Adaptive control charts for skew‐normal distribution

The standard Shewhart‐type chart, named FSS‐ chart, has been widely used to detect the mean shift of process by implementing fixed sample and sampling frequency schemes. The FSS‐ chart could be sensitive to the normality assumption and is inefficient to catch small or moderate shifts in the process mean. To monitor nonnormally distributed variables, Li et al [Commun Stat‐Theory Meth. 2014; 43(23):4908‐4924] extended the study of Tsai [Int J Reliab Qual Saf Eng. 2007; 14(1):49‐63] to provide a new skew‐normal FSS‐ (SN FSS‐ ) chart with exact control limits for the SN distribution. To enhance the sensitivity of the SN FSS‐ chart on detecting small or moderate mean shifts in the process, adaptive charts with variable sampling interval (VSI), variable sample size (VSS), and variable sample size and sampling interval (VSSI) are introduced for the SN distribution in this study. The proposed adaptive control charts include the normality adaptive charts as special cases. Simulation results show that all the proposed SN VSI‐ , SN VSS‐ , and SN VSSI‐ charts outperform the SN FSS‐ chart on detecting small or moderate shifts in the process mean. The impact of model misspecification on using the proposed adaptive charts and the sample size impact for using the FSS‐ chart to monitor the mean of SN data are also discussed. An example about single hue value in polarizer manufacturing process is used to illustrate the applications of the proposed adaptive charts.     

The steady‐state behavior of the synthetic and side‐sensitive synthetic double sampling charts

The steady‐state average run length is used to measure the performance of the recently proposed synthetic double sampling chart (synthetic DS chart). The overall performance of the DS chart in signaling process mean shifts of different magnitudes does not improve when it is integrated with the conforming run length chart, except when the integrated charts are designed to offer very high protection against false alarms, and the use of large samples is prohibitive. The synthetic chart signals when a second point falls beyond the control limits, no matter whether one of them falls above the centerline and the other falls below it; with the side‐sensitive feature, the synthetic chart does not signal when they fall on opposite sides of the centerline. We also investigated the steady‐state average run length of the side‐sensitive synthetic DS chart. With the side‐sensitive feature, the overall performance of the synthetic DS chart improves, but not enough to outperform the non‐synthetic DS chart. Copyright © 2014 John Wiley & Sons, Ltd.     

Charts with Variable Sample Size,Sampling Interval,and Warning Limits

The notion of variable warning limits is proposed for variable sample size and sampling interval (VSSI) charts. The basic purpose is to lower down the frequency of switches between the pairs of values of the sample sizes and sampling interval lengths of VSSI charts during their implementations. Expressions for performance measures for the variable sample size, sampling interval, and warning limits (VSSIWL) charts are developed. The performances of these charts are compared numerically with that of VSSI and VSSI (1, 3) charts, where VSSI (1, 3) charts are the VSSI charts with runs rule (1, 3) for switching between the pairs of values of sample sizes and sampling interval lengths. Runs rule (1, 3) greatly reduces the frequency of the switches; however, it slightly worsens the statistical performances of the VSSI charts in detecting moderate shifts in the process mean. It is observed that the out‐of‐control statistical performance and overall switching rate of VSSIWL charts are adaptive for the same in‐control statistical performances. These charts can be set to yield exactly similar performances as that of VSSI (1, 3) charts, to yield tradeoff performances between that of VSSI (1, 3) and VSSI charts, or to yield significantly lower switching rate than even that of VSSI (1, 3) charts at the cost of slightly inferior statistical performances than that of VSSI (1, 3) charts. Copyright © 2012 John Wiley & Sons, Ltd.     

