# Generalized Intuitionistic Fuzzy Normalized Weighted Optimized Geometric Bonferroni Mean and Their Application to MADM

Authors: Jin-Bo Zhang; Tian-Le Sun; Ming-Hua Shi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In an information age, people often need to face a lot of decision-making information when making decisions. Some indicators are on the high side and others are on the low side which is a common phenomenon in decision-making. So, it is difficult to make a correct and rational judgment. Long-term research has proved that information aggregation operator is an effective tool to solve this kind of problem. Bonferroni mean (BM) is an important information aggregation tool which has the main feature of capturing the interrelationships among aggregated arguments. Because the existing geometric Bonferroni mean (GBM) cannot reflect the two-layer average calculation and the weighted GBM do not feature reducibility, this paper develops the intuitionistic fuzzy normalized weighted optimized GBM (IFNWOGBM) and the generalized intuitionistic fuzzy normalized weighted optimized GBM (GIFNWOGBM) and also studies their desirable properties and special cases. In the end, based on the IFNWOGBM and GIFNWOGBM, a method to multiple attribute decision-making (MADM) problem is proposed. In order to verify the effectiveness of the method, it is used to select the location of the library.


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## Body

## 1. Introduction

As our society develops, situations human encounter when making decisions is becoming more and more complex, and data and information we rely upon in these situations are highly vague and uncertain. Under these new circumstances, in order to better utilize decision-making information in modeling, scholars have extended the fuzzy set to many other forms, such as triangular fuzzy set, vague set, and intuitionistic fuzzy set. The flexibility and efficacy that the intuitionistic fuzzy set demonstrate during actual decision-makings have won public attention with related theories which further enriched, developed, and applied extensively in intelligent algorithm, graphics and image processing, and other fields [1–4].Aggregation operator, as an important tool for information aggregation, has always been an academic focus of decision-making. To aggregate the intuitionistic fuzzy decision-making information, a large number of operators have been introduced. By analyzing the shortcomings of the existing weighted averaging (WA) operator, Kumar et al. proposed some improved WA operators and demonstrated their advantages in the field of intuitionistic fuzzy decision-making with a large number of decision-making cases [5]. In order to solve the problem of investment target selection represented by intuitionistic fuzzy information, Zou et al. improved the weighted geometric (WG) operator and proposed an effective method to solve this kind of problem [6]. Combined with the advantages of Choquet integral and arithmetic aggregation operator, Jia et al. proposed some novel operators to solve the intuitionistic fuzzy supplier selection decision-making problem [7].The operators in the above literature gather information from the perspective of independent evaluation indexes, but in reality, most decision indexes have certain relevance. In order to overcome this shortcoming, Yanger presents a new fuzzy information aggregation operator, known as Bonferroni mean (BM), which can increase reliability of decisions made when data and information is highly correlated. Some scholars seek to replace the arithmetic average with geometric average in the BM so as to generate geometric Bonferroni mean (GBM). More commonly seen nowadays are the GBM and the weighted GBM defined by Xia et al. [8]. Mahmoodi et al. introduced some GBMs to aggregate linguistic Z-number decision-making information [9]. Devaraj and Broumi defined several neutrosophic cubic fuzzy GBMs and proposed a method to solve the financial risk decision-making problem [10]. Huang et al. presented some GBMs to solve a hesitant fuzzy uncertain linguistic MADM problem [11]. Park et al. further propose the optimal weighted GBM and generalized optimal weighted GBM [12]. However, this kind of geometric operators cannot reflect the two-layer average calculation which is the key feature of BM. Moreover, the weighted GBMs mentioned above fail to bear a common feature of classic weighted operators, reducibility. This means when weights are equal, they cannot degenerate back to geometric Bonferroni mean.In order to get rid of the above shortcomings, based on the full analysis of the construction of geometric operators and BM operators, this paper proposes some improved GBM operators to deal with intuitionistic fuzzy MADM problems. This paper is organized as follows. In Section2, we review some necessary concepts and operators. In Section 3, we defined IFONWGBM and GIFONWGBM and discussed their properties and special cases. In Section 4, an example about location of the library is used to demonstrate the application of IFONWGBM and GIFONWGBM in MADM. In Section 5, we present the comparative analysis with other MADM methods. Finally, Section 6 gives the summary of the operators and methods proposed in this paper.


## 2. Preliminaries

### 2.1. Some Intuitionistic Fuzzy Concepts

Definition 1.
(see [13, 14]). Let X be a fixed set. Then, an intuitionistic fuzzy set on X can be defined as(1)A=x,μAx,υAxx|∈X,where μAx:X⟶0,1 and υAx:X⟶0,1 satisfy the condition 0≤μAx+υAx≤1, ∀x∈X, and μAx and υAx represent the membership degree and the nonmembership degree of x to A.
In order to facilitate discussion, Xu calls the pairα=μα,υα an intuitionistic fuzzy number (IFN) with the conditions:(2)μα∈0,1,υα∈0,1,μα+υα∈0,1.
Letαi=μαi,υαii=1,2 and α=μα,υα be three IFNs. Xu et al. defined the following operation laws [4]:(1)
α1⊕α2=μα1+μα2−μα1μα2,υα1υα2(2)
α1⊗α2=μα1μα2,υα1+υα2−υα1υα2(3)
λα=1−1−μαλ,υαλ,λ>0(4)
αλ=μαλ,1−1−υαλ,λ>0
The IFNs comparison method used in most literature is given by Xu et al. as follows.Definition 2.
(see [4]). Let αi=μαi,υαii=1,2 and α=μα,υα be three IFNs. sα=μα−υα is called the score function of α, and hα=μα+υα is called the accuracy degree function of α.(i)
Ifsα1>sα2, then α1>α2(ii)
Ifsα1=sα2, then
(i)
Ifhα1=hα2, then α1=α2(ii)
Ifhα1>hα2, then α1>α2(iii)
Ifhα1<hα2, then α1<α2


