# Approach for Multiattribute Decision-Making with an Interval Grey Number Based on Bonferroni Mean

Authors: Qiuyun Niu; Jinhai Guo
Journal: Journal of Mathematics
 (2022)

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## Abstract

Aiming at the multiattribute decision-making problem in the form of interval grey number, considering the interrelationship between the scheme attributes, this study defines the grey number weighted geometric Bonferroni meanGNWGBM operator for aggregating information. The feasibility of the proposed method is verified through a comparison of the example analysis and methods.


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## Body

## 1. Introduction

Multiattribute decision-making problems often occur in daily life; thus, they have attracted much attention with rapid development, and their methods are very important. Information forms are also diverse. Decision information with “poor information and few data” of grey uncertain characteristics [1] can often be expressed by grey number. A grey number has a certain range of values but cannot determine its specific value, and it can be operated to generate meaningful information.Using the grey number as the basic element, it has become increasingly extensive to establish the model under the grey information. The grey model has many applications in forecasting [2, 3] and decision-making [4] and has achieved good results. Scholars have attempted to analyze the grey number problems from multiple perspectives when solving the decision-making problems. There has been some discussion on the basic definition of kernel and grey degree. Dang et al. [5] established the fixed weight clustering model, and Liu et al. [6] proposed a decision method for grey correlation comprehensive closeness and similarity. An integration operator is used to process the information in the decision problem, which is beneficial for obtaining a better decision choice. For the information in the scheme, in [7], which reduced the influence of interrelationships of attributes based on the Choquet integral, the dynamic order weighted average operator was used for aggregation to reduce the influence of subjective factors [8]. In [9], the intuitionistic grey number was defined with its operator, and the grey decision model on the general grey number was studied. Although the integration operator is used in these decision models to try to reduce the influence of the relationship between attributes, the problem of correlation between attributes is not solved. The Bonferroni mean operator considers the interrelationship between attributes.So far, few studies have used the Bonferroni mean operator in grey decision information [10], but there have been many studies in uncertain linguistic and fuzzy information environments. Bonferroni [11] first proposed the Bonferroni mean (BM) operator, which can be used for the integration between the operators. Subsequently, the aggregation property [12–14] of the Bonferroni mean operator was explored and applied to uncertain information [15]. Dutta and Guha [16] made a decision regarding the scheme by aggregating the partition Bonferroni mean operator with partition attributes. In recent years, the development trend of the Bonferroni mean operator has been good in the hesitant and fuzzy environments. To overcome the limitation that operators consider only the real number and cannot handle other types of data, more systematic research results have been achieved on the theory and properties of Bonferroni mean operators, such as fuzzy Bonferroni mean operators [17, 18], fuzzy weighted average Bonferroni mean operators, weighted geometric Bonferroni mean operators, and generalized intuition fuzzy Bonferroni mean operators [19]. Meanwhile, that has proposed the generalized hesitation fuzzy Bonferroni mean operator [20], the interval hesitation fuzzy Bonferroni mean operator [21], and the interval hesitation fuzzy geometric Bonferroni mean operator [22] in a hesitation fuzzy environment. It is also feasible to use the Bonferroni mean operator to solve multiattribute decision-making problems [23–25].Under the excitation of the ideal Bonferroni mean and geometric Bonferroni mean, the study extends several Bonferroni mean operators under grey number information. Considering the possibility of interrelationship between the attributes, the operators are used to aggregate information on the alternatives. The scheme is selected according to the evaluation value.


## 2. Basic Concepts and Theories

In this section, we recall the concepts and definitions of interval grey numbers, and we reviewed the concepts ofBM, which are the foundations of the correlation operator methods and the decision-making methods we proposed.


