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

Authors: Mingwei Lin; Jiuhan Wei; Zeshui Xu; Riqing Chen
Journal: Complexity
 (2018)

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## Abstract

The partitioned Bonferroni mean (PBM) operator can efficiently aggregate inputs, which are divided into parts based on their interrelationships. To date, it has not been used to aggregate linguistic Pythagorean fuzzy numbers (LPFNs). In this paper, we extend the PBM operator and partitioned geometric Bonferroni mean (PGBM) operator to the linguistic Pythagorean fuzzy sets (LPFSs) and use them to develop a novel multiattribute group decision-making model under the linguistic Pythagorean fuzzy environment. We first define some novel operational laws for LPFNs, which take into consideration the interactions between the membership degree (MD) and nonmembership degree (NMD) from two different LPFNs. Based on these novel operational laws, we put forward the interaction PBM (LPFIPBM) operator, the weighted interaction PBM (LPFWIPBM) operator, the interaction PGBM (LPFIPGBM) operator, and the weighted interaction PGBM (LPFWIPGBM) operator. Then, we study some properties of these proposed operators and discuss their special cases. Based on the proposed LPFWIPBM and LPFWIPGBM operators, a novel multiattribute group decision-making model is developed to process the linguistic Pythagorean fuzzy information. Finally, some illustrative examples are introduced to compare our proposed methods with the existing ones.


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## Body

## 1. 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 [1–3]. 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 shows the distribution areas for IFNs and PFNs. The distribution 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 0.62+0.62<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].Figure 1
Possible distribution areas for IFNs and PFNs.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 decision-makers 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) [16–18].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 A1), the geographical position (attribute A2), the daylighting (attribute A3), and the apartment layout (attribute A4). In this case, there exists the relationship betweenA1 andA2, whileA3 is related withA4. These four attributes can be divided into two sets: P1=A1,A2 and P2=A3,A4. 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 Section2. 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.


## 2. 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.


### 2.1. 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 1 (see [19]).
LetY be a finite reference set and let S=sε∣ε∈0,τ with a positive integer τ be a continuous linguistic term set (LTS) [33]; then a LPFS on the finite set Y is expressed in a mathematical form as(1)L=y,smy,sny∣y∈Ywhere smy and sny are two functions, which are responsible for returning the MD and the NMD of the element y being the member of the LPFS L. Each two-tuple smy,sny in the LPFS is named as a LPFN or LPFV and it can be expressed in a simplified form as α=sm,sn satisfying that 0≤m≤τ, 0≤n≤τ, and 0≤m2+n2≤τ2. Let sπy=sτ2-m2-n2; then sπy can be referred to as the hesitance degree (HD) of the element y being the member of the LPFS L.
Garg [19] also presented the score value and the accuracy value for LPFNs and then developed a method for comparing two LPFNs.Definition 2 (see [19]).
There exist a continuous LTSS=sε∣ε∈0,τ and a LPFN α=sm,sn with sm, sn∈S; then the score value of the LPFN α is computed as(2)Dα=sτ2+m2-n2/2and the accuracy value is calculated as(3)Jα=sm2+n2The method for comparing two LPFNsα and β are described as follows:(i)
IfDα>Dβ, then α≻β, which means α is preferred to β.(ii)
IfDα=Dβ, then the accuracy values of the LPFNs α and β should be compared.
(a)
IfJα>Jβ, then α≻β.(b)
IfJα=Jβ, then α=β.Based on the t-conorm and t-norm, Garg gave several operational rules to compute two LPFNs as follows.Definition 3 (see [19]).
Assume a continuous LTSS=sε∣ε∈0,τ with a positive integer τ and two LPFNs α1=sm1,sn1 and α2=sm2,sn2 with sm1, sn1, sm2, sn2∈S; then(i)
α1⊕α2=sτm12/τ2+m22/τ2-m12m22/τ4,sτn1n2/τ2;(ii)
α1⊗α2=sτm1m2/τ2,sτn12/τ2+n22/τ2-n12n22/τ4;(iii)
λα1=sτ1-1-m12/τ2λ,sτn1/τλ;(iv)
α1λ=sτm1/τλ,sτ1-1-n12/τ2λ.


### 2.2. The 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.Definition 4 (see [34]).
An aggregation operator is a nondecreasing functionf:0,1n→0,1 satisfying that f0,0,…,0=0 and f1,1,…,1=1.
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 5 (see [36]).
Given a set of nonnegative input valuesα1,α2,…,αK, as well as two real numbers p,q≥0, the mathematical expression of the BM operator is described as(4)BMp,qα1,α2,…,αK=1KK-1∑u,v=1u≠vKαupαvq1/p+q
Its geometric version is also presented as follows.Definition 6 (see [37]).
Given a set of nonnegative input valuesα1,α2,…,αK, as well as two real numbers p,q≥0, the mathematical expression of the geometric BM (GBM) operator is described as(5)GBMp,qα1,α2,…,αK=1p+q∏u,v=1u≠vKpαu+qαv1/KK-1
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 7 (see [30]).
Given two nonnegative real numbersp,q≥0, as well as a set of nonnegative input values O=α1,α2,…,αK, which is divided into g different groups G1,G2,…,Gg, with ⋃t=1gGt=O, the mathematical expression of the PBM operator is described as(6)PBMp,qα1,α2,…,αK=1g∑t=1g1Gt∑αu∈Gtαup1Gt-1∑αv∈Gtαv≠αuαvq1/p+qwhere Gt is the number of elements in the group Gt and g denotes the number of groups ∑t=1gGt=K.Its geometric version is given as follows.Definition 8 (see [38]).
Given two nonnegative real numbersp,q≥0, as well as a set of nonnegative input values O=α1,α2,…,αK, which is divided into g different groups G1,G2,…,Gg, with ⋃t=1gGt=O, the mathematical expression of the PGBM operator is described as(7)PGBMp,qα1,α2,…,αK=∏t=1g1p+q∏αu,αv∈Gtαu≠αvpαu+qαv1/GtGt-11/gwhere Gt is the number of elements in the group Gt and g denotes the number of groups and ∑t=1gGt=K.


## 2.1. 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 1 (see [19]).
LetY be a finite reference set and let S=sε∣ε∈0,τ with a positive integer τ be a continuous linguistic term set (LTS) [33]; then a LPFS on the finite set Y is expressed in a mathematical form as(1)L=y,smy,sny∣y∈Ywhere smy and sny are two functions, which are responsible for returning the MD and the NMD of the element y being the member of the LPFS L. Each two-tuple smy,sny in the LPFS is named as a LPFN or LPFV and it can be expressed in a simplified form as α=sm,sn satisfying that 0≤m≤τ, 0≤n≤τ, and 0≤m2+n2≤τ2. Let sπy=sτ2-m2-n2; then sπy can be referred to as the hesitance degree (HD) of the element y being the member of the LPFS L.
Garg [19] also presented the score value and the accuracy value for LPFNs and then developed a method for comparing two LPFNs.Definition 2 (see [19]).
There exist a continuous LTSS=sε∣ε∈0,τ and a LPFN α=sm,sn with sm, sn∈S; then the score value of the LPFN α is computed as(2)Dα=sτ2+m2-n2/2and the accuracy value is calculated as(3)Jα=sm2+n2The method for comparing two LPFNsα and β are described as follows:(i)
IfDα>Dβ, then α≻β, which means α is preferred to β.(ii)
IfDα=Dβ, then the accuracy values of the LPFNs α and β should be compared.
(a)
IfJα>Jβ, then α≻β.(b)
IfJα=Jβ, then α=β.Based on the t-conorm and t-norm, Garg gave several operational rules to compute two LPFNs as follows.Definition 3 (see [19]).
Assume a continuous LTSS=sε∣ε∈0,τ with a positive integer τ and two LPFNs α1=sm1,sn1 and α2=sm2,sn2 with sm1, sn1, sm2, sn2∈S; then(i)
α1⊕α2=sτm12/τ2+m22/τ2-m12m22/τ4,sτn1n2/τ2;(ii)
α1⊗α2=sτm1m2/τ2,sτn12/τ2+n22/τ2-n12n22/τ4;(iii)
λα1=sτ1-1-m12/τ2λ,sτn1/τλ;(iv)
α1λ=sτm1/τλ,sτ1-1-n12/τ2λ.


## 2.2. The 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.Definition 4 (see [34]).
An aggregation operator is a nondecreasing functionf:0,1n→0,1 satisfying that f0,0,…,0=0 and f1,1,…,1=1.
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 5 (see [36]).
Given a set of nonnegative input valuesα1,α2,…,αK, as well as two real numbers p,q≥0, the mathematical expression of the BM operator is described as(4)BMp,qα1,α2,…,αK=1KK-1∑u,v=1u≠vKαupαvq1/p+q
Its geometric version is also presented as follows.Definition 6 (see [37]).
Given a set of nonnegative input valuesα1,α2,…,αK, as well as two real numbers p,q≥0, the mathematical expression of the geometric BM (GBM) operator is described as(5)GBMp,qα1,α2,…,αK=1p+q∏u,v=1u≠vKpαu+qαv1/KK-1
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 7 (see [30]).
Given two nonnegative real numbersp,q≥0, as well as a set of nonnegative input values O=α1,α2,…,αK, which is divided into g different groups G1,G2,…,Gg, with ⋃t=1gGt=O, the mathematical expression of the PBM operator is described as(6)PBMp,qα1,α2,…,αK=1g∑t=1g1Gt∑αu∈Gtαup1Gt-1∑αv∈Gtαv≠αuαvq1/p+qwhere Gt is the number of elements in the group Gt and g denotes the number of groups ∑t=1gGt=K.Its geometric version is given as follows.Definition 8 (see [38]).
Given two nonnegative real numbersp,q≥0, as well as a set of nonnegative input values O=α1,α2,…,αK, which is divided into g different groups G1,G2,…,Gg, with ⋃t=1gGt=O, the mathematical expression of the PGBM operator is described as(7)PGBMp,qα1,α2,…,αK=∏t=1g1p+q∏αu,αv∈Gtαu≠αvpαu+qαv1/GtGt-11/gwhere Gt is the number of elements in the group Gt and g denotes the number of groups and ∑t=1gGt=K.