Synthetic and Runs‐Rules Charts Combined With an Chart: Theoretical Discussion

A synthetic and a runs‐rules charts that are combined with a basic chart are called a Synthetic‐ and an improved runs‐rules charts, respectively. This paper gives the zero‐state and steady‐state theoretical results of the Synthetic‐ and improved runs‐rules monitoring schemes. The Synthetic‐ and improved runs‐rules schemes can each be classified into four different categories, that is, (i) non‐side‐sensitive, (ii) standard side‐sensitive, (iii) revised side‐sensitive, and (iv) modified side‐sensitive. In this paper, we first give the operation and, secondly, the general form of the transition probability matrices for each of the categories. Thirdly, in steady‐state, we show that for each of the categories, the three methods that are widely used in the literature to compute the initial probability vectors result in different probability expressions (or values). Fourthly, we derive the closed‐form expressions of the average run‐length (ARL) vectors for each of the categories, so that, by multiplying each of these ARL vectors with the zero‐state and steady‐state initial probability vectors, yield the zero‐state and steady‐state ARL expressions. Finally, we formulate the closed‐form expressions of the extra quadratic loss function for each of the categories. Copyright © 2016 John Wiley & Sons, Ltd.     

Variable Sample Size and Sampling Interval Charts with Runs Rules for Switching Between Sample Sizes and Sampling Interval Lengths

Variable sample size and sampling interval (VSSI) charts are substantially more efficient than are the static charts. However, the frequent switches between sample sizes and sampling interval lengths can be a complicating factor during the implementation of these charts. In this article, runs rules are proposed for switching between the sample sizes and the sampling interval lengths of these charts to reduce the frequency of switches. The expressions for the performance measures for the charts with these runs rules are developed. The methods presented are general and can be applied to other VSSI Shewhart control charts. The effects of different runs rules on the performances of the charts are compared numerically. The runs rules substantially reduce the frequency of switches. Some runs rules do not significantly alter the statistical performances of the charts; however, some adversely affect that in detecting large shifts in the process mean. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal Designs of the Variable Sample Size Chart Based on Median Run Length and Expected Median Run Length

The variable sample size (VSS) chart, devoted to the detection of moderate mean shifts, has been widely investigated under the context of the average run‐length criterion. Because the shape of the run‐length distribution alters with the magnitude of the mean shifts, the average run length is a confusing measure, and the use of percentiles of the run‐length distribution is considered as more intuitive. This paper develops two optimal designs of the VSS chart, by minimizing (i) the median run length and (ii) the expected median run length for both deterministic and unknown shift sizes, respectively. The 5th and 95th percentiles are also provided in order to measure the variation in the run‐length distribution. Two VSS schemes are considered in this paper, that is, when the (i) small sample size (n_S) or (ii) large sample size (n_L) is predefined for the first subgroup (n₁). The Markov chain approach is adopted to evaluate the performance of these two VSS schemes. The comparative study reveals that improvements in the detection speed are found for these two VSS schemes without increasing the in‐control average sample size. For moderate to large mean shifts, the optimal VSS chart with n₁ = n_L significantly outperforms the optimal EWMA chart, while the former is comparable to the latter when n₁ = n_S. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonparametric signed‐rank control charts with variable sampling intervals

Variable sampling interval (VSI) charts have been proposed in the literature for normal theory (parametric) control charts and are known to provide performance enhancements. In the VSI setting, the time between monitored samples is allowed to vary depending on what is observed in the current sample. Nonparametric (distribution‐free) control charts have recently come to play an important role in statistical process control and monitoring. In this paper a nonparametric Shewhart‐type VSI control chart is considered for detecting changes in a specified location parameter. The proposed chart is based on the Wilcoxon signed‐rank statistic and is called the VSI signed‐rank chart. The VSI signed‐rank chart is compared with an existing fixed sampling interval signed‐rank chart, the parametric VSI ‐chart, and the nonparametric VSI sign chart. Results show that the VSI signed‐rank chart often performs favourably and should be used.     

