SHEWHART CHARTS FOR THE DETECTION OF LINEAR TRENDS AND OF SHIFTS IN AN AUTOREGRESSION MODEL

The paper considers a univariate characteristic of a manufacturing process which is measured at discrete time points. The characteristic exhibits a linear trend under an AR(1) disturbance. If the slope of the linear trend and the autoregression coefficient are known, the process characteristic can be adjusted to vary as white noise around its target. However, the adjustment policy is very sensitive to departures from model assumptions and fails to achieve its objective in case of shifted model parameters, e.g., in case of biased estimates or external assignable causes which change the parameters. A discussion of the behaviour of the adjusted process shows that parameter shifts can have harmful consequences. As a protection against parameter shifts, additional statistical monitoring of the process is indispensable. The paper introduces various Shewhart control charts for the detection of shifts in the mean, the trend parameter, or the autoregression parameter. The performance of the charts is analyzed by the average run length criterion.

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Design of a Control Chart Using Extended EWMA Statistic

Technologies ◽

10.3390/technologies6040108 ◽

2018 ◽

Vol 6 (4) ◽

pp. 108 ◽

Cited By ~ 3

Author(s):

Muhammad Naveed ◽

Muhamma Azam ◽

Nasrullah Khan ◽

Muhammad Aslam

Keyword(s):

Control Chart ◽

Control Charts ◽

Average Run Length ◽

Moving Average ◽

Shewhart Control Charts ◽

Shewhart Control ◽

Application Data ◽

Mean And Variance ◽

The Mean ◽

Working Procedure

In the present paper, we propose a control chart based on extended exponentially weighted moving average (EEWMA) statistic to detect a quick shift in the mean. The mean and variance expression of the proposed EEWMA statistic are derived. The proposed EEWMA statistic is unbiased and simulation results show a smaller variance as compared to the traditional EWMA. The performance of the proposed control chart with the existing chart based on the EWMA statistic is evaluated in terms of average run length (ARL). Various tables were constructed for different values of parameters. The comparison of the EEWMA control chart with the traditional EWMA and Shewhart control charts illustrates that the proposed control chart performs better in terms of quick detection of the shift. The working procedure of the proposed control chart was also illustrated by simulated and application data.

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Single-Subgroup Performance Measures and Diagnostic Procedures for X-Bar Control Charts

Journal of Engineering for Industry ◽

10.1115/1.2901933 ◽

1994 ◽

Vol 116 (2) ◽

pp. 216-224

Author(s):

G. E. Rahn ◽

S. G. Kapoor ◽

R. E. DeVor

Keyword(s):

Performance Measures ◽

Control Charts ◽

Simulated Data ◽

Diagnostic Procedures ◽

Shewhart Control Charts ◽

Operational Characteristics ◽

Shewhart Control ◽

The Mean ◽

Measures Of Performance ◽

Tremendous Impact

Although Shewhart control charts have had a tremendous impact on quality improvement, the inability to precisely measure chart performance has limited their role, and subsequently overall effectiveness in the control of manufacturing processes. Measures of performance in terms of operational characteristics (OC) are defined on two distinct levels: (a) single-subgroup level, which examines the probability of a rule violation at any given subgroup (b) multiple-subgroup level, which considers the probability of one or more rule violations throughout process monitoring. Single-subgroup performance measures for X-bar charts that employ four rules are formulated. These measures are exact expressions of operational characteristics, except for the numerical approximation to the integral of the normal distribution. Applications of these models to simulated data demonstrate their accuracy in predicting chart performance. In addition, a diagnostic methodology is described which utilizes the derived performance measures to predict the mean of a shifted distribution. The proposed diagnostic procedure is illustrated in validation and application examples.

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On Shewhart Control Charts for Zero-Truncated Negative Binomial Distributions

Pakistan Journal of Engineering Technology & Science ◽

10.22555/pjets.v4i1.521 ◽

2016 ◽

Vol 4 (1) ◽

Author(s):

Anwer Khurshid ◽

Ashit B Chakraborty

Keyword(s):

Negative Binomial Distribution ◽

Binomial Distribution ◽

Control Charts ◽

Negative Binomial ◽

Average Run Length ◽

Shewhart Control Charts ◽

Binomial Distributions ◽

Shewhart Control ◽

Cusum Charts ◽

Number Of Individuals

The negative binomial distribution (NBD) is extensively used for the description of data too heterogeneous to be fitted by Poisson distribution. Observed samples, however may be truncated, in the sense that the number of individuals falling into zero class cannot be determined, or the observational apparatus becomes active when at least one event occurs. Chakraborty and Kakoty (1987) and Chakraborty and Singh (1990) have constructed CUSUM and Shewhart charts for zero-truncated Poisson distribution respectively. Recently, Chakraborty and Khurshid (2011 a, b) have constructed CUSUM charts for zero-truncated binomial distribution and doubly truncated binomial distribution respectively. Apparently, very little work has specifically addressed control charts for the NBD (see, for example, Kaminsky et al., 1992; Ma and Zhang, 1995; Hoffman, 2003; Schwertman. 2005). The purpose of this paper is to construct Shewhart control charts for zero-truncated negative binomial distribution (ZTNBD). Formulae for the Average run length (ARL) of the charts are derived and studied for different values of the parameters of the distribution. OC curves are also drawn.

