A Decision-Making Approach Based on New Aggregation Operators under Fermatean Fuzzy Linguistic Information Environment

: Fermatean fuzzy linguistic (FFL) set theory provides an efficient tool for modeling a higher level of uncertain and imprecise information, which cannot be represented using intuitionistic fuzzy linguistic (IFL)/Pythagorean fuzzy linguistic (PFL) sets. On the other hand, the linguistic scale function (LSF) is the better way to consider the semantics of the linguistic terms during the evaluation process. It is worth noting that the existing operational laws and aggregation operators (AOs) for Fermatean fuzzy linguistic numbers (FFLNs) are not valid in many situations, which can generate errors in real-life applications. The present study aims to define new robust operational laws and AOs under Fermatean fuzzy linguistic environment. To do so, first, we define some new modified operational laws for FFLNs based on LSF and prove some important mathematical properties of them. Next, the work defines several new AOs, namely, the FFL-weighted averaging (FFLWA) operator, the FFL-weighted geometric (FFLWG) operator, the FFL-ordered weighted averaging (FFLOWA) operator, the FFL-ordered weighted geometric (FFLOWG) operator, the FFL-hybrid averaging (FFLHA) operator and the FFL-hybrid geometric (FFLHG) operator under Fermatean fuzzy linguistic environment. Several properties of these AOs are investigated in detail. Further, based on the proposed AOs, a new decision-making approach with Fermatean fuzzy linguistic information is developed to solve group decision-making problems with multiple attributes. Finally, to illustrate the effectiveness of the present approach, a real-life supplier selection problem is presented where the evaluation information of the alternatives is given in terms of FFLNs. Com-pared to the existing methods, the salient features of the developed approach are (1) it can solve decision-making problems with qualitative information data using FFLNs; (2) It can consider the attitudinal character of the decision-makers during the solution process; (3) It has a solid ability to distinguish the optimal alternative.


Introduction
The intuitionistic fuzzy set (IFS) theory was introduced by Atanassov [1] in 1983 to accommodate uncertain and vague concepts more precisely in complex real-life situations. An IFS assigns each element a degree of membership (DM) and a degree of nonmembership (DNM), whose sum is always less than or equal to one. It has become an important and widely studied generalization of fuzzy sets [2]. Due to the applicability and effectiveness of the IFS theory, several researchers started work in this direction and established many significant results. For aggregating different intuitionistic fuzzy numbers (IFNs), a large number of AOs have been defined by considering various aspects of available information [3][4][5][6]. The Bonferroni mean operators were studied in [7][8][9][10] to capture the interrelationship between aggregated IFNs. Verma [11] proposed prioritized weighted aggregation operators with intuitionistic fuzzy information based on Einstein tnorms. Zhenghai and Xu [12][13][14] undertook a detailed study on intuitionistic fuzzy calculus and explored its utility in decision-making problems. Besides, several information measures have been proposed under an intuitionistic fuzzy environment, including distance measure [15][16][17][18], similarity measure [19][20][21][22], entropy measure [23][24][25][26], divergence measure [27][28][29], and inaccuracy measure [30] and applied them in different application areas including pattern recognition, medical diagnosis, and decision making.

