Design of double- and triple-sampling X-bar control charts using genetic algorithms

As today's manufacturing firms are moving towards agile manufacturing, quick and economic on-line statistical process control solutions are in high demand. Multiple sampling X-bar control charts are such an alternative. They can be designed to allow quick detection of a small shift in process mean and provide a quick response in an agile manufacturing environment. In this paper, the designs of double-sampling (DS) X-bar control charts are formulated and solved with a genetic algorithm. Based on the results in solving the DS chart design problems, triple sampling (TS) X-bar control charts are developed. The efficiency of the TS charts is compared with that of the DS charts. The results of the comparison show that TS charts are more efficient in terms of minimizing the average sample size.     

Construction of double sampling s‐control charts for agile manufacturing

Double sampling (DS) ‐control charts are designed to allow quick detection of a small shift of process mean and provides a quick response in an agile manufacturing environment. However, the DS ‐control charts assume that the process standard deviation remains unchanged throughout the entire course of the statistical process control. Therefore, a complementary DS chart that can be used to monitor the process variation caused by changes in process standard deviation should be developed. In this paper, the development of the DS s‐charts for quickly detecting small shift in process standard deviation for agile manufacturing is presented. The construction of the DS s‐charts is based on the same concepts in constructing the DS ‐charts and is formulated as an optimization problem and solved with a genetic algorithm. The efficiency of the DS s‐control chart is compared with that of the traditional s‐control chart. The results show that the DS s‐control charts can be a more economically preferable alternative in detecting small shifts than traditional s‐control charts. Copyright © 2002 John Wiley & Sons, Ltd.     

Note on ‘Design of double- and triple-sampling X-bar control charts using genetic algorithms’

He et al. formulated the designs of double-sampling (DS) X-bar control charts as optimization problems and solved these problems with a genetic algorithm. Based on the results in solving the DS chart design problems, triple-sampling (TS) X-bar control charts were developed. The efficiency of the TS charts was compared with that of the DS charts. They concluded that the TS charts are more efficient in terms of minimizing the average sample size. We explain that, since they only considered the average sample size when the process is in control, their conclusion is questionable. In fact, the question of which control chart (i.e. the standard Shewart X-bar control chart, DS chart or TS chart) is more efficient depends on both the probability of the process shifting from an in-control to an out-of-control state and the time the control chart will need to detect such a shift.     

Robust CUSUM charts for monitoring the process mean and variance

This paper considers the problem of obtaining robust control charts for detecting changes in the mean µ and standard deviation σ of process observations that have a continuous distribution. The standard control charts for monitoring µ and σ are based on the assumption that the process distribution is normal. However, the process distribution may not be normal in many situations, and using these control charts can lead to very misleading conclusions. Although some control charts for µ can be tuned to be robust to non‐normal distributions, the most critical problem with non‐robustness is with the control chart for σ. This paper investigates the performance of two CUSUM chart combinations that can be made to be robust to non‐normality. One combination consists of the standard CUSUM chart for µ and a CUSUM chart of absolute deviations from target for σ, where these CUSUM charts are tuned to detect relatively small parameter shifts. The other combination is based on using winsorized observations in the standard CUSUM chart for µ and a CUSUM chart of squared deviations from target for σ. Guidance is given for selecting the design parameters and control limits of these robust CUSUM chart combinations. When the observations are actually normal, using one of these robust CUSUM chart combination will result in some reduction in the ability to detect moderate and large changes in µ and σ, compared with using a CUSUM chart combination that is designed specifically for the normal distribution. Copyright © 2009 John Wiley & Sons, Ltd.     

A Modified S Chart for the Process Variance

Both the R and S charts are widely used in many manufacturing industries to monitor the process dispersion. The R chart is more popular among quality control practitioners especially when dealing with small sample sizes because of the simplicity of computing the range, R, from each sample. For larger sample sizes, the preferred choice is the S chart because it is slightly more effective than the R chart. The computation of the standard deviation, S, from each sample can now be made easily due to the availability of computers and scientific calculators. This article addresses the shortcomings of the conventional S chart and suggests a modified S chart to overcome these problems.     

