A Robust Control Chart

This article studies alternative standard deviation estimators that serve as a basis to determine the control chart limits used for real‐time process monitoring (phase II). Several existing (robust) estimation methods are considered. In addition, we propose a new estimation method based on a phase I analysis, that is, the use of a control chart to identify disturbances in a data set retrospectively. The method constructs a phase I control chart derived from the trimmed mean of the sample interquartile ranges, which is used to identify out‐of‐control data. An efficient estimator, namely the mean of the sample standard deviations, is used to obtain the final standard deviation estimate from the remaining data. The estimation methods are evaluated in terms of their mean squared errors and their effects on the performance of the phase II control chart. It is shown that the newly proposed estimation method is efficient under normality and performs substantially better than standard methods when disturbances are present in phase I. 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.     

VSSI median control chart with estimated parameters and measurement errors

An adaptive control chart called Shewhart chart with variable sample size and sampling interval (VSSI) is quicker than Shewhart chart, chart with variable sample size (VSS), and chart with variable sampling interval (VSI) in detecting the mean shifts of a normal process. In practice, the effects of measurement errors on control charts should be included. In this study, we present an VSSI median control chart with estimated parameters in the presence of measurement errors for a normal process. The average time to signal (ATS) is computed by using the Markov chain approach. The results show that the VSSI median control chart performs better than the Shewhart median, VSS median, and VSI median control charts in terms of ATS. The design parameters of the proposed chart are provided. Two examples are used to illustrate the application of the proposed control chart.     

A 2‐Stage Attribute‐Variable Control Chart to Monitor a Vector of Process Means

The aim of this paper is to propose a combined attribute‐variable control chart, namely M a x D  ? T ², to monitor a vector of process means μ  = [μ ₁,…,μ _q ] in a multivariate process control. The procedure consists of splitting a sample of size n into two sub‐samples of sizes n ₁ and n ₂(n  = n ₂ + n ₂), determined by an optimized process. Units of the first sub‐sample are evaluated by an attribute inspection. Using a device like a gauge ring, each unit of the first sub sample is considered approved related to the quality characteristic i if X _i ∈[ ; ]; otherwise, it is disapproved in the characteristic i , where and (obtained by an optimization) are respectively the lower and upper discriminating limits of the quality dimension X _i . If the number of disapproved items in any quality characteristic is higher than a control limit, then the measurement of the q quality characteristics is taken on each unit of the second sub‐sample and the statistic T ² is calculated. If T ² < L (L , the control limit) the process is judged as in control. The process will suffer intervention if both charts signal. The procedure has an advantage to not inspect the units of the second sub‐sample if the first sub‐sample indicates that the process is in control. This proposal shows a better performance than T ² control chart for a large number of scenarios. The two control limits and discriminant limits are optimized to reach a desired value of A R L ₀ and to minimize A R L ₁. Copyright © 2017 John Wiley & Sons, Ltd.     

Economic Design of the Variable Parameters Control Chart with a Corrected A&L Switching Rule

The variable parameters (VP) control chart is more efficient than the fixed parameters control chart. However, an obvious shortcoming of the VP control chart is its excessive number of switches among different design parameters, which significantly increases the complexity of operations for administer and the costs of maintaining the control chart. Amin and Letsinger (1991) proposed a switching rule, denoted by A&L switching rule, to reduce the number of switches. In this paper, we investigated the economic design of the VP control chart by using the corrected A&L switching rule scheme, which makes some corrections to the existing study of the A&L switching rule scheme. A three‐state Markov model is proposed to derive the performance measures of the suggested control chart. An expected cost model for the process is established with the cost function derived. The genetic algorithm is then employed to search for the solution for the economic design of the proposed control chart. The results obtained show that the developed model can significantly reduce the average number of the parameter's switches and save the expected cost in comparison with the conventional VP control scheme in detecting small and moderate mean shifts. A sensitivity analysis is also carried out to examine the effects of cost and model parameters on the solution of the economic design for the proposed control chart. The analysis demonstrates the expected cost per time unit is positively affected by the cost associated with a false signal, the average cost of repairs when a true signal is identified, the sampling cost and the cost associated with a switch, and is negatively affected by the cost associated with running the process in‐control per hour. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive multivariate double sampling and variable sampling interval Hotelling's T2 charts

