Effect of Inspection Error on Out-of-Control Average Run Length of CUSUM Charts

The cumulative sum (CUSUM) chart is commonly used for detecting small or moderate shifts in the fraction of defective manufactured items. However, its construction relies on the error-free inspection assumption, which can seldom be met in practice. In this article, we discuss the construction of an upward CUSUM chart in the presence of inspection error, study the effects of inspection error on the out-of-control ARL of the CUSUM chart, and present a formula for determining the sampling size that compensates for the effect of inspection error on the out-of-control ARL.     

Monitoring imprecise fraction of nonconforming items using p control charts

The quality characteristics, which are known as attributes, cannot be conveniently and numerically represented. Generally, the attribute data can be regarded as the fuzzy data, which are ubiquitous in the manufacturing process and cannot be measured precisely and often be collected by visual inspection. In this paper, we construct a p control chart for monitoring the fraction of nonconforming items in the process in which fuzzy sample data are collected from the manufacturing process. The resolution identity – a well-known theorem in the fuzzy set theory – is invoked to construct the control limits of fuzzy-p control charts using fuzzy data. In order to determine whether the plotted imprecise fraction of nonconforming items is within the fuzzy lower and upper control limits, we also propose a ranking method for a set of fuzzy numbers. Using the fuzzy-p control charts and the proposed acceptability function to classify the manufacturing process allows the decision-maker to make linguistic decisions such as rather in control or rather out of control. A practical example is provided to describe the applicability of the fuzzy set theory to a conventional p control chart.     

A new economic scheme for CCC charts with run rules based on average number of inspected items

New statistical techniques and procedures have been developed to control high-yield processes along with looking for process improvement opportunities and minimizing production cost. Cumulative count of conforming control chart is generally a technique for high-quality processes, when nonconforming items are rarely produced. The objective of this study is to design control chart based on cumulative count of conforming items and run rules that develops an economic model based on the average number of inspected items to design m-of-m CCC chart in order to facilitate minimum average cost per item produced. The optimal design parameters for different values of nonconforming fraction and different cost parameters in each scenario are determined. Finally, to analyze the behavior of optimal economic solutions, sensitivity analysis of the model parameters is performed.     

A synthetic control chart for monitoring the small shifts in a process mean based on an attribute inspection

Abstract

In this paper, a synthetic control chart is proposed by integrating the salient features of the np_x chart and the CRL chart. The synthetic chart achieves higher detection effectiveness on both small and large mean shifts while retaining the operational simplicity of the attribute charts owing to only using attribute inspection. Both statistical and economic design of the synthetic chart are considered and numerical tests have indicated that the synthetic chart has a higher power for detecting mean shifts than the np_x chart, MON chart and CUSUM chart. In addition, sensitivity analyses are also performed under both the statistical and economic design model.     

Bivariate copulas on the Hotelling's T2 control chart

In this paper, we propose five types of copulas on the Hotelling's T² control chart when observations are from exponential distribution and use the Monte Carlo simulation to compare the performance of the control chart, which is based on the Average Run Length (ARL) for each copula. Five types of copulas function for specifying dependence between random variables are used and measured by Kendall's tau. The results show that the copula approach can be fitted the observation and we can use copula as an option for application on Hotelling's T² control chart.     

Multi-attribute unit and group runs control chart to identify the process deterioration

Gadre and Rattihalli [Gadre, M.P. and Rattihalli, R.N., 2005a, A unit and group runs based chart to identify increases in fraction nonconforming. Journal of Quality Technology, 37, 199–209.] proposed a control chart called the unit and group runs (UGR) control chart to identify increases in fraction non-conforming. In this article, the concept of UGR chart is extended to the multi-attribute case to detect the process deterioration. It is illustrated that in multi-attribute cases also, the UGR chart gives a remarkable reduction in out-of-control average time to signal when compared with the multi-attribute np chart, the multi-attribute synthetic chart and the multi-attribute group runs chart recently developed by Gadre and Rattihalli [Gadre, M.P. and Rattihalli, R.N., 2005b, Some group inspection based multi-attribute control charts. Economic Quality Control, 20, 191–204.]. The steady state performance of the multi-attribute UGR chart is also excellent. The procedure of identifying the attributes causing signal is also described and illustrated.     

The RL2 chart versus the np chart for detecting upward shifts in fraction defective

The use of the np chart for monitoring fraction-defective is well-established, but there are a number of relatively simple alternatives based on run-lengths of conforming items. Here, the RL₂ chart, based on the moving sum of two successive conforming run-lengths, is investigated in order to provide SPC practitioners with clear-cut guidance on the comparative performance of these competing charts. Both sampling inspection and 100% inspection are considered here, and it is shown that the RL₂ chart can often be considerably more efficient than the np chart, but the comparative performance depends on the false-alarm rate used for the comparison. Graphs to aid parameter-choice for the RL₂ chart are also provided.     