### 2.1. The Concepts of Grey Number

Definition 1 (see [1]).
Letg±∈R be an unknown real number within a union set of closed or open grey intervals, where g±∈∪i=1nai¯,ai¯=∪i=1n⊗i; if i=1,2,⋯,n, n is an integer and 0<n<∞, ai¯,ai¯∈R, and ai−1¯≤ai¯≤ai¯≤ai+1¯, for any grey interval ⊗i∈ai¯,ai¯∈∪i=1nai¯,ai¯, then g± is called as a general grey number. g−=infai¯∈g±ai¯ and g+=supai¯∈g±ai¯ are called the lower and upper limits of g±, respectively, where the interval grey number and real number are special cases of general grey number.Definition 2 (see [1]).
A grey number which has both upper boundsa¯ and lower bounds a¯ is referred to as an interval grey number. For any interval number ⊗∈a¯,a¯ , it is obvious that a¯≤a¯. Assuming that the emergence background or universe of grey number ⊗ is Ω, u⊗ is the measurement of the number field of grey number ⊗. Then, we have(1)g∘⊗=u⊗uΩ.
The degree of greyness of the grey number⊗ satisfies properties given as follows:(1)
0≤g∘⊗≤1(2)
⊗∈a¯,a¯,a¯≤a¯,whena¯=a¯,g∘⊗=0.(3)
g∘⊗ is directly proportional to u⊗ and inversely proportional to uΩ.Definition 3 (see [1]).
Assume that⊗1∈a,b,a<b;⊗2∈c,d,c<d, then(2)⊗1∪⊗2=ξ|ξ∈a,borξ∈c,d.
The union of the grey numbers⊗1 and ⊗2, and their intersection, is(3)⊗1∩⊗2=ξ|ξ∈a,bandξ∈c,d.


### 2.2. The Bonferroni Mean (BM) Operator

Definition 4 (see [11]).
Ifp,q≥0 and cii=1,2,⋯,n is a collection of a non-negative real numbers. Then the aggregation functions(4)BMp,qc1,c2,⋯,cn=1nn−1∑i,j=1i≠jncipcjq1/p+q,is called the Bonferroni mean (BM) operator.


## 2.1. The Concepts of Grey Number

Definition 1 (see [1]).
Letg±∈R be an unknown real number within a union set of closed or open grey intervals, where g±∈∪i=1nai¯,ai¯=∪i=1n⊗i; if i=1,2,⋯,n, n is an integer and 0<n<∞, ai¯,ai¯∈R, and ai−1¯≤ai¯≤ai¯≤ai+1¯, for any grey interval ⊗i∈ai¯,ai¯∈∪i=1nai¯,ai¯, then g± is called as a general grey number. g−=infai¯∈g±ai¯ and g+=supai¯∈g±ai¯ are called the lower and upper limits of g±, respectively, where the interval grey number and real number are special cases of general grey number.Definition 2 (see [1]).
A grey number which has both upper boundsa¯ and lower bounds a¯ is referred to as an interval grey number. For any interval number ⊗∈a¯,a¯ , it is obvious that a¯≤a¯. Assuming that the emergence background or universe of grey number ⊗ is Ω, u⊗ is the measurement of the number field of grey number ⊗. Then, we have(1)g∘⊗=u⊗uΩ.
The degree of greyness of the grey number⊗ satisfies properties given as follows:(1)
0≤g∘⊗≤1(2)
⊗∈a¯,a¯,a¯≤a¯,whena¯=a¯,g∘⊗=0.(3)
g∘⊗ is directly proportional to u⊗ and inversely proportional to uΩ.Definition 3 (see [1]).
Assume that⊗1∈a,b,a<b;⊗2∈c,d,c<d, then(2)⊗1∪⊗2=ξ|ξ∈a,borξ∈c,d.
The union of the grey numbers⊗1 and ⊗2, and their intersection, is(3)⊗1∩⊗2=ξ|ξ∈a,bandξ∈c,d.


## 2.2. The Bonferroni Mean (BM) Operator

Definition 4 (see [11]).
Ifp,q≥0 and cii=1,2,⋯,n is a collection of a non-negative real numbers. Then the aggregation functions(4)BMp,qc1,c2,⋯,cn=1nn−1∑i,j=1i≠jncipcjq1/p+q,is called the Bonferroni mean (BM) operator.