In this article, two adaptive multivariate charts, which combine the double sampling (DS) and variable sampling interval (VSI) features, called the adaptive multivariate double sampling variable sampling interval T² (AMDSVSI T²) and the adaptive multivariate double sampling variable sampling interval combined T² (AMDSVSIC T²) charts, are proposed. The real purpose of using the proposed charts is to provide flexibility by enabling the sampling interval length of the DS T² chart to be varied so that the chart's sensitivity can be enhanced. The fundamental difference between the two proposed charts is that when a second sample is taken, the AMDSVSI T² chart uses the information of the combined sample mean vectors while the AMDSVSIC T² chart uses the information of the combined T² statistics, in deciding about the process status. This research is motivated by existing combined DS and VSI charts in the literature, which show convincing performance improvement over the standard DS chart. Consequently, it is believed that adopting this existing approach in the multivariate case will enable superior multivariate DS charts to be proposed. Numerical results show that the proposed charts outperform the existing standard T² and other adaptive multivariate charts, in detecting shifts in the mean vector, for the zero‐state and steady‐state cases. The performances of both charts when the shift sizes in the mean vector are unknown are also measured. The application of the AMDSVSI T² chart is illustrated with an example.     

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.     

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.     

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.     

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.     

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.     

Run Rules median control charts for monitoring process mean in manufacturing

Recent studies show that Shewhart median ( ) chart is simpler than the Shewhart chart and it is robust against outliers, but it is often rather inefficient in detecting small or moderate process shifts. The statistical sensitivity of a Shewhart control chart can be improved by using supplementary Run Rules. In this paper, we propose the Phase II median Run Rules type control charts. A Markov chain methodology is used to evaluate the statistical performance of these charts. Moreover, the performance of proposed charts is investigated in the presence of a measurement errors and modelled by a linear covariate error model. An extensive numerical analysis with several tables and figures to show the statistical performance of the investigated charts is provided for both cases of measurement errors and no measurement errors. An example illustrates the use of these charts.     

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.     

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.     

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.     

Economic Design Approach for an SPC Inspection Procedure Implementing The Adaptive C Chart

The present paper proposes a design approach for a statistical process control (SPC) procedure implementing a c control chart for non‐conformities, with the aim to minimize the hourly total quality‐related costs. The latter take into account the costs arising from the non‐conforming products while the process is in‐control and out‐of‐control, for false alarms, for assignable cause locations and system repairs, for sampling and inspection activities and for the system downtime. The proposed economic optimization approach is constrained by the expected hourly false alarms frequency, as well as the available labor resource level. A mixed integer non‐linear constrained mathematical model is developed to solve the treated optimization problem, whereas the Generalized Reduced Gradient Algorithm implemented on the solver of Microsoft Excel is adopted to resolve it. In order to illustrate the application of the developed procedure, a numerical analysis based on a fractional factorial design scheme, to investigate on the influence of several operating and costs parameters, is carried out, and the related considerations are given. Finally, the obtained results show that only few parameters have a meaningful effect on the selection of the optimal SPC procedure. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A Reevaluation of the Adaptive Exponentially Weighted Moving Average Control Chart When Parameters are Estimated

The performance of control charts can be adversely affected when based on parameter estimates instead of known in‐control parameters. Several studies have shown that a large number of phase I observations may be needed to achieve the desired in‐control statistical performance. However, practitioners use different phase I samples and thus different parameter estimates to construct their control limits. As a consequence, there would be in‐control average run length (ARL) variation between different practitioners. This kind of variation is important to consider when studying the performance of control charts with estimated parameters. Most of the previous literature has relied primarily on the expected value of the ARL (AARL) metric in studying the performance of control charts with estimated parameters. Some recent studies, however, considered the standard deviation of the ARL metric to study the performance of control charts. In this paper, the standard deviation of the ARL metric is used to study the in‐control and out‐of‐control performance of the adaptive exponentially weighted moving average (AEWMA) control chart. The performance of the AEWMA chart is then compared with that of the Shewhart and EWMA control charts. The simulation results show that the AEWMA chart might represent a good solution for practitioners to achieve a reasonable amount of ARL variation from the desired in‐control ARL performance. In addition, we apply a bootstrap‐based design approach that provides protection against frequent false alarms without deteriorating too much the out‐of‐control performance. Copyright © 2014 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.     

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.