**1. Introduction**

A group decision-making process, which is also called multi-person decision-making, is the problem of finding the best option accepted by the majority of decision makers among a list of possible alternatives. The basic act in the group decision-making is the process of consensus between different decision makers. Although a unanimous consensus is the ideal case, in real situations, full agreemen<sup>t</sup> rarely happens. Decision makers usually share and discuss their opinions about the alternatives to obtain a consensus or partial agreemen<sup>t</sup> for making the final decision. However, sometimes, they provide their priorities about the alternatives individually and then try to reach a consensus on them. Regardless which approach is applied, using aggregation operators is one of the most often used techniques to reach the process of consensus in group decision-making problems.

The aim of an aggregation operator of dimension K is to aggregate the K-tuple of objects into a single object by a bounded non-decreasing function *A* : A *<sup>K</sup>*∈<sup>N</sup>[0, 1] *K* → [0, 1]. The mean value, the median, the minimum, the maximum, the t-norms, and the t-conorms are commonly used to reach the process of consensus in group decision-making problems. To aggregate a sequence of inputs

with different importance degrees, the concept of weighting vector in aggregation operators has been studied, such as the weighted minimum and the weighted maximum [1–3]; ordered weighted average (OWA) [4], which is calculated based on the arithmetic mean; and ordered weighted geometric (OWG) [5], which is formulated based on the geometric mean. The main disadvantage of OWA and OWG operators, i.e., ignoring the importance of given arguments *x*1, ··· , *xK* for calculating the aggregated value, leads to definition of their extensions into the induced OWA (IOWA) operator [6], and induced OWG (IOWG) operator [7], respectively. However, these extensions have the inherent limitations from OWA operator and OWG operator, concerning the determination of associated weighting vector *w* for IOWA and IOWG operators. In practice, there is no unique strategy to find the associated weighting vector *w*. Usually, a quantifier function *Q* : [0, 1] → [0, 1] whose definition may change from one case to another one is applied to compute the associated weighting vector *w* = (*<sup>w</sup>*1, ··· , *wK*)*<sup>T</sup>* based on the formulation *wi* = *Q*( *iK* ) − *Q*(*<sup>i</sup>*−1*K* ) for all *i* [4,5,8,9].

Soft set theory (SS) [10], characterized by a set-valued function *f* : *P* → <sup>2</sup>*U*, is defined by a parameterization family of the universe *U*. Thus, in comparison with fuzzy set theory [11], it allows us to have a more comprehensive description of *U* based on any type and number of parameters *p* ∈ *P*. The concepts of fuzzy soft set (FSS) [12] and intuitionistic fuzzy soft set (IFSS) [13] were also studied to handle more complicated problems. Although soft set theory was originally developed to cope with the lack of parameterization tool in fuzzy sets, its flexibility to deal with set-valued functions makes it a powerful tool for providing a new methodology in decision-making problems. The first adaptivity of soft sets in decision-making was conducted by Maji et al. [14]. However, they did not discuss the aggregation methods. Roy and Maji [15] gave an algorithm to solve a group decision-making problem based on fuzzy soft sets. The FS minimum operator was applied for finding the consensus of multi-source parameter sets. Later, Alcantud [16] overcame the disadvantages of Roy and Maji's algorithm by applying FS product operator rather than FS minimum operator. Cagman et al. [17,18] applied FS minimum operator, FS maximum operator, and FS product operator to produce different fuzzy soft aggregation operators in a MAGDM problem. Guan et al. [19] applied the FS intersection operator for aggregating data. Then, a new ranking system of objects, in which the rate of objects is computed based on the full unanimity of experts with respect to all parameters rather than the number of parameters that are owned by each object, was constructed to rank alternatives in a group decision problem. Zhang et al. [20] used the IFS maximum operator to check the process of consensus in MAGDM problem. Mao et al. [21] extended IFSOWA and IFSOWG operators for intuitionistic fuzzy soft sets under three different cases of experts' weights called completely known, partly known, and completely unknown. Das and Kar [22] used the IFS product and the IFS sum operators of intuitionistic fuzzy soft matrices to obtain a collective opinion among different decision makers. Zhang and Zhang [23] utilized the FS union operation to reach the process of consensus in MAGDM problems based on trapezoidal interval type-2 fuzzy soft sets (TIT2FSSs). The TOPSIS approach was then applied to select the best option. However, recently, Pandey and Kumar [24] modified this procedure so that the complement operation that is used by Zhang and Zhang can only be applied for TIT2FSSs in which left and right heights are equal. Tao et al. [25] proposed four aggregation functions, including the soft maximum operator, soft minimum operator, soft average operator, and soft weighted average operator, to aggregate data in MAGDM problems. Two selection tools based on the SAW method and comparison matrix were also developed to obtain the optimal decision. Zhu and Zhan [26] developed the new concept fuzzy parameterized fuzzy soft sets (FPFSS), where the parameters are considered as fuzzy sets and then the consensus stage proceeds by using the t-norm and t-conorm products of FPFSSs. A new choice value function is also introduced to handle the process of selection. On the other hand, some researchers attempted to develop novel aggregation methods for FSS and IFSS. For example, Zahedi et al. [27,28] extended the concept of fuzzy soft topology to reach the consensus based on a collective preorder relation.

