Multiple Attribute Control Charts with False Discovery Rate Control

The statistical cumulative sum (CUSUM) chart is a powerful tool for monitoring the attribute quality variable in manufacturing industry. In this article, we studied the multiplicity problem caused by simultaneously monitoring more than one attribute quality variable. Multiple binomial and Poisson CUSUM charts incorporating a multiple hypothesis testing technique known as false discovery rate control were proposed. The procedures for establishing the new control schemes were presented, and the performance of the new methods was evaluated using Monte Carlo simulation. The approximation methods for obtaining the p‐values of the CUSUM statistics for conducting the new control schemes were also provided and evaluated. The new methods were also illustrated with a real example. Copyright © 2011 John Wiley & Sons, Ltd.     

A Comparison of Normal Approximation Rules for Attribute Control Charts

Control charts, known for more than 80 years, have been important tools for business and industrial manufactures. Among many different types of control charts, the attribute control chart (np‐chart or p‐chart) is one of the most popular methods to monitor the number of observed defects in products, such as semiconductor chips, automobile engines, and loan applications. The attribute control chart requires that the sample size n is sufficiently large and the defect rate p is not too small so that the normal approximation to the binomial works well. Some rules for the required values for n and p are available in the textbooks of quality control and mathematical statistics. However, these rules are considerably different, and hence, it is less clear which rule is most appropriate in practical applications. In this paper, we perform a comparison of five frequently used rules for n and p required for the normal approximation to the binomial. With this result, we also refine the existing rules to develop a new rule that has a reliable performance. Datasets are analyzed for illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

Performance Measures for ―X Charts with Asymmetric Control Limits

Theoretical and empirical justification is given for using asymmetric control limits for certain types of production processes. The following are also discussed: the sensitivity of the performance measures to the process and control parameters, the advantages and disadvantages of using asymmetric control limits, and the construction of tradeoff curves to characterize performance. The justification is given in terms of a collection of quantitative performance measures for ―X charts with asymmetric control limits. The performance measures quantify the false‐alarm frequency, the sensitivity to out‐of‐control conditions, and the resources required for sampling. Copyright © 2005 John Wiley & Sons, Ltd.     

Optimum Multiple and Multivariate Poisson Statistical Control Charts

This paper deals with the simultaneous statistical process control of several Poisson variables. The practitioner of this type of monitoring may employ a multiple scheme, i.e. one chart for controlling each variable, or may use a multivariate scheme, based on monitoring all the variables with a single control chart. If the user employs the multivariate schemes, he or she can choose from, for example, three options: (i) a control chart based on the sum of the different Poisson variables; (ii) a control chart on the maximum value of the different Poisson variables; and (iii) in the case of only two variables, a chart that monitors the difference between them. In this paper, the previous control charts are studied when applied to the control of p = 2, 3 and 4 variables. In addition, the optimization of a set of univariate Poisson control charts (multiple scheme) is studied. The main purpose of this paper is to help the practitioner to select the most adequate scheme for her/his production process. Towards this goal, a friendly Windows© computer program has been developed. The program returns the best control limits for each control chart and makes a complete comparison of performance among all the previous schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Attribute Charts for Monitoring the Mean Vector of Bivariate Processes

This article proposes two Shewhart charts, denoted np_xy and np_w charts, which use attribute inspection to control the mean vector (μ_x; μ_y)′ of bivariate processes. The units of the sample are classified as first‐class, second‐class, or third‐class units, according to discriminate limits and the values of their two quality characteristics, X and Y. When the np_xy chart is in use, the monitoring statistic is M = N₁ + N₂, where N₁ and N₂ are the number of sample units with a second‐class and third‐class classification, respectively. When the np_w chart is in use, the monitoring statistic is W = N₁ + 2N₂. We assume that the quality characteristics X and Y follow a bivariate normal distribution and that the assignable cause shifts the mean vector without changing the covariance matrix. In general, the synthetic np_xy and np_w charts require twice larger samples to outperform the T² chart. Copyright © 2014 John Wiley & Sons, Ltd.     

Geometric Charts with Estimated Control Limits

The geometric control chart has been shown to be more effective than p and np‐charts for monitoring the proportion of nonconforming items, especially for high‐quality Bernoulli processes. When implementing a geometric control chart, the in‐control proportion nonconforming is typically unknown and must be estimated. In this article, we used the standard deviation of the average run length (SDARL) and the standard deviation of the average number of inspected items to signal, SDARL*, to show that much larger phase I sample sizes are needed in practice than implied by previous research. The SDARL (or SDARL*) was used because practitioners would estimate the control limits based on different phase I samples. Thus, there would be practitioner‐to‐practitioner variability in the in‐control ARL (or ARL*). In addition, we recommend a Bayes estimator for the in‐control proportion nonconforming to take advantage of practitioners' knowledge and to avoid estimation problems when no nonconforming items are observed in the phase I sample. If the in‐control proportion nonconforming is low, then the required phase I sample size may be prohibitively large. In this case, we recommend an approach to identify a more informative continuous variable to monitor. Copyright © 2012 John Wiley & Sons, Ltd.     

