Memory‐type multivariate control charts with auxiliary information for process mean

In statistical process control, it is a common practice to increase the sensitivity of a control chart with the help of an efficient estimator of the underlying process parameter. In this paper, we consider an efficient estimator that requires information on several study variables along with one or more auxiliary variables when estimating the mean of a multivariate normally distributed process. Using this auxiliary‐information‐based (AIB) process mean estimator, we propose new multivariate EWMA (MEWMA), double MEWMA (DMEWMA), and multivariate CUSUM (MCUSUM) charts for monitoring the process mean, denoted by the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts, respectively. The run length characteristics of the proposed multivariate charts are computed using Monte Carlo simulations. The proposed charts are compared with their existing counterparts in terms of the run length characteristics. It turns out that the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts are uniformly and substantially better than the MEWMA, DMEWMA, and MCUSUM charts, respectively, when detecting different shifts in the process mean. A real dataset is considered to explain the implementation of the proposed and existing multivariate control charts.     

MEWMA Control Chart and Process Capability Indices for Simple Linear Profiles with Within‐profile Autocorrelation

A multivariate exponentially weighted moving average (MEWMA) control chart is proposed for detecting process shifts during the phase II monitoring of simple linear profiles (SLPs) in the presence of within‐profile autocorrelation. The proposed control chart is called MEWMA‐SLP. Furthermore, two process capability indices are proposed for evaluating the capability of in‐control SLP processes, and their utilization is demonstrated through examples. Intensive simulations reveal that the MEWMA‐SLP chart is more sensitive than existing control charts in detecting profile shifts. Copyright © 2016 John Wiley & Sons, Ltd.     

Attribute Charts for Monitoring a Dependent Process

For some repetitive production processes, the quality measure taken on the output is an attribute variable. An attribute variable classifies each output item into one of a countable set of categories. One of the simplest and most commonly used attribute variables is the one which classifies an item as either ‘conforming’ or ‘non‐conforming’. A tool used with a considerable amount of success in industry for monitoring the quality of a production process is the quality control chart. Generally a control charting procedure uses a sequence, of the quality measures to make a decision about the quality of the process. How this sequence is used to make a decision defines the control chart. In order to design a control chart one must consider how the underlying sequence, is modeled. The sequence is often modeled as a sequence of independent and identically distributed random variables. For many industrial processes, this model is appropriate, but in others it may not be. In this paper, a sequence of random variables, is used to classify an item as conforming or non‐conforming under a stationary Markov chain model and under 100% sequential sampling. Two different control charting schemes are investigated. Both schemes plot a sequence of measures on the control chart, that count the number of conforming items before a non‐conforming item. The first scheme signals as out‐of‐control if a value of falls below a certain lower limit. The second scheme signals as out‐of‐control if two out of two values of fall below a certain lower limit. The efficiency of both of the control charts is evaluated by the average run length (ARL) of the chart and the power of the chart to detect a shift in the process. The two out of two scheme is shown to have high power and a large ARL given certain parameter values of the process. An example of the two out of two scheme is provided for the interested reader. Copyright © 2006 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.     

Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

The average run length (ARL) is usually used as a sole measure of performance of a multivariate control chart. The Hotelling's T², multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts are commonly optimally designed based on the ARL. Similar to the case of univariate quality control, in multivariate quality control, the shape of the run length distribution changes in accordance to the magnitude of the shift in the mean vector, from highly skewed when the process is in‐control to nearly symmetric for large shifts. Because the shape of the run length distribution changes with the magnitude of the shift in the mean vector, the median run length (MRL) provides additional and more meaningful information about the in‐control and out‐of‐control performances of multivariate charts, not given by the ARL. This paper provides a procedure for optimal designs of the multivariate synthetic T² chart for the process mean, based on MRL, for both the zero and steady‐state modes. Two Mathematica programs, each for the zero state and steady‐state modes are given for a quick computation of the optimal parameters of the synthetic T² chart, designed based on MRL. These optimal parameters are provided in the paper, for the bivariate case with sample sizes, nin{4, 7, 10}. The MRL performances of the synthetic T², MEWMA and Hotelling's T² charts are also compared. Copyright © 2011 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.     

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.     

Scalable Chi‐Square Distance versus Conventional Statistical Distance for Process Monitoring with Uncorrelated Data Variables

Multivariate statistical process control charts are often used for process monitoring to detect out‐of‐control anomalies. However, multivariate control charts based on conventional statistical distance measures, such as the one used in the Hotelling's control chart, cannot scale up to large amounts of complex process data, e.g. data with a large number of variables and a high rate of data sampling. In our previous work we developed a multivariate statistical process monitoring procedure based on a more scalable chi‐square distance measure and tested this procedure for detecting out‐of control anomalies—intrusions—in a computer process using computer audit data. The testing results demonstrated the comparable performance of the scalable chi‐square procedure to that of Hotelling's control chart. To establish the chi‐square procedure as a generic, viable multivariate statistical processing monitoring procedure, we conduct a series of further studies to understand the detection power and limitations of the chi‐square procedure for processes with various kinds of data and various types of out‐of‐control anomalies in addition to the scalability and demonstrated performance of the chi‐square procedure for computer intrusion detection. This paper reports on one of these studies that investigates the effectiveness of the scalable chi‐square procedure in detecting out‐of‐control anomalies in processes with uncorrelated data variables, each of which has a normal probability distribution. The results of this study indicate that the chi‐square procedure is at least as effective as Hotelling's control chart for monitoring processes with uncorrelated data variables. Copyright © 2003 John Wiley & Sons, Ltd.     

