Design of adaptive s control charts

The Shewhart s chart has been widely used to monitor the standard deviation of a process. However, the main disadvantage of an s chart is its slowness to signal small increases in the variability. In this paper, ideas of adaptive control charts are extended to the Shewhart s chart for improving the efficiency in signalling increases in the standard deviation. A Markov chain model is applied to evaluate its performances and compares its performances with combined double sampling and variable sampling intervals s chart, variable parameters (VP) R chart, exponentially weighted moving average and Cusum charts. The statistical performances show that the VP s chart is more sensitive to increases in standard deviation.     

Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals

In this paper, we are concerned with pure statistical Shewhart control charts for the scale parameter of the three-parameter Weibull control variable, where, and are the location, the scale and the shape parameters, respectively, with fixed (FSI) and variable (VSI) sampling intervals. The parameters and are assumed to be known. We consider two-sided, and lower and upper one-sided Shewhart control charts and their FSI and VSI versions . They jointly control the mean and the variance of the Weibull control variable X. The pivotal statistic of those control charts is the maximum-likelihood estimator of for the Nth random sample XN=(X1N,X2N,...,XnN) of the Weibull control variable X. The design and performance of these control charts are studied. Two criteria, i.e. 'comparability criterion' (or 'matched criterion') under control and 'primordial criterion', are imposed on their design. The performance of these control charts is measured using the function average time to signal. For the VSI versions, the constant which defines the partition of the 'continuation region' is obtained through the 'comparability criterion' under control. The monotonic behaviour of the function average time to signal in terms of the parameters (magnitude of the shift suff ered by the target value 0), and is studied. We show that the function average time to signal of all the control charts studied in this paper does not depend on the value of the parameter or on 0, and, under control, does not depend on the parameter, when Delta (the probability of a false alarm) and n (sample size) are fixed. All control charts satisfy the 'primordial criterion' and, for fixed, on average, they all (except the two-sided VSI, for which we were not able to ascertain proof) are quicker in detecting the shift as increases. We conjecture - and we are not contradicted by the numerical example considered - that the same is true for the two-sided VSI control chart. We prove that, under the average time to signal criterion, the VSI versions are always preferable to their FSI versions. In the case of one-sided control charts, under the 'comparability criterion', the VSI version is always preferable to the FSI version, and this advantage increases with and the extent of the shift. Our one-sided control charts perform better and have more powerful statistical properties than does our two-sided control chart. The numerical example where n=5,0=1,=0.5, 1.0, 2.0, and Delta=1/370.4 is presented for the two-sided, and the lower and upper one-sided control charts. These numerical results are presented in tables and in figures. The joint influence of the parameters and in the function average time to signal is illustrated.     

Robustness of the EWMA and the combined X¯–EWMA control charts with variable sampling intervals to non-normality

The exponentially weighted moving average (EWMA) control charts with variable sampling intervals (VSIs) have been shown to be substantially quicker than the fixed sampling intervals (FSI) EWMA control charts in detecting process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be true for some processes. In the present paper, the performances of the EWMA and combined X¯–EWMA control charts with VSIs are evaluated under non-normality. It is shown that adding the VSI feature to the EWMA control charts results in very substantial decreases in the expected time to detect shifts in process mean under both normality and non-normality. However, the combined X¯–EWMA chart has its false alarm rate and its detection ability is affected if the process data are not normally distributed.     

A double exponentially weigiited moving average control procedure with variable sampling intervals

Shewhart, cumulative sum (CUSUM), and exponentially weighted moving average (EWMA) control procedures with variable sampling intervals (VSI) have been investigated in recent years for detecting shifts in the process mean. Such procedures have been shown to be more efficient when compared with the corresponding fixed sampling interval (FSI) charts with respect to the average time to signal (ATS) when the average run length (ARL) values of both types of procedures are held equal. Frequent switching between the different sampling intervals can be a complicating factor in the application of control charts with variable sampling intervals. In this article, we propose using a double exponentially weighted moving average control procedure with variable sampling intervals (VSI-DEWMA) for detecting shifts in the process mean. It is shown that the proposed VSI-DEWMA control procedure is more efficient when compared with the corresponding fixed sampling interval FSI-DEWMA chart with respect to the average time to signal (ATS) when the average run length (ARL) values of both types of procedures are held equal. It is also shown that the VSI-DEWMA procedure reduces the average number of switches between the sampling intervals and has similar ATS properties as compared to the VSI-EMTMA control procedure     

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.     

