2-Dimension Linguistic Bonferroni Mean Aggregation Operators and Their Application to Multiple Attribute Group Decision Making

Jianbin Zhao; Hua Zhu

doi:10.2991/ijcis.d.191125.001

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Volume 12, Issue 2, 2019, Pages 1557 - 1574

2-Dimension Linguistic Bonferroni Mean Aggregation Operators and Their Application to Multiple Attribute Group Decision Making

Authors

Jianbin Zhao¹^{, 2}, Hua Zhu¹^{, 2}^{, *}

¹School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, 450001, China

²Henan Key Laboratory of Financial Engineering, Zhengzhou University, Zhengzhou, Henan, 450001, China

^*Corresponding author. Email: zhuhua@zzu.edu.cn

Corresponding Author

Hua Zhu

Received 22 January 2019, Accepted 18 November 2019, Available Online 6 December 2019.

DOI: 10.2991/ijcis.d.191125.001 How to use a DOI?
Keywords: MAGDM; Bonferroni mean operator; 2-dimension linguistic weight Bonferroni mean aggregation operator; 2-dimension linguistic variable; 2-dimension linguistic lattice implication algebra
Abstract: The aim of this paper is to provide a multiple attribute group decision making (MAGDM) method based on the 2-dimension linguistic weight Bonferroni mean aggregation (2DLWBMA) operator. Firstly, the new operations of 2-dimension linguistic variables are defined. Then, the 2-dimension linguistic Bonferroni mean aggregation operator is proposed to describe the correlations of input arguments. Subsequently, the 2DLWBMA operator is investigated to consider the importance of attributes. Furthermore, a novel MAGDM method is introduced and two illustrative examples are given.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Multiple attribute group decision making (MAGDM) methods refer to ranking the limited alternatives or selecting the best one according to the evaluation results provided by different decision makers. MAGDM methods have been widely applied in many areas, such as economic, society, science and management and so on. Among some MAGDM problems, the decision makers prefer to evaluate the alternatives on the qualitative attributes by linguistic variables (LVs) rather than crisp numbers. Since Zadeh [1–3] introduced the notion of LV, researchers have developed many kinds of methods to rank the alternatives according to the values of LVs [4–10]. However, in some real MAGDM problems, besides the linguistic evaluation values of the attributes, the decision maker still uses LVs to represent his or her self-assessment on the given evaluation results. For describing such phenomena, Zhu et al. [11] proposed the concept of 2-dimension linguistic variable (2DLV). A 2DLV includes two classes of LVs, where the I class LV is used to represent the linguistic evaluation value of the attribute, and the II class LV is used to represent the decision maker's self-assessment.

The main advantage of 2DLVs is that it can distinguish indetermination between decision making problems and subjective understanding. By use of 2DLVs, the decision makers can better express their opinions. Up to now, the research achievements of 2-dimension linguistic information can be classified into three categories: 2DLVs [12–17], 2-dimension uncertain linguistic variables (2DULVs) [18–28], and hesitant fuzzy 2-dimension linguistic variables (HF2DLVs) [29,30]. Since there exist two classes of LVs in a 2DLV, the operations of 2DLVs become more difficult and complex than those of LVs. For simplicity of calculation, many existing 2DLV operations only take the minimum values of the II class LVs [12,19–21,23,25,26,31,32]. Although it is easy to operate by taking minimum values, many useful linguistic information may be lost or distorted. Because the I class LV of 2DLVs is used to describe the linguistic evaluation value of the attribute, the operation of the I class LV is consistent with that of the classical LV proposed by Zadeh [1]. On the other hand, the II class LV is used to represent the decision maker's self-assessment. At the same time, fuzzy number is a useful tool to describe the decision maker's subjective attitude. Hence we intend to convert the II class LV of 2DLVs into fuzzy number, which can more reasonably describe the decision maker's self-assessment. Concretely, the new operations of 2DLVs take the I class LV as the classical LV, and convert the II class LV into fuzzy number.

Bonferroni mean (BM) operator, as a useful aggregation operator, has the ability to capture the interrelationships between input arguments. So far, BM operator has been widely extended to fuzzy linguistic environment [6,33–38], intuitionistic fuzzy linguistic environment [39–44], hesitant fuzzy environment [45] and pythagorean fuzzy circumstance [46] for handling the increasingly complex fuzzy systems and decision systems. Wei et al. [6] developed the uncertain linguistic BM operator and the uncertain linguistic geometric BM operator for aggregating uncertain linguistic information. Liu et al. [41] extended BM operator to the intuitionistic linguistic environment. Dutta et al. [34] applied the extended Bonferroni Mean (EBM) in linguistic 2-tuple environment to capture heterogeneous interrelationship among the attributes. Tian et al. [35,36] combined BM operator with simplified neutrosophic linguistic numbers and gray linguistic numbers to deal with the corresponding MADM problems. Combing the generalized partitioned Bonferroni mean (PBM) operator with linguistic neutrosophic numbers, Wang et al. [37] proposed the linguistic neutrosophic generalized weighted partitioned Bonferroni mean (LNGWPBM) aggregation operator to solve MAGDM problems. However, in some 2-dimension linguistic MAGDM problems, because the evaluation values of attributes are extended to two classes of LVs, the relationships between input arguments may be more complicated. In order to capture the interrelationships between input 2DLVs, it is necessary to extend BM operator into 2-dimension linguistic decision making environment.

Further, we list some extended BM methods in different application environments to convenient the readers by using the following tabular (see Table 1).

Applications	Some Extended BM Methods
Fuzzy linguistic environment	Beliakov's method [33], Dutta's method [34], Tian's methods [35,36], Wang's method [37], Wei's method [6], Yager's method [38]
Intuitionistic fuzzy linguistic environment	Das's method [39], Heyd's method [40], Liu's methods [41,42], Zhang's method [43], Zhou's method [44]
Hesitant fuzzy linguistic environment	Zhu's method [45]
Pythagorean fuzzy linguistic environment	Nierx's method [46]
2-dimension fuzzy linguistic environment	Liu's method [25], Yin's method [32], the proposed method

BM = Bonferroni Mean.

Table 1

Some extended BM methods in different application environments.

In 2-dimension linguistic MAGDM problems, Yin et al. [32] have combined trapezoidal fuzzy 2-dimensional linguistic information with a PBM operator to address the 2-dimension linguistic decision making problems. Liu et al. [25] have proposed the 2-dimensional uncertain linguistic weighted Bonferroni mean (2DULWBM) operator and the 2-dimensional uncertain linguistic improved weighted Bonferroni Harmonic mean (2DULIWBHM) operator. However, this paper combine 2DLVs with BM operators for different aspects. Concretely, the operations of 2DLVs are improved as stated in the above, where the II class LV of 2DLVs is converted into a fuzzy number. Further, this paper considers BM operators rather than PBM operators. BM operators focus on the aggregated attributes to capture the interrelationships among them, which assumes that each attribute is related to the rest of the attributes. While PBM operators assume that attributes are partitioned into several unrelated classes and each attribute only has the interrelationship with rest of the attributes in the same class. Moreover, the comparison method of 2DLVs used in this paper [47] is improved more reasonably than many existing methods which can only deal with the total order of alternatives [15,25,32].

