Variable Sampling Interval Charts Research Articles

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.

Detecting a shift in variance using economically designed VSI control chart with combined attribute-variable inspection

Recently, a MIX S 2 control chart using both attribute and variable inspections is proposed by Ho and Quinino (2016) to monitor the process variability. This paper considers the variable sampling interval (VSI) MIX S 2 control chart in which the sampling interval between two successive sampling points is varied based on the value of the preceding complementary sample variance to improve the performance of the original MIX S 2 control chart. Additionally, a unified economical design model for the process is established with the cost function derived. The performance of the VSI chart is compared to that of the original static chart. The numerical results show that comparing to original MIX S 2 control chart, the VSI chart yields better performance in terms of less expected cost per unit of time.

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

ABSTRACTIn 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.

The variable sampling interval S2 chart with known or unknown in-control variance

The chart has been known as a powerful tool to monitor the variability of the normal process. This paper considers the variable sampling interval (VSI) chart to improve the performance of the original chart. A VSI chart based on the average run length (ARL) unbiased and average time to signal (ATS) unbiased criteria is developed when the in-control variance is known. The performance of the proposed chart is investigated in terms of the mean and standard deviation of the RL and total time to signal. The impact of parameter estimation on the proposed VSI chart is also evaluated. The results show that the performance of the control chart is seriously different from the known parameter case when the parameter is estimated. Then, a new VSI chart with the desired in-control ARL and ATS is considered when the in-control variance is estimated. In evaluating the usefulness of the VSI feature, the VSI charts are compared with the fixed sampling interval (FSI) versions in terms of their statistical properties. The comparison results show that the VSI chart can detect the shifts in the process variance quicker than the corresponding FSI chart, regardless of whether the in-control variance is known or unknown. Finally, two examples are used to illustrate the proposed control chart.

Comparison of two sampling intervals and three sampling intervals VSI charts for monitoring both means and variances

In industrial quality control, when engineers use VSI control procedure they should consider both required time to signal and switching behaviors together in the case of production process changed. Up to the present, many researchers have studied fixed sampling interval (FSI) chart and variable sampling interval (VSI) chart in the points of average number of samples to signal (ANSS) and average time to signal (ATS). However, ANSS and ATS do not provide any switching information between different sampling intervals of VSI schemes. In this study, performances of two sampling intervals VSI chart and three sampling intervals VSI chart are evaluated and compared. The numerical results show that ANSS and ATS values of two sampling intervals VSI chart and three sampling interval VSI chart are similar regardless the amount of shifts. However, the values of switching behaviors including ANSW are less efficient in three sampling intervals VSI charts than in two sampling intervals VSI chart.

The variable sampling interval control chart for finite-horizon processes

The requirement to be globally competitive requires companies to have a high level of flexibility to allow for the production of a large variety of products. To limit work-in-process, decision makers periodically schedule according to a make-to-order management strategy i.e. the production of finite batches of the same product code. Scheduling calls for frequent set up activities, which require the reconfiguration of a manufacturing process, and allows manufacturers to switch between different codes. This can limit the production horizon of one product code to a few hours or shifts. In this context, efficient online quality control monitoring using control charts is strategic to eliminate scrap or rework and to meet the demand at the time specified by the production plan. The design of control charts for a process with a limited production horizon is a challenge for statistical process control practitioners. Under this framework, this article investigates the issues related to the implementation of the Variable Sampling Interval (VSI) Shewhart control chart in a process with finite production horizon. When the production horizon is finite, the statistical properties of a control chart are known to be a function of the number of scheduled inspections. In the case of a VSI control chart, the quality practitioner cannot fix the number of inspections a priori due to the stochastic nature of the sampling interval selection. Therefore, the aim of this article is to propose a new Markov chain approach for the exact computation of the statistical performance of the VSI control chart in processes with an unknown and finite number of inspections. The proposed approach is general and does not depend on the monitored sample statistic. With reference to the process mean monitoring, an extensive numerical analysis compares the performance of the VSI chart to the Variable Sample Size and Fixed Sampling Rate z charts. Numerical results show that the VSI chart outperforms other charts for moderate to large shift sizes. An illustrative example shows the implementation of the VSI chart in a short run producing a finite batch of mechanical parts.

Economic design of attribute np control charts using a variable sampling policy

To use a control chart, the quality engineer should specify three decision variables, namely the sample size, the sampling interval and the critical region of the chart. A significant part of recent research relaxed the constraint of using fixed design parameters to open the way to a new type of control charts called adaptive ones where at least one of the decision variables may change in real time based on the last data information. These adaptive schemes have proven their effectiveness from economical and statistical point of views. In this paper, the economic design of an attribute np control chart using a variable sampling interval (VSI) is treated. A sensitivity analysis is conducted to search for optimal design parameters minimizing the expected total cost per hour and to reveal the impact of the process and cost parameters on the behavior of optimal solutions. An economic comparison between the classical np chart, variable sample size (VSS) np control chart and VSI chart is conducted. It is found that switching from the classical attribute chart to the VSI sampling strategy results in notable cost savings and in reduction of the average time to signal and the average number of false alarms. In most cases of the sensitivity analysis, the VSI np chart outperforms the VSS np chart based on economical and statistical considerations. Copyright © 2014 John Wiley & Sons, Ltd.

