On Designing a Triple Sampling Coefficient of Variation Chart

Control charts are essential in Statistical Process Control (SPC) to enhance the product quality by identifying assignable causes within the process. Extensive study on coefficient of variation (CV) charts has been conducted in recent years, indicating their growing significance in a variety of scientific applications. In order to monitor the quality characteristic, it should be noted that the statistical properties and economic perspectives are both indispensable factors. For the economic design, the optimal design parameters that minimise the expected cost function are obtained. However, this design has a major weakness, where it overlooks the statistical performance of the control chart. Therefore, it is of greater importance to examine the economic statistical design, which includes statistical constraints on the cost model. Although the existing variable sampling interval (VSI) CV chart has excellent statistical qualities, it is deficient in taking into account the economic aspects that are necessary for cost-effective process monitoring. In response, this research proposes the economic statistical design of the variable sampling interval (VSI) CV chart with known shift size. Moreover, the study includes sensitivity analyses of the optimal cost and the optimal design parameters for various input parameters. The effects of misspecification of the shift size on the performance of the VSI CV chart are investigated. Additionally, the comparisons in terms of the economic statistical performance between the VSI CV, Shewhart CV and EWMA CV charts are conducted. In comparison with Shewhart CV and EWMA CV charts, the findings clearly demonstrate that the VSI CV chart performs better in all numerical examples. As a result, this research advocates the use of the economic statistical design of the VSI CV chart in process monitoring and provides insightful information for quality practitioners seeking cost-effectiveness and optimal performance in their processes.

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.

The quality of products or services plays an important role towards customers’ level of satisfaction. An efficient way to maintain quality is by utilizing control charts, which is preferred by practitioners due to its ease of use and efficiency in detecting changes in a process. These changes affect the quality of products or services. Recently, control charts monitoring the coefficient of variation (CV) are proposed to monitor processes with an inconsistent mean and/or variance. These processes cannot be monitored by standard mean and/or standard deviation type charts. Since the first CV chart was proposed, numerous improvements were done, one of which is the Variable Sample Size (VSS) Exponentially Weighted Moving Average (EWMA) chart. The VSS EWMA CV chart is shown to be efficient in detecting changes in the CV and outperforms existing charts in the literature. However, for unknown shift sizes, τ, the existing VSS EWMA CV chart is only designed based on one possible range of (τmin, τmax). This results in a hindrance in implementation when practitioners have other possible range of (τmin, τmax) which are more appropriate. Hence, this paper proposes the VSS EWMA CV chart for a number of different (τmin,τmax) combinations. The expected average run length (EARL) will be optimized in this paper in order to obtain the optimal chart’s parameter. It is expected that the results of this paper will ease the implementation of the VSS EWMA CV chart.

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.

Multivariate process control has recently gained significant attention because it takes into account the correlation between process variables and offers a more comprehensive understanding of the overall process behavior. In line with this phenomenon, this paper proposes the multivariate triple sampling Hotelling’s T 2 (MTS) control chart as an efficient process monitoring method in multivariate statistical process control. The mathematical properties of the MTS chart are also derived. The performance of the MTS chart is assessed using key performance metrics, such as average run length (ARL), average number of observations to signal (ANOS) and expected average number of observations to signal (EANOS). The performance of the MTS chart is compared with that of the existing multivariate double sampling Hotelling’s T 2 (MDS) chart. The findings indicate that the MTS chart consistently surpasses the MDS chart across all performance metrics, demonstrating its superior capability in detecting shifts for a multivariate process. Notably, the MTS chart exhibits lower ARL, ANOS and EANOS values, emphasizing its effectiveness in promptly identifying process shifts. In terms of practical application, an example is given to elucidate the execution of the MTS chart.

Control charts for monitoring the coefficient of variation (γ) are useful for processes with an inconsistent mean (μ) and a standard deviation (σ) which changes with μ, by monitoring the consistency in the ratio σ over μ. The synthetic‐γ chart is one of the charts proposed to monitor γ, and its attractiveness lie in waiting until a second point to fall outside the control limits before a decision is made. However, existing synthetic‐γ charts do not differentiate between the points falling outside the upper control limit (UCL) and lower control limit (LCL). Hence, this paper proposes a side‐sensitive synthetic‐γ chart, where successive nonconforming samples must either fall above the UCL or below the LCL. Formulae to compute the average run length (ARL), the standard deviation of the run length (SDRL) and expected average run length (EARL) are derived using the Markov chain approach, and the algorithms to obtain the optimal charting parameters are proposed. Subsequently, the optimal charting parameters, ARL, SDRL and EARL values for various numerical examples are shown. Comparisons show that the side‐sensitive synthetic‐γ chart consistently outperforms the existing synthetic‐γ chart, especially for small shifts. The proposed chart also consistently outperforms the Shewhart‐γ chart, while showing comparable or better performance than the Exponentially Weighted Moving Average (EWMA) chart for most shift sizes, except for very small shifts. Finally, this paper shows the implementation of the proposed chart on an industrial example.

