# Multiattribute Decision-Making Method with Intuitionistic Fuzzy Archimedean Bonferroni Means

Authors: Hengzhi Zhao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The Bonferroni mean (BM) can portray the inter-relationships among the arguments, which is based on algebraic norm. Archimedean norm is the generalization of algebraic norm. In this work, the intuitionistic fuzzy (IF) BMs on the basis of the Archimedean norm are investigated, including the Archimedean norm-based IF arithmetic BM (AN-IFABM) and the Archimedean norm-based IF geometric BM (AN-IFGBM). Then, we discuss their typical properties and several particular cases of the AN-IFABM and AN-IFGBM in detail. Moreover, we design the Archimedean norm-based IF weighted arithmetic BM (AN-IFWABM) and the Archimedean norm-based IF weighted geometric BM (AN-IFWGBM) to consider the importance of each argument and their interconnections. Applying the proposed extensions, an approach is designed to cope with the multiattribute decision-making (MADM) problems. Finally, an efficiency evaluation problem of some public companies in China is analyzed by the proposed approach to demonstrate its applicability and validity.


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## Body

## 1. Introduction

Fuzzy sets (FSs) [1] have been widely applied and extended to a variety of fields. However, FS is a set in which each element is denoted by a real number in 0,1. To compensate for the deficiency, Atanassov [2–4] presented the intuitionistic fuzzy sets (IFSs). In IFSs, considering both membership degree and nonmembership degree, the data information is displayed in 2-tuples [5–12].The objective of the MADM problem is to select the most acceptable solution from a group of options with the decision makers’ (DMs) preference information. The widely adopted method is the priority ranking method [13–16]. The aggregation methods are utilized to fuse preference information by using the aggregation operators (AOs). Xu [17] defined the operational laws of IFSs and then designed a sequence of AOs with IFSs. By means of geometric mean, Xu and Yager [18] gave some geometric mean AOs with IFSs, and then they provided an application for the MADM problems. Zhao et al. [19] developed the generalized AOs, which are followed by the discussion of their properties. For the new operations in IFSs, Wei [20] constructed some operators and then investigated a novel IF model to cope with MADM problems. Under the probabilistic linguistic information and picture fuzzy environment, Lin et al. [21, 22] designed different types of MADM models to deal with practical problems.From the abovementioned analyze, we know that these AOs fail to capture the relationship among the aggregation arguments. To offset this flaw, several AOs have been generated. Under the IF information environment, Xu and Xia [23] investigated several new types of AOs on the basis of Choquet integral and Dempster–Shafer theory. Considering the Dempster–Shafer belief structure, Yang and Chen [24] proposed several new operators, which can capture the relationship among the input decision-making information. For the MADM problems with the prioritization ordering among the attributes, Yager [25] utilized the prioritized aggregation (PA) operators to develop a MADM method. In addition, Yu and Xu [26] proposed the prioritized IF aggregation (PIFA) operator to select the information security systems. Based on the power average operator [27], Xu [28] developed a novel IF MADM model. Zhou et al. [29, 30] proposed two generalized IF power aggregation algorithms, and then they drew up a new MADM method to assign the appropriate weights to the experts.The BM [31] can be utilized to capture the inter-relationships between input parameters. Some generalizations of BM were further proposed by Yager [32] to enhance their modeling capabilities. Furthermore, Xu and Yager [33] investigated the intuitionistic fuzzy BM (IFBM) and the weighted IFBM (WIFBM). To describe the inter-relationship between arguments, the IF geometric BM (IFGBM) and weighted IFGBM (WIFGBM) were introduced [34]. Under the linguistic Pythagorean fuzzy information environment, Lin et al. [35] developed a new MADM method by using several proposed linguistic Pythagorean fuzzy interaction partitioned Bonferroni mean aggregation operators. Xia et al. [36] defined the generalized IF weighted BM (GIFWBM) and Bonferroni geometric mean (GIFWBGM). Lin et al. [37] first introduced the concept of linguistic q-rung orthopair fuzzy set and then designed the linguistic q-rung orthopair fuzzy interactional PGHM operator with the help of Heronian mean.On the one hand, we can find that abovementioned BMs are all based on algebraic norm, which are special cases of Archimedean norm [38–40]. On the other hand, by using confidence level, Rahman et al. [41] proposed the confidence intuitionistic fuzzy Einstein hybrid averaging (CIFEHA) operator and confidence intuitionistic fuzzy Einstein hybrid geometric (CIFEHG) operator to fuse a group of IFNs into a collection IFN. However, the CIFEHA operator and CIFEHG operator [41] neglect the importance of each argument and their interconnections.Therefore, in order to cope with these shortcomings, we construct a MADM method with intuitionistic fuzzy Archimedean Bonferroni means. The main contributions of this paper are as follows:(i)
The Archimedean norm-based IF arithmetic BM (AN-IFABM) and the Archimedean norm-based IF geometric BM (AN-IFGBM) are designed(ii)
The typical properties and several particular cases of AN-IFABM and AN-IFGBM are discussed(iii)
The Archimedeant-norm-based IF weighted arithmetic BM (AN-IFWABM) and the Archimedean t-norm-based IF weighted geometric BM (AN-IFWGBM) are presented(iv)
A MADM method is developed and applied to evaluate the public companiesThe remainder of our work is structured as follows. In Section2 several concepts relevant to BM and IFSs are introduced. Section 3 investigates two IFBMs based on Archimedean norm, such as AN-IFABM and AN-IFGBM, and develops some specific IFBMs. In Section 4, we present the weighted versions of the AN-IFABM and the AN-IFGBM. Section 5 develops a method for intuitionistic fuzzy MADM with the proposed BMs. Section 6 shows a numerical example to validate our method. The major conclusions of our work are summarized in Section 7.


## 2. Preliminaries

In the following, the BM, geometric BM, IFSs, and Archimedean norm are briefly retrospected. Then, some IF operational laws are presented in terms of Archimedean norm.


### 2.1. BM and GBM

In order to capture the inter-relationships among attributes, BM [31] and geometric BM [34] are both valuable information aggregation methods, which are extensions of the arithmetic averaging (AA) [40] and the geometric averaging (GA) [42], respectively.Definition 1.
(see [31]). Assume that ηi for all i∈N+ are a set of numbers that are not negative and b,d≥0. If(1)BMb,dη1,η2,…,ηn=1nn−1∑i≠jηibηjdb+d,then BMb,d is called a BM.Definition 2.
(see [34]). Assume that ηi for all i∈N+ are a set of numbers that are not negative and b,d≥0. If(2)GBMb,dη1,η2,…,ηn=1b+d∏i≠jbηi+dηjnn−1,then GBMb,d is called a GBM.


### 2.2. Intuitionistic Fuzzy Operational Laws Based on Archimedean Norm

Definition 3.
(see [40]). Let a,b,c∈0,1; a function φ:0,1×0,1⟶0,1 is called a t-norm, if(1)
φ1,a=a(2)
φa,b=φb,a(3)
φa,φb,c=φφa,b,c(4)
If0≤a≤a′≤1 and 0≤b≤b′≤1, we have φa,b≤φa′,b′.
In addition, if a t-normφa,b is continuous and φa,a<a for all a∈0,1, then φa,b is an Archimedean t-norm.Definition 4.
(see [40]). Let a,b,c∈0,1; a function ψ:0,1×0,1⟶0,1 is called a t-conorm if it satisfies the following requirements:(1)
ψ0,a=a(2)
ψa,b=ψb,a(3)
ψa,ψb,c=ψψa,b,c(4)
If0≤a≤a′≤1 and 0≤b≤b′≤1, we have ψa,b≤ψa′,b′.
In addition, if a t-conormψa,b is continuous and ψa,a>a for all a∈0,1, then ψa,b is an Archimedean t-conorm.
According to [40, 43], we know that a strict Archimedean t-norm φa,b is expressed by an additive operator δ as φa,b=δ−1δa+δb. Analogously, the dual t-conorm denoted by ψa,b=ε−1εa+εb with εt=δ1−t, where δ:0,1⟶0,+∞ such that δ1=0; then, ε:0,1⟶0,+∞ is a strictly increasing function and ε0=0.Definition 5.
(see [2]). An IFS I in X is given by(3)I=x,μIx,νIx|x∈X,where the membership degree μI:X⟶0,1, the nonmembership degree νI:X⟶0,1, and(4)0≤μIx+νIx≤1,∀x∈X.
An intuitionistic fuzzy number (IFN) is denoted byμIx,νIx [18]. Xu and Yager [33] denoted an IFN by θ=μθ,νθ. Let H be the set of all IFNs.Definition 6.
(see [33]). Let θ=μθ,νθ be an IFN, the score function of θ is sθ=μθ−νθ, and the accuracy function of θ is fθ=μθ+νθ. Let θ1 and θ2 be two IFNs, then(1)
Ifsθ1>sθ2, then θ1>θ2(2)
Ifsθ1=sθ2, then(a)
Iffθ1>fθ2, then θ1>θ2(b)
Iffθ1=fθ2, then θ1=θ2Definition 7.
(see [40]). Let θ,θ1, and θ2 be three IFNs, then(1)
θ1⊕θ2=ψμθ1,μθ2,φνθ1,νθ2=ε−1εμθ1+εμθ2,δ−1δνθ1+δνθ2(2)
θ1⊗θ2=ψμθ1,μθ2,φνθ1,νθ2=δ−1δμθ1+δμθ2,ε−1ενθ1+ενθ2(3)
λθ=ε−1λεμθ,δ−1λδνθ,λ>0(4)
θλ=δ−1λδμθ,ε−1λενθ,λ>0Theorem 1.
Letθ,θ1, and θ2 be three IFNs, then we have(1)
θ1⊕θ2=θ2⊕θ1(2)
θ1⊗θ2=θ2⊗θ1(3)
λθ1⊕θ2=λθ1⊕λθ2,λ>0(4)
θ1⊗θ2λ=θ1λ⊗θ2λ,λ>0(5)
λ1θ⊕λ2θ=λ1+λ2θ,λ1,λ2>0(6)
θλ1⊗θλ2=θλ1+λ2,λ1,λ2>0


