Economic design of X charts for non-normally correlated data

When the X chart is applied to monitor a manufacturing process, three parameters should be determined: sample size, sampling interval between successive samples, and the control limits for the chart. In 1956, Duncan presented the first cost model to determine the three parameters for the X charts, which is called the economic design of X charts. This paper develops the economic design of X charts for non-normally correlated data. An example of juice production process is presented to illustrate the solution procedure. A sensitivity analysis is performed to show the effects of non-normality and correlation coefficient on the optimal design of the chart.     

EWMA control charts with variable sample sizes and variable sampling intervals

Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. Variable Sample Size (VSS) and Variable Sampling Interval (VSI) control charts vary the sampling rate from the process as a function of the data from the process. VSS and VSI control charts sample at a higher rate when there is evidence of a change in the process, and are thus able to detect process changes faster than traditional control charts. This paper considers general VSS and VSI control charts and develops integral equation and Markov chain methods for finding the statistical properties of these charts. EWMA charts with the VSS and/or the VSI features are studied in detail, and different ways of defining the EWMA control statistic are investigated. It is shown that using either the VSS or VSI feature in an EWMA control chart substantially improves the ability to detect all but very large shifts in the process mean. The VSI feature usually gives more improvement in detection ability than the VSS feature, and using both features together sometimes gives more improvement than using either one separately. Guidelines are given for choosing the possible sample sizes and the possible sampling intervals for these charts. EWMA charts with the VSS and/or VSI feature are compared to CUSUM charts and Shewhart X¯ charts with the VSS and/or VSI features.     

Economic statistical design of a T2 control chart with double warning lines

Recent studies have shown that enhancing the common T² control chart by using variable sample sizes (VSS) and variable sample intervals (VSI) sampling policies with a double warning line scheme (DWL) yields improvements in shift detection times over either pure VSI or VSS schemes in detecting almost all shifts in the process mean. In this paper, we look at this problem from an economical perspective, certainly at least as an important criterion as shift detection time if one considers what occurs in the industry today. Our method is to first construct a cost model to find the economic statistical design (ESD) of the DWL T² control chart using the general model of Lorenzen and Vance (Technometrics 1986; 28 :3–11). Subsequently, we find the values of the chart parameters which minimize the cost model using a genetic algorithm optimization method. Cost comparisons of Fixed ratio sampling, VSI, VSS, VSIVSS with DWL, and multivariate exponentially weighted moving average (MEWMA) charts are made, which indicate the economic efficacy of using either VSIVSS with DWL or MEWMA charts in practice if cost minimization is of interest to the control chart user. Copyright © 2010 John Wiley & Sons, Ltd.     

The Run Sum Hotelling's χ2 Control Chart with Variable Sampling Intervals

Control charts are widely used for process monitoring and quality control in manufacturing industries. Implementing variable sampling interval (VSI) control schemes on control charts rather than traditional fixed sampling interval procedure can significantly improve the control chart's efficiency. In this paper, the VSI run sum (RS) Hotelling's χ² chart is proposed. The optimal scores and parameters of the proposed chart are determined using an optimization technique to minimize the following: (i) out‐of‐control average time to signal (ATS); (ii) adjusted ATS (AATS), when the exact shift size can be specified; (iii) expected ATS; or (iv) expected AATS, when the exact shift size cannot be specified. The Markov chain method is used to evaluate the zero‐state ATS and expected ATS, and steady‐state AATS and expected AATS of the proposed chart. The results show that the VSI RS Hotelling's χ² chart significantly outperforms the standard RS Hotelling's χ² chart and the former also performs well compared with other competing charts. By adding more scoring regions, the efficiency of the VSI RS Hotelling's χ² chart can be further enhanced. An illustrative example using data from a manufacturing process is presented to demonstrate the application of the VSI RS Hotelling's χ² chart. The application of the proposed chart in a quality improvement program can be extended to management and service industries. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive ―X Control Charts with Sampling at Fixed Times

The idea of a variable sampling interval with sampling at fixed times (VSIFT) has been presented by Reynolds. This paper extends this idea to the other two adaptive ―X charts: the variable sampling rate with sampling at fixed times (VSRFT) ―X chart and the variable parameters with sampling at fixed times (VPFT) ―X chart. The VSIFT, VSRFT and VPFT ―X charts are inclusively called the adaptive with sampling at fixed times (AFT) ―X charts in this paper. The control scheme and the design issue are described and discussed for each of the AFT ―X charts. A comparative study shows that the AFT ―X charts have almost the same detection ability as the traditional adaptive ―X charts. However, from the practical viewpoint, the AFT ―X charts are considered to be more convenient to administer than the traditional adaptive ―X charts. Overall, this paper advances the application of ‘sampling at fixed times’ to the adaptive ―X control charts. Copyright © 2004 John Wiley & Sons, Ltd.     

