Process Mean Vector Research Articles

Simultaneous monitoring of mean vector and covariance matrix shifts in bivariate manufacturing processes using hybrid ensemble learning-based model

Many manufacturing processes are multivariate in nature because the quality of a given product is determined by several interrelated quality characteristics. Recently, various machine learning techniques (e.g., artificial neural network, support vector machine, support vector regression or decision tree) have been used as an effective tool to monitor process mean vector and covariance matrix shifts. However, most of these machine learning techniques-based approaches for process mean vector and covariance matrix have been developed separately in literature with the other parameter assumed to be under control. Little attention has been given to simultaneous monitoring of process mean vector and covariance matrix shifts. In addition, these approaches cannot provide more detailed shift information, for example the shift magnitude, which would be greatly useful for quality practitioners to search the assignable causes that give rise to the out-of-control situation. This study presents a hybrid ensemble learning-based model for simultaneous monitoring of process mean vector and covariance matrix shifts. The numerical results indicate that the proposed model can effectively detect and recognize not only mean vector or covariance matrix shifts but also mixed situations where mean vector and covariance matrix shifts exist concurrently. Meanwhile, the magnitude of the shift of each of the shifted quality characteristics can be accurately quantified. Empirical comparisons also show that the proposed model performs better than other existing approaches in detecting mean vector and covariance matrix shifts, while also providing the capability of recognition of shift types and quantification of shift magnitudes. A demonstrative example is provided.

Simultaneous monitoring of process mean vector and covariance matrix via penalized likelihood estimation

In recent years, some authors have incorporated the penalized likelihood estimation into designing multivariate control charts under the premise that in practice typically only a small set of variables actually contributes to changes in the process. The advantage of the penalized likelihood estimation is that it produces sparse and more focused estimates of the unknown population parameters which, when used in a control chart, can improve the performance of the resulting control chart. Nevertheless, the existing works focus on monitoring changes occurring only in the mean vector or only in the covariance matrix. Stemming from the ideas of the generalized likelihood ratio test and the multivariate exponentially weighted moving covariance, new control charts are proposed for simultaneously monitoring the mean vector and the covariance matrix of a multivariate normal process. The performance of the proposed charts is assessed by both Monte-Carlo simulations and a real example.

Attribute Charts for Monitoring the Mean Vector of Bivariate Processes

This article proposes two Shewhart charts, denoted npxy and npw charts, which use attribute inspection to control the mean vector (μx; μy)′ of bivariate processes. The units of the sample are classified as first‐class, second‐class, or third‐class units, according to discriminate limits and the values of their two quality characteristics, X and Y. When the npxy chart is in use, the monitoring statistic is M = N1 + N2, where N1 and N2 are the number of sample units with a second‐class and third‐class classification, respectively. When the npw chart is in use, the monitoring statistic is W = N1 + 2N2. We assume that the quality characteristics X and Y follow a bivariate normal distribution and that the assignable cause shifts the mean vector without changing the covariance matrix. In general, the synthetic npxy and npw charts require twice larger samples to outperform the T2 chart. Copyright © 2014 John Wiley & Sons, Ltd.

Application of Multivariate Process Control (MPC) in the Natural Rubber Industry

Statistical process control (SPC) monitors specified quality characteristics of a product orservice to detect whether the process has changed in a way that would affect product qualityand to measure the current quality of products or services. This study introduces severalapplications on multivariate process control in the rubber industry employing the factory datain the Dartonfield estate of the Rubber Research Institute of Sri Lanka (RRI). Multivariateprocess control was applied to consider two or more variables simultaneously. In this studymultivariate techniques were done for the application of bleaching agents, sodium bisulphateand acids in the process of crepe rubber production.Multivariable scatter diagrams and control ellipse has been used to show the process whenthe multivariate data composed of two variables. Hotelling T2 control chart was used tomonitor application of chemicals simultaneously in a single chart. Principal components wereused to interpret dimensions of components and star charts were applied to diagnostic fordetermining the relative contributions of the each chemical component. The multivariateEWMA (MEWMA) control chart for monitoring the process mean vector was astraightforward extension of the univariate EWMA chart. The graphical presentation is veryimportant and it is a quick way to see the advantages of MPC.Multivariate Statistical analysis is concerned with data that consist of sets of measurementson a number of variables. So, multivariate quality control provides a way for monitoring andevaluation of chemical application process. Scatter plots and control ellipse is very importantto detect out of control points and relationships as a graphical method. Hotelling T2 is animportant chart that can be decomposed into an overall measure of distance of the groupmeans from the target T2 and measure of variability. Principal components can be useful formultivariate quality control, especially to observe variation of dimensions of chemicalapplications. Multivariate EWMA procedure use additional information from recent historyof the process and it is more sensitive in monitoring chemical application process.Therefore multivariate process control provides way for manufacturer to test their products inan environment that provide many advantages over univariate models.