## 3. The Grey Number Bonferroni Mean Operator

### 3.1. The Concept of Grey Number Bonferroni Mean Operator

The BM operator usually operates in a real number environment and that under the interval grey number is defined as follows.Definition 5.
Assume thatαi⊗i=1,2,…,n, αi⊗=ai,bi is a set of interval grey numbers, for any p,q≥0, if(5)GNBMp,qα1⊗,α2⊗,⋯,αn⊗=1nn−1⊕i,j=1i≠jnαip⊗αjq1/p+q=1nn−1⊕i,j=1i≠jnaip⊗ajq1/p+q1nn−1⊕i,j=1i≠jnbip⊗bjq1/p+q,that the GNBMp,q is called the grey number Bonferroni mean operator.
TheGNBM operator has excellent properties such as idempotency, boundedness, and commutativity.Theorem 1 (Idempotency).
Letαj⊗=aj,bjj=1,2,⋯,n be a set of interval grey numbers. If all αj⊗=aj,bj are equal, αj⊗αj⊗=ai,bi=α⊗α⊗=a,b for all j, then(6)GNBMp,qα1⊗,α2⊗,⋯,αn⊗=α⊗.Theorem 2 (Boundedness).
Letαj⊗=aj,bjj=1,2,⋯,n be a set of interval grey numbers, and let(7)α−⊗=minjαj⊗=minjaj,minjbj,α+⊗=maxjαj⊗=maxjaj,maxjbj.
Then(8)α−⊗≤GNBMp,qα1⊗,α2⊗,⋯,αn⊗≤α+⊗.Theorem 3 (Commutativity).
Letαj⊗=aj,bj and αj′⊗=aj′,bj′j=1,2,⋯,n be two interval grey numbers, where αj′⊗=aj′,bj′j=1,2,⋯,n is any permutation of αj⊗=, aj,bjj=1,2,⋯,n, then(9)GNBMp,qα1⊗,α2⊗,⋯,αn⊗=GNBMp,qα1′⊗,α2′⊗,⋯,αn′⊗.
Now, we discuss some special cases of theGNBM with respect to the parameters p and q:Case 1.
Ifq⟶0, then from the GNBM, we get(10)limq⟶0GNBMp,qα1⊗,α2⊗,⋯,αn⊗=limq⟶01nn−1⊕i,j=1i≠jnαip⊗αjq1/p+q=limq⟶01nn−1⊕i,j=1i≠jnaip⊗ajq1/p+q,1nn−1⊕i,j=1i≠jnbip⊗bjq1/p+qq⟶0=1n⊕i,j=1i≠jnaip1/p,1n⊕i,j=1i≠jnbip1/p=1n⊕i,j=1i≠jnαip⊗1/p=GNBMp,0α1⊗,α2⊗,⋯,αn⊗,which we call as the interval grey number generalized mean operator.Case 2.
Ifp=2 and q⟶0, then by the GNBM, we have(11)GNBM2,0α1⊗,α2⊗,⋯,αn⊗=1n⊕i=1nαi2⊗1/2,which we call as the interval grey numbers square mean operator.Case 3.
Ifp=1 and q⟶0, then the GNBM reduces to interval grey numbers, which means the GNM operator is(12)GNBM1,0α1⊗,α2⊗,⋯,αn⊗=1n⊕i=1nαi⊗,which we call as the interval grey number mean GNM operator.Case 4.
Ifp=1 and q=1, then by the GNBM, we have(13)GNBM1,1α1⊗,α2⊗,⋯,αn⊗=1nn−1⊕i,j=1i≠jnαi⊗αj1/2,which we call as the interval grey number square mean GNSM operator.


