**1. Introduction**

In practical decision making environments, it is difficult for decision makers (DMs) to give evaluated information with exact real numbers. To overcome this disadvantage, Zadeh [1] has developed the fuzzy set (FS) theory which utilizes the function of membership degree to express decision making information instead of crisp results between 0 and 1. Based on studies of FS, Atanassov [2] further proposed another function, named the non-membership degree, as a supplementary function. Thus, the intuitionistic fuzzy set (IFS) was constructed; in the IFS each intuitionistic fuzzy set is characterized by the functions of membership degree and non-membership degree between 0 and 1, and the sum of these are limited to 1. Subsequent to these studies, more and more scholars have studied

the IFS in relation to many multiple attribute decision making (MADM) problems [3–16]. Xu [17] has defined some intuitionistic fuzzy weighted average operators. Xu and Yager [18] have proposed some aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, based on geometric operation laws and an intuitionistic fuzzy environment. Hung and Yang [19] studied the similarity measures of intuitionistic fuzzy sets based on an Lp-metric. Park et al. [20] have put forward some distance measures of interval-valued intuitionistic fuzzy sets. To handle intuitionistic fuzzy MADM problems with incomplete weight information, Wei [21] utilized the maximizing deviation method to build two intuitionistic fuzzy nonlinear programming models. By considering the entropy weight of an intuitionistic fuzzy set, Hung et al. [22] established a fuzzy TOPSIS decision making model. Luo [23] defined a projection method based on intuitionistic fuzzy information with uncertain attribute weights for MADM. Ye [24] has provided a cross-entropy method with which to handle decision making problems with interval-valued intuitionistic fuzzy information. On account of the indeterminacy of DMs, Zhang [25] has presented some interval-valued hesitant fuzzy aggregation operators and applied them to MADM problems. Liao and Xu [26] have defined some intuitionistic fuzzy hybrid weighted aggregation operators. To express fuzzy information more easily, Liu et al. [27] have developed the concepts of hesitant intuitionistic fuzzy linguistic elements (HLFLEs) and have defined some weighted aggregation operators. Peng et al. [28] have discussed the MADM approach under a hesitant interval-valued intuitionistic fuzzy environment. Chen and Huang [29] have given the definition of hesitant triangular intuitionistic fuzzy set (HTIFS) and investigated its applications in MADM problems.

In addition, some other fuzzy decision making approaches have been proposed by numerous scholars [30–33]. Hu et al. [34] have proposed a novel approach combining fuzzy data envelopment analysis (DEA) and the analytical hierarchical process (AHP) to rank units with multiple fuzzy criteria. Ziemba et al. [35] have studied the online comparison system with certain and uncertain criteria. Diouf and Kwak [36] have studied fuzzy AHP, DEA, and managerial analysis for supplier selection and development from the perspective of open innovation. Dong et al. [37] utilized the modified fuzzy VIKOR and scalable computing method to study the performance evaluation of residential demand responses. Kim and Kim [38] have developed a new model for the optimal LNG import portfolio. Chou et al. [39] have used fuzzy AHP and fuzzy TOPSIS to evaluate the human resource in science and technology (HRST) performance of Southeast Asian countries.

