Multiattribute Group Decision-Making Based on Linguistic Pythagorean Fuzzy Interaction Partitioned Bonferroni Mean Aggregation Operators

1College of Mathematics and Informatics, Fujian Normal University, Fuzhou, Fujian 350117, China 2Digital Fujian Internet-ofings Laboratory of Environmental Monitoring, Fujian Normal University, Fuzhou, Fujian 350117, China 3Business School, Sichuan University, Chengdu, Sichuan 610064, China 4School of Computer and So ware, Nanjing University of Information Science & Technology, Nanjing 210044, China 5Digital Fujian Institute of the Big Data for Agriculture and Forestry, Fujian Agriculture and Forestry University, Fuzhou, Fujian 350002, China


Introduction
Multiattribute group decision-making refers to a process where a group of decision-makers are invited to participate in the assessment of some given alternatives and then the optimal one is selected or all of them are ranked based on their assessment information [1][2][3].It can be commonly seen in our daily life.As the multiattribute group decision-making problems become more complex, it is very difficult for the decision-makers to assess the alternatives using real values [4].Zadeh [5] built the theory of fuzzy sets (FSs), which provides the decision-makers with an efficient means to model the fuzzy information.Nevertheless, FSs are incapable of expressing the nonmembership degree (NMD) of each element in the universe of discourse belonging to a FS [6].To deal with the shortcoming of FSs, Atanassov [7] gave the theory of intuitionistic fuzzy sets (IFSs) in which each intuitionistic fuzzy number (IFN) consists of MD and NMD and the sum of them is less than or equal to 1. Liu et al. [8] extended the Heronian mean to aggregate IFNs.
However, sometimes the sum of MD and NMD given by the decision-maker may be larger than 1, but their square sum is less than or equal to 1. Obviously, IFSs cannot model this kind of fuzzy information [9].In this case, Yager [10] defined the concept of Pythagorean fuzzy sets (PFSs) in which each Pythagorean fuzzy number (PFN) consists of MD and NMD whose square sum is less than or equal to 1. Figure 1  area refers to a set of all the possible IFNs or PFNs that are located in a two-dimensional coordinate system.It can be seen that all the possible IFNs only fall into the blue area and each PFN may appear in the blue area or yellow area.Namely, an IFN can also be a PFN, while a PFN may not be an IFN.
For instance, when a decision-maker provides a pair of MD and NMD ⟨0.6, 0.6⟩ to assess an alternative with respect to an attribute.It can be clearly seen that 0.6 + 0.6 > 1, but 0.6 2 + 0.6 2 < 1.Hence, ⟨0.6, 0.6⟩ can be only modeled as a PFN.Both IFSs and PFSs are the special cases of q-rung orthopair fuzzy sets [11,12].Both IFSs and PFSs model the uncertain and vague information using the quantitative form.However, in some complex decision-making environments, the decision-makers may prefer to use the qualitative form to give their preference information.For example, when evaluating the running speed of a central processing unit (CPU), if the decisionmakers do not know the detailed performance parameters of this CPU, they will prefer to utilize one of the linguistic terms such as "very slow," "slow," "medium," "fast," "very fast," and "perfect" to express their preference information [13].The fuzzy linguistic approach put forward by Zadeh [14] can model this kind of qualitative preference information.Xu et al [15] also devised a virtual linguistic model to redefine the syntax and semantics of existing linguistic computational models.All of the linguistic computational models are built based on the concept of linguistic term sets (LTSs) [16][17][18].
To introduce the qualitative modeling capability into IFSs, Zhang [20] gave the concept of linguistic intuitionistic fuzzy sets (LIFSs), which combines the theory of IFSs and the concept of LTSs.A LIFS is composed of linguistic intuitionistic fuzzy numbers (LIFNs) whose MD and NMD are expressed using the linguistic terms.Inspired by Zhang's idea, Garg [19] also combined the theory of PFSs with the concept of LTSs and proposed a novel qualitative assessment model called the linguistic Pythagorean fuzzy set (LPFS).Any linguistic Pythagorean fuzzy number (LPFN) in the LPFS consists of MD and NMD that are modeled utilizing the linguistic terms.The sum of squares of subscripts of MD and NMD is less than or equal to the square of the cardinality of LTS.
Aggregation operator [21,22], which is used to aggregate some pieces of data into one piece of data, plays an important role in the multiattribute decision-making models (MADMM) and multiattribute group decision-making models (MAGDMM).The functions and operations are two vital parts that have an impact on the aggregated results of aggregation operators.In the following part, the importance of functions and operations is discussed.
