Multivariate Synthetic |S| Control Chart with Variable Sampling Interval

A variable sampling interval (VSI) feature is introduced to the multivariate synthetic generalized sample variance |S| control chart. This multivariate synthetic control chart is a combination of the |S| sub-chart and the conforming run length sub-chart. The VSI feature enhances the performance of the multivariate synthetic control chart. The comparative results show that the VSI multivariate synthetic control chart performs better than other types of multivariate control charts for detecting shifts in the covariance matrix of a multivariate normally distributed process. An example is given to illustrate the operation of the VSI multivariate synthetic chart.     

Economic and economic–statistical designs of VSI Bayesian control chart using Monte Carlo method and ABC algorithm

The usual practice in using a Bayesian control chart to monitor a process is done by taking samples from the process with fixed sampling intervals. Recent studies on traditional control charts have shown that variable sampling interval (VSI) scheme compared to classical scheme (fixed ratio sampling, FRS) helps practitioners to detect process shifts more quickly. In this paper, the effectiveness of VSI scheme on performance of Bayesian control chart has been studied, based on economic (ED) and economic–statistical designs (ESD). Monte Carlo method and artificial bee colony algorithm have been utilized to obtain optimal design parameters of Bayesian control chart (sample size, sampling intervals, warning limit and control limit) since the statistic of this approach does not have any specified distribution. Finally, VSI Bayesian control chart has been compared to FRS Bayesian and VSI X-bar approaches based on ED and ESD, separately. According to the results, it has been found that the performance of VSI Bayesian scheme is better than FRS Bayesian and VSI X-bar approaches.     

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     

Economic design of VSI X̄ control chart with correlated non-normal data under multiple assignable causes

It has been recently revealed that the Shewhart control charts with variable sampling interval (VSI) perform better than the traditional Shewhart chart with the fixed sampling interval in detecting shifts in the process. In most of these research works, the normality and independency of the process data or measurements are assumed and that the process is subjected to only one assignable cause. While, in practice, these assumptions usually do not hold, some recent studies are focused on working with only one or two of these violations. In this paper, the situation in which the process data are correlated and follow a non-normal distribution and that there is multiplicity of assignable causes in the process is considered. For this case, a cost model for the economic design of the VSI X? control chart is developed, where the Burr distribution is employed to represent the non-normal distribution of the process data. To obtain the optimal values of the design parameters, a genetic algorithm is employed in which the response surface methodology is applied. A numerical example is presented to show the applicability and effectiveness of the proposed methodology. Sensitivity analysis is also carried out to evaluate the effects of cost and input parameters on the performance of the chart.     

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.     

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.     

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.     

On the Superiority of a Variable Sampling Interval Control Chart

The paper establishes the analytical grounds of the uniform superiority of a variable sampling interval (VSI) Shewhart control chart over the conventional fixed sampling interval (FSI) control chart, with respect to the zero-time performance, for a wide class of process distributions. We provide a sufficient condition on the distribution of a control chart statistic, and propose a criterion to determine the control limits and the regions in the in-control area of the VSI chart, corresponding to the different sampling intervals used by it. The condition and the criterion together ensure the uniform zero-time superiority of the VSI chart over the matched FSI chart, in detecting a process shift of any magnitude. It is shown that normal, Student's t and Laplace distributions satisfy the sufficient condition. In addition, chi-square, F and beta distributions satisfy it, provided that these are not extremely skewed. Further, it is illustrated that the superiority of the VSI feature is not trivial and cannot be assured if the sufficient condition is not satisfied or the control limits and the regions are not determined according to the proposed criterion. An application of the result to confirm the superiority of the VSI feature is demonstrated for the control chart for individual observations used to monitor a milk-pouch filling process.     