### 2.2. Optimized Geometric Bonferroni Mean

Definition 3.
(see [8]). Let p, q, and aii=1,2,…,n be nonnegative real numbers. If Bp,qa1,a2,…,an=1/nn−1∑i,j=1,i≠jnaipajq1/p+q and GBp,qa1,a2,…,an=1/p+q∏i,j=1,i≠jnpa1+qaj1/nn−1, then Bp,q is called Bonferroni mean (BM), and GBp,q is called geometric Bonferroni mean (GBM).
Due to the excellent nature of BM and GBM, their weighted forms have been studied in different fuzzy environments by many scholars. However these weighted functions do not have the reducibility. To solve this problem, Zhou introduced normalized weighted Bonferroni mean as follows.Definition 4.
(see [15]). Let p, q, and aii=1,2,…,n be nonnegative real numbers. If NWB(3)a1,a2,…,anp,q=∑i,j=1i≠jnwiwj1−wiaipajq1/p+q,then NWBp,q is called normalized weighted Bonferroni mean (NWB).
Obviously, ifw1=w2=⋯=wn=1/n, then NWBp,qa1,a2,…,an=Bp,qa1,a2,…,an. In particular, by rearranging the terms in NWBp,q, the NWB is expressed as follows:(4)NWBp,qa1,a2,…,an=∑i=1nwiaip∑j=1j≠inwj1−wiajq1/p+q.
Then, it is easy to see that NWB contains two weighted means.∑j=1,j≠inwj/1−wcajq is the weighted average of ajqj≠i; ∑i=1nwiaip∑j=1,j≠inwj/1−wiajq is the weighted average of aip and ∑j=1,j≠inwj/1−wiajq, that is, the distinguishing characteristic of the BM [9]. However, the above definition of GBM and its weighted forms cannot reflect two geometric means such as BM or NWB. Furthermore, to our knowledge, there has been no report concerning the reducible weighted geometric Bonferroni mean previously. So, based on the work of Zhou [15] and Xia et al. [8], we introduce a new GBM and its weighted forms.Definition 5.
Letp, q, and aii=1,2,…,n are nonnegative real numbers. If(5)OGBMp,qa1,a2,…,an=1p+q∏i=1npai+∏j=1j≠inqaj1/n−11/n,then OGBMp,q is called the optimized GBM (OGBM).
Obviously, the OGBM have the following properties:(1)
OGBMp,q0,0,…,0=0(2)
Ifai=ai=1,2,…,n, then OGBMp,qa,a,…,a=a(3)
Ifai≥dii=1,2,…,n, then OGBMp,qa1,a2,…,an≥OGBMp,qd1,d2,…,dn(4)
min1≤i≤nai≤GBMp,qa1,a2,…,an≤max1≤i≤nai(5)
Ifq=0, then OGBMp,qa1,a2,…,an=∏i=1nai1/nDefinition 6.
Letp≥0, q≥0, and aii=1,2,…,n be nonnegative real numbers with the weight wii=1,2,…,n, wi∈0,1, ∑i=1nwi=1. If(6)NWOGBMp,qa1,a2,…,an=1p+q∏i=1npai+∏j=1j≠inqajwj/1−wiwi,then NWOGBMp,q is called the normalized weighted OGBM (NWOGBM).Definition 7.
Letp, q, r, and aii=1,2,…,n be nonnegative real numbers. If(7)GOGBMp,q,ra1,a2,…,an=1p+q+r∏i=1npa1+∏j=1j≠inqaj+∏k=1k≠j≠inrak1/n−21/n−11/n.then GOGBMp,q,r is called the generalized OGBM(GOGBM).Definition 8.
Letp≥0, q≥0, r≥0, and aii=1,2,…,n be nonnegative real numbers with the weight wii=1,2,…,n, wi∈0,1, and ∑i=1nwi=1. If(8)GNWOGBMp,q,ra1,a2,…,an=1p+q+r∏i=1npai+∏j=1j≠inqaj+∏k=1k≠j≠inrakwk/1−wi−wjwj/1−wiwi,then GNWOGBMp,q,r is called the generalized normalized weighted OGBM (GNWOGBM).
It is obvious that GNOGBM reduces to NOGBM ifr=0. When the weights are the same,(9)GNWOGBMp,q,ra1,a2,…,an=GOGBMp,q,ra1,a2,…,an,NWOGBMp,qa1,a2,…,an=OGBMp,qa1,a2,…,an,which reflect GNWOGBM and NWOGBM have the reducibility.


## 2.1. Some Intuitionistic Fuzzy Concepts

Definition 1.
(see [13, 14]). Let X be a fixed set. Then, an intuitionistic fuzzy set on X can be defined as(1)A=x,μAx,υAxx|∈X,where μAx:X⟶0,1 and υAx:X⟶0,1 satisfy the condition 0≤μAx+υAx≤1, ∀x∈X, and μAx and υAx represent the membership degree and the nonmembership degree of x to A.
In order to facilitate discussion, Xu calls the pairα=μα,υα an intuitionistic fuzzy number (IFN) with the conditions:(2)μα∈0,1,υα∈0,1,μα+υα∈0,1.
Letαi=μαi,υαii=1,2 and α=μα,υα be three IFNs. Xu et al. defined the following operation laws [4]:(1)
α1⊕α2=μα1+μα2−μα1μα2,υα1υα2(2)
α1⊗α2=μα1μα2,υα1+υα2−υα1υα2(3)
λα=1−1−μαλ,υαλ,λ>0(4)
αλ=μαλ,1−1−υαλ,λ>0
The IFNs comparison method used in most literature is given by Xu et al. as follows.Definition 2.
(see [4]). Let αi=μαi,υαii=1,2 and α=μα,υα be three IFNs. sα=μα−υα is called the score function of α, and hα=μα+υα is called the accuracy degree function of α.(i)
Ifsα1>sα2, then α1>α2(ii)
Ifsα1=sα2, then
(i)
Ifhα1=hα2, then α1=α2(ii)
Ifhα1>hα2, then α1>α2(iii)
Ifhα1<hα2, then α1<α2


