An EWMA monitoring scheme with a single auxiliary variable for industrial processes
Introduction
Statistical process control (SPC) is a data-driven statistical method for monitoring and controlling a process in manufacturing industry. The Shewhart control chart (Shewhart, 1924) is an example of the memoryless-type control chart. The two most commonly used memory type control charts are the exponentially weighted moving average (EWMA) control chart and the cumulative sum (CUSUM) control chart, proposed by Page (1954) and Roberts (1959), respectively. The memory control charts utilize previous information with present information to yield a better result for detecting small to moderate shifts, unlike Shewhart-type charts that use only the current information. Research works related to the Shewhart-type charts and CUSUM-type charts can be found in Hawkins and Olwell, 1998, Mukherjee and Sen, 2015, Li et al., 2016, Chong et al., 2017, Lombard et al., 2017, Sanusi et al., 2017. Different EWMA-type schemes have also been proposed in the literature. Noorossana, Fathizadan, and Nayebpour (2016) investigated the joint effect of non-normality and parameter estimation on EWMA chart. Also, Tamirat and Wang (2016) introduced an acceptance sampling plan scheme based on an EWMA statistic. In the case of unknown parameters, EWMA median control chart with estimated parameters was introduced to monitor the location parameter of a normal process (Castagliola, Maravelakis, & Figueiredo, 2016). Also, Zhou, Shu, and Jiang (2016) suggested a one-sided EWMA scheme with varying sample sizes for monitoring rare events. Furthermore, EWMA control chart found early applications in economics (Muth, 1960) and in inventory control and forecasting (Dushman, Lafferty, & Brown, 1962). For more works on the improvement of EWMA chart, interested readers can see Liu et al., 2007, Sheu et al., 2009, Teh et al., 2011, Xie et al., 2011, Nishimura et al., 2015, Zwetsloot et al., 2016.
The traditional EWMA chart monitors the process mean (say ) of a process distribution. The is computed using the famous simple random sampling (SRS) approach. However, in the presence of an auxiliary variable (X) that is closely related to the study variable (Y), can be estimated more efficiently. Consequently, Cochran (1940) used the advantage of auxiliary information to propose a ratio mean estimator for estimating the population mean of Y. Choudhury and Singh (2012) noted that is most effective when there is a positive linear relationship (which passes through the origin) between Y and X and the mean square error (MSE) of Y is proportional to X. Murthy (1964) suggested the use of when , where , , and are, respectively, the correlation coefficient between Y and X, the coefficient of variation of the study variable (Y), and the coefficient of variation of X. Also, Adebola, Adegoke, and Sanusi (2015) introduced an efficient estimator, with cum-dual ratio estimator as intercept, for estimating the population mean.
Riaz (2008a) popularized the idea of using an auxiliary variable at the stage of estimating the plotting statistic of a monitoring chart. He presented a Shewhart-type scheme based on a regression-type estimator for detecting shifts in process variability and also showed the superiority of the scheme over the traditional Shewhart control chart. Also, a Shewhart-type control chart was suggested by Riaz (2008b) for monitoring process location. The chart is based on the regression mean estimator and it was shown that it outperforms the Shewhart’s X - bar chart and the cause-selecting charts. The work was later extended to the EWMA set-up for detecting small to moderate shifts in the process mean (Abbas, Riaz, & Does, 2014). It was revealed that the chart outperforms other existing univariate and bivariate charts Also, some efficient estimators with an auxiliary variable are used to improve the performance of the combined Shewhart-CUSUM chart for detecting both small and large shifts in the location parameter of a process (Sanusi, Riaz, & Abbas, 2017).
In this article, an auxiliary variable is introduced at the estimation stage to monitor the location parameter of a process distribution. The proposed chart, denoted as MrEWMA, is an EWMA-type control chart based on the ratio mean estimator. This motivation further enhances the sensitivity of the control chart, especially in detecting shifts of small magnitudes. The average run length (ARL) approach is used to evaluate the performance of the chart. Also, other performance measures such as the standard deviation of the run length (SDRL), the extra quadratic loss (EQL), the relative average run length (RARL), and the performance comparison index (PCI) are considered. The management perspective of the proposed scheme is also briefly discussed.
One of the three activities for the successful execution of an efficient management of a process is quality control. This requirement ensures that products are up to standard through continuous improvement. The proposed MrEWMA scheme would further help to continously improve the performance of a product, which will lead to a long-term reward for industries. Also, the early detection of shifts would avoid mass inspection in controlling quality, since a good quality is achieved by preventing defective items, instead of inspecting the items for bad products. Moreover, industry with a modern method of improving products quality, and can demonstrate process capability and control, has an edge over other competitors. These are in agreement with the Deming philosophy in improving management strategies (Montgomery, 2009).
The rest of this article is arranged as follows: The statistical preliminaries of the proposed scheme; the structural framework of the classical EWMA chart including its plotting statistic, control limits, and ARL; and the design of the proposed chart are presented in Section 2. The performance evaluations and the major findings of the proposed MrEWMA chart are provided in Section 3. The comparison of the MrEWMA control chart with its existing counterparts is given in Section 4. A real-life illustrative example is given in Section 5. Finally, the summary and conclusion of the findings are provided in Section 6.
Section snippets
The proposed monitoring scheme
The motivation of this work is to enhance the sensitivity of EWMA control chart in detecting shifts in the location parameter of a control process. This is achieved by introducing an auxiliary variable, in the form of a ratio estimator, to the charting scheme. In the next subsections, the mathematical preliminaries of the proposed scheme are explained, followed by a brief description of the classical EWMA, and then, the construction of the proposed MrEWMA chart.
Performance measures
Following Steiner (1999), a comprehensive assessment of the proposed MrEWMA chart in terms of ARL and SDRL is performed with the aid of R software (R Core Team, 2016). ARL is the average number of samples plotted until the first out-of-control signal (Montgomery, 2009), while SDRL is the standard deviation of the number of samples plotted until the first out-of-control signal. ARL0 is the ARL when a process is in-control, while ARL1 is the ARL when a process is out of control. It is expected
Comparisons with other location monitoring charts
A general comparison of the proposed scheme with the classical EWMA, MxEWMA, and the Mixed EWMA-CUSUM charts is provided in this section, by presenting a one by one comparison of MrEWMA chart with its existing counterparts.
Illustrative example
In this section, we provide an illustrative example to show the implementation of our proposed chart using a practical example. For this purpose, we have considered the dataset based on the nonisothermal continuous stirred tank chemical reactor model (namely CSTR process) originally proposed by Marlin (1995) and has been widely used as a benchmark in fault detection and diagnosis (Yoon and MacGregor, 2001, Shi et al., 2013). The CSTR process comprises of nine process variables, among which we
Summary and conclusion
Control charts are used in monitoring process to detect any special cause variations. Primarily, they are classified into memoryless control charts and memory control charts. Memoryless control charts aimed at detecting large shifts in a process parameter, while memory control charts have design structures that are efficient in detecting small to moderate shifts in a process parameter. We proposed an EWMA-type chart that uses an auxiliary variable in the form of ratio estimator. The auxiliary
Acknowledgement
The work described in this paper is partially supported by Hong Kong Research Grant Council (GRF project No. 9042327 and a theme-based project grant T32-101/15-R).
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