## 2.1. BM and GBM

In order to capture the inter-relationships among attributes, BM [31] and geometric BM [34] are both valuable information aggregation methods, which are extensions of the arithmetic averaging (AA) [40] and the geometric averaging (GA) [42], respectively.Definition 1.
(see [31]). Assume that ηi for all i∈N+ are a set of numbers that are not negative and b,d≥0. If(1)BMb,dη1,η2,…,ηn=1nn−1∑i≠jηibηjdb+d,then BMb,d is called a BM.Definition 2.
(see [34]). Assume that ηi for all i∈N+ are a set of numbers that are not negative and b,d≥0. If(2)GBMb,dη1,η2,…,ηn=1b+d∏i≠jbηi+dηjnn−1,then GBMb,d is called a GBM.


## 2.2. Intuitionistic Fuzzy Operational Laws Based on Archimedean Norm

Definition 3.
(see [40]). Let a,b,c∈0,1; a function φ:0,1×0,1⟶0,1 is called a t-norm, if(1)
φ1,a=a(2)
φa,b=φb,a(3)
φa,φb,c=φφa,b,c(4)
If0≤a≤a′≤1 and 0≤b≤b′≤1, we have φa,b≤φa′,b′.
In addition, if a t-normφa,b is continuous and φa,a<a for all a∈0,1, then φa,b is an Archimedean t-norm.Definition 4.
(see [40]). Let a,b,c∈0,1; a function ψ:0,1×0,1⟶0,1 is called a t-conorm if it satisfies the following requirements:(1)
ψ0,a=a(2)
ψa,b=ψb,a(3)
ψa,ψb,c=ψψa,b,c(4)
If0≤a≤a′≤1 and 0≤b≤b′≤1, we have ψa,b≤ψa′,b′.
In addition, if a t-conormψa,b is continuous and ψa,a>a for all a∈0,1, then ψa,b is an Archimedean t-conorm.
According to [40, 43], we know that a strict Archimedean t-norm φa,b is expressed by an additive operator δ as φa,b=δ−1δa+δb. Analogously, the dual t-conorm denoted by ψa,b=ε−1εa+εb with εt=δ1−t, where δ:0,1⟶0,+∞ such that δ1=0; then, ε:0,1⟶0,+∞ is a strictly increasing function and ε0=0.Definition 5.
(see [2]). An IFS I in X is given by(3)I=x,μIx,νIx|x∈X,where the membership degree μI:X⟶0,1, the nonmembership degree νI:X⟶0,1, and(4)0≤μIx+νIx≤1,∀x∈X.
An intuitionistic fuzzy number (IFN) is denoted byμIx,νIx [18]. Xu and Yager [33] denoted an IFN by θ=μθ,νθ. Let H be the set of all IFNs.Definition 6.
(see [33]). Let θ=μθ,νθ be an IFN, the score function of θ is sθ=μθ−νθ, and the accuracy function of θ is fθ=μθ+νθ. Let θ1 and θ2 be two IFNs, then(1)
Ifsθ1>sθ2, then θ1>θ2(2)
Ifsθ1=sθ2, then(a)
Iffθ1>fθ2, then θ1>θ2(b)
Iffθ1=fθ2, then θ1=θ2Definition 7.
(see [40]). Let θ,θ1, and θ2 be three IFNs, then(1)
θ1⊕θ2=ψμθ1,μθ2,φνθ1,νθ2=ε−1εμθ1+εμθ2,δ−1δνθ1+δνθ2(2)
θ1⊗θ2=ψμθ1,μθ2,φνθ1,νθ2=δ−1δμθ1+δμθ2,ε−1ενθ1+ενθ2(3)
λθ=ε−1λεμθ,δ−1λδνθ,λ>0(4)
θλ=δ−1λδμθ,ε−1λενθ,λ>0Theorem 1.
Letθ,θ1, and θ2 be three IFNs, then we have(1)
θ1⊕θ2=θ2⊕θ1(2)
θ1⊗θ2=θ2⊗θ1(3)
λθ1⊕θ2=λθ1⊕λθ2,λ>0(4)
θ1⊗θ2λ=θ1λ⊗θ2λ,λ>0(5)
λ1θ⊕λ2θ=λ1+λ2θ,λ1,λ2>0(6)
θλ1⊗θλ2=θλ1+λ2,λ1,λ2>0


## 3. IFBMs Based on Archimedean Norm

In MADM, the performance values of an alternative under an attribute are usually expressed with IFNs, which is a more objective reflection of the DM’s preference. An extension of the IFBM and IFGBM is given for the purpose of aggregating all performance values of the alternatives with respect to all attributes.