## 2.2. Optimized Geometric Bonferroni Mean

Definition 3.
(see [8]). Let p, q, and aii=1,2,…,n be nonnegative real numbers. If Bp,qa1,a2,…,an=1/nn−1∑i,j=1,i≠jnaipajq1/p+q and GBp,qa1,a2,…,an=1/p+q∏i,j=1,i≠jnpa1+qaj1/nn−1, then Bp,q is called Bonferroni mean (BM), and GBp,q is called geometric Bonferroni mean (GBM).
Due to the excellent nature of BM and GBM, their weighted forms have been studied in different fuzzy environments by many scholars. However these weighted functions do not have the reducibility. To solve this problem, Zhou introduced normalized weighted Bonferroni mean as follows.Definition 4.
(see [15]). Let p, q, and aii=1,2,…,n be nonnegative real numbers. If NWB(3)a1,a2,…,anp,q=∑i,j=1i≠jnwiwj1−wiaipajq1/p+q,then NWBp,q is called normalized weighted Bonferroni mean (NWB).
Obviously, ifw1=w2=⋯=wn=1/n, then NWBp,qa1,a2,…,an=Bp,qa1,a2,…,an. In particular, by rearranging the terms in NWBp,q, the NWB is expressed as follows:(4)NWBp,qa1,a2,…,an=∑i=1nwiaip∑j=1j≠inwj1−wiajq1/p+q.
Then, it is easy to see that NWB contains two weighted means.∑j=1,j≠inwj/1−wcajq is the weighted average of ajqj≠i; ∑i=1nwiaip∑j=1,j≠inwj/1−wiajq is the weighted average of aip and ∑j=1,j≠inwj/1−wiajq, that is, the distinguishing characteristic of the BM [9]. However, the above definition of GBM and its weighted forms cannot reflect two geometric means such as BM or NWB. Furthermore, to our knowledge, there has been no report concerning the reducible weighted geometric Bonferroni mean previously. So, based on the work of Zhou [15] and Xia et al. [8], we introduce a new GBM and its weighted forms.Definition 5.
Letp, q, and aii=1,2,…,n are nonnegative real numbers. If(5)OGBMp,qa1,a2,…,an=1p+q∏i=1npai+∏j=1j≠inqaj1/n−11/n,then OGBMp,q is called the optimized GBM (OGBM).
Obviously, the OGBM have the following properties:(1)
OGBMp,q0,0,…,0=0(2)
Ifai=ai=1,2,…,n, then OGBMp,qa,a,…,a=a(3)
Ifai≥dii=1,2,…,n, then OGBMp,qa1,a2,…,an≥OGBMp,qd1,d2,…,dn(4)
min1≤i≤nai≤GBMp,qa1,a2,…,an≤max1≤i≤nai(5)
Ifq=0, then OGBMp,qa1,a2,…,an=∏i=1nai1/nDefinition 6.
Letp≥0, q≥0, and aii=1,2,…,n be nonnegative real numbers with the weight wii=1,2,…,n, wi∈0,1, ∑i=1nwi=1. If(6)NWOGBMp,qa1,a2,…,an=1p+q∏i=1npai+∏j=1j≠inqajwj/1−wiwi,then NWOGBMp,q is called the normalized weighted OGBM (NWOGBM).Definition 7.
Letp, q, r, and aii=1,2,…,n be nonnegative real numbers. If(7)GOGBMp,q,ra1,a2,…,an=1p+q+r∏i=1npa1+∏j=1j≠inqaj+∏k=1k≠j≠inrak1/n−21/n−11/n.then GOGBMp,q,r is called the generalized OGBM(GOGBM).Definition 8.
Letp≥0, q≥0, r≥0, and aii=1,2,…,n be nonnegative real numbers with the weight wii=1,2,…,n, wi∈0,1, and ∑i=1nwi=1. If(8)GNWOGBMp,q,ra1,a2,…,an=1p+q+r∏i=1npai+∏j=1j≠inqaj+∏k=1k≠j≠inrakwk/1−wi−wjwj/1−wiwi,then GNWOGBMp,q,r is called the generalized normalized weighted OGBM (GNWOGBM).
It is obvious that GNOGBM reduces to NOGBM ifr=0. When the weights are the same,(9)GNWOGBMp,q,ra1,a2,…,an=GOGBMp,q,ra1,a2,…,an,NWOGBMp,qa1,a2,…,an=OGBMp,qa1,a2,…,an,which reflect GNWOGBM and NWOGBM have the reducibility.


