Interval-valued hesitant fuzzy Einstein prioritized aggregation operators and their applications to multi-attribute group decision making
Interval-valued intuitionistic fuzzy (IVIF) sets are useful to deal with fuzziness inherent in decision data and decisionmaking processes. The aim of this paper is to develop a nonlinearprogramming methodology that is based on the technique for order preference by similarity to ideal solution to solve multiattribute decision-making (MADM) problems with both ratings of alternatives on attributes and weights of attributes expressed with IVIF sets. In this methodology, nonlinear-programming models are constructed on the basis of the concepts of the relative-closeness coefficient and the weighted-Euclidean distance. Simpler auxiliary nonlinear-programming models are further deduced to calculate relative-closeness of IF sets of alternatives to the IVIF-positive ideal solution, which can be used to generate the ranking order of alternatives. The proposed methodology is validated and compared with other similar methods. A real example is examined to demonstrate the applicability and validity of the methodology proposed in this paper.