The exponentially weighted moving average (EWMA) control chart is a popular tool used to monitor and identify slight unnatural variations in the manufacturing, industrial, and service processes. In general, control charts operate under the assumption of normality observation of the attention quality feature, but it is not easy to maintain this assumption in practice. In such situations, the data of random processes are correlated data, such as stock price in the economic field or air pollution data in the environment field. The characteristics and performance of the control chart are measured by the average run length (ARL). In this article, we present the new explicit formula of ARL for EWMA control chart based on MAX(q,r) process. The proposed explicit formula of ARL for the MAX(q,r) process is proved using the Fredholm integral equation technique. Moreover, ARL values are also assessed using the numerical integral equations method based on Gaussian, midpoint, and trapezoidal rules. Banach's fixed point theorem guarantees the existence and uniqueness of the solution. Furthermore, the accuracy of the proposed explicit formula is assessed in absolute percentage relative error compared with the numerical integral equations method. The results found that the explicit formula's ARL values are similar to those obtained using the numerical integral equation method; the absolute percentage relative errors are less than 0.0001 percent. As a result, the essential conclusion is that the explicit formula outperforms the numerical method in computational time. Consequently, the proposed explicit formula and the numerical integral equation have been the alternative approaches for computing ARL values of the EWMA control chart. They would be applied in various fields, including economics, environment, biology, engineering, and others.

Quality control of products and services is essential across various industries and sectors, including manufacturing, engineering, public health, economics, finance, and medicine. It is critical to ensure that product characteristics align with customer requirements while maintaining consistent quality and standards. Control charts are essential tools in Statistical Process Control (SPC), used to detect changes in process means or variability, and play a critical role in monitoring, controlling, and enhancing quality within production processes. This study evaluates the Average Run Length (ARL) of the Double Homogeneously Weighted Moving Average (DHWMA) control chart for Moving Average processes (MA(q)) with exponential white noise, using Numerical Integral Equation (NIE) methods. These methods include Simpson’s Rule, the Midpoint Rule, the Trapezoidal Rule, and the Gauss–Legendre Quadrature. Additionally, the study compares the performance of the DHWMA and HWMA control charts. The results indicate that all four NIE methods yield similar ARL values. However, CPU time analysis shows that the Midpoint Rule generally requires less computation time than the other methods. Furthermore, when evaluating change-point detection effectiveness, the DHWMA control chart consistently outperforms the HWMA chart across all shift sizes, as demonstrated by its lower Expected Average Run Length (EARL).

A side‐sensitive modified group runs (SSMGR) chart is commonly examined under the assumption of known process parameters (denoted as Case‐K). However, in practical situations, the process parameters are seldom known, and it is essential to estimate them from in‐control (IC) phase I samples. From the study of the minimum amount of phase I samples, m that is required by the SSMGR chart when the process parameters are estimated (represented as Case‐U), to achieve practically the same performance as that of the Case‐K, it was found that a huge number of IC phase I samples are needed. Due to time and cost factors, accumulating a substantial amount of IC phase I samples is difficult. To circumvent this problem, we study the optimal designs of the Case‐U SSMGR chart, based on both the zero and steady states' average number of observations to signal (ANOS) and expected ANOS (EANOS). Tables containing optimal charting parameters are provided to ease the implementation of the Case‐U SSMGR chart. By using these optimal parameters specifically designed for the Case‐U SSMGR chart, we can obtain similar zero and steady states' ANOS and EANOS performances to that of the Case‐K. The use of the optimal Case‐U SSMGR chart is demonstrated with real data taken from a hard‐bake process in a manufacturing company.

Average run length (ARL) values are calculated for two X̄ control chart schemes and compared with those of a standard Shewhart chart. Both control charts are based on runs rules and are easily implemented. An out-of-control condition for one of the charts is a run of two of two successive points beyond a special control limit. The other chart uses a run of two of three successive points beyond a different control limit. Both schemes are shown to have better, that is, lower, ARL values than the standard Shewhart chart for process average shifts as large as 2.6 standard deviations from the mean.