(1) Regarding the functions of aggregation operators, the early aggregation operators are designed to fuse several real numbers into a single real number.To enhance the capability of fusing the complex data, various aggregation operators have been designed considering the importance of values or/and the relations among values.For example, the Choquet integral was introduced in [23] to study the correlated averaging operators and correlated geometric operators for aggregating the interval-valued intuitionistic fuzzy sets.To reduce the impact of unreasonable values on the decision results, several power aggregation operators [24] and dependent aggregation operators [25] have been proposed to fuse IFNs.In recent years, the Bonferroni mean (BM) operator [26] considering the interrelationships among input values is extended into the fuzzy environment [27,28].The BM operator can capture the interrelationship between each attribute value and the other ones.However, some attribute values may not have any relationship with the other ones in some practical decision-making problems [29].For instance, when a house is assessed, four common attributes are usually considered: the price (attribute A 1 ), the geographical position (attribute A 2 ), the daylighting (attribute A 3 ), and the apartment layout (attribute A 4 ).In this case, there exists the relationship between A 1 and A 2 , while A 3 is related with A 4 .These four attributes can be divided into two sets:  1 = { 1 ,  2 } and  2 = { 3 ,  4 }.The elements in the same set are related with each other, but the elements in the different sets do not have any relationship.Hence, the BM operator cannot capture this kind of relationship between the attribute values.To deal with this complex situation, the partitioned Bonferroni mean (PBM) operator is put forward in [30] to extend the capability of capturing relationships for BM operator.
(2) Regarding the operational rules of aggregation operators, until now, there have been very few studies reporting the aggregation operators for LPFNs.In [19], some averaging operators and geometric operators are put forward to fuse a set of LPFNs.However, the above study is put forward based on the operational rules, which are proposed by Garg [19].The operational rules for calculating two LPFNs do not contain the interactions between the MD and NMD from different LPFNs, which may result in unreasonable decision results.To the best of our knowledge, the study reporting the operators in [19] is the only one for LPFNs.
Through the above analysis, it can be seen that designing excellent aggregation operators should take the functions and operational rules into account at the same time [31,32].However, previous aggregation operators [19] put forward to fuse LPFNs have some drawbacks, which can be summarized as follows.
(1) Previous aggregation operators cannot deal with the situation that the attributes in the same set are related with each other, while the attributes in the different sets have no relationship with each other.They simply calculate the averaging value or geometric value of a set of LPFNs and do not consider the complex relationships among the attributes.
(2) Previous aggregation operators are put forward using the operational rules proposed by Garg [19], which may lead to unreasonable decision results, especially when the subscript of NMD in one of the LPFNs is zero.If the subscript of NMD in one of the LPFNs is zero, then the aggregated NMD would always be zero regardless of the value of the NMD of other LPFNs.
To deal with the above drawbacks, in this paper, we focus on proposing some linguistic Pythagorean fuzzy interaction partitioned Bonferroni mean (LPFIPBM) aggregation operators to aggregate LPFNs.Our contributions can be listed as follows: (1) The drawback of the operational rules for calculating two LPFNs is analyzed and the interactional operational rules considering the interaction between the MD and NMD from different LPFNs are devised for LPFNs.
(2) Some interaction PBM operators are developed for LPFNs based on the interactional operational rules.
(3) Several special cases of the proposed LPFIPBM aggregation operators are given and the desirable properties are also studied.
(4) The proposed LPFIPBM aggregation operators are used to develop a novel MAGDMM to fuse the LPFNs under the situation that the attributes in the same set are related with each other and the ones in the different sets are not related.
(5) An illustrative example concerning the selection of solid state drive (SSD) production is utilized to show the implementation process of the proposed MAGDMM based on the LPFIPBM aggregation operators and it is compared with the previous study under the linguistic Pythagorean fuzzy environment.
The remainder of this paper is arranged as follows: the concept of LPFSs, some operational rules of LPFSs, and the definitions of BM and PBM are briefly reviewed in Section 2. Several novel interactional operational rules are given for computing two LPFNs and then their properties are discussed in Section 3. In Section 4, the LPFIPBM, LPFWIPBM, LPFIPGBM, and LPFWIPGBM operators are devised to aggregate LPFNs.In Section 5, the LPFWIPBM and LPFWIPGBM operators are used to devise a novel MAGDMM with LPFNs.In Section 6, an illustrative example concerning the selection of SSD production is provided to demonstrate our proposed operators and a detailed comparison is performed between our proposed operators and previous study presented in [19].Finally, Section 7 presents the conclusions of our wok.