## 3. 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 Definition3 do not work well for all the LPFNs, especially when the MD or NMD of one of LPFNs is s0. Assume that there are two LPFNs, α1=sm1,s0 and α2=sm2,sn2. Based on the additive operation in Definition 3, the aggregated NMD of α1⊕α2 is always 0 regardless of the value of the NMD of α2. 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 s0. 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 LPFWAα1,α2 is s0. In a word, the value of NMD in the LPFN α2 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.Definition 9.
Suppose that a continuous LTSS=sε∣ε∈0,τ with a positive integer τ and two LPFNs α1=sm1,sn1 and α2=sm2,sn2 with sm1, sn1, sm2, sn2∈S; then(8)iα1⊕α2=sτ1-∏i=121-mi2/τ2,sτ∏i=121-mi2/τ2-∏i=121-mi2/τ2-ni2/τ2;iiα1⊗α2=sτ∏i=121-ni2/τ2-∏i=121-mi2/τ2-ni2/τ2,sτ1-∏i=121-ni2/τ2;iiiλα1=sτ1-1-m12/τ2λ,sτ1-m12/τ2λ-1-m12/τ2-n12/τ2λ;ivα1λ=sτ1-n12/τ2λ-1-m12/τ2-n12/τ2λ,sτ1-1-n12/τ2λ.Example 10.
Let a LTSS=sε∣ε∈0,8, as well as two LPFNs α1=s6,s0 and α2=s5,s2; if the additive operation in Definition 3 is used, then α1⊕α2=s6.85,0. If the additive operation in Definition 9 is used, then α1⊕α2=s6.85,s1.32. Obviously, the operational result derived from the operational rules in Definition 9 is more reasonable.Some theorems can be derived from Definition9 as follows.Theorem 11.
Assume that a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, as well as two LPFNs α1=sm1,sn1 and α2=sm2,sn2 with sm1, sn1, sm2, sn2∈S; then α1⊕α2 is also a LPFN.Proof.
(i)τ1-∏i=121-mi2/τ2, τ∏i=121-mi2/τ2-∏i=121-mi2/τ2-ni2/τ2∈0,τ should be proven.
According to Definition9, α1⊕α2=sτ1-∏i=121-mi2/τ2,sτ∏i=121-mi2/τ2-∏i=121-mi2/τ2-ni2/τ2.
Since0≤mi≤τ, then τ1-1-02/τ22≤τ1-∏i=121-mi2/τ2≤τ1-1-τ2/τ22⇒0≤τ1-∏i=121-mi2/τ2≤τ.
Letf=τ∏i=121-mi2/τ2-∏i=121-mi2/τ2-ni2/τ2 and u=∏i=121-mi2/τ2-∏i=121-mi2/τ2-ni2/τ2; then(9)dfdm1=τ2·u-1/2·dudm1=τ2·u-1/2·1-m22τ2·-2m1τ2-1-m22τ2-n22τ2·-2m1τ2=-m1n22τ3u-1/2≤0dfdm2=τ2·u-1/2·dudm2=τ2·u-1/2·1-m12τ2·-2m2τ2-1-m12τ2-n12τ2·-2m2τ2=-m2n12τ3u-1/2≤0dfdn1=τ2·u-1/2·dudn1=τ2·u-1/2·-1-m22τ2-n22τ2·-2n1τ2=n1τ2-m22-n22τ3u-1/2≥0dfdn2=τ2·u-1/2·dudn2=τ2·u-1/2·-1-m12τ2-n12τ2·-2n2τ2=n2τ2-m12-n12τ3u-1/2≥0Thus, the value of f decreases as m1 or m2 increases, while it increases as n1 or n2 increases.
Since0≤ni≤τ, then(10)τ1-τ2τ22-1-τ2τ2-02τ22≤f=τ∏i=121-mi2τ2-∏i=121-mi2τ2-ni2τ2≤τ1-02τ22-1-02τ2-τ2τ22⇒0≤f=τ∏i=121-mi2τ2-∏i=121-mi2τ2-ni2τ2≤τ
(ii)τ1-∏i=121-mi2/τ22+τ∏i=121-mi2/τ2-∏i=121-mi2/τ2-ni2/τ22≤τ2 should be proven.(11)τ1-∏i=121-mi2τ22+τ∏i=121-mi2τ2-∏i=121-mi2τ2-ni2τ22=τ2-∏i=12τ2-mi2-ni2≤τ2which completes the proof of Theorem 11.Theorem 12.
Assume that a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, as well as two LPFNs α1=sm1,sn1 and α2=sm2,sn2 with sm1, sn1, sm2, sn2∈S; then α1⊗α2 is also a LPFN.Proof.
The proof is similar to that of Theorem11.Theorem 13.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, three real numbers λ,λ1,λ2, as well as any three LPFNs α=sm,sn, α1=sm1,sn1, and α2=sm2,sn2 with sm, sn, sm1, sn1, sm2, sn2∈S, one has(i)
α1⊕α2=α2⊕α1;(ii)
λα1⊕α2=λα1⊕λα2;(iii)
λ1+λ2α=λ1α⊕λ2α;(iv)
α1⊗α2=α2⊗α1;(v)
α1⊗α2λ=α1λ⊗α2λ;(vi)
αλ1+λ2=αλ1⊗αλ2.Proof.
Letϕ1=1-m12/τ2, φ1=1-m12/τ2-n12/τ2, ϕ2=1-m22/τ2, φ2=1-m22/τ2-n22/τ2, ϕ=1-m2/τ2, φ=1-m2/τ2-n2/τ2, γ1=1-n12/τ2, γ2=1-n22/τ2, and γ=1-n2/τ2.(12)iα1⊕α2=sτ1-∏i=121-mi2/τ2,sτ∏i=121-mi2/τ2-∏i=121-mi2/τ2-ni2/τ2=sτ1-ϕ1ϕ2,sτϕ1ϕ2-φ1φ2=α2⊕α1;iiλα1⊕α2=λsτ1-ϕ1ϕ2,sτϕ1ϕ2-φ1φ2=sτ1-1-τ21-ϕ1ϕ2/τ2λ,sτ1-τ21-ϕ1ϕ2/τ2λ-1-τ21-ϕ1ϕ2/τ2-τ2ϕ1ϕ2-φ1φ2/τ2λ=sτ1-ϕ1ϕ2λ,sτϕ1ϕ2λ-φ1φ2λSince λα1=sτ1-ϕ1λ,sτϕ1λ-φ1λ and λα2=sτ1-ϕ2λ,sτϕ2λ-φ2λ, then(13)λα1⊕λα2=sτ1-∏i=121-τ1-ϕiλ2/τ2,sτ∏i=121-τ1-ϕiλ2/τ2-∏i=121-τ1-ϕiλ2/τ2-τϕiλ-φiλ2/τ2=sτ1-∏i=12ϕiλ,sτ∏i=12ϕiλ-∏i=12φiλ=λα1⊕α2iiiλ1+λ2α=sτ1-1-m2/τ2λ1+λ2,sτ1-m2/τ2λ1+λ2-1-m2/τ2-n2/τ2λ1+λ2=sτ1-ϕλ1+λ2,sτϕλ1+λ2-φλ1+λ2Since λ1α=sτ1-ϕλ1,sτϕλ1-φλ1 and λ2α=sτ1-ϕλ2,sτϕλ2-φλ2, then(14)λ1α⊕λ2α=sτ1-1-τ1-ϕλ12/τ21-τ1-ϕλ22/τ2,sτ1-τ1-ϕλ12/τ21-τ1-ϕλ22/τ2-1-τ1-ϕλ12/τ2-τϕλ1-φλ12/τ21-τ1-ϕλ22/τ2-τϕλ2-φλ22/τ2=sτ1-ϕλ1+λ2,sτϕλ1+λ2-φλ1+λ2=λ1+λ2αivα1⊗α2=sτγ1γ2-φ1φ2,sτ1-γ1γ2=α2⊗α1vα1⊗α2λ=sτ1-τ1-γ1γ22/τ2λ-1-τγ1γ2-φ1φ22/τ2-τ1-γ1γ22/τ2λ,sτ1-1-τ1-γ1γ22/τ2λ=sτγ1γ2λ-φ1φ2λ,sτ1-γ1γ2λSince α1λ=sτγ1λ-φ1λ,sτ1-γ1λ and α2λ=sτγ2λ-φ2λ,sτ1-γ2λ, then(15)α1⊗α2=sτ∏i=121-τ1-γiλ2/τ2-∏i=121-τγiλ-φiλ2/τ2-τ1-γiλ2/τ2,sτ1-∏i=121-τ1-γiλ2/τ2=sτ∏i=12γiλ-∏i=12φiλ,sτ1-∏i=12γiλ=α1⊗α2λwhich completes the proof of Theorem 13.


