Pythagorean Fuzzy Linguistic Power Generalized Maclaurin Symmetric Mean Operators and Their Application in Multiple Attribute Group Decision-Making

As an extension of Pythagorean fuzzy sets and linguistic term sets, Pythagorean fuzzy linguistic sets (PFSs) are powerful to describe decision-making information quantificational and qualitatively, which have received much scholars’ attention. The purpose of this paper is to propose a new multiple attribute group decision-making (MAGDM) approach with Pythagorean fuzzy linguistic (PFL) information. To this end, we firstly analyze the drawbacks of existing operations of PFL numbers and propose new operational rules based on linguistic scale function. The power average (PA) operator is famous for its capacity of reducing the negative influence of unreasonable evaluation values provided by prejudiced decision makers on the decision results. The generalized Maclaurin symmetric mean (GMSM) can not only capture the interrelationship among multiple inputs but also manipulate the effect of related properties by adjusting the parameters. When considering aggregation operators of PFL numbers, we combine PA with GMSM and propose the PFL power generalized Maclaurin symmetric and the PFL power generalized weighted Maclaurin symmetric operators. We also study important properties and special cases of these operators. We continue to investigate MAGDM problems with PFL decision information and propose a novel method to determine the optimal alternative. Finally, we conduct numerical examples to demonstrate the effectiveness of our proposed method. We also attempt to illustrate the advantages and superiorities of the proposed method via comparative analysis.