Note on ‘Construction of Double Sampling s‐Control Charts for Agile Manufacturing’

He and Grigoryan (Quality and Reliability Engineering International 2002; 18 :343–355) formulated the design of a double‐sampling (DS) s control chart as an optimization problem and solved it with a genetic algorithm. They concluded that the DS s control charts can be a more economically preferable alternative in detecting small shifts than traditional s control charts. We explain that, since they only considered the average sample size when the process is in control, their conclusion is questionable. Copyright © 2006 John Wiley & Sons, Ltd.     

Hotelling's T2 control chart with adaptive sample sizes

Some quality control schemes have been developed when several related quality characteristics are to be monitored: simultaneous X¯ charts, Hotelling's T² chart, multivariate CUSUM and multivariate EWMA. Hotelling's T² control chart has the advantage of its simplicity but it is slow in detecting small process shifts. The latest developments in variable sample sizes for univariate control charts are applied in this paper to define an adaptive sample sizes T² control chart. As occurs in the univariate case the ARL improvements are very important particularly for small process shifts. An example is given to illustrate the use of the proposed scheme.     

X¯ chart with adaptive sample sizes

Standard Shewhart control charts employ fixed sample sizes at equal sampling intervals. By varying the sample size depending on the current location of the process mean, the mean time to detect an off-target condition can be reduced. The adaptive-sample-size control chart is compared with the fixed-sample-size control chart in terms of average run length under shifts in the process mean of variable magnitude. Significant improvements have been obtained with the adaptive-sample-size charts, particularly for small shifts. These improvements are achieved without increasing the in-control average sample size beyond that of the fixed-sample-size approach. A fast initial response is suggested and advantages of the procedure over fixed-sample-size control are illustrated with two examples from discrete manufacturing processes.     

Guidelines for the application of adaptive control charting schemes

A comprehensive performance study and comparison of several adaptive statistical process control procedures is presented. These adaptive control chart procedures are modifications to standard Shewhart control charts that include changing the sampling interval, the sample size or both according to rules based on the value of the sample statistic. Adaptive control techniques are known to improve the performance of the standard Shewhart control charts. In this paper we develop a four-state adaptive sample size control chart and several variations of a three-state combined adaptive sample size and sampling interval control chart. We then compare these new schemes with the previously developed schemes, the two-state adaptive sampling interval, the two-state adaptive sample size and two-state combined adaptive sample size and sampling interval control chart, three-state adaptive sample size control chart and non-adaptive Shewhart control charts. These results show that the addition of the third and fourth states on the adaptive control chart schemes improve the control chart performance; however, the improvement is relatively modest.     

Improved one‐sided control charts for the mean of a positively skewed population using truncated saddlepoint approximations

This paper develops a truncated saddlepoint (TS) control chart for the mean of a population with a skewed density. This chart requires the estimation of only a few population cumulants from sample data and is relatively straightforward in construction. Performance is assessed at various levels of truncation in terms of type 1 and 2 error rates and compared with other heuristic control charts via the Monte Carlo simulation. The truncated saddlepoint (TS) chart is shown to generally outperform these charts, especially in cases with small sample sizes or high levels of population skewness. Copyright © 2011 John Wiley & Sons, Ltd.     

Using radial basis function neural networks to recognize shifts in correlated manufacturing process parameters

Traditional statistical process control (SPC) techniues of control charting are not applicable in many process industries because data from these facilities are autocorrelated. Therefore the reduction in process variability obtained through the use of SPC techniques has not been realized in process industries. Techniques are needed to serve the same function as SPC control charts, that is to identify process shifts, in correlated parameters. Radial basis function neural networks were developed to identify shifts in process parameter values from papermaking and viscosity data sets available in the literature. Time series residual control charts were also developed for the data sets. Networks were successful at separating data that were shifted 1.5 and 2 standard deviations from nonshifted data for both the papermaking and viscosity parameter values. The network developed on the basis of the papermaking data set was also able to separate shifts of 1 standard deviation from nonshifted data. The SPC control charts were not able to identify the same process shifts. The radial basis function neural networks can be used to identify shifts in process parameters, thus allowing improved process control in manufacturing processes that generate correlated process data.     