Control charts are one of the important tools to monitor quality. The coefficient of variation (CV) is a common measure of dispersion in many real-life applications. Recently, CV control charts are proposed to monitor processes which do not have a constant mean and a standard deviation which changes with the mean. These processes cannot be monitored by standard control charts which monitor the mean and/or standard deviation. This research proposes the monitoring of the multivariate coefficient of variation (MCV) by means of run rules (RR MCV) control charts, which is not available in the existing literature. The design of these charts is obtained using a Markov-chain approach. The proposed charts are simple to implement. The performance of the RR MCV and Shewhart MCV (SH MCV) charts are compared in terms of the average run length (ARL) and the expected average run length (EARL). An example is illustrated based on a real dataset. The findings revealed that the performance of the proposed charts surpasses the SH MCV chart for detecting small and moderate MCV shifts.

This paper presents the effect of measurement errors and learning on monitoring processes with individual Bernoulli observations. A cumulative sum control chart is considered to evaluate the possible impacts of measurement errors and learning. We propose a time‐dependent learning effect model along with measurement errors and incorporate them into the Bernoulli CUSUM control chart statistic. The performance of the Bernoulli CUSUM control chart is then merely assessed by comparing the average number of observations to signal (ANOS) under two proposed conditions with the condition of no possible errors. Thus, the ANOS values are obtained under different proportions of non‐conforming items, once considering errors due to measurement by inspectors, and once considering both errors and learning effect together. The experimental results show that the efficiency of the control chart to detect assignable causes deteriorates in the presence of measurement errors and enhances when learning affects operators' performance. The proposed approach has a potential to be used in monitoring high‐quality Bernoulli processes as well as disease diagnosis, and other health care applications with Bernoulli observations.

We improved the ability of the exponentially weighted moving average (EWMA) control chart to detect small shifts in the mean of a long-memory fractionally integrated autoregressive process with an exogenous variable under exponential white noise. We first designed the structure of the control chart and then evaluated its performance in terms of the average run length (ARL) via a simulation study. We first derived an analytical ARL using explicit formulas by solving integral equations and an approximated ARL derived by utilizing the numerical integral equation approach. Banach's fixed-point theorem proved that the analytical ARL exists and is unique. We then compared the out-of-control ARL values using both methods via a simulation study; the out-of-control ARL results for the analytical and approximated ARLs were similar. Moreover, the methods provided comparable accuracy in terms of the percentage difference in expected ARL and standard deviation of the run length. However, the explicit formula approach proved to be more advantageous in terms of faster computational speed and is thus recommended in this situation. An illustrative example using real data is also provided to demonstrate the practicability of the analytical ARL method.

In recent years, the manufacturing industry has tended to reduce mass production and produce in small quantities, which is called “Short Run Production”. In such a situation, the course of the production process is short, usually, the number of productions is less than 50. Therefore, a control chart for the short run production process is required. This paper discusses the comparison between multivariate control chart for short run production (V control chart) and T2 Hotelling control chart applied to sunergy glass data. Furthermore, a simulation of Average Run Length (ARL) was carried out to determine the performance of the two control charts. The results obtained are that the production process has not been statistically controlled using either the V control chart or the T2 Hotelling control chart. The number of out-of-control on the control chart V using the the EWMA test is more than the T2 Hotelling control chart. Based on the ARL value, it shows that the V control chart is more sensitive than the T2 Hotelling control chart.

The use of the auxiliary information (AI) method in control charts is gaining increasing attention. Many studies have shown that auxiliary information-based charts can boost the charts’ performances in the detection of out-of-control signals. In this study, a run sum chart for the mean based on auxiliary characteristics (abbreviated as the RS-AI chart) is proposed. The optimization designs of the RS-AI chart in minimizing the steady-state out-of-control average run length (ARL) and expected average run length (EARL) are developed. The formulae to compute the steady-state ARL and EARL of the RS-AI chart are derived using the Markov chain approach. The RS-AI chart is compared with the Shewhart AI, synthetic AI, and exponentially weighted moving average AI charts. The results show that the RS-AI chart outperforms the competing charts for all shift sizes when the correlation between the auxiliary and the study variable is large. A numerical example is given to demonstrate the implementation of the RS-AI chart.