# City Sustainable Development Evaluation Based on Hesitant Multiplicative Fuzzy Information

Authors: Xiaorong He; Yingyu Wu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Sustainable development evaluation is the basis of city sustainable development research, and effective evaluation is the foundation for guiding the formulation and implementation of sustainable development strategy. In this paper, we provided a new city sustainable development evaluation method called hesitant multiplicative fuzzy TODIM (HMF-TODIM). The main advantage of this method is that it can deal with the subjective preference information of the decision-makers. The comparison study of existing methods and HMF-TODIM is also carried out. Additionally, real case analysis is presented to show the validity and superiority of the proposed method. Research results in this paper can provide useful information for the construction of sustainable cities.


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## Body

## 1. Introduction

Urbanization brings a series of ecological and environmental problems while bringing about economic and social benefits [1]. For example, urbanization leads to water quality deterioration, resource depletion, air pollution, traffic congestion, and so on. The implementation of sustainable development is the only way for the development of urbanization and the urgent need of the international situation [2]. At present, the study on the evaluation of urban sustainable development is a universal issue [3]. At present, there are many researches on the evaluation of urban sustainable development at home and abroad. Most of the research has been carried out from the aspects of structure, coordination, and continuity of the cities [4]. In addition, most of the research has been using the traditional methods, such as AHP and Delphi. These methods cannot effectively deal with the subjective preferences of decision makers, and most models are difficult to apply to practice. This paper presents a new city sustainable development evaluation method based on hesitant multiplicative fuzzy information.Since Zadeh proposed the theory of fuzzy set [5], it has been extended to intuitionistic fuzzy sets [6–11], hesitant fuzzy sets [12–15], dual hesitant fuzzy set [16–18], and so on. And the above extended fuzzy set has been used in decision-making problems [19–23]. Recently, Xia et al. [24] presented the concept of intuitionistic multiplicative preference relation (IMPR). Xia and Xu [25] proposed the hesitant multiplicative preference relation (HMPR). These two kinds of preference relation are very useful in describing the preference relations of the decision-makers [26–28].As a useful tool for the multicriteria decision-making (MCDM) problem, the TODIM method was first introduced by Gomes and Lima [29, 30] and later widely spread in many fields for solving MCDM problems [31–33]. The TODIM method is based on the Prospect Theory [31] and takes the DMs’ psychological information and emotional preference into consideration. At present, the TODIM method has been extended to intuitionistic fuzzy [34, 35], interval-valued intuitionistic fuzzy [36], and hesitant fuzzy environment [37]. However, the TODIM method has not been studied with HMPR which is the focus of this paper.The remainder of this paper is organized as follows. In Section2, the concepts about hesitant fuzzy preference relation and hesitant multiplicative preference relation are presented; the aggregation operations for hesitant multiplicative values are also provided. In Section 3, we present a distance measure for the hesitant multiplicative elements. Section 4 proposes an extended TODIM method under hesitant multiplicative environment. In Section 5, an illustrative example about city sustainable development evaluation is shown to demonstrate the feasibility of the proposed method. Concluding remarks are presented in Section 6.