Motivated by the above ideas, we intend to

Define some new operations of 2DLVs.
Establish the 2-dimensional linguistic Boneferroni mean aggregation (2DLBMA) operator and the 2DLWBMA operator.
Investigate the properties and special cases of the 2DLBMA operator.
Propose a novel decision making method based on the 2DLWBMA operator.
Demonstrate the feasibility and practicality of the proposed method.

The remainder of this paper is established as follows: Section 2 reviews 2DLV and BM operator. In Section 3, some new operations of 2DLVs are developed and their properties are discussed. In Section 4, the 2DLBMA operator and the 2DLWBMA operator are proposed for 2DLVs. Section 5 provides a novel decision making approach based on the 2DLWBMA operator. In Section 6, two illustrative examples are given. Section 7 concludes this paper.

2. PRELIMINARIES

This section reviews some basic notions of 2DLV and BM operator.

2.1. 2DLV

Definition 1.

[11] Let S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht} be two linguistic term sets (LTSs), where g+1 is the cardinality of S and t+1 is the cardinality of H. ri=(sai,hbi) is called a 2DLV, in which hbi∈H is I class LV, which represents the assessment information about the alternative given by the decision maker, while sai∈S is II class LV, which represents the self-assessment of the decision maker.

For describing the complicated relationships between 2DLVs, Zhu et al. [47] proposed the concept of a 2-dimension linguistic lattice implication algebra (2DL-LIA). The method for comparing two 2DLVs is provided in a 2DL-LIA as follows.

Definition 2.

[47] Let (S×H,∨,∧,→,′) be a 2DL-LIA, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht} are two LTSs. ri=(sai,hbi) and rj=(saj,hbj) are any two 2DLVs of S×H, ai,aj∈{0,1,⋯,g} and bi,bj∈{0,1,⋯,t}, δ is a positive number in the real set R.

If ai<aj and bi⩽bj or ai⩽aj and bi<bj, then ri is less than rj, denoted by ri<rj;
If ai=aj and bi=bj, then ri is equivalent to rj, denoted by ri=rj;
If bi⩽bj and 0<ai−aj<δ, then ri is weakly less than rj, denoted by ri⩽Wrj;
If bi⩽bj and ai−aj⩾δ, then ri is incomparable to rj, denoted by ri∥rj.

Usually, the parameter δ takes value less than 1, which can be preseted by the decision maker's preferences.

2.2. BM Operator

Since BM operator can capture the interrelationship between input arguments, it has been widely applied in MAGDM problems.

Definition 3.

[48] Let p,q⩾0 and {a1,a2,⋯,an} be a set of non-negative real numbers. Then the mapping BMp,q:Rn→R such that

BMp,q(a1,⋯,an)=1n(n−1)∑i,j=1,i≠jnaipajq1p+q,

is called the BM operator.

BM operator satisfies the following properties:

If ai=a for all i, then BMp,q(a,a,⋯,a)=a;
If ai⩽ai′ for all i, then

BMp,q(a1,a2,⋯,an)⩽BMp,q(a1′,a2′,⋯,an′);
If a̲⩽ai⩽a¯ for all i, then

a̲⩽BMp,q(a1,a2,⋯,an)⩽a¯.

Especially,

BM1,0(a1,⋯,an)=1n∑i=1nai is the average mean;
limp→0BMp,0(a1,⋯,an)=∏i=1nai1n is the geometric mean.

3. NEW OPERATIONS OF 2DLVs

According to the Definition 1, in a 2DLV r=(sa,hb), hb represents the linguistic evaluation value of an attribute given by the decision maker, which is consistent with Zadeh's LV. So this paper assumes that the operation of hb is consistent with that of the classical LV [7]. While sa represents the decision maker's self-assessment, therefore it is reasonable to convert sa into fuzzy number ag, where g is the cardinality of S. By use of fuzzy number ag, the subjective attitude of the decision maker can be precisely expressed. Based on the above discussions, some new operational rules of 2DLVs are defined in this section. In the new operations of 2DLVs, the operation of sa is based on that of fuzzy number, while the operation of hb is unchanged.

Definition 4.

Let ri=(sai,hbi), rj=(saj,hbj) be two 2DLVs of a 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht}, then the operations are defined as follows:

ri⊕rj=(sai+aj−aiajg,hbi+bj);
ri⊗rj=saiajg,hbibj;
λri=(sg [1−1−aigλ],hλbi),λ>0;
riλ=(sg aigλ,hbiλ),λ>0.

Remark.

In Definition 4, there exist the mutual transformations between the II class LVs and fuzzy numbers. Concretely, we transform the II class LVs into fuzzy numbers, then implement the operational laws of fuzzy numbers, finally convert the operational results of fuzzy numbers into the II class LVs.

Example 1.

Let (S×H,∨,∧,→,′) be a 2DL-LIA, where S={s0,s1,s2,s3,s4}, H={h0,h1,h2,h3, h4,h5,h6}. Let r1=(s3,h2),r2=(s2,h3) and λ=2.

By Definition 4, we can compute that

r1⊕r2=(s3.5,h5), λr1=(s3.75,h4),

r1⊗r2=(s1.5,h6), r1λ=(s2.25,h4).

In the following, some propositions of the new operational rules are discussed.

Proposition 1.

Let ri=(sai,hbi), rj=(saj,hbj) and rk=(sak,hbk) be 2DLVs of a 2DL-LIA S×H, λ,λ1,λ2>0. Then the followings hold:

ri⊕rj=rj⊕ri;
λ(ri⊕rj)=λri⊕λrj;
λ1ri⊕λ2ri=(λ1+λ2)ri;
(ri⊕rj)⊕rk=ri⊕(rj⊕rk);
ri⊗rj=rj⊗ri;
(ri⊗rj)λ=ri λ⊗rjλ;
riλ1⊗riλ2=riλ1+λ2;
(ri⊗rj)⊗rk=ri⊗(rj⊗rk).

Proof.

The proof of (1) can be obtained according to Definition 4.

Firstly, we give the proof of (2). On one hand,

λ(ri⊕rj)

=λsai+aj−aiajg,hbi+bj

=sg 1−1−ai+ajg+aiajg2λ,hλ(bi+bj)

=sg 1−1−aigλ1−ajgλ,hλ(bi+bj);

On the other hand,

λri⊕λrj

=sg 1−1−aigλ,hλbi⊕sg 1−1−ajgλ,hλbj

=sg−g 1−aigλ1−ajgλ,hλ(bi+bj)

=sg 1−1−aigλ1−ajgλ,hλ(bi+bj);

Hence we have λ(ri⊕rj)=λri⊕λrj.

Then, the proof of (4) is given as follows. Since

(ri⊕rj)⊕rk

=sai+aj−aiajg,hbi+bj⊕sak,hbk

=sai+aj+ak−aiajg−aiakg−ajakg+aiajakg2,hbi+bj+bk;

On the other hand,

ri⊕(rj⊕rk)

=sai,hbi⊕saj+ak−ajakg,hbj+bk

=sai+aj+ak−aiajg−aiakg−ajakg+aiajakg2,hbi+bj+bk;

Hence (ri⊕rj)⊕rk=ri⊕(rj⊕rk).

Similarly, (3) and (5)–(8) can be proved, which are omitted here.

Proposition 2.