Preliminaries
In this section, the definition of LPFS and the operational laws of LPFSs, as well as the definitions of BM and PBM, are briefly reviewed.
. .Linguistic Pythagorean Fuzzy Sets (LPFSs).Garg [19] combined the theory of PFSs with the concept of LTSs and put forward a novel qualitative assessment model called the linguistic Pythagorean fuzzy sets (LPFSs) as follows.
Definition (see [19]).Let  be a finite reference set and let  = {  |  ∈ [0, ]} with a positive integer  be a continuous linguistic term set (LTS) [33]; then a LPFS on the finite set  is expressed in a mathematical form as where   () and   () are two functions, which are responsible for returning the MD and the NMD of the element  being the member of the LPFS .Each two-tuple (  (),   ()) in the LPFS is named as a LPFN or LPFV and it can be expressed in a simplified form as  = (  ,   ) satisfying that 0 ≤  ≤ , 0 ≤  ≤ , and 0 ≤  2 +  2 ≤  2 .Let   () =  √  2 − 2 − 2 ; then   () can be referred to as the hesitance degree (HD) of the element  being the member of the LPFS .
Garg [19] also presented the score value and the accuracy value for LPFNs and then developed a method for comparing two LPFNs.Definition (see [19]).There exist a continuous LTS  = {  |  ∈ [0, ]} and a LPFN  = (  ,   ) with   ,   ∈ ; then the score value of the LPFN  is computed as and the accuracy value is calculated as The method for comparing two LPFNs  and  are described as follows:

Complexity
Based on the t-conorm and t-norm, Garg gave several operational rules to compute two LPFNs as follows.
Definition (see [19]).Assume a continuous LTS  = {  |  ∈ [0, ]} with a positive integer  and two LPFNs  1 = (  1 ,   1 ) and . .e BM Operator and PBM Operator.Aggregation operators are an important tool introduced to fuse a set of input values into a single one.They are expressed in the form of mathematical functions as follows.
The BM operator, initially put forward by Bonferroni [35], is a valuable aggregation function that can capture the interrelationship between each attribute value and the other ones.Its mathematical expression is defined as follows.
Definition (see [36]).Given a set of nonnegative input values { 1 ,  2 , . . .,   }, as well as two real numbers ,  ≥ 0, the mathematical expression of the BM operator is described as Its geometric version is also presented as follows.
Definition (see [37]).Given a set of nonnegative input values { 1 ,  2 , . . .,   }, as well as two real numbers ,  ≥ 0, the mathematical expression of the geometric BM (GBM) operator is described as To capture the interrelationship that the attribute values in the same set are related and the ones in the different sets are not related, the PBM operator is put forward in [30] to extend the BM operator as follows.
Definition (see [30]).Given two nonnegative real numbers ,  ≥ 0, as well as a set of nonnegative input values  = { 1 ,  2 , . . .,   }, which is divided into  different groups  1 ,  2 , . . .,   , with ⋃  =1   = , the mathematical expression of the PBM operator is described as where |  | is the number of elements in the group   and  denotes the number of groups Its geometric version is given as follows.

Interactional Operational Rules for LPFNs
In this section, the drawbacks of the operational rules proposed by Garg [19] for LPFNs are analyzed and then some novel interactional operational rules are developed for the computations of LPFNs.Finally, their properties are discussed.
The operational rules presented in Definition 3 do not work well for all the LPFNs, especially when the MD or NMD of one of LPFNs is  0 .Assume that there are two LPFNs,  1 = (  1 ,  0 ) and  2 = (  2 ,   2 ).Based on the additive operation in Definition 3, the aggregated NMD of  1 ⊕  2 is always 0 regardless of the value of the NMD of  2 .Obviously, this operational result is unreasonable, which also happens in the situation that the multiplicative operation is performed to compute two LPFNs, of which the MD is  0 .Moreover, the aggregation operators based on the operational rules may also generate unreasonable results.For example, if the LPFWA operator presented in [19] is used to aggregate the above two LPFNs, the NMD obtained from ( 1 ,  2 ) is  0 .In a word, the value of NMD in the LPFN  2 does not influence the value of NMD of operational results and aggregation results.
To overcome the above drawback, in this section, several novel interactional operational rules are put forward to compute the LPFNs. , ) ; , ) ; ).Obviously, the operational result derived from the operational rules in Definition 9 is more reasonable.
Some theorems can be derived from Definition 9 as follows.
Thus, the value of  decreases as  1 or  2 increases, while it increases as (ii) which completes the proof of Theorem 11.