One-sided EWMA control charts for monitoring Poisson processes with varying sample sizes

ABSTRACT

Recently considerable research has been devoted to monitoring increases of incidence rate of adverse rare events. This paper extends some one-sided upper exponentially weighted moving average (EWMA) control charts from monitoring normal means to monitoring Poisson rate when sample sizes are varying over time. The approximated average run length bounds are derived for these EWMA-type charts and compared with the EWMA chart previously studied. Extensive simulations have been conducted to compare the performance of these EWMA-type charts. An illustrative example is given.     

Monitoring simple linear profiles using variable sample size schemes

ABSTRACT

Profile monitoring is one of the new research areas in statistical process control. Most of the control charts in this area are designed with fixed sampling rate which makes the control chart slow in detecting small to moderate shifts. In order to improve the performance of the conventional fixed control charts, adaptive features are proposed in which, one or more design parameters vary during the process. In this paper the variable sample size feature of EWMA₃ and MEWMA schemes are proposed for monitoring simple linear profiles. The EWMA₃ method is based on the combination of three exponentially weighted moving average (EWMA) charts for monitoring three parameters of a simple linear profile separately and the Multivariate EWMA (MEWMA) chart is based on the using a single chart to monitor the coefficients and variance of a general linear profile. Also a two-sided control chart is proposed for monitoring the standard deviation in the EWMA₃ method. The performance of the proposed charts is compared in terms of the average time to signal. Numerical examples show that using adaptive features increase the power of control charts in detecting the parameter shifts. Finally, the performance of the proposed variable sample size schemes is illustrated through a real case in the leather industry.     

A EWMA control chart based on an auxiliary variable and repetitive sampling for monitoring process location

ABSTRACT

This article develops an exponentially weighted moving average (EWMA) control chart using an auxiliary variable and repetitive sampling for efficient detection of small to moderate shifts in location. A EWMA statistic of a product estimator of the average (which utilities the information of auxiliary variables as well as repetitive sampling) is plotted on the proposed chart. The control chart coefficients of the proposed EWMA chart are determined for two strategic limits known as outer and inner control limits for the target in-control average run length. The performance of the proposed EWMA chart is studied using average run length when a shift occurs in the process average. The efficiency of the developed chart is compared with the competitive existing control charts. The results of the study revealed that proposed EWMA chart is more efficient than others to detect small changes in process mean.     

Double sampling |S| control chart with variable sample size and variable sampling interval

This study presents a control chart for monitoring shifts in the covariance matrix of a multivariate normally distributed process. This chart combines the double sampling, variable sample size and variable sampling interval features, and is called the DSVSSI |S| chart. A Markov chain approach is developed to design the DSVSSI |S| chart, by minimizing the average time to signal (ATS), for a specified shift size in the covariance matrix. The DSVSSI |S| chart has a better ATS performance compared to other existing charts. An example is given to illustrate the operation of the DSVSSI |S| chart.     

An alternative approach for analysis of performance of one-sided cusum charts with variable sampling intervals

An alternative approach for analyzing performance of one-sided Cusum charts with variable sampling intervals (VSI) is proposed. In this approach, a Markov chain with some dummy states is used. By this approach some dynamic performance measures of the VSI Cusum charts, such as the distribution of time to signal and the average time to signal against a change-point, can be determined. Some numerical results are shown, and from these results the dynamic performance of VSI Cusum charts is discussed.     

Economic design based on a multivariate Bayesian VSI chart with a dual control limit

In this paper, a multivariate Bayesian variable sampling interval (VSI) control chart for the economic design and optimization of statistical parameters is designed. Based on the VSI sampling strategy of a multivariate Bayesian control chart with dual control limits, the optimal expected cost function is constructed. The proposed model allows the determination of the scheme parameters that minimize the expected cost per time of the process. The effectiveness of the Bayesian VSI chart is estimated through economic comparisons with the Bayesian fixed sampling interval and the Hotelling's T² chart. This study is an in-depth study on a Bayesian multivariate control chart with variable parameter. Furthermore, it is shown that significant cost improvement may be realized through the new model.     