Let ri=(sai,hbi), ri′=(sai′,hbi′), rj=(saj,hbj), rj′=(saj′,hbj′) be 2DLVs of a 2DL-LIA S×H. If ri⩽ri′, rj⩽rj′, then the followings hold:

λri⩽λri′;
ri λ⩽ri′λ;
ri⊕rj⩽ri′⊕rj′;
ri⊗rj⩽ri′⊗rj′;
⊕i=1nri⩽⊕i=1nri′;
⊗i=1nri⩽⊗i=1nri′.

Proof.

Suppose ri⩽ri′, then according to Definition 2, we have ai⩽ai′ and bi⩽bi′.

(1) Because λ>0, we have

(1−aig)λ⩾(1−ai′g)λ, which follows that

g1−(1−aig)λ⩽g1−(1−ai′g)λ.

And we have λbi⩽λbi′.

Since λri=sg 1−(1−aig)λ,hλbi and

λri′=sg 1−(1−ai′g)λ,hλbi′,

then according to Definition 2, λri⩽λri′.

(2) Because λ>0, we have

g aigλ⩽g ai′gλ. And we have biλ⩽bi′λ.

Since ri λ=sg aigλ,hbiλ and

ri′λ=sg ai′gλ,hbi′λ, then according to Definition 2, ri λ⩽ri′λ.

(3) Suppose ri⩽ri′ and rj⩽rj′, then by Definition 2, we have ai⩽ai′, bi⩽bi′ and aj⩽aj′, bj⩽bj′.

Then we have

ai+aj−aiajg=g−g(1−aig)(1−ajg) and

ai′+aj′−ai′aj′g=g−g(1−ai′g)(1−aj′g).

Since ai⩽ai′ and aj⩽aj′, we have

0⩽1−ai′g⩽1−aig and 0⩽1−aj′g⩽1−ajg.

It follows that

0⩽(1−ai′g)(1−aj′g)⩽(1−aig)(1−ajg).

Therefore

g−g(1−aig)(1−ajg)⩽g−g(1−ai′g)(1−aj′g),

that is ai+aj−aiajg⩽ai′+aj′−ai′aj′g.

And bi+bj⩽bi′+bj′ holds obviously.

Since ri⊕rj=sai+aj−aiajg,hbi+bj and

ri′⊕rj′=sai′+aj′−ai′aj′g,hbi′+bj′, then according to Definition 2, we have ri⊕rj⩽ri′⊕rj′.

Similarly, (4) can be proved, which is omitted here.

(5) We can prove ⊕i=1nri⩽⊕i=1nri′ by the mathematical induction on n.

When n=2, we have r1⊕r2⩽r1′⊕r2′ by (3).

Assume n=k, the equation holds, that is

⊕i=1kri⩽⊕i=1kri′.

Then, when n=k+1, we have

⊕i=1k+1ri=⊕i=1kri⊕rk+1.

Because ⊕i=1kri⩽⊕i=1kri′ and rk+1⩽rk+1′,

then by (3), we have

⊕i=1kri⊕rk+1⩽⊕i=1kri′⊕rk+1′ =⊕i=1k+1ri′,

that is ⊕i=1k+1ri⩽⊕i=1k+1ri′.

Now, we have ⊕i=1nri⩽⊕i=1nri′ for all n.

(6) can be proved similarly by the mathematical induction.

Proposition 3.

Let ri=(sai,hbi) (i=1,2,⋯,n) be 2DLVs of a 2DL-LIA S×H, wi∈[0,1] and λ>0. Then the followings hold:

⊕i=1nri=sg−g∏i=1n(1−aig),h∑i=1nbi;
⊕i=1nwiri=sg−g∏i=1n(1−aig)wi,h∑i=1nwibi;
⊕i=1nriλ=sg 1−∏i=1n(1−aig)λ,h(∑i=1nbi)λ;
⊗i=1nri=sg∏i=1n(aig),h∏i=1nbi;
⊗i=1nriwi=sg∏i=1n(aig)wi,h∏i=1nbiwi;
⊗i=1nriλ=sg ∏i=1naigλ,h(∏i=1nbi)λ.

Proof.

(1) We can prove

⊕i=1nri=sg−g∏i=1n(1−aig),h∑i=1nbi,

by the mathematical induction on n.

When n=2, we have

r1⊕r2

=sa1+a2−a1a2g,hb1+b2

=sg−g 1−a1g−a2g+a1a2g2,hb1+b2

=sg−g∏i=12(1−aig),h∑i=12bi.

Assume n=k, the conclusion holds, that is

⊕i=1kri=sg−g∏i=1k(1−aig),h∑i=1kbi.

Then, when n=k+1,we have

⊕i=1k+1ri=⊕i=1kri⊕rk+1

=sg−g∏i=1k(1−aig),h∑i=1kbi⊕(sak+1,hbk+1)

=sg−g∏i=1k(1−aig)+∏i=1k(1−aig)ak+1,h∑i=1k+1bi

=sg−g∏i=1k+1(1−aig),h∑i=1k+1bi.

Now, we have ⊕i=1nri=sg−g∏i=1n(1−aig),h∑i=1nbi for all n.

Similarly, (2)–(6) can be proved by the mathematical induction, which are omitted.

4. 2DLBMA OPERATOR AND 2DLWBMA OPERATOR

In this section, based on BM operator and the new operational rules of 2DLVs, we develop the 2DLBMA operator to deal with MAGDM problems.

4.1. 2DLBMA Operator

Definition 5.

Let ri=(sai,hbi) (i=1,2,⋯,n) be 2DLVs of a 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht}, and p,q⩾0. Then the mapping 2DLBMAp,q:Ωn→Ω such that

2DLBMAp,q(r1,⋯,rn)=1n(n−1)⊕i,j=1,i≠jn(rip⊗rjq)1p+q,

is called the 2DLBMA operator.

The 2DLBMA operator is a useful tool to capture the interrelationship between the attributes in 2-dimension linguistic MAGDM environment.

Example 2.

Let (S×H,∨,∧,→,′) be a 2DL-LIA, where S={s0,s1,s2,s3,s4}, H={h0,h1,h2,h3,h4,h5,h6}. Let r1=(s3,h5),r2=(s3,h3),r3=(s1,h5),r4=(s2,h4) and p=1,q=1.

According to Definition 5, we can compute that

2DLBMA1,1(r1,r2,r3,r4)=(s2.26,h4.22).

The following theorem shows that the aggregation value of 2DLVs, which is aggregated by the 2DLBMA operator, is still a 2DLV.

Theorem 4.

Let ri=(sai,hbi) (i=1,2,⋯,n) be 2DLVs of a 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht}, and p,q⩾0. Then 2DLBMAp,q(r1,⋯,rn)

=(sg (1−(∏i,j=1,i≠jn(1−aipajqgp+q))1n(n−1))1p+q,h(1n(n−1)∑i,j=1,i≠jnbipbjq)1p+q).

Proof.

The proof of Theorem 4 includes three steps, which are given in the following.

Step 1 By Definition 4,

rip⊗rjq

=sg(aig)p,hbip⊗sg(ajg)q,hbjq

=sg(aig)p(ajg)q,hbipbjq.

then by Proposition 3, we have

⊕j=1,j≠in(rip⊗rjq)

=(sg (1−∏j=1,j≠in(1−aipajqgp+q)),h∑j=1,j≠inbipbjq).