## 4. 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.


### 4.1. The LPFIPBM Operator and the LPFWIPBM Operator

Definition 14.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the mathematical expression of the linguistic Pythagorean fuzzy interaction partitioned Bonferroni mean (LPFIPBM) aggregation operator is defined as(16)LPFIPBMp,qα1,α2,…,αK=1g⊕t=1g1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq1/p+qwhere Gt is the number of elements in the subset Gt and g denotes the number of subsets and ∑t=1gGt=K.According to the novel interactional operational rules developed for LPFNs in Definition9, several theorems can be derived from (16) as follows.Theorem 15.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the aggregated result obtained from (16) is still a LPFN, which is expressed as(17)LPFIPBMp,qα1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφupη1/Gt1/p+q1/gwhere εu=mu/τ,γu=nu/τ,εv=mv/τ,γv=nv/τ, 1-γu2=ϕu,1-γv2=ϕv, 1-εu2+γu2=φu,1-εv2+γv2=φv, and ρ=(∏αv∈Gtαv≠αu1-ϕvq+φvq)1/Gt-1,η=(∏αv∈Gtαv≠αuφvq)1/Gt-1.Proof.
According to Definition9, we have(18)αup=sτ1-nu2/τ2p-1-nu2/τ2+mu2/τ2p,sτ1-1-nu2/τ2p,αvq=sτ1-nv2/τ2q-1-nv2/τ2+mv2/τ2q,sτ1-1-nv2/τ2q.Let εu=mu/τ,γu=nu/τ,εv=mv/τ,γv=nv/τ; then(19)αup=sτ1-γu2p-1-εu2+γu2p,sτ1-1-γu2p,αvq=sτ1-γv2q-1-εv2+γv2q,sτ1-1-γu2q.Let 1-γu2=ϕu,1-γv2=ϕv and 1-εu2+γu2=φu,1-εv2+γv2=φv; then(20)αup=sτϕup-φup,sτ1-ϕup,αvq=sτϕvq-φvq,sτ1-ϕvq,⊕αv∈Gtαv≠αuαvq=sτ1-∏αv∈Gtαv≠αu1-ϕvq+φvq,sτ∏αv∈Gtαv≠αu1-ϕvq+φvq-∏αv∈Gtαv≠αuφvqAccording to Definition 9, we have(21)1Gt-1⊕αv∈Gtαv≠αuαvq=sτ1-1-τ1-∏αv∈Gtαv≠αu1-ϕvq+φvq2/τ21/Gt-1,sτ1-τ1-∏αv∈Gtαv≠αu1-ϕvq+φvq2/τ21/Gt-1-1-τ1-∏αv∈Gtαv≠αu1-ϕvq+φvq2/τ2-τ∏αv∈Gtαv≠αu1-ϕvq+φvq-∏αv∈Gtαv≠αuφvq2/τ21/Gt-1=sτ1-∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1,sτ∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1-∏αv∈Gtαv≠αuφvq1/Gt-1and (22)αup⊗1Gt-1⊕αv∈Gtαv≠αuαvq=sτϕup1-∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1+∏αv∈Gtαv≠αuφvq1/Gt-1-φup∏αv∈Gtαv≠αuφvq1/Gt-1,sτ1-ϕup1-∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1+∏αv∈Gtαv≠αuφvq1/Gt-1Let ρ=(∏αv∈Gtαv≠αu1-ϕvq+φvq)1/Gt-1,η=(∏αv∈Gtαv≠αuφvq)1/Gt-1; then the above equation can be transformed into(23)αup⊗1Gt-1⊕αv∈Gtαv≠αuαvq=sτϕup1-ρ+η-φupη,sτ1-ϕup1-ρ+ηThus, (24)⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq=sτ1-∏αu∈Gt1-τϕup1-ρ+η-φupη2/τ2,sτ∏αu∈Gt1-τϕup1-ρ+η-φupη2/τ2-∏αu∈Gt1-τϕup1-ρ+η-φupη2/τ2-τ1-ϕup1-ρ+η2/τ2=sτ1-∏αu∈Gt1-ϕup1-ρ+η+φupη,sτ∏αu∈Gt1-ϕup1-ρ+η+φupη-∏αu∈Gtφupηand (25)1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq=sτ1-1-τ1-∏αu∈Gt1-ϕup1-ρ+η+φupη2/τ21/Gt,sτ1-τ1-∏αu∈Gt1-ϕup1-ρ+η+φupη2/τ21/Gt-1-τ1-∏αu∈Gt1-ϕup1-ρ+η+φupη2/τ2-τ∏αu∈Gt1-ϕup1-ρ+η+φupη-∏αu∈Gtφupη2/τ21/Gt=sτ1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt,sτ∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt-∏αu∈Gtφupη1/GtThen, we have(26)1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq1/p+q=sτ1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q-∏αu∈Gtφupη1/Gt1/p+q,sτ1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+qand(27)⊕t=1g1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq1/p+q=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q,sτ∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q-∏t=1g∏αu∈Gtφupη1/Gt1/p+q
Thus,(28)1g⊕t=1g1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq1/p+q=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφupη1/Gt1/p+q1/gwhich completes the proof of Theorem 15.We continue to discuss the properties of the LPFIPBM operator as follows.Theorem 16 (idempotency).
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, if all αk are equal for any k, namely, αk=α=sm,sn, then(29)LPFIPBMp,qα1,α2,…,αK=α=sm,snProof.
Sincemu=mv=m and nu=nv=n, then we have(30)muτ=mvτ=εu=εv=ε,nuτ=nvτ=γu=γv=γ,1-γu2=1-γv2=ϕu=ϕv=ϕ,1-εu2+γu2=1-εv2+γv2=φu=φv=φ,and thus(31)ρ=∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1=1-ϕq+φq,η=∏αv∈Gtαv≠αuφvq1/Gt-1=φqBring the above equation into (17); then we have(32)LPFIPBMp,qα1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕp1-1-ϕq+φq+φq+φpφq1/Gt+∏αu∈Gtφpφq1/Gt1/p+q+∏αu∈Gtφpφq1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕp1-1-ϕq+φq+φq+φpφq1/Gt+∏αu∈Gtφpφq1/Gt1/p+q+∏αu∈Gtφpφq1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφpφq1/Gt1/p+q1/g=sτϕ-φ,sτ1-ϕ=sτ1-n2/τ2-1-m2/τ2-n2/τ2,sτ1-1-n2/τ2=sm,sn=αwhich finishes the proof of idempotency of the LPFIPBM operator.Theorem 17 (commutativity).
Given a continuous LTSS=sε∣0≤ε≤τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, if a set of LPFNs Gt′=αt1′,αt1′,…,αt1′ is any permutation of Gt=αt1,αt1,…,αt1 for each t, then(33)LPFIPBMp,qα1,α2,…,αK=LPFIPBMp,qα1′,α2′,…,αK′.Proof.
According to Theorem15, we have(34)LPFIPBMp,qα1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφupη1/Gt1/p+q1/gand (35)LPFIPBMp,qα1′,α2′,…,αK′=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕ′up1-ρ′+η′+φ′upη′1/Gt+∏αu∈Gtφ′upη′1/Gt1/p+q+∏αu∈Gtφ′upη′1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕ′up1-ρ′+η′+φupη1/Gt+∏αu∈Gtφ′upη′1/Gt1/p+q+∏αu∈Gtφ′upη′1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφ′upη′1/Gt1/p+q1/gFor each t, we have ∏αu∈Gt1-ϕ′up1-ρ′+η′+φ′upη′1/Gt=∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt and ∏αu∈Gtφ′upη′1/Gt=∏αu∈Gtφupη1/Gt.
Thus,LPFIPBMp,qα1,α2,…,αK=LPFIPBMp,qα1′,α2′,…,αK′, which finishes the proof of Theorem 17.By adjusting the values of the parametersp,q≥0, respectively, some special equations can be derived from the LPFIPBM operator as follows:(i)
Whenq→0, ρ=(∏αv∈Gtαv≠αu1-ϕvq+φvq)1/Gt-1=1 and η=(∏αv∈Gtαv≠αuφvq)1/Gt-1=1. Thus,(36)LPFIPBMp,0α1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup+φup1/Gt+∏αu∈Gtφup1/Gt1/p+∏αu∈Gtφup1/Gt1/p1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕup+φup1/Gt+∏αu∈Gtφup1/Gt1/p+∏αu∈Gtφup1/Gt1/p1/g-∏t=1g∏αu∈Gtφup1/Gt1/p1/g(ii)
Whenp=1 and q→0, the above equation can be transformed into(37)LPFIPBM1,0α1,α2,…,αK=sτ1-∏t=1g∏αu∈Gt1-ϕu+φu1/Gt1/g,sτ∏t=1g∏αu∈Gt1-ϕu+φu1/Gt1/g-∏t=1g∏αu∈Gtφu1/Gt1/g(iii)
Whenp→0, we have(38)LPFIPBM0,qα1,α2,…,αK=sτ1-∏t=1g1-1-ρ+η1/q+η1/q1/g,sτ∏t=1g1-1-ρ+η1/q+η1/q1/g-∏t=1gη1/q1/g(iv)
Whenp=1 and q=1, we have ρ=(∏αv∈Gtαv≠αu1-ϕv+φv)1/Gt-1 and η=(∏αv∈Gtαv≠αuφv)1/Gt-1. Thus, (39)LPFIPBM1,1α1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕu1-ρ+η+φuη1/Gt+∏αu∈Gtφuη1/Gt1/2+∏αu∈Gtφuη1/Gt1/21/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕu1-ρ+η+φuη1/Gt+∏αu∈Gtφuη1/Gt1/2+∏αu∈Gtφuη1/Gt1/21/g-∏t=1g∏αu∈Gtφuη1/Gt1/21/gAlthough 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.Definition 18.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the mathematical expression of the linguistic Pythagorean fuzzy weighted interaction partitioned Bonferroni mean (LPFWIPBM) aggregation operator is defined as(40)LPFWIPBMp,qα1,α2,…,αK=1g⊕t=1g1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq1/p+qwhere Gt is the number of LPFNs in the subset Gt, g denotes the number of subsets and ∑t=1gGt=K, ω=ω1,ω2,…,ωKT denotes the weight vector of the set of LPFNs O with ωi∈0,1, i=1,2,…,K, and ∑i=1Kωi=1.According to the novel interactional operational rules devised for LPFNs in Definition9, a theorem can be derived from (40) as follows.Theorem 19.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, a set of LPFNs O=α1,α2,…,αK, which is partitioned into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, and a weight vector ω=ω1,ω2,…,ωKT of LPFNs in O where ωi∈0,1, i=1,2,…,K, and ∑i=1Kωi=1, the aggregated result of (40) is still a LPFN, which is expressed as(41)LPFWIPBMp,qα1,α2,…,αK=sτ1-∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q1/g,sτ∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q1/g-∏t=1gη1/GtGt-1p+q1/gwhere εu=mu/τ, γu=nu/τ, εv=mv/τ, γv=nv/τ, 1-εu2ωu=ϕu, 1-εv2ωv=ϕv, 1-εu2-γu2ωu=φu, 1-εv2-γv2ωv=φv, ∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq=ρ, and ∏αu,αv∈Gtαu≠αvφupφvq=η.Proof.
According to Definition9, we have(42)ωuαu=sτ1-1-mu2/τ2ωu,sτ1-mu2/τ2ωu-1-mu2/τ2-nu2/τ2ωu,ωvαv=sτ1-1-mv2/τ2ωv,sτ1-mv2/τ2ωv-1-mv2/τ2-nv2/τ2ωvLet εu=mu/τ,γu=nu/τ,εv=mv/τ,γv=nv/τ; then the above equations can be transformed as(43)ωuαu=sτ1-1-εu2ωu,sτ1-εu2ωu-1-εu2-γu2ωu,ωvαv=sτ1-1-εv2ωv,sτ1-εv2ωv-1-εv2-γv2ωvThus,(44)ωuαup=sτ1-1-εu2ωu+1-εu2-γu2ωup-1-εu2-γu2ωup,sτ1-1-1-εu2ωu+1-εu2-γu2ωupand(45)ωvαvq=sτ1-1-εv2ωv+1-εv2-γv2ωvq-1-εv2-γv2ωvq,sτ1-1-1-εv2ωv+1-εv2-γv2ωvqLet 1-εu2ωu=ϕu,1-εv2ωv=ϕv and 1-εu2-γu2ωu=φu,1-εv2-γv2ωv=φv; then we have(46)ωuαup=sτ1-ϕu+φup-φup,sτ1-1-ϕu+φup,ωvαvq=sτ1-ϕv+φvq-φvq,sτ1-1-ϕv+φvqωuαup⊗ωvαvq=sτ1-ϕu+φup1-ϕv+φvq-φupφvq,sτ1-1-ϕu+φup1-ϕv+φvqThus,(47)⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq=sτ1-∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq,sτ∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq-∏αu,αv∈Gtαu≠αvφupφvqLet ∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq=ρ and ∏αu,αv∈Gtαu≠αvφupφvq=η; then the above equation can be transformed into(48)⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq=sτ1-ρ,sτρ-ηand we can have (49)1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq=sτ1-ρ1/GtGt-1,sτρ1/GtGt-1-η1/GtGt-1.Thus, (50)1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq1/p+q=sτ1-ρ1/GtGt-1+η1/GtGt-11/p+q-η1/GtGt-1p+q,sτ1-1-ρ1/GtGt-1+η1/GtGt-11/p+qand(51)⊕t=1g1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq1/p+q=sτ1-∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q,sτ∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q-∏t=1gη1/GtGt-1p+qFinally, we can have(52)1g⊕t=1g1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq1/p+q=sτ1-∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q1/g,sτ∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q1/g-∏t=1gη1/GtGt-1p+q1/gwhich finishes the proof of Theorem 19.


