Parameter estimation and design considerations in prospective applications of the � chart

The effects of parameter estimation on the in-control performance of the Shewhart X¯ chart are studied in prospective (phase 2 or stage 2) applications via a thorough examination of the attained false alarm rate (AFAR), the conditional false alarm rate (CFAR), the conditional and the unconditional run-length distributions, some run-length characteristics such as the ARL, the conditional ARL (CARL), some selected percentiles including the median, and cumulative run-length probabilities. The examination involves both numerical evaluations and graphical displays. The effects of parameter estimation need to be accounted for in designing the chart. To this end, as an application of the exact formulations, chart constants are provided for a specified in-control average run-length of 370 and 500 for a number of subgroups and subgroup sizes. These will be useful in the implementation of the X¯ chart in practice.     

A binomial CUSUM chart for detecting large shifts in fraction nonconforming

This article studies a unique feature of the binomial CUSUM chart in which the difference (d _t ?d ₀) is replaced by (d _t ?d ₀)² in the formulation of the cumulative sum C _t (where d _t and d ₀ are the actual and in-control numbers of nonconforming units, respectively, in a sample). Performance studies are reported and the results reveal that this new feature is able to increase the detection effectiveness when fraction nonconforming p becomes three to four times as large as the in-control value p ₀. The design of the new binomial CUSUM chart is presented along with the calculation of the in-control and out-of-control Average Run Lengths (ARL₀ and ARL₁).     

A study of economic design of control charts for cumulative count of conforming items

In this paper the economic design of Cumulative Count of Conforming (CCC) control charts to maintain the current control of fraction nonconforming of a process is studied. CCC chart is an attribute chart for monitoring high quality processes by plotting the cumulative count of conforming items between two nonconforming ones on a suitable chart. A process model is proposed to obtain an appropriate loss function. An alogorithm to search for the optimal setting of the sampling and control parameters is derived. Numerical illustrations of the method and some properties of the optimal economic design are provided.     

On-line monitoring data quality of high-dimensional data streams

ABSTRACT

In recent years, effective monitoring of data quality has increasingly attracted attention of researchers in the area of statistical process control. Among the relevant research on this topic, none used multivariate methods to control the multidimensional data quality process, but instead relied on multiple univariate control charts. Based on a novel one-sided multivariate exponentially weighted moving average (MEWMA) chart, we propose a conditional false discovery rate-adjusted scheme to on-line monitor the data quality of high-dimensional data streams. With thousands of input data streams, the average run length loses its usefulness because one will likely have out-of-control signals at each time period. Hence, we first control the percentage of signals that are false alarms. Then, we compare the power of the proposed MEWMA scheme with that of two alternative methods. Compared with two competitors, numerical results show that the proposed MEWMA scheme has higher average power.     

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.     

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.     

Variable Sampling Interval Cumulative Count of Conforming Chart with Runs Rules

The cumulative count of conforming (CCC) chart is effective in detecting very low fraction of nonconforming items for high yield manufacturing processes. In this study, a combination of runs rules and variable sampling interval feature is proposed to a lower sided CCC chart by inspecting the items one by one. The performance measures of the control chart are derived by using the Markov chain approach. The numerical comparisons show that the performance of the CCC chart can be improved by adding the runs rules and varying the sampling interval.     

A Distribution-Free Control Chart Based on Order Statistics

In this article, we introduce a new distribution-free Shewhart-type control chart that takes into account the location of a single order statistic of the test sample (such as the median) as well as the number of observations in that test sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution, and the average run length (ARL) are all derived. A key advantage of the chart is that, due to its nonparametric nature, the false alarm rate and in-control run length distribution are the same for all continuous process distributions, and so will be naturally robust. Tables are provided for the implementation of the chart for some typical ARL values and false alarm rates. The empirical study carried out reveals that the new chart is preferable from a robustness point of view in comparison to a classical Shewhart-type chart and also the nonparametric chart of Chakraborti et al. (2004 Chakraborti , S. , van der Laan , P. , van de Wiel , M. A. ( 2004 ). A class of distribution-free control charts . J. Roy. Statist. Soc. Ser. C-Appl. Statist. 53 ( 3 ): 443 – 462 .[Web of Science ®] , [Google Scholar]).     

The Effects of Sample Sizes in Phases I and II on p Control Chart Performance

Control charts designed for the properties of non conformities, also called p control charts, are powerful tools used for monitoring a performance of the fraction of non conforming units. Constructing a p chart is often based on the assumption that the in-control proportion of non conforming items (p ₀) is known. In practice, the value of p ₀ is rarely known and is frequently replaced by an estimate from an in-control reference sample in Phase I. This article investigates the effects of sample sizes in both Phase I and Phase II on the performance of p control charts. The conditional and marginal run length distributions are derived and the corresponding numerical studies are conducted. Moreover, the minimal sample sizes required in Phases I and II to ensure adequate statistical performance are proposed when p ₀ = 0.1 and 0.005.     