It follows that

⊕i,j=1,j≠in(rip⊗rjq)

=⊕i=1n⊕j=1,j≠in(rip⊗rjq)

=⊕i=1nsg 1−∏j=1,j≠in1−aipajqgp+q,h∑j=1,j≠inbipbjq

=(sg (1−∏i,j=1,j≠in(1−aipajqgp+q)),h∑i,j=1,j≠inbipbjq).

Step 2 By Definition 4, we can obtain that

1n(n−1)⊕i,j=1,i≠jn(rip⊗rjq)

=1n(n−1)sg 1−∏i,j=1,j≠in1−aipajqgp+q,h∑i,j=1,j≠inbipbjq

=sg 1−∏i,j=1,j≠in1−aipajqgp+q1n(n−1),h1n(n−1)∑i,j=1,j≠inbipbjq.

Step 3 Based on Step 2 and Definition 4, we have

2DLBMAp,q(r1,⋯,rn)

=1n(n−1)⊕i,j=1,i≠jn(rip⊗rjq)1p+q

=sg 1−∏i,j=1,j≠in1−aipajqgp+q1n(n−1),h1n(n−1)∑i,j=1,j≠inbipbjq1p+q

=sg 1−∏i,j=1,i≠jn1−aipajqgp+q1n(n−1)1p+q,h1n(n−1)∑i,j=1,i≠jnbipbjq1p+q.

Next, we can prove that the 2DLBMA operator satisfies the properties including idempotency, monotonicity, commutativity and boundedness. These properties are important to investigate the applications of 2DLVs.

Theorem 5.

(Idempotency) Let ri=(sai,hbi) (i=1,2,⋯,n) be 2DLVs of a 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht}, and p,q⩾0. If ri=r=(sa,hb) for all i, then

2DLBMAp,q(r1,⋯,rn)=r.

Proof.

Because ri=r=(sa,hb) for all i, from Definition 5 and Proposition 1, we have

2DLBMAp,q(r1,⋯,rn)

=1n(n−1)⊕i,j=1,i≠jn(rip⊗rjq)1p+q

=1n(n−1)⊕i,j=1,i≠jnrp+q1p+q

=r.

Theorem 6.

(Monotonicity) Let ri=(sai,hbi) and ri′=(sai′,hbi′) (i=1,2,⋯,n) be 2DLVs of a 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht}, and p,q⩾0. If ri⩽ri′ for all i, then

2DLBMAp,q(r1,⋯,rn)⩽2DLBMAp,q(r1′,⋯,rn′).

Proof.

Because ri⩽ri′ for all i, from Definition 5 and Proposition 2, we have

2DLBMAp,q(r1,⋯,rn)⩽2DLBMAp,q(r1′,⋯,rn′).

Theorem 7.

(Commutativity) Let ri=(sai,hbi) (i=1,2,⋯,n) be 2DLVs of a 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht}, and p,q⩾0. If (r~1,⋯,r~n) is a permutation of (r1,⋯,rn), then

2DLBMAp,q(r1,⋯,rn)=2DLBMAp,q(r̃1,⋯,r̃n).

Proof.

According to Proposition 1 (4), the conclusion can be obtained.

Theorem 8.

(Boundedness) Let ri=(sai,hbi) and ri′=(sai′,hbi′) (i=1,2,⋯,n) be 2DLVs of a 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht}, and p,q⩾0. Then

(s0,h0)=r0⩽2DLBMAp,q(r1,⋯,rn)⩽r∗=(sg,ht).

Proof.

Since for all i, r0⩽ri⩽r∗, then by Theorem 5 and Theorem 6, we have

(s0,h0)=r0=2DLBMAp,q(r0,⋯,r0)

⩽2DLBMAp,q(r1,⋯,rn)

⩽2DLBMAp,q(r∗,⋯,r∗)=r∗=(sg,ht).

Subsequently, some special cases of the 2DLBMA operator are discussed as follows:

Case 1 If q=0, then from Definition 5, we have

2DLBMAp,0(r1,⋯,rn)

=1n(n−1)⊕i,j=1,i≠jn(rip⊗rj0)1p+q

=1n(n−1)⊕i,j=1,i≠jnrip⊗(sg,h1)1p

=1n⊕i=1nrip1p,

which is called the 2-dimension linguistic generalized mean aggregation operator.

Case 2 If p=1,q=0, then from Definition 5 and Case 1, we have

2DLBMA1,0(r1,⋯,rn)=1n⊕i=1nri,

which is called the 2-dimension linguistic mean aggregation (2DLMA) operator.

Case 3 If p→0,q=0, then from Definition 5, we have

limp→02DLBMAp,0(r1,⋯,rn)=∏i=1nri1n,

which is called the 2-dimension linguistic geometric mean aggregation (2DLGMA) operator.

4.2. 2DLWBMA Operator

In real MAGDM environment, the weights of attributes are usually different, hence the 2DLWBMA operator will be discussed in the following.

Definition 6.

Let ri=(sai,hbi) (i=1,2,⋯,n) be 2DLVs of a 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht}, and p,q⩾0. Then the mapping 2DLWBMAp,q:Ωn→Ω such that

2DLWBMAp,q(r1,⋯,rn)

=1n(n−1)⊕i,j=1,i≠jn(βip⊗βjq)1p+q,

is called the 2-dimension linguistic weight Bonferroni mean aggregation (2DLWBMA) operator, where βi=nwiri, w=(w1,⋯,wn)T is the weight vector of ri satisfying wi∈[0,1] and ∑i=1nwi=1.

Obviously, if the weight vector w is equal to (1n,⋯,1n), then the 2DLWBMA operator is degenerated to the 2DLBMA operator.

Example 3.

Let (S×H,∨,∧,→,′) be a 2DL-LIA, where S={s0,s1,s2,s3,s4}, H={h0,h1,h2,h3,h4,h5,h6}. Let r1=(s3,h5),r2=(s3,h3),r3=(s1,h5),r4=(s2,h4), w=(0.2,0.27,0.3,0.23) and p=1,q=1.

According to Definition 6, we can compute that

2DLWBMA1,1(r1,r2,r3,r4)=(s2.21,h4.19).

The following theorem shows that the aggregated result of 2DLVs by the 2DLWBMA operator is still a 2DLV.

Theorem 9.

Let ri=(sai,hbi) (i=1,2,⋯,n) be 2DLVs of a 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht}, w=(w1,⋯,wn)T be the weight vector of ri satisfying wi∈[0,1] and ∑i=1nwi=1, p,q⩾0. Then

2DLWBMAp,q(r1,⋯,rn)

=sg 1−∏i,j=1,i≠jn1−(1−(1−aig)nwi)p(1−(1−ajg)nwj)q1n(n−1)1p+q, h1n(n−1)∑i,j=1,i≠jn(wibi)p(wjbj)q1p+q.

Proof.

This proof includes three steps.

Step 1 By Definition 4,

(nwiri)p⊗(nwjrj)q

=sg 1−(1−aig)nwi,hwibip⊗sg 1−(1−ajg)nwj,hnwjbjq

=sg 1−(1−aig)nwip,h(nwibi)p⊗sg 1−(1−ajg)nwjq,h(nwjbj)q

=sg 1−(1−aig)nwip1−(1−ajg)nwjq,h(nwibi)p(nwjbj)q.

