Next Article in Journal
Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients
Next Article in Special Issue
Fuzzy Model for Risk Assessment of Machinery Failures
Previous Article in Journal
A Multichannel Data Fusion Method Based on Multiple Deep Belief Networks for Intelligent Fault Diagnosis of Main Reducer
Previous Article in Special Issue
On Two Classes of Soft Sets in Soft Topological Spaces

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

# An EDAS Method for Multiple Attribute Group Decision-Making under Intuitionistic Fuzzy Environment and Its Application for Evaluating Green Building Energy-Saving Design Projects

by
Yuan Liang
School of Civil Engineering and Architecture, Northeast Petroleum University, Daqing 163318, China
Symmetry 2020, 12(3), 484; https://doi.org/10.3390/sym12030484
Submission received: 17 February 2020 / Revised: 7 March 2020 / Accepted: 15 March 2020 / Published: 23 March 2020

## Abstract

:
Multiple attribute group decision-making (MAGDM) methods have a significant influence on decision-making in a variety of strategic fields, including science, business and real-life studies. The problem of evaluation in green building energy-saving design projects could be regarded as a type of MAGDM problem. The evaluation based on distance from average solution (EDAS) method is one of the MAGDM methods, which simplifies the traditional decision-making process. Symmetry among some attributes that are known and unknown as well as between pure attribute sets and fuzzy attribute membership sets can be an effective way to solve MAGDM problems. In this paper, the classical EDAS method is extended to intuitionistic fuzzy environments to solve some MAGDM issues. First, some concepts of intuitionistic fuzzy sets (IFSs) are briefly reviewed. Then, by integrating the EDAS method with IFSs, we establish an IF-EDAS method to solve the MAGDM issues and present all calculating procedures in detail. Finally, we provide an empirical application for evaluating green building energy-saving design projects to demonstrate this novel method. Some comparative analyses are also made to show the merits of the method.

## 1. Introduction

There are various issues regarding uncertainty and vagueness that can impact the process of decision-making [1,2,3,4]. Thus, in order to improve the accuracy of decision-making, Zadeh [5] initially presented the theory of fuzzy sets (FSs). Atanassov [6] introduced the concept of intuitionistic fuzzy sets (IFSs). Gou et al. [7] pointed out a novel exponential operational law about IFNs (Intuitionistic Fuzzy Numbers) and offered a method used to aggregate intuitionistic fuzzy information. Li and Wu [8] presented a comprehensive decision method based on the intuitionistic fuzzy cross entropy distance and the grey correlation analysis. Khan, Lohani and Ieee [9] put forward a novel similarity measure about IFNs depending on the distance measure of a double sequence of a bounded variation. Li et al. [10] developed a grey target decision-making method in the form of IFNs on the basis of grey relational analysis [11]. Chen et al. [12] developed a novel MCDM (Multiple Criteria Decision Making) method on the basis of the TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method and similarity measures in the context of intuitionistic fuzzy. Gupta et al. [13] modified the superiority and inferiority ranking (SIR) method and combined it under IFSs. Lu and Wei [14] designed the TODIM (an acronym in Portuguese for Interactive Multi-criteria Decision Making) method for performance appraisal on social-integration-based rural reconstruction under IVIFSs (Interval-valued Intuitionistic Fuzzy Numbers). Wu et al. [15] provided the VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje) method for financing risk assessment of rural tourism projects under IVIFSs (Interval-values Intuitionistic Fuzzy Sets). Wu et al. [16] proposed some interval-valued intuitionistic fuzzy Dombi Heronian mean operators for evaluating the ecological value of forest ecological tourism demonstration areas. Wu et al. [17] designed the algorithms for competitiveness evaluation of tourist destinations with some interval-valued intuitionistic fuzzy Hamy mean operators.
Ghorabaee et al. [18] designed a novel method called evaluation based on distance from average solution (EDAS) to tackle multi-criteria inventory classification (MCIC) issues. Ghorabaee et al. [19] modified the EDAS method to tackle supplier selection issues. Zhang et al. [20] provided the EDAS method for MCGDM (Multi-Criteria Group Decision Making) issues with picture fuzzy information. Peng and Liu [21] designed the neutrosophic soft decision-making algorithms on the basis of EDAS and novel similarity measures. Feng et al. [22] integrated the EDAS method with an extended hesitant fuzzy linguistic environment. He et al. [23] designed the EDAS method for MAGDM with probabilistic uncertain linguistic information. Karasan and Kahraman [24] designed a novel interval-valued neutrosophic EDAS method. Li et al. [25] defined the EDAS method for MAGDM issues under a q-rung orthopair fuzzy environment. Wang et al. [26] proposed the EDAS method for MAGDM under a 2-tuple linguistic neutrosophic environment. Ghorabaee et al. [27] presented the EDAS method with normally distributed data to tackle stochastic issues. Zhang et al. [28] extended the EDAS method to picture a 2-tuple linguistic environment. Li et al. [29] developed a novel method by extending the traditional EDAS method to picture fuzzy environment.
To the authors’ knowledge, there is no research available which investigates the EDAS method based on the criteria importance using the CRiteria Importance Through Intercriteria Correlation (CRITIC) method with IFNs. Therefore, investigating an EDAS method with IFNs is a suitable research topic. The fundamental objective of our research was to develop an original method that could be used more effectively to address some MAGDM issues in the context of the EDAS method and IFNs. Thus, the main contribution of this paper can be outlined as follows: (1) The EDAS method was modified in the intuitionistic fuzzy environment; (2) the CRITIC method was used to derive the attributes’ weights; (3) the EDAS method under an intuitionistic fuzzy environment was proposed to solve the MAGDM issues; (4) an application for evaluating green building energy-saving design projects was provided to show the superiority of this novel method, and a comparative analysis between the IF-EDAS method and other methods was also used to further verify the merits of this method. Some fundamental knowledge of IFSs is concisely reviewed in Section 2. The extended EDAS method was integrated with IFNs and the calculating procedures are depicted in Section 3. An empirical application for evaluating green building energy-saving design projects is provided to show the superiority of this approach, and some comparative analyses are also offered to further show the merits of this method in Section 4. Finally, we provide an overall conclusion of our work in Section 5.