### 4.2. The 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 decision-makers 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.Definition 20.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the mathematical expression of the linguistic Pythagorean fuzzy interaction partitioned geometric Bonferroni mean (LPFIPGBM) operator is defined as(53)LPFIPGBMp,qα1,α2,…,αK=⊗t=1g1p+q⊗αu,αv∈Gtαu≠αvpαu⊕qαv1/GtGt-11/gwhere Gt is the number of elements in the subset Gt and g denotes the number of subsets and ∑t=1gGt=K.According to the novel interactional operational rules developed for LPFNs in Definition9, several theorems can be derived from (53) as follows.Theorem 21.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the aggregated result obtained from (53) is still a LPFN, which is expressed as(54)LPFIPGBMp,qα1,α2,…,αK=sτ∏t=1g1-1-ρ+η1/p+q+η1/p+q1/g-∏t=1gη1/p+q1/g,sτ1-∏t=1g1-1-ρ+η1/p+q+η1/p+q1/gwhere εu=mu/τ, γu=nu/τ, εv=mv/τ, γv=nv/τ, 1-εu2=ϕu, 1-εv2=ϕv, 1-εu2-γu2=φu, 1-εv2-γv2=φv, ρ=(∏αu,αv∈Gtαu≠αv1-ϕupϕvq+φupφvq)1/GtGt-1, and η=(∏αu,αv∈Gtαu≠αvφupφvq)1/GtGt-1.Proof.
The proof process is similar to that of Theorem15, which is omitted here.Then, some properties of the LPFIPGBM operator are given as follows.Theorem 22 (idempotency).
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, if all αk are equal for any k, namely, αk=α=sm,sn, then(55)LPFIPGBMp,qα1,α2,…,αK=α=sm,sn.Theorem 23 (commutativity).
Given a continuous LTSS=sε∣0≤ε≤τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, if a set of LPFNs Gt′=αt1′,αt1′,…,αt1′ is any permutation of Gt=αt1,αt1,…,αt1 for each t, then(56)LPFIPGBMp,qα1,α2,…,αK=LPFIPGBMp,qα1′,α2′,…,αK′.Proof.
The proof processes of Theorems22 and 23 are similar to those of Theorems 16 and 17, which are omitted here.By adjusting the values of the parametersp,q≥0, respectively, some special equations can be derived from the LPFIPGBM operator as follows:(i)
Whenp=1 and q→0, ρ=(∏αu,αv∈Gtαu≠αv1-ϕu+φu)1/GtGt-1 and η=(∏αu,αv∈Gtαu≠αvφu)1/GtGt-1. Thus,(57)LPFIPGBMp,qα1,α2,…,αK=sτ∏t=1g∏αu,αv∈Gtαu≠αv1-ϕu+φu1/GtGt-11/g-∏t=1g∏αu,αv∈Gtαu≠αvφu1/GtGt-11/g,sτ1-∏t=1g∏αu,αv∈Gtαu≠αv1-ϕu+φu1/GtGt-11/g(ii)
Whenp=1 and q=1, ρ=(∏αu,αv∈Gtαu≠αv1-ϕuϕv+φuφv)1/GtGt-1, and η=(∏αu,αv∈Gtαu≠αvφuφv)1/GtGt-1. Thus, (58)LPFIPGBMp,qα1,α2,…,αK=sτ∏t=1g1-1-∏αu,αv∈Gtαu≠αv1-ϕuϕv+φuφv1/GtGt-1+∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/2+∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/21/g-∏t=1g∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/21/g,sτ1-∏t=1g1-1-∏αu,αv∈Gtαu≠αv1-ϕuϕv+φuφv1/GtGt-1+∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/2+∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/21/g.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.Definition 24.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the mathematical expression of the linguistic Pythagorean fuzzy weighted interaction partitioned geometric Bonferroni mean (LPFWIPGBM) aggregation operator is defined as(59)LPFWIPGBMp,qα1,α2,…,αK=⊗t=1g1p+q⊗αu,αv∈Gtαu≠αvpαuωu⊕qαvωv1/GtGt-11/gwhere Gt is the number of LPFNs in the subset Gt, g denotes the number of subsets, and ∑t=1gGt=K, ω=ω1,ω2,…,ωKT denotes the weight vector of the set of LPFNs O with ωi∈0,1, i=1,2,…,K, and ∑i=1Kωi=1.According to the novel interactional operational rules devised for LPFNs in Definition9, a theorem can be derived from (59) as follows.Theorem 25.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, a set of LPFNs O=α1,α2,…,αK, which is partitioned into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, and a weight vector ω=ω1,ω2,…,ωKT of LPFNs in O where ωi∈0,1, i=1,2,…,K, and ∑i=1Kωi=1, the aggregated result of (59) is still a LPFN, which is expressed as(60)LPFWIPGBMp,qα1,α2,…,αK=sτ∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-11/p+q1/g-∏t=1gη1/GtGt-11/p+q1/g,sτ1-∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-11/p+q1/gwhere εu=mu/τ,γu=nu/τ, εv=mv/τ, γv=nv/τ, 1-γu2ωu=ϕu, 1-γv2ωv=ϕv, 1-εu2-γu2ωu=φu, 1-εv2-γv2ωv=φv, ∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq=ρ, and ∏αu,αv∈Gtαu≠αvφupφvq=η.Proof.
The proof process of this theorem is similar to that of Theorem19, which is omitted here.