In addition, as an e ffective MADM tool, the Pythagorean fuzzy set (PFS) [40,41] has emerged as a means to describe the indeterminacy and complexity of evaluation information. Similarly to the IFS, the PFS also consists of a membership degree and non-membership degree, the sum of the squares of which is restricted to 1. Thus, it is clear that the PFS is more widespread than the IFS and can express more decision-making information. For instance, the membership is given as 0.6 and the non-membership is given as 0.8; it is obvious that this problem is only valid for the PFS. In other words, all intuitionistic fuzzy decision-making problems are a special case of Pythagorean fuzzy decision-making problems, which means that the PFS is more e fficient in dealing with MADM problems. In previous literature, some research works have been studied by a large number of investigators. Zhang and Xu [42] defined the Pythagorean fuzzy TOPSIS model to deal with MADM problems. Peng and Yang [43] primarily proposed two Pythagorean fuzzy operations including the division and subtraction operations to better understand PFS. Reformat and Yager [44] handled the collaborative-based recommender system with Pythagorean fuzzy information. Gou et al. [45] have studied some important properties of continuous Pythagorean fuzzy information. Garg [46] has defined some new Pythagorean fuzzy aggregation operators, including Pthe ythagorean fuzzy Einstein weighted averaging (PFEWA) operator, the Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA) operator, the generalized Pythagorean fuzzy Einstein weighted averaging (GPFEWA) operator and the generalized Pythagorean fuzzy Einstein ordered weighted averaging (GPFEOWA) operator. Zeng et al. [47] have utilized the Pythagorean fuzzy ordered weighted averaging weighted

average distance (PFOWAWAD) operator to study Pythagorean fuzzy MADM issues. Ren et al. [48] built a Pythagorean fuzzy TODIM model. Liang et al. [49] investigated some Bonferroni mean operators with Pythagorean fuzzy information. Liang et al. [50] have presented Pythagorean fuzzy Bonferroni mean aggregation operators based on geometric averaging (GA) operations. Combining PFSs [40,41] and dual hesitant fuzzy sets (DHFSs) [51,52], Wei and Lu [53] introduced a definition of dual hesitant Pythagorean fuzzy sets (DHPFSs) and proposed some dual hesitant Pythagorean fuzzy Hamacher aggregation operators. Obviously, the DHPFSs have the advantages of considering the hesitance of DMs and expressing fuzzy information more e ffectively and reasonably.

However, in practical MADM problems, some relationships do exist between arguments being fused, and it is obvious that the dual hesitant Pythagorean fuzzy Hamacher aggregation operators defined by Wei and Lu [53] do not take the relationships between the arguments being fused into consideration. Thus, it is necessary to find another more e ffective method with which to fuse dual hesitant Pythagorean fuzzy information. To date, the Heronian mean (HM) [54] operator, which can effectively take the interrelationship between arguments into account, has drawn a large quantity of scholars' attention [55–59]. Based on intuitionistic fuzzy information and a geometric operator, Yu [54] developed the intuitionistic fuzzy geometric Heronian mean (IFGHM) operator and the intuitionistic fuzzy geometric weighted Heronian mean (IFGWHM) operator. Liu et al. [60] further proposed some Heronian mean operators under an intuitionistic uncertain linguistic environment for MADM. Yu et al. [61] have defined some linguistic hesitant fuzzy Heronian mean (LHFHM) operators. Li et al. [62] extended the Heronian mean operator to a single valued neutrosophic environment. Wei et al. [63] have presented some q-rung orthopair Heronian mean operators. Considering linguistic variables, Li et al. [64] developed some q-rung orthopair linguistic Heronian mean operators.

In this paper, based on the generalized Heronian mean (GHM) operator and generalized geometric Heronian mean (GGHM) operator, we develop some dual hesitant Pythagorean fuzzy generalized Heronian mean aggregation operators. The remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to the Pythagorean fuzzy set (PFS), the dual hesitant Pythagorean fuzzy set (DHPFS), and their operational laws. In Section 3, we propose some dual hesitant Pythagorean fuzzy Heronian mean aggregation operators such as: the dual hesitant Pythagorean fuzzy generalized weighted Heronian mean (DHPFGWHM) operator and the dual hesitant Pythagorean fuzzy generalized geometric weighted Heronian mean (DHPFGGWHM) operator. In Section 4, based on the DHPFGWHM and DHPFGGWHM operators, we propose some models for MADM problems with dual hesitant Pythagorean fuzzy information. In Section 5, we present a numerical example for supplier selection in supply chain managemen<sup>t</sup> with dual hesitant Pythagorean fuzzy information in order to illustrate the method proposed in this paper. Section 6 concludes the paper with some remarks.