Then by Proposition 3 (1), we have

⊕j=1,j≠in(nwiri)p⊗(nwjrj)q

=⊕j=1,j≠insg 1−(1−aig)nwip1−(1−ajg)nwjq,h(nwibi)p(nwjbj)q

=sg 1−∏j=1,j≠in1−(1−(1−aig)nwi)p(1−(1−ajg)nwj)q, h∑j=1,j≠in(nwibi)p(nwjbj)q.

Further, according to Proposition 3 (1), we have

⊕i,j=1,j≠in(nwiri)p⊗(nwjrj)q

=⊕i=1n⊕j=1,j≠in((nwiri)p⊗(nwjrj)q)

=⊕i=1nsg 1−∏j=1,j≠in1−(1−(1−aig)nwi)p(1−(1−ajg)nwj)q,h∑j=1,j≠in(nwibi)p(nwjbj)q

=sg 1−∏i,j=1,j≠in(1−(1−aig)nwi)p(1−(1−ajg)nwj)q, h∑i,j=1,j≠in(nwibi)p(nwjbj)q.

Step 2 By Definition 4, we can obtain that

1n(n−1)⊕i,j=1,i≠jn(nwiri)p⊗(nwjrj)q

=1n(n−1)sg 1−∏i,j=1,j≠in(1−(1−aig)nwi)p(1−(1−ajg)nwj)q, h∑i,j=1,j≠in(nwibi)p(nwjbj)q

=sg 1−∏i,j=1,j≠in(1−(1−aig)nwi)p(1−(1−ajg)nwj)q1n(n−1), h1n(n−1)∑i,j=1,j≠in(nwibi)p(nwjbj)q.

Step 3 Based on Step 2 and Definition 4, we have

2DLWBMAp,q(r1,⋯,rn)

=1n(n−1)⊕i,j=1,i≠jn((nwiri)p⊗(nwirj)q)1p+q

=sg 1−∏i,j=1,j≠in(1−(1−aig)nwi)p(1−(1−ajg)nwj)q1n(n−1), h1n(n−1)∑i,j=1,j≠in(nwibi)p(nwjbj)q1p+q

=sg 1−∏i,j=1,j≠in(1−(1−aig)nwi)p(1−(1−ajg)nwj)q1n(n−1)1p+q, h1n(n−1)∑i,j=1,i≠jn(wibi)p(wjbj)q1p+q.

5. A MAGDM METHOD BASED ON THE 2DLWBMA OPERATOR

In a MAGDM problem, let A={A1,⋯,An} be a set of alternatives and C={C1,⋯,Cm} be a set of attributes, where Ai denotes the ith alternative and Cj denotes the jth attribute. Let D={D1,⋯,Dl} be a set of experts, where Dk denotes the kth expert. Let w=(w1,⋯,wm)T be the weight vector of the attributes and γ=(γ1,⋯,γl)T be the weight vector of the experts, where 0⩽wj⩽1, ∑j=1mwj=1 and 0⩽γk⩽1, ∑k=1lγk=1. Suppose that there is a given 2DL-LIA S×H, where S={s0,s1,⋯,sg} and H={h0,h1,⋯,ht} be two LTSs.

The expert Dk expresses his (or her) evaluation on the alternative Ai with respect to the attribute Cj by a 2DLV rijk=(saijk,hbijk), which is an element of S×H. Suppose Rk=(rijk)n×m is the 2DLV decision matrix, where rijk=(saijk,hbijk)∈S×H means that the expert Dk gives the evaluation value for the alternative Ai with respect to the attribute Cj. Then, the ranking order of alternatives is required.

The proposed method involves the following steps:

Step 1 Normalize the 2DLV decision matrix.

Since the set of attributes C={C1,⋯,Cm} should be classified into the set of benefit attributes Cbenefit and the set of cost attributes Ccost. In order to achieve normalization, we generally transform the cost attribute values into benefit attribute values. Suppose Rk=(rijk)n×m is the normalized matrix of Zk=(zijk)n×m, where zijk=(saijk,hbijk). Then, the standardizing method is described as follows [16]:

rijk=(saijk,hbijk),zijk∈Cbenefit(saijk,ht−bijk),zijk∈Ccost,

where 1⩽i⩽n,1⩽j⩽m,1⩽k⩽l.

Step 2 Calculate the collective decision information by the 2DLWBMAp,q operator in the light of Definition 6.

Based on the normalized 2DLV decision matrixes Rk, (k=1,⋯,l), utilize the 2DLWBMAp,q operator to obtain the aggregated matrix R=(rij)n×m, where

rij=(saij,hbij)=2DLWBMAp,q(rij1,⋯,rijl)=

sg 1−∏k1,k2=1,k1≠k2l1−(1−(1−aijk1g)lγk1)p(1−(1−aijk2g)lγk2)q1l(l−1)1p+q, h1l(l−1)∑k1,k2=1,k1≠k2l(γk1bijk1)p(γk2bijk2)q1p+q,

where p,q⩾0, 1⩽i⩽n,1⩽j⩽m.

Step 3 Calculate the overall evaluation value of each alternative base on the 2DLWBMAp,q operator.

Based on the aggregated 2DLV decision matrix R=(rij)n×m, utilize the 2DLWBMAp,q operator to compute the overall evaluation values ri=(sai,hbi) as follows:

ri=(sai,hbi)=2DLWBMAp,q(ri1,⋯,rim)=

sg 1−∏j1,j2=1,j1≠j2m1−(1−(1−aij1g)mwj1)p(1−(1−aij2g)mwj2)q1m(m−1)1p+q, h1m(m−1)∑j1,j2=1,j1≠j2m(wj1bij1)p(wj2bij2)q1p+q.

where p,q⩾0, 1⩽i⩽n.

Step 4 Compare the obtained overall evaluation values ri=(sai,hbi) according to Definition 2.

Step 5 Rank all the alternatives or select the best one(s) in accordance with the ranking of ri. The prior the ri is, the best the alternative Ai is.

Step 6 Ends.

6. EXAMPLES ILLUSTRATIONS

In this section, we use the proposed MAGDM method based on the 2DLWBMA operator to solve the technological innovation ability evaluation problem [23] and the landfill site selection problem [49]. It shows that the effectiveness and advantage of the proposed method.

6.1. Application to the Technological Innovation Ability Evaluation Problem

For the technological innovation ability evaluation problem cited from literature [23], the proposed MAGDM method based on the 2DLWBMA operator is applied to rank the candidate enterprises.

Example 4.

[23] A practical use of the proposed approach involves the technological innovation ability evaluation of the four enterprises {A1,A2,A3,A4}. The set of four attributes is C={C1,C2,C3,C4}, which means that the ability of innovative resources input C1, the ability of innovation management C2, the ability of innovation tendency C3, the ability of research and development C4. The weight vector of the attributes is ω={0.25,0.27,0.25,0.23}T.

Let (S×H,∨,∧,→,′) be a 2DL-LIA, where S={s0,s1,s2,s3,s4}, H={h0,h1,h2,h3, h4,h5,h6}. Three experts Dk (k=1,2,3) are respectively required to evaluate the alternatives Ai on attributes Cj with 2DLVs rijk=(saijk,hbijk), where saijk∈S,hbijk∈H. The weight vector of the experts is γ={0.4,0.28,0.32}.