### 3.1. IFABM Based on Archimedean Norm

As defined in Section2, the operations can be utilized to integrate IF information. In this subsection, we shall investigate the AN-IFABM and then analyze some desirable properties of the AN-IFABM.Definition 8.
Letθi=μθi,νθi for all i∈N+=1,2,…,n be a group of IFNs and b,d≥0; the AN-IFABM is denoted as(5)AN−IFABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d.Theorem 2.
Letθi=μθi,νθi for all i∈N+ be a set of IFNs and b,d≥0, then the gathered value by using AN-IFABM is also an IFN, and(6)AN−IFABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d=δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj.Proof.
With the operations of IFN stated in Section2, we obtain that(7)θib=δ−1pδμθi,ε−1bενθi,θjd=δ−1dδμθj,ε−1dενθj,for alli,j,and then(8)θib⊗θjd=δ−1δδ−1bδμθi+δδ−1dδμθj,ε−1εε−1bενθi+εε−1dενθj=δ−1bδμθi+dδμθj,ε−1bενθi+dενθj.
According to Definition7, we can get that(9)⊕i,j=1i≠jnθib⊗θjd=ε−1∑i,j=1i≠jnεδ−1bδμθi+dδμθi,δ−1∑i,j=1i≠jnδε−1bενθi+dενθi.
Then, we have(10)1nn−1⊕i,j=1i≠jnθib⊗θjd=ε−11nn−1εε−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,δ−11nn−1δδ−1∑i,j=1i≠jnδε−1bενθi+dενθj=ε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,δ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj.
It follows that(11)1nn−1⊕i,j=1i≠jnηib⊗ηjd1/b+d=δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj,i.e., equation (6) holds.
In the following, we prove that the gathered value with AN-IFABM is also an IFN. Sinceδ:0,1⟶0,+∞ is a strictly decreasing function, and εt=δ1−t, then εt and ε−1t are two strictly increasing functions, δ−1t is a strictly decreasing function, and ε−1t=1−δ−1t. Therefore, we have(12)δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj≥0,ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj≥0.
Afterwards, we will demonstrate that(13)δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj+ε−11b+dεδ−11nn−1δ∑i,j=1i≠jnδε−1bενθi+dενθj≤1.
Sinceμθi≤1−νθi,i∈N+, then b⋅ενθi+d⋅ενθj=b⋅δ1−νθi+d⋅δ1−νθj≤b⋅δμθi+d⋅δμθj, for all i,j,and we have(14)ε−1b⋅ενθi+d⋅ενθj≤ε−1b⋅δμθi+d⋅δμθj=1−δ−1b⋅δμθi+d⋅δμθj.
Asδt is a strictly decreasing function, one can derive that(15)δε−1b⋅ενθi+d⋅ενθj≥δ1−δ−1b⋅δμθi+d⋅δμθj=εδ−1b⋅δμθi+d⋅δμθj,and then we have(16)∑i≠jδε−1b⋅ενθi+d⋅ενθj≥∑i≠jεδ−1b⋅δμθi+d⋅δμθj.
Thus, it follows that(17)δ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≤δ−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj=1−ε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj.
Hence,(18)1b+dεδ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≤1b+dε1−ε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj=1b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,and then(19)ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≤ε−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj=1−δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,i.e.,(20)ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj+δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj≤1,which completes the demonstration of Theorem 2.Theorem 3.
(idempotency). Letθi for all i∈N+ be a group of IFNs and b,d≥0. If all θi=θ=μθ,νθ for all i, then(21)AN−IFABMb,dθ1,θ2,…,θn=μθ,νθ=θ.Proof.
Sinceθ1=θ2=⋯=θn=θ=μθ,νθ, then we have(22)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dθ,θ,…,θ=δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθ+dδμθ,ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθ+dενθ=δ−11b+dδε−1εδ−1b+dδμθ,ε−11b+dεδ−1δε−1b+dενθ=δ−1δμθ,ε−1ενθ=μθ,νθ=θ.
The proof is completed.Remark 1.
Ifθi for all i∈N+ are a set of the smallest IFNs, i.e., θ1=θ2=⋯=θn=θ−=0,1, then(23)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dθ−,θ−,…,θ−=0,1.
Ifθi for all i∈N+are a group of the largest IFNs, i.e., θ1=θ2=⋯=θn=θ+=1,0, then(24)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dθ+,θ+,…,θ+=1,0.Theorem 4.
(monotonicity). Letθi=μθi,νθi and ϑi=μϑi,νϑi for all i∈N+ be two collections of IFNs, b,d≥0. If μθi≤μϑi and νθi≥νϑi, then(25)AN−IFABMb,dθ1,θ2,…,θn≤AN−IFABMb,dϑ1,ϑ2,…,ϑn.Proof.
Due toμθi≤μϑi and νθi≥νϑi for all i∈N+, εt and ε−1t are two strictly increasing functions, and δt and δ−1t are two strictly decreasing functions, we can obtain that(26)b⋅δμθi+d⋅δμθj≥b⋅εμϑi+d⋅εμϑj,b⋅ενθi+d⋅ενθj≥b⋅ενϑi+d⋅ενϑj,and then(27)δ−1b⋅δμθi+d⋅δμθj≤δ−1b⋅δμϑi+d⋅δμϑj,ε−1b⋅ενθi+d⋅ενθj≥ε−1b⋅ενϑi+d⋅ενϑj.
Thus,(28)δ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≥δ−11nn−1∑i,j=1i≠jnδε−1b⋅ενϑi+d⋅ενϑj.
Hence,(29)δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj≤δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμϑi+d⋅δμϑj,(30)ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≥ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1b⋅ενϑi+d⋅ενϑj.
Letθ=AN−IFABMb,dθ1,θ2,…,θn and ϑ=AN−IFABMb,dϑ1,ϑ2,…,ϑn, then equations (29) and (30) are equal to μθ≤μϑ and νθ≥νϑ. Hence, we have(31)sθ=μθ−νθ≤μϑ−νϑ=sϑ.Case 1.
Ifsθ<sϑ, then by using Definition 6, we can get that θ<ϑ, i.e.,(32)AN−IFABMb,dθ1,θ2,…,θn<AN−IFABMb,dϑ1,ϑ2,…,ϑn.Case 2.
Ifsθ=sϑ, i.e., μθ−νθ=μϑ−νϑ, then μθ+νϑ=μϑ+νθ. Since μθ≤μϑ and νθ≥νϑ, thus μθ=μϑ and νθ=νϑ, then we have(33)fθ=μθ+νθ=μϑ+νϑ=fϑ.
Then, by Definition6, we have θ=ϑ, i.e.,(34)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dϑ1,ϑ2,…,ϑn.
According to equations (32) and (34), we can obtain that equation (25) holds.Theorem 5.
(boundedness). Letθi for all i∈N+ be a set of IFNs and b,d≥0, and let(35)θL=miniμθi,maxiνθi,θU=maxiμθi,miniνθi,then we have(36)θL≤AN−IFABMb,dθ1,θ2,…,θn≤θU.Proof.
Asminiμθi≤μθi≤maxiμθi and maxiνθi≥νθi≥miniνθi, for all i, by Theorem 4, we obtain that(37)AN−IFABMb,dθL,θL,…,θL≤AN−IFABMb,dθ1,θ2,…,θn≤AN−IFABMb,dθU,θU,…,θU.
Furthermore, according to Theorem3, we have(38)AN−IFABMb,dθL,θL,…,θL=θL,AN−IFABMb,dθU,θU,…,θU=θU.
Hence, from equations (37) and (38), we get that(39)θL≤AN−IFABMb,dθ1,θ2,…,θn≤θU.Theorem 6.
(commutativity). Letθi=μθi,νθi for all i∈N+ be a group of IFNs and b,d≥0, then(40)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dθ˜1,θ˜2,…,θ˜n,where θ˜1,θ˜2,…,θ˜n is any permutation of θ1,θ2,…,θn.Proof.
Sinceθ˜1,θ˜2,…,θ˜n is any permutation of θ1,θ2,…,θn, then(41)AN−IFABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d=1nn−1⊕i,j=1i≠jnθ˜ib⊗θ˜jd1/b+d=AN−IFABMb,dθ˜1,θ˜2,…,θ˜n.Theorem 7.
Letθi for all i∈N+ be a set of IFNs and b,d≥0, then(42)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMd,bθ1,θ2,…,θn.Proof.
By using Theorem1, we get(43)AN−IFABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d=1nn−1⊕j,i=1j≠inθjd⊗θib1/d+b=AN−IFABMd,bθ1,θ2,…,θn.
If we assign the generatorδ with different forms, then AN-IFABM reduces to some specific intuitionistic fuzzy BMs.Remark 2.
Ifδt=−loget, the AN-IFABM converts to the IFBM defined by Xu and Yager [33]:(44)IFBMb,dθ1,θ2,…,θn=1−∏i,j=1i≠jn1−μθibμθjd1/nn−11/b+d,1−1−∏i,j=1i≠jn1−1−νθib1−νθjd1/nn−11/b+d.Remark 3.
Ifδt=loge2−t−loget, the AN-IFABM changes to the intuitionistic fuzzy Einstein BM (IFEBM):(45)IFEBMb,dθ1,θ2,…,θn=2Pμθi,μθj−Qμθi,μθj1/b+dPμθi,μθj+3Qμθi,μθj1/b+d−Pμθi,μθj−Qμθi,μθj1/b+d,Mνθi,νθj+3Nνθi,νθj1/b+d−Mνθi,νθj−Nνθi,νθj1/b+bMνθi,νθj+3Nνθi,νθj1/b+d+Mνθi,νθj−Nνθi,νθj1/b+d,where(46)Pμθi,μθj=∏i,j=1i≠jn2−μθib2−μθjd+3μθibμθjd1/nn−1,Qμθi,μθj=∏i,j=1i≠jn2−μθib2−μθjd−μθibμθjd1/nn−1,Mνθi,νθj=∏i,j=1i≠jn1+νθib1+νθjd+31−νθib1−νθjd1/nn−1,Nνθi,νθj=∏i,j=1i≠jn1+νθib1+νθjd−1−νθib1−νθjd1/nn−1.
If we take the parametersb and d of the AN-IFABM in various values, then several special conditions can be derived as follows:Remark 4.
Ifb⟶0 or b⟶0, then the AN-IFABM is transformed as the Archimedean norm-based generalized IF mean (AN-GIFM) (take d⟶0 for example):(47)AN−GIFMθ1,θ2,…,θn=limd⟶01nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d=1n⊕i=1nθib1/b=δ−11bδε−11n∑i=1nεδ−1b⋅δμθi,ε−11bεδ−11n∑i=1nδε−1b⋅ενθi.Remark 5.
Ifb=2 and d⟶0, then the AN-IFABM is transformed as the Archimedean norm-based IF square mean (AN-IFSM):(48)AN−IFSMθ1,θ2,…,θn=1n⊕i=1nθi21/2=δ−112δε−11n∑i=1nεδ−12δμθi,ε−112εδ−11n∑i=1nδε−12ενθi.Remark 6.
Ifb=1 and d⟶0, then the AN-IFABM is transformed as the Archimedean norm-based IF averaging (AN-IFA) [38]:(49)AN−IFAθ1,θ2,…,θn=1n⊕i=1nθi=ε−11n∑i=1nεμθi,δ−11n∑i=1nδνθi.Remark 7.
Ifb=d=1, then the AN-IFABM is transformed as the following:(50)AN−IFISMθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθi⊗θj1/2=δ−112δε−11nn−1∑i,j=1i≠jnεδ−1δμθi+δμθj,ε−112εδ−11nn−1∑i,j=1i≠jnδε−1ενθi+ενθj,which is named the Archimedean norm-based IF interrelated square mean (AN-IFISM).