Literature Review
In 2013, Yager [31] and Yager and Abbasov [32] proposed the notion of the Pythagorean fuzzy sets (PFSs) as a new generalization to IFSs. PFSs are more effective in modeling imperfect or vague information, which cannot be represented in terms of IFSs. For example: suppose an expert provides the DM of an alternative corresponding to a criterion as 0.8 and the DNM as 0.5. As we see, the sum of both degrees is 1.3, which does not satisfy the essential condition of IFS. Further, if we consider the sum of the squares of both the degrees, i.e., 0.8 2 + 0.5 2 then we obtain 0.89 < 1; hence, this information can be represented in the form of PFS, not in IFS. In a short span, the PFS theory has become an efficient tool for solving various real-life problems. Zhang and Xu [33] extended the TOPSIS method under the Pythagorean fuzzy environment. Yager [34] proposed some novel AOs for aggregating Pythagorean fuzzy numbers (PFNs). The power AOs were studied by Wei and Lu [35] under a Pythagorean fuzzy environment. Yang et al. [36] defined some Pythagorean fuzzy Bonferroni mean operators using t-norms. Akram et al. [37] developed a twophase group decision-making approach using the ELECTRE III method with Pythagorean fuzzy information. Ejegwa [38] defined a modified Zhang and Xu's distance measure for solving pattern recognition problems with Pythagorean fuzzy information. Molla et al. [39] extended the PROMETHEE method with PFSs and utilized them in medical diagnosis. Bakioglu and Atahan [40] conducted a detailed study on prioritizing risks in self-driving vehicles based on hybrid approaches with Pythagorean fuzzy information.
Again, let us assume the DM as 0.9 and the DNM as 0.6 in the above-discussed example. It is clear that we do not express this information by using IFS and PFS. To cope with this problem, Senapati and Yager [41] proposed the concept of Fermatean fuzzy set (FFS), where the DM and DNM are both real numbers that lie between 0 and 1 and satisfy the condition 0 ≤ (DM) 3 + (DNM) 3 ≤ 1. The main advantage of the FFS is that it provides a better tool than IFS and PFS for handling the higher level of uncertainties arising in many real-life decision-making problems. As we obtain 0.9 3 + 0.6 3 < 1, hence FFS is an appropriate tool to capture this uncertain information. Later on, Senapati and Yager [42] defined some operations on FFSs and discussed their application in decision-making. To aggregate different Fermatean fuzzy numbers (FFNs), Senapati and Yager [43] developed some weighted averaging/geometric AOs and utilized them to solve decision-making problems with multiple criteria. Aydemir and Yilmaz Gunduz [44] used the TOPSIS method with Dombi AOs for solving decision-making problems with Fermatean fuzzy information. Mishra et al. [45] formulated a Fermatean fuzzy CRITIC-EDAS approach to select sustainable logistics providers.
In many real-life situations, due to the increase in complexities and uncertainties in practical decision problems, an expert feels difficulty expressing his/her preference information by exact numerical values. Besides, many attributes and criteria can be evaluated quickly and effectively in terms of linguistic values. Firstly, Zadeh [46,47] developed the idea of the linguistic term set (LTS) in 1975. For example-suppose an expert evaluates the performance of a motorbike, then he/she may use the terms "good", "excellent", etc., to express his/her evaluation information because linguistic terms (LTs) are very close to human cognition. In 2010, Wang and Li [48] developed a hybrid set theory by combining the notions of LTS and IFS in a single formulation, which is known as the intuitionistic linguistic fuzzy sets (ILFSs). In the literature, several research studies have been conducted under the intuitionistic linguistic fuzzy environment. Liu [49] proposed some generalized dependent AOs with intuitionistic linguistic fuzzy numbers (ILFNs) and studied their application in decision-making. Liu and Wang [50] defined some intuitionistic linguistic generalized power aggregation operators. Su et al. [51] studied ordered weighted distance averaging operators with intuitionistic linguistic fuzzy information. Yu et al. [52] presented an extended TODIM method for solving MAGDM problems with ILFNs.
Recently, Liu et al. [53] generalized the notion of ILFSs and introduced Fermatean fuzzy linguistic sets (FFLS) by integrating the idea of LTS with FFS. Besides, a MCDM approach was formulated for solving decision problems with Fermatean fuzzy linguistic information. Further, Liu et al. [54] defined some new distance and similarity measures between FFLSs based on linguistic scale function (LSF) and utilized them in the development of TODIM and TOPSIS methods. In conclusion, Fermatean fuzzy linguistic set theory has a broader scope of applications in different practical areas. However, a limited investigation has been conducted on FFLSs and their applications. It is also worth noting that the operational laws defined by Liu et al. [54] for FFLNs are not valid in general. Therefore, it is significant to pay attention to the research studies under the Fermatean fuzzy linguistic environment.

