In many real-world decision-making problems, uncertainty extends beyond conventional fuzzy representations and often involves structural, temporal, and phase-dependent characteristics. Existing fuzzy set models, including their extensions, are generally limited in their ability to simultaneously capture both the magnitude and dynamic behavior of uncertain information. This paper develops a comprehensive framework of information measures for complex linear Diophantine fuzzy sets (CpLDyFS) and illustrates their utility through real-world applications. Although the CpLDyFS model provides an effective mechanism for representing multidimensional uncertainty, the development of associated information measures remains limited, with only a few similarity-based approaches currently available and no unified analytical structure established. To address this, we introduce a suite of measures for CpLDyFS, including the distance measure (DM), similarity measure (SM), cosine similarity (CSM), Jaccard similarity (JSM), exponential similarity (ESM), inclusion measure (IM), and entropy measure (EM). These measures establish a unified theoretical framework capable of capturing both magnitude and uncertainty through complex-valued and phase-dependent parameters, thereby extending the analytical capabilities of the existing model. The practical significance of the proposed framework is demonstrated through a medical case study on chronic kidney disease (CKD). In this context, the measures (SM, CSM, JSM, ESM) effectively discriminate among patient categories, highlighting subtle but clinically relevant differences in profiles. The EM quantifies the level of ambiguity within patient data, thereby improving the interpretability of uncertain information. The IM identifies containment and dominance relations among alternatives, while the DM provides reliable dissimilarity analysis. Comparative and superiority analyses further confirm the advantages of the proposed CpLDyFS framework over existing fuzzy set models, demonstrating stronger robustness and enhanced decision-making capability in the presence of imprecision.
Mustafa et al. (Mon,) studied this question.