Complexity
Theorem 12. Assume that a continuous LTS  = {  |  ∈ [0, ]} with a positive integer , as well as two LPFNs Proof.The proof is similar to that of Theorem 11.

Linguistic Pythagorean Fuzzy PBM Operators Considering the Interactional Operational Rules
In this section, the PBM operator and the novel interactional operational rules are combined to design the LPFIPBM operator, the LPFWIPBM operator, the LPFIPGBM operator, and the LPFWIPGBM operator for fusing the LPFNs.Then their special cases and properties are also discussed. .
where |  | is the number of elements in the subset   and  denotes the number of subsets and According to the novel interactional operational rules developed for LPFNs in Definition 9, several theorems can be derived from (16) where Proof.According to Definition 9, we have Let According to Definition 9, we have and 1) ; then the above equation can be transformed into Thus, , and Then, we have and Thus, which completes the proof of Theorem 15.
We continue to discuss the properties of the LPFIPBM operator as follows.

Theorem 16 (idempotency). Given a continuous LTS
Proof.Since   =  V =  and   =  V = , then we have Complexity and thus Bring the above equation into ( 17); then we have which finishes the proof of idempotency of the LPFIPBM operator.

Theorem 17 (commutativity). Given a continuous LTS
and For each , we have By adjusting the values of the parameters ,  ≥ 0, respectively, some special equations can be derived from the LPFIPBM operator as follows: (i) When  → 0,  = (∏ (ii) When  = 1 and  → 0, the above equation can be transformed into (iii) When  → 0, we have (iv) When  = 1 and  = 1, we have 1) .Thus, Although the LPFIPBM operator could capture the interrelationship between each attribute value and the other ones in each independent subset, it is not capable of considering the weights of attributes and the decision-makers.To deal with the drawback of the LPFIPBM operator, the LPFWIPBM operator is devised as follows.
According to the novel interactional operational rules devised for LPFNs in Definition 9, a theorem can be derived from (40) and we can have Thus, Finally, we can have which finishes the proof of Theorem 19.
. .e LPFIPGBM Operator and the LPFWIPGBM Operator.The aggregation operators are mainly divided into two categories: the arithmetic operators and the geometric operators [39].The aggregated results derived from the former ones are easily influenced by the extreme values, while the aggregated results derived from the latter ones could take the balance among all the values into consideration.Thus, the geometric operators perform better than the arithmetic operators, especially for the social economy data.The study presented in [40] reported that the aggregated results of the arithmetic operators are higher than those of the geometric ones when the same data set is fused.Hence, if the decisionmakers are optimistic, then the arithmetic operators are used.If they are pessimistic, then the geometric operators can be selected.To enhance the theory of aggregation operators, the PGBM operator and novel interactional operational rules are combined to devise the LPFIPGBM and LPFWIPGBM operators for LPFNs.
Then, some properties of the LPFIPGBM operator are given as follows.(56)

Theorem 22 (idempotency). Given a continuous LTS 𝑆 = {𝑠
Proof.The proof processes of Theorems 22 and 23 are similar to those of Theorems 16 and 17, which are omitted here.
By adjusting the values of the parameters ,  ≥ 0, respectively, some special equations can be derived from the LPFIPGBM operator as follows: Thus, 1) , and  = 1) .Thus, Similarly, the LPFIPGBM operator can capture the interrelationship between each attribute value and the other ones in each independent subset; it is not capable of taking the weights of attributes and decision-makers into account.To enhance the capability of the LPFIPGBM operator, the LPFWIPGBM operator is devised as follows.
According to the novel interactional operational rules devised for LPFNs in Definition 9, a theorem can be derived from (59) as follows.