## 4.1. The LPFIPBM Operator and the LPFWIPBM Operator

Definition 14.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the mathematical expression of the linguistic Pythagorean fuzzy interaction partitioned Bonferroni mean (LPFIPBM) aggregation operator is defined as(16)LPFIPBMp,qα1,α2,…,αK=1g⊕t=1g1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq1/p+qwhere Gt is the number of elements in the subset Gt and g denotes the number of subsets and ∑t=1gGt=K.According to the novel interactional operational rules developed for LPFNs in Definition9, several theorems can be derived from (16) as follows.Theorem 15.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the aggregated result obtained from (16) is still a LPFN, which is expressed as(17)LPFIPBMp,qα1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφupη1/Gt1/p+q1/gwhere εu=mu/τ,γu=nu/τ,εv=mv/τ,γv=nv/τ, 1-γu2=ϕu,1-γv2=ϕv, 1-εu2+γu2=φu,1-εv2+γv2=φv, and ρ=(∏αv∈Gtαv≠αu1-ϕvq+φvq)1/Gt-1,η=(∏αv∈Gtαv≠αuφvq)1/Gt-1.Proof.
According to Definition9, we have(18)αup=sτ1-nu2/τ2p-1-nu2/τ2+mu2/τ2p,sτ1-1-nu2/τ2p,αvq=sτ1-nv2/τ2q-1-nv2/τ2+mv2/τ2q,sτ1-1-nv2/τ2q.Let εu=mu/τ,γu=nu/τ,εv=mv/τ,γv=nv/τ; then(19)αup=sτ1-γu2p-1-εu2+γu2p,sτ1-1-γu2p,αvq=sτ1-γv2q-1-εv2+γv2q,sτ1-1-γu2q.Let 1-γu2=ϕu,1-γv2=ϕv and 1-εu2+γu2=φu,1-εv2+γv2=φv; then(20)αup=sτϕup-φup,sτ1-ϕup,αvq=sτϕvq-φvq,sτ1-ϕvq,⊕αv∈Gtαv≠αuαvq=sτ1-∏αv∈Gtαv≠αu1-ϕvq+φvq,sτ∏αv∈Gtαv≠αu1-ϕvq+φvq-∏αv∈Gtαv≠αuφvqAccording to Definition 9, we have(21)1Gt-1⊕αv∈Gtαv≠αuαvq=sτ1-1-τ1-∏αv∈Gtαv≠αu1-ϕvq+φvq2/τ21/Gt-1,sτ1-τ1-∏αv∈Gtαv≠αu1-ϕvq+φvq2/τ21/Gt-1-1-τ1-∏αv∈Gtαv≠αu1-ϕvq+φvq2/τ2-τ∏αv∈Gtαv≠αu1-ϕvq+φvq-∏αv∈Gtαv≠αuφvq2/τ21/Gt-1=sτ1-∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1,sτ∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1-∏αv∈Gtαv≠αuφvq1/Gt-1and (22)αup⊗1Gt-1⊕αv∈Gtαv≠αuαvq=sτϕup1-∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1+∏αv∈Gtαv≠αuφvq1/Gt-1-φup∏αv∈Gtαv≠αuφvq1/Gt-1,sτ1-ϕup1-∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1+∏αv∈Gtαv≠αuφvq1/Gt-1Let ρ=(∏αv∈Gtαv≠αu1-ϕvq+φvq)1/Gt-1,η=(∏αv∈Gtαv≠αuφvq)1/Gt-1; then the above equation can be transformed into(23)αup⊗1Gt-1⊕αv∈Gtαv≠αuαvq=sτϕup1-ρ+η-φupη,sτ1-ϕup1-ρ+ηThus, (24)⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq=sτ1-∏αu∈Gt1-τϕup1-ρ+η-φupη2/τ2,sτ∏αu∈Gt1-τϕup1-ρ+η-φupη2/τ2-∏αu∈Gt1-τϕup1-ρ+η-φupη2/τ2-τ1-ϕup1-ρ+η2/τ2=sτ1-∏αu∈Gt1-ϕup1-ρ+η+φupη,sτ∏αu∈Gt1-ϕup1-ρ+η+φupη-∏αu∈Gtφupηand (25)1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq=sτ1-1-τ1-∏αu∈Gt1-ϕup1-ρ+η+φupη2/τ21/Gt,sτ1-τ1-∏αu∈Gt1-ϕup1-ρ+η+φupη2/τ21/Gt-1-τ1-∏αu∈Gt1-ϕup1-ρ+η+φupη2/τ2-τ∏αu∈Gt1-ϕup1-ρ+η+φupη-∏αu∈Gtφupη2/τ21/Gt=sτ1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt,sτ∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt-∏αu∈Gtφupη1/GtThen, we have(26)1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq1/p+q=sτ1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q-∏αu∈Gtφupη1/Gt1/p+q,sτ1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+qand(27)⊕t=1g1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq1/p+q=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q,sτ∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q-∏t=1g∏αu∈Gtφupη1/Gt1/p+q
Thus,(28)1g⊕t=1g1Gt⊕αu∈Gtαup⊗1Gt-1⊕αv∈Gtαv≠αuαvq1/p+q=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφupη1/Gt1/p+q1/gwhich completes the proof of Theorem 15.We continue to discuss the properties of the LPFIPBM operator as follows.Theorem 16 (idempotency).
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, if all αk are equal for any k, namely, αk=α=sm,sn, then(29)LPFIPBMp,qα1,α2,…,αK=α=sm,snProof.
Sincemu=mv=m and nu=nv=n, then we have(30)muτ=mvτ=εu=εv=ε,nuτ=nvτ=γu=γv=γ,1-γu2=1-γv2=ϕu=ϕv=ϕ,1-εu2+γu2=1-εv2+γv2=φu=φv=φ,and thus(31)ρ=∏αv∈Gtαv≠αu1-ϕvq+φvq1/Gt-1=1-ϕq+φq,η=∏αv∈Gtαv≠αuφvq1/Gt-1=φqBring the above equation into (17); then we have(32)LPFIPBMp,qα1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕp1-1-ϕq+φq+φq+φpφq1/Gt+∏αu∈Gtφpφq1/Gt1/p+q+∏αu∈Gtφpφq1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕp1-1-ϕq+φq+φq+φpφq1/Gt+∏αu∈Gtφpφq1/Gt1/p+q+∏αu∈Gtφpφq1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφpφq1/Gt1/p+q1/g=sτϕ-φ,sτ1-ϕ=sτ1-n2/τ2-1-m2/τ2-n2/τ2,sτ1-1-n2/τ2=sm,sn=αwhich finishes the proof of idempotency of the LPFIPBM operator.Theorem 17 (commutativity).
Given a continuous LTSS=sε∣0≤ε≤τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, if a set of LPFNs Gt′=αt1′,αt1′,…,αt1′ is any permutation of Gt=αt1,αt1,…,αt1 for each t, then(33)LPFIPBMp,qα1,α2,…,αK=LPFIPBMp,qα1′,α2′,…,αK′.Proof.
According to Theorem15, we have(34)LPFIPBMp,qα1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt+∏αu∈Gtφupη1/Gt1/p+q+∏αu∈Gtφupη1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφupη1/Gt1/p+q1/gand (35)LPFIPBMp,qα1′,α2′,…,αK′=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕ′up1-ρ′+η′+φ′upη′1/Gt+∏αu∈Gtφ′upη′1/Gt1/p+q+∏αu∈Gtφ′upη′1/Gt1/p+q1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕ′up1-ρ′+η′+φupη1/Gt+∏αu∈Gtφ′upη′1/Gt1/p+q+∏αu∈Gtφ′upη′1/Gt1/p+q1/g-∏t=1g∏αu∈Gtφ′upη′1/Gt1/p+q1/gFor each t, we have ∏αu∈Gt1-ϕ′up1-ρ′+η′+φ′upη′1/Gt=∏αu∈Gt1-ϕup1-ρ+η+φupη1/Gt and ∏αu∈Gtφ′upη′1/Gt=∏αu∈Gtφupη1/Gt.
Thus,LPFIPBMp,qα1,α2,…,αK=LPFIPBMp,qα1′,α2′,…,αK′, which finishes the proof of Theorem 17.By adjusting the values of the parametersp,q≥0, respectively, some special equations can be derived from the LPFIPBM operator as follows:(i)
Whenq→0, ρ=(∏αv∈Gtαv≠αu1-ϕvq+φvq)1/Gt-1=1 and η=(∏αv∈Gtαv≠αuφvq)1/Gt-1=1. Thus,(36)LPFIPBMp,0α1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕup+φup1/Gt+∏αu∈Gtφup1/Gt1/p+∏αu∈Gtφup1/Gt1/p1/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕup+φup1/Gt+∏αu∈Gtφup1/Gt1/p+∏αu∈Gtφup1/Gt1/p1/g-∏t=1g∏αu∈Gtφup1/Gt1/p1/g(ii)
Whenp=1 and q→0, the above equation can be transformed into(37)LPFIPBM1,0α1,α2,…,αK=sτ1-∏t=1g∏αu∈Gt1-ϕu+φu1/Gt1/g,sτ∏t=1g∏αu∈Gt1-ϕu+φu1/Gt1/g-∏t=1g∏αu∈Gtφu1/Gt1/g(iii)
Whenp→0, we have(38)LPFIPBM0,qα1,α2,…,αK=sτ1-∏t=1g1-1-ρ+η1/q+η1/q1/g,sτ∏t=1g1-1-ρ+η1/q+η1/q1/g-∏t=1gη1/q1/g(iv)
Whenp=1 and q=1, we have ρ=(∏αv∈Gtαv≠αu1-ϕv+φv)1/Gt-1 and η=(∏αv∈Gtαv≠αuφv)1/Gt-1. Thus, (39)LPFIPBM1,1α1,α2,…,αK=sτ1-∏t=1g1-1-∏αu∈Gt1-ϕu1-ρ+η+φuη1/Gt+∏αu∈Gtφuη1/Gt1/2+∏αu∈Gtφuη1/Gt1/21/g,sτ∏t=1g1-1-∏αu∈Gt1-ϕu1-ρ+η+φuη1/Gt+∏αu∈Gtφuη1/Gt1/2+∏αu∈Gtφuη1/Gt1/21/g-∏t=1g∏αu∈Gtφuη1/Gt1/21/gAlthough 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.Definition 18.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the mathematical expression of the linguistic Pythagorean fuzzy weighted interaction partitioned Bonferroni mean (LPFWIPBM) aggregation operator is defined as(40)LPFWIPBMp,qα1,α2,…,αK=1g⊕t=1g1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq1/p+qwhere Gt is the number of LPFNs in the subset Gt, g denotes the number of subsets and ∑t=1gGt=K, ω=ω1,ω2,…,ωKT denotes the weight vector of the set of LPFNs O with ωi∈0,1, i=1,2,…,K, and ∑i=1Kωi=1.According to the novel interactional operational rules devised for LPFNs in Definition9, a theorem can be derived from (40) as follows.Theorem 19.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, a set of LPFNs O=α1,α2,…,αK, which is partitioned into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, and a weight vector ω=ω1,ω2,…,ωKT of LPFNs in O where ωi∈0,1, i=1,2,…,K, and ∑i=1Kωi=1, the aggregated result of (40) is still a LPFN, which is expressed as(41)LPFWIPBMp,qα1,α2,…,αK=sτ1-∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q1/g,sτ∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q1/g-∏t=1gη1/GtGt-1p+q1/gwhere εu=mu/τ, γu=nu/τ, εv=mv/τ, γv=nv/τ, 1-εu2ωu=ϕu, 1-εv2ωv=ϕv, 1-εu2-γu2ωu=φu, 1-εv2-γv2ωv=φv, ∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq=ρ, and ∏αu,αv∈Gtαu≠αvφupφvq=η.Proof.
According to Definition9, we have(42)ωuαu=sτ1-1-mu2/τ2ωu,sτ1-mu2/τ2ωu-1-mu2/τ2-nu2/τ2ωu,ωvαv=sτ1-1-mv2/τ2ωv,sτ1-mv2/τ2ωv-1-mv2/τ2-nv2/τ2ωvLet εu=mu/τ,γu=nu/τ,εv=mv/τ,γv=nv/τ; then the above equations can be transformed as(43)ωuαu=sτ1-1-εu2ωu,sτ1-εu2ωu-1-εu2-γu2ωu,ωvαv=sτ1-1-εv2ωv,sτ1-εv2ωv-1-εv2-γv2ωvThus,(44)ωuαup=sτ1-1-εu2ωu+1-εu2-γu2ωup-1-εu2-γu2ωup,sτ1-1-1-εu2ωu+1-εu2-γu2ωupand(45)ωvαvq=sτ1-1-εv2ωv+1-εv2-γv2ωvq-1-εv2-γv2ωvq,sτ1-1-1-εv2ωv+1-εv2-γv2ωvqLet 1-εu2ωu=ϕu,1-εv2ωv=ϕv and 1-εu2-γu2ωu=φu,1-εv2-γv2ωv=φv; then we have(46)ωuαup=sτ1-ϕu+φup-φup,sτ1-1-ϕu+φup,ωvαvq=sτ1-ϕv+φvq-φvq,sτ1-1-ϕv+φvqωuαup⊗ωvαvq=sτ1-ϕu+φup1-ϕv+φvq-φupφvq,sτ1-1-ϕu+φup1-ϕv+φvqThus,(47)⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq=sτ1-∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq,sτ∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq-∏αu,αv∈Gtαu≠αvφupφvqLet ∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq=ρ and ∏αu,αv∈Gtαu≠αvφupφvq=η; then the above equation can be transformed into(48)⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq=sτ1-ρ,sτρ-ηand we can have (49)1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq=sτ1-ρ1/GtGt-1,sτρ1/GtGt-1-η1/GtGt-1.Thus, (50)1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq1/p+q=sτ1-ρ1/GtGt-1+η1/GtGt-11/p+q-η1/GtGt-1p+q,sτ1-1-ρ1/GtGt-1+η1/GtGt-11/p+qand(51)⊕t=1g1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq1/p+q=sτ1-∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q,sτ∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q-∏t=1gη1/GtGt-1p+qFinally, we can have(52)1g⊕t=1g1GtGt-1⊕αu,αv∈Gtαu≠αvωuαup⊗ωvαvq1/p+q=sτ1-∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q1/g,sτ∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-1p+q1/g-∏t=1gη1/GtGt-1p+q1/gwhich finishes the proof of Theorem 19.


## 4.2. The 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 decision-makers 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.Definition 20.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the mathematical expression of the linguistic Pythagorean fuzzy interaction partitioned geometric Bonferroni mean (LPFIPGBM) operator is defined as(53)LPFIPGBMp,qα1,α2,…,αK=⊗t=1g1p+q⊗αu,αv∈Gtαu≠αvpαu⊕qαv1/GtGt-11/gwhere Gt is the number of elements in the subset Gt and g denotes the number of subsets and ∑t=1gGt=K.According to the novel interactional operational rules developed for LPFNs in Definition9, several theorems can be derived from (53) as follows.Theorem 21.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the aggregated result obtained from (53) is still a LPFN, which is expressed as(54)LPFIPGBMp,qα1,α2,…,αK=sτ∏t=1g1-1-ρ+η1/p+q+η1/p+q1/g-∏t=1gη1/p+q1/g,sτ1-∏t=1g1-1-ρ+η1/p+q+η1/p+q1/gwhere εu=mu/τ, γu=nu/τ, εv=mv/τ, γv=nv/τ, 1-εu2=ϕu, 1-εv2=ϕv, 1-εu2-γu2=φu, 1-εv2-γv2=φv, ρ=(∏αu,αv∈Gtαu≠αv1-ϕupϕvq+φupφvq)1/GtGt-1, and η=(∏αu,αv∈Gtαu≠αvφupφvq)1/GtGt-1.Proof.
The proof process is similar to that of Theorem15, which is omitted here.Then, some properties of the LPFIPGBM operator are given as follows.Theorem 22 (idempotency).
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, if all αk are equal for any k, namely, αk=α=sm,sn, then(55)LPFIPGBMp,qα1,α2,…,αK=α=sm,sn.Theorem 23 (commutativity).
Given a continuous LTSS=sε∣0≤ε≤τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, if a set of LPFNs Gt′=αt1′,αt1′,…,αt1′ is any permutation of Gt=αt1,αt1,…,αt1 for each t, then(56)LPFIPGBMp,qα1,α2,…,αK=LPFIPGBMp,qα1′,α2′,…,αK′.Proof.
The proof processes of Theorems22 and 23 are similar to those of Theorems 16 and 17, which are omitted here.By adjusting the values of the parametersp,q≥0, respectively, some special equations can be derived from the LPFIPGBM operator as follows:(i)
Whenp=1 and q→0, ρ=(∏αu,αv∈Gtαu≠αv1-ϕu+φu)1/GtGt-1 and η=(∏αu,αv∈Gtαu≠αvφu)1/GtGt-1. Thus,(57)LPFIPGBMp,qα1,α2,…,αK=sτ∏t=1g∏αu,αv∈Gtαu≠αv1-ϕu+φu1/GtGt-11/g-∏t=1g∏αu,αv∈Gtαu≠αvφu1/GtGt-11/g,sτ1-∏t=1g∏αu,αv∈Gtαu≠αv1-ϕu+φu1/GtGt-11/g(ii)
Whenp=1 and q=1, ρ=(∏αu,αv∈Gtαu≠αv1-ϕuϕv+φuφv)1/GtGt-1, and η=(∏αu,αv∈Gtαu≠αvφuφv)1/GtGt-1. Thus, (58)LPFIPGBMp,qα1,α2,…,αK=sτ∏t=1g1-1-∏αu,αv∈Gtαu≠αv1-ϕuϕv+φuφv1/GtGt-1+∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/2+∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/21/g-∏t=1g∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/21/g,sτ1-∏t=1g1-1-∏αu,αv∈Gtαu≠αv1-ϕuϕv+φuφv1/GtGt-1+∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/2+∏αu,αv∈Gtαu≠αvφuφv1/GtGt-11/21/g.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.Definition 24.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, and a set of LPFNs O=α1,α2,…,αK, which is divided into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, the mathematical expression of the linguistic Pythagorean fuzzy weighted interaction partitioned geometric Bonferroni mean (LPFWIPGBM) aggregation operator is defined as(59)LPFWIPGBMp,qα1,α2,…,αK=⊗t=1g1p+q⊗αu,αv∈Gtαu≠αvpαuωu⊕qαvωv1/GtGt-11/gwhere Gt is the number of LPFNs in the subset Gt, g denotes the number of subsets, and ∑t=1gGt=K, ω=ω1,ω2,…,ωKT denotes the weight vector of the set of LPFNs O with ωi∈0,1, i=1,2,…,K, and ∑i=1Kωi=1.According to the novel interactional operational rules devised for LPFNs in Definition9, a theorem can be derived from (59) as follows.Theorem 25.
Given a continuous LTSS=sε∣ε∈0,τ with a positive integer τ, two real numbers p,q≥0, a set of LPFNs O=α1,α2,…,αK, which is partitioned into g different subsets G1,G2,…,Gg, where αk=smk,snkk=1,2,…,K;smk,snk∈S and ⋃t=1gGt=O, and a weight vector ω=ω1,ω2,…,ωKT of LPFNs in O where ωi∈0,1, i=1,2,…,K, and ∑i=1Kωi=1, the aggregated result of (59) is still a LPFN, which is expressed as(60)LPFWIPGBMp,qα1,α2,…,αK=sτ∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-11/p+q1/g-∏t=1gη1/GtGt-11/p+q1/g,sτ1-∏t=1g1-1-ρ1/GtGt-1+η1/GtGt-11/p+q+η1/GtGt-11/p+q1/gwhere εu=mu/τ,γu=nu/τ, εv=mv/τ, γv=nv/τ, 1-γu2ωu=ϕu, 1-γv2ωv=ϕv, 1-εu2-γu2ωu=φu, 1-εv2-γv2ωv=φv, ∏αu,αv∈Gtαu≠αv1-1-ϕu+φup1-ϕv+φvq+φupφvq=ρ, and ∏αu,αv∈Gtαu≠αvφupφvq=η.Proof.
The proof process of this theorem is similar to that of Theorem19, which is omitted here.


