Optimal EWMA of linear combination of Poisson variables for multivariate statistical process control

In this paper, we propose a new process control chart for monitoring correlated Poisson variables, the EWMA LCP chart. This chart is the exponentially weighted moving average (EWMA) version of the recently proposed LCP chart. The latter is a Shewhart-type control chart whose control statistic is a linear combination of the values of the different Poisson variables (elements of the Poisson vector) at each sampling time. As a Shewhart chart, it is effective at signalling large process shifts but is slow to signal smaller shifts. EWMA charts are known to be more sensitive to small and moderate shifts than their Shewhart-type counterparts, so the motivation of the present development is to enhance the performance of the LCP chart by the incorporation of the EWMA procedure to it. To ease the design of the EWMA LCP chart for the end user, we developed a user-friendly programme that runs on Windows© and finds the optimal design of the chart, that is, the coefficients of the linear combination as well as the EWMA smoothing constant and chart control limits that together minimise the out-of-control ARL under a constraint on the in-control ARL. The optimization is carried out by genetic algorithms where the ARLs are calculated through a Markov chain model. We used this programme to evaluate the performance of the new chart. As expected, the incorporation of the EWMA scheme greatly improves the performance of the LCP chart.     

Design and implementation of qth quantile‐unbiased tr‐chart for monitoring times between events

The times between events control charts have been proposed in literature for statistical monitoring of high‐yield processes by observing the waiting times up to r ^th (r ≥ 1  ) non‐conforming items or defects. The average run length (ARL) is the most widely used performance measure to evaluate the chart's performance, but in recent years, it has been subjected to criticisms. Because the run length distribution is highly skewed and hence, the ARL is not necessarily a typical value of the run length. Thus, evaluation of the control chart based on ARL alone could be misleading. In this paper, the quantiles of run length distribution are considered, instead of ARL, to design the t_r ‐chart. Further, we eliminate the bias in q ^th quantile function of the t_r ‐chart for both the known and unknown parameter case. In particular, the MRL‐unbiased t_r ‐chart is discussed in detail and compared with the ARL‐unbiased t_r ‐chart. It is found that the MRL‐unbiased t_r ‐chart outperforms than the corresponding ARL‐unbiased chart in unknown parameter case. It is also found that the proposed chart requires less phase I observations than that of the earlier studies has been suggested.     

A new approach to setting control limits of cumulative count of conforming charts for high‐yield processes

Cumulative count of conforming (CCC‐r) charts are usually used to monitor non‐conforming fraction p in high‐yield processes. Existing approaches to setting the control limits may cause non‐maximal or biased in‐control average run length (ARL). Non‐maximal in‐control ARL implies that the chart might not quickly detect the upward shift of p from its nominal value p₀. On the other hand, biased in‐control ARL means that both the in‐control and out‐of‐control ARLs are inflated. This paper develops a new approach to setting control limits for CCC‐r charts with near‐maximal and near‐unbiased in‐control ARL. Experimental results show that the proposed approach is effective in terms of the maximization and unbiasedness of in‐control ARL. Copyright © 2009 John Wiley & Sons, Ltd.     

The Design of the S2 Control Charts Based on Conditional Performance via Exact Methods

In this paper, we consider the conditional performance of the equal‐tailed and average run lengths (ARL)‐unbiased two‐sided S² charts when the in‐control variance of a normal process is estimated. We derive the exact probability distributions of the conditional ARL for the two S² charts. Then we evaluate the performance of each S² chart in terms of the percentiles, mean and standard deviation of the conditional in‐control ARL distribution. Because the parameter estimation seriously affects the conditional performance of these S² charts, we propose an exact method to design the equal‐tailed and ARL‐unbiased S² charts with desired conditional in‐control performance. The results indicate that the new ARL‐unbiased S² chart has far smaller standard deviation ARL values and the unconditional ARL values are more close to the desired value than the corresponding new equal‐tailed S² chart. Copyright © 2017 John Wiley & Sons, Ltd.     

On the Use of the Expected ARL Metric for Control Charts Based on Estimated Parameters

At the origination of the Shewhart control chart, it was assumed that the process parameters were known. The in‐control Average Run Length (ARL) and the probability of having a false alarm (P) were introduced as metrics to indicate the in‐control performance. These two metrics are related when the process data are i.i.d. normally distributed: the ARL equals 1/P. When the process parameters are unknown and have to be estimated, a similar relation holds for each estimated control chart, but the relation between the expected ARL (the average of the ARLs of all possible estimated charts) and the expectedP is different. Control charts based on estimates are often designed such that the in‐control ARL equals a predefined value. This paper shows that the expected in‐control ARL is a less suitable design criterion. Copyright © 2016 John Wiley & Sons, Ltd.     