## 2. Preliminaries

### 2.1. Hesitant Fuzzy Preference Relation and Hesitant Multiplicative Preference Relation

Inspired by the HFS (hesitant fuzzy set) [12] and FPR (fuzzy preference relation) [38], Liao et al. [39] presented a new concept called HFPR (hesitant fuzzy preference relation).Definition 1 (see [39]).
In a HFPR, we have a set of fixed alternativesA=A1,A2,…,An and then derive a matrix B=hijn×n∈A×A, where hij=hijl∣l=1,2,…,lhij, through comparing every alternative with each other. And each hij is expressed with several hesitant fuzzy values so that DMs can provide their preferences for alternative hi over alternative hj in a hesitancy range. What is more, each hij should satisfy the following conditions:(1)hijσl+hjilhij-l+1=1,hii=0.5,lhij=lhji,where hij=0.5 means there is no difference between alternative i and alternative j, hij<0.5 means the alternative i is superior to alternative j, and the hij>0.5 means the alternative j is superior to alternative i.Based on the proposal of HFPR, an extended concept called HMPR (hesitant multiplicative preference relation), which utilizes a new information expressing method, is presented by Xia and Xu [25]. The only difference between the two concepts is that the HMPR uses Saaty’s 1–9 scale while the HFPR uses the 0.1–0.9 scale. Firstly, we give the definitions of HMS (hesitant multiplicative set) and HME (hesitant multiplicative element) proposed by Zhang and Wu [40]. Then we introduce the concept of HMPR.Definition 2 (see [40]).
A HMSC on a fixed set X is defined as follows:(2)C=x,hCx∣x∈X.hCx is a HME of x∈X and hCx satisfies the following conditions:(3)19≤ξ≤9,∀x∈X,∀ξ∈hCx.Definition 3 (see [25]).
In a HMPR, we have a set of fixed alternativesA~=A1~,A2~,…,An~ and then derive a matrix B~=hij~n×n∈A~×A~, where hij~=hijl~∣l=1,2,…,lhij~, through comparing alternative i with alternative j. And each hij~ should satisfy the following conditions:(4)hijσl~hjilhij-l+1~=1,hii~=1,lhij~=lhji~.We define thathij~ is a HME where hij~∈1/9,9 and hij~=1 means the alternative i is equally preferred to alternative j; 1/9≤hij~<1 means the alternative i is not preferred to alternative j; and 1<hij~≤9 means the alternative i is preferred to alternative j. What is more, hij~=1/9 presents the alternative i is extremely not preferred to alternative j and contrarily hij~=9 presents the alternative i is extremely preferred to alternative j.


### 2.2. Aggregation Operations for Hesitant Multiplicative Values

Aggregation operator is an important research topic in decision-making problems [41–44]. Hesitant multiplicative information aggregation operators were first proposed by Xia and Xu [25]. Then Yu [45] made some changes and presented some extended aggregation operators. In this paper, we use the following operator [45] to aggregate hesitant multiplicative information. The operator is stated as follows:(5)HMFWAρ1,ρ2,…,ρn=w1ρ1⊕w2ρ2⊕⋯⊕wnρn.This operator is called HMFWA (hesitant multiplicative fuzzy weighted averaging) operator andρ=ρ1,ρ2,…,ρn is a set of HMNs (hesitant multiplicative numbers). After that, Yu [45] presented a theorem with a condition that w=w1,w2,…,wn is a set of weight vectors with respect to the HMNs ρ=ρ1,ρ2,…,ρn, where wi∈0,1 and ∑i=1nwi=1. And ηi is the value in the HMN ρi. The theorem is provided below: (6)HMFWAρ1,ρ2,…,ρn=⋃ηi∈ρi∏i=1n1+2ηiwi-12.


## 2.1. Hesitant Fuzzy Preference Relation and Hesitant Multiplicative Preference Relation

Inspired by the HFS (hesitant fuzzy set) [12] and FPR (fuzzy preference relation) [38], Liao et al. [39] presented a new concept called HFPR (hesitant fuzzy preference relation).Definition 1 (see [39]).
In a HFPR, we have a set of fixed alternativesA=A1,A2,…,An and then derive a matrix B=hijn×n∈A×A, where hij=hijl∣l=1,2,…,lhij, through comparing every alternative with each other. And each hij is expressed with several hesitant fuzzy values so that DMs can provide their preferences for alternative hi over alternative hj in a hesitancy range. What is more, each hij should satisfy the following conditions:(1)hijσl+hjilhij-l+1=1,hii=0.5,lhij=lhji,where hij=0.5 means there is no difference between alternative i and alternative j, hij<0.5 means the alternative i is superior to alternative j, and the hij>0.5 means the alternative j is superior to alternative i.Based on the proposal of HFPR, an extended concept called HMPR (hesitant multiplicative preference relation), which utilizes a new information expressing method, is presented by Xia and Xu [25]. The only difference between the two concepts is that the HMPR uses Saaty’s 1–9 scale while the HFPR uses the 0.1–0.9 scale. Firstly, we give the definitions of HMS (hesitant multiplicative set) and HME (hesitant multiplicative element) proposed by Zhang and Wu [40]. Then we introduce the concept of HMPR.Definition 2 (see [40]).
A HMSC on a fixed set X is defined as follows:(2)C=x,hCx∣x∈X.hCx is a HME of x∈X and hCx satisfies the following conditions:(3)19≤ξ≤9,∀x∈X,∀ξ∈hCx.Definition 3 (see [25]).
In a HMPR, we have a set of fixed alternativesA~=A1~,A2~,…,An~ and then derive a matrix B~=hij~n×n∈A~×A~, where hij~=hijl~∣l=1,2,…,lhij~, through comparing alternative i with alternative j. And each hij~ should satisfy the following conditions:(4)hijσl~hjilhij-l+1~=1,hii~=1,lhij~=lhji~.We define thathij~ is a HME where hij~∈1/9,9 and hij~=1 means the alternative i is equally preferred to alternative j; 1/9≤hij~<1 means the alternative i is not preferred to alternative j; and 1<hij~≤9 means the alternative i is preferred to alternative j. What is more, hij~=1/9 presents the alternative i is extremely not preferred to alternative j and contrarily hij~=9 presents the alternative i is extremely preferred to alternative j.


