Variable Sample Size And Sampling Interval Research Articles

Weighted adaptive CUSUM mean chart with variable sample size and sampling interval

The classical CUSUM (C) chart is designed by assuming that the size of the shift in the process, in which a quick detection is important, is known a priori. However, this assumption is often violated in practice, as the actual shift size is rarely known even though it may vary within a certain interval. The adaptive C (AC) chart gives better protection against process shifts in a certain interval than the classical C chart. In this paper, we propose a new weighted AC (WAC) chart with the variable sample size and sampling interval (VSSI) feature, which encompasses the fixed sampling rate (FSR), variable sample size (VSS) and variable sampling interval (VSI) based WAC charts. The Monte Carlo simulations are adopted in the run-length computations of the considered charts, where the newly developed (VSS and VSSI based) charts outperform the existing (FSR and VSI) WAC charts when detecting small shifts in the process mean. An illustrative example shows the implementation of the existing and proposed WAC charts.

Variable sample size and sampling interval (VSSI) and variable parameters (VP) run sum charts for the coefficient of variation

ABSTRACT This paper improves the sensitivity of the existing run sum chart for shifts in the coefficient of variation by proposing two types of adaptive schemes, i.e. the variable sample size and sampling interval (VSSI) and variable parameter (VP) schemes. The VSSI run sum chart varies the sample size and sampling interval , while the VP run sum chart varies not only n and h but also the size of each region of the run sum chart. The operations, as well as formulae and algorithms to optimize the charts’ performance, are developed. Subsequently, tables of optimal charting parameters are obtained. The VP run sum chart is shown to outperform the VSSI run sum chart. The proposed charts are also shown to outperform the variable sampling interval (VSI) run sum, variable sample size (VSS) run sum, and the VSSI, VSS and VSI charts, as well as the VSS and VSI EWMA charts for moderate and large shift sizes. Implementation on an actual industrial example shows the sensitivity of the proposed charts towards small shifts.

An adaptive multivariate EWMA mean chart with variable sample sizes and/or variable sampling intervals

AbstractNew adaptive multivariate EWMA (AME) charts based on the variable sample size (VSS), variable sampling interval (VSI) and variable sample size and sampling interval (VSSI) techniques are proposed to reduce the time of the existing AME chart in detecting a mean shift in the multivariate normal process. A Monte Carlo simulation is conducted to compare the time to signal characteristics of the new VSS/VSI/VSSI AME and existing fixed sampling rate (FSR) AME and VSSI multivariate EWMA (ME) charts. It is found that the VSS/VSI/VSSI AME charts generally surpass the existing FSR AME and the non‐adaptive VSSI ME charts in detecting small and moderate shifts in the mean of a multivariate normal process, in terms of the expected weighted time‐to‐signal and expected relative average time‐to‐signal performance measures. An example is given to show the implementation of the existing and proposed charts.

An improved variable parameter mean square error control chart

In order to improve the efficiency of the mean square error control chart in detecting small to moderate shifts, an improved variable parameter mean square error (VPt MSE) control chart is proposed, which uses three different sampling intervals, two different sample sizes and two different control limits. Markov chain method is used to calculate the average time to signal and average number of observations to signal. The performance of the adaptive chart was compared with the variable sample size and sampling interval (VSSI), the improved variable sample size and sampling interval (VSSIt), the variable parameter (VP), the exponentially weighted moving average (EWMA), and the variable sampling interval exponentially weighted moving average (VSI EWMA) mean square error control chart. Numerical results show that the VPt MSE chart surpasses the other charts in detecting small to moderate shifts. At the same time, we also improve the VPt MSE chart, which greatly improves the statistical performance. An example is provided to illustrate the implementation of the VPt MSE chart.

A variable parameters auxiliary information based quality control chart with application in a spring manufacturing process: The Markov chain approach

In this paper, a new variable parameters chart for the process mean with a statistic that integrates information from the study and auxiliary variables is proposed. The proposed variable parameters chart with auxiliary information (AI) (abbreviated as VP-AI) is optimally designed to minimize the out-of-control steady-state (i) average time to signal (ATS1) and (ii) expected average time to signal (EATS1) values when the mean shift sizes are known and unknown, respectively. The Markov chain approach is adopted to derive the formulae of the performance measures ATS1, standard deviation of the time to signal (SDTS1) and EATS1. The VP-AI chart significantly outperforms the standard VP chart; thus, justifying the incorporation of auxiliary information to enhance the ability of the VP chart. The VP-AI chart is also compared with the Shewhart AI, synthetic AI, exponentially weighted moving average (EWMA) AI, run sum AI and variable sample size and sampling interval (VSSI) AI charts. The VP-AI chart significantly outperforms the Shewhart AI, synthetic AI and VSSI AI charts for all levels of shifts. Meanwhile, the VP-AI chart outperforms the EWMA AI and run sum AI charts for most shifts. The VP-AI chart is found to be more robust than the EWMA-AI chart when the correlation coefficient is misspecified or the bivariate normality assumption is violated as long as the size of the shift is moderate or large. A real application which monitors spring elasticity is used to illustrate the VP-AI chart’s implementation.