Control Charts with Variable Dimension for Linear Combination of Poisson Variables

This article analyzes the simultaneous control of several correlated Poisson variables by using the Variable Dimension Linear Combination of Poisson Variables (VDLCP) control chart, which is a variable dimension version of the LCP chart. This control chart uses as test statistic, the linear combination of correlated Poisson variables in an adaptive way, i.e. it monitors either p₁ or p variables (p₁ < p) depending on the last statistic value. To analyze the performance of this chart, we have developed software that finds the best parameters, optimizing the out‐of‐control average run length (ARL) for a shift that the practitioner wishes to detect as quickly as possible, restricted to a fixed value for in‐control ARL. Markov chains and genetic algorithms were used in developing this software. The results show performance improvement compared to the LCP chart. Copyright © 2015 John Wiley & Sons, Ltd.     

Control Charts for Binary Correlated Variables

To monitor the nonconforming fraction of a production process, usually np or p control charts are used for this purpose. However, in many practical situations, the binary variables are correlated but not easily perceived by practitioners. The aim of this article is to present the maximum likelihood and method of moment estimators of the correlation parameter ρ of an overdispersed binomial distribution. Inferential procedure is also introduced to test the null hypothesis H₀: ρ = 0 x H₁: ρ > 0. A Shewhart‐type control chart np_ρ and an Exponentiated Weighted Moving Average (EWMA)‐type control chart (EWMA np_ρ) are proposed to evaluate the nonconforming fraction when the binary variables are correlated. The traditional np chart is a particular case of the np_ρ control chart when ρ = 0. The misuse of control limits of np control in case of correlated binary variables will result a large number of false alarms. To have the same performance (in terms of average run length) of the traditional np control chart, the np_ρ control chart needs at least to double the sample size. Numerical example illustrates the proposal. Copyright © 2012 John Wiley & Sons, Ltd.     

Exponential CUSUM Charts with Estimated Control Limits

Exponential CUSUM charts are used in monitoring the occurrence rate of rare events because the interarrival times of events for homogeneous Poisson processes are independent and identically distributed exponential random variables. In these applications, it is assumed that the exponential parameter, i.e. the mean, is known or has been accurately estimated. However, in practice, the in‐control mean is typically unknown and must be estimated to construct the limits for the exponential CUSUM chart. In this article, we investigate the effect of parameter estimation on the run length properties of one‐sided lower exponential CUSUM charts. In addition, analyzing conditional performance measures shows that the effect of estimation error can be significant, affecting both the in‐control average run length and the quick detection of process deterioration. We also provide recommendations regarding phase I sample sizes. This sample size must be quite large for the in‐control chart performance to be close to that for the known parameter case. Finally, we provide an industrial example to highlight the practical implications of estimation error, and to offer advice to practitioners when constructing/analyzing a phase I sample. Copyright © 2013 John Wiley & Sons, Ltd.     

Attribute Charts for Monitoring a Dependent Process

For some repetitive production processes, the quality measure taken on the output is an attribute variable. An attribute variable classifies each output item into one of a countable set of categories. One of the simplest and most commonly used attribute variables is the one which classifies an item as either ‘conforming’ or ‘non‐conforming’. A tool used with a considerable amount of success in industry for monitoring the quality of a production process is the quality control chart. Generally a control charting procedure uses a sequence, of the quality measures to make a decision about the quality of the process. How this sequence is used to make a decision defines the control chart. In order to design a control chart one must consider how the underlying sequence, is modeled. The sequence is often modeled as a sequence of independent and identically distributed random variables. For many industrial processes, this model is appropriate, but in others it may not be. In this paper, a sequence of random variables, is used to classify an item as conforming or non‐conforming under a stationary Markov chain model and under 100% sequential sampling. Two different control charting schemes are investigated. Both schemes plot a sequence of measures on the control chart, that count the number of conforming items before a non‐conforming item. The first scheme signals as out‐of‐control if a value of falls below a certain lower limit. The second scheme signals as out‐of‐control if two out of two values of fall below a certain lower limit. The efficiency of both of the control charts is evaluated by the average run length (ARL) of the chart and the power of the chart to detect a shift in the process. The two out of two scheme is shown to have high power and a large ARL given certain parameter values of the process. An example of the two out of two scheme is provided for the interested reader. Copyright © 2006 John Wiley & Sons, Ltd.     

Adaptive ―X Control Charts with Sampling at Fixed Times

The idea of a variable sampling interval with sampling at fixed times (VSIFT) has been presented by Reynolds. This paper extends this idea to the other two adaptive ―X charts: the variable sampling rate with sampling at fixed times (VSRFT) ―X chart and the variable parameters with sampling at fixed times (VPFT) ―X chart. The VSIFT, VSRFT and VPFT ―X charts are inclusively called the adaptive with sampling at fixed times (AFT) ―X charts in this paper. The control scheme and the design issue are described and discussed for each of the AFT ―X charts. A comparative study shows that the AFT ―X charts have almost the same detection ability as the traditional adaptive ―X charts. However, from the practical viewpoint, the AFT ―X charts are considered to be more convenient to administer than the traditional adaptive ―X charts. Overall, this paper advances the application of ‘sampling at fixed times’ to the adaptive ―X control charts. Copyright © 2004 John Wiley & Sons, Ltd.     