### 3.2. The Concepts of GNWBM and GNGBM Operators

Considering that the input arguments may have different importance, we define the interval grey numbers weighted Bonferroni meanGNWBM operator.Definition 6.
Letαj⊗=aj,bjj=1,2,…,n be a set of interval grey numbers and p,q≥0, w=w1,w2,⋯,wnt be the weight vector of αj⊗=aj,bjj=1,2,⋯,n, where wj indicates the degree of importance of αj⊗, satisfying wj>0j=1,2,⋯,n and ∑j=1nwj=1. If(14)GNWBMwp,qα1⊗,α2⊗,⋯,αn⊗=1nn−1⊕i,j=1i≠jnwiαip⊗wjαjq1/p+q=1nn−1⊕i,j=1i≠jnwiaip⊗wjajq1/p+q,1nn−1⊕i,j=1i≠jnwibip⊗wjbjq1/p+q,then GNWBMwp,q is called as the interval grey numbers weighted Bonferroni mean GNWBM operator.Definition 7.
Letp,q≥0 and ci, i=1,2,⋯,n be collections of non-negative real numbers. Subsequently, the aggregation functions(15)GBMp,qc1,c2,⋯,cn=1p+q∏i,j=1i≠jnpci+qcj1/nn−1,is called as the geometric Bonferroni mean GBM operator.
However, the geometric Bonferroni mean (BM) operator has usually been used in situations where the input arguments are non-negative real numbers. We shall extend the GBM operators to accommodate the situations where the input arguments are interval grey numbers information. In this section, we shall investigate the GBM operator under interval grey numbers environments. Based on Definition7, we define the interval grey numbers geometric Bonferroni mean GNGBM operator as follows:Definition 8.
Letαj⊗=aj,bjj=1,2,⋯,n be a set of interval grey numbers and let p,q≥0. If(16)GNGBMp,qα1⊗,α2⊗,⋯,αn⊗=1p+q⊗i,j=1i≠jnpαi⊕qαj1/nn−1=1p+q⊗i,j=1i≠jnpai⊕qaj1/nn−1,1p+q⊗i,j=1i≠jnpbi⊕qbj1/nn−1,then GNGBMp,q is called the grey numbers geometric Bonferroni mean GNGBM operator.
Similar to theGNBM operator, the GNGBM operator also has properties such as idempotent, boundedness, and commutativity; therefore, it will not be detailed here.
Now, we discuss some special cases of theGNGBM with respect to the parameters p and q:Case 1. Ifq⟶0, then from the GNGBM, we get(17)limq⟶0GNGBMp,qα1⊗,α2⊗,⋯,αn⊗=limq⟶01p+q⊗i,j=1i≠jnpαi⊕qαj1/nn−1=limq⟶01p+q⊗i,j=1i≠jnpai⊕qaj1/nn−1,1p+q⊗i,j=1i≠jnpbi⊕qbj1/nn−1=1p⊗i,j=1i≠jnpai1/n,1p⊗i,j=1i≠jnpbi1/n=1p⊗i,j=1i≠jnpαi1/n=GNGBMp,0α1⊗,α2⊗,⋯,αn⊗,which we call as the interval grey numbers generalized geometric mean operator.Case 2. Ifp=2 and q⟶0, then by the GNGBM, we have(18)GNGBM2,0α1⊗,α2⊗,⋯,αn⊗=12⊗i=1n2αi⊗1/n,which we call as the interval grey numbers square geometric meanGNSGM operator.Case 3. Ifp=1 and q⟶0, then the GNGBM reduces to interval grey numbers, which means the GNGM operator is(19)GNGBM1,0α1⊗,α2⊗,⋯,αn⊗=⊗i=1nαi⊗1/n,which we call as the interval grey number geometric meanGNGM operator.Case 4. Ifp=1 and q=1, then by the GNGBM, we have(20)GNWGBM1,1α1⊗,α2⊗,⋯,αn⊗=12⊗i,j=1i≠jnαi⊕αj1/nn−1,which we call as the interval grey numbers square geometric mean GNSGM operator.


### 3.3. The Concept of GNWGBM Operator

Considering that the input arguments may have different importance, we define the interval grey numbers weighted geometric Bonferroni meanGNWGBM operator.Definition 9.
Letαj⊗=aj,bjj=1,2,…,n be a set of interval grey numbers and p,q≥0, w=w1,w2,⋯,…wnT be the weight vector of αj⊗=aj,bjj=1,2,⋯,n, where wj indicates the importance degree of αj⊗, satisfying wj>0j=1,2,⋯,n and ∑j=1nwj=1. If(21)GNWGBMwp,qα1⊗,α2⊗,⋯,αn⊗=1p+q⊗i,j=1i≠jnpαiwi⊕qαjwj1/nn−1=1p+q⊗i,j=1i≠jnpaiwi⊕qajwj1/nn−1,1p+q⊗i,j=1i≠jnpbiwi⊕qbjwj1/nn−1,then GNWGBMwp,q is called the interval grey numbers weighted geometric Bonferroni mean GNWGBM operator.