Economic-statistical design of simple linear profiles with variable sampling interval

Control charts are statistical tools to monitor a process or a product. However, some processes cannot be controlled by monitoring a characteristic; instead, they need to be monitored using profiles. Economic-statistical design of profile monitoring means determining the parameters of a profile monitoring scheme such that total costs are minimized while statistical measures maintain proper values. While varying sampling interval usually increases the effectiveness of profile monitoring, economic-statistical design of variable sampling interval (VSI) profile monitoring is investigated in this paper. An extended Lorenzen–Vance function is used for modeling total costs in VSI model where the average time to signal is employed for depicting the statistical measure of the obtained profile monitoring scheme. Two sampling intervals; number of set points and the parameters of control charts that are used in profile monitoring are the variables that are obtained thorough the economic-statistical model. A genetic algorithm is employed to optimize the model and an experimental design approach is used for tuning its parameters. Sensitivity analysis and numerical results indicate satisfactory performance for the proposed model.     

On the improvement of one-sided S control charts

The most common charting procedure used for monitoring the variance of the distribution of a quality characteristic is the S control chart. As a Shewhart-type control chart, it is relatively insensitive in the quick detection of small and moderate shifts in process variance. The performance of the S chart can be improved by supplementing it with runs rules or by varying the sample size and the sampling interval. In this work, we introduce and study one-sided adaptive S control charts, supplemented or not with one powerful runs rule, for detecting increases or decreases in process variation. The properties of the proposed control schemes are obtained by using a Markov chain approach. Furthermore, a practical guidance for the choice of the most suitable control scheme is also provided.     

Improved switching rules in control procedures using variable sampling intervals

Some properties of control procedures with variable sampling intervals (VSI) have been investigated in recent years by Amin, Renolds et al, and others. Such procedures have been shown to be more efficient when compared to the corresponding fixed sampling interval (FSI) charts with respect to the Average Time to Signal (ATS) when the Average Run Length (ARL) values for 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 (VSI). This problem is being addressed in this article, and improved switching rules are presented and evaluated for Shewhart, CUSUM, and EWMA control procedures. The proposed rules considerably reduce the average number of switches between the sampling intervals and also improve the ATS properties of the control procedures when compared to the conventional variable sampling interval procedures     

Optimization of design parameters for control charts with multiple assignable causes

Duncan's economic model of Shewhart's original x¯ chart has established its optimal and economic application for processes with the Markovian failure characteristic. As the sample statistics show some indications of process variations, the variable-sampling-interval (VSI) control charts perform more effectively than the fixed sampling interval (FSI) ones due to a higher frequency in the sampling rate. Regarding the economic design of control charts, most studies have been dedicated to the FSI scheme. In 1998, Bai & Lee considered the production process with a single assignable cause and proposed an economic VSI design for a general x¯ control chart. However, in real cases, there are multiple assignable causes in the production process. Therefore, concerning the operation characteristics of the real industry, this research develops an economic model for the VSI control chart with multiple assignable causes based on stochastic and statistics theory and determines the optimal design parameters of the chart. A numerical example is also provided to demonstrate the effectiveness of the proposed model and the result indicates that VSI performs more effectively than a FSI control chart.     

Runs rules schemes for monitoring process variability

To increase the sensitivity of Shewhart control charts in detecting small process shifts sensitizing rules based on runs and scans are often used in practice. Shewhart control charts supplemented with runs rules for detecting shifts in process variance have not received as much attention as their counterparts for detecting shifts in process mean. In this article, we examine the performance of simple runs rules schemes for monitoring increases and/or decreases in process variance based on the sample standard deviation. We introduce one-sided S charts that overcome the weakness of high false-alarm rates when runs rules are added to a Shewhart control chart. The average run length performance and design aspects of the charts are studied thoroughly. The performance of associated two-sided control schemes is investigated as well.     

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.     

Adaptive Hotelling's T 2 Control Charts with Run Rules

This study investigates the statistical properties of the adaptive Hotelling's T ² charts with run rules in which the sample size and sampling interval are allowed to vary according on the current and past sampling points. The adaptive charts include variable sample size (VSS), variable sampling interval (VSI), and variable sample size and sampling interval (VSSI) charts. The adaptive Hotelling's T ² charts with run rules are compared with the fixed sampling rate Hotelling's T ² chart with run rules. The numerical results show that the VSS, VSI, and VSSI features improve the performance of the Hotelling's T ² chart with run rules.     

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.     

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.     

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.