## 2.2. Aggregation Operations for Hesitant Multiplicative Values

Aggregation operator is an important research topic in decision-making problems [41–44]. Hesitant multiplicative information aggregation operators were first proposed by Xia and Xu [25]. Then Yu [45] made some changes and presented some extended aggregation operators. In this paper, we use the following operator [45] to aggregate hesitant multiplicative information. The operator is stated as follows:(5)HMFWAρ1,ρ2,…,ρn=w1ρ1⊕w2ρ2⊕⋯⊕wnρn.This operator is called HMFWA (hesitant multiplicative fuzzy weighted averaging) operator andρ=ρ1,ρ2,…,ρn is a set of HMNs (hesitant multiplicative numbers). After that, Yu [45] presented a theorem with a condition that w=w1,w2,…,wn is a set of weight vectors with respect to the HMNs ρ=ρ1,ρ2,…,ρn, where wi∈0,1 and ∑i=1nwi=1. And ηi is the value in the HMN ρi. The theorem is provided below: (6)HMFWAρ1,ρ2,…,ρn=⋃ηi∈ρi∏i=1n1+2ηiwi-12.


## 3. Distance Measure for HMEs

In this section, we propose a new distance measure for integrating the HMEs in order to combine with the TODIM method. We would set a nonideal HME and derive a series of corresponding values through calculating the distance between the nonideal HME and each HME we picked. According to the definition of the distance measure, the higher the score is, the bigger the difference between two objects is. What is more, the bigger the difference between the ideal HME and the HME we picked is, the better the HME is. So we can realize the target of evaluating each HME via the distance measure and the effect of it. And the effect is similar to the compatibility degree between a pair of intuitionistic multiplicative values presented by Jiang et al. [46]. We concisely represent the distance measure presented by Xu and Xia [47], called hesitant normalized Hamming distance.(7)dhnhM,N=1n∑i=1n1lxi∑j=1lxihMσjxi-hNσjxi.We need to calculate the distance between HMEs; however, this distance measure is for HFSs. In order to fit our method, we made a little change to this distance measure. And then we can calculate the distance between HMEs and present the measure as follows:(8)dh1,h2=1lx∑j=1lxh1σjx-h2σjx,where h1 and h2 are two HMEs. All the values in each HME are in a monotone increasing order and σj is the jth largest value in the element. What is more, lx represents the number of the values in the HME.HMEs and HFEs have many similar characteristics. For instance, they both have a membership degree containing several possible valves to express DMs’ preferences under hesitancy and uncertainty. The only difference is that HMEs are denoted by Saaty’s 1–9 scale so the distance measure is also applicable to the calculation under hesitant multiplicative fuzzy environment.


