Economic Statistical Design of the Variable Sampling Interval CV Chart with Known Shift Size

Process monitoring is essential for maintaining a stable production process. The triple sampling (TS) charts have been extensively investigated to monitor the process mean. However, the TS charts are unable to reliably detect process mean shifts when the mean and standard deviation are proportional to one another. To address this setback, in this study, a TS coefficient of variation (CV) chart which improves upon the double sampling (DS) CV chart's performance in detecting shifts in the process CV is developed. Optimization procedures in computing the optimal charting parameters by minimizing the out‐of‐control average number of observations to signal (ANOS), average run length (ARL), expected ANOS (EANOS) and expected ARL (EARL) values are provided. Tables of optimal parameters are given in this study to facilitate the quality engineer in implementing the TS CV chart. The findings reveal that the TS CV chart outperforms the DS CV chart in detecting all sizes of CV shifts. A real dataset involving the NPK fertilizer production process is used to illustrate the implementation of the TS CV chart.

An efficient process monitoring system is important for achieving sustainable manufacturing. The control charting technique is one of the most effective techniques to monitor process quality. In certain processes where the process mean and variance are not independent of one another, the coefficient of variation (CV), which measures the ratio of the standard deviation to the mean should be monitored. Castagliola et al. (2013a) proposed the two-sided run rules (RR) control charts for monitoring the CV and it is found that the RR CV charts revealed the problem of ARL-biased performances, especially when the monitored sample size is small, for detecting downward CV shifts. This paper alters the RR CV chart by suggesting the two one-sided run rules (ORR) CV charts achieve the unbiased ARL performances. Additionally, this paper also investigates the ORR CV charts in terms of the expected average run length (EARL) criterion, which is not discussed in Castagliola et al. (2013a). A Markov chain model is established for designing the proposed charts. The statistical performances of the ORR CV, RR CV and Shewhart CV (SH CV) charts are compared in terms of the average run length (ARL) and EARL criteria. The results show that the proposed charts surpass the RR CV and SH CV charts for detecting small and moderate upward and downward CV shifts. The implementation of the ORR CV charts is illustrated with an example using a real dataset.

A proper method of monitoring a stochastic system is to use the control charts of statistical process control in which a drift in characteristics of output may be due to one or several assignable causes. In the establishment of X charts in statistical process control, an assumption is made that there is no correlation within the samples. However, in practice, there are many cases where the correlation does exist within the samples. It would be more appropriate to assume that each sample is a realization of a multivariate normal random vector. Using three di erent loss functions in the concept of quality control charts with economic and economic statistical design leads to better decisions in the industry. Although some research works have considered the economic design of control charts under single assignable cause and correlated data, the economic statistical design of X control chart for multiple assignable causes and correlated data under Weibull shock model with three di erent loss functions have not been presented yet. Based on the optimization of the average cost per unit of time and taking into account the di erent combination values of Weibull distribution parameters, optimal design values of sample size, sampling interval and control limit coecient were derived and calculated. Then the cost models under non-uniform and uniform sampling scheme were compared. The results revealed that the model under multiple assignable causes with correlated samples with non-uniform sampling integrated with three di erent loss functions has a lower cost than the model with uniform sampling.

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.

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.

A proper monitoring of stochastic systems is the control charts of statistical process control and drift in characteristics of output may be due to one or several assignable causes. Although many research works have been done on the economic design of control charts with single assignable cause, the economic statistical design of T^2 control chart under Weibull shock model with multiple assignable causes and considering multivariate Taguchi loss function has not been presented yet. Using Taguchi loss function in the concept of quality control charts with economic and economic statistical design leads to better decisions in the industry. Based on the optimization of the average cost per unit of time and taking into account the different combination values of Weibull distribution parameters, optimal design values ??of sample size, sampling interval and control limit coefficient were derived and calculated. Then the cost models under non-uniform and uniform sampling scheme were compared. The results revealed that the model under multiple assignable causes with Taguchi loss function has a lower cost than single assignable cause model and integrated model with non-uniform sampling has a lower cost than that with uniform sampling.