Economic and economic-statistical designs of variable sample size and sampling interval coefficient of variation chart

In the literature, the variable sample size and sampling interval (VSSI) coefficient of variation (CV) chart was shown to have a better statistical performance than the Shewhart (SH) CV chart which uses fixed sample size and sampling interval. The VSSI CV chart adopts more frequent sampling and larger sample size when the process quality deteriorates for quicker detection of a process shift as there is a higher tendency for the said shift to occur. This paper investigates the economic and economic-statistical performances of the VSSI CV chart as the previous study only analyzes the chart’s statistical performance. By minimizing the cost via economic and economic-statistical designs, the VSSI CV chart can be implemented more economically. Statistical constraints are added in the economic-statistical design of the VSSI CV chart which makes it different from the economic design of the aforementioned chart. Numerical examples are given to illustrate the economic and economic-statistical performances of the VSSI CV chart, where a comparison with the SH CV chart is made. A study of misspecification of the shift size is also conducted to study the effect of wrongly specifying the shift size.

One-sided Downward Control Chart for Monitoring the Multivariate Coefficient of Variation with VSSI Strategy

In recent years, control charts monitoring the coefficient of variation (CV), denoted as the ratio of the variance to the mean, is attracting significant attention due to its ability to monitor processes in which the process mean and process variance are not independent of each other. However, very few studies have been done on charts to monitor downward process shifts, which is important since downward process shifts show process improvement. In view of the importance of today’s competitive manufacturing environment, this paper proposes a one-sided chart to monitor the downward multivariate CV (MCV) with variable sample size and sampling interval (VSSI), i.e. the VSSI D MCV chart. This paper monitors the MCV as most industrial processes simultaneously monitor at least two or more quality characteristics, while the VSSI feature is incorporated, as it is shown that this feature brings about a significant improvement of the chart. A Markov chain approach was adopted for designing a performance measure of the proposed chart. The numerical comparison revealed that the proposed chart outperformed existing MCV charts. The implementation of the VSSI D MCV chart is illustrated with an example.

An improved variable sample size and sampling interval S control chart

AbstractThe standard deviation chart (S chart) is used to monitor process variability. This paper proposes an upper‐sided improved variable sample size and sampling interval (VSSIt) S chart by improving the existing upper‐sided variable sample size and sampling interval (VSSI) S chart through the inclusion of an additional sampling interval. The optimal designs of the VSSIt S chart together with the competing charts under consideration, such as the VSSI S and exponentially weighted moving average (EWMA) S charts, by minimizing the out‐of‐control average time to signal (ATS1) and expected average time to signal (EATS1) criteria, are performed using the MATLAB programs. The performances of the standard S, VSSI S, EWMA S, and VSSIt S charts are compared, in terms of the ATS1 and EATS1 criteria, where the results show that the VSSIt S chart surpasses the other charts in detecting moderate and large shifts, while the EWMA S is the best performing chart in detecting small shifts. An illustrative example is given to explain the implementation of the VSSIt S chart.

New adaptive control charts for monitoring the multivariate coefficient of variation

Abstract An adaptive control chart is one of the most effective techniques in Statistical Process Control (SPC). The coefficient of variation (CV) is common in many real life applications, especially in manufacturing and materials engineering, finance, medical and biological sciences. This paper proposes three adaptive charts to monitor the multivariate coefficient of variation (MCV), in order to improve the sensitivity of the standard MCV chart, in detecting small and moderate MCV shifts. The proposed charts are designed using the Markov chain approach and they are compared with the existing standard MCV chart using the average time to signal (ATS), standard deviation of the time to signal (SDTS) and expected average time to signal (EATS) criteria. The performance comparison shows that the proposed adaptive MCV charts, particularly the variable sample size and sampling interval (VSSI) MCV chart, outperform the existing MCV charts, in terms of the ATS and EATS criteria. By allowing the sample size and sampling interval to be varied in the VSSI MCV chart, process engineers will have better control in process monitoring and at the same time are able to detect an out-of-control signal quicker. Illustrative examples are presented by considering the VSSI MCV chart to show the chart’s implementation.

