Generalized picture fuzzy distance and similarity measures on the complete lattice and their applications

Picture fuzzy sets (PFSs) with four dimensions of positive, neutral, negative, and rejection have more advantages in representing and evaluating ambiguous information than intuitionistic fuzzy sets. But, since the PFS’s introduction, its measure theory has not been sufficiently developed. The existing picture fuzzy distance measures (PFDMs) and picture fuzzy similarity measures (PFSMs) are sometimes invalid or irrational, as they were not constructed on a bounded lattice. To get over these restrictions, this paper seeks to develop a picture fuzzy measure system. First, their restrictions are determined by categorizing the PFDMs and PFSMs, the score functions. Second, the probability-based λ−1-precision score function for picture fuzzy numbers and its computational aspects are highlighted. Again, the proposed score function is used to depict the generalized PFDMs/PFSMs on a complete lattice. In comparison study, some numerical examples are provided to confirm the effectiveness of the proposed measures. Then, a framework for making decision is offered to address multi-criteria decision-making problems with uncertainty by combining the proposed generalized PFDM and the concept of the Vlsekriterumska Optimizaca I Kompromisno Resenje (VIKOR) method. Finally, the drug selection problem in medical diagnostics is addressed using the suggested decision-making framework. Sensitivity and comparative studies demonstrate the validity, adaptability, and superiority of the suggested method. •The shortcomings of existing score functions, PFDMs and PFSMs, are overwhelming.•A probability-based λ−1-precision score function is defined for PFNs.•Generalized PFDMs and PFSMs on a complete lattice are stated.•A new MCDM approach is defined to solve the problems.