Novel MAGDM Method Using the LPFWIPBM or LPFWIPGBM Operator
In the MAGDM problems, there exists a set of alternatives  = { 1 ,  2 , . . .,  ℎ } with a set of attributes  = { 1 ,  2 , . . .,   }, which can be split into two categories, namely, the benefit attributes and the cost attributes, respectively.A group of decision-makers  = { 1 ,  2 , . . .,   } are invited to select the linguistic terms from a continuous LTS  = {  |  ∈ [0, ]} with a positive integer  to construct the LPFNs for modeling the preference information.Suppose that   = (   ) ℎ× is a preference information matrix that is given by the decision-maker   , where    = (    ,     ) is a LPFN modeling the preference information given by the decision-maker   when the alternative   with respect to the attribute   is assessed.Suppose that  = ( 1 ,  2 , . . .,   )  represents the weight vector of the group of decision-makers , where   is the weight of the decision-maker   satisfying 0 ≤   ≤ 1 and ∑  =1   = 1.Let  = ( 1 ,  2 , . . .,   )  express the weight vector of the set of attributes , where   denotes the weight of the attribute   satisfying 0 ≤   ≤ 1 and ∑  =1   = 1.For the above MAGDM problems, a novel MAGDM method exploiting the LPFWIPBM operator or the LPFWIPGBM operator is currently devised as follows.
Step .Normalize the preference information matrix   = (   ) ℎ× that is provided by each decision-maker.If there exist cost attributes, then it is necessary to transform their values into the benefit-type ones.The normalized preference information matrix denoted as Ζ  = (   ) ℎ× ( = 1, 2, . . ., ) is derived according to the following rule: to compute the fused value of all the attribute values of each alternative   which are given by the decision-maker   .
Step .Exploit the LPFWIPBM operator to compute the fused value of the fused attribute value of each alternative   .
Step .Use Definition 2 to compute the score value and the accuracy value of all the fused value   as and Step .All the alternatives { 1 ,  2 , . . .,  ℎ } are ranked based on their score value and accuracy value, and the optimal one(s) is selected.

Illustrative Example and Comparison Analysis
In this section, a practical example is presented to illustrate the detailed implementation process of the proposed MAGDM method, and then it is compared with the existing study.