## 3. Intuitionistic Fuzzy Weighted OGBM

Definition 9.
Letp≥0, q≥0, and αii=1,2,…,n be intuitionistic fuzzy numbers with the weight wii=1,2,…,n, wi∈0,1, and ∑i=1nwi=1. If(10)IFNWOGBMp,qα1,α2,…,αn=1p+q⊗i=1npαi⊕⊗j=1,j≠inqαjwj/1−wiwi,then IFNWOGBMp,q is called the intuitionistic fuzzy normalized weighted OGBM (IFNWOGBM).
Ifwi=1/ni=1,2,…,n, then(11)IFNWOGBMp,qα1,α2,…,αn=1p+q⊗i=1npαi⊕⊗j=1,j≠inqαj1/n−1n,which we call the intuitionistic fuzzy OGBM (IFOGBM).Theorem 1.
Letp≥0, q≥0, and αi=μαi,υαii=1,2,…,n be intuitionistic fuzzy numbers with the weight wii=1,2,…,n, wi∈0,1, and ∑i=1nwi=1; then,(12)IFNWOGBMp,qα1,α2,…,αn=1−1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wiwi1/p+q,1−∏i=1n1−υaip1−∏j=1j≠in1−υajqwj/1−wiwi1/p+q,and IFNWOGBMp,qα1,α2,…,αn is also an IFN.Proof.
Sincepαi=1−1−μαip,υaip and qαj=1−1−μαjq,υajq, then(13)⊗j=1,j≠iNqαjwj/1−wi=⊗j=1,j≠iN1−1−μαjqwj/1−wi,1−1−υajqwj/1−wi=∏j=1,j≠in1−1−μαjqwj/1−wi,1−∏j=1,j≠in1−1−1−υajqwj/1−wi=∏j=1,j≠in1−1−μαjqwj/1−wi,1−∏j=1,j≠in1−υajqwj/1−wi.
Therefore,(14)pαi⊕⊗j=1,j≠iNqαjwj/1−wi=1−1−μαip+∏j=1,j≠in1−1−μαjqwj/1−wi−1−1−μαip∏j=1,j≠in1−1−μαjqwj/1−wi,υaip1−∏j=1,j≠in1−υajqwj/1−wi=1−1−μαip+1−μαip∏j=1,j≠in1−1−μαjqwj/1−wi,υaip1−∏j=1,j≠in1−υajqwj/1−wi=1−1−μαip1−∏j=1,j≠in1−1−μαjqwj/1−wi,υaip1−∏j=1,j≠in1−υajqwj/1−wipαi⊕⊗j=1,j≠inqαjwj/1−wiwi=1−1−μαip1−∏j=1,j≠in1−1−μαjqwj/1−wiwi,1−1−υaip1−∏j=1,j≠in1−υajqwj/1−wiwi.
Hence, we obtain(15)IFNWOGBMp,qα1,α2,…,αn=1p+q⊗i=1npαi⊕⊗j=1,j≠inqαjwj/1−wiwi=1p+q⊗i=1n1−1−μαip1−∏j=1,j≠in1−1−μαjqwj/1−wiwi,1−1−υaip1−∏j=1,j≠in1−υajqwj/1−wiwi=1p+q∏i=1n1−1−μαip1−∏j=1,j≠in1−1−μαjqwj/1−wiwi,1−∏i=1n1−υaip1−∏j=1,j≠in1−υajqwj/1−wiwi=1−1−∏i=1n1−1−μαip1−∏j=1,j≠in1−1−μαjqwj/1−wiwi1/p+q,1−∏i=1n1−υaip1−∏j=1,j≠in1−υajqwj/1−wiwi1/p+q.
Byμαi∈0,1, υαi∈0,1, and μαi+υαi∈0,1, we have(16)0≤1−1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wiwi1/p+q≤1,0≤1−∏i=1n1−υaip1−∏j=1j≠in1−υajqwj/1−wiwi1/p+q≤1.
In addition, sinceυαj≤1−μαj⇒1−υajq≥1−1−μαjq,(17)⇒1−υajqwj/1−wi≥1−1−μαjqwj/1−wi⇒∏j=1j≠in1−υajqwj/1−wi≥∏j=1j≠in1−1−μαjqwj/1−wi⇒1−∏j=1j≠in1−υajqwj/1−wi≤1−∏j=1j≠in1−1−μαjqwj/1−wi⇒υaip1−∏j=1j≠in1−υajqwj/1−wi≤1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wi⇒∏i=1n1−υaip1−∏j=1j≠in1−υajqwj/1−wiwi≥∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wiwi⇒1−∏i=1n1−υaip1−∏j=1j≠in1−υajqwj/1−wiwi1/p+q≤1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wiwi1/p+q⇒1−1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wiwi1/p+q+1−∏i=1n1−υaip1−∏j=1j≠in1−υajqwj/1−wiwi1/p+q≤1.
This completes the proof.Property 1.
(1)
Letαii=1,2,…,n be a collection of IFNs; if α1=α2=⋯=αn=α, then(18)IFNWOGBMp,qα1,α2,…,αn=α.(2)
Letαii=1,2,…,n be a collection of IFNs; if α1′,α2′,…,αN′ is any permutation of α1,α2,…,αn, then(19)IFNWOGBMp,qα1,α2,…,αn=IFNWOGBMp,qα1′,α2′,…,αn′.(3)
Letαi=μαi,υαi and βi=μβi,υβii=1,2,…,n be two collection of IFNs; if μαi≤μβi and υαi≥υβii=1,2,…,n, then(20)IFNWOGBMp,qα1,α2,…,αn≤IFNWOGBMp,qβ1,β2,…,βn.(4)
