# Optimal Design of One-Sided Exponential Adaptive EWMA Scheme Based on Median Run Length
## 1 Introduction  
Quality control is essential for manufacturing and service industries to ensure product qualify and enhance operational efficiency [1]. Statistical process control (SPC) serves as a critical tool in ensuring product quality by monitoring and controlling the state of processes [2]. Among various SPC tools, monitoring schemes are extensively used to identify shifts in process parameters. Conventional monitoring schemes, like the Shewhart-type scheme, are efficient at detecting large shifts but are typically less sensitive to small or moderate shifts. To address this limitation, advanced memory-type monitoring schemes such as the Exponentially Weighted Moving Average (EWMA) [3] and the Cumulative Sum (CUSUM) [4] schemes, which incorporate historical information from previous observations, have been developed. The EWMA scheme is especially adept at identifying minor to moderate changes in process parameters, owing to its method of integrating historical data with a focus on more recent measurements. The EWMA statistic calculates a weighted average of previous data values, giving higher weight to the most recent observations. The effectiveness of the EWMA scheme largely depends on the selection of the smoothing parameter, \(\lambda\), which is often fine-tuned for optimal detection of a particular magnitude of shift [5, 6, 7]. However, when the actual shift size differs from the expected one, the performance of the EWMA scheme can be significantly impacted. To overcome this limitation, adaptive EWMA (AEWMA) schemes have been proposed. AEWMA schemes adjust the smoothing parameter dynamically based on the observed data,thereby providing better performance across a range of shift sizes. This adaptability makes AEWMA schemes more robust and versatile in various industrial settings. For example, [PERSON] and [PERSON] (see [8]) introduced the concept of adaptive EWMA schemes, demonstrating their enhanced capability to identify various magnitudes of shifts. [PERSON] (see [9]) introduced an AEWMA scheme to surveil the shift in process variance, showing its effectiveness in various scenarios. [PERSON] et al. (see [10]) developed a variable sampling interval (VSI) AEWMA scheme, which further improved the detection performance by adjusting the sampling intervals dynamically. [PERSON] et al. (see [11]) presented an AEWMA-S\({}^{2}\) scheme with an adaptive smoothing parameter, aimed at overcoming the inefficiency of traditional schemes in detecting various shifts. [PERSON] et al. (see [12]) developed an AEWMA scheme based on the Hampel function, providing a robust approach to detect changes in the process location. High-quality processes are distinguished by minimal defect rates, often measured in parts per million (ppm) or as low as parts per billion (ppb) [13]. These processes are prevalent in industries such as semiconductor manufacturing, healthcare, and earthquake monitoring [14]. Traditional monitoring schemes, which assume a normal distribution, are not well-suited for monitoring processes with strictly positive variables due to issues such as high false alarm rates and the appearance of negative lower control limits (LCL). These issues can lead to misleading assessments of process stability and performance [15]. To address these challenges, specialized monitoring schemes have been developed. One common approach is to employ time-between-events (TBE) monitoring schemes, which focus on the time intervals between consecutive events, such as defects or failures. TBE data are typically modeled using a skewed distribution, such as the exponential distribution [16; 17], which is a natural fit for the inter-arrival times of a homogeneous Poisson process [18]. [PERSON] (see [19; 20]) proposed the one- and two-sided exponential EWMA schemes. [PERSON] and [PERSON] (see [21]) investigated the robustness of the exponential EWMA scheme when the TBE data deviate from an exponential distribution. [PERSON] et al. (see [22]) proposed an AEWMA scheme for monitoring the TBE based on the Gamma distribution.