가중표준편차를 이용한 가변표본채취간격 X 관 **리도

This article proposes a variable sampling interval (VSI) <TEX>$\bar{X}$</TEX> control chart using weighted standard deviation (WSD) method for skewed populations. The WSD method decomposes the standard deviation of a quality characteristic into upper and lower deviations and adjusts control limits and warning limits of a control chart in accordance with the direction and degree of skewness. A control chart constant is derived for estimating the standard deviation of skewed distributions with the mean of sample standard deviations. The proposed chart is compared with the conventional VSI <TEX>$\bar{X}$</TEX> control chart under some skewed distributions. Simulation study shows that the proposed WSD VSI chart can control the in-control average time to signal (ATS) as an adequate level better than the conventional VSI chart, and the proposed chart can detect a decrease in the process mean of a quality characteristic following a positively skewed distribution more quickly than the standard VSI chart.

Exact Results for Variable Sampling Interval Shewhart Control Charts with Runs Rules for Switching Between Sampling Interval Lengths

The exact expressions for the performance measures for variable sampling interval charts in which runs rules are used for switching between the sampling interval lengths are developed. The methods presented are general and can be applied to other Shewhart control charts. The effects of different runs rules on the performances of the charts are evaluated through numerical comparisons. Recommendations of some runs rules are made as per the requirements of the process monitoring.

A special variable sample size and sampling interval Hotelling’s T 2 chart

Among the variable sampling interval (VSI), variable sample size (VSS), and variable sample size and sampling interval (VSSI) T2 charts that use two values of each adaptive parameter, a VSI T2 chart is the most efficient chart to detect large shifts in the process mean vector while a VSSI T2 chart is the most efficient one to detect small shifts. The statistical performance of the T2 chart proposed in this paper very closely matches that of such VSI and VSSI T2 charts for detecting the large and small shifts, respectively. The proposed chart uses two values of the sampling interval and three values of the sample size.

Using VSI Loss Control Charts to Monitor a Process with Incorrect Adjustment

The article considers the optimum variables control scheme for a process with incorrect adjustment. Incorrect adjustment of a process may result in shifts in process mean and/or variance, ultimately affecting the quality of products and creating a loss. For the process with incorrect adjustment, we construct the variable sampling interval (VSI) and Z S 2 control charts to monitor the shifts in the process mean and variance and with minimum loss. The minimum loss variable sampling interval control charts is measured by the average loss per unit time derived by a Markov chain approach. An example is given to show the application and the performance of the proposed control charts. Furthermore, the comparison of the performance of the VSI loss control charts and the fixed sampling interval (FSI) loss control charts suggested that the VSI charts outperformed that of FSI charts.

A Special Variable Sample Size and Sampling Interval Chart

Among the variable sampling interval (VSI), variable sample size (VSS), and variable sample size and sampling interval (VSSI) charts that use two values of each adaptive parameter, a VSI chart is the best chart to detect large shifts in the process mean while a VSSI chart is the best one to detect small shifts. Neither of the two is uniformly superior to the other for detecting a shift of any magnitude over a wide range. We propose a special VSSI (SVSSI) chart whose statistical performance very closely matches that of such VSI and VSSI charts for detecting large and small shifts, respectively. The three charts perform almost similarly for detecting moderate shifts. An SVSSI chart uses two sampling intervals and three sample sizes. A reasonable procedure is described to obtain its optimal design. Guidelines are provided to choose a design that is fairly close to the optimal one. Besides this, we introduce a new measure of statistical performance of a control chart which unifies the two measures in the existing literature that measure the average shift detection time of a chart with different approaches.

A proposed switching rule for variable sampling interval charts using moving average

Theoretical studies demonstrated that the variable sampling interval (VSI) charts give substantially faster detection of most process mean shifts than traditional charts. The standard VSI chart allows the long and short sampling intervals to be interchangeably used in the practical operation of the chart based on the information of the last sample mean. However, frequent switching between the long and short sampling intervals can be a complicating factor in the application of control charts. Some switching rules have been presented and studied in the literature to improve the situation of frequent switching. In the present paper, a switching rule based on moving average (MA) is proposed for the VSI chart. The advantages of moving average are not only to involve the recent several sample means, but also to consider the magnitude of deviation of these sample means from the centerline, which is not included by the switching rules in the literature. A computer simulation approach is applied to evaluate the performance of the proposed VSI chart with MA rule. From the simulation results, it can be seen that for a small process mean shift, the VSI chart with MA rule is always superior to the other types of VSI charts.

