Comparisons of control schemes for monitoring the means of processes subject to drifts

Although statistical process control (SPC) techniques have been focused mostly on detecting step (constant) mean shift, drift which is a time-varying change frequently occurs in industrial applications. In this research, for monitoring drift change, the following five control schemes are compared: the exponentially weighted moving average (EWMA) chart and the cumulative sum (CUSUM) charts which are recommended detecting drift change in the literature; the generalized EWMA (GEWMA) chart proposed by Han and Tsung (2004) and two generalized likelihood ratio based schemes, GLR-S and GLR-L charts which are respectively under the assumption of step and linear trend shifts. Both the asymptotic estimation and the numerical simulation of the average run length (ARL) are presented. We show that when the in-control (IC) ARL is large (goes to infinity), the GLR-L chart has the best overall performance among the considered charts in detecting linear trend shift. From the viewpoint of practical IC ARL, based on the simulation results, we show that besides the GLR-L chart, the GEWMA chart offers a good balanced protection against drifts of different size. Some computational issues are also addressed.     

Variables control chart limits and tests for special causes

Control charts are used to detect problems in control such as outliers, shifts in levels or excess variability in subgroup means that may have a special cause. This paper addresses itself to deriving control chart limits based on past data and based on initial samples in a current control situation. We present a general setting for control charts. Furthermore, an overview is given of tests for special causes. The tests are standardized so that the asymptotic type I error does not exceed a fixed level. The distributions of the run lengths of the tests and combinations of tests are also evaluated. We propose to use a low percen-tile of the run length distribution, instead of the average run length, to study the performance of the tests. These indicate that, in particular when tests are combined, the run length percentiles may be too small for practical purposes. It is shown that (nearly) exact control chart limits for observations from a normal distribution exist. The traditional limits differ considerably from the proposed ones and correspond to even smaller run length percentiles.     

A new procedure for monitoring the range and standard deviation of a quality characteristic

The Shewhart and the Bonferroni-adjustment R and S chart are usually applied to monitor the range and the standard deviation of a quality characteristic. These charts are used to recognize the process variability of a quality characteristic. The control limits of these charts are constructed on the assumption that the population follows approximately the normal distribution with the standard deviation parameter known or unknown. In this article, we establish two new charts based approximately on the normal distribution. The constant values needed to construct the new control limits are dependent on the sample group size (k) and the sample subgroup size (n). Additionally, the unknown standard deviation for the proposed approaches is estimated by a uniformly minimum variance unbiased estimator (UMVUE). This estimator has variance less than that of the estimator used in the Shewhart and Bonferroni approach. The proposed approaches in the case of the unknown standard deviation, give out-of-control average run length slightly less than the Shewhart approach and considerably less than the Bonferroni-adjustment approach.     

Monitoring process mean level using auxiliary information

In this study, a Shewhart‐type control chart is proposed for the improved monitoring of process mean level (targeting both moderate and large shifts which is the major concern of Shewhart‐type control charts) of a quality characteristic of interest Y. The proposed control chart, namely the M_r chart, is based on the regression estimator of mean using a single auxiliary variable X. Assuming bivariate normality of (Y, X), the design structure of M_r chart is developed for phase I quality control. The comparison of the proposed chart is made with some existing control charts used for the same purpose. Using power curves as a performance measure, better performance of the proposedM_r chart is observed for detecting the shifts in mean level of the characteristic of interest.     

Hotelling’s <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> control chart with two adaptive sample sizes

Some quality control schemes have been developed when several related quality characteristics are to be monitored. The familiar multivariate process monitoring and control procedure is the Hotelling’s T ² control chart for monitoring the mean vector of the process. It is a direct analog of the univariate shewhart [`(x)]{\bar{x}} chart. As in the case of univariate, the ARL improvements are very important particularly for small process shifts. In this paper, we study the T ² control chart with two-state adaptive sample size, when the shift in the process mean does not occur at the beginning but at some random time in the future. Further, the occurrence time of the shift is assumed to be exponentially distributed random variable.     

