chart user to determine the optimal economic design parameters and minimum hourly costs for one production run according to different defective fractions, cost, time, and process parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust CUSUM charts for monitoring the process mean and variance

This paper considers the problem of obtaining robust control charts for detecting changes in the mean µ and standard deviation σ of process observations that have a continuous distribution. The standard control charts for monitoring µ and σ are based on the assumption that the process distribution is normal. However, the process distribution may not be normal in many situations, and using these control charts can lead to very misleading conclusions. Although some control charts for µ can be tuned to be robust to non‐normal distributions, the most critical problem with non‐robustness is with the control chart for σ. This paper investigates the performance of two CUSUM chart combinations that can be made to be robust to non‐normality. One combination consists of the standard CUSUM chart for µ and a CUSUM chart of absolute deviations from target for σ, where these CUSUM charts are tuned to detect relatively small parameter shifts. The other combination is based on using winsorized observations in the standard CUSUM chart for µ and a CUSUM chart of squared deviations from target for σ. Guidance is given for selecting the design parameters and control limits of these robust CUSUM chart combinations. When the observations are actually normal, using one of these robust CUSUM chart combination will result in some reduction in the ability to detect moderate and large changes in µ and σ, compared with using a CUSUM chart combination that is designed specifically for the normal distribution. Copyright © 2009 John Wiley & Sons, Ltd.     

Monitoring of time between events with a double generally weighted moving average control chart

Monitoring of time between events (TBE) instead of the number of events is used in high‐quality processes where the events occur rarely. This article presents a double generally weighted moving average control chart with a lower time‐varying control limit to monitor the TBE (regarded as DGWMA‐TBE chart). The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the DGWMA‐TBE chart is more effective than the DEWMA and GWMA charts in detecting moderate to large shifts. Furthermore, the DGWMA‐TBE chart is very robust for the same range of shifts when the TBE observations follow a Weibull or a lognormal distribution. Finally, examples are also presented to enhance the performance of the proposed chart.     

Optimization design of the CUSUM and EWMA charts for autocorrelated processes

The traditional control charts produce frequent false alarm signals in the presence of autocorrelation. The implementation of the modified chart scheme is a way of handling the problem of autocorrelation in control charts. In modified charts, the standard control limits of the traditional charts are adjusted to offset the influence because of the autocorrelation. The exponentially weighted moving average– and cumulative sum–modified charts are 2 widely used charts for monitoring autocorrelated data. These charts have design parameters in their formulation, and the choice of these parameters play significant roles in the detection of out‐of‐control situations. In reality, the magnitude of the mean shift is uncertain, and this leads to a difficulty in the choice of design parameters by the practitioner. The use of optimal parameters can enhance process performance in such situations. In this paper, we determine optimal design parameters for the charts using an exhaustive search procedure. In the optimization process, we determine the parameters that produce the smallest extra quadratic loss (EQL) value for each autocorrelation coefficient. This criterion measures the anticipated loss attributed to poor quality in the process. The loss in quality is lowered by minimizing the EQL and the combination of parameters in the chart that yields the smallest EQL has a better detection ability over the entire shift range. For the purpose of this work, we concentrate on autocorrelation that can be specifically modelled with autoregressive models. This article provides the practitioner with optimal parameters that can be used to enhance the overall effectiveness of the charts over an entire shift range.     

A DYNAMIC PROGRAMMING APPROACH TO THE ECONOMIC DESIGN OF X-CHARTS

Recent technological advances have rendered dynamic process control a viable alternative. A dynamic programming approach is proposed for the modeling and cost minimization of statistical process control activities. The decision parameters of the control chart are allowed to change dynamically as new information about the process becomes available. This general approach has been known as a theoretical possibility for many years, but its practical performance is explicitly investigated in this paper. It is shown with numerical examples that the dynamic programming solution can be much more economical than the conventional static solution with fixed control chart parameters. The substantial potential cost savings and the feasibility of a dynamic control procedure suggest that dynamic process control should replace standard statistical or economic design of control charts as the preferred method in automated production processes.     