Conventionally, soft-set-based aggregating tools have been developed for unipolar input data where the truth value belongs to [0, 1]. Some situations require multi-polar arguments to be aggregated. To handle the problem of multi-polarity in consensus process, some studies extended the aggregation operators from unipolar scale into the bipolar scale (i.e., the interval [−1, 1]) [29] and multi-polar scale (i.e., the space *K* × [0, 1] where *K* is a set of *m* different categories) [30,31]. These extensions allow dealing with inputs from different categories where the output is presented by a pair (*k*, *x*) that *k* shows the category of aggregated value *x*. However, in many decision-making problems, the attribute set contains multi-feature or multi-polar decision parameters where the final decision should reflect the best option based on all multi-polar attributes beyond their category. For example, in the hotel booking problem, "Location" is one of the most important parameters for finding a good hotel to stay, which is a multi-feature parameter depending on how close it is to the main road, city center, tourist attractions, etc. A hotel is selected if it has the best location in terms of all features from different categories not only one. In fact, there is no ideal category *k* in final solution. This issue becomes more complex when a group of people want to choose a hotel. In this case, different cases of agreemen<sup>t</sup> within a group including unanimous consensus and partial agreement, e.g., "almost all", "most", and "more than 50%", are considered to obtain the collective view based on individuals' opinions. Thus, an alternative aggregation operator is required to deal with multi-polar fuzzy attributes in group decision-making.

In a group decision process with multi-polar inputs, the performance of alternatives are judged by each decision maker with respect to each criterion. The main problem is to compare these judgements and reach a consensus among them. The existing aggregation methods usually consider weighted or unweighted cases under a unanimous agreemen<sup>t</sup> [15–23]. However, besides the importance degrees of experts, the consensus degree for a fuzzy majority of the experts and different choices of experts' judgments at a consensus level should be taken into account in the proposed alternative aggregation operator to reach more reliable methodology. Moreover, due to the extreme applicability of FSSs in MAGDM problems with multi-polar fuzzy soft input information, adaptability of the proposed aggregation operator for m-polar fuzzy soft sets should be studied. To do this, an extension of fuzzy soft sets into the m-polar fuzzy soft sets, where the values of membership functions *fp* are extended from the unit interval [0, 1] into the cubic [0, 1] *m*, needs to be developed.

Until now, weighted aggregation operators for multi-polar fuzzy soft arguments have not been considered. Thus, this study was carried out to develop some weighted m-polar aggregation operators which cover different scenarios at the consensus degree for a fuzzy majority expressed by linguistic variables, such as "most", "much more than 70%", and "more than half". The main goal is to design FS-set-based algorithms for finding the best solution in group decision-making with weighted multi-polar input information according to the proposed aggregation methods. To achieve this goal, the following problems are addressed in this study: (i) how to express the multi-polarity of input data under fuzzy soft environment; (ii) how to generate an aggregation method based on the fuzzy majority concept for weighted multi-polar inputs; (iii) how to apply the proposed aggregation method for finding the solution in group decision-making problems; and (iv) how to analyze the final result obtained by the proposed algorithm. Accordingly, there are four main contributions of this research as follows: (i) to define a new concept m-polar fuzzy soft set; (ii) to introduce m-polar fuzzy soft weighted aggregation operators based on the fuzzy majority concept; (iii) to design m-polar FS-based algorithms for finding the solution in group decision-making; and (iv) to give some illustrative examples for validating and comparing the results.

The rest of this paper is organized as the following. Section 2 represents some basic definitions and concepts from the related works. Section 3 gives a new aggregation operator, called M-pFSMWM, for weighted m-polar fuzzy soft data. This new procedure aggregates the experts' judgments based on their importance degrees, a linguistic or numerical consensus level between the experts, and different choices of experts' judgments at the consensus level. Some of its desirable properties as well as special families of M-pFSMWM operator according to different values of consensus degree *α* and weighting vector *ω* are studied. In Section 4, a new score value function is developed to design an algorithm for ranking alternatives in MAGDM problems based on the M-pFSMWM operator and a new m-polar fuzzy soft preference relation. To compare the proposed M-pFSMWM operator with some existing aggregation methodologies, in Section 5, the m-polar fuzzy soft induced ordered weighted average (M-pFSIOWA) operator and the m-polar fuzzy soft induced ordered weighted geometric (M-pFSIOWG) operator, which are the extensions of IOWA and IOWG operators, respectively, are developed and their properties are considered. We also present an algorithm to solve MAGDM problems based on M-pFSIOWA and M-pFSIOWG operators. Section 6 focuses on the efficiency of proposed techniques by some numerical examples. Finally, in Section 7, we discuss the advantages and limitations of our approach.