### 3.2. IFGBM Based on Archimedean Norm

Motivated by the geometric mean [42], we shall investigate the AN-IFGBM, and we also explore several adequate properties of the AN-IFGBM.Definition 9.
Letθi for all i∈N+ be a set of IFNs and b,d≥0; the AN-IFGBM is expressed as(51)AN−IFGBMb,dθ1,θ2,…,θn=1b+d⊗i,j=1i≠jnbθi⊕dθj1/nn−1.Theorem 8.
Letθi for all i∈N+ be a set of IFNs and b,d≥0, then the gathered value by using the AN-IFGBM is also an IFN, and(52)AN−IFGBMb,dθ1,θ2,…,θn=1b+d⊗i,j=1i≠jnbθi⊕dθj1/nn−1=ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bεμθi+dεμθj,δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδνθi+dδνθj.Similarly, the AN-IFGBM also satisfies the properties that the AN-IFABM has.Theorem 9.
(idempotency). Letθi for all i∈N+ be a group of IFNs and b,d≥0. If all θi are equal, i.e., θi=θ=μθ,νθ, for all i, then(53)AN−IFGBMb,dθ1,θ2,…,θn=μθ,νθ=θ.Remark 8.
Ifθi for all i∈N+ is a group of the smallest IFNs, i.e., θ1=θ2=⋯=θn=θ−=0,1, then(54)AN−IFGBMb,dθ1,θ2,…,θn=AN−IFGBMb,dθ−,θ−,…,θ−=0,1.
Ifθi for all i∈N+ are a group of the largest IFNs, i.e., θ1=θ2=⋯=θn=θ+=1,0, then(55)AN−IFGBMb,dθ1,θ2,…,θn=AN−IFGBMb,dθ+,θ+,…,θ+=1,0.Theorem 10.
(monotonicity). Let θ1 and ϑi for all i∈N+ be two collections of IFNs, b,d≥0. If μθi≤μϑi and νθi≥νϑi, for all i, then(56)AN−IFGBMb,dθ1,θ2,…,θn≤AN−IFGBMb,dϑ1,ϑ2,…,ϑn.Theorem 11.
(boundedness). Letθi for all i∈N+ be a set of IFNs and b,d≥0, and let(57)θL=miniμθi,maxiνθi,θU=maxiμθi,miniνθi,then we have(58)θL≤AN−IFGBMb,dθ1,θ2,…,θn≤θU.Theorem 12.
(commutativity). Letθi for all i∈N+ be a set of IFNs and b,d≥0, then(59)AN−IFGBMb,dθ1,θ2,…,θn=AN−IFGBMb,dθ˜1,θ˜2,…,θ˜n,where θ˜1,θ˜2,…,θ˜nθ˙1,θ˙2,…,θ˙n is any permutation of θ1,θ2,…,θn.Moreover, if we assign the additive generatorδ with different forms, then the AN-IFGBM reduces to some specific intuitionistic fuzzy geometric BMs.Remark 9.
Ifδt=−loget, the AN-IFGBM is transformed as the IFGBM defined by Xia et al. [34]:(60)IFGBMb,dθ1,θ2,…,θn=1−1−∏i,j=1i≠jn1−1−μθib1−μθjd1/nn−11/b+d,1−∏i,j=1i≠jn1−νθibνθjd1/nn−11/b+d.Remark 10.
Ifδt=loge2−t−loget, the AN-IFGBM is transformed as the intuitionistic fuzzy Einstein geometric BM (IFEGBM):(61)IFEGBMb,dθ1,θ2,…,θn=Mμθi,μθj+3Nμθi,μθj1/b+d−Mμθi,νθj−Nμθi,μθj1/b+dMμθi,μθj+3Nμθi,μθj1/b+d+Mμθi,μθj−Nμθi,νθj1/b+d,2Pνθi,νθj−Qνθi,νθj1/b+dPνθi,νθj+3Qνθi,νθj1/b+d−Pνθi,νθj−Qνθi,νθj1/b+d,where(62)Mμθi,μθj=∏i,j=1i≠jn1+μθib1+μθjd+31−μθib1−μθjd1/nn−1,Nμθi,μθj=∏i,j=1i≠jn1+μθib1+μθjd−1−μθib1−μθjd1/nn−1,Pνθi,νθj=∏i,j=1i≠jn2−νθib2−νθjd+3νθibνθjd1/nn−1,Qνθi,νθj=∏i,j=1i≠jn2−νθib2−νθjd−νθibνθjd1/nn−1.
In the following, we will consider several exceptional cases of the AN-IFGBM with diverse values of the parametersb and d.Remark 11.
Ifb⟶0 or d⟶0, then the AN-IFGBM is transformed as the Archimedean norm-based generalized IF geometric mean (AN-GIFGM) (take d⟶0 for example):(63)AN−GIFGMθ1,θ2,…,θn=limd⟶01b+d⊗i,j=1i≠jnbθi⊕dθj1/nn−1=1b⊗i=1nbθi1/n=ε−11bεδ−11n∑i=1nδε−1bεμθi,δ−11bδε−11n∑i=1nεδ−1bδνθi.Remark 12.
Ifb=2 and d⟶0, then the AN-IFGBM is transformed as(64)AN−IFSGMθ1,θ2,…,θn=12⊗i=1n2θi1/n=ε−112εδ−11n∑i=1nδε−12εμθi,δ−112δε−11n∑i=1nεδ−12δνθi,which we call the Archimedean norm-based IF square geometric mean (AN-IFSGM).Remark 13.
Ifb=1 and d⟶0, the AN-IFGBM is transformed as the Archimedean norm-based intuitionistic fuzzy geometric averaging (AN-IFGA) [40]:(65)AN−IFGAθ1,θ2,…,θn=⊗i=1nθi1/n=ε−1εδ−11n∑i=1nδε−1εμθi,δ−1δε−11n∑i=1nεδ−1δνθi=δ−11n∑i=1nδμθi,ε−11n∑i=1nενθi.Remark 14.
Ifb=d=1, then the AN-IFGBM is transformed as the Archimedean norm-based IF interrelated geometric square mean (AN-IFIGSM):(66)AN-IFIGSMθ1,θ2,…,θn=12⊗i,j=1i≠jnθi⊕θj1/nn-1=ε-112⋅εδ-11nn-1∑i,j=1i≠jnδε-1εμθi+εμθj,δ-112⋅δε-11nn-1∑i,j=1i≠jnεδ-1δνθi+δνθj.


