Some novel probabilistic dual hesitant fuzzy information measures and their applications to multi-attribute decision-making

Abstract

Probabilistic dual hesitant fuzzy set (PDHFS), as an extension of dual hesitant fuzzy set (DHFS) and probabilistic hesitant fuzzy set (PHS), can better describe the uncertainty and fuzziness in the real world through membership degree (MD) and non-membership degree (NMD) and their respective probability information. Therefore, it can collect more comprehensive fuzzy information in decision-making issues, which makes PDHFS has a wide range of applications in practice. The distance, similarity and entropy measures (EMs) of PDHFSs were studied. Due to the particularity of PDHFS, the length of MD and NMD of two probabilistic dual hesitation fuzzy elements (PDHFEs) are often different. To define the distance measure more expediently, many current studies need to add fresh elements to ensure that the length of corresponding elements in MD and NMD are the same, which will destroy the original information of the elements. In order to overcome this disadvantage, this paper defined some novel PDHF distance measures without adding elements. Some novel PDHF similarity measures were defined through the complementary relationship between distance and similarity measures based on the newly proposed PDHF distance measures. The fuzziness of a given PDHFS can be measured by the similarity measure between it and its complement. Based on the property, some novel PDHF EMs were proposed. Finally, three numerical examples are given to illustrate the effectiveness and rationality of these novel information measures.