## 4. Extended TODIM Method under Hesitant Multiplicative Environment

### 4.1. The Classical TODIM Method

Suppose there is a MCDM problem [19, 48–64]; the set of alternatives were represented as A=(A1,A2,…,Am) as well as a set of criteria as shown as C=(C1,C2,…,Cn). Assume one of the criteria as the reference criterion and give the weight vectors denoted as w=(w1,w2,…,wn) with respect to the set of criteria and the weight vectors satisfy wi∈0,1 and ∑i=1nwi=1. Then experts are asked to estimate each alternative. After obtaining the values of the alternatives, each value should be divided by the sum of all the values and then obtain a matrix where the values are normalized [65–73]. The matrix of the values is called the normalized alternatives’ scores against criteria and the matrix is denoted as A=[amn] in Table 1.Table 1
Normalized alternatives’ scores against criteria.
C
1
C
2
⋯
C
n
A
1
a
11
a
12
⋯
a
1
n
A
2
a
21
a
22
⋯
a
2
n
⋮
⋮
⋮
⋱
⋮
A
m
a
m
1
a
m
2
⋯
a
m
nAfter assigning the proper values to the weights and its normalization, DMs have to select a reference criterionwr which is the most vital compared with the other criteria. wrc=wc/wr, wc is the weight of criterion c.Each alternative’s performance should be transformed into the same dimension via usingwrc. And we use the following mathematical expression to calculate the dominance of alternative ai over another alternative aj.(9)δai,aj=∑c=1mΦcai,aj,∀i,jwhen(10)Φcai,aj=wrcaic-ajc∑c=1mwrcaic-ajc>00aic-ajc=0-1θ∑c=1mwrcajc-aicwrcaic-ajc<0.θ is a parameter of the attenuation factor of the losses.Then we need to calculate the global value of alternativeai using the following expression:(11)ξi=∑j=1nδai,aj-mini⁡∑j=1nδai,ajmaxi⁡∑j=1nδai,aj-mini⁡∑j=1nδai,aj.Finally, we can rank the alternatives varying from 1 tom according to the global value, respectively.


### 4.2. An Extended TODIM Method

We plan to present a new TODIM method synthesizing the aggregation operations, distance measures, and classical TODIM method mentioned above. Our innovation is to utilize the TODIM method’s advantages such as considering the DMs’ psychological information and emotional preference, to solve MCDM problems under hesitant multiplicative environment. Zhang and Wu [40] had solved a MCMD problem using some aggregation operations with interval-valued hesitant information. Nevertheless, they only relied on aggregation operations to deal with the alternatives’ performances but not thinking about something psychological. So our method is necessary and valid for this black space.Step 1.
We provide a set of alternatives denoted asA=(A1,A2,…,Am) and several criteria C=(C1,C2,…,Cn). Besides we set weight vectors w=(w1,w2,…,wn) for the criteria with wi∈0,1 and ∑i=1nwi=1. What is more, we need a group of experts E=(e1,e2,…,ek) to give their evaluation using hesitant multiplicative information. According to these, we obtain a HMPR called B=bijm×m constructed by Saaty’s 1–9 scale, where bij represents the dominance degree of alternative i over alternative j. Compared with the ordinary HMPR, there is a little difference in the following step which we use a set of criteria to replace the role of decision organization. In general, we obtain some HMPRs associated with each decision organization, respectively. However, in order to accomplish the new method combined with the TODIM method, we need experts to provide their preferences for each alternative with respect to the criteria. Furthermore due to the restriction of professions, experts are not permitted to evaluate certain alternative associated with a criterion if they know little about it. Note that experts are independent of each other. Then we can obtain a hesitant multiplicative preference relation called Rn=rijnn×n, where rijn denotes all the values given by experts comparing alternative i with alternative j with respect to the criterion n.Step 2.
We use (6) to aggregate every rijnj=1,2,…,m in the HMPR associated with each criterion, respectively, and then derive a series of rin for the alternative i under the criterion n.Step 3.
Set a nonideal element with the minimum hesitant multiplicative value (HMV) calledA∗, where A∗=1/9. Then we calculate the distance between the nonideal alternative and the target element rin we calculated in the last step, via utilizing (8). It is obvious that the bigger the distance value is, the better the performance of the alternative is. According to this, we can obtain a new matrix called D=dinn×n composed of the distance grades din between the nonideal element and each alternative i associated with the criterion n, respectively. And this matrix was presented in Table 2.Table 2
The distance values between alternatives and the nonideal alternative.
C
1
C
2
⋯
C
n
A
1
d
1
1
d
1
2
⋯
d
1
n
A
2
d
2
1
d
2
2
⋯
d
2
n
⋮
⋮
⋮
⋱
⋮
A
m
d
m
1
d
m
2
⋯
d
m
nIn Table2, A=(A1,A2,…,Am) and C=(C1,C2,…,Cn) indicate the alternatives and criteria we selected, respectively.Step 4.
We utilize the TODIM method to deal with the matrixD=dinn×n. Through the process, we can obtain the global values of the alternatives and get the rank ordering for these alternatives. Specific processes are as follows:(i)
Calculate the dominance of alternativedin over another alternative djn corresponding to each criterion after determining the reference criterion wr, via the expression(12)δdi,dj=∑c=1mΦcdi,dj,∀i,jwhen(13)Φcdi,dj=wrcdin-djn∑c=1mwrcdin-djn>00din-djn=0-1θ∑c=1mwrcdjn-dinwrcdin-djn<0.θ is a parameter to reflect the attenuation of the losses.(ii)
Calculate the global values of each alternative using the following formula:(14)ξi=∑j=1nδdi,dj-mini⁡∑j=1nδdi,djmaxi⁡∑j=1nδdi,dj-mini⁡∑j=1nδdi,dj.