I. INTRODUCTION
The Pythagorean fuzzy sets (PFSs) originated by Prof. Yager [1] are an efficient to portray decision makers' (DMs) fuzzy and complicated judgements in realistic multi-attribute group decision-making (MAGDM) process. The prominent characteristic of PFS is µ 2 + v 2 ≤ 1, where µ and v represent the membership grade (MG) and non-membership grade (NMG) respectively. Due to this feature, PFSs have been extensively employed in expressing fuzzy decision information and PFSs based MAGDM has been a hot The associate editor coordinating the review of this manuscript and approving it for publication was Giovanni Pau . research topic in modern decision-making science. Generally, recent researches on PFSs in decision-making can be roughly divided three categories. The first category is utility values-based PF-MAGDM method. For handling different decision-making situations, scholars proposed different AOs to integrate PF numbers (PFNs) to compute the overall preference information of alternatives. For example, to capture the interrelationship among attributes the PF Bonferroni mean [2], [3] and Maclaurin symmetric mean [4], [5] operators were proposed. To improve the reliability of the final decision results by reducing the bad influence of DMs' extreme evaluation values, Wei and Lu [6] proposed a series of PF power average operators. To effectively deal with the VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ heterogeneous interrelationship among PFNs, Liang et al. [7] proposed a set of PF partitioned Bonferroni mean operators.
The third category is based on information measures. Scholars investigated the distance and similarity measures [24]- [27], entropy [28], and correlation coefficient of PFSs [29], and studied their applications in PF-MAGDM problems.
Besides, some scholars have focused on extensions of traditional PFSs to improve their efficiency in depicting fuzzy information, among which the Pythagorean fuzzy linguistic set (PFLS) [30] is the one the representative. The PFLS is a combination of PFSs with linguistic term set so that it can portray both DMs' quantitative and qualitative evaluation information. Afterwards, in Ref. [30]- [32], the authors proposed the PFL weighted average (PFLWA), Muirhead mean, and power MSM (PMSM) operators and employed them in MAGDM. Additionally, the authors discussed their advantages and superiorities through numerical examples. However, the decision-making methods proposed in [30]- [32] still have limitations. First, the operations of PFL numbers (PFLNs) proposed in [30]- [32] are not so reasonable, as they are simply calculated by using the subscript of the linguistic terms (LTs). For example, let S = {s θ |θ = 0, 1, . . . , 6 } be a predefined LTS and α 1 = s 3 , (0.4, 0.5) , and α 2 = s 4 , (0.6, 0.7) be two PFLN, then according to the PFL operations proposed by Peng et al. [30] we have α 1 ⊕ α 2 = s 7 , (0.68, 0.38) . Obviously, the linguistic part of the result exceeds the upper bound of the given LTS, which is irrational, counterintuitive and meaningless. Second, the MAGDM method introduced by Liu and Qin has the ability of considering the interrelationship among attributes, but it is powerless to effectively handle decision makers' extreme evaluation values. Third, although the method proposed by Liu et al. [32] based on the PMSM operator can reduce the negative influence of DMs' unreasonable evaluation values and capture the interrelationship among multiple attributes meanwhile, it fails to reflect the individual importance of aggregated arguments. Hence, existing MAGDM methods based on PFLSs still have flaws when dealing with realistic decision-making problems.
Considering that existing MAGDM methods under PFLSs still have some limitations, it is necessary to propose novel PFLSs based MAGDM method, which is the main motivation of this study. To this end, we conduct our research from following aspects. First, to overcome drawback of existing PFL operations, we introduce the linguistic scale function (LSF) into PFLSs and propose novel operational rues of PFLNs. The new operations not only have good closure but also can flexibly adapt to the semantic changes of DMs. As a matter of fact, LSFs has been widely applied in intuitionistic uncertain linguistic sets [33], interval-valued intuitionistic uncertain linguistic sets [34] and picture fuzzy linguistic sets [35]. Hence, it is necessary and worth extending LSFs into PFL sets. Second, to overcome the drawbacks of existing the aggregation operators when fusing Pythagorean fuzzy linguistic information, some novel aggregation operators are developed. Based on the new operational rules, the power average (PA) [36] operator is extended to PFL environment, and the Pythagorean fuzzy linguistic power average (PFLPA) operator and the Pythagorean fuzzy linguistic power weighted average (PFLPWA) operator are introduced. These two operators can be used to aggregate individual decision matrix to calculate the comprehensive decision matrix. In addition, the generalized Maclaurin symmetric mean (GMSM) operator is a powerful aggregation function proposed by Wang et al. [37], which not only reduces the bad effect of decision makers' extreme values and capture the interrelationship among attributes, but also reflects individual importance of aggregated values. Due to these advantages, the GMSM is has been widely used to solve MAGDM problems under q-rung orthopair fuzzy sets [38], Pythagorean fuzzy sets [4], intuitionistic fuzzy soft set [39], and probabilistic linguistic terms set [40]. Motivated by these researches, in this study we combine PA with GMSMS under Pythagorean fuzzy linguistic context and propose the Pythagorean fuzzy linguistic power generalized Maclaurin symmetric mean (PFLPGMSM) operator and the Pythagorean fuzzy linguistic power weighted generalized Maclaurin symmetric mean (PFLPWGMSM) operator. These two operators can be applied in calculating the final overall evaluation values of alternatives. Obviously, these two operators can overcome the drawbacks of the operators proposed in [31], [32]. Final, we propose a new MAGDM method with PFL information. In the proposed method, the PFLPWA operator is employed to compute the collective decision matrix and the PFLPWGMSM is employed to calculate the comprehensive evaluation value of each alternative. Hence, the final decision results are more reasonable and reliable. The main contribution of this study is to propose a novel MAGDM method under PFLSs, which can overcome shortcomings of some existing decision-making method. More specifically, contributions of this paper contain the following four aspects. 1) Novel operational rules for PFLNs based on LSF are proposed. The proposed novel operations not only are closed but also can flexibly adapt to the semantic changes of DMs, making them more powerful and reasonable than existing operations. 2) Based on the new operations, the PFLPA and PFLPWA operators are proposed, which have the ability of effectively dealing with DMs' unduly high or low evaluation values can be applied in computing the overall evaluation matrix. 3) Some novel aggregation operators, i.e., PFLPGMSM and PFLPWGMSM operators are developed, which integrate PA and GMSM under PFL sets. These two operators absorb the advantages of both PA and GMSM, making them more powerful than some existing operators. 4) A new MAGDM method is presented based on the new operational rules as well as aggregation operators. Moreover, our proposed method is applied to solve realistic MAGDM problems to illustrate its effectiveness. To better illustrate the main findings of this study, we present the rest of our paper as follows. Section 2 reviews basic concepts and proposes novel operations of PFLNs based on LSF. Section 3 introduces some new AOs of PFLNs and discusses their properties. Section 4 introduces a new MAGDM method and gives their detailed steps. Section 5 illustrates the performance of the new method and analyzes its advantages. Conclusions are presented in Section 6.