A moving average control chart using a robust scale estimator for process dispersion

Control charts are one of the most powerful tools used to detect and control industrial process deviations in statistical process control. In this paper, a moving average control chart based on a robust scale estimator of standard deviation, namely, the sample median absolute deviation (MAD) statistic, for monitoring process dispersion, is proposed. A simulation study is conducted to evaluate the performance of the proposed moving average median absolute deviation (MA‐MAD) chart, in terms of average run length for various distributions. The results show that the moving average MAD chart performs well in detecting small and moderate shifts in process dispersion, especially when the normality assumption is violated. In addition, this chart is very efficient, especially when the quality characteristic follows a skewed distribution. Numerical and simulated examples are given at the end of the paper.     

The development and evaluation of CUSUM-based control charts for an AR(1) process

An important component of the quality program of many manufacturing operations is the use of control chart for variables. Inherent in the construction of these control charts is the assumption that the sampled process is a normal distribution whose observations are independent and identically distributed (iid). Many processes such as those found in chemical manufacturing, refinery operations, smelting operations, wood product manufacturing, waste-water processing and the operation of nuclear reactors have been shown to have autocorrelated observations. Autocorrelation, which violates the independence assumption of standard control charts, is known to have an adverse effect on the average run length (ARL) performance of control charts. This paper will consider a statistical testing procedure for the change-point problem for monitoring the level parameter of the AR(1) process. This test is shown to result in a CUSUM-based control chart. Two different solutions of the change-point problem are given which result in slightly different control charts. The average run length of each of these CUSUM control charts is found via the Markov chain approach. A methodology for designing the CUSUM-based control chart is presented and the performance of these control charts is compared to other approaches in the literature.     

SHORT-RUN STATISTICAL PROCESS CONTROL: MULTICRITERIA PART FAMILY FORMATION

Owing to customer demands and short product life cycles, manufacturing trends have shifted towards a wide variety of mixed products with small batch sizes. It is difficult to apply traditional control charts efficiently and effectively in such environments, and it is not necessary to plot a control chart for each individual part. In this study, we propose a multicriteria part family formation technique and algorithm for implementing short-run SPC charting. We first carry out simulation to obtain ARLs for various shifts, and then use a maximin approach to help obtain a compromise or satisfactory ratio of standard deviations allowable within part families—type I and type II errors of Shewhart X control charts are considered simultaneously. This research establishes Shewhart X control charts for each part family to examine the quality status of all part types in the same family. We also provide a numerical example for purposes of illustration. © 1997 by John Wiley & Sons, Ltd.     

On developing an exponentially weighted moving average chart under progressive setup: An efficient approach to manufacturing processes

The exponentially weighted moving average (EWMA) control chart is a memory chart that is widely used in process monitoring to spot small and persistent disturbances in the process parameter(s). This chart requires normality of the quality characteristic(s) of interest and a smaller choice of smoothing parameter. Any deviations from these conditions affect its performance in terms of efficiency and robustness. For the said two concerns, this study develops a new mixed EWMA chart under progressive setup (mixed EWMA–progressive mean [MEP] chart). The proposed MEP chart combines the advantages of robustness (under nonnormal scenarios) and high sensitivity to small and persistent shifts in the process mean. The performance of the proposed MEP control chart is evaluated in terms of average run length and some other characteristics of run length distribution. The assessment of the proposed chart is made under standard normal, student's t, gamma, Laplace, logistic, exponential, contaminated normal and lognormal distributions. The performance of the proposed MEP chart is also compared with some existing competitors including the classical EWMA, the classical cumulative sum (CUSUM), the homogenously weighted moving average, the mixed EWMA–CUSUM, the mixed CUSUM–EWMA and the double EWMA charts. The analysis reveals that the proposal of this study offers a superior design structure relative to its competing counterparts. An application from substrates manufacturing process (in which flow width of the resist is the key quality characteristic) is also provided in the study.     