EWMA chart with variable sample size and sampling interval for monitoring Pareto data with estimated parameters

AbstractThe Pareto distribution has been widely used to model data in numerous areas including actuarial science, climatology, economics, finance, and reliability. The exponential weighted moving average (EWMA) chart with variable sample size and sampling interval (VSSI) is quicker than EWMA chart, EWMA chart with variable sample size (VSS), and EWMA chart with variable sampling interval (VSI) in detecting the mean shifts of a normal process. In this study, we present an EWMA‐VSSI chart to monitor the Pareto data with estimated parameters. The average time to signal (ATS) is computed by using the Markov chain approach. The results indicate that the proposed chart performs better than the other EWMA charts with fixed sample size, VSS, and VSI in terms of ATS. The design parameters of the proposed control charts are given in order to expedite the use of the chart by practitioners. Two examples are used to illustrate the application of the proposed control charts.

VSSI median control chart with estimated parameters and measurement errors

AbstractAn adaptive control chart called Shewhart chart with variable sample size and sampling interval (VSSI) is quicker than Shewhart chart, chart with variable sample size (VSS), and chart with variable sampling interval (VSI) in detecting the mean shifts of a normal process. In practice, the effects of measurement errors on control charts should be included. In this study, we present an VSSI median control chart with estimated parameters in the presence of measurement errors for a normal process. The average time to signal (ATS) is computed by using the Markov chain approach. The results show that the VSSI median control chart performs better than the Shewhart median, VSS median, and VSI median control charts in terms of ATS. The design parameters of the proposed chart are provided. Two examples are used to illustrate the application of the proposed control chart.

Economic-statistical Design of NP Control Chart with Variable Sample Size and Sampling Interval

The control charts are graphical tools and proven techniques to improve the performance of a process. Usually, the processes are not naturally controlled, so the use of control charts will help to reduce the variability and increase the stability of the process. In the traditional approach, control charts with fix sample size and constant sampling intervals were used to identify the changes in the process. While, control charts can be examined from different statistical and economical directions using different sampling schemes for achieving a better result. In past studies, it has been shown that np control charts show better results in detecting shifts by using variable sampling schemes but it is also important to consider how the cost of using variable sampling schemes will be, since the cost of a process depends on the parameters of the control chart. In this paper, the economic-statistical design of np control chart with variable sample size and sampling interval (VSSI) is formulated and then we compare and analyze the resulted obtained with other schemes. Results show significant improvement in terms of economic and statistical performance.

Adaptive control charts for skew‐normal distribution

AbstractThe standard Shewhart‐type chart, named FSS‐ chart, has been widely used to detect the mean shift of process by implementing fixed sample and sampling frequency schemes. The FSS‐ chart could be sensitive to the normality assumption and is inefficient to catch small or moderate shifts in the process mean. To monitor nonnormally distributed variables, Li et al [Commun Stat‐Theory Meth. 2014; 43(23):4908‐4924] extended the study of Tsai [Int J Reliab Qual Saf Eng. 2007; 14(1):49‐63] to provide a new skew‐normal FSS‐ (SN FSS‐ ) chart with exact control limits for the SN distribution. To enhance the sensitivity of the SN FSS‐ chart on detecting small or moderate mean shifts in the process, adaptive charts with variable sampling interval (VSI), variable sample size (VSS), and variable sample size and sampling interval (VSSI) are introduced for the SN distribution in this study. The proposed adaptive control charts include the normality adaptive charts as special cases. Simulation results show that all the proposed SN VSI‐ , SN VSS‐ , and SN VSSI‐ charts outperform the SN FSS‐ chart on detecting small or moderate shifts in the process mean. The impact of model misspecification on using the proposed adaptive charts and the sample size impact for using the FSS‐ chart to monitor the mean of SN data are also discussed. An example about single hue value in polarizer manufacturing process is used to illustrate the applications of the proposed adaptive charts.

A variable sample size and sampling interval control chart for monitoring the process mean using auxiliary information

A variable sample size and sampling interval (VSSI) control chart using the auxiliary information (VSSI AI) is proposed for efficiently monitoring the process mean. The control chart’s statistics, optimal design and implementation are discussed. The average time to signal (ATS) and expected ATS (EATS) criteria are adopted to evaluate the performance of the VSSI AI chart. The ATS and EATS performances of the VSSI AI chart, exponentially weighted moving average (EWMA) chart, EWMA chart with AI (EWMA AI) and the synthetic chart with AI (Syn AI) are compared and explored for common patterns. The proposed VSSI AI chart shows a better performance than the EWMA, EWMA AI and Syn AI charts, in terms of the ATS and EATS criteria. An example is also presented to illustrate the implementation of the VSSI AI chart.

