Bi-polar preference based weights allocation with incomplete fuzzy relations

Normalized weight vector determination under bi-polar preferences is important in multicriteria decision making and its related evaluation problems. In order to determine weights for the elements in partially ordered set which can embody bi-polar preferences, some new methods such as the ordered weighted averaging (OWA) aggregation on lattice using three-set formulation have been proposed. However, when there are no posets and orders but fuzzy relations available, some new effective generalized methods should be proposed. This work differentiates two types of special fuzzy relations, called incomplete fuzzy relation and contradictive fuzzy relation. Two objective methods to derive incomplete fuzzy relation from a set of vectors and basic uncertain information (BUI) granules are introduced. Two scaling methods to transform contradictive fuzzy relation into incomplete fuzzy relation are suggested. Based on those derived fuzzy relations and given convex/concave basic unit monotonic (BUM) functions, some weights allocation methods are proposed which can well embody the bi-polar preferences of decision makers. The method further generalizes the OWA aggregation on lattice. Some mathematical properties, four different instances and some numerical examples with application backgrounds or potentials are also provided.