A synthetic double sampling control chart for process mean using auxiliary information

A control chart is a simple yet powerful tool that is extensively adopted to monitor shifts in the process mean. In recent years, auxiliary‐information–based (AIB) control charts have received considerable attention as these control charts outperform their counterparts in monitoring changes in the process parameter(s). In this article, we integrate the conforming run length chart with the existing AIB double sampling (AIB DS) chart to propose an AIB synthetic DS chart for the process mean. The AIB synthetic DS chart also encompasses the existing synthetic DS chart. A detailed discussion on the construction, optimization, and evaluation of the run length profiles is provided for the proposed control chart. It is found that the optimal AIB synthetic DS chart significantly outperforms the existing AIB Shewhart, optimal AIB synthetic, and AIB DS charts in detecting various shifts in the process mean. An illustrative example is given to demonstrate the implementation of the existing and proposed AIB control charts.     

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.     

Shewhart control charts to detect mean and standard deviation shifts based on grouped data

A Shewhart control chart is proposed based on gauging theoretically continuous observations into multiple groups. This chart is designed to monitor the process mean and standard deviation for deviations from stability. By assuming an underlying normal distribution, we derive the optimal grouping criterion that maximizes the expected statistical information available in a sample. Control charts based on grouped observations are superior to standard control charts based on variables, such as X and R charts, when the quality characteristic is difficult or expensive to measure precisely, but economical to gauge.     

Comparisons of Some Exponentially Weighted Moving Average Control Charts for Monitoring the Process Mean and Variance

An exponentially weighted moving average (EWMA) control chart for monitoring the process mean μ may be slow to detect large shifts in μ when the EWMA tuning parameter λ is small. An additional problem, sometimes called the inertia problem, is that the EWMA statistic may be in a disadvantageous position on the wrong side of the target when a shift in μ occurs, which may significantly delay detection of a shift in μ. Options for improving the performance of the EWMA chart include using the EWMA chart in combination with a Shewhart chart or in combination with an EWMA chart based on squared deviations from target. The EWMA chart based on squared deviations from target is designed to detect increases in the process standard deviation σ, but it is also very effective for detecting large shifts inμ. Capizzi and Masarotto recently proposed the option of an adaptive EWMA control chart in which λ is a function of the data. With the adaptive feature, the EWMA chart behaves like a standard EWMA chart when the current observation is close to the previous EWMA statistic, and like a Shewhart chart otherwise. Here we extend the use of the adaptive feature to EWMA charts based on squared deviations from target, and also consider an alternate way of defining the adaptive feature. We discuss performance measures that we believe are appropriate for assessing the effects of inertia, and compare the performance of various charts and combinations of charts. Standard practice is to simultaneously monitor both μ and σ, so we consider control chart performance when the objective is to detect small or large changes in μ or increases in σ. We find that combinations of EWMA control charts that include a chart based on squared deviations from target give good overall performance whether or not these charts have the adaptive feature.     

Using an MQE chart based on a self-organizing map NN to monitor out-of-control signals in manufacturing processes

Control charts are recognized as one of the most important tools for statistical process control (SPC), used for monitoring any abnormal deviations in the state of manufacturing processes. However, the effectiveness of control charts is strictly dependent on statistical assumptions that in real applications are frequently violated. In contrast, neural networks (NNs) have excellent noise tolerance in real time, requiring no hypothesis on the statistical distribution of monitored processes. This feature makes NNs promising tools for quality control. In this paper, a self-organizing map (SOM)-based monitoring approach is proposed for enhancing the monitoring of processes. It is capable of providing a comprehensive and quantitative assessment value for the current process state, achieved by minimum quantization error (MQE) calculation. Based on MQE values over time series, a novel MQE chart is developed for monitoring process changes. The aim of this research is to analyse the performance of the MQE chart under the assumption that predictable abnormal patterns are not available. To this aim, the performance of the MQE chart in manufacturing processes (including non-correlated, auto-correlated and multivariate processes) is evaluated. The results indicate that the MQE chart may be a promising tool for quality control.