## 5. Novel MAGDM Method Using the LPFWIPBM or LPFWIPGBM Operator

In the MAGDM problems, there exists a set of alternativesA=A1,A2,…,Ah with a set of attributes B=B1,B2,…,Bw, which can be split into two categories, namely, the benefit attributes and the cost attributes, respectively. A group of decision-makers M=M1,M2,…,Mr are invited to select the linguistic terms from a continuous LTS S=sε∣ε∈0,τ with a positive integer τ to construct the LPFNs for modeling the preference information. Suppose that Ll=αijlh×w is a preference information matrix that is given by the decision-maker Ml, where αijl=smijl,snijl is a LPFN modeling the preference information given by the decision-maker Ml when the alternative Ai with respect to the attribute Bj is assessed. Suppose that κ=κ1,κ2,…,κrT represents the weight vector of the group of decision-makers M, where κk is the weight of the decision-maker Mk satisfying 0≤κk≤1 and ∑k=1rκk=1. Let ω=ω1,ω2,…,ωwT express the weight vector of the set of attributes B, where ωt denotes the weight of the attribute Bt satisfying 0≤ωt≤1 and ∑t=1wωt=1.For the above MAGDM problems, a novel MAGDM method exploiting the LPFWIPBM operator or the LPFWIPGBM operator is currently devised as follows.Step 1.
Normalize the preference information matrixLl=αijlh×w 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 Ζl=zijlh×wl=1,2,…,r is derived according to the following rule:(61)zijl=smijl,snijl=smijl,snijlforbenefit-typeattributeBjsnijl,smijlforcost-typeattributeBjStep 2.
Exploit the LPFWIPBM operator(62)zil=smil,snil=LPFWIPBMzi1l,zi2l,…,ziwlor the LPFWIPGBM operator(63)zil=smil,snil=LPFWIPGBMzi1l,zi2l,…,ziwlto compute the fused value of all the attribute values of each alternative Ai which are given by the decision-maker Ml.Step 3.
Exploit the LPFWIPBM operator(64)zi=smi,sni=LPFWIPBMzi1,zi2,…,ziror the LPFWIPGBM operator(65)zi=smi,sni=LPFWIPGBMzi1,zi2,…,zirto compute the fused value of the fused attribute value of each alternative Ai.Step 4.
Use Definition2 to compute the score value and the accuracy value of all the fused value zi as(66)Dzi=sτ2+mi2-ni2/2and(67)Jzi=smi2+ni2Step 5.
All the alternativesA1,A2,…,Ah are ranked based on their score value and accuracy value, and the optimal one(s) is selected.Step 6.
End.


## 6. 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.


### 6.1. Illustrative Example

In this subsection, we introduce a practical case concerning the selection of SSD productions to show the implementation process of the proposed MAGDM method.Example 26.
A company plans to purchase a batch of SSDs to upgrade their computer systems. Currently, there are four kinds of SSD productionsA=A1,A2,A3,A4 available to be chosen. Three decision-makers M=M1,M2,M3 are invited to assess these SSD productions considering four attributes, which are the basic requirements B1, physical characteristics B2, brand B3, and overall performance B4. Naturally, the former two attributes are benefit-type, while the latter two attributes are cost-type. Therefore, these four attributes are split into two groups, G1=B1,B2 and G2=B3,B4. The weight vector of the attributes is ω=0.25,0.25,0.20,0.30T and the weight vector of the decision-makers is κ=0.40,0.25,0.35T. Based on the continuous LTS S=sε∣ε∈0,8, the decision-makers utilize the LPFNs to evaluate each SSD production Ai with respect to each attribute Bj and then construct the preference information matrices Ll=αijl4×4l=1,2,3 that are presented in Tables 1–3.Table 1
The linguistic Pythagorean fuzzy preference information matrixL1 provided by the decision-maker M1.
B 1 B 2 B 3 B 4 A 1 s 1 , s 6 s 3 , s 7 s 6 , s 2 s 7 , s 2 A 2 s 4 , s 3 s 5 , s 4 s 1 , s 3 s 1 , s 5 A 3 s 7 , s 3 s 7 , s 1 s 2 , s 6 s 1 , s 6 A 4 s 3 , s 7 s 4 , s 3 s 2 , s 6 s 6 , s 4Table 2
The linguistic Pythagorean fuzzy preference information matrixL2 provided by the decision-maker M2.
B 1 B 2 B 3 B 4 A 1 s 1 , s 7 s 2 , s 6 s 5 , s 2 s 7 , s 2 A 2 s 4 , s 3 s 7 , s 1 s 2 , s 6 s 1 , s 5 A 3 s 7 , s 3 s 6 , s 1 s 2 , s 6 s 2 , s 5 A 4 s 7 , s 1 s 3 , s 5 s 2 , s 3 s 7 , s 2Table 3
The linguistic Pythagorean fuzzy preference information matrixL3 provided by the decision-maker M3.
B 1 B 2 B 3 B 4 A 1 s 2 , s 7 s 1 , s 6 s 7 , s 3 s 6 , s 2 A 2 s 6 , s 1 s 4 , s 1 s 2 , s 3 s 3 , s 5 A 3 s 6 , s 2 s 7 , s 2 s 1 , s 7 s 4 , s 5 A 4 s 5 , s 2 s 2 , s 3 s 6 , s 1 s 4 , s 6( 1 ) Rank all the SSD productions using the LPFWIPBM operatorStep 1.
Here, there exist cost-type attributesG2=B3,B4. Hence, (61) is used to normalize the linguistic Pythagorean fuzzy preference information matrices shown in Tables 1–3. The normalized ones are listed in Tables 4–6.Table 4
The normalized linguistic Pythagorean fuzzy preference information matrixZ1 provided by the decision-maker M1.
B 1 B 2 B 3 B 4 A 1 s 1 , s 6 s 3 , s 7 s 2 , s 6 s 2 , s 7 A 2 s 4 , s 3 s 5 , s 4 s 3 , s 1 s 5 , s 1 A 3 s 7 , s 3 s 7 , s 1 s 6 , s 2 s 6 , s 1 A 4 s 3 , s 7 s 4 , s 3 s 6 , s 2 s 4 , s 6Table 5
The normalized linguistic Pythagorean fuzzy preference information matrixZ2 provided by the decision-maker M2.
B 1 B 2 B 3 B 4 A 1 s 1 , s 7 s 2 , s 6 s 2 , s 5 s 2 , s 7 A 2 s 4 , s 3 s 7 , s 1 s 6 , s 2 s 5 , s 1 A 3 s 7 , s 3 s 6 , s 1 s 6 , s 2 s 5 , s 2 A 4 s 7 , s 1 s 3 , s 5 s 3 , s 2 s 2 , s 7Table 6
The normalized linguistic Pythagorean fuzzy preference information matrixZ3 provided by the decision-maker M3.
B 1 B 2 B 3 B 4 A 1 s 2 , s 7 s 1 , s 6 s 3 , s 7 s 2 , s 6 A 2 s 6 , s 1 s 4 , s 1 s 3 , s 2 s 5 , s 3 A 3 s 6 , s 2 s 7 , s 2 s 7 , s 1 s 5 , s 4 A 4 s 5 , s 2 s 2 , s 3 s 1 , s 6 s 6 , s 4Step 2.
Letp=1,q=1; we utilize the LPFWIPBM operator in (62) to compute the fused value of all the attribute values of each alternative Ai which are given by the decision-maker Ml as(68)z11=s1.1383,s4.3667,z21=s2.3993,s1.6564,z31=s3.9857,s1.8209,z41=s2.3005,s3.7101z12=s0.9214,s4.0755,z22=s3.2940,s1.2182,z32=s3.5949,s1.8665,z42=s2.5429,s3.0728z13=s1.0700,s4.2451,z23=s2.6405,s1.2113,z33=s3.7648,s1.8823,z43=s2.4596,s2.5731Step 3.
Use the LPFWIPBM operator in (64) to compute the fused value of the fused attribute value of each alternative Ai as (69)z1=s0.6175,s2.6090,z2=s1.6129,s0.8502,z3=s2.2977,s1.1792,z4=s1.4199,s1.9703Step 4.
Use (66) to compute the score value of all the fused value zi as(70)Dz1=s5.3654,Dz2=s5.7393,Dz3=s5.8262,Dz4=s5.5738Step 5.
According to their score values, these SSD productions can be ranked as(71)A3≻A2≻A4≻A1Thus, the optimal one is A3.Step 6.
End.( 2 ) Rank all the SSD productions using the LPFWIPGBM operatorStep 1.
Here, there exist cost-type attributesG2=B3,B4. Hence, (61) is utilized to normalize the linguistic Pythagorean fuzzy preference information matrices shown in Tables 1–3. The normalized ones are listed in Tables 4–6.Step 2.
Letp=1,q=1; we utilize the LPFWIPGBM operator in (63) to compute the fused value of all the attribute values of each alternative Ai which are given by the decision-maker Ml as(72)z11=s1.9546,s4.0674,z21=s2.5722,s1.3727,z31=s4.2678,s0.9934,z41=s2.9957,s3.1754z12=s1.3856,s3.9419,z22=s3.3852,s0.9351,z32=s3.8904,s1.1276,z42=s2.7264,s2.9112z13=s1.8911,s3.9484,z23=s2.7107,s1.0449,z33=s3.9817,s1.3648,z43=s2.8544,s2.1267Step 3.
Utilize the LPFWIPGBM operator in (65) to compute the fused value of the fused attribute value of each alternative Ai as (73)z1=s1.1601,s2.4170,z2=s1.6939,s0.6746,z3=s2.4921,s0.6776,z4=s1.7840,s1.6478Step 4.
Use (66) to compute the score value of the fused value zi as(74)Dz1=s5.4545,Dz2=s5.7626,Dz3=s5.9056,Dz4=s5.6775Step 5.
According to their score values, these SSD productions can be ranked as(75)A3≻A2≻A4≻A1Thus, the optimal one is A3.Step 6.
End.From the above implementation process, it can be seen that both the LPFWIPBM and LPFWIPGBM operators get the same ranking result.


