Novel measures for linguistic hesitant Pythagorean fuzzy sets and improved TOPSIS method with application to contributions of system-of-systems

Abstract

Assessment of the contribution rates of weapon system-of-systems is an extremely uncertain and complex problem that should be addressed by domain experts. Addressing these problems, which is often hesitant, fuzzy, and linguistic, requires evaluation of multiple indices of different types. Hesitant Pythagorean fuzzy sets combined with linguistic term sets are powerful tools to handle this kind of problem. In this paper, recent studies on hesitant Pythagorean fuzzy sets (HPFSs) are first reviewed and linguistic HPFSs (LHPFSs) are proposed. To efficiently solve the evaluation problem, the axiomatic properties of entropies for HPFSs and LHPFSs are defined. According to these properties, novel entropy measures for HPFSs and LHPFSs are introduced. A novel method that can better reveal the characteristics of HPFSs and LHPFS to represent the HPF and LHPF numbers is also introduced. On this foundation, the definitions of distance measures for HPFSs and LPFSs are proposed, and a set of distance measures for HPFSs and LHPFSs are developed. Moreover, the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method is modified through hierarchical thought by using the presented entropy and distance measures for LHPFSs; the improved TOPSIS method is applied preferably to the case of contribution of weapon system-of-systems and shows better validity and distinguishability than other methods under the LHPFS environment in the examples. The hierarchical TOPSIS method can better solve the problem of weight allocation in case of a lot of indices.