City Sustainable Development Evaluation Based on Hesitant Multiplicative Fuzzy Information

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.


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.
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][32][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 Section 2, 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
Definition 1 (see [39]).In a HFPR, we have a set of fixed alternatives  = ( 1 ,  2 , . . .,   ) and then derive a matrix  = (ℎ  ) × ∈  × , where ℎ  = {ℎ   |  = 1, 2, . . .,  ℎ  }, through comparing every alternative with each other.And each ℎ  is expressed with several hesitant fuzzy values so that DMs can provide their preferences for alternative ℎ  over alternative ℎ  in a hesitancy range.What is more, each ℎ  should satisfy the following conditions: where ℎ  = 0.5 means there is no difference between alternative  and alternative , ℎ  < 0.5 means the alternative  is superior to alternative , and the ℎ  > 0.5 means the alternative  is superior to alternative .
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 0.1-0.9scale.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 HMS  on a fixed set  is defined as follows: ℎ  () is a HME of  ∈  and ℎ  () satisfies the following conditions: Definition 3 (see [25]).In a HMPR, we have a set of fixed alternatives Ã = ( Ã1 , Ã2 , . . ., Ã ) and then derive a matrix B = ( h ) × ∈ Ã × Ã, where h = { h  |  = 1, 2, . . ., lℎ  }, through comparing alternative  with alternative .And each h should satisfy the following conditions: We define that h is a HME where h ∈ [1/9, 9] and h = 1 means the alternative  is equally preferred to alternative ; 1/9 ≤ h < 1 means the alternative  is not preferred to alternative ; and 1 < h ≤ 9 means the alternative  is preferred to alternative .What is more, h = {1/9} presents the alternative  is extremely not preferred to alternative  and contrarily h = {9} presents the alternative  is extremely preferred to alternative .

Aggregation Operations for Hesitant Multiplicative Values.
Aggregation operator is an important research topic in decision-making problems [41][42][43][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: This operator is called HMFWA (hesitant multiplicative fuzzy weighted averaging) operator and  = ( 1 ,  2 , . . .,   ) is a set of HMNs (hesitant multiplicative numbers).After that, Yu [45] presented a theorem with a condition that  = ( 1 ,  2 , . . .,   ) is a set of weight vectors with respect to the HMNs  = ( 1 ,  2 , . . .,   ), where   ∈ [0, 1] and ∑  =1   = 1.And   is the value in the HMN   .The theorem is provided below:

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.
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: where ℎ 1 and ℎ 2 are two HMEs.All the values in each HME are in a monotone increasing order and () is the th largest value in the element.What is more,   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.
After assigning the proper values to the weights and its normalization, DMs have to select a reference criterion   which is the most vital compared with the other criteria.  =   /  ,   is the weight of criterion .
Each alternative's performance should be transformed into the same dimension via using   .And we use the following mathematical expression to calculate the dominance of alternative   over another alternative   .

𝛿 (𝑎
is a parameter of the attenuation factor of the losses.
Then we need to calculate the global value of alternative   using the following expression: Finally, we can rank the alternatives varying from 1 to  according to the global value, respectively.

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.
(2)    ⋅ ⋅ ⋅  ()  Saaty's 1-9 scale, where   represents the dominance degree of alternative  over alternative .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   = (  () ) × , where   () denotes all the values given by experts comparing alternative  with alternative  with respect to the criterion .
Step 2. We use (6) to aggregate every   () ( = 1, 2, . . ., ) in the HMPR associated with each criterion, respectively, and then derive a series of  ()  for the alternative  under the criterion .
Step 3. Set a nonideal element with the minimum hesitant multiplicative value (HMV) called  * , where  * = {1/9}.Then we calculate the distance between the nonideal alternative and the target element  ()   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  = (  () ) × composed of the distance grades  ()  between the nonideal element and each alternative  associated with the criterion , respectively.And this matrix was presented in Table 2.
Step 4. We utilize the TODIM method to deal with the matrix  = (  () ) × .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 alternative   () over another alternative   () corresponding to each criterion after determining the reference criterion   , via the expression when is a parameter to reflect the attenuation of the losses.
(ii) Calculate the global values of each alternative using the following formula:

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 ( 1 ), society ( 2 ), and resources and environment ( 3 ), the sustainable development level and ability of three cities  = ( 1 ,  2 ,  3 ) in Zhejiang province, China, are evaluated.
We invite three experts  = ( 1 ,  2 ,  3 ) 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 called  = ( 1 ,  2 ,  3 ).The matrices  1 ,  2 ,  3 are the HMPRs associated with the three cities under  1 (economy),  2 (society), and  3 (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.(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  ()   , where  means the alternative  and  means the criterion .
(1)  1 = {1.4574,1.6897, 1.8811, 1.6086, 1.8588, 2.0650} , {0.7131, 0.7856, 0.8662, 0.7440, 0.8184, 0.9010, 0.8795, 0.9620, 1.0536} , (ii) In general cases of group decision-making (GDM) problem, lots of researchers use a weighted aggregation operation to aggregate  ()  and obtain   which means the global value of alternative  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  * = {1/9}.So we can illustrate whether an alternative's value is bigger than another one through calculating their distance value  ()   .And we obtain a matrix  = { ()  } × composed of the distance values in Table 6.
(iii) The distance values of  1 are bigger than those of  2 and  3 associated with  1 and  3 .And the distance value of  3 associated with  2 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  1 associated with  1 and  3 is best, while the performance of  3 related to  2 is better than the other two.
(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.
From the result, it is obvious that  1 is ranked first.We can roughly derive that  1 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  1 is relatively manifest.And in Table 7, the superiority of  1 is more obvious because only the dominance degree of  1 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  1 is the best from the result of calculating the dominance degree.

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   () ( = 1, 2, . . ., ) in the HMPRs associated with each criterion in Tables 3-5, respectively.And then derive a series of  ()   for the alternative  under the criterion .This step is the same as the HMF-TODIM method.
Step 2. Through utilizing (6) again, we aggregate each  ()  to derive every alternative's global value   .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: Noting that Δℎ is the number of the values of , if ( 1 ) > ( 2 ), then  1 >  2 .
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 of  ()   : Then we aggregate the set of  ()  to obtain the global values   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.

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][76][77][78] and supplier evaluation and selection [79][80][81].

Table 6 :
The distance values between the three companies and the nonideal alternative.

Table 7 :
The dominance degrees of the three companies over each other.

Table 8 :
The global values of the three companies and their rank ordering.

Table 9 :
The score values and the rank ordering concerning the three alternatives.