Correction factors for Shewhart and control charts to achieve desired unconditional ARL

In this paper we derive correction factors for Shewhart control charts that monitor individual observations as well as subgroup averages. In practice, the distribution parameters of the process characteristic of interest are unknown and, therefore, have to be estimated. A well-known performance measure within Statistical Process Monitoring is the expectation of the average run length (ARL), defined as the unconditional ARL. A practitioner may want to design a control chart such that, in the in-control situation, it has a certain expected ARL. However, accurate correction factors that lead to such an unconditional ARL are not yet available. We derive correction factors that guarantee a certain unconditional in-control ARL. We use approximations to derive the factors and show their accuracy and the performance of the control charts – based on the new factors – in out-of-control situations. We also evaluate the variation between the ARLs of the individually estimated control charts.     

A head-to-head comparative study of the conditional performance of control charts based on estimated parameters

Implementation of the Shewhart, CUSUM, and EWMA charts requires estimates of the in-control process parameters. Many researchers have shown that estimation error strongly influences the performance of these charts. However, a given amount of estimation error may differ in effect across charts. Therefore, we perform a pairwise comparison of the effect of estimation error across these charts. We conclude that the Shewhart chart is more strongly affected by estimation error than the CUSUM and EWMA charts. Furthermore, we show that the general belief that the CUSUM and EWMA charts have similar performance no longer holds under estimated parameters.     

Estimating the time of step change with Poisson CUSUM and EWMA control charts

Knowing when a process has changed would simplify the search for and identification of the special cause. Consequently, having an estimate of the process change point following a control chart signal would be useful to process engineers. Much of the literature on change point models and techniques for statistical process control applications consider processes well modelled by the normal distribution. However, the Poisson distribution is commonly used in industrial quality control applications for modelling attribute-based process quality characteristics (e.g., counts of non-conformities). Some commonly used control charts for monitoring Poisson distributed data are the Poisson cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts. In this paper, we study the effect of changes in the design of the control chart on the performances of the change point estimators offered by these procedures. In particular, we compare root mean square error performances of the change point estimators offered by the Poisson CUSUM and EWMA control charts relative to that achieved by a maximum likelihood estimator for the process change point. Results indicate that the relative performance achieved by each change point estimator is a function of the corresponding control chart design. Relative mean index plots are provided to enable users of these control charts to choose a control chart design and change point estimator combination that will yield robust change point estimation performance across a range of potential change magnitudes.     

An exact method for designing Shewhart and S2 control charts to guarantee in-control performance

The in-control performance of Shewhart and S² control charts with estimated in-control parameters has been evaluated by a number of authors. Results indicate that an unrealistically large amount of Phase I data is needed to have the desired in-control average run length (ARL) value in Phase II. To overcome this problem, it has been recommended that the control limits be adjusted based on a bootstrap method to guarantee that the in-control ARL is at least a specified value with a certain specified probability. In this article we present simple formulas using the assumption of normality to compute the control limits and therefore, users do not have to use the bootstrap method. The advantage of our proposed method is in its simplicity for users; additionally, the control chart constants do not depend on the Phase I sample data.     

Identifying the period of a step change in high‐yield processes

Quality control charts have proven to be very effective in detecting out‐of‐control states. When a signal is detected a search begins to identify and eliminate the source(s) of the signal. A critical issue that keeps the mind of the process engineer busy at this point is determining the time when the process first changed. Knowing when the process first changed can assist process engineers to focus efforts effectively on eliminating the source(s) of the signal. The time when a change in the process takes place is referred to as the change point. This paper provides an estimator for a period of time in which a step change in the process non‐conformity proportion in high‐yield processes occurs. In such processes, the number of items until the occurrence of the first non‐conforming item can be modeled by a geometric distribution. The performance of the proposed model is investigated through several numerical examples. The results indicate that the proposed estimator provides a reasonable estimate for the period when the step change occurred at the process non‐conformity level. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimating the drift time for processes subject to linear trend disturbance using fuzzy statistical clustering

Although control charts can notify the state of out-of-control in a process by generating a signal, the indication is usually followed by a considerable amount of delay. Identifying the real time of the change in a process would provide a starting point for further investigation of an assignable cause. This paper addresses the problem of detecting the change point in different processes when the quality characteristics drift steadily away from an in-control state. For this purpose, a fuzzy statistical clustering (FSC) method is used to estimate the drift time in different processes. Since the application of an FSC method requires both in- and out-of-control values of the process parameter, a linear regression model is utilised to estimate the trend rate and then calculate the out-of-control process parameter. Through extensive simulations, the performance of the proposed change point estimation method is analysed and compared with the most recent estimators for several control charts. The results demonstrate that the proposed method is more effective in detecting the drift time through a wide range of trend rates. Furthermore, it is shown that the proposed method offers a higher estimation precision compared to conventional statistical methods.     

