Abstract: With the continuous improvement of the market economy system, the focus on production and scale in the market competition has been gradually shifted to one on quality and service. Therefore, to have an advantage in the fierce market competition, it is crucial to implement quality strategy. Quality management focuses on the effective control of the production process, and quality inspection is one of the key aspects. How to implement quality inspection strategies on the production to achieve the dual goals of quality improvement and cost saving is a challenge that modern manufacturing managers need to solve. Statistical Process Control (SPC), as an important quality management technique, provides many statistical tools to monitor the production process, among which the control chart is considered to be one of the most widely used tools. The control chart is often used to monitor and analyze the quality characteristics of products in the process, improve the product quality and ultimately bring down production costs for enterprises. In recent years, the research on the control chart for monitoring the ratio of two normal variables (RZ) is one of the important directions of SPC, which plays a significant part in the actual production process.
In production scenarios, when the product specification is related to the relative ratio of two components in a mixture, or when the ratio represents the quality characteristics of the product, or the difference between a product’s quality measurement before and after an operation (such as a chemical reaction), the control chart for monitoring RZ can be applied to ensure the stability of the process and make the product quality meet the production expectations.
The traditional Shewhart chart is weak in monitoring small or moderate shifts in the process, while the Exponentially Weighted Moving Average (EWMA) chart can improve the performance of the Shewhart chart by making full use of previous samples’ information. To further improve the sensitivity of the traditional RZ chart to small or moderate process shifts, based on the traditional EWMA-RZ chart, this paper puts forward several new EWMA schemes for monitoring RZ. First, by weighting the smoothing coefficient of the EWMA-RZ control chart twice, this paper puts forward the Double EWMA (DEWMA) RZ control chart. Second, the performance of the control chart can be improved by adopting the Variable Sampling Interval (VSI) strategy in the actual production process, while the traditional RZ control chart usually adopts the Fixed Sampling Intervals (FSI) strategy. Therefore, this paper further introduces the VSI strategy into the DEWMA-RZ control chart, and puts forward the VSI-DEWMA-RZ control chart. Third, the Monte-Carlo (MC) simulation is used to simulate the run length (RL) distribution of the proposed control charts in this paper. Moreover, a bisection search algorithm is used to calculate the control limit coefficient and warning limit coefficient. Under different combinations of the chart parameters, the performances of VSI-DEWMA-RZ, VSI-EWMA-RZ and DEWMA-RZ charts are analyzed and compared in this paper. The results show that the VSI-DEWMA-RZ chart is superior to the DEWMA-RZ chart, and superior to the existing VSI-EWMA-RZ control chart for monitoring small process shifts. Finally, this paper uses food formulation data to illustrate the practical application of the VSI-DEWMA-RZ and DEWMA-RZ control charts. The weight of “pumpkin seeds” and “flaxseeds” in food processing is monitored by an example of the food processing plant, which further illustrates the superiority of the proposed control charts.
The major goal of this paper is to construct an improved EWMA control chart model for monitoring RZ and to further improve the sensitivity of the traditional RZ control chart to small process shifts. Therefore, this paper has a theoretical and practical significance. Theoretically, this paper improves the quality of the control chart for monitoring RZ in small process shifts, enriches a theoretical study of the RZ control chart, and provides new ideas and reference bases for improving the performance of the control chart for monitoring RZ. In practice, the effective continuous monitoring of RZ by improving the performance of the EWMA-RZ control chart ensures the stability of the production process. When the process is out of control, it can detect small process shifts, and triggers an out-of-control signal more quickly. Then, the quality engineer should take actions to find and remove the potential assignable causes, and bring the process back in control. Therefore, this study is of great practical importance to improve the production efficiency and product quality for enterprises.
Future work can investigate the effect of measurement error on the monitoring process for the proposed control charts in this paper. Moreover, the current studies on RZ control charts are basically based on the assumption of independent normal observations of the two quality characteristics. However, due to the high frequency of sensor data collection, autocorrelation may exist between consecutive observations of X and Y. Therefore, subsequent studies can be centered on constructing an autocorrelated RZ control chart.

Key words: ratio of two normal variables, EWMA chart, DEWMA chart, VSI

摘要： 针对两个正态随机变量比率(Ratio of Two Normal Random Variables, RZ)监控的研究是近年来统计过程控制的重要方向之一。为了进一步提高传统休哈特型RZ控制图对过程中较小或中等参数偏移的灵敏度,本文以指数加权移动平均(Exponentially Weighted Moving Average, EWMA)RZ控制图为基础,提出了一种新的RZ控制图。首先,对EWMA-RZ控制图的平滑系数进行两次加权,提出了二次指数加权移动平均(Double EWMA,DEWMA)RZ控制图,并进一步引入了变采样间隔(Variable Sampling Interval, VSI)特性,提出了VSI-DEWMA-RZ控制图;其次,采用蒙特卡罗(Monte-Carlo,MC)仿真模拟所提出控制图的运行链长分布特征,并详细分析了控制图的性能;再次,针对不同的控制图参数,比较了VSI-DEWMA-RZ控制图与DEWMA-RZ和VSI-EWMA-RZ控制图的性能。仿真结果表明,本文提出的VSI-DEWMA-RZ控制图优于DEWMA-RZ控制图,且其对过程中较小和中等偏移的监控效果优于现有的VSI-EWMA-RZ控制图。最后,通过监控食品加工过程中“南瓜籽”和“亚麻籽”的重量,进一步说明了所提出控制图的优越性。

关键词: 正态随机变量比率, EWMA控制图, DEWMA控制图, VSI