On-Line Process Control using Attributes with Misclassification Errors: An Economical Design for Short-Run Production

On-line process control consists of inspecting a single item for every m (integer and m ≥ 2) produced items. Based on the results of the inspection, it is decided whether the process is in-control (the fraction of conforming items is p ₁; State I) or out-of-control (the fraction of conforming items is p ₂ < p ₁; State II). If the inspected item is non conforming, it is determined that the process is out-of-control, and the production process is stopped for an adjustment; otherwise, production continues. As most designs of on-line process control assume a long-run production, this study can be viewed as an extension because it is concerned with short-run production and the decision regarding the process is subject to misclassification errors. The probabilistic model of the control system employs properties of an ergodic Markov chain to obtain the expression of the average cost of the system per unit produced, which can be minimised as a function of the sampling interval, m. The procedure is illustrated by a numerical example.     

Estimation of multiple correlation coefficient when observations are missing on one of the variables

This paper proposes a control chart with variable sampling intervals (VSI) to detect increases in the expected value of the number of defects in a random sample of constant size n the upper one-sided c-VSI chart

The performance of this chart is evaluated by means of the average time to signal (ATS).The comparisons made between the standard FSI (fixed sampling intervals) and the VSI upper one-sided c - charts indicate that using variable sampling intervals can substantially reduce the average time to signal. Using stochastic ordering we prove that this reduction always occurs.

Special attention is given to the choice of the proposed control chart parameters and to the chart graphical display.     

Online control by attributes in the presence of classification errors with variable inspection interval

ABSTRACT

The procedure for online control by attribute consists of inspecting a single item at every m items produced (m ≥ 2). On each inspection, it is determined whether the fraction of the produced conforming items decreased. If the inspected item is classified as non conforming, the productive process is adjusted so that the conforming fraction returns to its original status. A generalization observed in the literature is to consider inspection errors and vary the inspection interval. This study presents an extension of this model by considering that the inspected item can be rated independently r (r ≥ 1) times. The process is adjusted every time the number of conforming classifications is less than a, 1 ≤ a ≤ r. This method uses the properties of an ergodic Markov chain to obtain the expression for the average cost of this control system. The genetic algorithm methodology is used to search for the optimal parameters that minimize the expected cost. The procedure is illustrated by a numerical example.     

The effect of autocorrelation on the retrospective X-chart

Quality control chart interpretation is usually based on the assumption that successive observations are independent over time. In this article we show the effect of autocorrelation on the retrospective Shewhart chart for individuals, often referred to as the X-chart, with the control limits based on moving ranges. It is shown that the presence of positive first lag autocorrelation results in an increased number of false alarms from the control chart. Negative first lag autocorrelation can result in unnecessarily wide control limits such that significant shifts in the process mean may go undetected. We use first-order autoregressive and first-order moving average models in our simulation of small samples of autocorrelated data.     

Conditional design of the EWMA median chart with estimated parameters

The exponentially weighted moving average (EWMA) chart is often designed assuming the process parameters are known. In practice, the parameters are rarely known and need to be estimated from Phase I samples. Different Phase I samples are used when practitioners construct their own control chart's limits, which leads to the “Phase I between-practitioners” variability in the in-control average run length (ARL) of control charts. The standard deviation of the ARL (SDARL) is a good alternative to quantify this variability in control charts. Based on the SDARL metric, the performance of the EWMA median chart with estimated parameters is investigated in this paper. Some recommendations are given based on the SDARL metric. The results show that the EWMA median chart requires a much larger amount of Phase I data in order to reduce the variation in the in-control ARL up to a reasonable level. Due to the limitation of the amount of the Phase I data, the suggested EWMA median chart is designed with the bootstrap method which provides a good balance between the in-control and out-of-control ARL values.     

Sequential testing with interval censored data ∗

Let X ₁,…,X _n be the lifetimes of n items put on testing at the same time. It is not possible to observe the actual lifetimes. However, it is possible to inspect the items at a finite number of time intervals. At each time of inspection, the number of failures can be recorded. Only these numbers of failures at times of inspections will be available to make decision on the distribution of the lifetimes. Decision can be made at the time of inspection. A “sequential” statistical test is developed to test the mean levels of the lifetimes when the probability distribution is assumed to be exponential. Some numerical results will be presented. The power and the expected time for the decision are compared with those for the idealized situation when each and every actual lifetime is recorded. They are also compared with those for the case when one and only one inspection is allowed to make the decision.     