II. PRELIMINARIES
In the present section, we will briefly review some fundamental notions which will be used in the following sections.

A. PYTHAGOREAN LINGUISTIC SETS AND THEIR NOVEL OPERATIONS
Definition 1 [30]: Let X be an ordinary set and s θ(x) ∈ S, then a Pythagorean fuzzy linguistic set A defined on X is expressed as , denoting the MD and NMD of x ∈ X belong to the linguistic term s θ(x) , satisfying is called a PFLN for convenience, which can be denoted as α = s θ , (µ, v) for simplicity.
Definition 3 [41]: Let S = {s i |i = 0, 1, . . . , 2t } be a linguistic term se, s i ∈ S be a linguistic term and τ i ∈ [0, 1] be a real number. A linguistic scale function (LSF) f is a mapping from is mapping from s i to τ i (i = 1, 2, . . . , 2t) such that where 0 ≤ τ 0 < τ 1 < . . . < τ 2t . Hence, f is a strictly monotonically increasing function with regard to linguistic subscript i. Generally, there are three types of LSFs and we give a brief review in the following.
(1) The most widely used LSF is expressed as which is a simple average calculation of the subscripts of linguistic terms.
(2) The second type of LSF is expressed s follows (3) The third type of LSF is expressed as Based on the LSF, we introduce new operational rules for PFLNs.
Definition 5: Let α = s θ , (µ, v) be a PFLN, then the SF of α is expressed as and the AF is given as Based on the SF and AF of PFLNs, in the following we propose a comparison method of PFLNs.

B. POWER AVERAGE OPERATOR AND GENERALIZED MACLAURIN SYMMETRIC MEAN
Yager [36] initiated the concept of power average (PA) operator, which is presented as follows. Definition 8 [36]: Let a i (i = 1, 2, . . . , n) be a collection of non-negative crisp numbers, then the PA operator is defined as Sup a i , a j , Sup a i , a j denotes the support for a i from a j , satisfying the conditions The Definition of PGMSM operator is provided as follows. Definition 9 [37]: Let a i (i = 1, 2, . . . , n) be a set of non-negative real numbers, k ∈ [1, n] be an integer, and then GMSM (k,p 1 ,p 2 ,...,p k ) is called the generalized Maclaurin symmetric mean (GMSMS) operator, where (i 1 , i 2 , . . . , i k ) traversals all the k-tuple combination of (1, 2, . . . , n) and C k n is the binominal coefficient.

III. SOME NEW AGGREGATION OPERATORS OF PYTHAGOREAN FUZZY LINGUISTIC NUMBERS AND THEIR PROPERTIES
Based on the new operational rules of PFLNs, we propose some AOs to fuse Pythagorean fuzzy linguistic information.

A. THE PYTHAGOREAN FUZZY LINGUISTIC POWER AVERAGE OPERATOR
The Pythagorean fuzzy linguistic power average (PFLPA) operator is expressed as , and d (α, β) is the Hamming distance between α and β.
If we let then Eq. (11) can be simplified as where 0 ≤ ω i ≤ 1 and According to Definition4, we can obtain the following theorem.
Theorem 2: . . , n) be a collection of PFLNs, then the aggregated value by the PFLPA operator is still a PFLN and Theorem 2 is trivial and we omit its proof. In addition, the proposed PFLPA also has the following properties.
The proofs of Property 1 and 2 are trivial and we omit them.

