Adaptive CUSUM control chart with variable sampling intervals

The standard cumulative sum chart (CUSUM) is widely used for detecting small and moderate process mean shifts, and its optimal detection ability for any pre-specified mean shift has been demonstrated by its equivalence to continuous sequential tests. In real practice, the assumption of knowing the true mean shift in prior cannot be always met. So it is desirable to design a procedure that is efficient for detecting a range of future expected but unknown mean shifts. Adaptive CUSUM control chart, which can continuously adjust itself by a one-step forecasting operator, has been proposed to detect efficiently and robustly for a range of mean shifts in the early literature. Moreover, in terms of sampling time to signal, control chart with the VSI (variable sampling intervals) feature can detect the process changes more quickly than the traditional FSI (fixed sample intervals) chart. In this paper, a new CUSUM control chart which is based on both adaptive and VSI features is discussed. Also, a two-dimensional Markov chain model is developed to evaluate its run-time performance.     

Process control using VSI cause selecting control charts

The article considers the variable process control scheme for two dependent process steps with incorrect adjustment. Incorrect adjustment of a process may result in shifts in process mean, process variance, or both, ultimately affecting the quality of products. We construct the variable sampling interval (VSI) Z_{[`(X)]</sub>-Z<sub>SX<sup>2</sup></sub>{Z_{\overline{X}}-Z_{S_X^2}} and Z<sub>[`(e)]}-Z_Se²{Z_{\bar{{e}}}-Z_{S_e^2}} control charts to effectively monitor the quality variable produced by the first process step with incorrect adjustment and the quality variable produced by the second process step with incorrect adjustment, respectively. The performance of the proposed VSI control charts is measured by the adjusted average time to signal derived using a Markov chain approach. An example of the cotton yarn producing system shows the application and performance of the proposed joint VSI Z_{[`(X)]</sub> -Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}} -Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts in detecting shifts in mean and variance for the two dependent process steps with incorrect adjustment. Furthermore, the performance of the VSI Z_{[`(X)]</sub>-Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}}-Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts and the fixed sampling interval Z_{[`(X)]</sub> -Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}} -Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts are compared by numerical analysis results. These demonstrate that the former is much faster in detecting small and median shifts in mean and variance. When quality engineers cannot specify the values of variable sampling intervals, the optimum VSI Z_{[`(X)]</sub>-Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}}-Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts are also proposed by using the Quasi-Newton optimization technique.     

Using neural networks to detect the bivariate process variance shifts pattern

Most of the research in statistical process control has been focused on monitoring the process mean. Typically, it is also important to detect variance changes as well. This paper presents a neural network-based approach for detecting bivariate process variance shifts. Some important implementation issues of neural networks are investigated, including analysis window size, number of training examples, sample size, training algorithm, etc. The performance of the neural network, in terms of the ARL and run length distribution, is compared with that of traditional multivariate control charts. Through rigorous evaluation and comparison, our research results show that the proposed neural network performs substantially better than the traditional generalized variance chart and might perform better than the adaptive sizes control charts in the case that the out-of-control covariance matrix is not known in advance.     

Mean shifts detection and classification in multivariate process: a neural-fuzzy approach

For monitoring multivariate quality control process, traditional multivariate control charts have been proposed to detect mean shifts. However, a persistent problem is that such charts are unable to provide any shift-related information when mean shifts occur in the process. In fact, the immediate classification of the magnitude of mean shifts can greatly narrow down the set of possible assignable causes, hence facilitating quick analysis and corrective action by the technician before many nonconforming units are manufactured. In this paper, we propose a neural-fuzzy model for detecting mean shifts and classifying their magnitude in multivariate process. This model is divided into training and classifying modules. In the training module, a neural network (NN) model is trained to detect various mean shifts for multivariate process. Then, in the classifying module, the outputs of NN are classified into various decision intervals by using a fuzzy classifier and an additional two-point-in-an-interval decision rule to determine shift status. An example is presented to illustrate the application of the proposed model. Simulation results show that it outperforms the multivariate T²control chart in terms of out-of-control average run length under fixed type I error. In addition, the correct classification percentages are also studied and the general guidelines are given for the proper use of the proposed model.     