## 4.1. The Classical TODIM Method

Suppose there is a MCDM problem [19, 48–64]; the set of alternatives were represented as A=(A1,A2,…,Am) as well as a set of criteria as shown as C=(C1,C2,…,Cn). Assume one of the criteria as the reference criterion and give the weight vectors denoted as w=(w1,w2,…,wn) with respect to the set of criteria and the weight vectors satisfy wi∈0,1 and ∑i=1nwi=1. Then experts are asked to estimate each alternative. After obtaining the values of the alternatives, each value should be divided by the sum of all the values and then obtain a matrix where the values are normalized [65–73]. The matrix of the values is called the normalized alternatives’ scores against criteria and the matrix is denoted as A=[amn] in Table 1.Table 1
Normalized alternatives’ scores against criteria.
C
1
C
2
⋯
C
n
A
1
a
11
a
12
⋯
a
1
n
A
2
a
21
a
22
⋯
a
2
n
⋮
⋮
⋮
⋱
⋮
A
m
a
m
1
a
m
2
⋯
a
m
nAfter assigning the proper values to the weights and its normalization, DMs have to select a reference criterionwr which is the most vital compared with the other criteria. wrc=wc/wr, wc is the weight of criterion c.Each alternative’s performance should be transformed into the same dimension via usingwrc. And we use the following mathematical expression to calculate the dominance of alternative ai over another alternative aj.(9)δai,aj=∑c=1mΦcai,aj,∀i,jwhen(10)Φcai,aj=wrcaic-ajc∑c=1mwrcaic-ajc>00aic-ajc=0-1θ∑c=1mwrcajc-aicwrcaic-ajc<0.θ is a parameter of the attenuation factor of the losses.Then we need to calculate the global value of alternativeai using the following expression:(11)ξi=∑j=1nδai,aj-mini⁡∑j=1nδai,ajmaxi⁡∑j=1nδai,aj-mini⁡∑j=1nδai,aj.Finally, we can rank the alternatives varying from 1 tom according to the global value, respectively.


## 4.2. An Extended TODIM Method

We plan to present a new TODIM method synthesizing the aggregation operations, distance measures, and classical TODIM method mentioned above. Our innovation is to utilize the TODIM method’s advantages such as considering the DMs’ psychological information and emotional preference, to solve MCDM problems under hesitant multiplicative environment. Zhang and Wu [40] had solved a MCMD problem using some aggregation operations with interval-valued hesitant information. Nevertheless, they only relied on aggregation operations to deal with the alternatives’ performances but not thinking about something psychological. So our method is necessary and valid for this black space.Step 1.
We provide a set of alternatives denoted asA=(A1,A2,…,Am) and several criteria C=(C1,C2,…,Cn). Besides we set weight vectors w=(w1,w2,…,wn) for the criteria with wi∈0,1 and ∑i=1nwi=1. What is more, we need a group of experts E=(e1,e2,…,ek) to give their evaluation using hesitant multiplicative information. According to these, we obtain a HMPR called B=bijm×m constructed by Saaty’s 1–9 scale, where bij represents the dominance degree of alternative i over alternative j. Compared with the ordinary HMPR, there is a little difference in the following step which we use a set of criteria to replace the role of decision organization. In general, we obtain some HMPRs associated with each decision organization, respectively. However, in order to accomplish the new method combined with the TODIM method, we need experts to provide their preferences for each alternative with respect to the criteria. Furthermore due to the restriction of professions, experts are not permitted to evaluate certain alternative associated with a criterion if they know little about it. Note that experts are independent of each other. Then we can obtain a hesitant multiplicative preference relation called Rn=rijnn×n, where rijn denotes all the values given by experts comparing alternative i with alternative j with respect to the criterion n.Step 2.
We use (6) to aggregate every rijnj=1,2,…,m in the HMPR associated with each criterion, respectively, and then derive a series of rin for the alternative i under the criterion n.Step 3.
Set a nonideal element with the minimum hesitant multiplicative value (HMV) calledA∗, where A∗=1/9. Then we calculate the distance between the nonideal alternative and the target element rin we calculated in the last step, via utilizing (8). It is obvious that the bigger the distance value is, the better the performance of the alternative is. According to this, we can obtain a new matrix called D=dinn×n composed of the distance grades din between the nonideal element and each alternative i associated with the criterion n, respectively. And this matrix was presented in Table 2.Table 2
The distance values between alternatives and the nonideal alternative.
C
1
C
2
⋯
C
n
A
1
d
1
1
d
1
2
⋯
d
1
n
A
2
d
2
1
d
2
2
⋯
d
2
n
⋮
⋮
⋮
⋱
⋮
A
m
d
m
1
d
m
2
⋯
d
m
nIn Table2, A=(A1,A2,…,Am) and C=(C1,C2,…,Cn) indicate the alternatives and criteria we selected, respectively.Step 4.
We utilize the TODIM method to deal with the matrixD=dinn×n. Through the process, we can obtain the global values of the alternatives and get the rank ordering for these alternatives. Specific processes are as follows:(i)
Calculate the dominance of alternativedin over another alternative djn corresponding to each criterion after determining the reference criterion wr, via the expression(12)δdi,dj=∑c=1mΦcdi,dj,∀i,jwhen(13)Φcdi,dj=wrcdin-djn∑c=1mwrcdin-djn>00din-djn=0-1θ∑c=1mwrcdjn-dinwrcdin-djn<0.θ is a parameter to reflect the attenuation of the losses.(ii)
Calculate the global values of each alternative using the following formula:(14)ξi=∑j=1nδdi,dj-mini⁡∑j=1nδdi,djmaxi⁡∑j=1nδdi,dj-mini⁡∑j=1nδdi,dj.