The 2DLV decision matrixes Rk=(rijk)4×4 given by three experts are listed as follows:

C1 C2 C3 C4R1=A1A2A3A4(s3,h5)(s3,h3)(s4,h5)(s2,h4)(s3,h4)(s3,h5)(s4,h3)(s2,h4)(s3,h3)(s3,h4)(s4,h4)(s2,h5)(s3,h6)(s3,h2)(s4,h3)(s2,h4)

C1 C2 C3 C4R2=A1A2A3A4(s4,h4)(s3,h4)(s3,h4)(s4,h6)(s4,h5)(s3,h3)(s3,h5)(s4,h3)(s4,h4)(s3,h4)(s3,h3)(s4,h4)(s4,h5)(s3,h5)(s3,h2)(s4,h4)

C1 C2 C3 C4R3=A1A2A3A4(s3,h5)(s2,h3)(s4,h4)(s1,h5)(s3,h4)(s2,h5)(s4,h2)(s1,h3)(s3,h4)(s2,h5)(s4,h1)(s1,h4)(s3,h3)(s2,h3)(s4,h5)(s1,h5).

6.1.1. Illustration of the proposed method

The proposed method based on the 2DLWBMAp,q operator is applied to rank the alternatives Ai (i=1,2,3,4) in Example 4. The main process is shown as follows:

Step 1 Normalize the 2DLV decision matrixes.

Since all the attributes are the benefit attributes, the evaluation values of 2DLVs do not need normalizing.

Step 2 Aggregate the evaluation values of each expert by the 2DLWBMAp,q operator.

Based on the 2DLV decision matrixes Rk=(rij)4×4, (k=1,2,3), we utilize the 2DLWBMA1,0 operator to obtain the 2DLV aggregated matrix R as follows:

C1 C2 C3 C4A1A2A3A4(s4,h4.72)(s2.75,h3.28)(s4,h4.4)(s4,h4.88)(s4,h4.28)(s2.75,h4.44)(s4,h3.24)(s4,h3.4)(s4,h3.6)(s2.75,h4.32)(s4,h2.76)(s4,h4.4)(s4,h4.76)(s2.75,h3.16)(s4,h3.36)(s4,h4.32).

Step 3 Calculate the overall evaluation value of each alternative.

Based on the 2DLV aggregated decision matrix R=(rij)4×4, we utilize the 2DLWBMAp,q operator to obtain the overall 2-dimension linguistic evaluation value ri of the alternative Ai.

Here let p=3,q=3, then we can compute that r1=(s4,h4.31), r2=(s4,h3.92), r3=(s4,h3.83), r4=(s4,h3.91).

Step 4 Rank ri according to Definition 2.

According to Definition 2, we can obtain the ranking order of ri as r3<r4<r2<r1.

Step 5 Rank all the alternatives in accordance with the ranking of ri. The prior the ri is, the best the alternative Ai is.

Thus the ranking order of the alternatives is A3<A4<A2<A1, shown as Figure 1.

6.1.2. Exploration of the parameters' influence

Here in this example, for illustrating the influences of the parameters p and q on the ranking order of alternatives, we take different values p and q in the 2DLWBMAp,q operator. The ranking order of alternatives solved by the the proposed method with different p and q, are shown in Table 2 and Figure 2.

The Values of p and q	The Evaluation Values r_i of Alternatives	Ranking Results
p→0,q=0	r1=(s4,h4.26),r2=(s4,h3.80), r3=(s4,h3.70), r4=(s4,h3.84).	A3<A2<A4<A1
p=1,q=0	r1=(s4,h4.29), r2=(s4,h3.86), r3=(s4,h3.77), r4=(s4,h3.88).	A3<A2<A4<A1
p=1,q=1	r1=(s4,h4.28), r2=(s4,h3.84), r3=(s4,h3.75), r4=(s4,h3.86).	A3<A2<A4<A1
p=3,q=3	r1=(s4,h4.31), r2=(s4,h3.92), r3=(s4,h3.83), r4=(s4,h3.91).	A3<A4<A2<A1
p=5,q=5	r1=(s4,h4.34), r2=(s4,h4), r3=(s4,h3.91), r4=(s4,h3.96).	A3<A4<A2<A1
p=10,q=10	r1=(s4,h4.41), r2=(s4,h4.17), r3=(s4,h4.04), r4=(s4,h4.06).	A3<A4<A2<A1

Table 2

Ranking orders of the alternatives under different values of p and q.

From Table 2, we can see that the best alternative is always A1 and the worst alternative is A3, and the relationships between A4 and A2 may be different with various values p and q.

When p=1 and q=0, we obtain that A2 is less than A4, that is A2<A4. In such case, the 2DLWBMAp,q operator is degenerated to the 2DLWMA operator.

When p→0 and q=0, we obtain that A2 is less than A4, that is A2<A4. In such case, the 2DLWBMAp,q operator is degenerated to the 2DLWGMA operator.

When p=1 and q=1, we obtain that A2<A4. While in other cases, (including p=3,q=3, p=5,q=5 and p=10,q=10), we obtain that A4 is less than A2, that is A4<A2.

According to the outcomes of Table 2, it shows that the 2DLWBMAp,q operator plays a key role for considering the interrelationships between the attributes. Obviously, when p→0,q=0 and p=1,q=0, we assume that the attributes are independent from each other. However, it can not guarantee that the attributes are always independent from each others in all decision making environment. In this example, the four attributes Cj may be pairwise correlated, and the degree of correlation can be regulated by the parameters p and q. If p=1,q=1, then the ranking order is still A3<A2<A4<A1. While when p=3,q=3, p=5,q=5, or p=10,q=10, the ranking order is A3<A4<A2<A1. It shows that the ranking order of alternatives changes with the values of the parameters p and q. In fact, the decision makers can select the values of the parameters p and q according to the real decision making conditions or their experience.

6.1.3. Sensitivity analysis of the weight vector

Sensitivity analysis of the weight vector, as an important part of MAGDM, has been developed to assess the stability of the ranking [16,50,51].

When we conduct the weight sensitivity analysis, only one criterion is focused at a time. That is, if a criterion changes, the other criteria are assumed to be changed uniformly in order to remain normalized.

Memariani et al. [51] provided a new method for the weight sensitivity analysis in MADM. Let w=(w1,w2,⋯,wm)T be the original weight vector of attributes. If the weight of the pth attribute changes from wp to wp′, then the new vector of weights transforms into w′=(w1′,w2′,⋯,wm′)T. It can prove that wj′=1−wp′1−wpwj, where j=1,2,⋯,m and j≠p.

By use of Memariani's method [51], we only conduct the weight sensitivity analysis of attributes in Example 4. In addition, we only analyse the 2DLWBMAp,q operator with p=3,q=3 in Step 3. The outcomes of the weight sensitivity analysis are shown in Table 3.

Attribute	Original Weights	Stability Intervals of Weight
		Min	Max
C1	0.25	0.23	0.26
C2	0.27	0.27	0.29
C3	0.25	0.19	0.28
C4	0.23	0	0.23

Table 3

Sensitivity analysis for the weight vector of attributes.

In Table 3, it indicates that the ranking order solved by the proposed method is resistant to change in Example 4 for the different stability intervals of weights. The weight stability interval of the attribute C1 is in [0.23,0.26], C2 in [0.27,0.29], C3 in [0.19,0.28] and C4 in [0,0.23].