The Effect of Measurement Errors on the Performance of Variable Sample Size and Sampling Interval Control Chart

The effect of measurement errors on adaptive and non-adaptive control charts has been occasionally considered by researchers throughout the years. However, that effect on the variable sample size and sampling interval (VSSI) control charts has not so far been investigated. In this paper, we evaluate the effect of measurement errors on the VSSI control charts. After a model development, the effect of measurement errors and multiple measurements on the performance of VSSI scheme are evaluated in terms of the out-of-control average time to signal (ATS) criterion, which is obtained using a Markov Chain approach. At last, a real case is presented to show the application of the proposed scheme.

Monitoring the coefficient of variation using a variable sample size and sampling interval control chart

ABSTRACTThis article proposes a CV chart by using the variable sample size and sampling interval (VSSI) feature to improve the performance of the basic CV chart, for detecting small and moderate shifts in the CV. The proposed VSSI CV chart is designed by allowing the sample size and the sampling interval to vary. The VSSI CV chart's statistical performance is measured by using the average time to signal (ATS) and expected average time to signal (EATS) criteria and is compared with that of existing CV charts. The Markov chain approach is employed in the design of the chart.

A modified variable sample size and sampling interval controlchart

In this article, a modified version of variable sample size and sampling interval chart using three sampling intervals and two sample sizes, called the VSSIt control chart, is proposed to overcome the weakness of the standard Shewhart (SS) control chart in detecting small and moderate shifts. In order to evaluate the performance of the proposed scheme, average time to signal (ATS), average number of observations to signal (ANOS), and average number of samples to signal (ANSS) are used. Using this control chart can reduce the ATS value around 68, 27, 7, and 1.83 time unit (sampling interval of the SS chart) in detecting shift of size 0.25, 0.5, 0.75, and 1 standard deviation, respectively. The performance of the constructed adaptive chart is compared with various adaptive charts such as variable sample size (VSS), variable sampling interval (VSI), variable sample size and sampling interval (VSSI), and special variable sample size and sampling interval (SVSSI) \( \overline{x} \) control chart. It is shown that the proposed chart always outperforms SS, VSI, VSS, and VSSI schemes. It also can detect shift of size 0.25 standard deviation quicker than SVSSI scheme. Finally, the optimal parameters of the proposed chart are recommended.

A New EWMA Loss Control Chart with Adaptive Control Scheme

A single chart, instead of and R charts or and S charts, to simultaneously monitor the process mean and variability would reduce the required time and effort. A number of studies have attempted to find such charts. Moreover, a number of studies demonstrated that the adaptive control charts may detect process shifts faster than the fixed control charts. This paper proposes the EWMA loss chart with variable sample sizes and sampling intervals (VSSI) to effectively monitor the difference of process measurements and target. An example is used to illustrate the application and performance of the proposed control chart in detecting the changes in the difference of the process measurements and target. Numerical analyses demonstrated that the VSSI EWMA loss chart outperforms the fixed sampling interval EWMA average loss chart and the Shewhart joint and S charts. Therefore, the VSSI EWMA loss chart is recommended.

VARIABLE SAMPLE SIZE AND SAMPLING INTERVAL HOTELLING'S T2 CHARTS WITH RUNS RULES FOR SWITCHING BETWEEN SAMPLE SIZES AND SAMPLING INTERVAL LENGTHS

Variable sample size and sampling interval (VSSI) T2 charts are substantially more efficient than static T2 charts. However, the frequent switches between sample sizes and sampling interval lengths can be a complicating factor during the implementation of these charts. In this paper, runs rules are proposed for switching between sample sizes and sampling interval lengths of VSSI T2 charts in order to reduce the frequency of switches. The expressions for performance measures for the charts with these runs rules are developed. The effects of different runs rules on performances of the charts are evaluated numerically. In general, runs rules substantially reduce the frequency of switches. However, some runs rules significantly affect statistical performances of the charts.

The economic design of multivariate binomial EWMA VSSI control charts

Since multi-attribute control charts have received little attention compared with multivariate variable control charts, this research is concerned with developing a new methodology to employ the multivariate exponentially weighted moving average (MEWMA) charts for m-attribute binomial processes; the attributes being the number of nonconforming items. Moreover, since the variable sample size and sampling interval (VSSI) MEWMA charts detect small process mean shifts faster than the traditional MEWMA, an economic design of the VSSI MEWMA chart is proposed to obtain the optimum design parameters of the chart. The sample size, the sampling interval, and the warning/action limit coefficients are obtained using a genetic algorithm such that the expected total cost per hour is minimized. At the end, a sensitivity analysis has been carried out to investigate the effects of the cost and the model parameters on the solution of the economic design of the VSSI MEWMA chart.