与控制点方法论相结合的V-mask累积和控制图

V-mask累积和控制图虽然能够有效地监控过程中发生的微小偏移,但是因为它需要存储大量统计量且计算时间较长,所以在计算机中实施起来比较困难.为了解决这一问题,介绍了将控制点方法论应用于V-mask累积和控制图这一方法,并通过实例来进一步说明.结果表明,与控制点方法论结合的控制图减少了存储量,缩短了计算时间,而且将在顾客满意度控制中得到广泛应用.     

Improved Control Charts for Attributes

The classic control charts for attribute data (p-charts, u-charts, etc.,), are based on assumptions about the underlying distribution of their data (binomial or Poisson). Inherent in those assumptions is the further assumption that the “parameter” (mean) of the distribution is constant over time. In real applications, this is not always true (some days it rains and some days it does not). This is especially noticeable when the subgroup sizes are very large. Until now, the solution has been to treat the observations as variables in an individual's chart. Unfortunately, this produces flat control limits even if the subgroup sizes vary. This article presents a new tool, the p'-chart, which solves that problem. In fact, it is a universal technique that is applicable whether the parameter is stable or not.     

Modified U Charts for Monitoring and Controlling Poisson Attribute Processes

Control charts in general and U charts in particular are commonly used in most industries. This article presents a method of modifying the U chart when the usual assumption of Poisson rate data is not valid. The method consists of partitioning the data into Poisson and non-Poisson sources and using this partitioning to construct a modified U chart. Examples of the common U chart for Poisson data and the common U chart for data that are not purely Poisson are presented. In addition, the conventional individuals chart method of dealing with the violation of the Poisson assumption is discussed. Finally, the partitioning method is presented and an example given.     

基于APL的偏态控制图优化设计

以平均产品长度(APL)为评价控制图性能的标准,研究了偏态控制图的优化设计问题.针对一般控制图无法有效解决偏态总体的不对称性的情况,采用赋权方差法来构造非对称的偏态控制图,并获得其最优设计模型;最后给出了模型的灵敏度分析及算例.     

Control Charts for Dispersed Count Data: An Overview

Control charts based on the Poisson distribution are commonly used to monitor count data in attributes. However, the Poisson distribution is based on the underlying equidispersion assumption that is limiting as discussed by different researchers in the literature. Therefore, a generalized control chart is required that can be used to monitor both overdispersed and underdispersed count data. This article reviews the methods to implement for dispersed count data and present ideas for future work in this area. A comprehensive literature review for researchers and practitioners is presented in this article. Copyright © 2014 John Wiley & Sons, Ltd.     

Calculating the (Almost) Exact Control Limits for a C-Chart

A power transformation using an exponent of 2/3 for Poisson-distributed data, with a small constant added, achieves symmetry for improved statistical process control (SPC) applications whether it is for an individual, a cumulative sum, or an exponentially weighted moving average chart. Two simple equations are proposed for calculating the lower control limit (LCL) and the upper control limit (UCL) for Poisson type data. Agreement between the exact LCL and UCL, as determined by the lower and upper tail area, is excellent. The square-root transformation that stabilizes the variance produces a negatively skewed distribution and tends to give false SPC signals.     

Effective Control Charts for Monitoring the NGINAR(1) Process

In recent years, there has been a growing interest in the control of autocorrelated count data. Existing results focus on the Poisson integer‐valued autoregressive (INAR) process, but this process cannot deal with overdispersion (variance is greater than mean), which is a common phenomenon in count data. We propose to control the autocorrelated count data based on a new geometric INAR (NGINAR) process, which is an alternative to the Poisson one. In this paper, we use the combined jumps chart, the cumulative sum chart, and the combined exponentially weighted moving average chart to detect the shift of parameters in the process. We compare the performance of these charts for the case of an underlying NGINAR(1) process in terms of the average run lengths. One real example is presented to demonstrate good performances of the charts. Copyright © 2015 John Wiley & Sons, Ltd.     

Attribute control charts using generalized zero‐inflated Poisson distribution

This paper presents a control charting technique to monitor attribute data based on a generalized zero‐inflated Poisson (GZIP) distribution, which is an extension of ZIP distribution. GZIP distribution is very flexible in modeling complicated behaviors of the data. Both the technique of fitting the GZIP model and the technique of designing control charts to monitor the attribute data based on the estimated GZIP model are developed. Simulation studies and real industrial applications illustrate that the proposed GZIP control chart is very flexible and advantageous over many existing attribute control charts. Copyright © 2008 John Wiley & Sons, Ltd.