B. THE PYTHAGOREAN FUZZY LINGUISTIC POWER WEIGHTED AVERAGE OPERATOR
be a collection of PFLNs, and w = (w 1 , w 2 , . . . , w n ) T be the weight vector of α i (i = 1, 2, . . . , n), such that 0 ≤ w i ≤ 1 and n i=1 w i = 1. The Pythagorean fuzzy linguistic power weighted average (PFLPWA) operator is expressed as denotes the support for α i from α j , satisfying the properties in Definition 9 (37). Similarity, if we assume then Eq. (12) can be written as where 0 ≤ η i ≤ 1 and The aggregated value by the PFLPWA operator can be obtained on the basis of Definition4.
Theorem 3: Let α i = s θ i , (µ i , v i ) be a collection of PFLNs, then the aggregated value by the PFLPWA operator is still a PFLN and It is easy to prove that PFLPWA operator has the properties of idempotency and boundedness.

C. THE PYTHAGOREAN FUZZY LINGUISTIC POWER GENERALIZED MACLAURIN SYMMETRIC MEAN OPERATOR
Definition 12: Let be a collection of PFLNs, be an integer, and The Pythagorean fuzzy linguistic power generalized Maclaurin symmetric mean (PFLPGMSM) operator is defined as for α i from α j , satisfying the conditions presented in Definition 11. To simplify Eq. (21), we assume then Eq. (21) can be written as be an integer, and p 1 , p 2 , . . . , p k ≥ 1. Then the aggregated value by the PFLPGMSM operator is still a PFLN and (24), as shown at the bottom of the next page, Proof: According to the operations, we can obtain Therefore, 1 the bottom of the next page.
Property 4 (Boundedness): Proof: As the LSF f is a strictly monotonically increasing function, then it is easy to prove (at the bottom of page 8).
In the followings, we discuss special cases of the PFLPGMSM operator with respect to the parameters k and p 1 , p 2 , . . . , p k .
Special Case 1: When k = 1, then the PFLPGMSM operator reduces to In this case, if Sup α i , α j = t (t > 0), then the PFLPGMSM operator reduces to the generalized Pythagorean fuzzy linguistic average (GPFLA) operator, i.e.
Special Case 2: When k = 2, then the PFLPGMSM operator reduces to (29), as shown at the bottom of page 9, which is the Pythagorean fuzzy linguistic power Bonferroni mean (PFLPBM) operator with the parameters p 1 and p 2 .
In this case, if Sup α i , α j = t (t > 0), then the PFLPGMSM operator reduces to the Pythagorean fuzzy linguistic Bonferroni mean (PFLBM) operator, i.e. (30), as shown at the bottom of page 9.
Special Case 3: When k = 3, then the PFLPGMSM operator reduces to (31), as shown at the bottom of page 10, which is the Pythagorean fuzzy linguistic power generalized Bonferroni mean (PFLPGBM) operator with the parameters p 1 , p 2 , and p 3 .
In this case, if Sup α i , α j = t (t > 0), then the PFLPGMSM operator reduces to the generalized Pythagorean fuzzy linguistic Bonferroni mean (GPFLBM) operator, i.e. (32), as shown at the bottom of page 10.
Special Case 4: When p 1 = p 2 = . . . = p k = 1, then the PFLPGMSM operator reduces to (33), as shown at the bottom of page 11, which is the Pythagorean fuzzy linguistic power Maclaurin symmetric mean (PFLPMSM) operator with the parameter k.
In this case, if Sup α i , α j = t (t > 0), then the PFLPGMSM operator reduces to the Pythagorean fuzzy linguistic Maclaurin symmetric mean (PFLMSM) operator, i.e. (34), as shown at the bottom of page 11. Special Case 5: When p 1 = p 2 = . . . = p k = 1 n, then the PFLPGMSM operator reduces to In this case, if Sup α i , α j = t (t > 0), then the PFLPGMSM operator reduces to the Pythagorean fuzzy linguistic geometric (PFLG) operator, i.e.
PFLNs, k ∈ [1, n] be an integer, and p 1 , p 2 , . . . , p k ≥ 1. Then the aggregated value by the PFLPWGMSM operator is still a PFLN and (40), as shown at the bottom of the next page.
Similarly, the proposed PFLPWGMSM operator has the properties of boundedness.