## 3.1. IFABM Based on Archimedean Norm

As defined in Section2, the operations can be utilized to integrate IF information. In this subsection, we shall investigate the AN-IFABM and then analyze some desirable properties of the AN-IFABM.Definition 8.
Letθi=μθi,νθi for all i∈N+=1,2,…,n be a group of IFNs and b,d≥0; the AN-IFABM is denoted as(5)AN−IFABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d.Theorem 2.
Letθi=μθi,νθi for all i∈N+ be a set of IFNs and b,d≥0, then the gathered value by using AN-IFABM is also an IFN, and(6)AN−IFABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d=δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj.Proof.
With the operations of IFN stated in Section2, we obtain that(7)θib=δ−1pδμθi,ε−1bενθi,θjd=δ−1dδμθj,ε−1dενθj,for alli,j,and then(8)θib⊗θjd=δ−1δδ−1bδμθi+δδ−1dδμθj,ε−1εε−1bενθi+εε−1dενθj=δ−1bδμθi+dδμθj,ε−1bενθi+dενθj.
According to Definition7, we can get that(9)⊕i,j=1i≠jnθib⊗θjd=ε−1∑i,j=1i≠jnεδ−1bδμθi+dδμθi,δ−1∑i,j=1i≠jnδε−1bενθi+dενθi.
Then, we have(10)1nn−1⊕i,j=1i≠jnθib⊗θjd=ε−11nn−1εε−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,δ−11nn−1δδ−1∑i,j=1i≠jnδε−1bενθi+dενθj=ε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,δ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj.
It follows that(11)1nn−1⊕i,j=1i≠jnηib⊗ηjd1/b+d=δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj,i.e., equation (6) holds.
In the following, we prove that the gathered value with AN-IFABM is also an IFN. Sinceδ:0,1⟶0,+∞ is a strictly decreasing function, and εt=δ1−t, then εt and ε−1t are two strictly increasing functions, δ−1t is a strictly decreasing function, and ε−1t=1−δ−1t. Therefore, we have(12)δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj≥0,ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj≥0.
Afterwards, we will demonstrate that(13)δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj+ε−11b+dεδ−11nn−1δ∑i,j=1i≠jnδε−1bενθi+dενθj≤1.
Sinceμθi≤1−νθi,i∈N+, then b⋅ενθi+d⋅ενθj=b⋅δ1−νθi+d⋅δ1−νθj≤b⋅δμθi+d⋅δμθj, for all i,j,and we have(14)ε−1b⋅ενθi+d⋅ενθj≤ε−1b⋅δμθi+d⋅δμθj=1−δ−1b⋅δμθi+d⋅δμθj.
Asδt is a strictly decreasing function, one can derive that(15)δε−1b⋅ενθi+d⋅ενθj≥δ1−δ−1b⋅δμθi+d⋅δμθj=εδ−1b⋅δμθi+d⋅δμθj,and then we have(16)∑i≠jδε−1b⋅ενθi+d⋅ενθj≥∑i≠jεδ−1b⋅δμθi+d⋅δμθj.
Thus, it follows that(17)δ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≤δ−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj=1−ε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj.
Hence,(18)1b+dεδ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≤1b+dε1−ε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj=1b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,and then(19)ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≤ε−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj=1−δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj,i.e.,(20)ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθi+dενθj+δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθi+dδμθj≤1,which completes the demonstration of Theorem 2.Theorem 3.
(idempotency). Letθi for all i∈N+ be a group of IFNs and b,d≥0. If all θi=θ=μθ,νθ for all i, then(21)AN−IFABMb,dθ1,θ2,…,θn=μθ,νθ=θ.Proof.
Sinceθ1=θ2=⋯=θn=θ=μθ,νθ, then we have(22)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dθ,θ,…,θ=δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδμθ+dδμθ,ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bενθ+dενθ=δ−11b+dδε−1εδ−1b+dδμθ,ε−11b+dεδ−1δε−1b+dενθ=δ−1δμθ,ε−1ενθ=μθ,νθ=θ.
The proof is completed.Remark 1.
Ifθi for all i∈N+ are a set of the smallest IFNs, i.e., θ1=θ2=⋯=θn=θ−=0,1, then(23)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dθ−,θ−,…,θ−=0,1.
Ifθi for all i∈N+are a group of the largest IFNs, i.e., θ1=θ2=⋯=θn=θ+=1,0, then(24)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dθ+,θ+,…,θ+=1,0.Theorem 4.
(monotonicity). Letθi=μθi,νθi and ϑi=μϑi,νϑi for all i∈N+ be two collections of IFNs, b,d≥0. If μθi≤μϑi and νθi≥νϑi, then(25)AN−IFABMb,dθ1,θ2,…,θn≤AN−IFABMb,dϑ1,ϑ2,…,ϑn.Proof.
Due toμθi≤μϑi and νθi≥νϑi for all i∈N+, εt and ε−1t are two strictly increasing functions, and δt and δ−1t are two strictly decreasing functions, we can obtain that(26)b⋅δμθi+d⋅δμθj≥b⋅εμϑi+d⋅εμϑj,b⋅ενθi+d⋅ενθj≥b⋅ενϑi+d⋅ενϑj,and then(27)δ−1b⋅δμθi+d⋅δμθj≤δ−1b⋅δμϑi+d⋅δμϑj,ε−1b⋅ενθi+d⋅ενθj≥ε−1b⋅ενϑi+d⋅ενϑj.
Thus,(28)δ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≥δ−11nn−1∑i,j=1i≠jnδε−1b⋅ενϑi+d⋅ενϑj.
Hence,(29)δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμθi+d⋅δμθj≤δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1b⋅δμϑi+d⋅δμϑj,(30)ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1b⋅ενθi+d⋅ενθj≥ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1b⋅ενϑi+d⋅ενϑj.
Letθ=AN−IFABMb,dθ1,θ2,…,θn and ϑ=AN−IFABMb,dϑ1,ϑ2,…,ϑn, then equations (29) and (30) are equal to μθ≤μϑ and νθ≥νϑ. Hence, we have(31)sθ=μθ−νθ≤μϑ−νϑ=sϑ.Case 1.
Ifsθ<sϑ, then by using Definition 6, we can get that θ<ϑ, i.e.,(32)AN−IFABMb,dθ1,θ2,…,θn<AN−IFABMb,dϑ1,ϑ2,…,ϑn.Case 2.
Ifsθ=sϑ, i.e., μθ−νθ=μϑ−νϑ, then μθ+νϑ=μϑ+νθ. Since μθ≤μϑ and νθ≥νϑ, thus μθ=μϑ and νθ=νϑ, then we have(33)fθ=μθ+νθ=μϑ+νϑ=fϑ.
Then, by Definition6, we have θ=ϑ, i.e.,(34)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dϑ1,ϑ2,…,ϑn.
According to equations (32) and (34), we can obtain that equation (25) holds.Theorem 5.
(boundedness). Letθi for all i∈N+ be a set of IFNs and b,d≥0, and let(35)θL=miniμθi,maxiνθi,θU=maxiμθi,miniνθi,then we have(36)θL≤AN−IFABMb,dθ1,θ2,…,θn≤θU.Proof.
Asminiμθi≤μθi≤maxiμθi and maxiνθi≥νθi≥miniνθi, for all i, by Theorem 4, we obtain that(37)AN−IFABMb,dθL,θL,…,θL≤AN−IFABMb,dθ1,θ2,…,θn≤AN−IFABMb,dθU,θU,…,θU.
Furthermore, according to Theorem3, we have(38)AN−IFABMb,dθL,θL,…,θL=θL,AN−IFABMb,dθU,θU,…,θU=θU.
Hence, from equations (37) and (38), we get that(39)θL≤AN−IFABMb,dθ1,θ2,…,θn≤θU.Theorem 6.
(commutativity). Letθi=μθi,νθi for all i∈N+ be a group of IFNs and b,d≥0, then(40)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMb,dθ˜1,θ˜2,…,θ˜n,where θ˜1,θ˜2,…,θ˜n is any permutation of θ1,θ2,…,θn.Proof.
Sinceθ˜1,θ˜2,…,θ˜n is any permutation of θ1,θ2,…,θn, then(41)AN−IFABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d=1nn−1⊕i,j=1i≠jnθ˜ib⊗θ˜jd1/b+d=AN−IFABMb,dθ˜1,θ˜2,…,θ˜n.Theorem 7.
Letθi for all i∈N+ be a set of IFNs and b,d≥0, then(42)AN−IFABMb,dθ1,θ2,…,θn=AN−IFABMd,bθ1,θ2,…,θn.Proof.
By using Theorem1, we get(43)AN−IFABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d=1nn−1⊕j,i=1j≠inθjd⊗θib1/d+b=AN−IFABMd,bθ1,θ2,…,θn.
If we assign the generatorδ with different forms, then AN-IFABM reduces to some specific intuitionistic fuzzy BMs.Remark 2.
Ifδt=−loget, the AN-IFABM converts to the IFBM defined by Xu and Yager [33]:(44)IFBMb,dθ1,θ2,…,θn=1−∏i,j=1i≠jn1−μθibμθjd1/nn−11/b+d,1−1−∏i,j=1i≠jn1−1−νθib1−νθjd1/nn−11/b+d.Remark 3.
Ifδt=loge2−t−loget, the AN-IFABM changes to the intuitionistic fuzzy Einstein BM (IFEBM):(45)IFEBMb,dθ1,θ2,…,θn=2Pμθi,μθj−Qμθi,μθj1/b+dPμθi,μθj+3Qμθi,μθj1/b+d−Pμθi,μθj−Qμθi,μθj1/b+d,Mνθi,νθj+3Nνθi,νθj1/b+d−Mνθi,νθj−Nνθi,νθj1/b+bMνθi,νθj+3Nνθi,νθj1/b+d+Mνθi,νθj−Nνθi,νθj1/b+d,where(46)Pμθi,μθj=∏i,j=1i≠jn2−μθib2−μθjd+3μθibμθjd1/nn−1,Qμθi,μθj=∏i,j=1i≠jn2−μθib2−μθjd−μθibμθjd1/nn−1,Mνθi,νθj=∏i,j=1i≠jn1+νθib1+νθjd+31−νθib1−νθjd1/nn−1,Nνθi,νθj=∏i,j=1i≠jn1+νθib1+νθjd−1−νθib1−νθjd1/nn−1.
If we take the parametersb and d of the AN-IFABM in various values, then several special conditions can be derived as follows:Remark 4.
Ifb⟶0 or b⟶0, then the AN-IFABM is transformed as the Archimedean norm-based generalized IF mean (AN-GIFM) (take d⟶0 for example):(47)AN−GIFMθ1,θ2,…,θn=limd⟶01nn−1⊕i,j=1i≠jnθib⊗θjd1/b+d=1n⊕i=1nθib1/b=δ−11bδε−11n∑i=1nεδ−1b⋅δμθi,ε−11bεδ−11n∑i=1nδε−1b⋅ενθi.Remark 5.
Ifb=2 and d⟶0, then the AN-IFABM is transformed as the Archimedean norm-based IF square mean (AN-IFSM):(48)AN−IFSMθ1,θ2,…,θn=1n⊕i=1nθi21/2=δ−112δε−11n∑i=1nεδ−12δμθi,ε−112εδ−11n∑i=1nδε−12ενθi.Remark 6.
Ifb=1 and d⟶0, then the AN-IFABM is transformed as the Archimedean norm-based IF averaging (AN-IFA) [38]:(49)AN−IFAθ1,θ2,…,θn=1n⊕i=1nθi=ε−11n∑i=1nεμθi,δ−11n∑i=1nδνθi.Remark 7.
Ifb=d=1, then the AN-IFABM is transformed as the following:(50)AN−IFISMθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnθi⊗θj1/2=δ−112δε−11nn−1∑i,j=1i≠jnεδ−1δμθi+δμθj,ε−112εδ−11nn−1∑i,j=1i≠jnδε−1ενθi+ενθj,which is named the Archimedean norm-based IF interrelated square mean (AN-IFISM).


