On the Use of Copula for Quality Control Based on an AR(1) Model

Manufacturing for a multitude of continuous processing applications in the era of automation and ‘Industry 4.0′ is focused on rapid throughput while producing products of acceptable quality that meet customer specifications. Monitoring the stability or statistical control of key process parameters using data acquired from online sensors is fundamental to successful automation in manufacturing applications. This study addresses the significant problem of positive autocorrelation in data collected from online sensors, which may impair assessment of statistical control. Sensor data collected at short time intervals typically have significant autocorrelation, and traditional statistical process control (SPC) techniques cannot be deployed. There is a plethora of literature on techniques for SPC in the presence of positive autocorrelation. This paper contributes to this area of study by investigating the performance of ‘Copula’ based control charts by assessing the average run length (ARL) when the subsequent observations are correlated and follow the AR(1) model. The conditional distribution of yt given yt−1 is used in deriving the control chart limits for three different categories of Copulas: Gaussian, Clayton, and Farlie-Gumbel-Morgenstern Copulas. Preliminary results suggest that the overall performance of the Clayton Copula and Farlie-Gumbel-Morgenstern Copula is better compared to other Archimedean Copulas. The Clayton Copula is the more robust with respect to changes in the process standard deviation as the correlation coefficient increases.

Change point detection in process control with robust individuals control chart

It is crucial to realize when a process has changed and to what extent it has changed, then it would certainly ease the task. On occasion that practitioners could determine the time point of the change, they would have a smaller search window to pursue for the special cause. As a result, the special cause can be discovered quicker and the necessary actions to improve quality can be triggered sooner. In this paper, we had demonstrated the use of so-called exploratory data analysis robust modified individuals control chart incorporating the M-scale estimator and had made some comparisons to the existing charts. The proposed modified robust individuals control chart which incorporates the M-scale estimator in order to compute the process standard deviation offers substantial improvements over the existing median absolute deviation framework. With respect to the application in real data set, the proposed approach appears to perform better than the typical robust control chart, and outperforms other conventional charts particularly in the presence of contamination. Thus, it is for these reasons that the proposed modified robust individuals control chart is preferred especially when there is a possible existence of outliers in data collection process.

A Combined Upper-Sided Synthetic S2 Chart for Monitoring the Process Variance

In order to monitor the process variance, this paper proposes a combined upper-sided synthetic S2 chart for monitoring the process standard deviation of a normally distributed process. This combined upper-sided synthetic S2 chart comprises a synthetic chart and an upper-sided S2 chart. The design and performance of the proposed chart are presented, and the steady-state average run length comparisons show that the combined upper-sided synthetic S2 chart outperforms the standard synthetic S2 chart as well as several run rules S2 charts, especially for larger shifts in the process variance.

An Enhanced Auxiliary Information-Based EWMA-t Chart for Monitoring the Process Mean

The exponentially weighted moving average t chart using auxiliary information (AIB-EWMA-t chart) is an effective approach for monitoring small process mean shifts when the process standard deviation is unstable or poorly estimated. To further enhance the sensitivity of the AIB-EWMA-t chart, in this study, we propose an AIB generally weighted moving average (GWMA) t chart (AIB-GWMA-t chart) to monitor the process mean. The existing EWMA-t, GWMA-t, and AIB-EWMA-t charts are special cases of the AIB-GWMA-t chart. Numerical simulation studies indicate that the AIB-GWMA-t chart performs uniformly and substantially better than the EWMA-t and GWMA-t charts in terms of average run length. Moreover, the AIB-GWMA-t chart with large design and adjustment parameters also outperforms the AIB-EWMA-t chart when the correlation coefficients are within a certain range. An illustrative example is provided to highlight the efficiency of the proposed AIB-GWMA-t chart in detecting small process mean shifts.