Letαi=μαi,υαii=1,2,…,n be a collection of IFNs, and(21)α−=min1≤i≤nμαi,max1≤i≤nυαi,α+=max1≤i≤nμαi,min1≤i≤nυαi,then(22)α−≤IFNWOGBMp,qα1,α2,…,αn≤α+.Proof.
(1)
(23)IFNWOGBMp,qα1,α2,…,αn=1p+q⊗i=1npαi⊕⊗j=1,j≠inqαjwj/1−wiwi=1p+q⊗i=1npα⊕⊗j=1,j≠inqαwj/1−wiwi=1p+q⊗i=1npα⊕qαwi=1p+q⊗i=1np+qαwi=α.(2)
(24)IFNWOGBMp,qα1,α2,…,αn=1p+q⊗i=1npαi⊕⊗j=1,j≠inqαjwj/1−wiwi=1p+q⊗i=1npαi′⊕⊗j=1,j≠inqαj′wj/1−wiwi=IFNWOGBMp,qα1′,α2′,…,αn′.(3)
According toμαi≤μβi and υαi≥υβii=1,2,…,n, we can obtain(25)∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wiwi≤∏i=1n1−1−μβip1−∏j=1j≠in1−1−μβjqwj/1−wiwi.
Thus,(26)1−1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wiwi1/p+q≤1−1−∏i=1n1−1−μβip1−∏j=1j≠in1−1−μβjqwj/1−wiwi1/p+q.
Similarly, we can obtain(27)1−∏i=1n1−υaip1−∏j=1j≠in1−υajqwj/1−wiwi1/p+q≥1−∏i=1n1−υβip1−∏j=1j≠in1−υβjqwj/1−wiwi1/p+q.
Then,(28)1−1−∏i=1n1−1−μβip1−∏j=1j≠in1−1−μβjqwj/1−wiwi1/p+q−1−∏i=1n1−υβip1−∏j=1j≠in1−υβjqwj/1−wiwi1/p+q≥1−1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wiwi1/p+q−1−∏i=1n1−υaip1−∏j=1j≠in1−υajqwj/1−wiwi1/p+q∗.
Let the score and accuracy degree values ofIFNWOGBMp,qα1,α2,…,αn and IFNWOGBMp,qβ1,β2,…,βn be sα, sβ and hα, hβ, respectively. Then, equation ∗ can be denoted as sβ≥sα.(1)
Ifsβ>sα, then, by Definition 2, we have IFNWOGBMp,qα1,α2,…,αn<IFNWOGBMp,qβ1,β2,…,βn.(2)
Ifsα=sβ, then, by μαi≤μβi and υαi≥υβii=1,2,…,n, we have(29)1−1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjqwj/1−wiwi1/p+q=1−1−∏i=1n1−1−μβip1−∏j=1j≠in1−1−μβjqwj/1−wiwi1/p+q1−∏i=1n1−υaip1−∏j=1j≠in1−υajqwj/1−wiwi1/p+q=1−∏i=1n1−υβip1−∏j=1j≠in1−υβjqwj/1−wiwi1/p+q.
Thus,hα=hβ and(30)IFNWOGBMp,qα1,α2,…,αn=IFNWOGBMp,qβ1,β2,…,βn.
Therefore, (1) and (2) indicate that(31)IFNWOGBMp,qα1,α2,…,αn≤IFNWOGBMp,qβ1,β2,…,βn.
(4) Based on (1) and (3), we have(32)IFNWOGBMp,qα1,α2,…,αn≤IFNWOGBMp,qα+,α+,…,α+=α+,α−=IFNWOGBMp,qα−,α−,…,α−≤IFNWOGBMp,qα1,α2,…,αn.
Then,α−≤IFNWOGBMp,qα1,α2,…,αn≤α+.
This completes the property.
Let us now further consider some specials of IFNWOGBM with respect to the parametersp and q.Case 1: ifq⟶0, then IFNWOGBM can be converted to GIFWGBM [16]:(33)IFNWOGBMp,qα1,α2,…,αn=1p⊗i=1npαiwi=1p⊗i=1n1−1−μαip,υaipwi=1p⊗i=1n1−1−μαipwi,1−1−υaipwi=1p∏i=1n1−1−μαipwi,1−∏i=1n1−υaipwi=1−1−∏i=1n1−1−μαipwi1/p,1−∏i=1n1−υaipwi1/p.Case 2: ifp=1 and q⟶0, then IFNWOGBM can be converted to IFWGM [16]:(34)IFNWOGBMp,qα1,α2,…,αn=⊗i=1Nαiwi=∏i=1nμαiwi1−∏i=1n1−υaiwi.Case 3: ifp=2 and q⟶0, then IFNWOGBM can be converted to IFWSGM [3]:(35)IFNWOGBMp,qα1,α2,…,αn=1−1−∏i=1n1−1−μαi2wi1/2,1−∏i=1n1−υai2wi1/2.Case 4: ifp=1 and q=1, then IFNWOGBM can be converted to intuitionistic fuzzy normalized weighted optimized square GBM (IFNWOSGBM):(36)IFNWOGBMp,qα1,α2,…,αn=1−1−∏i=1n1−1−μαi1−∏j=1j≠inμαjwj/1−wiwi1/2,.1−∏i=1n1−υai1−∏j=1j≠in1−υajwj/1−wiwi1/2