Monitoring autocorrelated processes using variable sampling schemes at fixed‐times

AbstractTraditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. A recent development in control charts is the use of adaptive control charts, i.e. the variable sampling interval (VSI) and variable sampling size charts. This paper extends this idea to the autocorrelated process. We consider a time series model which is a first‐order autoregressive process plus a random error. With variable intervals, the sampling time may be inconvenient, so using only two intervals, referred to as ‘variable sampling interval at fix times’ makes the method easier to use in practice. The sampling rate can also be adjusted by the number of samples collected, VSRFT, for ‘variable sampling rate at fixed times’. We study what we call ‘variable sampling at fixed times’, VSFT, which includes both VSIFT and VSRFT schemes, using a Markov chain model and integral equations. We show that our methods detect process shifts faster, on average, than fixed sampling X‐bar charts, and at least comparable detection ability with the less practical VSI charts. Copyright © 2007 John Wiley &amp; Sons, Ltd.

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.

Optimal Design of VSI ―X Control Charts for Monitoring Correlated Samples

This paper develops an economic design of variable sampling interval (VSI)―X control charts in which the next sample is taken sooner than usual if there is an indication that the process is off-target. When designing VSI―X control charts, the underlying assumption is that the measurements within a sample are independent. However, there are many practical situations that violate this hypothesis. Accordingly, a cost model combining the multivariate normal distribution model given by Yang and Hancock with Bai and Lee's cost model is proposed to develop the design of VSI charts for correlated data. An evolutionary search method to find the optimal design parameters for this model is presented. Also, we compare VSI and traditional ―X charts with respect to expected cost per unit time, utilizing hypothetical cost and process parameters as well as various correlation coefficients. The results indicate that VSI control charts outperform the traditional control charts for larger mean shift when correlation is present. In addition, there is a difference between the design parameters of VSI charts when correlation is present or absent. Copyright © 2005 John Wiley & Sons, Ltd.

Optimal Design of the Adaptive Sample Size and Sampling Interval np Control Chart

AbstractRecent research has shown that the adaptive control charts are quicker than the traditional static charts in detecting process shifts. This paper develops the algorithm for the optimization designs of the adaptive np control charts for monitoring the process fraction non‐conforming p. It includes the variable sample size chart, the variable sampling interval chart, and the variable sample size and sampling interval chart. The performance of the adaptive np charts is measured by the average time to signal under the steady‐state mode, which allows the shift in p to occur at any time, even during the sampling inspection. By studying the performance of the adaptive np charts systematically, it is found that they do improve effectiveness significantly, especially for detecting small or moderate process shifts. Copyright © 2004 John Wiley &amp; Sons, Ltd.

An evolutionary economic-statistical design for VSI X control charts under non-normality

Since Reynolds et al. [1] proposed the variable sampling interval (VSI) X control chart, the statistical properties of the chart have been discussed and shown to be better than the traditional control chart in which the sampling intervals length are fixed. However, the VSI charts are, like the traditional charts, still costly when they are used for the prevention of defective products. For this reason, an appropriate design is necessary before it is used. In this paper, a VSI X control chart is used for monitoring non-normal process data, which is widely encountered in practice. Then, a cost-quality model based on the Burr distribution is proposed for constructing an economic-statistical design for the VSI X chart. The design of the chart has been developed considering the optimisation of the cost function in this model posed by statistical constraints. The design parameters can be derived through the evolutionary search method, and the overall finding indicates that the designed VSI chart always outperforms the traditional control chart with respect to the expected cost per unit time. This model is also suitable for normally distributed data; thus, it is suitable for a general application.

An enhancement of DSI X� control charts using a fuzzy-genetic approach

A traditional control chart used to monitor a process draws the process data at a fixed sampling rate, while a variable sampling interval (VSI) control chart varies the sampling rate as a function of on-line process data. In such a sampling policy, a higher sampling rate is adopted when there is suspicion of a change in a process. Therefore, it is able to detect the process change faster than traditional control chart, and thus has been much accepted for use. Nevertheless, the binary suspicious grade used in VSI policy to specify the sampling rate is not detailed enough to explain the acquired information from process data. As a result, this paper aims to refine the suspicious grade and sampling interval lengths to increase the detection ability of VSI charts. This study first establishes a composition function on two sides of the control chart by introducing the concept of fuzzily suspicious grade to specify the sampling rate. Then, genetic algorithms (GAs) is used to adjust the values of the parameters in this composition function to enhance the dual-sampling-interval (DSI) charts-one type of the VSI charts in common use-in terms of average time to signal (ATS) for process mean shift. In addition, some statistical properties of the enhanced DSI charts as well as performance comparison to traditional DSI charts are provided and analysed.

Steady-state-optimal adaptive control charts based on variable sampling intervals

Variable sampling interval (VSI) control charts vary the sampling rate adaptively as a function of the data coming from the process in order to reduce the detection delay of process changes. Zero-time performance refers to the detection delay of a process change that is present during the onset of the chart at time zero. Steady-state performance refers to the detection delay of a process change that occurs after the chart has been operating for some time. The zero-time performance of a VSI control chart can differ considerably from the chart's steady-state performance, which is generally more important than the zero-time performance. We develop an efficient quadratic-programming algorithm for the construction and investigation of steady-state-optimal sampling policies for various VSI charts. We show that a steady-state-optimal VSI scheme is fundamentally different from the respective zero-time-optimal VSI scheme, and recommend VSI policies based on two sampling intervals for the various types of control charts considered.