An Extension of Economic Design of x-Bar Control Charts for Non Normally Distributed Data Under Weibull Shock Models

Three parameters—sample size, sampling intervals, and the control limits—must be determined when the x bar chart to monitor a manufacturing process. The constant sampling intervals were widely employed because of its administrative simplicity. However, the variable sampling interval (VSI) has recently been shown to give substantially faster detection of most process shifts than fixed-sampling-interval (FSI) for x-bar charts. In addition, these measurements in the subgroup are assumed to be normally distributed. That assumption may not be tenable. This investigation compares the economic design of x-bar control charts for non normal data under Weibull shock models with various sampling avenues.     

Hotelling’s T2 control chart with double warning lines

Recent studies have shown that the T ² control chart with variable sampling intervals (VSI) and/or variable sample sizes (VSS) detects process shifts faster than the traditional T ² chart. This article extends these studies for processes that are monitored with VSI and VSS using double warning lines (T ² —DWL). It is assumed that the length of time the process remains in control has exponential distribution. The properties of T ² —DWL chart are obtained using Markov chains. The results show that the T ² —DWL chart is quicker than VSI and/or VSS charts in detecting almost all shifts in the process mean.     

Properties and performance of one-sided cumulative count of conforming chart with parameter estimation in high-quality processes

The one-sided cumulative count of conforming (CCC) chart is a useful method to monitor nonconforming fraction in high-quality manufacturing processes. The nonconforming fraction parameter is assumed to be known when implementing a one-sided CCC chart. In this study, we investigated the impact of estimated nonconforming fraction, [pcirc] ₀, in a one-sided CCC chart. The run length distribution is derived as well as the conditional probability of a false alarm rate (CFAR), conditional average run length (CARL) and its standard deviation (CSDRL). Simulation results are conducted to evaluate the effect of [pcirc] ₀ in a one-sided CCC chart. The results show that values of CFAR, CARL and CSDRL are close to the nominal values for a large sample. The impact of estimation errors was also studied. We find that CFAR decreases for large [pcirc] ₀. Thus, a large value of [pcirc] ₀ is suggested for fewer false alarms.     

Double-objective economic statistical design of the VP T2 control chart: Wald's identity approach

Research has shown that applying the T² control chart by using a variable parameters (VP) scheme yields rapid detection of out-of-control states. In this paper, the problem of economic statistical design of the VP T²control chart is considered as a double-objective minimization problem with the statistical objective being the adjusted average time to signal and the economic objective being expected cost per hour. We then find the Pareto-optimal designs in which the two objectives are met simultaneously by using a multi-objective genetic algorithm. Through an illustrative example, we show that relatively large benefits can be achieved by applying the VP scheme when compared with usual schemes, and in addition, the multi-objective approach provides the user with designs that are flexible and adaptive.     

Combined double sampling and variable sampling interval np chart

A combined double sampling and variable sampling interval (DSVSI) np chart is investigated in this study. The optimal design of the DSVSI np chart is based on minimizing the out-of-control average time to signal. From the numerical results, the DSVSI np chart performs reasonably well in comparison with the standard np chart, double sampling np chart, synthetic double sampling np chart, and other existing np type control charts for detecting increases in the process of fraction non conforming, based on the zero-state case. An example is provided to illustrate the application of the DSVSI np chart.     

Optimal Variable Sample Size and Sampling Interval MSE Chart Based on Improved Square Root Transformation

In the field of statistical process control (SPC), control charts for attributes are widely used to detect the out-of-control condition by checking the number of nondefective units or nondefective in a sample. In this article, we use the average time to signal (ATS) and the average number of observations to signal (ANOS) to evaluate the performance of the optimal variable sample size and sampling interval (VSSI) improved square root transformation (ISRT) mean square error (MSE) (VSSI_ ISRT_ MSE) control chart for attribute data. In addition, this control chart will be used to monitor: (1) the difference between the process mean and the target value, and (2) the process variance shifts. We found that the optimal VSSI_ ISRT_ MSE chart performs better than the specific VSSI, the optimal variable sampling interval (VSI), and the fixed parameters (FP) ISRT_MSE charts. An example is given to illustrate this new proposed approach.