On monitoring shift in the mean processes with vector autoregressive residual control charts of individual observation

In this paper, we purpose a monitoring scheme of individual observation that can be modeled as a first-order vector autoregressive process and develop it to mean square successive difference matrix based on residuals. Due to the effect of shift in mean process on VAR residual charts, we investigate the T 2 -type charts performance of residual and control charts as an individual observation. Then we compare both T 2 -type charts using upper control limit approximation with α=0.005 (fix) based simulation. From the results of comparison, it shows that the chart for a small to moderate level of shift in the process mean vector gives a better performance than the charts.

Multivariate process parameter change identification by neural network

Whenever there is an out-of-control signal in process parameter control charts, maintenance engineers try to diagnose the cause near the time of the signal which does not always lead to prompt identification of the source(s) of the out-of-control condition, and this in some cases yields to extremely high monetary loses for the manufacturer owner. This paper applies multivariate exponentially weighted moving average (MEWMA) control charts and neural networks to make the signal identification more effective. The simulation of this procedure shows that this new control chart can be very effective in detecting the actual change point for all process dimension and all shift magnitudes considered. This methodology can be used in manufacturing and process industries to predict change points and expedite the search for failure causing parameters, resulting in improved quality at reduced overall cost. This research shows development of MEWMA by usage of neural network for identifying the step change-point and the variable responsible for the change in the process mean vector.

An Optimal Control Procedure based on Multivariate Synthetic Cumulative Sum

A multivariate synthetic cumulative sum (MSCUSUM) control chart is presented in this study. The MSCUSUM control chart consists of a multivariate cumulative sum (MCUSUM) control chart and a conforming run length (CRL) control chart, and it combines the strengths of both cumulative sum and CRL procedures. Some properties of this new scheme are investigated, and implementation procedure is shown with real data example. Simulation studies are also carried out to show that the MSCUSUM control chart is more efficient than the MCUSUM, synthetic T2 and Hotelling's T2 control charts, for detecting shifts in the process mean vector. The MSCUSUM chart also performs slightly better than the multivariate synthetic exponentially weighted moving average chart in detecting moderate and large shifts. Copyright © 2013 John Wiley & Sons, Ltd.

Multivariate setup adjustment with fixed adjustment cost

For a discrete-part manufacturing process, adjustment is necessary to bring the process to target when the setup of the machine is improperly done. This paper investigates the adjustment scheme for a multivariate process with initial offset in the presence of fixed adjustment cost. The objective is to derive a solution for the case that the cost for different responses can be different. Based on the state process-control model, expressions of optimal adjustment scheme are derived by the dynamic programming and congruent transformation of quality loss matrix. This scheme belongs to a deadband form and calls for an adjustment only when the norm of the linear transformation of perceived process mean vector is at the outside of the deadband. A simulation study is conducted to compare the scheme proposed in this paper with two other schemes.

Variable Sampling Rate Multivariate Exponentially Weighted Moving Average Control Chart with Double Warning Lines

This study investigates the variable sample sizes and variable sampling intervals multivariate exponentially weighted moving average control chart using double warning lines (MEWMA-DWL). The steady-state average time to signal of the MEWMA-DWL control chart is obtained by using the Markov chain approach. Then the performance of the MEWMA-DWL control chart is compared with the corresponding fixed sampling rate (FSR) MEWMA control chart. From the numerical results, the MEWMA-DWL control chart is faster than the corresponding FSR MEWMA control chart in detecting small and moderate shifts in the process mean vector.