The np Chart with Guaranteed In‐control Average Run Lengths

We evaluate the in‐control performance of the np‐control chart with estimated parameter conditional on the Phase I sample. We then apply the bootstrap method to adjust the control chart limits to guarantee the desired in‐control average run length (ARL₀) value in the monitoring stage. The adjusted limits ensure that the ARL₀ would take a value greater than the desired value (say, B ) with a certain specified probability, that is, Pr(ARL ₀ > B ) = 1 ? ρ . The results indicate that adjusting control limits is not always necessary. We present a method to design control charts such that in control and out of control run lengths are guaranteed with pre specified probabilities. This method is an improvement of the classical statistical design approach employing constraints on in control and out of control ARL because, with this approach, there is a substantial probability that the actual run length in control may be too small. In addition, using the ARL approach may result in an actual out of control run length that is too large. Some numerical examples illustrate the efficacy of this design method. Copyright © 2016 John Wiley & Sons, Ltd.     

Control Charts for Binary Correlated Variables

To monitor the nonconforming fraction of a production process, usually np or p control charts are used for this purpose. However, in many practical situations, the binary variables are correlated but not easily perceived by practitioners. The aim of this article is to present the maximum likelihood and method of moment estimators of the correlation parameter ρ of an overdispersed binomial distribution. Inferential procedure is also introduced to test the null hypothesis H₀: ρ = 0 x H₁: ρ > 0. A Shewhart‐type control chart np_ρ and an Exponentiated Weighted Moving Average (EWMA)‐type control chart (EWMA np_ρ) are proposed to evaluate the nonconforming fraction when the binary variables are correlated. The traditional np chart is a particular case of the np_ρ control chart when ρ = 0. The misuse of control limits of np control in case of correlated binary variables will result a large number of false alarms. To have the same performance (in terms of average run length) of the traditional np control chart, the np_ρ control chart needs at least to double the sample size. Numerical example illustrates the proposal. Copyright © 2012 John Wiley & Sons, Ltd.     

A Control Chart for COM–Poisson Distribution Using Resampling and Exponentially Weighted Moving Average

In this paper, we propose a control chart using exponentially weighted moving average (EWMA) statistic for count data based on the Conway–Maxwell–Poisson (called COM–Poisson) distribution. Repetitive sampling is considered by constructing two pairs of control limits for the proposed control chart. The performance of the proposed control chart is evaluated using the average run length (ARL) for various values of specified parameters. It has been observed that the proposed control chart is more efficient in terms of ARLs as compared to the existing control charts. The tables are provided and explained with the help of example. Copyright © 2015 John Wiley & Sons, Ltd.     

A Two‐Sided Cumulative Sum Chart for First‐Order Integer‐Valued Autoregressive Processes of Poisson Counts

Count data processes are often encountered in manufacturing and service industries. To describe the autocorrelation structure of such processes, a Poisson integer‐valued autoregressive model of order 1, namely, Poisson INAR(1) model, might be used. In this study, we propose a two‐sided cumulative sum control chart for monitoring Poisson INAR(1) processes with the aim of detecting changes in the process mean in both positive and negative directions. A trivariate Markov chain approach is developed for exact evaluation of the ARL performance of the chart in addition to a computationally efficient approximation based on bivariate Markov chains. The design of the chart for an ARL‐unbiased performance and the analyses of the out‐of‐control performances are discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

A New Distribution‐free Control Chart for Joint Monitoring of Unknown Location and Scale Parameters of Continuous Distributions

While the assumption of normality is required for the validity of most of the available control charts for joint monitoring of unknown location and scale parameters, we propose and study a distribution‐free Shewhart‐type chart based on the Cucconi 1 statistic, called the Shewhart‐Cucconi (SC) chart. We also propose a follow‐up diagnostic procedure useful to determine the type of shift the process may have undergone when the chart signals an out‐of‐control process. Control limits for the SC chart are tabulated for some typical nominal in‐control (IC) average run length (ARL) values; a large sample approximation to the control limit is provided which can be useful in practice. Performance of the SC chart is examined in a simulation study on the basis of the ARL, the standard deviation, the median and some percentiles of the run length distribution. Detailed comparisons with a competing distribution‐free chart, known as the Shewhart‐Lepage chart (see Mukherjee and Chakraborti 2 ) show that the SC chart performs just as well or better. The effect of estimation of parameters on the IC performance of the SC chart is studied by examining the influence of the size of the reference (Phase‐I) sample. A numerical example is given for illustration. Summary and conclusions are offered. Copyright © 2013 John Wiley & Sons, Ltd.     