Multivariate multiple sampling charts

A new multivariate statistical process control scheme, the Multivariate Multiple Sampling (MMS) control chart scheme, is proposed in this paper. A MMS chart is a multivariate extension of a double sampling X-bar control chart with at least two sampling stages. In the paper, a statistical design optimization procedure to design the MMS chart is presented and the performance of the MMS chart is investigated. The statistical efficiency in terms of average run length of the MMS chart is compared with that of the Hotelling chart both with and without variable sampling schemes, a multivariate CUMulative SUM (CUSUM) chart, and a multivariate Exponentially Weighted Moving Average (EWMA) chart. The ability of the MMS chart to handle the worst-case scenario is also investigated and compared with that of the multivariate EWMA and CUSUM charts. The results of the investigation show that even with only two sampling stages, the MMS chart provides an improvement in efficiency in detecting small shifts over the Hotelling chart without variable sampling schemes. When the number of sampling stages is equal to two, the MMS chart is better in detecting large shifts and the multivariate EWMA and CUSUM charts are better in detecting relatively small shifts. As the number of sampling stages is increased beyond two, the improvement in sensitivity of the MMS chart in detecting the small shifts increases. When the number of sampling stages ≥3, the MMS chart begins to give a better performance than a Hotelling chart with a variable sampling scheme for small shifts and is also better than a multivariate EWMA chart for both small and large shifts. As the number of sampling stages ≥4, the MMS chart begins to give a better performance than a multivariate CUSUM chart for both small and large shifts. The results of the investigation also show that the MMS chart outperforms the multivariate EWMA and CUSUM charts in the worst-case scenario situation.     

Estimating the time of step change with Poisson CUSUM and EWMA control charts

Knowing when a process has changed would simplify the search for and identification of the special cause. Consequently, having an estimate of the process change point following a control chart signal would be useful to process engineers. Much of the literature on change point models and techniques for statistical process control applications consider processes well modelled by the normal distribution. However, the Poisson distribution is commonly used in industrial quality control applications for modelling attribute-based process quality characteristics (e.g., counts of non-conformities). Some commonly used control charts for monitoring Poisson distributed data are the Poisson cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts. In this paper, we study the effect of changes in the design of the control chart on the performances of the change point estimators offered by these procedures. In particular, we compare root mean square error performances of the change point estimators offered by the Poisson CUSUM and EWMA control charts relative to that achieved by a maximum likelihood estimator for the process change point. Results indicate that the relative performance achieved by each change point estimator is a function of the corresponding control chart design. Relative mean index plots are provided to enable users of these control charts to choose a control chart design and change point estimator combination that will yield robust change point estimation performance across a range of potential change magnitudes.     

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.     

Cusum of Q chart with variable sampling intervals for monitoring the process mean

Recently, adaptive control charts (that is, with variable sample sizes and/or sampling intervals) for univariate or multivariate quality characteristics have received considerable attention in Phase II analysis in the literature. Due to insufficient samples to obtain good knowledge of the parameters in the start-up process, adding adaptive features to self-starting control charts remains an open problem. In this paper, we propose an adaptive Cusum of Q chart with variable sampling intervals for monitoring the process mean of normally distributed variables. A Fortran program is available to assist in the design of the control chart with different parameters. The effect of the control chart parameters on the performance is studied in detail. The control chart is further enhanced by finding adaptive reference values. Due to the powerful properties of the proposed control chart, the Monte Carlo simulation results show that it provides quite satisfactory performance in various cases. The proposed control chart is applied to a real-life data example to illustrate its implementation.     

Control charts for simultaneously monitoring of process mean and coefficient of variation with and without auxiliary information

A control chart is very useful to control assignable causes which detect the shifted process parameters (eg, mean and dispersion). Simultaneous monitoring of the process parameters is a well‐known approach utilized for the bilateral processes. In the current study, we proposed the blended control chart that monitors the process mean and process coefficient of variation simultaneously. Further, the sensitivity of control chart is enhanced by incorporating an auxiliary variable. We have utilized the concept of EWMA chart and also the log transformation to transform the distribution of sample coefficient of variation to the normal distribution for structuring a joint monitoring control chart. The performance comparison among proposed control charts is presented. On the basis of ARLs and SDRLs, several advantages of the proposed control charts are diagnosed. The empirical evidence is also provided to support proposed control chart with a real‐life dataset.     

Design of exponential control charts using a sequential sampling scheme

Control charts for monitoring the time between events can be applied in various areas. In this study, we focus on the exponential control chart and consider the phase II problem (when process parameters are known) as well as the phase I problem (when process parameters are unknown). An exponential chart designed with the conventional approach has the disadvantage that the Average Run Length (ARL) value may increase when the process deviates from the nominal state. An ARL-unbiased design approach is therefore proposed for both phase II and phase I exponential charts. A sequential sampling scheme is adopted for the phase I exponential chart. The proposed ARL-unbiased design approach has several advantages over the conventional one, as it provides a self-starting feature and can significantly improve the ARL performance. Specific guidelines are suggested regarding the time to stop updating the estimates of parameters and control limits based on the actual false alarm rate. The phase I exponential chart can be calibrated to a constant in-control ARL value for each successive event accumulated to date. Simulated and real data examples are given to demonstrate the use and efficiency of the proposed design approach.     