### 6.2. The Impact of the Parameters on the Decision Results

Figures2–9 present the ranking results derived from the LPFWIPBM and LPFWIPGBM operators when the values of the parameters p,q change between 0 and 10.Figure 2
Score values of alternative A1 derived from LPFWIPBM whenp,q∈0,10.Figure 3
Score values of alternative A2 derived from LPFWIPBM whenp,q∈0,10.Figure 4
Score values of alternative A3 derived from LPFWIPBM whenp,q∈0,10.Figure 5
Score values of alternative A4 derived from LPFWIPBM whenp,q∈0,10.Figure 6
Score values of alternative A1 derived from LPFWIPGBM whenp,q∈0,10.Figure 7
Score values of alternative A2 derived from LPFWIPGBM whenp,q∈0,10.Figure 8
Score values of alternative A3 derived from LPFWIPGBM whenp,q∈0,10.Figure 9
Score values of alternative A4 derived from LPFWIPGBM whenp,q∈0,10.As depicted in Figures2–9, it can be seen that the ranking results derived from the LPFWIPBM and LPFWIPGBM operators remain unchanged as the values of the parameters p,q change and the optimal alternative is always A3.


### 6.3. The 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 Example26. For fair comparison, the parameters p and q, the weight vectors of the attributes, and the decision-makers are set as p=1 and q=1, ω=0.25,0.25,0.20,0.30T, and κ=0.40,0.25,0.35T. Then the ranking results are listed in Table 7.Table 7
The ranking results of our proposed operators and the operators in [19].
Methods Score values Ranking results The LPFWA operator in [19] D z 1 = s 3.6553, Dz2=s6.5676, Dz3=s7.1377, Dz4=s5.9919 X 3 ≻ X 2 ≻ X 4 ≻ X 1 The LPFWG operator in [19] D z 1 = s 3.4561, Dz2=s6.3172,  Dz3=s6.9701, Dz4=s5.0140 X 3 ≻ X 2 ≻ X 4 ≻ X 1 Our proposed LPFWIPBM operator D z 1 = s 5.3654, Dz2=s5.7393,  Dz3=s5.8262, Dz4=s5.5738 X 3 ≻ X 2 ≻ X 4 ≻ X 1 Our proposed LPFWIPGBM operator D z 1 = s 5.4545, Dz2=s5.7626,  Dz3=s5.9056, Dz4=s5.6775 X 3 ≻ X 2 ≻ X 4 ≻ X 1As shown in Table7, 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.


### 6.4. The 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 27.
Suppose that an investment corporation plans to expand the investment scale. There are four investment solutionsA=A1,A2,A3,A4 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 B1, the size of market B2, the investment risk B3, and the competitive power B4. The weight vector of these factors is ω=0.25,0.25,0.25,0.25T. 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, G1=B1,B2 and G2=B3,B4. Based on the continuous LTS S=sε∣ε∈0,8, the decision-maker utilizes the LPFNs to evaluate each project Ai with respect to each attribute Bj and then construct the preference information matrix L=αij4×4, which are presented in Table 8.Table 8
The linguistic Pythagorean fuzzy preference information matrixL provided by the decision-maker M.
B 1 B 2 B 3 B 4 A 1 s 4 , s 3 s 6 , s 4 s 0 , s 3 s 1 , s 7 A 2 s 6 , s 2 s 1 , s 0 s 2 , s 5 s 3 , s 7 A 3 s 5 , s 2 s 4 , s 2 s 1 , s 6 s 0 , s 5 A 4 s 4 , s 1 s 5 , s 3 s 0 , s 2 s 5 , s 6In Example27, there exist two cost-type attributes G2=B3,B4. 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.Table 9
The normalized linguistic Pythagorean fuzzy preference information matrixZ provided by the decision-maker M.
B 1 B 2 B 3 B 4 A 1 s 4 , s 3 s 6 , s 4 s 3 , s 0 s 7 , s 1 A 2 s 6 , s 2 s 1 , s 0 s 5 , s 2 s 7 , s 3 A 3 s 5 , s 2 s 4 , s 2 s 6 , s 1 s 5 , s 0 A 4 s 4 , s 1 s 5 , s 3 s 2 , s 0 s 6 , s 5In the real applications, it usually happens that the MD or NMD of one of the aggregated LPFNs iss0. 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 s0, 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 between the MD and NMD from different LPFNs. Hence, they may result in unreasonable decision results, which has been discussed in Section 3. However, the LPFWA operator in [19] with novel interactional operational rules and the LPFWG operator in [19] with novel interactional operational rules utilize our proposed novel interactional operational rules, which can overcome the weakness of the operational rules in [19].Table 10
The ranking results of our proposed operators and the operators in [19].
Method Score value Ranking results The LPFWA operator in [19] with novel operational rules D A 1 = s 6.6182, DA2=s6.6330, DA3=s6.6355, DA4=s5.7542 A 3 ≻ A 2 ≻ A 1 ≻ A 4 The LPFWG operator in [19] with novel operational rules D A 1 = s 6.6984, DA2=s6.9134, DA3=s6.6243, DA4=s6.5580 A 2 ≻ A 1 ≻ A 3 ≻ A 4 The LPFWA operator in [19] D A 1 = s 6.9119, DA2=s6.9328, DA3=s6.7152, DA4=s6.5532 A 2 ≻ A 1 ≻ A 3 ≻ A 4 The LPFWG operator in [19] D A 1 = s 6.3075, DA2=s6.0891, DA3=s6.5658, DA4=s5.9043 A 3 ≻ A 1 ≻ A 2 ≻ A 4 Our proposed LPFWIPBM operator D A 1 = s 5.9453, DA2=s5.9648, DA3=s5.9623, DA4=s5.6419 A 2 ≻ A 3 ≻ A 1 ≻ A 4 Our proposed LPFWIPGBM operator D A 1 = s 6.0780, DA2=s6.1478, DA3=s5.9807, DA4=s6.0086 A 2 ≻ A 1 ≻ A 4 ≻ A 3Although 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 A1,A3,A4 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 Table11.Table 11
The differences between our studies and existing studies for LPFNs.
Methods Whether considering the interrelationships between input values Whether considering the interactions between the MD and NMD Whether considering the partition of input values The LPFWA operator in [19] No No No The LPFWG operator in [19] No No No Our proposed LPFWIPBM operator Yes Yes Yes Our proposed LPFWIPGBM operator Yes Yes YesAs presented in Table11, 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.