## 5. Case Studies

Sustainable development evaluation is the basis of urban sustainable development research, and correct evaluation is an important basis for guiding the formulation and implementation of sustainable development strategy. At the same time, due to the specificity and complexity of the city, the sustainable development of different cities is difficult to use the same indicators. Therefore, the sustainable development evaluation is both a hot and difficult point in the field of research.Starting from the three aspects of economy (c1), society (c2), and resources and environment (c3), the sustainable development level and ability of three cities X=(x1,x2,x3) in Zhejiang province, China, are evaluated.We invite three expertsE=(e1,e2,e3) to give their assessment with respective to the three cities associated with different criteria. What is more, each expert is not disturbed by each other.First we obtain three matrices denoted by the HMPR for each criterion calledR=R1,R2,R3. The matrices R1,R2,R3 are the HMPRs associated with the three cities under c1 (economy), c2 (society), and c3 (resources and environment), respectively. The values in the HMPRs are the primary data provided by the experts and we present them in Tables 3–5.Table 3
The hesitant multiplicative preference relationR1.
x
1
x
2
x
3
x
1
{
1
}
{
3
/
2,2
}
{
2,3
,
4
}
x
2
{
1
/
2,2
/
3
}
{
1
}
{
3,4
}
x
3
{
1
/
4,1
/
3,1
/
2
}
{
1
/
4,1
/
3
}
{
1
}Table 4
The hesitant multiplicative preference relationR2.
x
1
x
2
x
3
x
1
{
1
}
{
3
/
2,3
/
4,5
/
6
}
{
3
/
2,2
,
3
}
x
2
{
6
/
5,4
/
3,2
}
{
1
}
{
1
/
5,1
/
3,1
/
2
}
x
3
{
1
/
3,1
/
2,2
/
3
}
{
2,3
,
5
}
{
1
}Table 5
The hesitant multiplicative preference relationR3.
x
1
x
2
x
3
x
1
{
1
}
{
2,3
}
{
6,7
,
8
}
x
2
{
1
/
3,1
/
2
}
{
1
}
{
2,3
,
4
}
x
3
{
1
/
8,1
/
7,1
/
6
}
{
1
/
4,1
/
3,1
/
2
}
{
1
}(i) Then we use the aggregation operations to deal with the primary data via (6) and derive nine results of each alternative with respect to the criteria denoted as rin, where i means the alternative i and n means the criterion n.(15)r11=1.4574,1.6897,1.8811,1.6086,1.8588,2.0650,r21=1.2380,1.3899,1.3297,1.4895,r31=0.4449,0.4787,0.4787,0.5317,0.5400,0.5772,r12=1.3333,1.4574,1.6898,1.0536,1.1736,1.3722,1.0874,1.2100,1.4129,r22=0.7131,0.7856,0.8662,0.7440,0.8184,0.9010,0.8795,0.9620,1.0536,r32=0.9620,1.1356,1.4015,1.0536,1.2380,1.5206,1.1355,1.3297,1.6272,r13=2.3994,2.5411,2.6707,2.7436,2.99020,3.0470,r23=0.9620,1.1355,1.2784,1.0536,1.2380,1.3900,r33=0.3892,0.4210,0.4787,0.3976,0.4297,0.4880,0.4086,0.4410,0.500.(ii) In general cases of group decision-making (GDM) problem, lots of researchers use a weighted aggregation operation to aggregaterin and obtain ri which means the global value of alternative i over the other alternatives. Even Zhang and Wu [40] did the same work in a MCMD problem based on the interval-valued hesitant multiplicative preference relation. However we do not use this operation but lead a distance measure (8) into this step to calculate the distance between the alternatives we picked and a nonideal alternative A∗=1/9. So we can illustrate whether an alternative’s value is bigger than another one through calculating their distance value din. And we obtain a matrix D=dinn×n composed of the distance values in Table 6.Table 6
The distance values between the three companies and the nonideal alternative.
x
1
x
2
x
3
x
1
1.6490
1.1971
2.6062
x
2
1.2507
0.7470
1.0651
x
3
0.3944
1.5596
0.3282(iii) The distance values ofx1 are bigger than those of x2 and x3 associated with c1 and c3. And the distance value of x3 associated with c2 ranks first. As we mentioned in Section 4.2, the bigger the distance value is, the better the alternative’s performance is. Therefore the result we obtained in Table 7 means the performance of x1 associated with c1 and c3 is best, while the performance of x3 related to c2 is better than the other two.Table 7