A Distribution-Free Shewhart Quality Control Chart Based on Signed-Ranks

Since their inception by Walter Shewhart in the late 1920s, most control chart developments have been distribution-based procedures in the sense that the process output is assumed to follow a specified probability distribution (normal for continuous measurements and binomial or Poisson for attribute data). Due to Deming's influence and their widespread adoption as one of the seven basic tools of total quality management (TQM), control charts have been applied to processes where data may be markedly nonnormal. In this article, we propose a distribution-free (or nonparametric) statistical quality control chart for monitoring a process center. The proposed chart is of the Shewhart type and is based on the signed-ranks of grouped observations. The exact false alarm rate and the in-control average run length of the proposed chart are computed by using the null distribution of the well-known Wilcoxon signed-rank statistic. The out-of-control run lengths are computed exactly for normal underlying distributions and by simulation for uniform, double exponential, and Cauchy shift alternatives. Efficiency studies show that the proposed chart is more efficient than the traditional Shewhart X-bar chart under heavy-tailed distributions (the double exponential and the Cauchy) but is less efficient under light-tailed distributions (the uniform and the normal).     

Properties of the Markov‐Dependent Attribute Control Chart with Estimated Parameters

The performance of attribute control charts that monitor Markov‐dependent data is usually evaluated under the assumption of known process parameters, that is, known values of a the probability an item is nonconforming given the previous item is conforming and b the probability an item is conforming given the previous item is nonconforming. In practice, these parameters are usually not known and are calculated from an in‐control Phase I‐data set. In this paper, a comparison of the in‐control ARL (average run length) properties of the attribute chart for Markov‐dependent data with known and estimated parameters is presented. The probability distribution of the estimators is developed and used to calculate the in‐control ARL and standard deviation of the run length of the chart with estimated parameters. For particular values of a and b, the in‐control ARL values of the charts with estimated parameters may be very different than those with known parameters. The size of the Phase‐I data set needed for charts with estimated parameters to exhibit the same in‐control ARL properties as those with known parameters may vary widely depending on the parameters of the process, but in general, large samples are needed to obtain accurate estimates. As the Phase‐I sample size increases, the in‐control ARL values of the charts with estimated parameters approach that of the known parameter case but not in a monotonic fashion as in the case of the X‐bar chart. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal design of one-sided exponential cumulative sum charts with known and estimated parameters based on the median run length

As a useful tool in statistical process control (SPC), the exponential control chart is more and more popular for monitoring high-quality processes. Considering both known and estimated parameter cases, the one-sided exponential cumulative sum (CUSUM) charts are studied in this paper through a Markov chain approach. Because the shape of the run length (RL ) distribution of the one-sided exponential CUSUM charts is skewed and it also changes with the mean shift size and the number of Phase I samples used to estimate the process parameter, the median run length (MRL ) is employed as a good alternative performance measure for the charts. The optimal design procedures based on MRL of the one-sided exponential CUSUM charts with known and estimated parameters are discussed. By comparing the MRL performance of the chart with known parameters with the one of the chart with estimated parameters, we investigate the effect of estimated process parameters on the properties of the chart. Finally, an application is illustrated to show the implementation of the chart.     

The density control chart: a general approach for constructing a single chart for simultaneously monitoring multiple parameters

The majority of the existing literature on simultaneous control charts, i.e. control charting mechanisms that monitor multiple population parameters such as mean and variance on a single chart, assume that the process is normally distributed. In order to adjust and maintain the overall type-I error probability, these existing charts rely largely on the property that the sample mean and sample variance are independent under the normality assumption. Furthermore, the existing charting procedures cannot be readily extended to non-normal processes. In this article, we propose and study a general charting mechanism which can be used to construct simultaneous control charts for normal and non-normal processes. The proposed control chart, which we call the density control chart, is essentially based on the premise that if a sample of observations is from an in-control process, then another sample of observations is no less likely to be also from the in-control process if the likelihood of the latter is no less than the likelihood of the former. The density control chart is developed for normal and non-normal processes where the distribution of the plotting statistic of the density control chart can be explicitly derived. Real examples are given and discussed in these cases. We also discuss how the density control chart can be constructed in cases when the distribution of the plotting statistic cannot be determined. A discussion of potential future research is also given.