New similarity measure of Pythagorean fuzzy sets based on the Jaccard index with its application to clustering

Abstract

A similarity measure is being widely used as a tool to create the degree of similarity between two different objects or sets. Several similarity measures between two Pythagorean fuzzy sets (PFSs) are suggested in literature. As far as our knowledge is concerned no one has considered similarity measure between two Pythagorean fuzzy sets (PFSs) based on Jaccard index (JI) so far. Therefore, in this manuscript a novel similarity measure based on JI between two PFSs is suggested. The JI is statics used to compare similarity as well as variety of simple sets. The advantages of proposed similarity based on Jaccard index includes: (i) It can be utilized to compare any two sets and it can also be applied to text data, numerical data and even categorical data. (ii) It is robust to outliers, which means that it is not significantly affected by presence of rare or unusual items exist in the data. This makes it useful in variety of applications including clustering and anomaly detection (iii) It is simple and efficient in computation which make it right choice in many applications where fast processing time is important. Then we gave several numerical examples to show the reliability of suggested similarity measures. Our proposed similarity measure between PFSs based on JI give better properties with respect to numerical results. Based on the numerical analysis, our proposed method is reliable, reasonable and better than the existing ones. Furthermore, we utilized proposed similarity measure with Pythagorean Fuzzy Technique of Order Preference by Similarity to Ideal Solution (PF-TOPSIS) in handling complex daily life issues involving multicriteria decision-making (MCDM) processes. Furthermore, similarity measure between two PFSs is also very useful in clustering. Therefore, we also apply our proposed similarity measure in an application to clustering based on Pythagorean fuzzy data set. Final results revealed that our suggested methods are well suited and reasonable in multicriteria decision-making and clustering environment.

Keywords

• Pythagorean Fuzzy Sets
• Similarity measure
• Jaccard index
• PF-TOPSIS
• Clustering