Objective and Contributions of the Work
The main objective of this work is to define the modified operational laws for Fermatean fuzzy linguistic numbers (FFLNs) and study different AOs based on them to aggregate Fermatean fuzzy linguistic information. To fulfill the aim of the proposed study, firstly, the work defines some new modified operational laws for FFLNs based on LSF, which overcome the drawbacks of the existing operational laws. We also study several essential properties of the proposed modified operational laws. Then, the paper develops several new AOs for aggregating different FFLNs and discusses several properties associated with them. Finally, a decision-making approach is formulated to solve MAGDM problems under the Fermatean fuzzy linguistic environment. The contributions of this paper can be summarized as follows: 1. New and improved operational laws are introduced for FFLNs with their properties.

3.
A MAGDM method based on the proposed AOs is constructed to support the decision-making problems under Fermatean fuzzy linguistic environment.

4.
A sensitivity analysis is also conducted to analyze the impact of different AOs on the ranking of the alternatives.

Organization of the Paper
The rest of the manuscript is organized as follows: In Section 2 we briefly review some preliminary results on linguistic variables (LVs), LSF, FFS, FFLS and discuss some significant drawbacks of the Fermatean fuzzy linguistic operational laws defined by Liu et al. [54]. Section 3 presents modified algebraic operational laws for FFLNs based on LSF and proves several important properties of FFLNs using proposed operation laws. Then, we define the FFLWA, FFLWG, FFLOWA, FFLOWG, FFLHA, and FFLHG AOs to aggregate different FFLNs. In Section 4, based on the developed AOs, a MAGDM approach is formulated for solving real-life decision problems with Fermatean fuzzy linguistic information. Then, a real-life supplier selection problem is given to illustrate the decision-making steps and effectiveness of the developed approach. In Section 5 we conclude the paper and discuss some future works.

Linguistic Variables
The linguistic variable provides a useful tool to represent qualitative information in terms of linguistic values. According to Herrera and Martínez [55], the linguistic variable can be defined as follows: where neg denotes the negation operator.
For example, a well-known set of seven linguistic terms can be defined as:

Linguistic Scale Function
In the evaluation process, an expert uses LTs directly rather than their corresponding semantics. In general, the simplest way to deal with LTs is to use the levels of LTs directly. However, in different semantics decision-making environments, LTs have some differences in expressing evaluations. To resolve these issues, Wang et al. [57] defined the LSF to deal with linguistic information. According to the decision-making environment, experts can consider different LSFs, which express available linguistic information more flexibly and precisely in different semantic situations.

 
where  is a threshold, which can be determined by a subjective method according to the specific problem, and it should be greater than or equal to 1. If the LTS is a set of seven terms, then   LTS with seven terms. Figures 1-3 show the balanced distribution of L , unbalanced distribution of L in an increasing trend and the unbalanced distribution of L in a decreasing trend, respectively. Besides, Figure 4 represents the relationships between LTs of L and their corresponding semantics under different situations.    Meanwhile, to avoid an information loss and to facilitate the calculation process, the LSF  can be further generalized to an extended continuous LTS as follows: x X  , the corresponding degree of hesitancy (DH) is defined as In the interest of simplicity, Senapati and Yager [41] called the pair be an extended continuous LTS, then a FFLS can be defined as Note that when Based on the above analysis, we conclude that the operational laws defined in Definition 7 are not suitable for FFLNs. Therefore, in order to nullify the above shortcomings, it is necessary to modify these operational laws. In the next section, we first define some new modified operational laws for FFLNs based on LSF and discuss their properties in detail. Then, we introduce some aggregation operators for aggregating different FFLNs.

Improved Operational Laws for FFLNs Based on LSF
Here, we define some improved operational laws for FFLNs, which overcome the shortcomings of the existing operations.  3  3  3  3  3  3  3  3  3  3  3  3  3  3  1  2 , , For any The following results hold: The results follow directly from Definition 9, so we omit the proofs of them. □ be three FFLNs and 1 2 , , 0 Proof. Here, we only prove the parts (i), (iii), (v), (vii), and (ix); the others can be proved similarly.

FFL-Weighted Average (FFLWA) Operator
The weighted average (WA) is the most commonly used mean operator in a wide range of application areas. Here, we extend the idea of WA to the Fermetean fuzzy linguistic information environment and propose the following formal definition.
Proof. The first result directly holds from Theorem 1. Using the principle of mathematical induction, we shall prove the result stated in Equation (21). Firstly, for 2 n  , by Definition 9, we obtain Hence, Hence, the result is valid for 2 n  .