## 4. Generalized Intuitionistic Fuzzy Weighted OGBM

Definition 10.
Letp≥0, q≥0, r≥0, and αii=1,2,…,n be intuitionistic fuzzy numbers with the weight wii=1,2,…,n, wi∈0,1, and ∑i=1nwi=1. If(37)GIFNWOGBMp,q,rα1,α2,…,αn=1p+q+r⊗i=1Npαi⊕⊗j=1,j≠inqαj⊕⊗k=1,k≠j≠inrαkwk/1−wi−wjwj/1−wiwi,then GIFNWOGBMp,q is called the generalized intuitionistic fuzzy normalized weighted OGBM (GIFNWOGBM).
Ifwi=1/ni=1,2,…,n, then(38)GIFNWOGBMp,q,rα1,α2,…,αn=1p+q+r⊗i=1npαi⊕⊗j=1,j≠inqαj⊕⊗k=1,k≠j≠inrαk1/n−21/n−11/n,which we call the generalized intuitionistic fuzzy OGBM (GIFOGBM).Theorem 2.
Letp≥0, q≥0, r≥0, and αi=μαi,υαii=1,2,…,n are intuitionistic fuzzy numbers with the weight wii=1,2,…,n, wi∈0,1, and ∑i=1nwi=1; then,(39)GIFNWOGBMp,q,rα1,α2,…,αn=1−1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjq1−∏k=1k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wiwi1/p+q+r,1−∏i=1n1−υaip1−∏j=1j≠in1−υajq1−∏k=1k≠j≠in1−υαkrwk/1−wi−wjwj/1−wiwi1/p+q+r,and GIFNWOGBMp,q,rα1,α2,…,αn is also an IFN.Proof.
Sincepαi=1−1−μαip,υaip, qαj=1−1−μαjq,υajq, and rαk=1−1−μαkr,υakr, then(40)⊗k=1,k≠j≠inrαkwk/1−wi−wj=⊗k=1,k≠j≠in1−1−μαkrwk/1−wi−wj,1−1−υαkrwk/1−wi−wj=∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wj,1−∏k=1,k≠j≠in1−υαkrwk/1−wi−wjqαj⊕⊗k=1,k≠j≠inrαkwk/1−wi−wj=1−1−μαjq+∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wj−1−1−μαjq∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wj,υajq1−∏k=1,k≠j≠in1−υαkrwk/1−wi−wj=1−1−μαjq1−∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wj,υajq1−∏k=1,k≠j≠in1−υαkrwk/1−wi−wj⊗j=1,j≠inqαj⊕⊗k=1,k≠j≠inrαkwk/1−wi−wjwj/1−wi=⊗j=1,j≠in1−1−μαjq1−∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wi,1−1−υajq1−∏k=1,k≠j≠in1−υαkrwk/1−wi−wjwj/1−wi=∏j=1,j≠in1−1−μαjq1−∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wi,1−∏j=1,j≠in1−υajq1−∏k=1,k≠j≠in1−υαkrwk/1−wi−wjwj/1−wipαi⊕⊗j=1,j≠inqαj⊕⊗k=1,k≠j≠inrαkwk/1−wi−wjwj/1−wi=1−1−μαip1−∏j=1,j≠in1−1−μαjq1−∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wi,υaip1−∏j=1,j≠in1−υajq1−∏k=1,k≠j≠in1−υαkrwk/1−wi−wjwj/1−wi.
Therefore,(41)GIFNWOGBMp,qα1,α2,…,αn=1p+q+r⊗i=1npαi⊕⊗j=1,j≠inqαj⊕⊗k=1,k≠j≠inrαkwk/1−wi−wjwj/1−wiwi=1p+q+r⊗i=1n1−1−μαip1−∏j=1,j≠in1−1−μαjq1−∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wiwi,1−1−υaip1−∏j=1,j≠in1−υajq1−∏k=1,k≠j≠in1−υαkrwk/1−wi−wjwj/1−wiwi=1p+q+r∏i=1n1−1−μαip1−∏j=1,j≠in1−1−μαjq1−∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wiwi,1−∏i=1n1−υaip1−∏j=1,j≠in1−υajq1−∏k=1k≠j≠in1−υαkrwk/1−wi−wjwj/1−wiwi=1−1−∏i=1n1−1−μαip1−∏j=1,j≠in1−1−μαjq1−∏k=1,k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wiwi1/p+q+r,1−∏i=1n1−υaip1−∏j=1,j≠in1−υajq1−∏k=1,k≠j≠in1−υαkrwk/1−wi−wjwj/1−wiwi1/p+q+r.
Sinceμαi∈0,1, υαi∈0,1, and μαi+υαi∈0,1, then(42)0≤1−1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjq1−∏k=1k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wiwi1/p+q+r≤1,0≤1−∏i=1n1−υaip1−∏j=1j≠in1−υajq1−∏k=1k≠j≠in1−υαkrwk/1−wi−wjwj/1−wiwi1/p+q+r≤1.
In addition, sinceυαi≤1−μαi, υαj≤1−μαj, and υαk≤1−μαk, we obtain(43)1−∏i=1n1−υaip1−∏j=1j≠in1−υajq1−∏k=1k≠j≠in1−υαkrwk/1−wi−wjwj/1−wiwi1/p+q+r≤1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjq1−∏k=1k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wiwi1/p+q+r,and thus,(44)1−1−∏i=1n1−1−μαip1−∏j=1j≠in1−1−μαjq1−∏k=1k≠j≠in1−1−μαkrwk/1−wi−wjwj/1−wiwi1/p+q+r+1−∏i=1n1−υaip1−∏j=1j≠in1−υajq1−∏k=1k≠j≠in1−υαkrwk/1−wi−wjwj/1−wiwi1/p+q+r≤1.
This completes the proof.Property 2.
(1)
Letαii=1,2,…,n be a collection of IFNs; if α1=α2=⋯=αn=α, then(45)GIFNWOGBMp,qα1,α2,…,αn=α.(2)
Letαii=1,2,…,n be a collection of IFNs; if α1′,α2′,…,αn′ is any permutation of α1,α2,…,αn, then(46)GIFNWOGBMp,qα1,α2,…,αn=GIFNWOGBMp,qα1′,α2′,…,αn′.(3)
Letαi=μαi,υαi and βi=μβi,υβii=1,2,…,n be two collection of IFNs; if μαi≤μβi and υαi≥υβii=1,2,…,n, then(47)GIFNWOGBMp,qα1,α2,…,αn≤GIFNWOGBMp,qβ1,β2,…,βn.(4)
Letαi=μαi,υαii=1,2,…,n be a collection of IFNs, and(48)α−=min1≤i≤nμαi,max1≤i≤nυαi,α+=max1≤i≤nμαi,min1≤i≤nυαi,then, α−≤GIFNWOGBMp,qα1,α2,…,αn≤α+
The proof of Property2 is similar to Property 1 and is not displayed.
Let us now further consider some specials of GIFNWOGBM.Case 1: ifr⟶0, then GIFNWOGBM can be converted to intuitionistic fuzzy normalized weighted OGBM (IFNWOGBM):(49)GIFNWOGBMp,q,rα1,α2,…,αn=1−1−∏i=1n1−1−μαip1−∏j=1,j≠in1−1−μαiqwj/1−wiwi1/p+q,1−∏i=1n1−υa1p1−∏j=1,j≠in1−υa1pwj/1−wiwi1/p+q.Case 2: ifp=1, q⟶0, and r⟶0, then GIFNWOGBM can be converted to IFWGM [16]:(50)GIFNWOGBMp,q,rα1,α2,…,αn=⊗i=1Nαiwi=∏i=1nμαiwi,1−∏i=1n1−υaiwi.Case 3: ifp=2, q⟶0, and r⟶0, then GIFNWOGBM can be converted to IFWSGM [3]:(51)GIFNWOGBMp,q,rα1,α2,…,αn=1−1−∏i=1n1−1−μαi2wi1/2,1−∏i=1n1−υai2wi1/2.Case 4: ifp=1, q=1, and r⟶0, then GIFNWOGBM can be converted to generalized intuitionistic fuzzy normalized weighted optimized triple GBM (GIFNWOTGBM):(52)GIFNWOGBMp,q,rα1,α2,…,αn=1−1−∏i=1N1−1−μαi1−∏j=1,j≠iN1−1−μαj1−∏k=1,k≠j≠in1−1−μαkwk/1−wi−wjwj/1−wiwi1/3,1−∏i=1n1−υai1−∏j=1,j≠in1−υaj1−∏k=1,k≠j≠in1−υakwk/1−wi−wjwj/1−wiwi1/3.