6.1.4. Comparison analysis

In this subsection, different decision making approaches are applied to solve Example 4. These decision making approaches include respectively the decision making approach based on the generalized triangle fuzzy number (TFN) [15], the decision making approach based on the extended Interactive and Multiple Attribute Decision Making (TODIM) method [23], the decision making approach based on the power operator [20] and the decision making approach based on the 2DLV with two 2-tuples [17].

Since the decision making approach based on the generalized TFN [15] can only handle MADM problems in 2-dimension linguistic environment, it may as well select the second decision maker's decision matrix R2 to rank the alternatives.

Moreover, the decision making approach based on the extended TODIM method [23] mainly considers 2DULVs. Since a 2DLV is the special case of the corresponding 2DULV, we can replace 2DULVs with 2DLVs in this 2-dimension linguistic computational model.

Similarly, the decision making approach based on the power operator [20] considers 2DULVs. we also replace 2DULVs with 2DLVs in this 2-dimension linguistic computational model.

Then we use the decision making approach based on the 2DLV with two 2-tuples [17] to solve Example 4, it should transform the 2DLV decision matrix into the 2DLV decision matrix with two 2-tuples.

The ranking results by different approaches are shown in Table 4.

Methods	Ranking Results
Yu's method [15] (based on
the 2DLWA operator)	A3<A2<A4<A1
Zhu's method [17] (based
on the 2DLWAA operator)	A3<A2<A4<A1
The proposed method
with p=1,q=0	A3<A2<A4<A1
Liu's method [23] (based on
the extended TODIM method)	A3<A4<A2<A1
Liu's method [20] (based on
the power operator)	A3<A4<A2<A1
The proposed method with p=3,q=3	A3<A4<A2<A1

2DLWA = 2-dimension linguistic weighted averaging; 2DLWAA = 2-dimension linguistic weighted arithmetic aggregation.

Table 4

Ranking results by different methods.

From Table 4, the best alternative is A1 and the worst alternative is A3. While the ranking order of alternatives A2 and A4 may be different from various methods. Obviously, the ranking order of alternatives is the same by Yu's method [15], Zhu's method [17] and the proposed method with p=1,q=0. On the other hand, the ranking order of alternatives is the same by Liu's method [23], Liu's method [20] and the proposed method with p=3,q=3. The reasons are given in the following.

Because Yu's method [15], Zhu's method [17] and the proposed method with p=1,q=0 do not consider the correlations of attributes, it assume that the attributes are independent from each other. Hence the results solved by these three methods are the same.
Liu's method [20] based on the power operator considers the correlations of attributes. Moreover, Liu's method [23] based on the extended TODIM method considers the bounded rationality of decision makers. Furthermore, the proposed method can not only capture the correlations of attributes, but also describe the rationality of decision makers. Therefore, the results solved by Liu's method [20], Liu's method [23] and the proposed method with p=3,q=3 are the same.

In a word, the proposed method with 2DLVs for MAGDM problems has the following advantages. Firstly, the new operations of 2DLVs may be more reasonable than the existing operations. This new operations, that is, the II class LV of 2DLV is regarded as fuzzy number, can precisely describe the decision maker's self-assessment. Secondly, the proposed 2DLWBMA operator can not only capture the interrelationships between attributes, but also describe the rationality of decision makers in MAGDM problems. Moreover, the proposed 2DLWBMA operator can also deal with the attributes which are independent from each other. When taking special parameter values p=1 and q=0 (or p=0 and q=1), the 2DLWBMA operator can be degenerated to the 2-dimension linguistic weighted averaging (2DLWA) operator.

In fact, the decision makers can select suitable parameter values in the proposed method according to the actual decision situations and their experience or knowledge. In other words, taking different parameter values of p and q in the proposed method, the decision makers can flexibly express the degree of correlation between attributes.

6.2. Application to the Landfill Site Selection Problem

By use of the proposed MAGDM method, we can solve the landfill site selection problem which is cited from literature [49]. It concludes that the proposed method can not only handle MAGDM problems within 2-dimension linguistic environment, but also deal with MAGDM problems within fuzzy linguistic environment.

Example 5.

[49] The practical example involves a landfill siting problem in KS City. Considering the dense population of KS City, there are four candidate locations A1,A2,A3,A4. The seven evaluation attributes for landfill site selection are transportation convenience (C1), terrain suitability (C2), community equity (C3), environmental impact (C4), ecological impact (C5), construction cost (C6) and historic impact (C7) respectively, where the attributes C1,C2,C3 are maximizing benefit attributes, and the attributes C4,C5,C6,C7 are minimizing cost attributes. The set of seven evaluation attributes is denoted by C={C1,C2,C3,C4,C5,C6,C7}, with Cbenefit={C1,C2,C3} and Ccost={C4,C5,C6,C7}.

The LTS H={Absolutely low (AL),Very low(VL),

Low(L),Medium low(ML),Medium(M),

Medium high(MH),High(H),Very High(VH),

Absolutely high (AH)} ={h0,h1,h2,h3,h4,h5,h6,h7,h8} is applied to evaluate the four candidate locations A1,A2,A3,A4 with respective to the seven attributes C1,C2,C3,C4,C5,C6,C7 by decision makers. The weight of the seven attributes, also expressed by linguistic terms in H, is w={H,MH,H,MH,MH,M,ML}={h6,h5,h6,h5,h5,h4,h3}. The overall linguistic evaluation decision matrix R=(rij)4×7 is listed as follows:

C1 C2 C3 C4 C5 C6 C7R=A1A2A3A4LAHALLLAHAHALVHHMMLMLLAHALMHAHAHMVHHLHVHMHMH.

6.2.1. Illustration of the proposed method

Example 5 is solved by the proposed method based on the 2DLWBMAp,q operator to rank the candidate locations. The main process is shown as follows:

Step 1 Normalize the 2DLV decision matrix.

In order to apply the proposed MAGDM method to rank the four candidate locations, we assume that the decision maker selects the same linguistic term s2 from the LTS S={Notfamiliar,Familiar,Veryfamilar}={s0,s1,s2} to give his (or her) self-assessment. Here we denote the original 2DLV decision making matrix as Z=(zij)4×7, where zij=(s2,hj).

C1 C2 C3 C4 C5 C6 C7Z=A1A2A3A4(s2,h2)(s2,h8)(s2,h0)(s2,h2)(s2,h2)(s2,h8)(s2,h8)(s2,h0)(s2,h7)(s2,h6)(s2,h4)(s2,h3)(s2,h3)(s2,h2)(s2,h8)(s2,h0)(s2,h5)(s2,h8)(s2,h8)(s2,h4)(s2,h7)(s2,h6)(s2,h2)(s2,h6)(s2,h7)(s2,h5)(s2,h4)(s2,h6)

C1 C2 C3 C4 C5 C6 C7R=A1A2A3A4(s2,h2)(s2,h8)(s2,h0)(s2,h6)(s2,h6)(s2,h0)(s2,h0)(s2,h0)(s2,h7)(s2,h6)(s2,h4)(s2,h5)(s2,h5)(s2,h6)(s2,h8)(s2,h0)(s2,h5)(s2,h0)(s2,h0)(s2,h4)(s2,h1)(s2,h6)(s2,h2)(s2,h6)(s2,h1)(s2,h3)(s2,h4)(s2,h2).

