Linguistic Fuzzy Process Capability Analysis Based on Pythagorean and q-Rung Orthopair Fuzzy Sets

Process Capability Analysis (PCA) is a widely used statistical method for assessing whether a process can meet customer-defined specification limits (SLs). However, traditional PCA faces limitations when dealing with rigidly defined SLs or uncertain data types. To address these challenges, fuzzy set theory (FST) has been extensively applied, enhancing PCA sensitivity by capturing imprecision in human judgments. This study proposes a novel PCA method based on Pythagorean fuzzy sets (PFSs) and q-rung orthopair fuzzy sets (q-ROFSs), incorporating two linguistic terms (LTs) for quality evaluation and hesitation. Unlike previous studies, our method replaces traditional α-cuts with a hesitancy-based approach and employs q -ROFSs to represent uncertainty comprehensively. In a real-world case study in the meat industry, the proposed method demonstrated a long-term capability index ( Ppk ) ranging from 0.42 to 0.73 and a short-term capability index ( Cpk ) ranging from 1.01 to 1.43. At the same time, traditional approaches showed wider and less informative ranges. These results indicate enhanced sensitivity and robustness in process performance assessment. The novelty of this approach lies in combining dual linguistic terms and the hesitancy estimation via q-ROFSs, offering a more nuanced and accurate depiction of process capability, particularly in contexts involving imprecise human evaluations.