The concept of Pythagorean fuzzy sets (PFSs) was initially developed by Yager in 2013, which provides a novel way to model uncertainty and vagueness with high precision and accuracy compared to intuitionistic fuzzy sets (IFSs). The concept was concretely designed to represent uncertainty and vagueness in mathematical way and to furnish a formalized tool for tackling imprecision to real problems. In the present paper, we have used both probabilistic and nonprobabilistic types to calculate fuzzy entropy of PFSs. Firstly, a probabilistic-type entropy measure for PFSs is proposed and then axiomatic definitions and properties are established. Secondly, we utilize a nonprobabilistic-type with distances to construct new entropy measures for PFSs. Then a min–max operation to calculate entropy measures for PFSs is suggested. Some examples are also used to demonstrate suitability and reliability of the proposed methods, especially for choosing the best one/ones in structured linguistic variables. Furthermore, a new method based on the chosen entropies is presented for Pythagorean fuzzy multicriterion decision making to compute criteria weights with ranking of alternatives. A comparison analysis with the most recent and relevant Pythagorean fuzzy entropy is conducted to reveal the advantages of our developed methods. Finally, this method is applied for ranking China-Pakistan Economic Corridor (CPEC) projects. These examples with applications demonstrate practical effectiveness of the proposed entropy measures.

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The increasing use of social media and information sharing has given major benefits to humanity. However, this has also given rise to a variety of challenges including the spreading and sharing of hate speech messages. Thus, to solve this emerging issue in social media sites, recent studies employed a variety of feature engineering techniques and machine learning algorithms to automatically detect the hate speech messages on different datasets. However, to the best of our knowledge, there is no study to compare the variety of feature engineering techniques and machine learning algorithms to evaluate which feature engineering technique and machine learning algorithm outperform on a standard publicly available dataset. Hence, the aim of this paper is to compare the performance of three feature engineering techniques and eight machine learning algorithms to evaluate their performance on a publicly available dataset having three distinct classes. The experimental results showed that the bigram features when used with the support vector machine algorithm best performed with 79% off overall accuracy. Our study holds practical implication and can be used as a baseline study in the area of detecting automatic hate speech messages. Moreover, the output of different comparisons will be used as state-of-art techniques to compare future researches for existing automated text classification techniques.