Economic design of T<Superscript>2</Superscript> control charts with the VSSI sampling scheme

T ² charts are used to monitor a process when more than one quality variable associated with process is being observed. Recent studies have shown that the T ² chart with variable sampling size and sampling interval (VSSI) detects a small shift in the process mean vector faster than the traditional T ² chart. The paper considers an economic design of the VSSI T ² chart, in which the expected hourly loss is constructed and regarded as an objective function for optimally determining the design parameters (i.e. the maximum/minimum sample size, the longest/shortest sampling interval, and the warning/action limits) in sampling-and-charting. Furthermore, the effects of process parameters and cost parameters upon the expected hourly loss and design parameters are examined.     

A process control method based on five-parameter generalized lambda distribution

The use of control charts in statistical quality control, which are statistical measures of quality limits, is based on several assumptions. For instance, the process output distribution is assumed to follow a specified probability distribution (normal for continuous measurements and binomial or Poisson for attribute data) and the process supposed to be for large production runs. These assumptions are not always fulfilled in practice. This paper focuses on the problem when the process monitored has an output which has unknown distribution, or/and when the production run is short. The five-parameter generalized lambda distributions (GLD) which are subject to estimating data distributions, as a very flexible family of statistical distributions is presented and proposed as the base of control parameters estimation. The proposed chart is of the Shewhart type and simple equations are proposed for calculating the lower and upper control limits (LCL and UCL) for unknown distribution type of data. When the underlying distribution cannot be modeled sufficiently accurately, the presented control chart comes into the picture. We develop a computationally efficient method for accurate calculations of the control limits. As the vital measure of performance of SPC methods, we compute ARL’s and compare them to show the explicit excellence of the proposed method.     

Asymmetric control limits for small samples

When designing control charts, it is usually assumed that the measurement in the subgroups are normally distributed. The assumption of normality implies that the control limits for a chart for sample averages will be symmetrical about the centerline of the chart. However, the assumption of an underlying normal distribution of the data may not hold in some processes. If the measurements are asymmetrically distributed then the decision maker may choose different actions. One thing that can be done is to consider the degree of skewness. If the nature of the underlying distribution is skewed, then the traditional Shewhart individuals chart may not be valid. This paper presents a technique for constructing appropriate asymmetric control limits when the distribution of data cannot be assumed to be a normal distribution. Meanwhile, it proposes a skewness correction method for the generated Burr, lognormal and exponential distributions. Some numerical calculations are generated for n  =  2, 3, 4 by using MATLAB.     

Run length distributions and economic design of [`(X)]\bar X charts with unknown process variance

The run length distribution of charts with unknown process variance is analized using numerical integration. Both traditional chart limits and a method due to Hillier are considered. It is shown that setting control limits based on the pooled standard deviation, as opposed to the average sample standard deviation, provides better run length performance due to its smaller mean square error. The effect of an unknown process variance is shown to increase the area under both tails of the run length distribution. If Hillier’s method is used instead, only the right tail of the run length distribution is increased. Collani’s model for the economic design of charts is extended to the case of unknown process variance by writing his standardized objective function in terms of average run lengths.     

Run length analysis of Shewhart charts applied to drifting processes under an integrative SPC/EPC model

Engineering Process Controllers (EPC) are frequently based on parametrized models. If process conditions change, the parameter estimates used by the controllers may become biased, and the quality characteristics will be affected. To detect such changes it is adequate to use Statistical Process Control (SPC) methods. The run length statistic is commonly used to describe the performance of an SPC chart. This paper develops approximations for the first two moments of the run length distribution of a one-sided Shewhart chart used to detect two types of process changes in a system that is regulated by a given EPC scheme: i) changes in the level parameter; ii) changes in the drift parameter. If the drift parameter shifts, it is further assumed that the form of the drift process changes from a linear trend under white noise (the in-control drift model) into a random walk with drift model. Two different approximations for the run length moments are presented and their accuracy is numerically analyzed. Received: August 1998     

