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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

A Note on “Capability Assessment for Processes with Multiple Characteristics: A Generalization of the Popular Index Cpk”

The generalized yield index establishes the relationship between the manufacturing specifications and the actual process performance, which provides a lower bound on process yield for two‐sided processes with multiple characteristics. The results attended are very practical for industrial application. In this article, we extended the results in cases with one‐sided specification and multiple characteristics. The generalized index was considered, and the asymptotic distribution of the natural estimator was developed. Then, we derived the lower confidence bounds as well as the critical values of index . We not only provided some tables but also presented an application example. Copyright © 2012 John Wiley & Sons, Ltd.     

Control Chart Limits for Monitoring Process Mean Based on Downton's Estimator

Control charts are important tools in statistical process control used to monitor shift in process mean and variance. This paper proposes a control chart for monitoring the process mean using the Downton estimator and provides table of constant factors for computing the control limits for sample size (n ≤ 10). The derived control limits for process mean were compared with control limits based on range statistic. The performance of the proposed control charts was evaluated using the average run length for normal and non‐normal process situations. The obtained results showed that the control chart, using the Downton statistic, performed better than Shewhart chart using range statistic for detection of small shift in the process mean when the process is non‐normal and compares favourably well with Shewhart chart that is normally distributed. Copyright © 2015 John Wiley & Sons, Ltd.     

Process Yield for Multiple Stream Processes with Individual Observations and Subsamples

A multiple stream process consists of several identical process streams. We present the process yield index for multiple stream processes with individual observations. An approximate distribution of the estimator of is derived. A simulation study is conducted to evaluate the performance of the confidence interval using the proposed method and the existing method. The simulation results show that the proposed method outperforms the existing method regarding interval length. We extend this process yield index for the case of subsamples. An approximate distribution for the estimator of is also derived. Two real examples are used to demonstrate the performance of the proposed approach. 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.     

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.     

The Effect of Measurement Errors on the Synthetic Chart

Measurement errors often exist in quality control applications. In this paper, the performance of the synthetic chart is investigated when measurement errors exist using a linearly covariate error model. It is shown that the performance of the synthetic chart is significantly affected in the presence of measurement errors. The effect of taking multiple measurements for each item in a subgroup on the performance of synthetic chart is also investigated in this paper. An example is provided in order to illustrate the application of the synthetic chart with measurement errors. Copyright © 2014 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.     

The Performance of Variable Sample Size Chart with Measurement Errors

The variable sample size (VSS) chart has been investigated by several researchers under the assumption of no measurement error. However, in practice, measurement errors may exist in quality control applications. In this paper, the overall performance of the VSS chart is investigated when measurement errors exist using a linearly covariate error model, and a methodology is proposed for choosing optimal parameters by considering measurement errors. It is shown that the overall performance of the VSS chart is significantly affected by the presence of measurement errors. The effect of taking multiple measurements for each item in a subgroup on the performance of VSS chart is also investigated in this paper. An example is provided to illustrate the application of the VSS chart with measurement errors. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

Filter‐based compressed sensing MRI reconstruction

Compressed sensing (CS) enables to reconstruct MR images from highly undersampled k‐space data by exploiting the sparsity which is implicit in the images. In this article, an MR image as a combination of a high‐frequency component and a low‐frequency component through a pair of filters has been proposed to express. Since exhibits a sparser representation in the wavelet transform domain, reconstructing and separately yields a better result than reconstructing directly. Two parameters, normalized sparsity (NS) and power ratio (PR), are defined to design the filters, that is, the high‐pass filter H _HP and the low‐pass filter H _LP. H _HP is applied to pick out high‐frequency k‐space data for the reconstruction of high‐frequency image ; while H _LP is used for filtering , which is reconstructed from the entire undersampled k‐space data to obtain the low‐frequency reconstruction . Summing and leads to the final reconstruction of . Experimental results demonstrate that the proposed method outperforms the conventional CS‐MRI method. It provides 2–4 dB improvement in peak signal to noise ratio (PSNR) value and preserves more edges and details in the images. © 2016 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 26, 173–178, 2016