On the Performance of Shewhart‐type Synthetic and Runs‐rules Charts Combined with an Chart

A synthetic chart is a combination of a conforming run‐length chart and an chart, or equivalently, a 2‐of‐(H + 1) runs‐rules (RR) chart with a head‐start feature. However, a synthetic chart combined with an chart is called a Synthetic‐ chart. In this article, we build a framework for Shewhart Synthetic‐ and improved RR (i.e., 1‐of‐1 or 2‐of‐(H + 1) without head‐start) charts by conducting an in‐depth zero‐state and steady‐state study to gain insight into the design of different classes of these schemes and their average run‐length performance using the Markov chain imbedding technique. More importantly, we propose a modified side‐sensitive Synthetic‐ chart, and then using overall performance measures (i.e., the extra quadratic loss, average ratio of average run‐length, and performance comparison index), we show that this new chart has a uniformly better performance than its Shewhart competitors. We also provide easy‐to‐use tables for each of the chart's design parameters to aid practical implementation. Moreover, a performance comparison with their corresponding counterparts (i.e., synthetic and RR charts) is conducted. Copyright © 2015 John Wiley & Sons, Ltd.     

Exponentially weighted moving average control charts based on the moving average statistic and lnS2 for monitoring a Weibull process with subgroups

In this paper, we propose 2 new exponentially weighted moving average (EWMA) control charts based on the moving average (MA) statistic and lnS2 to monitor the process mean and variability of a Weibull process with subgroups. The inverse error function is used to transform the Weibull‐distributed data to a standard normal distribution. The Markov chain approach is used to derive the average run length (ARL). Subsequently, the performances of the proposed charts with other existing control charts are provided. The comparison shows that the EWMA‐MA outperforms the and EWMA‐ control charts for monitoring the process mean of ARL values. The comparison also shows that the EWMA‐lnS2 outperforms the S2 and S2‐MA control charts for monitoring the process variability of ARL value. Two examples are used to illustrate the application of the proposed control charts.     

Control charts for paired differences: and Sd charts

Control chart procedures for monitoring paired variables are sparse in the literature. After considering the average run length properties of the  chart, which monitors the mean of the differences between paired variables, we propose a new chart based on S_d, the subgroup standard deviations of the differences. Our findings show that the  chart is powerful for monitoring the changes in the means and the S_d chart is suitable for monitoring changes in the covariance structure. Furthermore, we show that the and S_d charts perform better than existing bivariate control charts for detecting shifts in mean and variance/covariance, respectively, when standards are known. The difference charts also performed well compared to common alternatives when the standards are unknown arising from a limited amount of Phase I data. An application of these difference charts in a finance context is illustrated using the returns of Apple Inc's stock and the S&P 500 index.     

Re‐evaluation of the VSI‐ chart performance with estimated parameters

The performance of the variable sampling interval‐ (VSI‐ ) chart with estimated parameters has been investigated on the basis of the average time to signal (ATS) and standard deviation of time to signal (SDTS) in past research studies. Since the values of ATS and SDTS vary from practitioner to practitioner, the use of these 2 measures is not reliable. The use of different historical data sets in phase I results in varying parameter estimates, control limits, warning limits, ATS, and SDTS values. In this study, we use the standard deviation of average time to signal (SDATS) to evaluate and compare the performance of the VSI‐ chart with known parameters and estimated parameters. This study shows that variation reduction in ATS values requires a larger than previously recommended phase I data. Also, detection of up to moderate shifts in the process mean with the desired ATS value would be achievable with the number of samples recommended in the past, but the in‐control performance of the chart would not be reliable. Furthermore, we evaluate the effect of using large and small desired values of ATS₀ on the performance of in‐control and out‐of‐control VSI‐ chart. We also study the effects of estimating the mean and standard deviation on the ATS values using numerical simulation. Finally, we present a method based on warning and control limits coefficients for the estimated parameters case to reduce the number of samples required in phase I.     