Control charts for health care monitoring under overdispersion

An attractive way to control attribute data from high quality processes is to wait till r ≥ 1 failures have occurred. The choice of r in such negative binomial charts is dictated by how much the failure rate is supposed to change during Out-of-Control. However, these results have been derived for the case of homogeneous data. Especially in health care monitoring, (groups of) patients will often show large heterogeneity. In the present paper we will show how such overdispersion can be taken into account. In practice, typically neither the average failure rate, nor the overdispersion parameter(s), will be known. Hence we shall also derive and analyze the estimated version of the new chart.     

On the monitoring of multi-attributes high-quality production processes

Over the last decade, there have been an increasing interest in the techniques of process monitoring of high-quality processes. Based upon the cumulative counts of conforming (CCC) items, Geometric distribution is particularly useful in these cases. Nonetheless, in some processes the number of one or more types of defects on a nonconforming observation is also of great importance and must be monitored simultaneously. However, there usually exist some correlations between these two measures, which obligate the use of multi-attribute process monitoring. In the literature, by assuming independence between the two measures and for the cases in which there is only one type of defect in nonconforming items, the generalized Poisson distribution is proposed to model such a problem and the simultaneous use of two separate control charts (CCC & C chats) is recommended. In this paper, we propose a new methodology to monitor multi-attribute high-quality processes in which not only there exist more than one type of defects on the observed nonconforming item but also there is a dependence structure between the two measures. To do this, first we transform multi-attribute data in a way that their marginal probability distributions have almost zero skewnesses. Then, we estimate the transformed mean vector and covariance matrix and apply the well-known χ² control chart. In order to illustrate the proposed method and evaluate its performance, we use two numerical examples by simulation and compare the results. The results of the simulation studies are encouraging.     

CUMIN charts

Classical control charts are very sensitive to deviations from normality. In this respect, nonparametric charts form an attractive alternative. However, these often require considerably more Phase I observations than are available in practice. This latter problem can be solved by introducing grouping during Phase II. Then each group minimum is compared to a suitable upper limit (in the two-sided case also each group maximum to a lower limit). In the present paper it is demonstrated that such MIN charts allow further improvement by adopting a sequential approach. Once a new observation fails to exceed the upper limit, its group is aborted and a new one starts right away. The resulting CUMIN chart is easy to understand and implement. Moreover, this chart is truly nonparametric and has good detection properties. For example, like the CUSUM chart, it is markedly better than a Shewhart X-chart, unless the shift is really large.     

变样本容量比例控制图设计及应用

针对单边休哈特比例控制图失控状态下平均报警时间长的问题,提出一种单边变样本容量比例控制图来监控两个正态变量间比例的偏移。采用马尔可夫链方法计算了该控制图的平均运行链长。在保证过程处于受控状态的统计性能基础上,使失控状态下的平均运行链长最小,从而获得控制图的最优决策变量和最优性能指标。仿真结果表明,单边变样本容量比例控制图的性能明显优于单边休哈特比例控制图。最后将该方法应用于港口皮带机配煤比例监控的问题,通过仿真实验说明了该方法能够对煤炭配比过程中的比例偏移做出有效反应。     

Robustness issues regarding content-corrected tolerance limits

This article reviews the content-corrected method for tolerance limits proposed by Fernholz and Gillespie (2001) and addresses some robustness issues that affect the length of the corresponding interval as well as the corrected content value. The content-corrected method for k-factor tolerance limits consists of obtaining a bootstrap corrected value p ^* that is robust in the sense of preserving the confidence coefficient for a variety of distributions. We propose several location/scale robust alternatives to obtain robust corrected-content k-factor tolerance limits that produce shorter intervals when outlying observations are present. We analyze the Hadamard differentiability to insure bootstrap consistency for large samples. We define the breakdown point for the particular statistic to be bootstrapped, and we obtain the influence function and the value of the breakdown point for the traditional and the robust statistics. Two examples showing the advantage of using these robust alternatives are also included.     

