**1. Introduction**

Multi-criteria decision-making (MCDM) problems with different kinds of fuzzy information is handled by utilizing Zadeh's fuzzy set [1] and their various extensions, including the interval-valued fuzzy set [2], intuitionistic fuzzy set [3,4], Pythagorean fuzzy set [5,6], Type-2 fuzzy set [7,8], fuzzy multi set [9], and hesitant fuzzy set (HFS) [10,11]. However, these fuzzy tools are only suitable to deal with quantitative situations rather than qualitative situations. The Fuzzy linguistic method (FLM) [2,12,13], which decision makers prefer to provide an evaluation for using a linguistic term, is a more suitable approach than the above fuzzy set to handle qualitative situations and has been extensively applied in various fields and applications [14–18]. In some cases, the modeling capacity of fuzzy linguistic is also quite limited because simple linguistic terms find it hard to express the hesitation of decision makers. For instance, a customer is invited to evaluate the satisfying degree of a service product with respect to a given criterion. Suppose *S* = {*<sup>s</sup>*−<sup>2</sup> = *very low*, *s*−1 = *low*, *s*0 = *medium*, *s*1 = *high*, *s*2 = *very high*} is a linguistic term set (LTS). The customer regards *s*0 or *s*1 as the evaluation value of the satisfying degree for a service product, but he/she quietly finds it difficult to choose one of them as the final evaluation value. In this situation, an effective method is that the evaluation value of the satisfying degree provided by the customer should consist of the two possible values. To handle this situation, Rodríguez et al. [19] proposed the concept of hesitant fuzzy linguistic term set (HFLTS), which uses

a linguistic term to replace the numerical elements of HFS. Subsequently, Liao et al. [20] gave the mathematical form of the HFLTS according to the concept of HFLTS and utilized the hesitant fuzzy linguistic element (HFLE) to represent the elements of HFLTS. For the above example, the customer's evaluation can be expressed by an HFLE {*<sup>s</sup>*0,*s*1}. The HFLTS, which is a combination of HFS and FLM, has the advantages of HFS and FLM at the same time. Therefore, it is a useful tool for a decision maker to express his/her judgment under the hesitation and fuzziness environment.

Recently, HFLTS has been used by more and more researchers to handle MCDM problems with uncertain information [21]. In this situation, the hesitant fuzzy linguistic (HFL) aggregation operator that is applied to aggregate the criteria's value into a comprehensive value of the alternative is one of the core issues. Therefore, the investigation of HFL aggregation operator is one of the hot topics. Various HFL aggregation operators have been developed from four respects as follows (1) Rodríguez et al. [19] defined the operational rules on HFLTS and proposed the min\_upper and max\_lower operators to select the worst of the superior values and the best of the inferior values, respectively; (2) based on the likelihood-based comparison relation between two HFLEs, Wei et al. [22] proposed the HFL weighted averaging (HFLWA) and HFL order weighted averaging (HFLOWA) operators, and Lee and Chen [23] presented the HFLWA, HFLOWA, and HFL weighted geometric (HFLWG), and HFL order weighted geometric operators; (3) according to the operational laws defined on HFLTS in [24,25], Zhang and Wu [24] proposed a family of operators for HFLEs, such as HFLWA, HFLWG, and generalized HFLWA operators. Wang [25] developed an extending HFLTS according to the definition of HFLTS, and defined the extending HFLWA, extending HFLWG and their ordered weighted forms. Shi and Xiao [26] presented the HFL reducible weighted Bonferroni mean, HFL generalized the reducible weighted Bonferroni mean, and HFL weighted power Bonferroni mean operators. Xu et al. [27] proposed an HFL order weighted distance operator and utilized to deal with multi-attribute group decision-making (MAGDM) problems. Liu et al. [28] developed the HFLWA, HFLWG, and HFL harmonic operators and their order weighted and hybrid weighted forms; (4) Based on the equivalent transformation function between HFLE and hesitant fuzzy element (HFE), Zhang and Qi [29] presented the HFLWA and HFLWG operators, and applied to solve a production strategy decision-making problem; Gou et al. [30] introduced the Bonferroni mean operator into the HFLTS environment and defined the HFL Bonferroni mean and HFL weighted Bonferroni mean operators.