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SIMULATED SHEWHART CONTROL CHART FOR MONITORING VARIANCE COMPONENTS

International Journal of Reliability Quality and Safety Engineering ◽

10.1142/s0218539309003265 ◽

2009 ◽

Vol 16 (01) ◽

pp. 1-22 ◽

Cited By ~ 4

Author(s):

TZONG-RU TSAI ◽

YI-WEI HSIEH

Keyword(s):

Variance Components ◽

Control Charts ◽

Average Run Length ◽

Random Effect ◽

Random Effect Model ◽

Simulation Method ◽

Shewhart Control Charts ◽

Effect Model ◽

Shewhart Control ◽

Simulation Results

Shewhart control charts based on the simulation method are proposed for monitoring separate variance components of the single-factor random effect model. Monte Carlo simulation results show that the proposed control charts have competitive performance relative to the approximate Shewhart regression control charts (ARSCCs) proposed by Chang and Gan2 in terms of the average run length (ARL). Compared with the ARSCCs, our control charts can be constructed easily with less sample resources. The application of our proposed method is illustrated with two examples.

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The Enhanced EWMA Control Chart with Autocorrelation

Jurnal Teknologi ◽

10.11113/jt.v71.3862 ◽

2014 ◽

Vol 71 (5) ◽

Author(s):

Abbas Umar Farouk ◽

Ismail Mohamad

Keyword(s):

Control Chart ◽

Control Charts ◽

Average Run Length ◽

Independence Assumption ◽

Shewhart Control Charts ◽

Ewma Chart ◽

Ewma Control Chart ◽

Shewhart Control ◽

Autocorrelated Data ◽

The Given

Control charts are effective tool with regard to improving process quality and productivity, Shewhart control charts are efficiently good at detecting large shifts in a given process but very slow in detecting small and moderate shifts, such problem could be tackled through design of sensitizing rules. It has been observed that autocorrelation has an advert effect on the control charts developed under the independence assumption [1]. In this article a new EWMA control chart has been introduced with autocorrelation and some run rule schemes were introduced to enhanced the performance of the EWMA control chart when autocorrelated. The three-out-of three EWMA scheme and three-out-of- four EWMA schemes were introduced and the generated data with induced autocorrelation were used to construct the EWMA chart to sensitize the shifts presence. Simulation of autocorrelated data were carried out for the proposed schemes which detects the shifts as soon as it occurs in the given process, the performance were evaluated using the ARL (average run length) and the results were compared with the published results of Steiner (1991) and the Saccucci (1990) which were designed for large, small and moderate shift. The results indicates that the proposed schemes are more sensitive to the shifts at ARL0=500, 300 and 200 with autocorrelation of 0.2, 0.5 and 0.9 considered in the study.

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PERFORMANCE OF SENSITIZING RULES ON SHEWHART CONTROL CHARTS WITH AUTOCORRELATED DATA

International Journal of Reliability Quality and Safety Engineering ◽

10.1142/s0218539301000438 ◽

2001 ◽

Vol 08 (02) ◽

pp. 159-171 ◽

Cited By ~ 3

Author(s):

SANDY D. BALKIN ◽

DENNIS K. J. LIN

Keyword(s):

Time Series ◽

Control Charts ◽

Moving Average ◽

Correlated Data ◽

Autoregressive Moving Average ◽

Shewhart Control Charts ◽

Shewhart Control ◽

The Mean ◽

Autocorrelated Data ◽

Using Data

Sensitizing Rules are commonly applied to Shewhart Charts to increase their effectiveness in detecting shifts in the mean that may otherwise go unnoticed by the usual "out-of-control" signals. The purpose of this paper is to demonstrate how well these rules actually perform when the data exhibit autocorrelation compared to non-correlated data. Since most control chart data are collected as time series, it is of interest to examine the performance of Shewhart's [Formula: see text] Chart using data generated from typical time series models. In this paper, measurements arising from autoregressive (AR), moving average (MA) and autoregressive moving average (ARMA) processes are examined using Shewhart Control Charts in conjunction with several sensitizing rules. The results indicate that the rules work well when there are strong autocorrelative relationships, but are not as effective in recognizing small to moderate levels of correlation. We conclude with the recommendation to practitioners that they use a more definitive measure of autocorrelation such as the Sample Autocorrelation Function correlogram to detect dependency.

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