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Similarity measures are very effective and meaningful tool used for evaluating the closeness between any two attributes which are very important and valuable to manage awkward and complex information in real-life problems. Therefore, for better handing of fuzzy information in real life, Ullah et al. (Complex Intell Syst 6(1): 15–27, 2020) recently introduced the concept of complex Pythagorean fuzzy set (CPyFS) and also described valuable and dominant measures, called various types of distance measures (DisMs) based on CPyFSs. The theory of CPyFS is the essential modification of Pythagorean fuzzy set to handle awkward and complicated in real-life problems. Keeping the advantages of the CPyFS, in this paper, we first construct an example to illustrate that a DisM proposed by Ullah et al. does not satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Then, combining the 3D Hamming distance with the Hausdorff distance, we propose a new DisM for CPyFSs, which is proved to satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Moreover, similarly to some DisMs for intuitionistic fuzzy sets, we present some other new complex Pythagorean fuzzy DisMs. Finally, we apply our proposed DisMs to a building material recognition problem and a medical diagnosis problem to illustrate the effectiveness of our DisMs. Finally, we aim to compare the proposed work with some existing measures is to enhance the worth of the derived measures.

Being a pair of dual concepts, the normalized distance and similarity measures are important tools fordecision-making and pattern recognition under an intuitionistic fuzzy set framework. A good normalizeddistance measure should ensure that its dual similarity measure satisfies the axiomatic definition in orderto be more effective for problems. In this paper, we first construct some counterexamples to illustrate thattwo existing measures do not meet the axiomatic definition of intuitionistic fuzzy similarity measures.We then show that (1) these two measures cannot effectively distinguish some intuitionistic fuzzy values(IFVs), which can be directly judged by our intuition; (2) except for the endpoints, there exist infinitelymany pairs of IFVs, where the maximum distance 1 can be achieved under these two distances, leading tocounter-intuitive results. To overcome these drawbacks, we introduce the concept of strict intuitionisticfuzzy distance measure (SIFDisM) and strict intuitionistic fuzzy similarity measure (SIFSimM), andpropose an improved intuitionistic fuzzy distance measure based on Jensen-Shannon divergence. Weprove that (1) it is a SIFDisM; (2) its dual similarity measure is a SIFSimM; (3) its induced entropyis an intuitionistic fuzzy entropy. Comparative analysis and numerical examples demonstrate that ourproposed distance measure is completely superior to the existing ones.