A Comparison of MCUSUM‐based and MEWMA‐based Spatiotemporal Surveillance Under Non‐homogeneous Populations

Motivated by the application in health‐care surveillance under non‐homogeneous populations, this paper compares the performance of multivariate exponentially weighted moving average (MEWMA)‐based methods with that of multivariate cumulative sum (MCUSUM)‐based methods for spatiotemporal chronic disease surveillance. We perform the comprehensive simulation studies of the MEWMA‐based methods for the selection of weight parameters. Under temporally or spatially non‐homogeneous population trends, we compare the MEWMA methods with the MCUSUM methods, which are specific forms of the spatiotemporal scan statistics if the baseline rate is known. The performance of the MCUSUM methods has extreme variations depending on whether the change occurs on a small population or a large population. When the change occurs in the time period or within the spatial region with a small population, the weighted likelihood ratio‐based MCUSUM has better detection speed, but worse identification of detection clusters than the likelihood ratio‐based MCUSUM. On the other hand, when the change occurs in the time period or within the region with a large population, the weighted likelihood ratio‐based MCUSUM has worse detection speed, but better identification than the likelihood ratio‐based MCUSUM. Unlike the MCUSUM methods, the MEWMA‐based methods show relatively stable and robust performance in terms of detection speed, and they show better identification than the MCUSUM‐based methods under most cases. 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.     

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.     

Design Strategies for the Multivariate Exponentially Weighted Moving Average Control Chart

The multivariate exponentially weighted moving average (MEWMA) control chart has received significant attention from researchers and practitioners because of its desirable properties. There are several different approaches to the design of MEWMA control charts: statistical design; economic–statistical design; and robust design. In this paper a review and comparison of these design strategies is provided.Copyright © 2004 John Wiley & Sons, Ltd.     

Effect of non‐normality on the monitoring of simple linear profiles

In some statistical process control (SPC) applications, it is assumed that a quality characteristic or a vector of quality characteristics of interest follows a univariate or multivariate normal distribution, respectively. However, in certain applications this assumption may fail to hold and could lead to misleading results. In this paper, we study the effect of non‐normality when the quality of a process or product is characterized by a linear profile. Skewed and heavy‐tailed symmetric non‐normal distributions are used to evaluate the non‐normality effect numerically. The results reveal that the method proposed by Kimtextitet al. (J. Qual. Technol. 2003; 35 :317–328) can be designed to be robust to non‐normality for both highly skewed and heavy‐tailed distributions. Copyright © 2010 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.     

On‐line Identification and Quantification of Mean Shifts in Bivariate Processes using a Neural Network‐based Approach

Many statistical process control (SPC) problems are multivariate in nature because the quality of a given process or product is determined by several interrelated variables. Various multivariate control charts (e.g. Hotelling's , multivariate cumulative sum and multivariate exponentially weighted moving average charts) have been designed for detecting mean shifts. However, the main shortcoming of such charts is that they can detect an unusual event but do not directly provide the information required by a practitioner to determine which variable or group of variables has caused the out‐of‐control signal. In addition, these charts cannot provide more detailed shift information, for example the shift magnitude, which would be very useful for quality practitioners to search the assignable causes that give rise to the out‐of‐control situation. This work proposes a neural network‐based model that can identify and quantify the mean shifts in bivariate processes on‐line. The performance evaluation performed by the simulation demonstrates that the proposed model outperforms the conventional multivariate control schemes in terms of average run length, and can accurately estimate the magnitude of the shift of each of the shifted variables in a real‐time mode. Extensive simulation is also carried out to examine the effects of correlation on the performance of the proposed model. A numerical example is presented to illustrate the usage of the proposed model. Although a mean shift identification and quantification tool for bivariate SPC is the particular application presented here, the proposed neural network‐based methodology can be applied to multivariate SPC in general. Copyright © 2006 John Wiley & Sons, Ltd.     

A sensitivity analysis of an integrated model for joint determination of economic design of $\overline{x}$‐control charts,economic production quantity and production run length for a deteriorating production system

This study presents an integrated model for the joint economic design of ‐control charts and maintenance schedules and, simultaneously, determines the economic production quantity and production run length for a deteriorating production system. The operating state of the production process is classified as either in control or out of control. In the latter state, the process produces some defective items. An ‐control chart is used to monitor the process mean. Both uniform and non‐uniform inspection schemes are adopted. Inspection and maintenance are performed simultaneously. Replacement cost is assumed to be very high. The process failure mechanism is assumed to follow a general probability distribution with an increasing failure rate. The concept of a truncated production cycle is introduced. The production cycle begins when a new component is installed and ends with a repair after the detection of a failure or after a specified number of inspection intervals, , whichever occurs first. The effects of preventive maintenance on quality control are discussed. Numerical examples are provided to evaluate the performance of the model. Sensitivity analyses are conducted to study the effects of various model parameters. Copyright © 2002 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.     

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.     

Distributional properties of estimated capability indices based on subsamples

Under the assumption of normality, the distribution of estimators of a class of capability indices, containing the indices , , and , is derived when the process parameters are estimated from subsamples. The process mean is estimated using the grand average and the process variance is estimated using the pooled variance from subsamples collected over time for an in‐control process. The derived theory is then applied to study the use of hypothesis testing to assess process capability. Numerical investigations are made to explore the effect of the size and number of subsamples on the efficiency of the hypothesis test for some indices in the studied class. The results for and indicate that, even when the total number of sampled observations remains constant, the power of the test decreases as the subsample size decreases. It is shown how the power of the test is dependent not only on the subsample size and the number of subsamples, but also on the relative location of the process mean from the target value. As part of this investigation, a simple form of the cumulative distribution function for the non‐central ‐distribution is also provided. Copyright © 2003 John Wiley & Sons, Ltd.