Distance measures play a crucial role in fuzzy decision-making, pattern recognition, and uncertainty modeling. However, some existing distance measures for Complex Picture Fuzzy Sets (CPiFSs) have shown limitations and may produce counterintuitive results in certain cases. Moreover, only a few studies have explored such measures. To overcome these issues, in this study, some novel measures of distance for CPiFSs are proposed to effectively handle two-dimensional uncertainty characterized by amplitude and phase components. The proposed measures are developed by integrating both magnitude and phase information in a unified mathematical framework, ensuring improved discrimination capability and structural consistency. We rigorously prove that the suggested measures fulfill the essential properties of a distance function. Additionally, the normalization characteristics and stability behavior are analytically examined to ensure robustness in practical implementations. The proposed measure of distance is then applied to a multi-criteria decision-making (MCDM) case study, where alternatives are evaluated under Complex Picture Fuzzy information to demonstrate its practical effectiveness and ranking consistency. Using a CPiFS-based TOPSIS framework, distances from the positive and negative ideal solutions are computed via the developed metric, and the relative closeness coefficient is employed to obtain a stable and discriminative ranking of alternatives. Furthermore, comparative analysis with several existing distance measures demonstrates the stability and superiority of the proposed method in distinguishing complex fuzzy information.
Alhussain et al. (Tue,) studied this question.
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