Joint ―X and R Charts with Two‐stage Samplings

When joint ―X and R charts are in use, samples of fixed size are regularly taken from the process, and their means and ranges are plotted on the ―X and R charts, respectively. In this article, joint ―X and R charts have been used for monitoring continuous production processes. The sampling is performed, in two stages. During the first stage, one item of the sample is inspected and, depending on the result, the sampling is interrupted if the process is found to be in control; otherwise, it goes on to the second stage, where the remaining sample items are inspected. The two‐stage sampling procedure speeds up the detection of process disturbances. The proposed joint ―X and R charts are easier to administer and are more efficient than the joint ―X and R charts with variable sample size where the quality characteristic of interest can be evaluated either by attribute or variable. Copyright © 2004 John Wiley & Sons, Ltd.     

Optimisation of a set of or principal components control charts using genetic algorithms

When a multivariate process is to be monitored, there are the options of employing a set of univariate control charts or a single multivariate chart. This paper shows how to effectively design a multivariate control scheme consisting of two or three X charts, using genetic algorithms to optimise the charts parameters. The procedure is implemented using software tools, which we designed. A complete performance comparison of the scheme with the Hotelling's T ² control chart can be made in order to help the user in choosing the most adequate option for the process under consideration. Also, if the user prefers to employ charts based on principal components rather than on the original variables, the software can be used in the same way to optimise a set of two or three control charts based on these components, and to compare their performance with the performance of the T ² chart on the principal components.     

An economic design model for X¯ charts with random sampling policies

Recent literature contains many papers on the study of variable (but deterministic) sampling interval (VSI) policies for X¯ control charts. VSI policies perform better than their corresponding classical fixed-sampling-interval (FSI) policies. Results are also available in the literature for an adaptive VSI policy, an enhancement of the VSI policy, where the sample size also changes with the sampling interval. However, for many industrial (especially continuous) processes where X¯ charts are used, maintaining deterministic time intervals between samples is not possible owing to inherent randomness of the processes. In this paper we propose a new variable sampling policy, named random sampling policy (RSP), for X¯ charts in which the sampling intervals and the sample sizes (considered a function of the sampling intervals) are treated as random variables. An economic modeling framework for design of RSP with run rules is developed. Within the proposed modeling framework, special cases of RSP, such as VSI and FSI policies with and without run rules, can also be evaluated. An application of the economic model is demonstrated through design of RSP parameters for two numerical example problems.     

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.     

Detecting changes in autoregressive processes with X¯ and EWMA charts

The traditional use of control charts necessarily assumes the independence of data. It is now recognized that many processes are autocorrelated thus violating the fundamental assumption of independence. As a result, there is a need for a broader approach to SPC when data are time-dependent or autocorrelated. This paper utilizes control charts with fixed control limits for residuals to monitor the performance of a process yielding time-dependent data subject to shifts in the mean and the autocorrelation structure. The effectiveness of the framework is evaluated by an average run length study of both X¯ and EWMA charts using analytical and simulation techniques. Average run lengths are tabulated for various process disturbance scenarios, and recommendations for the most effective monitoring tool are made. The findings of this research present motivation to extend the traditional paradigms of a shifted process (e.g., mean and/or variance). The results show that decreases in the underlying time series parameters are practically impossible to detect with standard control charts. Furthermore, the practitioner is motivated to employ runs rules since the runs are more likely with time-dependent observations.     

An economic model for \font\twelveit=cmti10 scaled 1600$\overline{\kern‐0.85ex\hbox{\twelveit X}}$\nopagenumbers\end and R charts with time‐varying parameters

This paper proposes an economic model for the selection of time‐varying control chart parameters for monitoring on‐line the mean and variance of a normally distributed quality characteristic. The process is subject to two independent assignable causes. One cause changes the process mean and the other changes the process variance. The occurrence times of these assignable causes are described by Weibull distributions having increasing failure rates. The paper combines two existing models: (I) the model of Ohta and Rahim (IIE Transactions 1997; 29 :481–486) for a dynamic economic design of $\overline{X}$\nopagenumbers\end control charts, where a single assignable cause occurs according to a Weibull distribution and all design parameters are time varying; (II) the model of Costa and Rahim (QRE International 2000; 16 :143–156) for the joint economic design of $\overline{X}$\nopagenumbers\end and R control charts where two assignable causes occur independently according to Weibull distribution, with variable sampling intervals. The advantages of the proposed model over traditional $\overline{X}$\nopagenumbers\end and R control charts with fixed parameters are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

Economic designs of constrained EWMA and combined EWMA-X¯ control schemes

This research presents a comparison between the cost performance of the Exponentially Weighted Moving Average (EWMA) and the combined EWMA-x¯ control chart schemes. In particular, we explore the impact of constraining the in-control average run length on the optimal cost performance of both schemes. Methodologically, we incorporate traditional expected cost models and study the robustness of the two approaches. In general, there appears to be minimal motivation to combine the use of both charts within the same application. The cost model for the combined chart is not a well-behaved function, and yields varying optimal parameters when the in-control average run length is constrained.     