## 3.2. IFGBM Based on Archimedean Norm

Motivated by the geometric mean [42], we shall investigate the AN-IFGBM, and we also explore several adequate properties of the AN-IFGBM.Definition 9.
Letθi for all i∈N+ be a set of IFNs and b,d≥0; the AN-IFGBM is expressed as(51)AN−IFGBMb,dθ1,θ2,…,θn=1b+d⊗i,j=1i≠jnbθi⊕dθj1/nn−1.Theorem 8.
Letθi for all i∈N+ be a set of IFNs and b,d≥0, then the gathered value by using the AN-IFGBM is also an IFN, and(52)AN−IFGBMb,dθ1,θ2,…,θn=1b+d⊗i,j=1i≠jnbθi⊕dθj1/nn−1=ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bεμθi+dεμθj,δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδνθi+dδνθj.Similarly, the AN-IFGBM also satisfies the properties that the AN-IFABM has.Theorem 9.
(idempotency). Letθi for all i∈N+ be a group of IFNs and b,d≥0. If all θi are equal, i.e., θi=θ=μθ,νθ, for all i, then(53)AN−IFGBMb,dθ1,θ2,…,θn=μθ,νθ=θ.Remark 8.
Ifθi for all i∈N+ is a group of the smallest IFNs, i.e., θ1=θ2=⋯=θn=θ−=0,1, then(54)AN−IFGBMb,dθ1,θ2,…,θn=AN−IFGBMb,dθ−,θ−,…,θ−=0,1.
Ifθi for all i∈N+ are a group of the largest IFNs, i.e., θ1=θ2=⋯=θn=θ+=1,0, then(55)AN−IFGBMb,dθ1,θ2,…,θn=AN−IFGBMb,dθ+,θ+,…,θ+=1,0.Theorem 10.
(monotonicity). Let θ1 and ϑi for all i∈N+ be two collections of IFNs, b,d≥0. If μθi≤μϑi and νθi≥νϑi, for all i, then(56)AN−IFGBMb,dθ1,θ2,…,θn≤AN−IFGBMb,dϑ1,ϑ2,…,ϑn.Theorem 11.
(boundedness). Letθi for all i∈N+ be a set of IFNs and b,d≥0, and let(57)θL=miniμθi,maxiνθi,θU=maxiμθi,miniνθi,then we have(58)θL≤AN−IFGBMb,dθ1,θ2,…,θn≤θU.Theorem 12.
(commutativity). Letθi for all i∈N+ be a set of IFNs and b,d≥0, then(59)AN−IFGBMb,dθ1,θ2,…,θn=AN−IFGBMb,dθ˜1,θ˜2,…,θ˜n,where θ˜1,θ˜2,…,θ˜nθ˙1,θ˙2,…,θ˙n is any permutation of θ1,θ2,…,θn.Moreover, if we assign the additive generatorδ with different forms, then the AN-IFGBM reduces to some specific intuitionistic fuzzy geometric BMs.Remark 9.
Ifδt=−loget, the AN-IFGBM is transformed as the IFGBM defined by Xia et al. [34]:(60)IFGBMb,dθ1,θ2,…,θn=1−1−∏i,j=1i≠jn1−1−μθib1−μθjd1/nn−11/b+d,1−∏i,j=1i≠jn1−νθibνθjd1/nn−11/b+d.Remark 10.
Ifδt=loge2−t−loget, the AN-IFGBM is transformed as the intuitionistic fuzzy Einstein geometric BM (IFEGBM):(61)IFEGBMb,dθ1,θ2,…,θn=Mμθi,μθj+3Nμθi,μθj1/b+d−Mμθi,νθj−Nμθi,μθj1/b+dMμθi,μθj+3Nμθi,μθj1/b+d+Mμθi,μθj−Nμθi,νθj1/b+d,2Pνθi,νθj−Qνθi,νθj1/b+dPνθi,νθj+3Qνθi,νθj1/b+d−Pνθi,νθj−Qνθi,νθj1/b+d,where(62)Mμθi,μθj=∏i,j=1i≠jn1+μθib1+μθjd+31−μθib1−μθjd1/nn−1,Nμθi,μθj=∏i,j=1i≠jn1+μθib1+μθjd−1−μθib1−μθjd1/nn−1,Pνθi,νθj=∏i,j=1i≠jn2−νθib2−νθjd+3νθibνθjd1/nn−1,Qνθi,νθj=∏i,j=1i≠jn2−νθib2−νθjd−νθibνθjd1/nn−1.
In the following, we will consider several exceptional cases of the AN-IFGBM with diverse values of the parametersb and d.Remark 11.
Ifb⟶0 or d⟶0, then the AN-IFGBM is transformed as the Archimedean norm-based generalized IF geometric mean (AN-GIFGM) (take d⟶0 for example):(63)AN−GIFGMθ1,θ2,…,θn=limd⟶01b+d⊗i,j=1i≠jnbθi⊕dθj1/nn−1=1b⊗i=1nbθi1/n=ε−11bεδ−11n∑i=1nδε−1bεμθi,δ−11bδε−11n∑i=1nεδ−1bδνθi.Remark 12.
Ifb=2 and d⟶0, then the AN-IFGBM is transformed as(64)AN−IFSGMθ1,θ2,…,θn=12⊗i=1n2θi1/n=ε−112εδ−11n∑i=1nδε−12εμθi,δ−112δε−11n∑i=1nεδ−12δνθi,which we call the Archimedean norm-based IF square geometric mean (AN-IFSGM).Remark 13.
Ifb=1 and d⟶0, the AN-IFGBM is transformed as the Archimedean norm-based intuitionistic fuzzy geometric averaging (AN-IFGA) [40]:(65)AN−IFGAθ1,θ2,…,θn=⊗i=1nθi1/n=ε−1εδ−11n∑i=1nδε−1εμθi,δ−1δε−11n∑i=1nεδ−1δνθi=δ−11n∑i=1nδμθi,ε−11n∑i=1nενθi.Remark 14.
Ifb=d=1, then the AN-IFGBM is transformed as the Archimedean norm-based IF interrelated geometric square mean (AN-IFIGSM):(66)AN-IFIGSMθ1,θ2,…,θn=12⊗i,j=1i≠jnθi⊕θj1/nn-1=ε-112⋅εδ-11nn-1∑i,j=1i≠jnδε-1εμθi+εμθj,δ-112⋅δε-11nn-1∑i,j=1i≠jnεδ-1δνθi+δνθj.