The Design of the Multivariate Synthetic Exponentially Weighted Moving Average Control Chart

A multivariate synthetic exponentially weighted moving average (MSEWMA) control chart is presented in this study. The MSEWMA control chart consists of a multivariate exponentially weighted moving average (MEWMA) control chart and a conforming run length control chart. The average run length of the MSEWMA control chart is obtained using a Markov chain approach. From the numerical comparisons, it is shown that the MSEWMA control chart is more efficient than the multivariate synthetic T 2 control chart and the MEWMA control chart for detecting shifts in the process mean vector.

A Bayesian Approach to Change Point Estimation in Multivariate SPC

A Bayesian procedure is developed to estimate the time of a change in the process mean vector for a multivariate process, given that an out-of-control signal was raised on a multivariate control chart. In addition, we can infer simultaneously which variable(s) had a change in mean value, when the change occurred, and the value of the changed mean. All three problems (inferring change point time, variables that shifted, and new values for the shifted variables) are addressed in a single statistical model. Markov chain Monte Carlo (MCMC) methods, through the software WinBUGS, are used to estimate parameters of the change point models. To identify the mean shift in a process with more than two variables, we propose a branch-and-bound search algorithm so that MCMC can be carried out with a predictable computing time in each search step. A simulation study shows that the Bayesian approach has similar performance compared to the maximum likelihood estimation (MLE) in terms of identifying the true change point location when a noninformative prior is assumed; however, it can perform better when proper prior knowledge is incorporated into the estimation procedure. The Bayesian approach provides full posterior distributions for the model and change point, which can contain information that is not available in a likelihood analysis.

Methods for identifying influential variables in an out-of-control multivariate normal process

Hotelling's T 2 is a well-known statistic for testing the mean vector of a multivariate normal distribution. Control charts based on T 2 have been widely used in statistical process control for monitoring a multivariate process. Although it is a powerful tool, the T 2 statistic has a practical problem, namely, that a significant T 2 -value that normally signals an overall out-of-control condition in the process mean vector does not provide direct information about which variable or group of variables may have caused this out-of-control condition. We propose a diagnostic method to identify the influential variable(s) for cases with and without a speci- fied out-of-control mean vector. Our approach, based on the likelihood principle, computes the conditional likelihood of a variable or sub-group of variables causing or not causing the overall out-of-control condition. Unlike many existing meth- ods, our method assumes that an out-of-control condition already exists; hence, all conditional likelihoods in this paper are based on non-central distributions of the monitoring/testing statistics. By comparing these conditional likelihoods, we iden- tify the influential variable(s). We use an example from the literature to illustrate our method and to demonstrate its effectiveness.

Designing the optimal process mean vector for mixed multiple quality characteristics

For the manufacturing community, determining the optimal process mean can often lead to a significant reduction in waste and increased opportunity for monetary gain. Given the process specification limits and associated rework or rejection costs, the traditional method for identifying the optimal process mean involves assuming values for each of the process distribution parameters prior to implementing an optimization scheme. In contrast, this article proposes integrating response surface methods into the framework of the problem, thus removing the need to make assumptions on the parameters. Furthermore, whereas researchers have studied models to investigate this research problem for a single quality characteristic and multiple nominal-the-best type characteristics, this article specifically examines the mixed multiple quality characteristic problem. A non-linear programming routine with economic considerations is established to facilitate the identification of the optimal process mean vector. An analysis of the sensitivity corresponding to the cost structure, tolerance, and quality loss settings is also provided to illustrate their effect on the solutions.

Detecting and estimating the time of a step-change in multivariate Poisson processes

In multi-attribute process monitoring, when a control chart signals an out-of-control condition indicating the existence of a special cause, knowing when the process has really changed (the change point) accelerates the identification of the source of the special cause and makes the corrective measures to be employed sooner. This, of course, results in a considerable amount of savings in time and money. Since many real world multi-attribute processes are Poisson and most process changes are step-change, a new method is proposed, in this paper, to derive the maximum likelihood estimator of the time of a step-change in the mean vector of multivariate Poisson processes. In this method, two transformations are first employed to almost remove the inherent skewness involved in multi-attribute processes and make them almost multivariate normal, and also to almost diminish correlations between the attributes. Then, a T2 control chart is employed for out-of-control detection and a maximum likelihood estimator is used to estimate the change point. The performance of the proposed methodology is illustrated using some simulation experiments in which we show that the proposed procedure is relatively accurate and reliable in detecting and estimating the change point.