Monitoring the Process Mean When Standards Are Unknown: A Classic Problem Revisited

One of the most common applications in statistical process monitoring is the use of control charts to monitor a process mean. In practice, this is often performed with a Shewhart chart along with a Shewhart R (or an S) chart. Thus, two charts are typically used together, as a scheme, each using the 3‐sigma limits. Moreover, the process mean and standard deviation are often unknown and need to be estimated before monitoring can begin. We show that there are three major issues with this monitoring scheme described in most textbooks. The first issue is not accounting for the effects of parameter estimation, which is known to degrade chart performance. The second issue is the implicit assumption that the charting statistics are both normally distributed and, accordingly, using the 3‐sigma limits. The third issue is multiple charting, because two charts are used, in this scheme, at the same time. We illustrate the deleterious effects of these issues on the in‐control properties of the charting scheme and present a method for finding the correct charting constants taking proper account of these issues. Tables of the new charting constants are provided for some commonly used nominal in‐control average run length values and different sample sizes. This will aid in implementing the charting scheme correctly in practice. Examples are given along with a summary and some conclusions. Copyright © 2015 John Wiley & Sons, Ltd.     

An Efficient Shewhart‐Type Control Chart to Monitor Moderate Size Shifts in the Process Mean in Phase II

According to Shewhart, control charts are not very sensitive to small and moderate size process shifts that is why those are less likely to be effective in Phase II. So to monitor small or moderate size process shifts in Phase II, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are considered as alternate of Shewhart control charts. In this paper, a Shewhart‐type control chart is proposed by using difference‐in‐difference estimator in order to detect moderate size shifts in process mean in Phase II. The performance of the proposed control chart is studied for known and unknown cases separately through a detailed simulation study. For the unknown case, instead of using reference samples of small sizes, large size reference sample(s) is used as we can see in some of nonparametric control chart articles. In an illustrative example, the proposed control charts are constructed for both known and unknown cases along with Shewhart ‐chart, classical EWMA, and CUSUM control charts. In this application, the proposed chart is found comprehensively better than not only Shewhart ‐chart but also EWMA and CUSUM control charts. By comparing average run length, the proposed control chart is found always better than Shewhart ‐chart and in general better than classical EWMA and CUSUM control charts when we have relatively higher values of correlation coefficients and detection of the moderate shifts in the process mean is concerned. Copyright © 2015 John Wiley & Sons, Ltd.     

Run sum X¯ control chart with estimated process parameters

The run sum $\overline{X}$ control chart is usually investigated under the assumption of known process parameters. In practice, process parameters are rarely known and they need to be estimated from an in‐control Phase I dataset. However, different practitioners use different numbers of Phase I samples to estimate the process parameters. As a result, the commonly used performance measure, ie, the average run length becomes a random variable. In this study, we present a run sum $\overline{X}$ control chart with estimated process parameters and use the standard deviation of the average run length to evaluate the average of the average run length performance of the run sum $\overline{X}$ chart when process parameters are estimated. Based on the standard deviation of the average run length criterion, the number of Phase I samples required by the estimated process parameter–based run sum $\overline{X}$ chart to have an average of the average run length performance close to that obtained under the assumption of known process parameters is recommended.     

Empirical Comparisons of X‐bar Charts when Control Limits are Estimated

A control chart is a very common tool used to monitor the quality of business processes. An estimator of the process variability is generally considered to obtain the control limits of a chart when parameters of the process are unknown. Assuming Monte Carlo simulations, this paper first compares the efficiency of the various estimators of the process variability. Two empirical measures used to analyze the performance of control charts are defined. Results derived from various empirical studies reveal the existence of a linear relationship between the performance of the various estimators of the process variability and the performance of charts. The various Monte Carlo simulations are conducted under the assumption that the process is in both situations of in‐control and out‐of‐control. Copyright © 2015 John Wiley & Sons, Ltd.     