## 4. IFWBMs Based on Archimedean Norm

The AN-IFABM and AN-IFGBM mentioned in Section3 take into consideration the connection between the attributes, in which each element gives the same contribution to AN-IFABM and AN-IFGBM. While, sometimes, different attributes have an unequal importance in real-life problems, thus, each attribute produces different effects on the results. To evaluate the impact of each argument, we develop two IFWBMs with Archimedean norm.Definition 10.
Letθi for all i∈N+ be a group of IFNs and b,d≥0; an AN-IFWABM is a mapping AN−IFWABM:Hn⟶H, satisfying(67)AN−IFWABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnυiθib⊗υjθjd1/b+d,where υ=υ1,υ2,…,υnΤ is the weight vector (WV), and υi>0,∑i=1nυi=1.Theorem 13.
Letθi for all i∈N+ be a group of IFNs, whose WV is υ=υ1,υ2,…,υnΤ, and satisfy υi>0,∑i=1nυi=1. Let b,d≥0, then the gathered value using the AN-IFWABM is also an IFN, and(68)AN−IFWABMb,dθ1,θ2,…,θn=1nn−1⊕i,j=1i≠jnυiθib⊗υjθjd1/b+d=δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδε−1υiεμθi+dδε−1υjεμθj,ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bεδ−1υiδνθi+dεδ−1υjδνθj.If we assign the generatorδ with various forms, then one can derive several specific IF weighted BMs.Remark 15.
Ifδt=−loget, the AN-IFWABM is transformed as the IF weighted BM (IFWBM) [33]:(69)IFWBMb,dθ1,θ2,…,θn=1−∏i,j=1i≠jn1−1−1−μθiυib1−μθjυjd1/nn−11/b+d,1−1−∏i,j=1i≠jn1−1−νθiυib1−νθjυjd1/nn−11/b+d.Remark 16.
Ifδt=loge2−t−loget, the AN-IFWABM is transformed as the IF Einstein weighted BM (IFEWBM):(70)IFEBMb,dθ1,θ2,…,θn=2Aμθi,μθj−Bμθi,μθj1/b+dAμθi,μθj+3Bμθi,μθj1/b+d−Aμθi,μθj−Bμθi,μθj1/b+d,Cνθi,νθj+3Dνθi,νθj1/b+d−Cνθi,νθj−Dνθi,νθj1/b+dCνθi,νθj+3Dνθi,νθj1/b+d+Cνθi,νθj−Dνθi,νθj1/b+d,where(71)Aμθi,μθj=∏i,j=1i≠jn1+μθiυi+31−μθiυib1+μθjυj+31−μθjυjd+31+μθiυi−1−μθiυib1+μθjυj−1−μθjυjd1/nn−1,Bμθi,μθj=∏i,j=1i≠jn1+μθiυi+31−μθiυib1+μθjυj+31−μθjυjd−1+μθiυi−1−μθiυib1+μθjυj−1−μθjυjd1/nn−1,Cνθi,νθj=∏i,j=1i≠jn2−νθiυi+3νθiυib2−νθjυj+3νθjυjd+32−νθiυi−νθiυib×2−νθjυj−νθjυjd1/nn−1,Dνθi,νθj=∏i,j=1i≠jn2−νθiυi+3νθiυib2−νθjυj+3νθjυjd−2−νθiυi−νθiυib×2−νθjυj−νθjυjd1/nn−1.Definition 11.
Letθi=μθi,νθi for all i∈N+ be a group of IFNs and b,d≥0; an AN-IFWGBM is defined as(72)AN−IFWGBMb,dθ1,θ2,…,θn=1b+d⊗i,j=1i≠jnbθiυi⊕dθjυj1/nn−1,where υ=υ1,υ2,…,υnΤ is the WV, and υi>0,∑i=1nυi=1.Theorem 14.
Letθi=μθi,νθi for all i∈N+ be a collection of IFNs, whose WV is υ=υ1,υ2,…,υnΤ, and satisfy υi>0,∑i=1nυi=1. Let b,d≥0, then the assembled value using the AN-IFWGBM is also an IFN, and(73)AN−IFWGBMb,dθ1,θ2,…,θn=1b+d⊗i,j=1i≠jnbθiυi⊕dθjυj1/nn−1=ε−11b+dεδ−11nn−1∑i,j=1i≠jnδε−1bεδ−1υiδμθi+dεδ−1υjδμθj,δ−11b+dδε−11nn−1∑i,j=1i≠jnεδ−1bδε−1υiενθi+dδε−1υjενθj.Proof.
The demonstration of Theorem14 is analogous to Theorem 8.
Moreover, given different forms for the additive generatorδ, several generalized IFWGBMs can be obtained as follows.Remark 17.
Ifδt=−loget, then the AN-IFWGBM is transformed as the IFWGBM [34]:(74)IFWGBMb,dθ1,θ2,…,θn=1−1−∏i,j=1i≠jn1−1−μθiυib1−μθjυjd1/nn−11/b+d,1−∏i,j=1i≠jn1−1−1−νθiυib1−νθjυjd1/nn−11/b+d.Remark 18.
Ifδt=loge2−t−loget, then the AN-IFWGBM is transformed as the IF Einstein weighted geometric BM (IFEWGBM):(75)IFEWGBMb,dθ1,θ2,…,θn=Cμθi,μθj+3Dμθi,μθj1/b+d−Cμθi,μθj−Dμθi,μθj1/b+dCμθi,μθj+3Dμθi,μθj1/b+d+Cμθi,μθj−Dμθi,μθj1/b+d,2Aνθi,νθj−Bνθi,νθj1/b+dAνθi,νθj+3Bνθi,νθj1/b+d−Aνθi,νθj−Bνθi,νθj1/b+d,where(76)Cμθi,μθj=∏i,j=1i≠jn2−μθiυi+3μθiυib2−μθjυj+3μθjυjd+32−μθiυi−μθiυib×2−μθjυj−μθjυjd1/nn−1,Dμθi,μθj=∏i,j=1i≠jn2−μθiυi+3μθiυib2−μθjυj+3μθjυjd−2−μθiυi−μθiυib×2−μθjυj−μθjυjd1/nn−1,Aνθi,νθj=∏i,j=1i≠jn1+νθiυi+31−νθiυib1+νθjυj+31−νθjυjd+31+νθiυi−1−νθiυib1+νθjυj−1−νθjυjd1/nn−1,Bνθi,νθj=∏i,j=1i≠jn1+νθiυi+31−νθiυib1+νθjυj+31−νθjυjd−1+νθiυi−1−νθiυib1+νθjυj−1−νθjυjd1/nn−1.


## 5. An Approach for MADM under IF Environment

Here, an IF MADM method is discussed based on AN-IFWABM and AN-IFWGBM.For MADM problems with IFNs, letY=Y1,Y2,…,Ym be a discrete set of options and ℝ=ℝ1,ℝ2,…,ℝn be a group of attributes, whose WV is υ=υ1,υ2,…,υnΤ, satisfying υj>0 and ∑j=1nυj=1, where υj denotes the impact value of the attribute ℝj. Assume that the DMs are demanded to provide the evaluation information of alternative Yi under the attribute ℝj with IFFs θij=μθij,νθij, where μθij,νθij∈0,1, and μθij+νθij∈0,1. With the performance of all alternatives available, we can construct an IF decision matrix D=θijm×n.Next, AN-IFWABM and AN-IFWGBM are availed to handle the IF MADM problems.Step 1.
For all attributesℝj that are benefit type, they need not to be normalized. Hence, we convert the IF decision matrix D=θijm×n to the normalized IF decision matrix B=ϑijm×n, where(77)ϑij=θij,for benefit attributeℝj,θijc,for cost attributeℝj,i=1,2,…,m,j=1,2,…,n,and θijc=νθij,μθij.Step 2.
Utilize AN-IFWABM,(78)ϑi=AN−IFWABMb,dϑi1,ϑi2,…,ϑin=1nn−1⊕j,k=1j≠knυjϑijb⊗υkϑikd1/b+d,or AN-IFWGBM,(79)ϑi=AN−IFWGBMb,dϑi1,ϑi2,…,ϑin=1b+d⊗j,k=1j≠knbϑijυj⊕dϑikυk1/nn−1,to gather all the individual IFN ϑijinto the overall IFN ϑi, where b,d≥0.Step 3.
The values ofsϑi and fϑi are calculated, respectively. Then, we sort the overall IFVs ϑi by Definition 6.Step 4.
Select the best oneYi with the highest priority by the sorting of ϑi.Step 5.
End.