Percentile‐based control chart design with an application to Shewhart X̅ and S2 control charts

We present a method to design control charts such that in‐control and out‐of‐control run lengths are guaranteed with prespecified probabilities. We call this method the percentile‐based approach to control chart design. This method is an improvement over the classical and popular statistical design approach employing constraints on in‐control and out‐of‐control average run lengths since we can ensure with prespecified probability that the actual in‐control run length exceeds a desired magnitude. Similarly, we can ensure that the out‐of‐control run length is less than a desired magnitude with prespecified probability. Some numerical examples illustrate the efficacy of this design method.     

Estimating the change point of a normal process mean with a monotonic change

One of the essential steps for process improvement is to quickly recognize the starting time or the change point of a process disturbance. In this paper, we describe the behavior model of process mean and then obtain a maximum‐likelihood estimator (MLE) for the change point of the normal process mean without requiring the exact knowledge of the change type. Instead, we assume that the type of change present belongs to a family of monotonic changes. Finally, we study the performance of the proposed change‐point estimator relative to the MLEs for the process mean change point derived under a simple step change and linear trend change assumption. We do this for a number of monotonic change types following a signal from a Shewhart X̄ control chart. Copyright © 2008 John Wiley & Sons, Ltd.     

A Self‐Starting CUSUM Chart Combined with a Maximum Likelihood Estimator for the Time of a Detected Shift in the Process Mean

Detection of a special cause of variation and the identification of the time it occurs are two important activities in any quality improvement strategy. Detection of changes in a process can be done using control charts. One of these charts, the self‐starting CUSUM chart, was created to detect small sustained changes and be implemented without a Phase I or a priori knowledge of the parameters of the process. To estimate the time of a detected change, a CUSUM‐based change‐point estimator can be used, but experiments show that the corresponding MLE has smaller bias and standard error. This paper proposes the sequential use of the self‐starting CUSUM chart and the MLE of a change point in series of independent normal observations. Performance is studied with Monte Carlo simulations showing that the use of the MLE reduces the bias of the change‐point estimation. It is also shown how extra observations after a change is detected can be used to improve estimation of the change‐point time. Copyright © 2013 John Wiley & Sons, Ltd.     

A magnitude‐robust control chart for monitoring and estimating step changes for normal process means

Statistical process control charts are intended to assist operators of a usually stable system in monitoring whether a change has occurred in the process. When a change does occur, the control chart should detect it quickly. If the operator can also be provided information that aids in the search for the special cause, then critical off‐line time can be saved. We investigate a process‐monitoring tool that not only provides speedy detection regardless of the magnitude of the process shift, but also supplies useful change point statistics. A likelihood ratio approach can be used to develop a control chart for permanent step change shifts of a normal process mean. The average run length performance for this chart is compared to that of several cumulative sum (CUSUM) charts. Our performance comparisons show that this chart performs better than any one CUSUM chart over an entire range of potential shift magnitudes. The likelihood ratio approach also provides point and interval estimates for the time and magnitude of the process shift. These crucial change‐point diagnostics can greatly enhance special cause investigation. Copyright © 2002 John Wiley & Sons, Ltd.     

Directed control charts for detecting the shape changes from linear profiles to quadratic profiles

In profile monitoring, control charts are constructed to detect any unanticipated departures from the statistical stability of product quality over time, where product quality is characterised by a function. In many situations, due to the characteristics of a system or an operation, certain process signals can be anticipated. Thus, when a kind of departure specifically feared is identified in advance, a directed process monitoring approach can be developed. Motivated by the monitoring of cylindrical surfaces, this paper focuses on quickly detecting the shape changes from a straight line to a second-order polynomial curve. Based on the hypothesis testing on the quadratic term, two directed control charts and a combined scheme are proposed to surveillance the sampled linear shape. The performance of our proposed methods is studied and compared with the alternative charts by numerical simulations. Simulation studies show that the two proposed directed charts are almost the same, and outperform the alternative methods in some cases. Moreover, the combined scheme is robust for all the parameter combinations.     

Identifying the time of polynomial drift in the mean of autocorrelated processes

Control charts are used to detect changes in a process. Once a change is detected, knowledge of the change point would simplify the search for and identification of the special ause. Consequently, having an estimate of the process change point following a control chart signal would be useful to process engineers. This paper addresses change point estimation for covariance‐stationary autocorrelated processes where the mean drifts deterministically with time. For example, the mean of a chemical process might drift linearly over time as a result of a constant pressure leak. The goal of this paper is to derive and evaluate an MLE for the time of polynomial drift in the mean of autocorrelated processes. It is assumed that the behavior in the process mean over time is adequately modeled by the kth‐order polynomial trend model. Further, it is assumed that the autocorrelation structure is adequately modeled by the general (stationary and invertible) mixed autoregressive‐moving‐average model. The estimator is intended to be applied to data obtained following a genuine control chart signal in efforts to help pinpoint the root cause of process change. Application of the estimator is demonstrated using a simulated data set. The performance of the estimator is evaluated through Monte Carlo simulation studies for the k=1 case and across several processes yielding various levels of positive autocorrelation. Results suggest that the proposed estimator provides process engineers with an accurate and useful estimate for the last sample obtained from the unchanged process. Copyright © 2009 John Wiley & Sons, Ltd.     