## 5. A Method of Intuitionistic Fuzzy MADM

Based on the IFNWOGBM and GIFNWOGBM below, we present a method to aggregate multicriteria information under intuitionistic fuzzy environment.LetX=x1,x2,…,xn be a set of n alternatives and G=g1,g2,…,gl be a set of attributes with the weight vector wii=1,2,…,nT, wi∈0,1, and ∑i=1nwi=1. Suppose that the decision maker provides the intuitionistic fuzzy evaluated values under the attribute gj∈G for the alternative xi∈X, denoted by a IFN αij=μαij,υαij, and constructs the intuitionistic fuzzy decision matrix H=αijn×l.Step 1: transform matrixH=αijn×l into the positive matrix H¯=α¯ijn×l, where α¯ij=αij, for positive index (the bigger the number, the better the evaluation) gj; α¯ij=Nαij=υαij,μαij for negative index (the smaller the number, the better the evaluation) gj, i=1,2,…,n and j=1,2,…,l.Step 2: utilize the IFNWOGBM (or GIFNWOGBM) to aggregate theith line of H¯=α¯ijn×l:(53)αi=IFNWOGBMα¯i1,α¯i2,…,α¯il,i=1,2,…,n,orαi=GIFNWOGBMα¯i1,α¯i2,…,α¯il,i=1,2,…,n,and get the comprehensive evaluation valueαi of alternative xi.Step 3: rank all the alternativesxii=1,2,…,n according to αii=1,2,…,n in descending order by the binary relation described in Definition 2.Example 1.
(see [17]). In human history, library is the product of the development of human civilization to a certain stage. The development of culture, science, and technology has led to the emergence of books, the record carrier of knowledge and information, and the increase of books has produced an early library whose main function is to preserve books. It was the preservation function of the early library that preserved the excellent cultural and scientific achievements of mankind. In modern times, the function of the library has changed from book collection to borrowing, and the range of readers has expanded from senior intellectuals to ordinary people. Library plays an irreplaceable role in the continuous progress of mankind and the sustainable development of society. Libraries have gradually become the basis of sustainable social development. At present, more and more local governments are establishing public libraries.
Suppose a city wants to build a large, comprehensive, and modern public library. The city administrators need to decide what brand of air conditioning products used in library. Three aspects of alternative (whose weighting vector isw=0.3,0.5,0.2T) are evaluated by experts, which are as follows:g1: economicg2: functionalg3: operational
The five feasible alternativesxii=1,2,…,5 are to be evaluated using IFNs under the above three criteria, and construct the following matrix, see Table 1:Step 1: in this example, all criteriagii=1,2,3 are benefit-type criteria and do not need normalizationStep 2: utilize the IFNWOGBM (or GIFNWOGBM) to obtain the comprehensive evaluation valueαi of alternative xi (see Tables 2 and 3).Step 3: rank all the alternativesxii=1,2,…,5 according to αii=1,2,…,5 in the descending order by the binary relation described in Definition 2 (see Table 4)
From Table4, we can see that the decision-making results derived by the IFNWOGBM or GIFNWOGBM depend on the choice of parameters p,q,r. Moreover, the optimal alternative given by the IFNWOGBM is different from that of the GIFNWOGBM. This is principally because the IFNWOGBM captures the interrelationship between any two aggregated arguments, but the GIFNWOGBM captures the interrelationship of any three aggregated arguments. Therefore, the GIFNWOGBM is more focused on the overall performance of aggregated arguments. As can be seen from Tables 2 and 3, the intuitionistic fuzzy comprehensive evaluation values derived by the IFNWOGBM or GIFNWOGBM are influenced by the input parameters p, q, and r. With larger or smaller parameters, it may lead to the degradation of recognition ability in decision-making process. For this reason, we recommend taking p=q=r=1, which not only makes the calculations easier but also interrelationship of the aggregated arguments can be fully take into account.Table 1
Decision matrix.
g1g2g3x1α11=0.3,0.4α12=0.7,0.2α13=0.5,0.3x2α21=0.5,0.2α22=0.4,0.1α23=0.7,0.1x3α31=0.4,0.5α32=0.7,0.2α33=0.4,0.4x4α41=0.2,0.6α42=0.8,0.1α43=0.8,0.2x5α51=0.9,0.1α52=0.6,0.3α53=0.2,0.5Table 2
The comprehensive evaluation value by IFNWOGBM.
p, qx1x2x3x4x5p=10, q=10α1=0.459,0.317α2=0.470,0.000α3=0.452,0.408α4=0.608,0.316α5=0.465,0.366p=6, q=6α1=0.485,0.304α2=0.480,0.137α3=0.481,0.386α4=0.612,0.299α5=0.483,0.353p=2, q=2α1=0.522,0.288α2=0.501,0.132α3=0.527,0.344α4=0.625,0.259α5=0.547,0.305p=1, q=1α1=0.530,0.284α2=0.507,0.130α3=0.536,0.335α4=0.633,0.256α5=0.588,0.279p=1, q=0.001α1=0.508,0.286α2=0.478,0.131α3=0.529,0.344α4=0.529,0.310α5=0.544,0.294Hs1, q=1α1=0.463,0.309α2=0.510,0.139α3=0.484,0.382α4=0.476,0.358α5=0.511,0.301Table 3
The comprehensive evaluation value by GIFNWOGBM.
p, q, rx1x2x3x4x5p=10, q=10, r=10α1=0.460,0.560α2=0.393,0.503α3=0.482,0.573α4=0.508,0.697α5=0.683,0.570p=1, q=5, r=10α1=0.318,0.560α2=0.368,0.503α3=0.356,0.573α4=0.297,0.697α5=0.486,0.570p=1, q=1, r=1α1=0.476,0.413α2=0.454,0.355α3=0.489,0.427α4=0.556,0.434α5=0.683,0.443p=1, q=0.5, r=1α1=0.483,0.386α2=0.454,0.308α3=0.490,0.412α4=0.584,0.407α5=0.662,0.409p=0.5, q=1, r=1α1=0.440,0.445α2=0.454,0.429α3=0.462,0.446α4=0.486,0.449α5=0.669,0.481p=1, q=1, r=0.5α1=0.493,0.397α2=0.457,0.314α3=0.504,0.415α4=0.586,0.401α5=0.674,0.419Table 4
Decision-making results.
IFNWOGBM operatorGIFNWOGBM operatorp, qResultpq, rResultp=10, q=10x2≻x4≻x1≻x5≻x3p=10q=10, r=10x5≻x3≻x1≻x2≻x4p=6, q=6x2≻x4≻x1≻x5≻x3p=1, q=1, r=1x5≻x2≻x3≻x1≻x4p=2, q=2x2≻x4≻x5≻x1≻x3p=1, q=0.001, r=0.001x5≻x4≻x2≻x1≻x3p=1, q=1x2≻x4≻x5≻x1≻x3p=0.001, q=1, r=1x5≻x4≻x2≻x1≻x3p=1, q=0.001x2≻x5≻x1≻x4≻x3p=1, q=0.001, r=1x5≻x4≻x2≻x1≻x3Hs1, q=1x2≻x5≻x1≻x4≻x3p=0.5, q=0.5, r=0.5x5≻x4≻x2≻x1≻x3