Statistical Process Control Based on Optimum Gages

Classical statistical process control (SPC) by attributes is based on counts of nonconformities. However, process quality has greatly improved with respect to past decades, and the vast majority of samples taken from high‐quality processes do not exhibit defective units. Therefore, control charts by variables are the standard monitoring scheme employed. However, it is still possible to design an effective SPC scheme by attributes for such processes if the sample units are classified into categories such as ‘large’, ‘normal’, or ‘small’ according to limits that are different from the specification limits. Units classified as ‘large’ or ‘small’ will most likely still be conforming (within the specifications), but such a classification allows monitoring the process with attributes charts. In the case of dimensional quality characteristics, gages can be built for this purpose, making inspection quick and easy and reducing the risk of errors. We propose such a control chart, optimize it, compare its performance with the traditional and S charts and with another chart in the literature that is also based in classifying observations of continuous variables through gaging, and present a brief sensitivity analysis of its performance. The new chart is shown to be competitive with the use of –S charts, with the operational advantage of simpler, faster, and less costly inspection. Copyright © 2017 John Wiley & Sons, Ltd.     

EWMA chart based on Bayes‐conditional pivotal quantity for Weibull percentiles under complete data and type‐II censoring

Bayes‐conditional control chart has been used for monitoring the Weibull percentiles with complete data and type‐II censoring. Firstly, the Weibull data are transformed to the smallest extreme value (SEV) distribution. Secondly, the posterior median of quantiles is used as a monitoring statistic. Finally, a pivotal quantity based on the monitoring statistic with its conditional distribution function is derived for obtaining the control limits. This control chart is denoted as Shewhart‐SEV‐ . In this study, we extend this work based on an exponential weighted moving average model named exponential weighted moving average‐SEV‐ for monitoring the Weibull percentiles. We provide the statistical properties of the monitoring statistic. The average run length and the standard deviation of run lengths, computed by the integral equation approach, are used as performance measures. The results indicate that the proposed chart performs better than the Shewhart‐SEV‐ . The breaking strength of carbon fibers is used to illustrate the application of the proposed control chart.     

Shewhart Control Charts for Monitoring Reliability with Weibull Lifetimes

In this paper, we present Shewhart‐type and S² control charts for monitoring individual or joint shifts in the scale and shape parameters of a Weibull distributed process. The advantage of this method is its ease of use and flexibility for the case where the process distribution is Weibull, although the method can be applied to any distribution. We illustrate the performance of our method through simulation and the application through the use of an actual data set. Our results indicate that and S² control charts perform well in detecting shifts in the scale and shape parameters. We also provide a guide that would enable a user to interpret out‐of‐control signals. Copyright © 2014 John Wiley & Sons, Ltd.     

Process Yield for Multivariate Linear Profiles with One‐sided Specification Limits

The new investigation of profile monitoring is focused mainly on a process with multiple quality characteristics. Process yield has been used widely in the manufacturing industry, as an index for measuring process capability. In this study, we present two indices and to measure the process capability for multivariate linear profiles with one‐sided specification limits under mutually independent normality. Additionally, two indices and are proposed to measure the process capability for multivariate linear profiles with one‐sided specification limits under multivariate normality. These indices can provide an exact measure of the process yield. The approximate normal distributions for and are constructed. A simulation study is conducted to assess the performance of the proposed approach. The simulation results show that the estimated value of performs better as the number of profiles increases. Two illustrative examples are used to demonstrate the applicability of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Guaranteed in‐control performance of the synthetic chart with estimated parameters

Control charts are usually investigated under the assumption of known process parameters. In practice, however, the process parameters are rarely known and they have to be estimated from different Phase I data sets. The properties of control charts with estimated parameters are usually investigated with the unconditional average of the average run length. Control chart's performance is known to vary among practitioners because of the use of different Phase I data sets. Considering the between‐practitioners variability in control chart's performance, the standard deviation of the average run length is developed to reevaluate the properties of the synthetic chart with estimated parameters. Because of the limited amount of Phase I data in practice, the bootstrap method is used as a good adjustment technique for the synthetic chart's parameters.