**1. Introduction**

Decision-making is a common activity in daily life, aiming to select the best alternative from several candidates. As one of the most important branches of modern decision-making theory, multi-attribute group decision-making (MAGDM) has been widely investigated and successfully applied to economics and managemen<sup>t</sup> due to its high capacity to model the fuzziness and uncertainty of information [1–15]. In actual decision-making problems, decision-makers usually rely on their intuition and prior expertise to make decisions. Owing to the complicacy of decision-making problems, the precondition is to represent the fuzzy and vague information appropriately in the process of MAGDM. Atanassov [16] originally proposed the concept of intuitionistic fuzzy set (IFS), characterized by a membership degree and a non-membership degree. Since its appearance, IFS has received substantial attention and has been studied by thousands of scientists worldwide in theoretical and practical aspects [17–20]. Thereafter, Yager [21] proposed the concept of Pythagorean fuzzy set (PYFS). The constraint of PYFS is that the square sum of membership and non-membership degrees is less than or equal to one, making PYFS more effective and powerful than IFS. Due to its merits and advantages, PYFS has been widely applied to decision-making [22–25].

PYFSs can effectively address some real MAGDM problems. However, there are quite a few cases that PYFSs cannot deal with. For instance, the membership and non-membership degrees provided by a decision-maker are 0.7 and 0.8 respectively. Evidently, the ordered pair (0.7, 0.8) cannot be represented by Pythagorean fuzzy numbers (PYFNs), as 0.7<sup>2</sup> + 0.8<sup>2</sup> = 1.13 > 1. In other words, PYFSs do not work for some circumstances in which the square sum of membership and non-membership degrees is greater than one. To effectively deal with these cases, Yager [26] proposed the concept of *q*-ROFS, whose constraint is the sum of *q*th power of membership degree and *q*th power of the degree of non-membership is less than or equal to one. Thus, *q*-ROFSs relax the constraint of PYFSs and widen the information range. In other words, all intuitionistic fuzzy membership degrees and Pythagorean fuzzy membership degrees are a part of *q*-rung orthopair fuzzy membership degrees. This characteristic makes *q*-ROFSs more powerful and general than IFSs and PYFSs. Subsequently, Liu and Wang [27] developed some simple weighted averaging operators to aggregate *q*-rung orthopair fuzzy numbers (*q*-ROFNs) and applied these operators to MAGDM. Considering these operators cannot capture the interrelationship among aggregated *q*-ROFNs, Liu P.D. and Liu J.L. [28] proposed a family of *q*-rung orthopair fuzzy Bonferroni mean operators.

As real decision-making problems are too complicated, we may face the following issues. The first issue is that although IFSs, PYFSs and *q*-ROFSs have been successfully applied in decision-making, there are situations that cannot be addressed by IFSs, PYFSs and *q*-ROFSs. For example, human voters may be divided into groups of those who: vote for, abstain, refusal of in a voting. In other words, in a voting we have to deal with more answers of the type: yes, abstain, no, refusal. Evidently, IFSs, PYFSs and *q*-ROFSs do not work in this case. Recently, Cuong [29] proposed the concept of PIFS, characterized by a positive membership degree, a neutral membership degree, and a negative membership degree. Since its introduction, PIFSs have drawn much scholars' attention and have been widely investigated [30–37]. Therefore, motivated by the ideas of *q*-ROFS and PIFS, we propose the concept of *q*-rung picture fuzzy set (*q*-RPFS), which takes the advantages of both *q*-ROFS and PIFS. The proposed *q*-RPFS can not only express the degree of neutral membership, but also relax the constraint of PIFS that the sum of the three degrees must not exceed 1. The lax constraint of *q*-RPFS is that the sum of *q*th power of the positive membership, neutral membership and negative membership degrees is equal to or less than 1. In other words, the proposed *q*-RPFS enhances Yager's [26] *q*-ROFS by taking the neutral membership degree into consideration. For instance, if a decision-maker provides the degrees of positive membership, neutral membership and negative membership as 0.6, 0.3, and 0.5 respectively. Then the ordered pair (0.6, 0.3, 0.5) is not valid for *q*-ROFSs or PIFSs, whereas valid for the proposed *q*-RPFSs. This instance reveals that *q*-RPFS has a higher capacity to model fuzziness than *q*-ROFSs and PIFSs. The second issue is that, in some situations, decision-makers prefer to make qualitative decisions instead of quantitative decisions due to time shortage and a lack of prior expertise. Zadeh's [38] linguistic variables are powerful tools to model these circumstances. However, Wang and Li [39] pointed out that linguistic variables can only express decision-makers' qualitative preference but cannot consider the membership and non-membership degrees of an element to a particular concept. And subsequently, they proposed the concept of intuitionistic linguistic set. Other extensions are interval-valued Pythagorean fuzzy linguistic set proposed by Du et al. [40] and picture fuzzy linguistic set proposed by Liu and Zhang [41]. Therefore, this paper proposes the concept of *q*-RPLS by combining linguistic variables with *q*-RPFSs. The third issue is that in most real MAGDM problems, attributes are dependent, meaning that the interrelationship among aggregated values should be considered. The Bonferroni mean (BM) [42] and Heronian mean (HM) [43] are two effective aggregation technologies which can capture the interrelationship among fused arguments. However, Yu and Wu [44] pointed out that HM has some advantages over BM. Therefore, we utilize HM to aggregate *q*-rung picture linguistic information.

The main contribution of this paper is that a novel decision-making model is proposed. In the proposed model, attribute values take the form *q*-RPLNs, and weights of attributes take the form of crisp numbers. The motivations and aims of this paper are: (1) to provide the definition of *q*-RPLS and

operations for *q*-RPLNs; (2) to develop a family of *q*-rung picture linguistic Heronian mean operators; (3) to put forward a novel approach to MAGDM with *q*-rung picture linguistic information on the basis of the proposed operators. In order to do this, the rest of this paper is organized as follows. Section 2 briefly recalls some basic concepts. In Section 3, we develop some *q*-rung picture linguistic aggregation operators. In addition, we present and discuss some desirable properties of the proposed operators. In Section 4, we introduce a novel method to MAGDM problems based on the proposed operators. In Section 5, a numerical instance is provided to show the validity and superiority of the proposed method. The conclusions are given in Section 6.