Sample size and the Binomial CUSUM Control Chart: the case of 100% inspection

The Binomial CUSUM is used to monitor the fraction defective (p) of a repetitive process, particularly for detecting small to moderate shifts. The number of defectives from each sample is used to update the monitoring CUSUM. When 100% inspection is in progress, the question arises as to how many sequential observations should be grouped together in forming successive samples. The tabular form of the CUSUM has three parameters: the sample size n, the reference value k, and the decision interval h, and these parameters are usually chosen using statistical or economic-statistical criteria, which are based on Average Run Length (ARL). Unlike earlier studies, this investigation uses steady-state ARL rather than zero-state ARL, and the occurrence of the shift can be anywhere within a sample. The principal finding is that there is a significant gain in the performance of the CUSUM when the sample size (n) is set at one, and this CUSUM might be termed the Bernoulli CUSUM. The advantage of using n=1 is greater for larger shifts and for smaller values of in-control ARL. First version: September 1998/Third revision: September 2000     

Economically optimalc- andnp-control charts

In v. Collani (1981, 1986, 1987a and 1987b) simple procedures are developed to determine the approximately optimal economic design of control charts for measurements. Applying these procedures to the case of control charts for attributes, nomograms are obtained from which the approximately optimal design ofc-charts, i.e. charts for defects, is readily available. Furthermore it is shown that this method also provides good approximately optimalnp-charts, i.e. charts for defectives. Research supported by the DFG (Deutsche Forschungsgemeinschaft).     

Ratio cum product method of estimation

Summary In this paper methods of estimation which may be considered as combination of ratio and product methods have been suggested. The mean square errors of these estimators utilizing two supplementary variables are compared with (i) simple unbiased estimator (p=0), (ii) usual ratio and product methods of estimation (p=1) and (iii) multivariate ratio and multivariate product estimators (p=2), wherep is the number of supplementary variables utilized. Conditions for their efficient use have been obtained for each case. Extension to general case ofp-variables has been briefly discussed. A new criteria for the efficient use of product estimator have been obtained.     

Control Charts for One-sided Capability Indices

There are many industrial product characteristics are desired to be the bigger the best and the smaller the best. The two well-know processes capability indices C _pl and C _pu, which measure larger-the-better and smaller-the-better process capabilities. Obviously, the formulae for the two indices C _pl and C _pu are easy to understand and straightforward to apply. Thus, indices C _pl and C _pu have been utilized by a number of Japanese companies and the U.S. automotive industry by Ford Motor Company. Boyles (1991, Journal of Quality Technology. 23: 17–26) and Spring (1995, Total Quality Management 6(3): 427–438.) point out that as soon as and S control charts are in statistical control, the control charts of process capability indices can be used to monitor the quality of process. In the previous, we know that if the process is not in control, the process capability index control chart can be used to monitor the differences of process capability, and as soon as the process is in control the stable process capability can be identified. Therefore, process capability index control chart not only can be used to monitor the stability of process’s quality but also can be used to monitor the quality of process. Since Boyles (1991, Journal of Quality Technology 23: 17–26.) and Spiring (1995, Total Quality Management 6(1): 21–33.) had had research about control chart of the bilateral specification index C _pm., but there are many kinds of products, which meet unilateral quality specification. Therefore, we will construct the control chart of unilateral specification index C _pl and C _pu to monitor and evaluate the stability of process and process capability.     

NONSENSE REGRESSIONS BETWEEN INTEGRATED PROCESSES OF DIFFERENT ORDERS

Abstract Herein we develop an analytical study of the asymptotic distributions obtained when we run linear regressions in the levels of stochastically independent integrated time series when the orders of integration of the dependent and independent variables are different. These theoretical findings largely explain the Monte Carlo results recently reported in Banerjee et al. (1993).