Control charts for fraction nonconforming in a bivariate binomial process

Many multivariate quality control techniques are used for multivariate variable processes, but few work for multivariate attribute processes. To monitor multivariate attributes, controlling the false alarms (type I errors) and considering the correlation between attributes are two important issues. By taking into account these two issues, a new control chart is presented to monitor a bivariate binomial process. An example is illustrated for the proposed method. To evaluate the performance of the proposed method, a simulation study is conducted to compare the results with those using both the multivariate np chart and skewness reduction approaches. The results show that the correlation is taken into account in the designed chart and the overall false alarm is controlled at the nominal value. Moreover, the process shift can be quickly detected and the variable that is responsible for a signal can be determined.     

A Phase I nonparametric Shewhart-type control chart based on the median

A nonparametric Shewhart-type control chart is proposed for monitoring the location of a continuous variable in a Phase I process control setting. The chart is based on the pooled median of the available Phase I samples and the charting statistics are the counts (number of observations) in each sample that are less than the pooled median. An exact expression for the false alarm probability (FAP) is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits for a specified nominal FAP value (of 0.01, 0.05 and 0.10, respectively) and for some values of the sample size (n) and the number of Phase I samples (m). Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. A simulation study shows that the proposed chart performs as well as, and in some cases better than, an existing Shewhart-type chart based on the normal distribution. Numerical examples are given to demonstrate the implementation of the new chart.     

The performance of multivariate CUSUM control charts with estimated parameters

In this paper, we study the effect of estimating the vector of means and the variance–covariance matrix on the performance of two of the most widely used multivariate cumulative sum (CUSUM) control charts, the MCUSUM chart proposed by Crosier [Multivariate generalizations of cumulative sum quality-control schemes, Technometrics 30 (1988), pp. 291–303] and the MC1 chart proposed by Pignatiello and Runger [Comparisons of multivariate CUSUM charts, J. Qual. Technol. 22 (1990), pp. 173–186]. Using simulation, we investigate and compare the in-control and out-of-control performances of the competing charts in terms of the average run length measure. The in-control and out-of-control performances of the competing charts deteriorate significantly if the estimated parameters are used with control limits intended for known parameters, especially when only a few Phase I samples are used to estimate the parameters. We recommend the use of the MC1 chart over that of the MCUSUM chart if the parameters are estimated from a small number of Phase I samples.     

A Multivariate Synthetic Control Chart for Monitoring the Process Mean Vector of Skewed Populations Using Weighted Standard Deviations

This article proposes a multivariate synthetic control chart for skewed populations based on the weighted standard deviation method. The proposed chart incorporates the weighted standard deviation method into the standard multivariate synthetic control chart. The standard multivariate synthetic chart consists of the Hotelling's T ² chart and the conforming run length chart. The weighted standard deviation method adjusts the variance–covariance matrix of the quality characteristics and approximates the probability density function using several multivariate normal distributions. The proposed chart reduces to the standard multivariate synthetic chart when the underlying distribution is symmetric. In general, the simulation results show that the proposed chart performs better than the existing multivariate charts for skewed populations and the standard T ² chart, in terms of false alarm rates as well as moderate and large mean shift detection rates based on the various degrees of skewnesses.     

Conditional design of the CUSUM median chart for the process position when process parameters are unknown

In practice, different practitioners will use different Phase I samples to estimate the process parameters, which will lead to different Phase II control chart's performance. Researches refer to this variability as between-practitioners-variability of control charts. Since between-practitioners-variability is important in the design of the CUSUM median chart with estimated process parameters, the standard deviation of average run length (SDARL) will be used to study its properties. It is shown that the CUSUM median chart requires a larger amount of Phase I samples to sufficiently reduce the variation in the in-control ARL of the CUSUM median chart. Considering the limitation of the amount of the Phase I samples, a bootstrap approach is also used here to adjust the control limits of the CUSUM median chart. Comparisons are made for the CUSUM and Shewhart median charts with estimated parameters when using the adjusted- and unadjusted control limits and some conclusions are made.     

Properties and performance of the c-chart for attributes data

The effects of parameter estimation are examined for the well-known c-chart for attributes data. The exact run length distribution is obtained for Phase II applications, when the true average number of non-conformities, c, is unknown, by conditioning on the observed number of non-conformities in a set of reference data (from Phase I). Expressions for various chart performance characteristics, such as the average run length (ARL), the standard deviation of the run length (SDRL) and the median run length (MDRL) are also obtained. Examples show that the actual performance of the chart, both in terms of the false alarm rate (FAR) and the in-control ARL, can be substantially different from what might be expected when c is known, in that an exceedingly large number of false alarms are observed, unless the number of inspection units (the size of the reference dataset) used to estimate c is very large, much larger than is commonly used or recommended in practice. In addition, the actual FAR and the in-control ARL values can be very different from the nominally expected values such as 0.0027 (or ARL₀=370), particularly when c is small, even with large amounts of reference data. A summary and conclusions are offered.     