Complexity
Score values for A1 derived from LPFWIPBM (74) Step .According to their score values, these SSD productions can be ranked as Thus, the optimal one is  3 .
From the above implementation process, it can be seen that both the LPFWIPBM and LPFWIPGBM operators get the same ranking result.
. .e Impact of the Parameters on the Decision Results.Figures 2-9 present the ranking results derived from the LPFWIPBM and LPFWIPGBM operators when the values of the parameters ,  change between 0 and 10.
As depicted in Figures 2-9, it can be seen that the ranking results derived from the LPFWIPBM and LPFWIPGBM operators remain unchanged as the values of the parameters ,  change and the optimal alternative is always  3 .
. .e Validation of Effectiveness of Our Studies.For verifying the effectiveness of our proposed LPFWIPBM and LPFWIPGBM operators, we replay the existing LPFWA and LPFWG operators, which are the only ones that are put forward to fuse the LPFNs to date, to deal with the linguistic Pythagorean fuzzy information in Example 26.For fair  Complexity 21 The ranking results of our proposed operators and the operators in [19].
As shown in Table 7, the ranking results derived from the LPFWA and LPFWG operators in [19] are the same as those derived from our proposed LPFWIPBM and LPFWIPGBM operators.This comparison result can verify the effectiveness of our proposed LPFWIPBM and LPFWIPGBM operators.
. .e Validation of Advantages of Our Studies.In this subsection, we further validate the advantages of our proposed LPFWIPBM and LPFWIPGBM operators by introducing an extra example.
Example .Suppose that an investment corporation plans to expand the investment scale.There are four investment solutions  = { 1 ,  2 ,  3 ,  4 } available for selection, which are the real estate project, the biological medicine project, the children education project, and the IT project.Regarding these investment projects, four factors should be considered: the national policy  1 , the size of market  2 , the investment risk  3 , and the competitive power  4 .The weight vector of these factors is  = (0.25, 0.25, 0.25, 0.25)  .The former two factors are benefit-type attributes, while the latter two ones are cost-type attributes.Hence, these four factors are divided into two groups,  1 = { 1 ,  2 } and  2 = { 3 ,  4 }.Based on the continuous LTS  = {  |  ∈ [0, 8]}, the decisionmaker utilizes the LPFNs to evaluate each project   with respect to each attribute   and then construct the preference information matrix  = (  ) 4×4 , which are presented in Table 8.
In Example 27, there exist two cost-type attributes  2 = { 3 ,  4 }.Hence, (61) is used to normalize the linguistic Pythagorean fuzzy preference information matrix shown in Table 8.The normalized one is listed in Table 9.
In the real applications, it usually happens that the MD or NMD of one of the aggregated LPFNs is  0 .To verify the superiority of the proposed interactional operational rules over the existing operational rules that are put forward by Garg [19] when the MD or NMD of one of the fused LPFNs is  0 , the LPFWA operator in [19] with novel interactional operational rules and the LPFWG operator in [19] with novel interactional operational rules are developed for comparison with the LPFWA operator in [19] and the LPFWG operator in [19].To validate the superiority of the PBM operator over the existing averaging and geometric operators, our proposed LPFWIPBM and LPFWIPGBM operators are compared to the LPFWA operator in [19] with novel interactional operational rules and the LPFWG operator in [19] with novel interactional operational rules.As shown in Table 10, the LPFWA operator in [19] with novel operational rules and the LPFWA operator in [19] get different ranking results, and the LPFWG operator in [19] with novel operational rules and the LPFWG operator in [19] also obtain different ranking results.That is because the LPFWA and LPFWG operators in [19] utilize the operational rules, which do not consider the interactions Table 10: The ranking results of our proposed operators and the operators in [19].
Although our proposed LPFWIPBM and LPFWIPGBM operators obtain the same optimal solution as the LPFWG operator in [19] with novel operational rules, their ranking results of the alternatives  1 ,  3 ,  4 are different.Moreover, our proposed LPFWIPBM and LPFWIPGBM operators show the different ranking results from the LPFWA operator in [19] with novel interactional operational rules.The reason is that our proposed LPFWIPBM and LPFWIPGBM operators are capable of capturing the interrelationship that the attribute values in the same set are related and the ones in the different sets are not related, which exists in Example 27.However, the LPFWA operator in [19] with novel operational rules and the LPFWG operator in [19] with novel operational rules do not consider the interrelationship existing in Example 27.
The comparison results verify the superiority of our proposed operational rules and the LPFWIPBM and LPFWIPGBM operators.
As analyzed above, the differences between our studies and existing study reported for LPFNs could be summarized and presented in Table 11.
As presented in Table 11, it can be seen that our proposed LPFWIPBM operator and LPFWIPGBM operator adopt the novel interactional operational rules, which could consider the interactions between the MD and NMD from different LPFNs.Their PBM function can also capture the interrelationships between input values in the MAGDM problems that the input values in the same set are related and the ones in the different sets are not related.In a word, our proposed LPFWIPBM and LPFWIPGBM operators could deal with these complex MAGDM problems.

Conclusions
In this paper, we firstly devise some novel operational rules to overcome the drawbacks of the existing operational rules for LPFNs.Then, the novel operational rules and PBM/PGBM operators are combined to propose the LPFIPBM, LPFWIPBM, LPFIPGBM, and LPFWIPGBM operators for fusing LPFNs.At the same time, the properties of these operators are studied and some special cases are discussed.Based on the LPFWIPBM and LPFWIPGBM operators, a novel MAGDM method is developed to process the MAGDM problems with LPFNs.Finally, a practical example with respect to the selection of SSD (solid state drive) productions is presented to demonstrate the implementation process of the proposed MAGDM method.The effect of the parameters  and  on the aggregated results is also discussed and the comparison analysis between the proposed LPFWIPBM and LPFWIPGBM operators and existing study reported for LPFNs is conducted to verify the effectiveness and superiority of the proposed interactional operational rules as well as the LPFWIPBM and LPFWIPGBM operators.
In the future research, we plan to put forward some efficient operators to fuse the LPFNs or preference relations

Table 8 :
The linguistic Pythagorean fuzzy preference information matrix  provided by the decision-maker .

Table 9 :
The normalized linguistic Pythagorean fuzzy preference information matrix  provided by the decision-maker .

Table 11 :
The differences between our studies and existing studies for LPFNs.