## 3.1. The Concept of Grey Number Bonferroni Mean Operator

The BM operator usually operates in a real number environment and that under the interval grey number is defined as follows.Definition 5.
Assume thatαi⊗i=1,2,…,n, αi⊗=ai,bi is a set of interval grey numbers, for any p,q≥0, if(5)GNBMp,qα1⊗,α2⊗,⋯,αn⊗=1nn−1⊕i,j=1i≠jnαip⊗αjq1/p+q=1nn−1⊕i,j=1i≠jnaip⊗ajq1/p+q1nn−1⊕i,j=1i≠jnbip⊗bjq1/p+q,that the GNBMp,q is called the grey number Bonferroni mean operator.
TheGNBM operator has excellent properties such as idempotency, boundedness, and commutativity.Theorem 1 (Idempotency).
Letαj⊗=aj,bjj=1,2,⋯,n be a set of interval grey numbers. If all αj⊗=aj,bj are equal, αj⊗αj⊗=ai,bi=α⊗α⊗=a,b for all j, then(6)GNBMp,qα1⊗,α2⊗,⋯,αn⊗=α⊗.Theorem 2 (Boundedness).
Letαj⊗=aj,bjj=1,2,⋯,n be a set of interval grey numbers, and let(7)α−⊗=minjαj⊗=minjaj,minjbj,α+⊗=maxjαj⊗=maxjaj,maxjbj.
Then(8)α−⊗≤GNBMp,qα1⊗,α2⊗,⋯,αn⊗≤α+⊗.Theorem 3 (Commutativity).
Letαj⊗=aj,bj and αj′⊗=aj′,bj′j=1,2,⋯,n be two interval grey numbers, where αj′⊗=aj′,bj′j=1,2,⋯,n is any permutation of αj⊗=, aj,bjj=1,2,⋯,n, then(9)GNBMp,qα1⊗,α2⊗,⋯,αn⊗=GNBMp,qα1′⊗,α2′⊗,⋯,αn′⊗.
Now, we discuss some special cases of theGNBM with respect to the parameters p and q:Case 1.
Ifq⟶0, then from the GNBM, we get(10)limq⟶0GNBMp,qα1⊗,α2⊗,⋯,αn⊗=limq⟶01nn−1⊕i,j=1i≠jnαip⊗αjq1/p+q=limq⟶01nn−1⊕i,j=1i≠jnaip⊗ajq1/p+q,1nn−1⊕i,j=1i≠jnbip⊗bjq1/p+qq⟶0=1n⊕i,j=1i≠jnaip1/p,1n⊕i,j=1i≠jnbip1/p=1n⊕i,j=1i≠jnαip⊗1/p=GNBMp,0α1⊗,α2⊗,⋯,αn⊗,which we call as the interval grey number generalized mean operator.Case 2.
Ifp=2 and q⟶0, then by the GNBM, we have(11)GNBM2,0α1⊗,α2⊗,⋯,αn⊗=1n⊕i=1nαi2⊗1/2,which we call as the interval grey numbers square mean operator.Case 3.
Ifp=1 and q⟶0, then the GNBM reduces to interval grey numbers, which means the GNM operator is(12)GNBM1,0α1⊗,α2⊗,⋯,αn⊗=1n⊕i=1nαi⊗,which we call as the interval grey number mean GNM operator.Case 4.
Ifp=1 and q=1, then by the GNBM, we have(13)GNBM1,1α1⊗,α2⊗,⋯,αn⊗=1nn−1⊕i,j=1i≠jnαi⊗αj1/2,which we call as the interval grey number square mean GNSM operator.


