Multivariate Control Charts for Monitoring the Process Mean and Variability Using Sequential Sampling

Abstract

Traditional control charts for process monitoring are based on taking samples of fixed size at equally spaced sampling points. As an alternative to these traditional fixed sampling rate (FSR) control charts, variable sampling rate (VSR) control charts change the sampling rate as a function of the data from the process. With VSR control charts the sampling rate is increased whenever there is some indication of a problem with the process, and decreased when there is no indication of a problem. This paper investigates a type of VSR control chart based on sequential sampling in which the sample size used at a sampling point is a function of the data from the current and past sampling points. Sequential sampling is investigated in the context of simultaneously monitoring both the mean and variability of a multivariate normal process. Multivariate exponentially weighted moving average (MEWMA) control charts based on sample means and on the sum of the squared deviations from the target are considered. When sequential sampling is used in a combination of the MEWMA charts based on sample means and on squared deviations from the target, the average performance is much better than that of the standard FSR multivariate control charts that have traditionally been used.     

A Note on the Aging Properties of the Run Length of Markov-Type Control Charts

Abstract

A change in a production process must be detected quickly so that a corrective action can be taken. Thus, it comes as no surprise that the run length (RL) is usually used to describe the performance of a quality control chart.

This popular performance measure has a phase-type distribution when dealing with Markov-type charts, namely, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts, as opposed to a geometric distribution, when standard Shewhart charts are in use.

In this article, we briefly discuss sufficient conditions on the associated probability transition matrix to deal with run lengths with aging properties such as new better than used in expectation, new better than used, and increasing hazard rate.

We also explore the implications of these aging properties of the run lengths, namely when we decide to confront the in control and out-of-control variances of the run lengths of matched in control Shewhart and Markov-type control charts.     

An ARL-unbiased thinning-based EWMA chart to monitor counts

Abstract

Shewhart control charts are known to be somewhat insensitive to shifts of small and moderate size. Expectedly, alternative control schemes such as the exponentially weighted moving average (EWMA) charts have been proposed to speed up the detection of such shifts. Unfortunately, applying the ordinary EWMA recursion to count data leads to a control statistic no longer with a fixed discrete range. Therefore, we propose a novel chart which relies on a EWMA control statistic where the usual scalar product is replaced by a thinning operation. Actually, we use the new fractional binomial thinning to avoid the typical over-smoothing ascribable to ceiling, rounding, and flooring operations. The properties of this discrete statistic are similar to the ones of its continuous EWMA counterpart and the run length (RL) performance of the associated chart can be computed exactly using the Markov chain approach for independent and identically distributed (i.i.d.) counts. Moreover, this chart is set in such way that: the average run length (ARL) curve attains a maximum in the in-control situation, i.e., the chart is ARL-unbiased; and the in-control ARL is equal to a pre-specified value. We use the R statistical software to provide compelling illustrations of this unconventional EWMA chart and to compare its RL performance with the ones of a few competing control charts for the mean of i.i.d. Poisson counts.     

Control charts applying a general sequential test at each sampling point

A control chart is proposed which applies a generalized sequential probability ratio test (SPRT) at fixed sampling intervals.the GSPRT chart, to monitor a parameter θ.which indexes a variable X with a genrnal distribution. Expression in terms of integral equations are developed for the moments of the zero-time and steady time to signal and number of observation to signal of the GSPRT chart. Comparsion are made between the performance of the GSPRT chart and the fixed sampling interval (FSI) and variable sampling interval (vsI) X and CUSUM charts. The GSPRT chart is found to be highly efficient and to have administrative advantages General Guidelines are provided for the design of GSPRT charts.     

EWMA Charts for Monitoring the Mean and the Autocovariances of Stationary Gaussian Processes

Abstract

In this article simultaneous individual control charts for the mean and the autocovariances of a stationary process are introduced. All control schemes are EWMA (exponentially weighted moving average) charts. A multivariate quality characteristic is considered. It describes the behavior of the mean and the autocovariances. This quantity is transformed to a one-dimensional variable by using the Mahalanobis distance. The control statistic is obtained by exponentially smoothing these variables. Another control procedure is based on a multivariate EWMA recursion applied directly to our multivariate quality characteristic. After that the resulting statistic is transformed to a univariate random variable. Besides modified control charts we consider residual charts. For the residual charts the same procedure is used but the original observations are replaced by the residuals. In an extensive simulation study all control schemes are compared with each other. The target process is assumed to be an ARMA(1, 1) process.     

Microencapsulation in foam spray drying

Abstract

The article presents the results of the experiments of gas-admixing foam spray drying microencapsulation of oil in co-current spray drying tower. The introduction of a foaming gas resulted in a decrease of encapsulation efficiency: for high foaming gas/liquid ratio (GLR) 6.43·10^?3 kg/kg and T_air,0=215?°C by over 50% relative to the non-foaming process due to damage of the microcapsule structure and leakage of the oil phase to the particle surface. For moderate drying temperatures and GLR, 80% of microencapsulation efficiency might be achieved with simultaneous control of selected product properties like apparent and bulk density or angle of repose. Abbreviations GLR gas-to-liquid ratio

HR Hausner ratio

MDX maltodextrin

PSD particle size distribution

SEM scanning electron microscope.