It's worth noting that these existing HFL aggregation operators are constructed by the algebraic product and algebraic sum operational laws of HFLEs, which are a pair of special t-norm and t-conorm. A generalized intersection and union on HFLEs can be constructed by a generalized t-norm and t-conorm. For an intersection and union, a good alternative and approximation to the algebraic product and algebraic sum are the Einstein product and Einstein sum, respectively [31,32]. Recently, Wang and Liu [31,32] proposed the intuitionistic fuzzy Einstein weighted averaging and intuitionistic fuzzy Einstein weighted geometric operators. Further, Zhang [33] presented the intuitionistic fuzzy Einstein hybrid weighted averaging and intuitionistic fuzzy Einstein hybrid weighted geometric operators and their quasi-forms. Yu [34] introduced the Einstein operations into the HFS and developed the hesitant fuzzy Einstein weighted averaging and hesitant fuzzy Einstein weighted geometric operators and their ordered forms. Jin et al. [35] derived some interval-valued hesitant fuzzy Einstein prioritized operators and applied to solve MAGDM problems. On the other hand, Hamacher [36] presented a Hamacher t-norm and Hamacher t-conorm, which can be transformed into the algebraic and Einstein t-norms and t-conorms when the parameter *υ* = 1 and *υ* = 2 in Hamacher t-norm and t-conorm, respectively. Therefore, as general and flexible continuous triangular norms, Hamacher t-norm and t-conorm have been explored by many researchers in various fuzzy environments. Tan et al. [37] defined some hesitant fuzzy operational laws based on Hamacher operations and presented a family of hesitant fuzzy Hamacher aggregation operators, such as hesitant fuzzy Hamacher weighted averaging and hesitant fuzzy Hamacher weighted geometric operators. Ju et al. [38] proposed the dual hesitant fuzzy Hamacher weighted averaging and dual hesitant fuzzy Hamacher weighted geometric operators, and their order and hybrid forms. Liu et al. [39] proposed the improved interval-valued hesitant

fuzzy Hamacher ordered weighted averaging and improved interval-valued hesitant fuzzy Hamacher ordered weighted geometric operators. Moreover, Hamacher operations are also introduced to other fuzzy environments, such as the intuitionistic fuzzy set [40], interval-valued intuitionistic fuzzy set [41], Pythagorean fuzzy set [42], and single-valued neutrosophic 2-tuple linguistic set [43]. From the above analysis, we can see that it is of important theoretical significance to explore the aggregation operators of HFLTS based on Hamacher operational laws and their application to MCDM problems, which is justly the first focus of this paper.

In practical MCDM process, it is extensively important to employ a suitable aggregation operator to drive the comprehensive preference value of each alternative. Various aggregation operators have been developed by many researchers to perform this process in MCDM problems. In these operators, the power average (PA) operator was originally presented by Yager [44], which allows the input data to support and strengthen one another, and the weight vectors in PA operator are associated with the input arguments. Inspired by the PA operator, Xu and Yager [45] presented a power geometric (PG) operator and a power ordered weighted geometric operator. The prominent characteristic of PA and PG operators is that they consider the relationships between the input arguments. Based on this advantages, many extending forms of PA and PG operators have been proposed, such as Xu [46] developing the intuitionistic fuzzy power weighted averaging and intuitionistic fuzzy power weighted geometric operators and their ordered forms. Further, Wei and Liu [47] introduced the PA and PG operators into a Pythagorean fuzzy environment and proposed a family of Pythagorean fuzzy power aggregation operators, including the Pythagorean fuzzy power averaging and Pythagorean fuzzy power geometric operators and their weighted, ordered weighted, and hybrid weighted forms. Zhang [48] presented a series of hesitant fuzzy power aggregation operators, such as hesitant fuzzy power averaging and hesitant fuzzy power geometric operators, and their ordered, weighted, and generalized forms. Furthermore, PA and PG operators have also been extended to other fuzzy environments to propose some new operators, such as intuitionistic fuzzy power aggregation operators based on entropy [49], linguistic hesitant fuzzy power aggregation operators [50], linguistic intuitionistic fuzzy power aggregation operators [51], dual hesitant fuzzy power aggregation operators based on Archimedean t-norm and t-conorm [52], and simplified neutrosophic power aggregation operators [53]. However, there is no one has explored the power aggregation operators on HFLTS, especially based on the Hamacher operations. Therefore, extending the power aggregation operators to HFLTS environments, especially based on Hamacher operational laws, is also very meaningful work and another focus of this paper.

According to the analysis above, this paper extends the Hamacher t-norm and t-conorm to an HFL environment and presents several new HFL aggregation operators to handle MCDM problem with HFL information. The main advantage of these operators is that they provide a good compensation to the existing HFL aggregation operators, and the HFL power aggregation operators can capture the relationships between the input arguments. The organization of this paper is arranged as follows. In Section 2, we briefly introduce the Hamacher t-norm and t-conorm and review some basic concepts of an HFL term set. We develop some HFL Hamacher aggregation operators and some HFL Hamacher power aggregation operators in Sections 3 and 4, respectively, and also discuss their special cases and investigate their basic properties. Section 5 utilizes these proposed operators to present two methods to handle MCDM problems with HFL information. We perform the developed methods on a numerical example and compare them with some existing HFL MCDM approaches in Section 6. Section 7 provides the conclusions of this paper.