Monitoring Process Variance Using an ARL‐unbiased EWMA‐p Control Chart

Control charts are effective tools for signal detection in manufacturing processes. As much of the data in industries come from processes having non‐normal or unknown distributions, the commonly used Shewhart variable control charts cannot be appropriately used, because they depend heavily on the normality assumption. The average run length (ARL) is generally used to measure the detection performance of a process when using a control chart, but it is biased for the monitoring statistic with an asymmetric distribution. That is, the ARL‐biased control chart leads to take longer to detect the shifts in parameter than to trigger a false alarm. To overcome this problem, we herein propose an ARL‐unbiased exponentially weighted moving average proportion (EWMA‐p) chart to monitor the process variance for process data with non‐normal or unknown distributions. We further explore the procedure to determine the control limits and to investigate the out‐of‐control variance detection performance of the ARL‐unbiased EWMA‐p chart. With a numerical example involving non‐normal service times from a bank branch in Taiwan, we illustrate the applications of the proposed ARL‐unbiased EWMA‐p chart and also compare the out‐of‐control detection performance of the ARL‐unbiased EWMA‐p chart, the arcsin transformed symmetric EWMA variance, and other existing variance charts. The proposed ARL‐unbiased EWMA‐p chart shows superior detection performance. Thus, we recommend the ARL‐unbiased EWMA‐p chart for process data with non‐normal or unknown distributions. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimum Multiple and Multivariate Poisson Statistical Control Charts

This paper deals with the simultaneous statistical process control of several Poisson variables. The practitioner of this type of monitoring may employ a multiple scheme, i.e. one chart for controlling each variable, or may use a multivariate scheme, based on monitoring all the variables with a single control chart. If the user employs the multivariate schemes, he or she can choose from, for example, three options: (i) a control chart based on the sum of the different Poisson variables; (ii) a control chart on the maximum value of the different Poisson variables; and (iii) in the case of only two variables, a chart that monitors the difference between them. In this paper, the previous control charts are studied when applied to the control of p = 2, 3 and 4 variables. In addition, the optimization of a set of univariate Poisson control charts (multiple scheme) is studied. The main purpose of this paper is to help the practitioner to select the most adequate scheme for her/his production process. Towards this goal, a friendly Windows© computer program has been developed. The program returns the best control limits for each control chart and makes a complete comparison of performance among all the previous schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Phase II Shewhart‐type Control Charts for Monitoring Times Between Events and Effects of Parameter Estimation

Monitoring times between events (TBE) is an important aspect of process monitoring in many areas of applications. This is especially true in the context of high‐quality processes, where the defect rate is very low, and in this context, control charts to monitor the TBE have been recommended in the literature other than the attribute charts that monitor the proportion of defective items produced. The Shewhart‐type t‐chart assuming an exponential distribution is one chart available for monitoring the TBE. The t‐chart was then generalized to the t_r‐chart to improve its performance, which is based on the times between the occurrences of r (≥1) events. In these charts, the in‐control (IC) parameter of the distribution is assumed known. This is often not the case in practice, and the parameter has to be estimated before process monitoring and control can begin. We propose estimating the parameter from a phase I (reference) sample and study the effects of estimation on the design and performance of the charts. To this end, we focus on the conditional run length distribution so as to incorporate the ‘practitioner‐to‐practitioner’ variability (inherent in the estimates), which arises from different reference samples, that leads to different control limits (and hence to different IC average run length [ARL] values) and false alarm rates, which are seen to be far different from their nominal values. It is shown that the required phase I sample size needs to be considerably larger than what has been typically recommended in the literature to expect known parameter performance in phase II. We also find the minimum number of phase I observations that guarantee, with a specified high probability, that the conditional IC ARL will be at least equal to a given small percentage of a nominal IC ARL. Along the same line, a lower prediction bound on the conditional IC ARL is also obtained to ensure that for a given phase I sample, the smallest IC ARL can be attained with a certain (high) probability. Summary and recommendations are given. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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 hybrid homogeneously weighted moving average control chart for process monitoring

Several modifications and enhancements to control charts in increasing the performance of small and moderate process shifts have been introduced in the quality control charting techniques. In this paper, a new hybrid control chart for monitoring process location is proposed by combining two homogeneously weighted moving average (HWMA) control charts. The hybrid homogeneously weighted moving average (HHWMA) statistic is derived using two smoothing constants λ₁ and λ₂ . The average run length (ARL) and the standard deviation of the run length (SDRL) values of the HHWMA control chart are obtained and compared with some existing control charts for monitoring small and moderate shifts in the process location. The results of study show that the HHWMA control chart outperforms the existing control charts in many situations. The application of the HHWMA chart is demonstrated using a simulated data.     