## 3.2. The Concepts of GNWBM and GNGBM Operators

Considering that the input arguments may have different importance, we define the interval grey numbers weighted Bonferroni meanGNWBM operator.Definition 6.
Letαj⊗=aj,bjj=1,2,…,n be a set of interval grey numbers and p,q≥0, w=w1,w2,⋯,wnt be the weight vector of αj⊗=aj,bjj=1,2,⋯,n, where wj indicates the degree of importance of αj⊗, satisfying wj>0j=1,2,⋯,n and ∑j=1nwj=1. If(14)GNWBMwp,qα1⊗,α2⊗,⋯,αn⊗=1nn−1⊕i,j=1i≠jnwiαip⊗wjαjq1/p+q=1nn−1⊕i,j=1i≠jnwiaip⊗wjajq1/p+q,1nn−1⊕i,j=1i≠jnwibip⊗wjbjq1/p+q,then GNWBMwp,q is called as the interval grey numbers weighted Bonferroni mean GNWBM operator.Definition 7.
Letp,q≥0 and ci, i=1,2,⋯,n be collections of non-negative real numbers. Subsequently, the aggregation functions(15)GBMp,qc1,c2,⋯,cn=1p+q∏i,j=1i≠jnpci+qcj1/nn−1,is called as the geometric Bonferroni mean GBM operator.
However, the geometric Bonferroni mean (BM) operator has usually been used in situations where the input arguments are non-negative real numbers. We shall extend the GBM operators to accommodate the situations where the input arguments are interval grey numbers information. In this section, we shall investigate the GBM operator under interval grey numbers environments. Based on Definition7, we define the interval grey numbers geometric Bonferroni mean GNGBM operator as follows:Definition 8.
Letαj⊗=aj,bjj=1,2,⋯,n be a set of interval grey numbers and let p,q≥0. If(16)GNGBMp,qα1⊗,α2⊗,⋯,αn⊗=1p+q⊗i,j=1i≠jnpαi⊕qαj1/nn−1=1p+q⊗i,j=1i≠jnpai⊕qaj1/nn−1,1p+q⊗i,j=1i≠jnpbi⊕qbj1/nn−1,then GNGBMp,q is called the grey numbers geometric Bonferroni mean GNGBM operator.
Similar to theGNBM operator, the GNGBM operator also has properties such as idempotent, boundedness, and commutativity; therefore, it will not be detailed here.
Now, we discuss some special cases of theGNGBM with respect to the parameters p and q:Case 1. Ifq⟶0, then from the GNGBM, we get(17)limq⟶0GNGBMp,qα1⊗,α2⊗,⋯,αn⊗=limq⟶01p+q⊗i,j=1i≠jnpαi⊕qαj1/nn−1=limq⟶01p+q⊗i,j=1i≠jnpai⊕qaj1/nn−1,1p+q⊗i,j=1i≠jnpbi⊕qbj1/nn−1=1p⊗i,j=1i≠jnpai1/n,1p⊗i,j=1i≠jnpbi1/n=1p⊗i,j=1i≠jnpαi1/n=GNGBMp,0α1⊗,α2⊗,⋯,αn⊗,which we call as the interval grey numbers generalized geometric mean operator.Case 2. Ifp=2 and q⟶0, then by the GNGBM, we have(18)GNGBM2,0α1⊗,α2⊗,⋯,αn⊗=12⊗i=1n2αi⊗1/n,which we call as the interval grey numbers square geometric meanGNSGM operator.Case 3. Ifp=1 and q⟶0, then the GNGBM reduces to interval grey numbers, which means the GNGM operator is(19)GNGBM1,0α1⊗,α2⊗,⋯,αn⊗=⊗i=1nαi⊗1/n,which we call as the interval grey number geometric meanGNGM operator.Case 4. Ifp=1 and q=1, then by the GNGBM, we have(20)GNWGBM1,1α1⊗,α2⊗,⋯,αn⊗=12⊗i,j=1i≠jnαi⊕αj1/nn−1,which we call as the interval grey numbers square geometric mean GNSGM operator.


## 3.3. The Concept of GNWGBM Operator

Considering that the input arguments may have different importance, we define the interval grey numbers weighted geometric Bonferroni meanGNWGBM operator.Definition 9.
Letαj⊗=aj,bjj=1,2,…,n be a set of interval grey numbers and p,q≥0, w=w1,w2,⋯,…wnT be the weight vector of αj⊗=aj,bjj=1,2,⋯,n, where wj indicates the importance degree of αj⊗, satisfying wj>0j=1,2,⋯,n and ∑j=1nwj=1. If(21)GNWGBMwp,qα1⊗,α2⊗,⋯,αn⊗=1p+q⊗i,j=1i≠jnpαiwi⊕qαjwj1/nn−1=1p+q⊗i,j=1i≠jnpaiwi⊕qajwj1/nn−1,1p+q⊗i,j=1i≠jnpbiwi⊕qbjwj1/nn−1,then GNWGBMwp,q is called the interval grey numbers weighted geometric Bonferroni mean GNWGBM operator.