## 6.1. Illustrative Example

In this subsection, we introduce a practical case concerning the selection of SSD productions to show the implementation process of the proposed MAGDM method.Example 26.
A company plans to purchase a batch of SSDs to upgrade their computer systems. Currently, there are four kinds of SSD productionsA=A1,A2,A3,A4 available to be chosen. Three decision-makers M=M1,M2,M3 are invited to assess these SSD productions considering four attributes, which are the basic requirements B1, physical characteristics B2, brand B3, and overall performance B4. Naturally, the former two attributes are benefit-type, while the latter two attributes are cost-type. Therefore, these four attributes are split into two groups, G1=B1,B2 and G2=B3,B4. The weight vector of the attributes is ω=0.25,0.25,0.20,0.30T and the weight vector of the decision-makers is κ=0.40,0.25,0.35T. Based on the continuous LTS S=sε∣ε∈0,8, the decision-makers utilize the LPFNs to evaluate each SSD production Ai with respect to each attribute Bj and then construct the preference information matrices Ll=αijl4×4l=1,2,3 that are presented in Tables 1–3.Table 1
The linguistic Pythagorean fuzzy preference information matrixL1 provided by the decision-maker M1.
B 1 B 2 B 3 B 4 A 1 s 1 , s 6 s 3 , s 7 s 6 , s 2 s 7 , s 2 A 2 s 4 , s 3 s 5 , s 4 s 1 , s 3 s 1 , s 5 A 3 s 7 , s 3 s 7 , s 1 s 2 , s 6 s 1 , s 6 A 4 s 3 , s 7 s 4 , s 3 s 2 , s 6 s 6 , s 4Table 2
The linguistic Pythagorean fuzzy preference information matrixL2 provided by the decision-maker M2.
B 1 B 2 B 3 B 4 A 1 s 1 , s 7 s 2 , s 6 s 5 , s 2 s 7 , s 2 A 2 s 4 , s 3 s 7 , s 1 s 2 , s 6 s 1 , s 5 A 3 s 7 , s 3 s 6 , s 1 s 2 , s 6 s 2 , s 5 A 4 s 7 , s 1 s 3 , s 5 s 2 , s 3 s 7 , s 2Table 3
The linguistic Pythagorean fuzzy preference information matrixL3 provided by the decision-maker M3.
B 1 B 2 B 3 B 4 A 1 s 2 , s 7 s 1 , s 6 s 7 , s 3 s 6 , s 2 A 2 s 6 , s 1 s 4 , s 1 s 2 , s 3 s 3 , s 5 A 3 s 6 , s 2 s 7 , s 2 s 1 , s 7 s 4 , s 5 A 4 s 5 , s 2 s 2 , s 3 s 6 , s 1 s 4 , s 6( 1 ) Rank all the SSD productions using the LPFWIPBM operatorStep 1.
Here, there exist cost-type attributesG2=B3,B4. Hence, (61) is used to normalize the linguistic Pythagorean fuzzy preference information matrices shown in Tables 1–3. The normalized ones are listed in Tables 4–6.Table 4
The normalized linguistic Pythagorean fuzzy preference information matrixZ1 provided by the decision-maker M1.
B 1 B 2 B 3 B 4 A 1 s 1 , s 6 s 3 , s 7 s 2 , s 6 s 2 , s 7 A 2 s 4 , s 3 s 5 , s 4 s 3 , s 1 s 5 , s 1 A 3 s 7 , s 3 s 7 , s 1 s 6 , s 2 s 6 , s 1 A 4 s 3 , s 7 s 4 , s 3 s 6 , s 2 s 4 , s 6Table 5
The normalized linguistic Pythagorean fuzzy preference information matrixZ2 provided by the decision-maker M2.
B 1 B 2 B 3 B 4 A 1 s 1 , s 7 s 2 , s 6 s 2 , s 5 s 2 , s 7 A 2 s 4 , s 3 s 7 , s 1 s 6 , s 2 s 5 , s 1 A 3 s 7 , s 3 s 6 , s 1 s 6 , s 2 s 5 , s 2 A 4 s 7 , s 1 s 3 , s 5 s 3 , s 2 s 2 , s 7Table 6
The normalized linguistic Pythagorean fuzzy preference information matrixZ3 provided by the decision-maker M3.
B 1 B 2 B 3 B 4 A 1 s 2 , s 7 s 1 , s 6 s 3 , s 7 s 2 , s 6 A 2 s 6 , s 1 s 4 , s 1 s 3 , s 2 s 5 , s 3 A 3 s 6 , s 2 s 7 , s 2 s 7 , s 1 s 5 , s 4 A 4 s 5 , s 2 s 2 , s 3 s 1 , s 6 s 6 , s 4Step 2.
Letp=1,q=1; we utilize the LPFWIPBM operator in (62) to compute the fused value of all the attribute values of each alternative Ai which are given by the decision-maker Ml as(68)z11=s1.1383,s4.3667,z21=s2.3993,s1.6564,z31=s3.9857,s1.8209,z41=s2.3005,s3.7101z12=s0.9214,s4.0755,z22=s3.2940,s1.2182,z32=s3.5949,s1.8665,z42=s2.5429,s3.0728z13=s1.0700,s4.2451,z23=s2.6405,s1.2113,z33=s3.7648,s1.8823,z43=s2.4596,s2.5731Step 3.
Use the LPFWIPBM operator in (64) to compute the fused value of the fused attribute value of each alternative Ai as (69)z1=s0.6175,s2.6090,z2=s1.6129,s0.8502,z3=s2.2977,s1.1792,z4=s1.4199,s1.9703Step 4.
Use (66) to compute the score value of all the fused value zi as(70)Dz1=s5.3654,Dz2=s5.7393,Dz3=s5.8262,Dz4=s5.5738Step 5.
According to their score values, these SSD productions can be ranked as(71)A3≻A2≻A4≻A1Thus, the optimal one is A3.Step 6.
End.( 2 ) Rank all the SSD productions using the LPFWIPGBM operatorStep 1.
Here, there exist cost-type attributesG2=B3,B4. Hence, (61) is utilized to normalize the linguistic Pythagorean fuzzy preference information matrices shown in Tables 1–3. The normalized ones are listed in Tables 4–6.Step 2.
Letp=1,q=1; we utilize the LPFWIPGBM operator in (63) to compute the fused value of all the attribute values of each alternative Ai which are given by the decision-maker Ml as(72)z11=s1.9546,s4.0674,z21=s2.5722,s1.3727,z31=s4.2678,s0.9934,z41=s2.9957,s3.1754z12=s1.3856,s3.9419,z22=s3.3852,s0.9351,z32=s3.8904,s1.1276,z42=s2.7264,s2.9112z13=s1.8911,s3.9484,z23=s2.7107,s1.0449,z33=s3.9817,s1.3648,z43=s2.8544,s2.1267Step 3.
Utilize the LPFWIPGBM operator in (65) to compute the fused value of the fused attribute value of each alternative Ai as (73)z1=s1.1601,s2.4170,z2=s1.6939,s0.6746,z3=s2.4921,s0.6776,z4=s1.7840,s1.6478Step 4.
Use (66) to compute the score value of the fused value zi as(74)Dz1=s5.4545,Dz2=s5.7626,Dz3=s5.9056,Dz4=s5.6775Step 5.
According to their score values, these SSD productions can be ranked as(75)A3≻A2≻A4≻A1Thus, the optimal one is A3.Step 6.
End.From the above implementation process, it can be seen that both the LPFWIPBM and LPFWIPGBM operators get the same ranking result.


## 6.2. The Impact of the Parameters on the Decision Results

Figures2–9 present the ranking results derived from the LPFWIPBM and LPFWIPGBM operators when the values of the parameters p,q change between 0 and 10.Figure 2
Score values of alternative A1 derived from LPFWIPBM whenp,q∈0,10.Figure 3
Score values of alternative A2 derived from LPFWIPBM whenp,q∈0,10.Figure 4
Score values of alternative A3 derived from LPFWIPBM whenp,q∈0,10.Figure 5
Score values of alternative A4 derived from LPFWIPBM whenp,q∈0,10.Figure 6
Score values of alternative A1 derived from LPFWIPGBM whenp,q∈0,10.Figure 7
Score values of alternative A2 derived from LPFWIPGBM whenp,q∈0,10.Figure 8
Score values of alternative A3 derived from LPFWIPGBM whenp,q∈0,10.Figure 9
Score values of alternative A4 derived from LPFWIPGBM whenp,q∈0,10.As depicted in Figures2–9, it can be seen that the ranking results derived from the LPFWIPBM and LPFWIPGBM operators remain unchanged as the values of the parameters p,q change and the optimal alternative is always A3.


## 6.3. The 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 Example26. For fair comparison, the parameters p and q, the weight vectors of the attributes, and the decision-makers are set as p=1 and q=1, ω=0.25,0.25,0.20,0.30T, and κ=0.40,0.25,0.35T. Then the ranking results are listed in Table 7.Table 7
The ranking results of our proposed operators and the operators in [19].
Methods Score values Ranking results The LPFWA operator in [19] D z 1 = s 3.6553, Dz2=s6.5676, Dz3=s7.1377, Dz4=s5.9919 X 3 ≻ X 2 ≻ X 4 ≻ X 1 The LPFWG operator in [19] D z 1 = s 3.4561, Dz2=s6.3172,  Dz3=s6.9701, Dz4=s5.0140 X 3 ≻ X 2 ≻ X 4 ≻ X 1 Our proposed LPFWIPBM operator D z 1 = s 5.3654, Dz2=s5.7393,  Dz3=s5.8262, Dz4=s5.5738 X 3 ≻ X 2 ≻ X 4 ≻ X 1 Our proposed LPFWIPGBM operator D z 1 = s 5.4545, Dz2=s5.7626,  Dz3=s5.9056, Dz4=s5.6775 X 3 ≻ X 2 ≻ X 4 ≻ X 1As shown in Table7, 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.


## 6.4. The 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 27.
Suppose that an investment corporation plans to expand the investment scale. There are four investment solutionsA=A1,A2,A3,A4 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 B1, the size of market B2, the investment risk B3, and the competitive power B4. The weight vector of these factors is ω=0.25,0.25,0.25,0.25T. 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, G1=B1,B2 and G2=B3,B4. Based on the continuous LTS S=sε∣ε∈0,8, the decision-maker utilizes the LPFNs to evaluate each project Ai with respect to each attribute Bj and then construct the preference information matrix L=αij4×4, which are presented in Table 8.Table 8
The linguistic Pythagorean fuzzy preference information matrixL provided by the decision-maker M.
B 1 B 2 B 3 B 4 A 1 s 4 , s 3 s 6 , s 4 s 0 , s 3 s 1 , s 7 A 2 s 6 , s 2 s 1 , s 0 s 2 , s 5 s 3 , s 7 A 3 s 5 , s 2 s 4 , s 2 s 1 , s 6 s 0 , s 5 A 4 s 4 , s 1 s 5 , s 3 s 0 , s 2 s 5 , s 6In Example27, there exist two cost-type attributes G2=B3,B4. 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.Table 9
The normalized linguistic Pythagorean fuzzy preference information matrixZ provided by the decision-maker M.
B 1 B 2 B 3 B 4 A 1 s 4 , s 3 s 6 , s 4 s 3 , s 0 s 7 , s 1 A 2 s 6 , s 2 s 1 , s 0 s 5 , s 2 s 7 , s 3 A 3 s 5 , s 2 s 4 , s 2 s 6 , s 1 s 5 , s 0 A 4 s 4 , s 1 s 5 , s 3 s 2 , s 0 s 6 , s 5In the real applications, it usually happens that the MD or NMD of one of the aggregated LPFNs iss0. 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 s0, 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 between the MD and NMD from different LPFNs. Hence, they may result in unreasonable decision results, which has been discussed in Section 3. However, the LPFWA operator in [19] with novel interactional operational rules and the LPFWG operator in [19] with novel interactional operational rules utilize our proposed novel interactional operational rules, which can overcome the weakness of the operational rules in [19].Table 10
The ranking results of our proposed operators and the operators in [19].
Method Score value Ranking results The LPFWA operator in [19] with novel operational rules D A 1 = s 6.6182, DA2=s6.6330, DA3=s6.6355, DA4=s5.7542 A 3 ≻ A 2 ≻ A 1 ≻ A 4 The LPFWG operator in [19] with novel operational rules D A 1 = s 6.6984, DA2=s6.9134, DA3=s6.6243, DA4=s6.5580 A 2 ≻ A 1 ≻ A 3 ≻ A 4 The LPFWA operator in [19] D A 1 = s 6.9119, DA2=s6.9328, DA3=s6.7152, DA4=s6.5532 A 2 ≻ A 1 ≻ A 3 ≻ A 4 The LPFWG operator in [19] D A 1 = s 6.3075, DA2=s6.0891, DA3=s6.5658, DA4=s5.9043 A 3 ≻ A 1 ≻ A 2 ≻ A 4 Our proposed LPFWIPBM operator D A 1 = s 5.9453, DA2=s5.9648, DA3=s5.9623, DA4=s5.6419 A 2 ≻ A 3 ≻ A 1 ≻ A 4 Our proposed LPFWIPGBM operator D A 1 = s 6.0780, DA2=s6.1478, DA3=s5.9807, DA4=s6.0086 A 2 ≻ A 1 ≻ A 4 ≻ A 3Although 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 A1,A3,A4 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 Table11.Table 11
The differences between our studies and existing studies for LPFNs.
Methods Whether considering the interrelationships between input values Whether considering the interactions between the MD and NMD Whether considering the partition of input values The LPFWA operator in [19] No No No The LPFWG operator in [19] No No No Our proposed LPFWIPBM operator Yes Yes Yes Our proposed LPFWIPGBM operator Yes Yes YesAs presented in Table11, 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.


## 7. 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 parametersp and q 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 [41, 42] in the MAGDM problems with heterogeneous relationship among attributes.



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*Source: 9531064-2018-11-01.xml*