The dominance degrees of the three companies over each other.
x
1
x
2
x
3
x
1
1
1.3689
1.5781
x
2
-
4.8933
1
-
0.1293
x
3
-
5.3293
-
2.8779
1(iv) We use the TODIM method to deal with the matrix calculated above and gain the dominance degree of each alternative via (12). We present the dominance degree in Table 7. And then we can get the global valves of all the alternatives which are presented in Table 8.Table 8
The global values of the three companies and their rank ordering.
Global value
Rank
x
1
1
1
x
2
0.2855
2
x
3
0
3From the result, it is obvious thatx1 is ranked first. We can roughly derive that x1 possesses the biggest distance value with the nonideal alternative compared with the other two in Table 7. As we mentioned above, the bigger the distance value is, the better the alternative’s performance is. Therefore, after calculating the distance values between the three companies and the nonideal alternative, the dominant position of x1 is relatively manifest. And in Table 7, the superiority of x1 is more obvious because only the dominance degree of x1 is a positive number while the dominance degrees of the other two are negative numbers. And the definition of the dominance degree is the comparison of two alternatives’ advantages and disadvantages. Therefore we can infer that x1 is the best from the result of calculating the dominance degree.


### 5.1. Comparison between the New Method and an Aggregation Operation Method

In this subsection, we plan to do a comparison between our new method, HMF-TODIM, and an existing method based on aggregation operations proposed by Yu [45] which is provided as follows.Step 1.
Use (6) to aggregate every rijnj=1,2,…,m in the HMPRs associated with each criterion in Tables 3–5, respectively. And then derive a series of rin for the alternative i under the criterion n. This step is the same as the HMF-TODIM method.Step 2.
Through utilizing (6) again, we aggregate each rin to derive every alternative’s global value ri. The global value indicates the performance of each alternative related to the all criteria under the hesitant multiplicative environment.Step 3.
According to a score function defined by Xia [74], identify the sorting of the alternatives’ values. And the score function is given as follows:(16)sr=∏η∈hη1/Δh.Noting thatΔh is the number of the values of r, if sr1>sr2, then r1>r2.It is obvious to see that Step  1 of this method is the same as Step  2 of the HMF-TODIM method. Therefore we can obtain the same values to the set ofrin:(17)r11=1.4574,1.6897,1.8811,1.6086,1.8588,2.0650,r21=1.2380,1.3899,1.3297,1.4895,r31=0.4449,0.4787,0.4787,0.5317,0.5400,0.5772,r12=1.3333,1.4574,1.6898,1.0536,1.1736,1.3722,1.0874,1.2100,1.4129,r22=0.7131,0.7856,0.8662,0.7440,0.8184,0.9010,0.8795,0.9620,1.0536,r32=0.9620,1.1356,1.4015,1.0536,1.2380,1.5206,1.1355,1.3297,1.6272,r13=2.3994,2.5411,2.6707,2.7436,2.99020,3.0470,r23=0.9620,1.1355,1.2784,1.0536,1.2380,1.3900,r33=0.3892,0.4210,0.4787,0.3976,0.4297,0.4880,0.4086,0.4410,0.500.Then we aggregate the set ofrin to obtain the global values ri via (6). Considering the enormous calculation process, we do not put the calculation result of Step  2 in our paper.Finally, we identify the sorting of the three alternatives via the score function.s
r
1
=
1.7822, sr2=1.1136, and sr3=0.6681. Since sr1>sr2>sr3, then r1>r2>r3, and we present the score values related to the alternatives in Table 9.Table 9
The score values and the rank ordering concerning the three alternatives.
Score value
Rank
x
1
1.7822
1
x
2
1.1136
2
x
3
0.6681
3