FFL-Ordered Weighted Average (FFLOWA) Operator
The ordered weighted averaging (OWA) operator [59] is an aggregation operator that provides a parameterized family of aggregation operators between the minimum and the maximum. In this subsection, we extend the idea of the FFLWA operator into the FFLOWA operator based on the OWA operator.

FFL-Ordered Weighted Geometric (FFLOWG) Operator
The ordered weighted geometric (OWG) operator [60] is a common aggregation operator in the field of information fusion. However, the existing OWG operator cannot aggregate FFLNs. Now, we define the FFLOWG operator based on the notion of the OWG operator to aggregate FFLNs.
Proof. We can derive the proof similar to Theorem 4, so we omit it here. □ Moreover, the FFLOWG operator also satisfies properties such as idempotency, monotonicity, boundedness, and commutativity.

FFL-Hybrid Average (FFLHA) Operator and FFL-Hybrid Geometric (FFLHG) Operator
From Definitions 10 to 13, we know that the FFLWA and FFLWG AOs only weight the FFLNs, while the FFLOWA and FFLOWG AOs weight the ordered position of the FFLNs instead of weighting the FFLNs itself. In both cases, the weights address different aspects during the aggregation process of the FFLNs. However, the developed aggregation operators for FFLNs consider only one of them. The hybrid averaging (HA) operator [61] is an aggregation operator that uses the weighted average (WA) and the ordered weighted averaging (OWA) operator in the same formulation. In the following, we propose the FFL-hybrid average (FFLHA) operator and the FFL-hybrid geometric (FFLHG) operator. Proof.
Proof. In the next section, we formulate a new decision-making method to solve MAGDM problems under the Fermatean fuzzy linguistic environment. Then, we consider a real-life supplier selection problem to demonstrate the decision-making steps.

MAGDM Problem Description
For a MAGDM problem, let   is a collection of s experts whose weight vector is represents an attribute evaluation value, given by the expert q E , for the alternative 1, 2, , j n   . Then, the ranking of the alternatives is required to obtain the best alternative(s).

Decision Method
The decision method comprises the following steps.
Step 1: To nullify the effect of the different attributes, transform the decision matrices . The elements of the normalized decision matrices   q B can be obtained as follows: Step or FFLOWG operator where EP(extremly poor), VP very poor , P poor , SP slightly poor , F fair , SG slightly good , .
G good , VG very good , G extremly good The experts provide the following Fermetean fuzzy linguistic decision matrices , as listed in Tables 3-6, respectively.  Tables 7-10).     Table 12.
Step 5: The final ranking order of the suppliers following the score values   i   S is , thus, the most desirable supplier is 2 F . Additionally, if we use the FFLOWG operator in Step 2 and the FFLWG operator in Step 3 instead of FFLOWA and FFLWA operators, respectively, in the developed method, then the procedure steps are as follows: Step 1: Same as above.
Step 2: Utilizing the FFLOWG operator to aggregate all the normalized decision ma- , the obtained results corresponding to each alternative are shown in Table 13.  Table 14. It is worth noting that a decision-maker can choose the appropriate aggregation operator based on his/her behavior towards the aggregation procedure. If a decision-maker has optimistic behavior towards the aggregation of experts' preference information and pessimistic behavior towards the final decision, then he/she uses the FFLOWA and FFLWG operators in Step 2 and Step 3, respectively, of the developed approach. A complete analysis has been conducted to analyze the effect of the decision-maker's behavioral attitude on the final ranking. The results are summarized in Table 15, along with the suppliers' ranking order. The results shown in Table 15 indicate that when we use the FFLOWA (or the FFLOWG) operator in Step 2 and the FFLWA (or FFLHA) operator in Step 3 then the ranking order of the alternatives is always On the other hand, if we use the FFLOWA (or the FFLOWG) operator in Step 2 and the FFLWG (or FFLHG) operator in Step 3 then the ranking order of the alternatives is obtained as , which is slightly different from the previous ones. It shows the effect of the nature of varying aggregation operators on the final ranking order of the alternatives.
Apart from the above analysis, to examine the influence of the different LSFs on the alternatives' ranking order, we have employed different LSFs in the calculation process of the developed decision-making approach. Then, after applying the steps, the corresponding results are summarized in Table 16.  Table 16, it has been observed that although the score values of the alternatives are entirely different when we use different LSFs; however, the best alternative is always 2 F for the considered problem. Note that the use of different LSFs shows an influence on the final ranking order of the alternatives. It is also worth noting that, in other real-life decision problems, the best alternative may be different depending on the use of different aggregation operators. Our developed method provides an ability to the DMs for choosing the appropriate LSF according to his/her personal choice and the actual semantic environment.