The Capability Index when Some Assumptions are not Satisfied: Analysis and Empirical Comparisons

The process capability index (PCI) evaluates the ability of a process to produce items with certain quality requirements. The PCI depends on the process standard deviation, which is usually unknown and estimated by using the sample standard deviation. The construction of confidence intervals for the PCI is also an important topic. The usual estimator of the PCI and its corresponding confidence interval are based on various assumptions, such as normality, the fact that the process is under control, or samples selected from infinite populations. The main aim of this paper is to investigate the empirical properties of estimators of the PCI, and analyze numerically the effect on confidence intervals when such assumptions are not satisfied, since these situations may arise in practice.

Automated Iterative Tolerance Value Allocation and Analysis

Dimensional variation is inherent in manufacturing and it is impossible to attain exact nominal dimensions. Ideally, designer should accommodate this variation likelihood in design stage and define an allowable variation. This allowable variation is represented as tolerances and is either considered a bounded zone or a scaled allowable process standard deviation (e.g. 6 times standard deviation). The allowable tolerances are usually constrained by assemblability, functionality and manufacturing economics. Meeting these constraints simultaneously becomes a paramount task and designers usually never consider production economics when specifying tolerances. As a consequence rarely do the tolerances specified by product designer match that of the process designer. Automated tolerance value allocation can empower the product designer to include all the constraints at the design phase and reduce the overall time line between development to production. Automated tolerance allocation method described in this paper is intended for use by the designer and encompasses only 1st order tolerancing. The Critical stack detection/loop detection tool extracts the assembly level stacks that dictate assemblability. The allocation tool utilizes these stacks to distribute tolerance budget by a rule of thumb. These stacks are subjected to variation analysis to compute acceptance rates and used as feedback for iterative reallocation using a hill climbing optimization algorithm till statistical fit requirements are satisfied or allocation gets exhausted. The algorithm is tested on some case studies and presented in the paper.

THE EXACT RUN LENGTH DISTRIBUTION AND DESIGN OF THE S2 CHART WHEN THE IN-CONTROL VARIANCE IS ESTIMATED

When monitoring the process variability, it is a common practice that a Phase I data set is used to estimate the unknown in-control process standard deviation σ0 or variance [Formula: see text] to set up the control limits, then monitoring proceeds. Once the process is considered to be in-control, the estimated control limits are assumed as fixed. This practice ignores the effect of estimating the unknown in-control process variance [Formula: see text]. In this paper, we derive the exact run length distribution of the S2 control chart when the in-control process variance [Formula: see text] is estimated and find that m = 200 or more Phase I samples are needed to neglect the effect of using estimated control limits. New control limits when m is small are also derived.

Monte Carlo Simulations Assessing the Risk of Salmonellosis from Consumption of Almonds

Recent outbreaks of salmonellosis associated with raw almonds have raised awareness of this food as a vector for foodborne illness. We performed a quantitative assessment of the risk of contracting salmonellosis from consumption of raw almonds, accounting for factors that become important after almonds reach the processor. We estimated the risk associated with the consumption of raw almonds and the risk reduction associated with almonds treated with a theoretical 5-log reduction process or treated with propylene oxide using a standard commercial process. Probability distributions were chosen to describe the chance of almond contamination and the effects of storage time, storage temperature, and processing from currently available data. A β-Poisson model for the dose-response relationship for Salmonella was obtained from published literature. The simulation estimated a 78% chance of one or more cases of salmonellosis per year from consumption of raw almonds. The application of a commercial propylene oxide treatment reduced this risk to 0.01%. Hypothetical 5-log reduction treatments with different standard deviations (±1, ±0.5, ±0.1, and ±0) reduced the predicted yearly risk of salmonellosis to 0.69, 0.35, 0.30, and 0.21%, respectively. These results suggest that the risk of one or more U.S. cases of salmonellosis per year from consumption of raw almonds can be reduced from 78% to less than 1% by using a process achieving a 5-log reduction in Salmonella with a process standard deviation as large as 1 log unit or by using a commercial propylene oxide treatment.