New generalization of fuzzy soft sets: $(a,b)$-Fuzzy soft sets

Many models of uncertain knowledge have been designed that combine expanded views of fuzziness (expressions of partial memberships) with parameterization (multiple subsethood indexed by a parameter set). The standard orthopair fuzzy soft set is a very general example of this successful blend initiated by fuzzy soft sets. It is a mapping from a set of parameters to the family of all orthopair fuzzy sets (which allow for a very general view of acceptable membership and non-membership evaluations). To expand the scope of application of fuzzy soft set theory, the restriction of orthopair fuzzy sets that membership and non-membership must be calibrated with the same power should be removed. To this purpose we introduce the concept of $(a,b)$-fuzzy soft set, shortened as $(a,b)$-FSS. They enable us to address situations that impose evaluations with different importances for membership and non-membership degrees, a problem that cannot be modeled by the existing generalizations of intuitionistic fuzzy soft sets. We establish the fundamental set of arithmetic operations for $(a,b)$-FSSs and explore their main characteristics. Then we define aggregation operators for $(a,b)$-FSSs and discuss their main properties and the relationships between them. Finally, with the help of suitably defined scores and accuracies we design a multi-criteria decision-making strategy that operates in this novel framework. We also analyze a decision-making problem to endorse the validity of $(a,b)$-FSSs for decision-making purposes.