Sensitivity Analysis
The decision-making approach developed in this paper uses various AOs with different LSFs to aggregate information data as provided by decision-makers. This section discusses the influence of these AOs and LSFs on the decision result.
A sensitivity analysis has been conducted to better understand the impact of the various AOs on the ranking order of the alternatives with different LSFs. Figures 5-10 have been plotted, portraying the sensitivity of the results based on various AOs taking different LSFs. As shown in Figures 5-10, changes in the AOs and LSFs significantly influence the ranking of alternatives.

A Comparative Overview of the Results Based on Different AOs
In order to compare the performance of the developed AOs with existing ones, this paper uses the AOs and decision method formulated by Li et al. [53] to solve the above considered MAGDM problem. The obtained values of the closeness coefficient and ranking order of the suppliers are summarized in Table 17.  Table 17, it is inferred that the best alternative obtained by the proposed approach agrees with the method formulated by Li et al. [53] which confirms the consistency of the proposed operators. It is also clear from the results summarized in Table 17 that the best alternative   2 F and the worst alternative   3 F are the same by all the approaches.
However, the complete ranking order of the suppliers is slightly different from our ranking order because of the use of different aggregation operators for aggregating the preference information of the decision-makers.
In addition, Spearman's rank correlation test [63] is used for the subsequent analyses to identify the differences between the ranking orders of the alternatives obtained by using different AOs in the formulated approach. Spearman's rank correlation measures the strength of association of two variables. It can be defined as follows:      20 show that the rankings obtained by the FFLOWA+ FFLWG and FFLOWA+ FFLHG also have a perfect positive relationship.

Some Advantages and Limitations of the Proposed Approach
1. The formulated approach can solve decision-making problems with qualitative information data in terms of FFLNs very efficiently. 2. We can also apply this method to solve many existing decision problems with intuitionistic fuzzy linguistic and Pythagorean fuzzy linguistic information [48][49][50]. 3. The main limitation of this study is that it cannot handle the situations in which the attributes have some interaction and prioritization relationship between them. 4. The developed study opens a new door for further research under a qualitative information environment. 5. There can be some adverse situations in which the defined operational laws for FFLNs may not work. Then, we will need to undertake a further detailed investigation of the operational laws of FFLNs.

Conclusions
In this paper, we studied the MAGDM problems under a Fermatean fuzzy linguistic environment. To overcome the shortcomings of the existing operational laws of FFLNs, the paper has defined four new algebraic operational laws for FFLNs based on the idea of LSF. Several mathematical properties of the proposed operational laws have been studied in detail. Next, we formulated some AOs, including the FFLWA operator, the FFLWG operator, the FFLOWA operator, the FFLOWG operator, the FFLHA operator, and the FFLHG operator to aggregate different FFLNs. Furthermore, the work has proved many important properties of the proposed AOs, such as idempotency, monotonicity, commutativity, and boundedness, to establish their applicability in different areas. Utilizing the proposed AOs, the paper has developed a new decision-making approach to solve MAGDM problems with Fermatean fuzzy linguistic information. Finally, a real-life supplier selection problem has been considered to illustrate the steps of the proposed method.
In the future, we will employ the proposed AOs to solve some other decision-making problems such as medical diagnosis, pattern recognition, and image processing. Further, we will also develop some new aggregation operators such as Bonferroni mean operator, Heronian mean operator, and Hamy mean operator to aggregate the correlative Fermatean fuzzy linguistic information in future work.