## 4. Approach for Multiattribute Decision-Making with an Interval Grey Number Based on Bonferroni Mean

A multiattribute evaluation system usually has different dimensions and orders of magnitude owing to different properties of each evaluation index. Dimensionless data processing mainly deals with the comparability of the data. Therefore, it is necessary to standardize the raw index data to ensure the reliability of the results and eliminate its dimensionality so that a comprehensive evaluation and analysis can be carried out.Definition 10 (see [26]).
LetA⊗=⊗aijm×n be the matrix of interval grey number matrix, where ⊗aij∈aij¯,aij¯ is transformed into a dimensionless normalized matrix X⊗=⊗xijm×n, in which the standardized method of ⊗xij∈xij¯,xij¯0≤xij¯≤xij¯≤1 is given as follows.
For benefit attributes,(22)xij¯=aij¯−miniaij¯maxiaij¯−miniaij¯xij¯=aij¯−miniaij¯maxiaij¯−miniaij¯,i=1,2,⋯,m;j=1,2,⋯,n.
For cost attributes,(23)xij¯=maxiaij¯−aij¯maxiaij¯−miniaij¯xij¯=maxiaij¯−aij¯maxiaij¯−miniaij¯,i=1,2,⋯,m;j=1,2,⋯,n.Definition 11 (see [15]).
Letα1⊗=a1,b1 and α2⊗=a2,b2 be two interval grey numbers, and let lenα1⊗=b1−a1,lenα2⊗=b2−a2, then the degree of possibility of α1⊗≥α2⊗ is defined as(24)pα1⊗≥α2⊗=max0,lenα1+lenα2−maxb2−a1,0lenα1+lenα2.
From the Definition10, we can easily get the results as follows:(1)
0≤pα1⊗≥α2⊗≤1,(2)
pα1⊗≥α2⊗+pα2⊗≥α1⊗=1.where especially pα1⊗≥α1⊗=pα2⊗≥α2⊗=0.5.
In this section, we utilize the interval grey numbers weighted Bonferroni meanGNWBM operator (interval grey numbers weighted geometric Bonferroni mean GNWGBM operator) to develop an approach for multiple attribute decision-making problems with interval grey numbers information, which can be described as follows:Step 1.
Utilize the decision information given in matrixR, and the GNWBM operator (in general, we can take p=1,q=1)Step 2.
Normalizing the interval grey numbers
In general, owing to the different attributes and representative meanings of information data, we normalize the attribute information to maintain unified dimensions and avoid evaluation criteria. We performed different treatments for the different types of attribute information.Step 3.
Integrating the normalization information matrix, we utilize the normalization matrix within theGNWBM and the GNWGBM operators as(25)ri=ai,bi=GNWBMwp,qri1,ri2,⋯,rin=1nn−1⊕k,l=1k≠lnwkrikp⊗wlrilq1/p+qi=1,2,⋯,m,ri=ai,bi=GNWGBMwp,qri1,ri2,⋯,rin=1p+q⊗k,l=1k≠lnprikwk⊕qrilwl1/nn−1,to derive the overall interval grey numbers preference values rii=1,2,⋯,m of the alternative xi.Step 4.
For simplicity, we letpij=pri≥rj and then we develop a complementary matrix as P=pijm×m, where pij≥0,pij+pji=1,pii=0.5,i,j=1,2,⋯m. Then, we have(26)pi=∑j=1mpij,i=1,2,⋯,m.
We then rank the collective overall preference valuesrii=1,2,⋯,m in the descending order in accordance with the values of pii=1,2,⋯,m.Step 5.
Get the best scheme according to the sort.