To shrink or not to shrink: Hotelling's T2 chart based on shrunken covariance estimates

Several authors have studied the effect of parameter estimation on the performance of Phase II control charts and shown that large in‐control reference samples are necessary for the Phase II control charts to perform as desired. For higher dimensional data, even larger reference samples are required to achieve stable estimation of the in‐control parameters. Shrinkage estimation has been widely studied as a method to achieve stable estimation of the covariance matrix for high‐dimensional data. We investigate the average run length (ARL) distribution of the Hotelling T² chart when using a shrunken covariance matrix. Specifically, we explore the following questions: (1) Does the use of a shrinkage estimator of the covariance matrix result in reduced variability in the ARL performance of the T² chart? (2) Does the use of a shrinkage estimator of the covariance matrix result in a reduced occurrence of “strictly multivariate” false alarms on the T²chart? (3) How does shrinkage of the covariance matrix affect the out‐of‐control performance of the T² chart? We use a simulation study to investigate the use of shrinkage estimation with the Hotelling T² chart in Phase II. Our results indicate that, while shrinkage estimation affects the ARL performance of the T² chart, the benefits are small and occur in fairly specific circumstances. The benefits of shrinking may not justify the use of more advanced techniques.     

The Effect of Parameter Estimation on Upper‐sided Bernoulli Cumulative Sum Charts

The Bernoulli cumulative sum (CUSUM) chart has been shown to be effective for monitoring the rate of nonconforming items in high‐quality processes where the in‐control proportion of nonconforming items (p₀) is low. The implementation of the Bernoulli CUSUM chart is often based on the assumption that the in‐control value p₀ is known; therefore, when p₀ is unknown, accurate estimation is necessary. We recommend using a Bayes estimator to estimate the value of p₀ to incorporate practitioner knowledge and to avoid estimation issues when no nonconforming items are observed in phase I. We also investigate the effects of parameter estimation in phase I on the upper‐sided Bernoulli CUSUM chart by using the expected value of the average number of observations to signal (ANOS) and the standard deviation of the ANOS. It is found that the effects of parameter estimation on the Bernoulli CUSUM chart are more significant than those on the Shewhart‐type geometric chart. The low p₀ values inherent to high‐quality processes imply that a very large, and often unrealistic, sample size may be needed to accurately estimate p₀. A methodology to identify a continuous variable to monitor is highly recommended when the value of p₀ is low and the required phase I sample size is impractically large. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimum variable-dimension EWMA chart for multivariate statistical process control

The variable-dimension T² control chart (VDT² chart) was recently proposed for monitoring the mean of multivariate processes in which some of the quality variables are easy and inexpensive to measure while other variables require substantially more effort or expense. The chart requires most of the times that only the inexpensive variables be sampled, switching to sampling all the variables only when a warning is triggered. It has good ARL performance compared with the standard T² chart, while significantly reducing the sampling cost. However, like the T² chart, it has limited sensitivity to small and moderate mean shifts. To detect such shifts faster, we developed an exponentially weighted moving average (EWMA) version of the VDT² chart, along with Markov chain models for ARL calculation, and software (made available) for optimizing the chart design. The optimization software, which is based on genetic algorithms, runs in Windows^© and has a friendly user interface. The performance analysis shows the great gain in performance achieved by the incorporation of the EWMA procedure.     

The Design of the ARL‐Unbiased S2 Chart When the In‐Control Variance Is Estimated

The S² chart has been known as a powerful tool to monitor the variability of the normal process. When the variance of the process is unknown, it needs to be estimated by Phase I samples. It is well known that there are serious effects of parameter estimation on the performance of the S² chart based on known parameter assumption. If the effects of parameter estimation are not considered, it can lead to an increase in the number of false alarms and a reduction in the ability of the chart to detect process changes except for very small shifts in the variance. Based on the criterion of average run length (ARL) unbiased, a S² control chart is developed when the in‐control variance is estimated. The performance of the proposed control chart is also evaluated in terms of the ARL and standard deviation of the run length. Finally, an example is used to illustrate the proposed control chart. Copyright © 2013 John Wiley & Sons, Ltd.