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 δ≥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 (δ_min,δ_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.