Multicriteria Pythagorean fuzzy decision analysis: A hierarchical QUALIFLEX approach with the closeness index-based ranking methods

Pythagorean fuzzy set initially developed by Yager (2014) is a new tool to model imprecise and ambiguous information in multicriteria decision making problems. In this paper, we propose a novel closeness index for Pythagorean fuzzy number (PFN) and also introduce a closeness index-based ranking method for PFNs. Next, we extend the Pythagorean fuzzy set to present the concept of interval-valued Pythagorean fuzzy set (IVPFS) which is parallel to interval-valued intuitionistic fuzzy set. The elements in IVPFS are called interval-valued Pythagorean fuzzy numbers (IVPFNs). We further introduce the basic operations of IVPFNs and investigate their desirable properties. Meanwhile, we also explore the ranking method and the distance measure for IVPFNs. Afterwards, we develop a closeness index-based Pythagorean fuzzy QUALIFLEX method to address hierarchical multicriteria decision making problems within Pythagorean fuzzy environment based on PFNs and IVPFNs. This hierarchical decision problem includes the main-criteria layer and the sub-criteria layer in which the relationships among main-criteria are interdependent, the relationships among sub-criteria are independent and the weights of sub-criteria take the form of IVPFNs. Therefore, in the developed method we first define the concept of concordance/discordance index based on the closeness index-based ranking methods and compute the sub-weighted concordance/discordance indices by employing the weighted averaging aggregation operator based on the closeness indices of IVPFNs. In order to take main-criteria interactions into account, we further employ Choquet integral to calculate the main-weighted concordance/discordance indices. By investigating all possible permutations of alternatives with the level of concordance and discordance of the complete preference order, we finally obtain the optimal rankings of alternatives. The proposed method is implemented in a risk evaluation problem in order to demonstrate its applicability and superiority. The salient features of the proposed method, compared to the state-of-the-art QUALIFLEX-based methods, are: (1) it can take the interactive phenomena among criteria into account; (2) it can manage simultaneously the PFN and IVPFN decision data; (3) it can deal effectively with the hierarchal structure of criteria. The proposed method provides us with a useful way for hierarchical multicriteria decision making problems within Pythagorean fuzzy contexts. In addition, we also extend the proposed method to manage heterogeneous information which includes five different types of information such as real numbers, interval numbers, fuzzy numbers, PFNs and hesitant fuzzy elements. (C) 2015 Elsevier Inc. All rights reserved.