A boosting approach for understanding out-of-control signals in multivariate control charts

The most widely used tools in statistical quality control are control charts. However, the main problem of multivariate control charts, including Hotelling's T ² control chart, lies in that they indicate that a change in the process has happened, but do not show which variable or variables are the source of this shift. Although a number of methods have been proposed in the literature for tackling this problem, the most usual approach consists of decomposing the T ² statistic. In this paper, we propose an alternative method interpreting this task as a classification problem and solving it through the application of boosting with classification trees. The classifier is then used to determine which variable or variables caused the change in the process. The results prove this method to be a powerful tool for interpreting multivariate control charts.     

统计过程控制在微电脑控制器生产中的应用:案例研究

以某电子集团生产的某类型控制器的实际生产为例,采用因果分析法对其生产过程中造成质量缺陷的原因从5M1E(人、机、料、法、测、环)方面进行综合分析.通过对其测量系统进行监控,在其稳定可靠的情况下采集数据.运用SPC技术对波峰焊工序进行控制图监控,对失控原因进行分析并在线调整.有效地保证了控制器的质量,为其质量改善指明了方向.     

Cuscore control charts for generalized feedback‐control systems

The cumulative score (Cuscore) statistic is devised to ‘resonate’ with deviations or signals of an expected type. When a process signal subject to feedback control occurs, it results in a fault signature in the output error. In this paper, Cuscore statistics are designed to monitor process parameters and characteristics measured by a generalized minimum variance (GMV) feedback‐control system sensitive to the fault signature of a spike, step, and bump signal. In this study, the GMV considered is a first‐order dynamic system with autoregressive moving average (ARMA) noise. We show theoretically that the performance of Cuscore charts is independent of the amount of variability transferred from the output quality characteristic to the adjustment actions in the GMV control system. Simulation is used to test the performance using the Cuscore charts. In general, the Cuscore can detect signals over a broad range of system parameter values. However, areas of low detection capability occur for certain fault signatures. In these cases, a tracking signal test is combined with the Cuscore statistics to improve detection performance. This study provides several illustrations of the underlying behavior and shows how the methodology developed can be easily applied in practice. Copyright © 2006 John Wiley & Sons, Ltd.     

面向质量目标的Q控制图设计

针对大批量生产开始阶段的过程监控,提出了一种基于预定质量目标的Q控制图监控方法.其基本思路是利用面向质量目标的统计公差技术与Q控制图相结合应用,以实现大批量过程开始阶段均值和方差未知时面向质量目标的过程监控.基于质量目标建立统计公差(CP*,k*),并将该统计公差转化为基于给定置信概率的对CP和k的估计值的判定条件.通过案例分析,面向质量目标的Q控制图能够在过程保持受控状态的前提下以一定置信概率保证质量目标.     

Monitoring a fraction with easy and reliable settings of the false alarm rate

Monitoring a fraction arises in many manufacturing applications and also in service applications. The traditional p‐chart is easy to use and design but is difficult to achieve the desired false alarm rate. We propose a two‐sided CUSUM Arcsine method that achieves both large and small desired false alarm rates for an in‐control probability anywhere between 0 and 1. The parameters of the new method are calculated easily, without tables, simulation, or Markov chain analysis used by many of the existing methods. The proposed method detects increases and decreases and works for constant and Poisson distributed sample sizes. The CUSUM Arcsine also has a superior sensitivity compared with other easily designed existing methods for monitoring Binomial distributed data. This paper includes an extensive literature review and a taxonomy of the existing monitoring methods for a fraction. Copyright © 2009 John Wiley & Sons, Ltd.     

Economic design of Xbar control charts with continuously variable sampling intervals

In the context of economic design, a two‐stage model is proposed that utilizes continuously variable sampling intervals. Specifically, each successive sampling interval is determined by the extremity of the latest sample. Modeling the situation as a Markov chain, the hourly cost is developed for any arbitrary set of design parameters. This proposed approach is found to be more economically desirable, possessing a smaller average out‐of‐control production time, when compared with a standard two‐stage approach where the sampling interval alternates between two fixed values. Copyright © 2009 John Wiley & Sons, Ltd.     

Using economically designed Shewhart and adaptive ―X charts for monitoring the quality of tiles

This paper presents the economic design of ―X control charts for monitoring a critical stage of the main production process at a tile manufacturer in Greece. Two types of ―X charts were developed: a Shewhart‐type chart with fixed parameters and adaptive charts with variable sampling intervals and/or sample size. Our prime motivation was to improve the statistical control scheme employed for monitoring an important quality characteristic of the process with the objective of minimizing the relevant costs. At the same time we tested and confirmed the applicability of the theoretical models supporting the economic design of control charts with fixed and variable parameters in a practical situation. We also evaluated the economic benefits of moving from the broadly used static charts to the application of the more flexible and effective adaptive control charts. The main result of our study is that, by redesigning the currently employed Shewhart chart using economic criteria, the quality‐related cost is expected to decrease by approximately 50% without increasing the implementation complexity. Monitoring the process by means of an adaptive ―X chart with variable sampling intervals will increase the expected cost savings by about 10% compared with the economically designed Shewhart chart at the expense of some implementation difficulty. Copyright © 2006 John Wiley & Sons, Ltd.