IV. A NOVEL METHOD TO MULTIPLE ATTRIBUTE GROUP DECISION-MAKING WITH PYTHAGOREAN FUZZY LINGUISTIC NUMBERS
This section introduces the main steps of solving multiple attribute group decision-making (MAGDM) problems in which DMs' judgements over alternatives are expressed by Pythagorean fuzzy linguistic numbers (PFLNs). Suppose there are m feasible alternative which are to be evaluated by DMs under n attributes. For the convenience of description, let the alternative set be A = {A 1 , A 2 , . . . , A m } and attribute set be G = {G 1 , G 2 , . . . , G n }. The weight vector of attributes is w = (w 1 , w 2 , . . . , w n ) T , such that 0 ≤ w j ≤ 1 and PFLPGMSM (k,1,1,...,1) (α 1 , α 2 , . . . , α n ) linguistic decision matrices are obtained and in the following we present a method to determine the optimal based on the proposed aggregation operators.
Step 1: Normalize the original Pythagorean fuzzy linguistic decision matrices. Due to the existence of benefit and cost types of attributes, to make alternatives more comparable, it is necessary to normalize the original decision matrices according to the following formula where T 1 and T 2 denote the benefit and cost types of attributes, respectively. Go to the next step.
Step 2: Compute the support between α h ij and α g ij with respective to DMs, where d α h ij , α g ij is the Hamming distance between α h ij and α g ij . Go to the next step.
Step 3: Compute the weighted overall supports T α h ij of the PFLN by Step 4: Based on T α h ij and the weight vector of DMs, compute the power weight associated with PFLN α h ij given by DM D h . Go to the next step.
Step 5: Aggregate individual decision matrix by the PFLPWA operator to compute the comprehensive decision matrix, i.e.
Then, a collective decision matrix is derived. Go to the next step.
Step 7: Compute the weighted overall supports T α ij according to the following formula and go the next step.
Step 8: Compute the power weight associated with the PFLN α ij by the following formula and go to the next step.

V. NUMERICAL EXAMPLE
Example 1: Let's consider a low-carbon tourism destination selection (LCTDS) problem. With the increasing popularity of environmental protection concepts, low-carbon tourism has gradually gained widespread attention. The socalled low-carbon tourism is a kind of tourism that reduces carbon. In other words, in tourism activities, tourists try to reduce carbon dioxide emissions as much as possible. Lowcarbon tourism is green travel based on low energy consumption and low pollution, advocating to minimize carbon footprint and carbon dioxide emissions during travel, which is also a deep-level expression of environmental protection tourism. In low-carbon tourism, one of the most important problems is LCTDS, i.e., choosing a suitable destination which is low-carbon. When evaluating the performance of low carbon of tourism destinations, decision makers usually have to consider multiple aspects. Generally, when considering a LCTDS problem, the following four attributes should be taken into consideration, i.e., traffic conditions (G 1 ), attractions of the low carbon tourism destination (G 2 ); tourist consumption satisfaction (G 3 ), and environmental quality (G 4 ). Let's consider a realistic LCTDS problems. Suppose there are four destinations (alternatives), which can be denoted as A 1 , A 2 , A 3 , and  Tables 1-3. The weight vector of the three decision experts is λ = (0.4, 0.32, 0.28) T . In the following sections, our proposed novel MAGDM method is employed to select the best low-carbon destination.

A. THE PROCEDURE OF CHOOSING THE OPTIMAL ALTERNATIVE
Step 1: As all the attributes are benefit type, the original decision matrix does need to be normalized.
Step 2: Compute the support between two PFLNs α h ij and α g ij , where i, j = 1, 2, 3, 4, h, g = 1, 2, 3 and h = g according  to Eq. (42). For the facility of expression, we S h g to represent the support of α k ij from α d ij and we obtain the following results.   Step 4: Based on DMs' weight vector and the weighted overall supports T α h ij , compute the power weight associated with the PFLN α h ij by Eq. (44), and we obtain Step 5: Compute the overall decision matrix by the PFLPWA operator and the results are shown in Table 4.
Step 6: Calculate the support between Sup α ij , α is by Eq. (46). Similarly, we employ Sup j s to denote Sup α ij , α is , and we have Step 7: Compute the weighted overall supports T α ij by Eq. (47)    Step 9: Compute the collective evaluation values of alternatives by the PFLPWGMSM operator. Without loss of generality, we assume k = 2 and p 1 = p 2 = 1, then we can obtain Step 10: Compute the scores of the alternatives and we have Step 11: Rank alternatives according to their scores and we have A 1 A 4 A 2 A 3 , and A 4 is the optimal alternative.

