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

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.

In order to improve the monitoring efficiency of the multivariable autocorrelation process and reduce the cost of process control, an economic design method of residual Multivariate Exponentially Weighted Moving Average (MEWMA) control charts with variable sample size and sampling intervals (VSSIs) was proposed. First, residual MEWMA control charts with VSSI were designed for monitoring multivariable autocorrelation processes. Second, the economic model of the VSSI residual MEWMA control chart was established, and a data example was given. The optimal design parameters of the control chart were obtained by using a genetic algorithm to minimize the expected cost per unit time. Third, using orthogonal experimental design and regression analysis, a sensitivity analysis of the economic model was carried out to get the relationship between the model parameters and the design parameters of the control chart. Finally, through the optimality analysis, it was verified that the proposed economic model was superior to the models established by the variable sampling intervals strategy and the variable sample size strategy and had less expected cost per unit time.

The 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.

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.

On the design of variable sample size and sampling intervals [formula omitted] charts under non-normality

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.

Adaptive sampling enhancement for Hotelling’s T2 charts

The deployment of statistical process control (SPC) in service process is a prominent global phenomenon in recent years. In this paper, we proposed the optimal variable sample size and sampling interval (VSSI) mean square error (MSE) control chart to monitor the difference between the process mean and the target value and the process variation shifts, and its performance is measured by the average time to signal (ATS). From data analyses, we found the optimal VSSI MSE chart that performs better than the specific VSSI and optimal variable sampling interval (VSI) and fixed sample size and sampling interval (FSSI) MSE chart. An example was given to illustrate this new proposed approach.

Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. Variable sample size (VSS) and variable sampling interval (VSI) control charts vary the sampling rate from the process as a function of the data from the process. By sampling at a higher rate when there is an indication of a change in the process, VSS and VSI control charts can detect process changes faster than traditional control charts. Previous research has considered the properties of CUSUM charts which use the VSI feature. This paper considers CUSUM charts with the VSS feature and with both the VSS and VSI features. Two ways of developing the control statistic of these charts are considered. It is shown that using either the VSS or VSI feature in a CUSUM control chart will improve the ability to detect all but very large process shifts. The VSI feature usually gives more improvement in detection ability than the VSS feature, but using both features together will give more improvement than either one separately. Guidelines are given for choosing the possible sample sizes and the possible sampling intervals for these charts. Methods for setting up these charts for practical applications are also given.

Recent 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 & Sons, Ltd.

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.

The deployment of statistical process control in the service process is a prominent global phenomenon in recent years. In this paper, we proposed the optimal variable sample size and sampling interval (VSSI) mean square error (MSE) control chart to monitor the difference between the process mean and the target value and the process variation shifts, and its performance is measured by the average time to signal. From data analyses, we found the optimal VSSI MSE chart that performs better than the specific VSSI and the optimal variable sampling interval and the fixed parameters MSE charts. An example was given to illustrate this new proposed approach.

In this paper, improved Shewhart control charts based on hybrid adaptive and run rule schemes are introduced to enhance the statistical performances of the traditional static scheme, designed with consideration given to the fixed values of sample size, the width of the control limits and the sampling frequency. The proposed hybrid adaptive schemes consider both variable sampling interval and variable sample size combined with run rules. The objective of this research is to develop a statistical comparison between adaptive schemes, charts with run rules and hybrid adaptive schemes with run rules to help decision-makers in the selection of the best performing chart for an expected value of shift in the mean of a controlled parameter. An extensive set of numerical results is presented to test the effectiveness of the proposed models in detecting small and moderate shifts in the process mean. The optimal statistical designs of the charts are obtained through a heuristic algorithm, properly modified to cope with the problem.

Traditionally, the statistical quality control techniques utilize either an attributes or variables product quality measure. Recently, some methods such as three-level control chart have been developed for monitoring multi attribute processes. Control chart usually has three design parameters: the sample size (n), the sampling interval (h) and the control limit coefficient (k).The design parameters of the control chart are generally specified according to statistical or/and economic criteria. The variable sampling interval (VSI) control scheme has been shown to provide an increase to the detecting efficiency of the control chart with fixed sampling rate (FRS). In this paper a method is proposed to conduct the economic-statistical design for variable sampling interval of the three-level control charts. We use the cost model developed by Costa and Rahim and optimize this model by genetic algorithm approach. We compare the expected cost per unit time of the VSI and FRS 3-level control charts. Results indicate that the proposed chart has improved performance.

The ability of a control chart to detect a shift in the process can be determined by the average run length (ARL), a widely used performance indicator for single sampling interval (SSI) control charts. When it comes to variable sampling interval (VSI) schemes, average time to signal (ATS) is utilized instead of ARL. Although analysis of ARL has been extensively studied for SSI schemes, calculation of the ATS for VSI control charts is still in its infancy due to several theoretical challenges. In this context, the paper derives closed-form expressions for time to signal distribution and the ATS for a spatial design of VSI multivariate Bayesian control chart referred to as the dual sampling scheme (DSS). In a DSS strategy, the process is being monitored initially with a longer sampling interval, which is changed to the shorter sampling interval at a designed switching point. Unlike previous approximate developments where the switching point is ignored, the paper derives exact expressions for computation of time to signal distribution and ATS within such a Bayesian framework. Derivations of the ATS are based on an intuitively pleasing definition of an artificial absorbing state and coded values of the posterior probability using Markov chain theory.