## 5. Application Example

Example 1.
(Housing purchase problem) There are four types of residential options, denoted byU=x1,x2,x3,x4. House buyers mainly consider the following indicators: housing area A1m2, equipment A2 (level), environmental A3 (level) and price A4 (per unit area: yuan /m2), and distance A5 (mile). Among them, A1,A2,A3 is the benefit index and A4,A5 is the cost index. The weight of each indicator is wi=0.2087,0.2225,0.2256,0.2046,0.1386,where i=1,2,⋯,5. The case data are selected from reference [26].Step 1: To construct the decision matrix interval grey number in Table1Step 2: Normalize the decision matrix of the interval grey number using formulas (22) and (23) in Table 2Step 3: Utilize the normalization matrix and theGNWBM operator using formula (14) (in general, we can take p=1,q=1) as(27)r1=GNWBMw11,1r11,r12,⋯,r1n=1nn−1⊕k,l=1k≠lnwkr1k⊗wlr1l1/2=0.0637,0.0865,r2=0.0260,0.0533,r3=0,0,r4=0.0353,0.0481.Alternatively, theGNWGBM operator using formula (21) (in general, we can take p=1,q=1) is(28)r1=GNWGBMw11,1r11,r12,⋯,r1n=12⊗k,l=1k≠lnr1kwk⊕r1lwl1/nn−1=0.9059,0.9654,r2=0,0.6861,r3=0,0.6379,r4=0.8444,0.9020.Step 4: The possibility is calculated from the overall preference values using formula (24). Then we have(29)pi=∑j=1mpij,,i=1,2,⋯,m.The possibility is calculated from the preference values obtained by theGNWBM operator:(30)p12=max0,lenr1+lenr2−maxb2−a1,0lenr1+lenr2=1,p13=1,p14=1,p1=p12+p13+p14=3,p2=1.4253,p3=0,p4=1.5747.Possibility is calculated from the preference values obtained by theGNWGBM operator:(31)p1=3,p2=0.5182,p3=0.4818,p4=2.We know that the order we get from both operators isp1>p4>p2>p3, namely, r1>r4>r2>r3.Step 5: Get the best scheme according to the sort.
Sox1>x4>x2>x3, namely, the scheme x1 is the best one.
It can be seen from the above analysis in Table3 that the order of the two results is exactly consistent. The optimal scheme is x1.Table 1
Direct observation of quantitative indicators.
UA1A2A3A4A5x1[90, 120][7, 7][6, 8][2900, 3100][10, 10]x2[70, 100][3, 3][4, 6][3000, 3200][8, 8]x3[50, 70][5, 5][11, 12][3500, 3700][20, 20]x4[60, 80][5, 5][8, 9][3000, 3200][12, 12]Table 2
Normalized decision matrix.
UA1A2A3A4A5x1[0.5714, 1.0000][1.0000, 1.0000][0.2500, 0.5000][0.7500, 1.0000][0.8333, 0.8333]x2[0.2857, 0.7143][0, 0][0, 0.2500][0.6250, 0.8750][1.0000, 1.0000]x3[0, 0.2857][0.5000, 0.5000][0.8750, 1.0000][0, 0.2500][0, 0]x4[0.1429, 0.4286][0.5000, 0.5000][0.5000, 0.6250][0.6250, 0.8750][0.6667, 0.6667Table 3
Comparison of the results with [26].
Scheme rankingThe paper [26]x1>x4>x2>x3The method in this paperx1>x4>x2>x3Example 2.
(The example is taken from [8]) Assume that a company needs to hire a branch manager and that there are currently four candidates x1,x2,x3,x4. This study selected the first three interviews. In order to recruit the most suitable candidates, the decision-makers mainly consider the following aspects: confidence c1, interpersonal communication c2, work enthusiasm c3, and expected salary c4. The interview scores are presented in Table 4. The weight of each attribute is the w1=0.2363,w2=0.2254,w3=0.2806,w1=0.2577.Step 1: To construct the decision matrix interval grey number in Table4Step 2: To normalize the decision matrix of interval grey number by formulas (22) and (23) in Table 5Step3, Step4, and Step5 are the same as in Example1, and the possibility is calculated from the preference values obtained by the GNWGBM operator:(32)p1=1.54,p2=1.71,p3=0.80,p4=1.95.Sox4>x2>x1>x3, namely, the scheme x4 is the best one.
This paper is the same as the best candidate obtained at the different moments of the three interviews in the literature [8].
The method used in this study is simpler, and it is reasonable. It can be observed that the proposed method is effective.Table 4
Interview scores.
Candidatec1c2c3c4x1[0.85, 0.95][83, 92][70, 78][12000, 15000]x2[0.82, 0.90][84, 89][78, 84][11000, 14500]x3[0.78, 0.88][86, 90][82, 88][10800, 13000]x4[0.75, 0.85][88, 94][84, 90][12000, 15500]Table 5
The normalized interview scores.
CandidateA1A2A3A4x1[0.50, 1.00][0, 0.81][0, 0.40][0.25, 0.89]x2[0.35, 0.75][0.09, 0.54][0.40, 0.70][0.04, 0.78]x3[0.15, 0.65][0.27, 0.63][0.60, 0.90][0, 0.46]x4[0, 0.50][0.45, 1.00][0.70, 1.00][0.25, 1.00]


## 6. Conclusion

In the case of information acquisition, the interval grey number is used to describe the decision information, which is more in line with the actual situation of management, engineering, and so on. Considering the mutual relationship between the attributes of schemes is also necessary for the development of practical problems. This study defines the interval grey number weighted geometric Bonferroni meanGNWGBM operator, which is used to aggregate grey number information. Two examples are provided to demonstrate the effectiveness of the proposed method, and its practicability and effectiveness are also proved.



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