## 5.1. Comparison between the New Method and an Aggregation Operation Method

In this subsection, we plan to do a comparison between our new method, HMF-TODIM, and an existing method based on aggregation operations proposed by Yu [45] which is provided as follows.Step 1.
Use (6) to aggregate every rijnj=1,2,…,m in the HMPRs associated with each criterion in Tables 3–5, respectively. And then derive a series of rin for the alternative i under the criterion n. This step is the same as the HMF-TODIM method.Step 2.
Through utilizing (6) again, we aggregate each rin to derive every alternative’s global value ri. The global value indicates the performance of each alternative related to the all criteria under the hesitant multiplicative environment.Step 3.
According to a score function defined by Xia [74], identify the sorting of the alternatives’ values. And the score function is given as follows:(16)sr=∏η∈hη1/Δh.Noting thatΔh is the number of the values of r, if sr1>sr2, then r1>r2.It is obvious to see that Step  1 of this method is the same as Step  2 of the HMF-TODIM method. Therefore we can obtain the same values to the set ofrin:(17)r11=1.4574,1.6897,1.8811,1.6086,1.8588,2.0650,r21=1.2380,1.3899,1.3297,1.4895,r31=0.4449,0.4787,0.4787,0.5317,0.5400,0.5772,r12=1.3333,1.4574,1.6898,1.0536,1.1736,1.3722,1.0874,1.2100,1.4129,r22=0.7131,0.7856,0.8662,0.7440,0.8184,0.9010,0.8795,0.9620,1.0536,r32=0.9620,1.1356,1.4015,1.0536,1.2380,1.5206,1.1355,1.3297,1.6272,r13=2.3994,2.5411,2.6707,2.7436,2.99020,3.0470,r23=0.9620,1.1355,1.2784,1.0536,1.2380,1.3900,r33=0.3892,0.4210,0.4787,0.3976,0.4297,0.4880,0.4086,0.4410,0.500.Then we aggregate the set ofrin to obtain the global values ri via (6). Considering the enormous calculation process, we do not put the calculation result of Step  2 in our paper.Finally, we identify the sorting of the three alternatives via the score function.s
r
1
=
1.7822, sr2=1.1136, and sr3=0.6681. Since sr1>sr2>sr3, then r1>r2>r3, and we present the score values related to the alternatives in Table 9.Table 9
The score values and the rank ordering concerning the three alternatives.
Score value
Rank
x
1
1.7822
1
x
2
1.1136
2
x
3
0.6681
3


## 6. Conclusion

In this paper, we proposed a new TODIM method called HMF-TODIM method and applied it to the field of city sustainable development evaluation. The use of asymmetric information can solve more practical problems in the real world. Expressing the DMs’ preferences with asymmetric information is more scientific and closer to the reality because the DMs’ preference degrees may increase irregularly. For instance, the growth range from the extremely nonpreferred one to the equally preferred one may be smaller than that from the extremely preferred one to the equally preferred one. In addition, we also considered the hesitancy and uncertainty during decision-making process. A real case analysis is presented to show the validity and superiority of the proposed method. In the future study, we intend to apply the proposed methods to the field of energy [75–78] and supplier evaluation and selection [79–81].



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*Source: 8306508-2017-10-08.xml*