1) THE IMPACT OF K ON THE RESULTS
The parameter k in the PFLPWGMSM operator has important impact on the decision results and in this subsection we investigate its influence. To this end, we assign different value to k in Step 9 and present the scores and ranking results in Table 5. Without loss of generality, we assume p 1 = . . . = p k = 1 and the LST is taken as f (θ ) = θ 2t (t = 3). As we can see from Table 5, different scores and ranking orders are obtained with different parameter values k in the PFLPWGMSM operator. This is because the value k determines the number of dependent attributes among which the interrelationship among them is taken into consideration. When k = 1, our method does not consider the interrelationship between attributes. When k = 2 and 3, the method captures the interrelationship between any two or three attributes. When k = 4, then the interrelationship among all the four attributes. As there is usually interrelationship among attributes, our proposed method is flexible to deal with practical MAGDM problems. Moreover, we find out that the increase of the value k leads to the decrease of the score values of each alternative. Hence, the value of k can be regarded as DMs' attitude towards the performance of alternatives. If DMs are optimistic to the alternatives, then they should choose a smaller value of k. If DMs are pessimistic to the alternatives, then they can select a larger value of k. If DMs are neutral, then they can set k = n 2 , where [] is the round function and n is the number of attributes.

2) THE EFFECT OF THE PARAMETER VECTOR ON THE RESULTS
In this section, we investigate the influence of the parameter vector (p 1 , p 2 , . . . , p k ) on the final decision results. The score values of alternatives and the final ranking orders with different parameters are presented in Tables 6 and 7.As seen from  Tables 6 and 7, different score values and ranking orders are derived with different values of the parameters. In Table 6, we find out that let p 1 be fixed and then the increase of the value of parameter p 2 leads to the decrease of the score values of each alternative. Similarly, if p 2 is fixed then the score values of alternatives will also decrease if the value of p 1 increase. However, the ranking orders are always the same. In addition, when both the values of p 1 and p 2 increase, the score values of alternatives also decrease. Hence, we can determine appropriate values according to actual needs and basically we should choose the value of p 1 and p 2 , such that 0 ≤ p 1 , p 2 ≤ 1. Because if p 1 = 0(p 2 = 1), then the interrelationship between attributes is not taken into account, which is usually inconsistent with the reality. In Table 7, we can find the similar phenomenon, i.e., if any two of the parameters p 1 , p 2 , and p 3 are fixed, then the increase of the other parameter leads to the increase of the score values of each alternative. In addition, the parameters p 1 , p 2 , and p 3 should not be assigned zero, otherwise the proposed method fails to consider the interrelationship among multiple attributes.

C. ADVANTAGES AND SUPERIORITIES ANALYSIS
In this section, we attempt to demonstrate the advantages and superiorities of our proposed MAGDM method by comparing to some existing PFL sets based MAGDM method. These methods involve that proposed by Peng and Yang [30] based on the PFL weighted average (PFLWA) operator, that developed by Liu et al. [31] based on PFL weighted Muirhead mean (PFLWMM) operator, and that presented by Teng et al. [32] based on PFL power weighted MSM (PFLPWMSM) operator.

1) THE RATIONALITY AND FLEXIBILITY OF THE PROPOSED OPERATIONAL RULES
It is noted that the MAGDM methods presented in [30]- [32] are based on the basic algebraic operational rules. However, as pointed out in Introduction, the main shortcoming of these operational rules is that they are not closed. In other word, the calculation process may exceed the bound of the predefined LTS. In addition, these operational rules may cause contrary to the subjective intuition of the decision makers in the process of MAGDM. Our proposed method is based on new operational rules for PFLNs, i.e., LSF based operations. Advantages of these new operational rules are obvious. First of all, the new operational laws proposed in this study have properties of closure and can solve the cross-border problems of the operational rules used in [30]- [32]. Second, our proposed operational rules can flexibly adapt to the semantic changes of DMs, which is consistent with realistic decisionmaking process. Hence, our proposed method is more powerful, useful and flexible than those MAGDM approaches presented in [30]- [32].