Optimal Statistical Design of a Multivariate EWMA Chart Based on ARL and MRL

Statistical design is applied to a multivariate exponentially weighted moving average (MEWMA) control chart. The chart parameters are control limit H and smoothing constant r. The choices of the parameters depend on the number of variables p and the size of the process mean shift δ. The MEWMA statistic is modeled as a Markov chain and the Markov chain approach is used to determine the properties of the chart. Although average run length has become a traditional measure of the performance of control schemes, some authors have suggested other measures, such as median and other percentiles of the run length distribution to explain run length properties of a control scheme. This will allow a thorough study of the performance of the control scheme. Consequently, conclusions based on these measures would provide a better and comprehensive understanding of a scheme. In this article, we present the performance of the MEWMA control chart as measured by the average run length and median run length. Graphs are given so that the chart parameters of an optimal MEWMA chart can be determined easily.     

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.     

A test for suitability of the preliminary samples for constructing control limits of X¯ chart

Unless the preliminary m subgroups of small samples are drawn from a stable process, the estimated control limits of X¯ chart in phase I can be erroneous, due to which the performance of the chart in phase II can be significantly affected. In this work, a quantitative approach based on extraction of the shape features of control chart patterns in the X¯ chart is proposed for evaluating the stability of the process mean, while the preliminary samples were drawn and thus, the subjectivity associated with the visual analysis of the patterns is eliminated. The effectiveness of the test procedure is evaluated using simulated data. The results show that the proposed approach can be very effective for m≥48. The power of the test can be improved by identifying a new feature that can more efficiently discriminate the cyclic pattern of smaller periodicity from the natural pattern and by redefining the test statistic.     

DDMA-charts: Nonparametric multivariate moving average control charts based on data depth

Summary: This paper studies the DDMA–chart, a data depth based moving–average control chart for monitoring multivariate data. This chart is nonparametric and it can detect simultaneously location and scale changes in the process. It improves upon the existing r– and Q–chart in the efficiency of detecting location changes. Both theoretical justifications and simulation studies are provided. Comparisons with some existing multivariate control charts via simulation results are also provided. Some applications of the DDMA–chart to the analysis of airline performance data (collected by the FAA) are demonstrated. The results indicate that the DDMA–chart is an effective nonparametric multivariate control chart.*Research supported in part by grants from the National Science Foundation, the National Security Agency, and the Federal Aviation Administration. The discussion on aviation safety in this paper reects the views of the authors, who are solely responsible for the accuracy of the analysis results presented herein, and does not necessarily reect the official view or policy of the FAA. The dataset used in this paper has been partially masked in order to protect confidentiality.     

Copula-Based Bivariate ZIP Control Chart for Monitoring Rare Events

One difficulty with developing multivariate attribute control charts is the lack of the related joint distribution. So, if it would be possible to generate the joint distribution of two (or more) attribute characteristics, then a bivaraite (or multivariate) attribute control chart can be developed based on Types I and II errors. Copula function is a solution to the matter. In this article, applying the copula function approach, we achieve the joint distribution of two correlated zero inflated Poisson (ZIP) distributions. Then, using this joint distribution, we develop a bivaraite control chart which can be used for monitoring correlated rare events. This copula-based bivariate ZIP control chart is compared with the simultaneous use of two separate univariate ZIP control charts. Based on the average run length (ARL) measure, it is shown that the proposed control chart is much better than the simultaneous use of two separate univariate charts. In addition, a real case study related to the environmental air in a sterilization process is investigated to show the applicability of the developed control 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.     

Monitoring of zero-inflated binomial processes with a DEWMA control chart

Control charts are widely used for monitoring quality characteristics of high-yield processes. In such processes where a large number of zero observations exists in count data, the zero-inflated binomial (ZIB) models are more appropriate than the ordinary binomial models. In ZIB models, random shocks occur with probability θ, and upon the occurrence of random shocks, the number of non-conforming items in a sample of size n follows the binomial distribution with proportion p. In the present article, we study in more detail the exponentially weighted moving average control chart based on ZIB distribution (ZIB-EWMA) and we also propose a new control chart based on the double exponentially weighted moving average statistic for monitoring ZIB data (ZIB-DEWMA). The two control charts are studied in detecting upward shifts in θ or p individually, as well as in both parameters simultaneously. Through a simulation study, we compare the performance of the proposed chart with the ZIB-Shewhart, ZIB-EWMA and ZIB-CUSUM charts. Finally, an illustrative example is also presented to display the practical application of the ZIB charts.