A modified economic-statistical design of the T2 control chart with variable sample sizes and control limits

Recent studies have shown that using variable sampling size and control limits (VSSC) schemes result in charts with more statistical power than variable sampling size (VSS) when detecting small to moderate shifts in the process mean vector. This paper presents an economic-statistical design (ESD) of the VSSC T2 control chart using the general model of Lorenzen and Vance [22]. The genetic algorithm approach is then employed to search for the optimal values of the six test parameters of the chart. We then compare the expected cost per unit of time of the optimally designed VSSC chart with optimally designed VSS and FRS (fixed ratio sampling) T2 charts as well as MEWMA charts.

Statistical Merits and Economic Evaluation of T2 Control Charts with the VSSC Scheme

T2 control charts are used to monitor a process when more than one quality variable associated with the process is being observed. Recent studies have shown that using variable sampling size (VSS) schemes results in charts with more statistical power for detecting small to moderate shifts in the process mean vector. This paper presents an economic-statistical design of T2 control charts with variable sample size and control limits (VSSC). We build a cost model of a VSSC T2 control chart for the purpose of economic-statistical design using the model of Costa and Rahim (J. Appl. Stat. 28:875–885, 2001). This cost model constructed involves the cost of false alarms, the cost of finding and eliminating an assignable cause, the cost associated with production in an out-of-control state, and the cost of sampling and testing. We optimize this model using a genetic algorithm approach. Furthermore, VSSC and VSS T2 charts are compared with respect to the expected cost per unit time.

Diagnosing the source(s) of a monotonic change in the process mean vector

When a control chart detects any departure of an underlying process from in-control state, a root cause analysis should be initiated by process engineer to identify and eliminate the source(s) of contributing factors. In a multivariate environment, such information is crucial and plays an essential role for an effective root cause analysis. Different schemes have been proposed in the literature which could be used for this purpose when change type is known a priori. However, in practice, when the change type deviates from the assumed change type, the performance of these schemes deteriorates considerably. In this paper, a model based on artificial neural network is introduced which helps to diagnose the source(s) of assignable cause without requiring the exact knowledge of the change type in the mean vector of the process. A case study in chemical industry is considered to evaluate the performance of the proposed schemes when process experiences different change types. Also, the capabilities of the proposed scheme are compared to the two existing models. The correct classification percentage is used as the measure of the performance evaluation. Several other numerical examples are also used to evaluate the robustness of the model.

On the advantages of economically designed variable sample sizes and sampling intervals T2 control chart: double warning lines scheme

Faraz and Parsian (Stat Pap 47:569–593, 2006) have shown that the double warning lines (DWL) scheme detects process shifts more quickly than the other variable ratio sampling schemes such as variable sample sizes (VSS), variable sampling intervals (VSI) and variable sample sizes and sampling intervals (VSSVSI). In this paper, the DWL T2 control chart for monitoring the process mean vector is economically designed. The cost model proposed by Costa and Rahim (J Appl Stat 28:875–885, 2001) is used here and is minimized through a genetic algorithm (GA) approach. Then the effects of the model parameters on the chart parameters and resulting operating loss is studied and finally a comparison between all possible variable ratio sampling (VRS) schemes are made to choose the best option economically.

Variable sampling interval hotelling's T2 charts with runs rules for switching between sampling interval lengths

The use of runs rules is proposed for switching between the sampling interval lengths of variable sampling interval Hotelling's T2 charts. The purpose of applying these rules is to reduce the frequency of the switches which causes inconvenience in the administration of the charts. The expressions for the performance measures for the charts with these rules are derived. The effects of different runs rules on the performances are evaluated through numerical comparisons. The runs rules substantially reduce the frequency of switches during the in‐control period and during the out‐of‐control periods due to the small to moderate shifts in the process mean vector. They also fairly improve the statistical performances of the charts in detecting the small shifts and do not affect that in detecting the large shifts. However, some runs rules slightly worsen the statistical performances in detecting the moderate shifts. Copyright © 2011 John Wiley & Sons, Ltd.

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

AbstractThe 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.