The Steady-State Performance of Cumulative Count of a Conforming Control Chart Based On Runs Rules

Cumulative count of conforming control chart is usually used to monitor fraction nonconforming in high-yield processes. In this article, we propose m-of-m control chart based on cumulative count of conforming units for high-yield processes. The steady-state properties of the m-of-m control chart are investigated. We compare performance of the m-of-m control chart with control chart based on cumulative count of conforming units. We present Markov chain model of the m-of-m control chart to evaluate average run length, standard deviation of run length and quartiles.     

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.     

Detecting mean increases in Poisson INAR(1) processes with EWMA control charts

Processes of serially dependent Poisson counts are commonly observed in real-world applications and can often be modeled by the first-order integer-valued autoregressive (INAR) model. For detecting positive shifts in the mean of a Poisson INAR(1) process, we propose the one-sided s exponentially weighted moving average (EWMA) control chart, which is based on a new type of rounding operation. The s-EWMA chart allows computing average run length (ARLs) exactly and efficiently with a Markov chain approach. Using an implementation of this procedure for ARL computation, the s-EWMA chart is easily designed, which is demonstrated with a real-data example. Based on an extensive study of ARLs, the out-of-control performance of the chart is analyzed and compared with that of a c chart and a one-sided cumulative sum (CUSUM) chart. We also investigate the robustness of the chart against departures from the assumed Poisson marginal distribution.     

Confidence intervals for the largest mean from k correlated normal populations

Let X= (X₁,…, X_k)’ be a k-variate (k ≥ 2) normal random vector with unknown population mean vector μ = (μ₁ ,…, μ_k)’ and covariance matrix Σ of order k and let μ_[1] ≤ … ≤ μ_[k] be the ordered values of the μ ’ s. No prior knowledge of the pairing of the μ_[i] with the X_j. (or μ_[i] with the σ_j ²) is assumed for any i and j (1 ≤ i, j ≤ k). Based on a random sample of N independent vector observations on X, this paper considers both upper and lower (one-sided) and two-sided 100γ% (0 < γ < 1) confidence intervals for μ_[k] and μ_[1], the largest and the smallest mean, respectively, when Σ is known and when Σ is equal to σ²R with common unknown variance σ² > 0 and correlation matrix R known, respectively. An optimum two-sided confidence interval via finding the shortest length from this class is also considered. Necessary tables and computer program to actually apply these procedures are provided.     

A Nonparametric Shewhart-Type Synthetic Control Chart

In this article, we provide a nonparametric Shewhart-type synthetic control chart based on the signed-rank statistic to monitor shifts in the known in-control process median. The synthetic control chart is a combination of a signed-rank chart due to Bakir (2004 Bakir , S. T. ( 2004 ). A distribution-free Shewhart quality control chart based on signed-ranks . Quality Engineering 16 : 613 – 623 .[Taylor & Francis Online] , [Google Scholar]) and a conforming run length chart due to Bourke (1991 Bourke , P. D. ( 1991 ). Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection . Journal of Quality Technology 23 : 225 – 238 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The operation and design of the chart are discussed and the performance of the chart has been studied. The chart has an attractive average run length behavior as compared to the parametric control chart for a class of symmetric continuous process distributions. The proposed chart performs better than the nonparametric signed-rank chart given by Bakir (2004 Bakir , S. T. ( 2004 ). A distribution-free Shewhart quality control chart based on signed-ranks . Quality Engineering 16 : 613 – 623 .[Taylor & Francis Online] , [Google Scholar]) and Chakraborti and Eryilmaz (2007 Chakraborti , S. , Eryilmaz , S. (2007). A nonparametric Shewhart-type signed-rank control chart based on runs. Communications in Statistics—Simulation and Computation 36:335–356.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

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.     

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.     

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.     

The random intrinsic fast initial response of one-sided CUSUM charts

This article analyses the performance of a one-sided cumulative sum (CUSUM) chart that is initialized using a random starting point following the natural or intrinsic probability distribution of the CUSUM statistic. By definition, this probability distribution remains stable as the chart is used. The probability that the chart starts at zero according to this intrinsic distribution is always smaller than one, which confers on the chart a fast initial response feature. The article provides a fast and accurate algorithm to compute the in-control and out-of-control average run lengths and run-length probability distributions for one-sided CUSUM charts initialized using this random intrinsic fast initial response (RIFIR) scheme. The algorithm also computes the intrinsic distribution of the CUSUM statistic and random samples extracted from this distribution. Most importantly, no matter how the chart was initialized, if no level shifts and no alarms have occurred before time τ?>?0, the distribution of the run length remaining after τ is provided by this algorithm very accurately, provided that τ is not too small.