Online monitoring and fault identification of mean shifts in bivariate processes using decision tree learning techniques

With modern data collection system and computers used for on-line process monitoring and fault identification in manufacturing processes, it is common to monitor more than one correlated process variables simultaneously. The main problems in most multivariate control charts (e.g., T ² charts, MCUSUM charts, MEWMA charts) are that they cannot give direct information on which variable or subset of variables caused the out-of-control signals. A Decision Tree (DT) learning based model for bivariate process mean shift monitoring and fault identification is proposed in this paper under the assumption of constant variance-covariance matrix. Two DT classifiers based on the C5.0 algorithm are built, one for process monitoring and the other for fault identification. Simulation results show that the proposed model can not only detect the mean shifts but also give information on the variable or subset of variables that cause the out-of-control signals and its/their deviate directions. Finally a bivariate process example is presented and compared with the results of an existing model.     

A new auxiliary information based cumulative sum median control chart for location monitoring

Control charts are commonly used tools in statistical process control for the detection of shifts in process parameters. Shewhart-type charts are efficient for large shift values, whereas cumulative sum (CUSUM) charts are effective in detecting medium and small shifts. Control chart use commonly assumes that data are free of outliers and parameters are known or correctly estimated based on an in-control process. In practice, these assumptions are not often true because some processes occasionally have outliers. Monitoring the location parameter is usually based on mean charts, which are seriously affected by violations of these assumptions. In this paper we propose several CUSUM median control charts based on auxiliary variables, and offer comparisons with their corresponding mean control charts. To monitor the location parameter, we examined the performance of mean and median control charts in the presence and absence of outliers. Both symmetric and non-symmetric processes were studied to examine the properties of the proposed control charts to monitor the location parameter using CUSUM control charts. We used different run length measures to study in-control and out-of-control performances of CUSUM charts. Results revealed that our proposed control charts perform much better than the traditional charts in the presence of outliers. A real application of our study was provided using data on concrete compressive strength as it relates to the quality of cement manufacturing.

     

Economic design of two-stage non-central chi-square charts for dependent variables

Non-central chi-square charts are more effective than the joint and R charts in detecting small mean shifts or variance changes of a performance variable. However, the cost may be high to monitor a primary quality characteristic, such as the weight of each bag in a cement filling process. It is more economical to monitor a surrogate variable, for example, the milliampere of the load cell. When the correlation of the performance variable of surrogate variable exists, this article proposes a two-stage charting design to monitor either the performance variable or its surrogate variable in an alternating fashion rather than monitoring the performance variable alone. The proposed method simplifies process monitoring when users only concern about whether a process is in control or not. The application of the proposed method and the advantages of the proposed chart over the existing methods are presented through an example. Numerical results show that the proposed chart is insensitive on the correlation of the performance variable and surrogate variable even when the historical information on the correlation coefficient is not very accurate.     

Optimal confidence interval for the largest normal mean under heteroscedasticity

A two-stage sampling procedure for obtaining an optimal confidence interval for the largest or smallest mean of k independent normal populations is proposed, where the population variances are unknown and possibly unequal. The optimal confidence interval is obtained by maximizing the coverage probability with a fixed width at a least favorable configuration of means. Then, the sample sizes can be determined by this procedure. It has been shown that the optimal interval is globally optimal over all possible choices of symmetric and asymmetric intervals. In situations where the two-stage sampling procedure cannot be completely carried through, a one-stage sampling procedure can be implemented, and their relationship is discussed. A numerical example to demonstrate the use of these sampling procedures is given.     

A neural network-based procedure for the monitoring of exponential mean

Control charts are widely used for both manufacturing and service industries. Cumulative sum (CUSUM) charts are known to be very sensitive in detecting small shifts in the mean. In this paper, we propose a neural network as an alternative approach to CUSUM charts when monitoring exponential mean. The performance of neural network was evaluated by estimating the average run lengths (ARLs) using simulation. The results obtained with simulated data suggest that control scheme based on neural network is significantly more sensitive to process shifts than CUSUM charts. This research also examines the feasibility of using CUSUM chart and neural network together in detecting process mean shifts. The results indicate that using the two methods in combination is more effective than using the methods separately.     