## 6. Comparative Analysis with Other Methods

To verify the effectiveness of the proposed method, it is compared with the classical MADM method based on the aggregation operator. Comparative analysis results are shown in Table5.Table 5
Comparison with other methods.
[17][8][12]Proposed methodInformationIntuitionistic fuzzy numberIntuitionistic fuzzy numberIntuitionistic fuzzy numberIntuitionistic fuzzy numberAggregationIFWBMIFWGBMIFOWGBMIFNWOGBM/GIFNWOGBMConditionsIntuitionistic fuzzy number, attribute weights are knownIntuitionistic fuzzy number, attribute weights are knownIntuitionistic fuzzy number, attribute weights are knownIntuitionistic fuzzy number, attribute weights are knownResultRanking of alternativesRanking of alternativesRanking of alternativesRanking of alternativesExample1p=q=1x2≻x5≻x1≻x4≻x3x2≻x4≻x5≻x1≻x3x2≻x4≻x5≻x1≻x3x2≻x4≻x1≻x5≻x3As can be seen from Table5, the primary difference in the above four methods is in the aggregation operators. Xu and Yager [17] use IFWBM based on average mean, but the operators of other three kinds of methods based on geometric mean. The IFWBM and IFWGBM do not mine the interrelationship between attributes in the process of information aggregation, while the IFOWGBM and IFNWOGBM (GIFNWOGBM) take this into account. Moreover, unlike the other three kinds of operators, the IFNWOGBM and GIFNWOGBM have the reducibility. However, any kind of operators’ flaws and certain restrictions would exist in the aggregation of decision information. Therefore, the decision makers should choose the proper operators according to practical circumstances for further improving the effect of decision-making.


## 7. Conclusions

Based on the normalized weighted BM and geometric mean, this paper introduces the OGBM, the NWOGBM, and the GNWOGBM. The three kinds of operators designed in this paper have strong operability and can effectively capture the correlation between decision evaluation indexes.To deal with the MADM problem that the criteria evaluation information is intuitionistic fuzzy numbers, an approach has been proposed on the basis of the IFNWOGBM and GIFNWOGBM. Finally, the practicability and validity of the approach are verified with a case of library location.Hopefully, we will use the proposed operators to solve the problems of criminal identification, optimal financial scheme selection, optimal driving path selection, and so on and extend the operators proposed in this paper to other fuzzy environments, such as hesitant fuzzy language environment.



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