Construction of the Shewhart Median Chart Based on the Expected Percentile Run Length

A recent study explored the construction of the Shewhart median (S-M) chart by utilizing the percentile-based (PL) method. By using the PL method, practitioners can ascertain that the required conditions on run length (RL) performances for both in-control (IC) and out-of-control (OC) cases are met with desired probabilities. They conducted a comprehensive investigation on the IC and OC capabilities of the S-M chart utilizing the average run length (ARL), median run length (MRL), and PL methods for known shift sizes, and concluded that by adopting the PL method, the S-M chart demonstrated a noteworthy enhancement in the IC performance in comparison to the ARL method without causing much deterioration to the OC performance when $\delta\geq 1.5$. However, in various practical scenarios, the exact shift size is commonly unknown. Hence, we propose a new performance measure called the expected percentile run length (EPRL), where 50^th EPRL is equivalent to expected MRL (EMRL). It can be inferred that by adopting the PL method, the S-M chart demonstrated a noteworthy enhancement in the IC performance in comparison to the ARL method without causing much deterioration to the OC performance when $(\delta_{\mathrm{m}\ln},\delta_{\max})=(1.5,2.0)$ for unknown shift sizes. We illustrate the application of the proposed method using a real-life example based on the hard-baking process.