A novel decision-making approach using three-way decision in fractional fuzzy environment

Multi-criteria decision-making approach serves as a prominent method for addressing authentic real-world scenarios and exploring optimal solutions by taking into account relevant criteria or attributes. Decision making in the context of uncertainty pertains to scenarios characterized by a significant absence of information, wherein the functional relationships remain entirely unspecified. Data serves as a crucial component in the decision-making process encompassing varying degrees of uncertainty that require distinct methodologies to ensure sound and valid. Therefore, in this paper we investigate the multi criteria decision-making problem with fractional fuzzy information. Then, we first develop two types of operations on fractional fuzzy set (FFS) such as Einstein sum and Einstein product and further develop novel operational laws under fractional fuzzy environment. Based on these operational laws, we develop a new set of Einstein aggregation operators along with some of its properties like Idempotency, monotonicity, and boundedness. After that the TODIM (Portuguese acronym for Interactive Multi-Criteria Decision Making) method under fractional fuzzy information are briefly analyzed. Furthermore, the three-way decision process was discussed to give the best optimal results. In three-way decision, process the conditional probability, expected loss function and decision rule were studied. To illustrate the suitability of the suggested technique to resolve multi-criteria decision-making issues an example was provided. Finally, we compare our proposed work with existing methods to judge the stability and applicability of the work.