Introducing Two New Classes of Hesitant Fuzzy Soft Sets

A soft set can be defined as a parameterised family of subsets defined over a universal set. While the soft set theory has been demonstrated to be effective in a number of scenarios, in many real-world applications the parameters involved are of greater complexity. This has led to the extension of soft set theory into the framework of fuzzy sets. The employment of fuzzy membership functions has become a common practice in modelling the approximate nature of parameters for fuzzy elements. Beyond ambiguity and uncertainty, practical challenges also arise from experts' hesitancy in precisely defining sets and parameters. This has led to the development of hesitant fuzzy soft sets, which are able to account for such indeterminacy. However, real-world decision-making frequently demands more nuanced modelling. To mitigate errors in representation, it is imperative to address dual perspectives within parameters, concurrently capturing ambiguity and uncertainty, and the conflicting evaluations of decision-makers (e.g., integrating both positive and negative judgments). To address this gap, the paper introduces two advanced frameworks: complex hesitant fuzzy soft sets and bipolar complex hesitant fuzzy soft sets. These models generalise existing concepts by incorporating multidimensional hesitancy and bipolarity into parameter definitions. The paper then goes on to rigorously discuss key definitions, foundational theorems, and practical numerical examples in order to demonstrate the applicability and advantages of these models in decision-making contexts.