## 6. Application to Efficiency Assessment Problem

As a demonstration of the application of the designed approach, in this section, a test of our method is presented for the efficiency evaluation problem. Then, a comparison is made between our approach and the methods in Xia et al. [34] and Rahman et al. [41].The transformation efficiency of scientific and technological achievements for the following 6 companies is evaluated: Chinasoft International (Y1), Geely Automobile Company (Y2), GWT Co., Ltd. (Y3), Tencent Holdings Ltd. (Y4), iFlyteK Co., Ltd. (Y5), and Alibaba (Y6). Assisted by 10 experts from different fields, a DM assessed these companies depending on the following three attributes: technical staff ℝ1, research spending ℝ2, and invention patents ℝ3. After reviewing the documents related to the three attributes in the assessment process, this DM provides the WV of the attributes ℝ1,ℝ2,ℝ3 as υ=0.30,0.50,0.20Τ. Suppose that the features of the public companies Y1,Y2,…,Y6 for the attributes ℝ1,ℝ2,ℝ3 are expressed by IFNs θij=μθij,νθij and all θij=μθij,νθij are included the IF decision matrix D=θij6×3 (see Table 1).Table 1
The IF decision matrixD=θij6×3.
ℝ1ℝ2ℝ3Y1(0.30, 0.40)(0.70, 0.20)(0.50, 0.30)Y2(0.50, 0.20)(0.40, 0.10)(0.70, 0.10)Y3(0.40, 0.50)(0.70, 0.20)(0.40, 0.40)Y4(0.20, 0.30)(0.80, 0.10)(0.80, 0.20)Y5(0.90, 0.10)(0.60, 0.30)(0.20, 0.50)Y6(0.60, 0.20)(0.70, 0.20)(0.50, 0.50)To choose the most preferred company, we utilize the AN-IFWABM and AN-IFWGBM to solve the IF MADM problems, respectively. The following are the major steps:(i)Step 1. Given that all attributes ℝj are revenue types that do not require normalization, then go to the next step.(ii)Step 2. For the matrix D=θij6×3 in IFSs, we utilize the AN-IFWABM (78) and AN-IFWGBM (79) (we take b=d=1 and let the additive generator δt=loge2−t−loget) to aggregate all the individual IFNs θij. The overall IFNs θi of the companies Yi are obtained, which are indicated in Table 2.(iii)Step 3. The values of sθi with the overall θi are calculated, which are indicated in Table 3.(iv)Step 4. Sort all Yi by the sθi; the ordering of the alternatives is presented in Table 4.Table 2
The overall IFNsθi.
θ1θ2θ3θ4θ5θ6AN-IFWABM (b = d = 1)(0.2224, 0.7196)(0.2208, 0.5945)(0.2284, 0.7511)(0.2998, 0.6901)(0.3314, 0.5912)(0.3012, 0.6551)AN-IFWGBM (b = d = 1)(0.8316, 0.1048)(0.8270, 0.0439)(0.8255, 0.1681)(0.8678, 0.0922)(0.8575, 0.1012)(0.8446, 0.0817)AN-IFWABM (b = d = 2)(0.2380, 0.7085)(0.2216, 0.5842)(0.2468, 0.7387)(0.3383, 0.6479)(0.4229, 0.5345)(0.3204, 0.6122)AN-IFWGBM (b = d = 2)(0.8270, 0.1076)(0.8099, 0.0468)(0.8144, 0.1682)(0.8533, 0.0984)(0.8331, 0.1142)(0.8459, 0.1106)Table 3
The scores of the overall IFNs.
sθ1sθ2sθ3sθ4sθ5sθ6AN-IFWABM (b = d = 1)−0.4972−0.3746−0.5227−0.3903−0.2598−0.3539AN-IFWGBM (b = d = 1)0.72680.78310.65740.77560.75630.7629AN-IFWABM (b = d = 2)−0.4705−0.3626−0.4919−0.3096−0.1116−0.2918AN-IFWGBM (b = d = 2)0.71640.76310.64620.75490.71690.7353Table 4
Ordering of six corporations.
OrderingAN-IFWABM (b = d = 1)Y5≻Y6≻Y2≻Y4≻Y1≻Y3AN-IFWGBM (b = d = 1)Y2≻Y4≻Y6≻Y5≻Y1≻Y3AN-IFWABM (b = d = 2)Y5≻Y6≻Y4≻Y2≻Y1≻Y3AN-IFWGBM (b = d = 2)Y2≻Y4≻Y6≻Y5≻Y1≻Y3One can observe that the ranking of the companies is slightly different as per the BMs deployed. Thus, the results may produce diverse decision-making results by the various BMs used.Now, if we takeb=d=2, then by the AN-IFWABM and AN-IFWGBM (let the additive generator δt=loge2−t−loget), we get the results as shown in Tables 2–4.In what follows, in view of the same IF decision matrixD=θij5×3, the methods established by Xia et al. [34] and Rahman et al. [41] are applied to get the most desirable company. With the method by Xia et al. [34], the following process steps are obtained.(i) Step [1]: see Step 1.(ii) Step [2]: applying the IFWBM equation (69) and IFWGBM equation (74) (we take b=d=1 and b=d=2, respectively) to gather all the individual IFNs θij, respectively, and then get the overall IFNs θi of the alternatives Yi, which are shown in Table 5.(iii) Step [3]: the values ofsθi with the overall IFVs θi are calculated in Table 6.(iv) Step [4]: sort allYi with the scores sθi; the ordering of the Yi is expressed in Table 7.Table 5
The overall IFNsθi.
θ1θ2θ3θ4θ5θ6IFWBM (b = d = 1)(0.1982, 0.6814)(0.2091, 0.5309)(0.2023, 0.7229)(0.2658, 0.6684)(0.2782, 0.6544)(0.2768, 0.6907)IFWGBM (b = d = 1)(0.8084, 0.1034)(0.8101, 0.0438)(0.8112, 0.1269)(0.8561, 0.0915)(0.8381, 0.1006)(0.8430, 0.1015)IFWBM (b = d = 2)(0.2091, 0.6728)(0.2089, 0.5248)(0.2141, 0.7129)(0.3022, 0.6467)(0.3279, 0.6248)(0.2984, 0.6312)IFWGBM (b = d = 2)(0.8039, 0.1056)(0.7924, 0.0465)(0.8100, 0.1290)(0.8406, 0.0971)(0.8092, 0.1130)(0.8259, 0.0805)Table 6
The scores of theθi.
sθ1sθ2sθ3sθ4sθ5sθ6IFWBM (b = d = 1)−0.5022−0.3218−0.5206−0.4026−0.3762−0.4139IFWGBM (b = d = 1)0.70500.76640.68430.76470.73750.7415IFWBM (b = d = 2)−0.4637−0.3159−0.4988−0.3445−0.2969−0.3328IFWGBM (b = d = 2)0.69820.74590.68100.74350.69620.7454Table 7
Ordering of six corporations.
OperatorsOrderingIFWBM (b = d = 1)Y2≻Y5≻Y4≻Y6≻Y1≻Y3IFWGBM (b = d = 1)Y2≻Y4≻Y6≻Y5≻Y1≻Y3IFWBM (b = d = 2)Y5≻Y2≻Y6≻Y4≻Y1≻Y3IFWGBM (b = d = 2)Y2≻Y6≻Y4≻Y1≻Y5≻Y3By using confidence level, Rahman et al. [41] proposed the CIFEHA operator and CIFEHG operator to fuse a group of IFNs into a collection IFN and then developed a MADM method. With the help of the method by Rahman et al. [41], the most desirable company can be derived as follows:(i) Step A: see Step1.(ii) Step B: based on the provided IF decision matrixD=θij6×3, we can determine the confidence level preference matrix Β=βij6×3 (where βij=nυjθij) in Table 8.(iii) Step C: utilizing the CIFEHA operator and CIFEHG operator [41] to fuse the confidence level preference values βijj=1,2,3 into the overall confidence level preference values βii=1,2,…,6 is shown in Table 9.(iv) Step D: by using Definition6, we can get the score functions of βii=1,2,…,6, and the ranking order of six companies can be derived, which are shown in Tables 10 and 11.Table 8
The IF decision matrixΒ=βij6×3.
ℝ1ℝ2ℝ3Y1(0.4728, 0.4559)(0.6452, 0.2257)(0.4515, 0.2013)Y2(0.6419, 0.3625)(0.4325, 0.0615)(0.7121, 0.2009)Y3(0.4256, 0.1526)(0.4456, 0.1254)(0.5569, 0.3749)Y4(0.3325, 0.5959)(0.8112, 0.1232)(0.7855, 0.2004)Y5(0.8945, 0.0922)(0.5525, 0.2254)(0.3456, 0.4521)Y6(0.5546, 0.3434)(0.6542, 0.1254)(0.5546, 0.4125)Table 9
The overall IFNsβii=1,2,…,6.
β1β2β3β4β5β6CIFEHA operator(0.4518, 0.5126)(0.4859, 0.4015)(0.4225, 0.5050)(0.4523, 0.4698)(0.3956, 0.4125)(0.4411, 0.4208)CIFEHG operator(0.6915, 0.1100)(0.7246, 0.1334)(0.7364, 0.2634)(0.7030, 0.1224)(0.7416, 0.2425)(0.7154, 0.1643)Table 10
The scores ofβii=1,2,…,6.
sβ1sβ2sβ3sβ4sβ5sβ6CIFEHA operator−0.06080.0844−0.0825−0.0175−0.0169−0.0203CIFEHG operator0.58150.59120.47300.58060.49910.5511Table 11
Ordering of six corporations.
OperatorsOrderingCIFEHA operatorY2≻Y5≻Y4≻Y6≻Y1≻Y3CIFEHG operatorY2≻Y1≻Y4≻Y6≻Y5≻Y3Obviously, the proposed method in this work yields the slightly different ranking result with the methods by Xia et al. [34] and Rahman et al. [3]. However, the method by Xia et al. [34] is designed to carry combination processes with the algebraic rules of IFNs, which are different from the limit case of general fuzzy sets [36]. On the other hand, the algebraic norm is a special case of Archimedean norm. Therefore, it is understandable that the decision-making process in our approach would be more efficient and common than the model established by Xia et al. [34]. In addition, a slightly different ranking of Yi was found which varies in parameters b and d, which mirrors the risk preferences of DMs. Moreover, with the variation of parameter values, the results of the alternatives may change. Yet, we cannot say which is the perfect order, it relies on the DMs’ optimism or pessimism and just reflects the DMs’ attitude.During the decision-making process by Rahman et al. [41], the CIFEHA operator and CIFEHG operator neglect the importance of each argument and their interconnections. Furthermore, according to the original IF decision matrix D=θij6×3 provided by DM, it is observed that θ4j>θ1j,j=1,2,3, which indicates that Tencent Holdings Ltd. Y4 is preferred to Chinasoft International Y1, i.e., Y4≻Y1. Therefore, the developed MADM method in the paper is more reliable than Rahman et al.’s [41] method.


## 7. Conclusions

As the Archimedean norm can capture the internal relationship between decision-making information, thus, it is widely used in the design of information fusion and the construction of decision model. This paper focuses on the construction of MADM method with intuitionistic fuzzy Archimedean Bonferroni means. Firstly, we introduced the intuitionistic fuzzy operations based on Archimedean norm. Then, we developed two new kinds of intuitionistic fuzzy BMs with the help of Archimedean norm, such as AN-IFABM and AN-IFGBM. We further discussed their desirable properties, including commutativity, idempotency, monotonicity, and boundedness. Furthermore, an intuitionistic fuzzy MADM method is developed by the proposed intuitionistic fuzzy BM. Finally, the new MADM method is applied to evaluate the efficiency of the conversion of scientific and technological achievements of six public companies in China.However, the proposed MADM method with intuitionistic fuzzy Archimedean Bonferroni means does not consider the cooperation consensus among DMs. Therefore, in the future, we will investigate the cooperation MADM methods for intuitionistic fuzzy information. The proposed intuitionistic fuzzy Archimedean Bonferroni means can also be expanded to other uncertainty areas, such ass Pythagorean fuzzy information and Pythagorean picture fuzzy information [44, 45].



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