Discrete Pseudo-Quasi Overlap Functions and Their Applications in Fuzzy Multi-Attribute Group Decision-Making

The overlap function, a continuous aggregation function, is widely used in classification, decision-making, image processing, etc. Compared to applications, overlap functions have also achieved fruitful results in theory, such as studies on the fundamental properties of overlap functions, various generalizations of the concept of overlap functions, and the construction of additive and multiplicative generators based on overlap functions. However, most of the research studies on the overlap functions mentioned above contain commutativity and continuity, which can limit their practical applications. In this paper, we remove the symmetry and continuity from overlap functions and define discrete pseudo-quasi overlap functions on finite chains. Meanwhile, we also discuss their related properties. Then, we introduce pseudo-quasi overlap functions on sub-chains and construct discrete pseudo-quasi overlap functions on finite chains using pseudo-quasi overlap functions on these sub-chain functions. Unlike quasi-overlap functions on finite chains generated by the ordinal sum, discrete pseudo-quasi overlap functions on finite chains constructed through pseudo-quasi overlap functions on different sub-chains are dissimilar. Eventually, we remove the continuity from pseudo-automorphisms and propose the concept of pseudo-quasi-automorphisms. Based on this, we utilize pseudo-overlap functions, pseudo-quasi-automorphisms, and integral functions to obtain discrete pseudo-quasi overlap functions on finite chains, moreover, we apply them to fuzzy multi-attribute group decision-making. The results indicate that compared to overlap functions and pseudo-overlap functions, discrete pseudo-quasi overlap functions on finite chains have stronger flexibility and a wider range of practical applications.