Performance of the zone control chart

Average run lengths of the zone control chart are presented, The performance of this chart is compared with that of several Shewhart charts with and without runs rules, It is shown that the standard zone control chart has performance similar to some even simpler charts and a much higher false alarm rate than the Shewhart chart with all of the common runs rules. It is also shown that a slightly modified zone control chart outperforms the Shewhart chart with the common runs rules.     

An NP Control Chart Using Double Inspections

The np control chart is used widely in Statistical Process Control (SPC) for attributes. It is difficult to design an np chart that simultaneously satisfies a requirement on false alarm rate and has high detection effectiveness. This is mainly because one is often unable to make the in-control Average Run Length ARL₀ of an np chart close to a specified or desired value. This article proposes a new np control chart which is able to overcome the problems suffered by the conventional np chart. It is called the Double Inspection (DI) np chart, because it uses a double inspection scheme to decide the process status (in control or out of control). The first inspection decides the process status according to the number of non-conforming units found in a sample; and the second inspection makes a decision based on the location of a particular non-conforming unit in the sample. The double inspection scheme makes the in-control ARL₀ very close to a specified value and the out-of-control Average Run Length ARL₁ quite small. As a result, the requirement on a false alarm rate is satisfied and the detection effectiveness also achieves a high level. Moreover, the DI np chart retains the operational simplicity of the np chart to a large degree and achieves the performance improvement without requiring extra inspection (testing whether a unit is conforming or not).     

An investigation of the effects of inspection errors on the run-length control charts

For continuous inspection schemes in an automated manufacturing environment, a useful alternative to the traditional p or np chart is the Run-Length control chart, which is based on plotting the run lengths (the number of conforming items) between successive nonconforming items. However, its establishment relies on the error-free inspection assumption, which can seldom be met in practice. In this paper, the effects of inspection errors on the Run-Length chart are investigated based on that these errors are assumed known. The actual false alarm probability and the average number inspected (ANI) in the presence of inspection errors are studied. This paper also presents the adjusted control limits for the Run-Length chart, which can provide much closer ANI curves to the ones obtained under error-free inspection.     

Synthetic exponential control charts with unknown parameter

The existing synthetic exponential control charts are based on the assumption of known in-control parameter. However, the in-control parameter has to be estimated from a Phase I dataset. In this article, we use the exact probability distribution, especially the percentiles, mean, and standard deviation of the conditional average run length (ARL) to evaluate the effect of parameter estimation on the performance of the Phase II synthetic exponential charts. This approach accounts for the variability in the conditional ARL values of the synthetic chart obtained by different practitioners. Since parameter estimation results in more false alarms than expected, we develop an exact method to design the adjusted synthetic charts with desired conditional in-control performance. Results of known and unknown in-control parameter cases show that the control limit of the conforming run length sub-chart of the synthetic chart should be as small as possible.     

New control charts for simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles

Simultaneous monitoring of the mean vector and covariance matrix in multivariate processes allows practitioners to avoid the inflated false alarm rate that results from using two independent control charts. In this paper, we extend exponentially weighted moving average semicircle and generally weighted moving average semicircle control charts to monitor the mean vector and covariance matrix of multivariate multiple linear regression profiles in Phase II simultaneously. These new control charts are compared with the existing control charts in the literature in terms of the average run length criterion. Finally, a case is considered to show the application of the proposed charts.     

Attribute Control Chart for Multivariate Poisson Distribution

In this study, a control chart is constructed to monitor multivariate Poisson count data, called the MP chart. The control limits of the MP chart are developed by an exact probability method based on the sum of defects or non conformities for each quality characteristic. Numerical examples are used to illustrate the MP chart. The MP chart is evaluated by the average run length (ARL) in simulation. The result indicates that the MP chart is more appropriate than the Shewhart-type control chart when the correlation between variables exists.     

Accounting for phase I sampling variability in the performance of the MEWMA control chart with estimated parameters

In this article, we assess the performance of the multivariate exponentially weighted moving average (MEWMA) control chart with estimated parameters while considering the practitioner-to-practitioner variability. We evaluate the chart performance in terms of the in-control average run length (ARL) distributional properties; mainly the average (AARL), the standard deviation (SDARL), and some percentiles. We show through simulations that using estimates in place of the in-control parameters may result in an in-control ARL distribution that almost completely lies below the desired value. We also show that even with the use of larger amounts of historical data, there is still a problem with the excessive false alarm rates. We recommend the use of a recently proposed bootstrap-based design technique for adjusting the control limits. The technique is quite effective in controlling the percentage of short in-control ARLs resulting from the estimation error.     