A general fuzzy-statistical clustering approach for estimating the time of change in variable sampling control charts

Despite their capability in monitoring the variability of the processes, control charts are not effective tools for identifying the real time of such changes. Identifying the real time of the change in a process is recognized as change-point estimation problem. Most of the change-point models in the literature are limited to fixed sampling control charts which are only a special case of more effective charts known as variable sampling charts. In this paper, we develop a general fuzzy-statistical clustering approach for estimating change-points in different types of control charts with either fixed or variable sampling strategy. For this purpose, we devise and evaluate a new similarity measure based on the definition of operation characteristics and power functions. We also develop and examine a new objective function and discuss its relation with maximum-likelihood estimator. Finally, we conduct extensive simulation studies to evaluate the performance of the proposed approach for different types of control charts with different sampling strategies.     

Design and optimization of EWMA control charts for in-control,indifference, and out-of-control regions

The design of quality control charts is normally carried out considering a process shift size that is considered important to be detected. The EWMA control chart is one of the best available options to use when good performance is needed to detect small process shifts. This paper presents a method for design of EWMA charts for control processes, in which the detection of small shifts is not necessary, and at the same time is effective in detecting important shifts. In such cases the EWMA control chart can also be designed successfully to deal with these requirements. A Markov chain approach is also applied to determine the ARL of the modified EWMA control chart. The implementation and interpretations are provided and numerical examples are used to illustrate the application procedure. We also investigate some basic properties of the proposed scheme. Genetic algorithms have been used to carry out this design.     

A combined synthetic and np scheme for detecting increases in fraction nonconforming

The applications of attribute control charts cover a wide variety of manufacturing processes in which quality characteristics cannot be measured on a continuous numerical scale or even a quantitative scale. The np control chart is an attribute chart used to monitor the fraction nonconforming p of a process. This chart is effective for detecting large process shifts in p. The attribute synthetic chart is also proposed to detect p shifts. It utilizes the information about the time interval or the Conforming Run Length (CRL) between two nonconforming samples. During the implementation of a synthetic chart, a sample is classified as nonconforming if the number d of nonconforming units falls beyond a warning limit. Unlike the np chart, the synthetic chart is more powerful to detect small and moderate p shifts. This article proposes a new scheme, the Syn-np chart, which comprises a synthetic chart and an np chart. Since the Syn-np chart has both the strength of the synthetic chart for quickly detecting small p shifts and the advantage of the np chart of being sensitive to large p shifts, it has a better and more uniform overall performance. Specifically, it is more effective than the np chart and synthetic chart by 73% and 31%, respectively, in terms of Weighted Average of Average Time to Signal (WAATS) over a wide range of p shifts under different conditions.     

A power function model for determining sample sizes for the operations of multivariate control charts

Using the hypothesis-testing approach, we develop a model for determining sample sizes for the operation of multivariate control charts. A simple solution procedure that can be processed on any personal or small computer is also developed. The effect of correlation between pairs of variables on the performance of the model is studied. The performances of multivariate and univariate control charts are compared under the model. Before the development of the model, a brief review of multivariate test of hypothesis and multivariate control charts was done. The model is recommended for any quality control engineer who may like to specify a desired level of protection against inferior quality.     

A hybrid fuzzy adaptive sampling - Run rules for Shewhart control charts

In crisp run control rules, usually it is stated that a process moves very sharply from in-control condition to out-of-control act. This causes an increase in both false-alarm rate and control chart sensitivity. Moreover, the classical run control rules are not implemented on an intelligent sampling strategy that changes control charts’ parameters to reduce error probability when the process appears to have a shift in parameter values. This paper presents a new hybrid method based on a combination of fuzzified sensitivity criteria and fuzzy adaptive sampling rules, which make the control charts more sensitive and proactive while keeping false alarms rate acceptably low. The procedure is based on a simple strategy that includes varying control chart parameters (sample size and sample interval) based on current fuzzified state of the process and makes inference about the state of process based on fuzzified run rules. Furthermore, in this paper, the performance of the proposed method is examined and compared with both conventional run rules and adaptive sampling schemes.     

A control chart based on likelihood ratio test for detecting patterned mean and variance shifts

Control charts based on generalized likelihood ratio test (GLRT) are attractive from both theoretical and practical points of view. Most of the existing works in the literature focusing on the detection of the process mean and variance are almost based on the assumption that the shifts remain constant over time. The case of the patterned mean and variance changes may not be well discussed. In this research, we propose a new control chart which integrates the exponentially weighted moving average (EWMA) procedure with the GLRT statistics to monitor the process with patterned mean and variance shifts. The attractive advantage of our control chart is its reference-free property. Due to the good properties of GLRT and EWMA procedures, our simulation results show that the proposed chart provides quite effective and robust detecting ability for various types of shifts. The implementation of our proposed control chart is illustrated by a real data example from chemical process control.     

