### Journals Information

Mathematics and Statistics Vol. 10(3), pp. 624 - 635
DOI: 10.13189/ms.2022.100319
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## Exact Run Length Computation on EWMA Control Chart for Stationary Moving Average Process with Exogenous Variables

Wannaphon Suriyakat 1, Kanita Petcharat 2,*
1 Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok, Thailand
2 Faculty of Applied Science, King Mongkutâ€™s University of Technology North Bangkok, Bangkok, Thailand

ABSTRACT

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.

KEYWORDS
EWMA, Moving Average, Average Run Length, Explanatory Variable, Exponential White Noise

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Wannaphon Suriyakat , Kanita Petcharat , "Exact Run Length Computation on EWMA Control Chart for Stationary Moving Average Process with Exogenous Variables," Mathematics and Statistics, Vol. 10, No. 3, pp. 624 - 635, 2022. DOI: 10.13189/ms.2022.100319.

(b). APA Format:
Wannaphon Suriyakat , Kanita Petcharat (2022). Exact Run Length Computation on EWMA Control Chart for Stationary Moving Average Process with Exogenous Variables. Mathematics and Statistics, 10(3), 624 - 635. DOI: 10.13189/ms.2022.100319.