Monitoring the process mean and variance using a weighted loss function CUSUM scheme with variable sampling intervals

Recent research has shown that adaptive control charts and the CUmulative SUM (CUSUM) schemes are quicker in detecting process shifts than traditional static Shewhart charts. This article proposes a weighted loss function CUSUM (WLC) scheme with Variable Sampling Intervals (VSI). It simultaneously monitors both mean shifts and an increasing variance shift by manipulating a single CUSUM chart. Most importantly, this VSI WLC scheme is much easier to operate and design than a VSI CCC scheme which comprises of three CUSUM charts (two of them monitoring the increasing and decreasing mean shifts and one monitoring the increasing variance shift). In terms of detection efficiency, the VSI WLC scheme is a much more powerful tool than the static X&S chart, the VSI X&S chart and the static WLC scheme. It is even more powerful than the VSI CCC scheme for many different combinations of mean and increasing variance shifts.     

An Adaptive CUSUM Chart with Single Sample Size for Monitoring Process Mean and Variance

This article proposes an adaptive absolute cumulative sum chart (called the adaptive ACUSUM chart) for statistical process control. The new development includes the variable sampling interval (VSI), variable sample size (VSS) and VSS and interval (VSSI) versions, all of which are highly effective for monitoring the mean and variance of a variable x by inspecting the absolute sample shift (where μ₀ is the in‐control mean or target value of x). While the adaptive ACUSUM chart is a straightforward extension of the ABS CUSUM chart developed by Wu, et al., it is much more effective than all other adaptive CUSUM charts. Noteworthily, the superiority of VSI ACUSUM chart over the best adaptive CUSUM chart in literature is about 35% from an overall viewpoint. Moreover, the design and implementation of the adaptive ACUSUM chart are much simpler than that of all other adaptive CUSUM schemes. All these desirable features of the adaptive ACUSUM chart may be attributable to the use of a single sample size (n = 1). Another quite interesting finding is that the simpler VSI ACUSUM chart works equally well as the more complicated VSSI ACUSUM chart. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Optimal process control for multiple dependent subprocesses

A renewal theory approach is proposed to derive the cost model for multiple dependent subprocesses. The optimal individual Y control chart and multiple cause-selecting control chart are thus constructed to monitor the overall product quality and specific product quality contributed by the multiple dependent subprocesses. They can be used to maintain the process with minimum cost and effectively distinguish which component of the subprocesses is out of control. The optimal design parameters of the proposed control charts can be determined by minimizing the cost model using a simple grid search method. An example is given to illustrate the application of the optimal individual Y control chart and multiple cause-selecting control chart. © 1998 John Wiley & Sons, Ltd.     

Constrained optimization model for the design of an adaptive X chart

An economic-statistical model is developed for variable parameters (VP) X charts in which all design parameters vary adaptively, that is, each of the design parameters (sample size, sampling interval and control-limit width) vary as a function of the most recent process information. The cost function due to controlling the process quality through a VP X chart is derived. During the optimization of the cost function, constraints are imposed on the expected times to signal when the process is in and out of control. In this way, required statistical properties can be assured. Through a numerical example, the proposed economicstatistical design approach for VP X charts is compared to the economic design for VP X charts and to the economic-statistical and economic designs for fixed parameters (FP) X charts in terms of the operating cost and the expected times to signal. From this example, it is possible to assess the benefits provided by the proposed model. Varying some input parameters, their effect on the optimal cost and on the optimal values of the design parameters was analysed.     

The joint economic‐statistical design of X and R charts for nonnormal data

We consider the joint economic‐statistical design of X and R control charts under the assumption that the quality measurement and the in‐control time have Johnson and Weibull distributions. The Johnson distribution is general in that it can be made to fit all possible values of skewness and kurtosis. The four parameters—the sample size n, time h between successive samples, and the control factors k₁ and k₂ for the X and R charts—are determined so that the mean hourly loss‐cost is minimized under constraints on the Type I and II error probabilities. We have generalized the Costa model to accommodate the Johnson and Weibull distributions. Sensitivity to nonnormality, shift, and Weibull scale parameter is considered in our analysis. Our sensitivity analysis shows that the optimal design parameters are sensitive to nonnormality. Comparisons of the fully economic and economic‐statistical designs are given. Copyright © 2010 John Wiley & Sons, Ltd.     

Short-run statistical process control: Q-chart enhancements and alternative methods

In processes where the length of the production run is short, data to estimate the process parameters and control limits may not be available prior to the start of production, and because of the short run time, traditional methods for establishing control charts cannot be easily applied. Recently, Q charts have been proposed to address this problem. We study the average run length (ARL) of Q charts for a normally distributed variable assuming that a sustained shift occurs in the quality characteristic. It is shown that in some cases Q charts do not exhibit adequate ARL performance. Modifications that enhance the ARL properties of Q charts are presented. Some alternatives to Q charts are also discussed. For the case of a known process target two alternative methods are presented: an exponentially weighted moving average (EWMA) method and an adaptive Kalman filtering method. It is shown that both methods have better ARL performance than Q charts for that case. For the case of both process parameters unknown, an adaptive Kalman filtering method used with a tracking signal provides an ARL performance that improves as better estimates of the process mean and variance are given. A practical example illustrates the tracking signal method for the case when the process parameters are unknown.