     

Improved process control for the plastics industry

One of the valuable tools for organizations involved in a total quality improvement effort is statistical process control (SPC). When implementing SPC in their processes, many of these organizations have equated SPC only with X & R control charts. These control charts, though important, may not be the best tool to monitor and control every process. Other charts such as an X , R_w, R_m control chart, which monitors the location of the process outpout, the instantaneous variation, and the process variation over time, may often be a better tool. This is especially true where manufacturing companies are processing materials in batches or using batches of materials such as in polymer production and molding.     

Counting by Weighing: An Alternative Sampling Scheme

Abstract

In the last decades, various methods are applied for evaluating different performance measures of change-point detection schemes. In only some cases—recent examples are Chandrasekaran et al. (Chandrasekaran, S.; English, J.R.; Disney, R.L. Modeling and analysis of EWMA control schemes with variance-adjusted control limits. IIE Transactions 1995, 277, 282–290) and Steiner (Steiner, S.H. EWMA control charts with time-varying control limits and fast initial response. Journal of Quality Technology 1999, 31 (1), 75–86)—EWMA schemes with varying control limits are considered. However, EWMA charts with just these limits are sometimes more appropriate than those with fixed limits. Here, a computational approach is presented which allows to compute the usual performance measures with high precision. The main idea is connected to earlier results of Madsen and Conn (Madsen, R.W.; Conn, P.S. Ergodic behavior for nonnegative kernels. Ann. Probab. 1973, 1, 995–1013), Woodall (Woodall, W.H. The distribution of the run length of one-sided CUSUM procedures for continuous random variables. Technometrics 1983, 25, 295–301), and Waldmann (Waldmann, K.-H. Bounds for the distribution of the run length of geometric moving average charts. J. R. Stat. Soc., Ser. C, Appl. Stat. 1986a, 35, 151–158). Additionally, quantities as the steady-state ARL and the steady-state distribution of the chart statistic can be computed very precisely.     

Evaluating properties of variable sampling interval control charts

Standard fixed sampling interval (FSI) control charts take samples from a process at fixed length sampling intervals for purposes of detecting changes in the peocess that may affect the quality of the output. Variable sampling interval (VSI) control charts vary the sampling interval as a function of what is observed from the process and can detect process changes faster than FSI control charts. Evaluation of properties of VSI control charts requires extensions of methods that have been developed for FSI control Control charts. A unified treatment of two widely used methods, the Markov chain method and the integral equation method. is given for VSI control charts.This unified treatment includes some results which are new in the FSI case. The new methods are used for the numerical evaluation of propertics of exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts.Some general optimality results for the choice fo the sampling intervals are also given.     

Shewhart Revisited

Abstract

The Shewhart control chart was first to monitor an ongoing process and raise an alarm when it appears that the level has changed. We show that the Shewhart chart is optimal for the criterion of maximizing the probability of detecting a change upon its occurrence subject to an average run length to false alarm. It is remarkable, particularly when the change is of moderate size, that Shewhart's procedure is better than cumulative sum (CUSUM). In the multivariate setting, applying the Shewhart procedure to each process separately is suboptimal. We create a generalized Shewhart procedure that is optimal for the aforementioned criterion. The results are illustrated in common settings.     

The Generalized Synthetic Chart

Abstract

The Generalized Synthetic chart is presented and mathematical expressions for its average run length and variance of the run length are developed. The methodology is applied to the EWMA and CUSUM charts and near-optimization procedures are discussed. The synthetic EWMA and CUSUM charts are compared with their standard counterparts, the original synthetic chart, and the Shewhart chart. Significant improvements in detecting power are reported.     

Ewma charts for multivariate time series

In this paper a muIt,ivariat.e EWMA chart for time series is introduced. In principle, it is a generalization of the control scheme of Lowry et al. (I992) for multivarite indendent observations.

The autocovariances of the EWMA recursion are derived for stationary multivariate time series. IYsing tllese reslllts a co11t.rol chart hased or1 t11 illt.ivariate EWMA recursion is proposed. For a multivariate autoregressive process of order 1, a sufficient. condition is given such that the in-control average run length (ARL) is invariant, withrespect to the covariance of the White Noise process. This scheme is compared with the MEWMA control chart of Lowry et al. (1992) applied to the residuals.

By an extensive Monte carlo study the ARL of both charts are determined for several multivariate autoregressive processses.     