Control chart for monitoring multivariate COM-Poisson attributes

Statistical process control of multi-attribute count data has received much attention with modern data-acquisition equipment and online computers. The multivariate Poisson distribution is often used to monitor multivariate attributes count data. However, little work has been done so far on under- or over-dispersed multivariate count data, which is common in many industrial processes, with positive or negative correlation. In this study, a Shewhart-type multivariate control chart is constructed to monitor such kind of data, namely the multivariate COM-Poisson (MCP) chart, based on the MCP distribution. The performance of the MCP chart is evaluated by the average run length in simulation. The proposed chart generalizes some existing multivariate attribute charts as its special cases. A real-life bivariate process and a simulated trivariate Poisson process are used to illustrate the application of the MCP chart.     

New V control chart for the Maxwell distribution

Control charts have been popularly used as a user-friendly yet technically sophisticated tool to monitor whether a process is in statistical control or not. These charts are basically constructed under the normality assumption. But in many practical situations in real life this normality assumption may be violated. One such non-normal situation is to monitor the process variability from a skewed parent distribution where we propose the use of a Maxwell control chart. We introduce a pivotal quantity for the scale parameter of the Maxwell distribution which follows a gamma distribution. Probability limits and L-sigma limits are studied along with performance measure based on average run length and power curve. To avoid the complexity of future calculations for practitioners, factors for constructing control chart for monitoring the Maxwell parameter are given for different sample sizes and for different false alarm rate. We also provide simulated data to illustrate the Maxwell control chart. Finally, a real life example has been given to show the importance of such a control chart.     

Reduction of control-chart signal variablity for high-quality processes

The design of a control chart is often based on the statistical measure of average run length (ARL). A longer in-control ARL is ensured by the design, but the variance run length distribution may also be large for such a design. In practical terms, the variability in false alarms and true signals may be large. If the sample size for plotting a point is not constant, then the focus is on the average number inspected as against the ARL. This article considers two well-known attribute control chart procedures for monitoring high quality based on the number inspected, and shows how the variability in false alarms and correct signals can be reduced.     

Multivariate Synthetic |S| Control Chart with Variable Sampling Interval

A variable sampling interval (VSI) feature is introduced to the multivariate synthetic generalized sample variance |S| control chart. This multivariate synthetic control chart is a combination of the |S| sub-chart and the conforming run length sub-chart. The VSI feature enhances the performance of the multivariate synthetic control chart. The comparative results show that the VSI multivariate synthetic control chart performs better than other types of multivariate control charts for detecting shifts in the covariance matrix of a multivariate normally distributed process. An example is given to illustrate the operation of the VSI multivariate synthetic chart.     

Monitoring Nonlinear Profiles with Random Effects by Nonparametric Regression

The monitoring of process/product profiles is presently a growing and promising area of research in statistical process control. This study is aimed at developing monitoring schemes for nonlinear profiles with random effects. We utilize the technique of principal components analysis to analyze the covariance structure of the profiles and propose monitoring schemes based on principal component (PC) scores. The number of the PC scores used in constructing control charts is crucial to the detecting power. In the Phase I analysis of historical data, due to the dependency of the PC-scores, we adopt the usual Hotelling T ² chart to check the stability. For Phase II monitoring, we study individual PC-score control charts, a combined chart scheme that combines all the PC-score charts, and a T ² chart. Although an individual PC-score chart may be perfect for monitoring a particular mode of variation, a chart that can detect general shifts, such as the T ² chart and the combined chart scheme, is more feasible in practice. The performances of the schemes under study are evaluated in terms of the average run length.     

On the Run Length of a State-Space Control Chart for Multivariate Autocorrelated Data

The literature on statistical process control (SPC) describes the negative effects of autocorrelation in terms of the increase in false alarms. This has been treated by the individual modeling of each series or the application of VAR models. In the former case, the analysis of the cross correlation structure between the variables is altered. In the latter, if the cross correlation is not strong, the filtering process may modify the weakest relations. In order to improve these aspects, state-space models have been introduced in multivariate statistical process control (MSPC). This article presents a proposal for building a control chart for innovations, estimating its average run length to highlight its advantages over the VAR approach mentioned above.     

Run rules based phase II c and np charts when process parameters are unknown

Abstract

The performance of attributes control charts is usually evaluated under the assumption of known process parameters (i.e., the nominal proportion of non conforming units or the nominal average number of nonconformities). However, in practice, these process parameters are rarely known and have to be estimated from an in-control Phase I data set. The major contributions of this paper are (a) the derivation of the run length properties of the Run Rules Phase II c and np charts with estimated parameters, particularly focusing on the ARL, SDRL, and 0.05, 0.5, and 0.95 quantiles of the run length distribution; (b) the investigation of the number m of Phase I samples that is needed by these charts in order to obtain similar in-control ARLs to the known parameters case; and (c) the proposition of new specific chart parameters that allow these charts to have approximately the same in-control ARLs as the ones obtained in the known parameters case.