Cumulative conformance count charts with variable sampling intervals for correlated samples

The cumulative conformance count (CCC) control chart is often employed to monitor the fraction nonconforming of high-yield processes. Traditional CCC chart is used when the items from a process are inspected one-at-a-time following the production order. In recent years, the CCC chart has been generalized to accommodate some industrial practices where items from a process are inspected sample by sample and not according to the production order. In order to increase the sensitivity of the generalized CCC (GCCC) chart to changes in fraction nonconforming, the variable sampling interval (VSI) scheme is used in this study. The output characteristic within each sample is assumed with correlation. The statistical properties of the GCCC chart with the VSI scheme are deduced using the Markov chain method. In evaluating the usefulness of the VSI feature, GCCC charts with VSI and fixed sampling interval (FSI) schemes are compared in terms of their statistical properties. The comparison results show that using the VSI scheme can improve the speed of GCCC chart in detecting changes in fraction nonconforming. Finally, according to the comparison results, a design procedure is applied to an industrial example to validate its practicability.     

Synthetic double sampling np control chart for attributes

The quality of a product, based on the number of non-conforming items can be controlled using the np chart. This paper proposes a synthetic double sampling (DS) np chart which comprises two sub-charts, i.e. the DS np and conforming run length (CRL) sub-charts. For the zero-state case, the synthetic DS np chart surpasses its standard counterpart, i.e. the synthetic np and the basic DS np chart, and other np type charts like the standard np, combined synthetic and np (Syn-np), variable sample size (VSS) np, exponentially weighted moving average (EWMA) np and cumulative sum (CUSUM) np charts, for detecting increases in the fraction of non-conforming items p, for most shift sizes. The synthetic DS np chart also performs reasonably well in the steady-state case in comparison with other charts mentioned above. Thus, among the competing charts, the synthetic DS np chart stands out as one of the best charts.     

Adaptive EWMA procedures for monitoring processes subject to linear drifts

The conventional Statistical Process Control (SPC) techniques have been focused mostly on the detection of step changes in process means. However, there are often settings for monitoring linear drifts in process means, e.g., the gradual change due to tool wear or similar causes. The adaptive exponentially weighted moving average (AEWMA) procedures proposed by Yashchin (1995) have received a great deal of attention mainly for estimating and monitoring step mean shifts. This paper analyzes the performance of AEWMA schemes in signaling linear drifts. A numerical procedure based on the integral equation approach is presented for computing the average run length (ARL) of AEWMA charts under linear drifts in the mean. The comparison results favor the AEWMA chart under linear drifts. Some guidelines for designing AEWMA charts for detecting linear drifts are presented.     

An exponentially weighted moving average scheme with variable sampling intervals for monitoring linear profiles

This paper proposes an exponentially weighted moving average scheme with variable sampling intervals for monitoring linear profiles. A computer program in Fortran is available to assist in the design of the control chart and the algorithm of the Fortran program is also given. Some useful guidelines are also provided to aid users in choosing parameters for a particular application. Simulation results on the detection performance of the proposed control chart, compared with some other competing methods show that it provides quite robust and satisfactory performance in various cases, including intercept shifts, slope shifts and standard deviation shifts. A real data example from an optical imaging system is employed to illustrate the implementation and the use of the proposed control scheme.     

An exponentially weighted moving average scheme with variable sampling intervals for monitoring linear profiles

This paper proposes an exponentially weighted moving average scheme with variable sampling intervals for monitoring linear profiles. A computer program in Fortran is available to assist in the design of the control chart and the algorithm of the Fortran program is also given. Some useful guidelines are also provided to aid users in choosing parameters for a particular application. Simulation results on the detection performance of the proposed control chart, compared with some other competing methods show that it provides quite robust and satisfactory performance in various cases, including intercept shifts, slope shifts and standard deviation shifts. A real data example from an optical imaging system is employed to illustrate the implementation and the use of the proposed control scheme.