Since the attributes C4,C5,C6,C7 are the minimizing cost attributes, we need to convert the 2DLV decision making matrix Z=(zij)4×7 into R=(rij)4×7, where the elements (s2,hj) in the 4th, 5th, 6th and 7th columns of Z are changed into (s2,h8−j).

The original 2DL decision matrix Z and the normalized 2DL decision matrix R are shown as follows:

Step 2 Calculate the overall evaluation value of each alternative.

Based on the normalized 2DLV decision matrix R=(rij)4×7, we utilize the 2DLWBMAp,q operator to obtain the overall 2-dimension linguistic evaluation values ri of the alternative Ai.

Here let p=3,q=3, then we can compute that r1=(s2,h5.02), r2=(s2,h5.26), r3=(s2,h4.84), r4=(s2,h4.71).

Step 3 Rank ri according to Definition 2.

According to Definition 2, we can obtain the ranking order of ri as r4<r3<r1<r2.

Step 4 Rank all the alternatives and select the best one(s) in accordance with the ranking of ri. The prior the ri is, the best the alternative Ai is.

Thus the ranking order of the alternatives is A4<A3<A1<A2, shown as Figure 3.

6.2.2. Exploration of the parameters influence

For illustrating the influences of the parameters p and q, we take different values p and q in the 2DLWBMAp,q operator in this example. The ranking order of alternatives solved by the the proposed method with different p and q, is shown in Table 5.

The Values of p and q	The Evaluation Values r_i of Alternatives	Ranking Results
p→0,q=0	r1=(s2,h4.84), r2=(s2,h5.75), r3=(s2,h3.06), r4=(s2,h2.83).	A4<A3<A1<A2
p=1,q=0	r1=(s2,h3.29), r2=(s2,h4.53), r3=(s2,h2.85), r4=(s2,h3.65).	A3<A1<A4<A2
p=1,q=1	r1=(s2,h3.01), r2=(s2,h4.43), r3=(s2,h2.44), r4=(s2,h3.5).	A3<A1<A4<A2
p=3,q=3	r1=(s2,h5.02), r2=(s2,h5.26), r3=(s2,h4.84), r4=(s2,h4.71).	A4<A3<A1<A2
p=5,q=5	r1=(s2,h5.7), r2=(s2,h5.68), r3=(s2,h5.79), r4=(s2,h5.5).	A4<A2<A1<A3
p=10,q=10	r1=(s2,h6.35), r2=(s2,h6.3), r3=(s2,h6.71), r4=(s2,h6.37).	A2<A1<A4<A3

Table 5

Ranking orders of the four candidate locations under different values of p and q.

From Table 5, we can see that the relationships among alternatives Ai (i=1,2,3,4) change with various values p and q.

When the 2DLWBMAp,q operator is degenerated to the 2DLWGMA operator, that is p→0 and q=0, the ranking order is A4<A3<A2<A1.

When the 2DLWBMAp,q operator is degenerated to the 2DLWMA operator, that is p=1 and q=0, the ranking order of alternatives is A3<A1<A4<A2, which is the same with when p=1 and q=1.

When p=3 and q=3, we obtain that A4<A3<A2<A1. Similarly, when p=5 and q=5, the ranking order is A4<A2<A1<A3. Finally, when p=10 and q=10, the ranking order is A2<A1<A4<A3.

According to the outcomes of Table 5, it shows that the 2DLWBMAp,q operator plays a key role for considering the interrelationship between the attributes. It shows that the ranking order of alternatives changes with the values of the parameters p and q.

6.2.3. Sensitivity analysis of the weight vector

In order to illustrate the influences of linguistic weight vector on the ranking results, a sensitivity analysis is conducted in Example 5.

In Subsection 6.2.3, we have discussed Memariani's method [51]. Applying Memariani's method, we can also normalize the linguistic weight vector of attributes.

We only discuss the 2DLWBMAp,q operator with p=1,q=1. Table 6 lists the outcomes of the sensitivity analysis for the linguistic weight vector.

Attribute	The Original Linguistic Weight	Stability Intervals of Weight
		Min	Max
C1	H (h6)	L (h2)	AH (h8)
C2	MH (h5)	ML (h3)	AH (h8)
C3	H (h6)	M (h4)	AH (h8)
C4	MH (h5)	ML (h3)	AH (h8)
C5	MH (h5)	ML (h3)	AH (h8)
C6	M (h4)	VL (h1)	VH (h7)
C7	ML (h3)	AL (h0)	AH (h8)

Table 6

Sensitivity analysis for the weight vector of attributes.

From Table 6, we find that the ranking order solved by the proposed method is fully resistant to change in the importance of the attribute C7. Moreover, we obtain that the linguistic weight stability intervals of attributes C2, C4, C5 are the same interval [h3,h8]. While the linguistic weight stability intervals of attributes C1, C3, C6 are different from each other: C1 is in [h2,h8], C3 in [h4,h8], C6 in [h1,h7].

7. CONCLUSIONS

Since a 2DLV adds a class of LV to express the decision maker's self-assessment, it can better express fuzzy information. In order to embody the innate character of the decision maker's self-assessment, this paper defined the new operations of 2DLVs. In the new operations of 2DLVs, the operations of the II class LV which represents the decision maker's self-assessment are based on fuzzy numbers. Moreover, BM operator has a prominent characteristic which can consider the interrelationships of decision arguments. Through adjusting the parameters p and q, the relationships of decision arguments can be flexibly described by BM operator. Based on BM operator and the new operational rules of 2DLVs, this paper developed the 2DLBMA operator. Further, the 2DLWBMA operator was introduced to consider the importance of attribute weights. Subsequently, a novel MAGDM method based on the 2DLWBMA operator was proposed to deal with the 2-dimension linguistic MAGDM problems evaluated by 2DLVs. Finally, two practical examples were given to illustrate the steps of the proposed method. Then the sensitivity analysis of attributes were conducted to obtain the stability intervals of weights in two examples. Moreover, the proposed method was compared with the other relevant methods to show the effectiveness and flexibility of the proposed method.

In future, based on this new operational rules of 2DLVs, other kinds of aggregation operators should be proposed to deal with the corresponding situations in real 2-dimension linguistic decision making. Moreover, the proposed 2DLWBMA operator should be considered to extend into other fuzzy environments for solving MAGDM problems.

CONFLICT OF INTEREST

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

AUTHORS' CONTRIBUTIONS

Jianbin Zhao did the formal analysis, editing, and Software. Hua Zhu was involved with the resources and writing original draft. Both the authors were involved with the conceptualization, methodology, validation, investigation and writing-review.

ACKNOWLEDGMENTS

This work is partially supported by the Natural Science Foundation of China (Grant No. 61673320 and 61772476). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 12 - 2
Pages: 1557 - 1574
Publication Date: 2019/12/06
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.191125.001 How to use a DOI?
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TY  - JOUR
AU  - Jianbin Zhao
AU  - Hua Zhu
PY  - 2019
DA  - 2019/12/06
TI  - 2-Dimension Linguistic Bonferroni Mean Aggregation Operators and Their Application to Multiple Attribute Group Decision Making
JO  - International Journal of Computational Intelligence Systems
SP  - 1557
EP  - 1574
VL  - 12
IS  - 2
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.191125.001
DO  - 10.2991/ijcis.d.191125.001
ID  - Zhao2019
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