A mixed control chart using process capability index

Control charts are considered as a compulsory tool for the monitoring and improvement of the quality of products. The commonly used attribute and the variable control charts are unsatisfactory for process monitoring. To overcome this di?culty, we proposed a monitoring scheme that is an improved technique based on the joint merits of the attribute as well as the variable control charts. The control chart coe?cients have been determined for in-control and out-of-control situations for specified average run length values using simulation. The e?ciency of the proposed control chart has been evaluated using average run length criteria for different values of the process settings. A practical example has been studied for implementation of the proposed technique. It has been observed that the proposed chart performs well for the mixed technique under the process capability index.     

Sequentially Updated Residuals and Detection of Stationary Errors in Polynomial Regression Models

Abstract

The question whether a time series behaves as a random walk or as a stationary process is an important and delicate problem, particularly arising in financial statistics, econometrics, and engineering. This article studies the problem to detect sequentially that the error terms in a polynomial regression model no longer behave as a random walk but as a stationary process. We provide the asymptotic distribution theory for a Monitoring procedure given by a control chart; i.e., a stopping time, which is related to a well-known unit root test statistic calculated from sequentially updated residuals. We provide a functional central limit theorem for the corresponding stochastic process that implies a central limit theorem for the control chart. The finite sample properties are investigated by a simulation study.     

A comparative study of a fluidised bed reactor and a gas lift loop reactor for the ibe process: Part III. Reactor performances and scale Up

Continuous fermentations using Clostridium spp. DSM 2152 immobilised in calcium alginate beads to produce butanol and isopropanol from glucose were carried out in a fluidised bed reactor with liquid recycle (FBR, 10.9 dm³ working volume, 41 % solids) and in a gas lift loop reactor (GLR, 11.4 dm³ working volume, 32% solids). In the FBR in-situ produced non-coalescing gas bubbles had a negligible influence on the fluidisation pattern and the steady state results of the fermentation were in accordance with those predicted by a reactor model. The FBR was operated reliably for 5 weeks without decrease of activity. The GLR behaved as a three phase reactor because of the recycled fermentation gas. The steady state fermentation results were as predicted by the GLR model. Scale up to a 50 m³ FBR and a 65 m³ GLR led to development of a plug flow with recycle model for the FBR and a stirred tank model for the GLR. On the basis of overall reactor performance and ease of integration with a simultaneous product recovery the FBR is preferred to the GLR for application in a large scale butanol/isopropanol process using immobilised Clostridia spp.     

利用控制图准确测定丁二烯橡胶的凝胶含量

对性能稳定质量优良的国产丁二烯橡胶(BR)9000样品,进行凝胶含量测定,根据测定结果各统计量绘制凝胶含量测定的X-R(或S)控制图。根据X图、R图和S图,可以综合判断出实验室丁二烯橡胶样品凝含量的测定结果处于良好的受控状态,实验室具有较高的系统和随机控制的能力,测定结果准确可靠。在测定丁二烯橡胶凝胶含量时,系统误差控制在±0.02%范围内,随机误差应小于0.05%,标准偏差应小于0.05。     

Sampling Performance of Impactors for Fungal Spores and Yeast Cells

Impactor than impinger and filter is more commonly used for bioaerosol sampling because of direct collection without post-collection sample processing. In this study, sampling fungal performance of microbial impactors with different sampling velocities and cut-point diameters was assessed. It was demonstrated that biological recoveries of Penicillium citrinum spores were ob served to be similar to those of two yeast cells, Candida famata var. flareri and Rhodotorula mucilaginosa . In addition, it was indicated that sampling performance of Burkard samplers was relatively better than those of Andersen 1-STG and MAS-100 samplers. Moreover, sampling time was indicated to not significantly influence fungal recoveries of the three evaluated impactors.     

Critical evaluation of approaches for on-line batch process monitoring

Since the introduction of batch process monitoring using component models in 1992, different approaches for statistical batch process monitoring have been suggested in the literature. This is the first evaluation of five proposed approaches so far. The differences and similarities between the approaches are highlighted. The derivation of control charts for these approaches are discussed. A control chart should give a fast and reliable detection of disturbances in the process. These features are evaluated for each approach by means of two performance indices. First, the action signal time for various disturbed batches is tested. Secondly, the probability of a false warning in a control chart is computed. In order to evaluate the five approaches, five different data sets are studied: one simulation of a batch process, three batch processes obtained from industry and one laboratory spectral data set. The obtained results for the performance indices are summarised and discussed. Recommendations helpful for practical use are given.     

Fault diagnosis and fault tolerant control using reduced order models

In this work, we have developed a reduced order model relevant for fault diagnosis and control. This model is combined with a generalized likelihood ratio (GLR) method and integrated with fault tolerant control schemes developed earlier. Simulation studies of an ideal binary distillation column show that the use of reduced order model improves the diagnostic performance thereby leading to improved fault tolerance. Furthermore, the proposed strategy is also computational more efficient. In inferential control, such as the use of temperature measurements to infer product compositions, the proposed fault tolerant control scheme performs better than the conventional control scheme when various soft faults occur.