The coefficient of variation (CV) is one of the most important quality characteristics in industrial applications. There have been some control charts for monitoring this dimensionless and normalized characteristic. This paper presents three double exponentially weighted moving average (DEWMA) charts for monitoring the CV. Many numerical simulations are performed and the results are given in several tables to show the properties of the proposed DEWMA CV charts. Through comparisons with other existing exponentially weighted moving average (EWMA) CV charts using the average run length (ARL) metric, it is shown that the proposed DEWMA CV charts perform better than the existing EWMA CV charts for small shifts, especially when the smoothing parameter increases. In addition, a detailed comparison among the proposed three DEWMA CV charts is also presented. A real date example from industry is used to show the implementation of the proposed charts.

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.

In many cases, a $\overline X $ control chart based on a performance variable is used in industrial fields. Typically, the control chart monitors the measurements of a performance variable itself. However, if the performance variable is too costly or impossible to measure, and a less expensive surrogate variable is available, the process may be more efficiently controlled using surrogate variables. In this paper, we present a model for the economic statistical design of a VSI (Variable Sampling Interval) $\overline X $ control chart using a surrogate variable that is linearly correlated with the performance variable. We derive the total average profit model from an economic viewpoint and apply the model to a Very High Temperature Reactor (VHTR) nuclear fuel measurement system and derive the optimal result using genetic algorithms. Compared with the control chart based on a performance variable, the proposed model gives a larger expected net income per unit of time in the long-run if the correlation between the performance variable and the surrogate variable is relatively high. The proposed model was confined to the sample mean control chart under the assumption that a single assignable cause occurs according to the Poisson process. However, the model may also be extended to other types of control charts using a single or multiple assignable cause assumptions such as VSS (Variable Sample Size) $\overline X $ control chart, EWMA, CUSUM charts and so on.

The cumulative conformance count (CCC) charts have been shown to be useful in high-quality processes. The CCC-charts regarded every individual item inspected as a sample and called the time between two successive samples the sampling interval. Typically, the length of sampling interval they employed is constant and fixed (FSI). Recently, the use of CCC-charts with the variable sampling interval (VSI) scheme has been proposed to enhance the efficiency of CCC-charts without increasing the rate of inspected items and false alarm occurrences. This paper presents an economic model for designing the VSI CCC-chart. In the economic design, a cost function is constructed, involving the cost of sampling and testing, the cost of false alarms, the cost to detect and remove the assignable cause, and the cost when the process is operating out-of-control. A heuristic method of finding optimal values of adaptive design parameters for minimizing the cost function is also presented. The VSI and FSI CCC-charts are compared with respect to the expected loss per unit time. In addition, the effects of process and cost parameters upon the operating cost and the design parameters for charting are discussed.

Controlling over-adjusted process means and variances using VSI cause selecting control charts

In this paper, a new adaptive control chart is presented to study the relative process variability, which is called the coefficient of variation (CV). The CV study is the monitoring of a process with an unstable mean value over time. So, a function‐based adaptive exponentially weighted moving average (FAEWMA) CV control chart is proposed to detect the infrequent changes in the process CV, named as “FAEWMA CV” control chart. The function is suggested to adapt the smoothing constant (SC) of the plotting statistic as per arising process shift size in CV. The efficiency is examined through the run‐length values, which are computed through the Monte Carlo (MC) simulation method. The FAEWMA CV chart outperformed the existing AEWMA CV charts. A real‐life example is studied to explain implementation details in the example section.

ABSTRACTControl charts show the distinction between the random and assignable causes of variation in a process. The real process may be affected by many characteristics and several assignable causes. Therefore, the economic statistical design of multiple control chart under Burr XII shock model with multiple assignable causes can be an appropriate candidate model. In this paper, we develop a cost model based on the optimization of the average cost per unit of time. Indeed, the cost model under the influence of a single match case assignable cause and multiple assignable causes under a same cost and time parameters were compared. Besides, a sensitivity analysis was also presented in which the changeability of loss-cost and design parameters were evaluated based on the changes in cost, time and Burr XII distribution parameters.

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 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.