2) THE ABILITY OF CAPTURING THE INTERRELATIONSHIP AMONG ANY NUMBERS OF ATTRIBUTES
Our proposed method is based on the PFLPWGMSM operator and hence it can considers the interrelationship that widely exists in practical MAGDM problems. In addition, our method is capable to consider the different important of different aggregated values of attributes. Given these advantages, our proposed method is more powerful than some existing PFL sets based decision-making methods. First, Peng and Yang's [30] method is based on the simply weighted average operator. As it is known that the simple weighted operator fails to consider the interrelationship among attributes. In other word, Peng and Yang's [30] method is based on the assumption that attributes are independent, which is somewhat inconsistent with the reality. In most real MAGDM problems, attributes are correlated and such interrelationship among attributes should be taken into consideration. Hence, the MAGDM method introduced by Peng and Yang's [30] is somewhat defective. Our proposed method is able to consider the interrelationship when calculating the final decision-making results and hence our proposed method is better and more powerful than that proposed by Peng and Yang. In addition, Teng et al.'s [32] method based on the PFLPWMSM operators can also take the interrelationship among attributes into account, which is the same as our method. However, its flaw is also obvious, i.e., it assumes that all input evaluation values have the same importance, which is somewhat inconsistent with real cases. Our proposed method is effective to consider the different importance of input aggregated values. By assigning different values in the vector P, the importance degrees of aggregated values are manipulated. Moreover, the PMSM operator is a special case of our proposed PGMSM operator. Hence, our method is more powerful than Teng et al.'s [32] method.

3) THE ABILITY OF REDUCING THE BAD INFLUENCE OF EXTREME EVALUATION VALUES
In many practical MAGDM problems, in order to make a smart decision, a group of DMs instead of only one are invited to evaluate the performance of alternatives. DMs usually come from different fields and have different expertise and experience. In addition, due to time shortage and complexity of decision-making problems, it is difficult for DMs to acquire all information related to decision-making problems. Hence it is common that DMs maybe provide unduly high or low evaluation values, which obviously have negative impact on the final decision results. To make the final decision results reasonable and acceptable, such kind of bad influence of extreme evaluation values should be reduced or eliminated. Peng and Yang's [30] simple weighted average operator fails to effectively handle DMs' extremely high or low evaluation values. In other word, the reliability of decision results produced by Peng

VI. CONCLUSION
Recently PFLSs are have been regarded as an efficient tool to express fuzzy information, which have been extensively investigated and applied in MAGDM procedures. The main contribution of this paper is to propose a new MAGDM method wherein attribute values are given in the forms of PFLNs. In order to do this, we firstly introduced new operational rules of PFLNs based on LSFs. Then, we presented novel AOs to fuse Pythagorean fuzzy linguistic information, i.e. PFLPA, PFLPWA, PFLPGMSM, and PFLP-WGMSM operators. Thirdly, we introduced an approached to objectively determine the weighs information. Finally, based on the newly developed AOs and weights determination approach we presented a novel MAGDM method. Afterwards, we proved the effectiveness and advantages of our method via numerical examples. In the future, we shall continue our study form the following two aspects. First, we shall investigate applications our proposed method in more practical MAGDM problems, such as evaluation of offshore oil spill response waste management strategies [42], sustainable supplier evaluation [43], plan selection of urban integrated energy systems [44], evaluation of groundwater quality [45], etc. Second, our study does not consider whether the final decision results are accepted by DMs. Actually, consensus reaching process is an important and interesting research topic in group decision-making and large scale group decision-making [46]- [50]. Hence, we shall study consensus reaching process in group decision-making under PLF sets.