Multiple Criteria Decision Making with Probabilities in Interval-Valued Pythagorean Fuzzy Setting

Interval-valued Pythagorean fuzzy set (IVPFS), as an extension of interval-valued intuitionistic fuzzy set (IVIFS) and Pythagorean fuzzy set to deal with uncertainty, has attracted much attention since its introduction, in both theory and application aspects. The present work aims at investigating new distance measures in the IVPFSs and then employing them into multiple criteria decision-making application. To begin with, generalized interval-valued Pythagorean fuzzy weighted distance measure and generalized interval-valued Pythagorean fuzzy ordered weighted distance measure are firstly introduced in the IVPFSs. Afterward, we propose generalized probabilistic interval-valued Pythagorean fuzzy weighted averaging distance (P-GIVPFWAD) operator, generalized probabilistic interval-valued Pythagorean fuzzy order weighted averaging distance (P-GIVPFOWAD) operator and immediate generalized probabilistic interval-valued Pythagorean fuzzy ordered weighted averaging distance (IP-GIVPFOWAD) operator which are new distance measures and are able to integrate the ordered weighted averaging operator, probabilistic weight and individual distance of two interval-valued Pythagorean fuzzy numbers (IVPFNs) in the same formulation. These distance measures are very suitable to deal with the situation where the input data are represented in IVPFNs. Then we present a kind of multiple criteria decision-making method with interval-valued Pythagorean fuzzy information based on the developed distance measures. Finally, a numerical example is provided to explain the feasibility of the proposed method and the validity of the developed method is also analyzed according to the validity criterion of multiple criteria decision making.