## 2. Preliminaries

#### Intuitionistic Fuzzy Sets

Definition 1
[6]. An intuitionistic fuzzy set (IFS) on the universe $X$ is an object of the form
$I = { 〈 x , μ I ( x ) , ν I ( x ) 〉 | x ∈ X }$
where $μ I ( x ) ∈ [ 0 , 1 ]$ is called the “degree of membership of $I$” and $ν I ( x ) ∈ [ 0 , 1 ]$ is called the “degree of non-membership of $I$”, and $μ I ( x )$, $ν I ( x )$ satisfy the following condition: $0 ≤ μ I ( x ) + ν I ( x ) ≤ 1$, .
Definition 2
[30]. Let $I 1 = ( μ 1 , ν 1 )$ and $I 2 = ( μ 2 , ν 2 )$ be two intuitionistic fuzzy numbers (IFNs); the operation formula can then be defined as:
$I 1 ⊕ I 2 = ( μ 1 + μ 2 − μ 1 μ 2 , ν 1 ν 2 )$
$I 1 ⊗ I 2 = ( μ 1 μ 2 , ν 1 + ν 2 − ν 1 ν 2 )$
$λ I 1 = ( 1 − ( 1 − μ 1 ) λ , ν 1 λ ) , λ > 0$
$I 1 λ = ( μ 1 λ , 1 − ( 1 − ν 1 ) λ ) , λ > 0$
Definition 3
[31]. Let $I 1 = ( μ 1 , ν 1 )$ and $I 2 = ( μ 2 , ν 2 )$ be IFNs; the score and accuracy functions of $I 1$ and $I 2$ can then be expressed as:
$S ( I 1 ) = μ 1 + μ 1 ( 1 − μ 1 − ν 1 ) , S ( I 2 ) = μ 2 + μ 2 ( 1 − μ 2 − ν 2 )$
$H ( I 1 ) = μ 1 + ν 1 , H ( I 2 ) = μ 2 + ν 2$
For the two IFNs $I 1$ and $I 2$, regarding Definition 3, then:
Under the context of the IFSs, some aggregation operators are introduced in this section, including an intuitionistic fuzzy weighted averaging (IFWA) operator and an intuitionistic fuzzy weighted geometric (IFWG) operator.
Definition 4.
[30]. Let $I j = ( μ I j , ν I j ) ( j = 1 , 2 , ⋯ , n )$ be a collection of IFNs; the intuitionistic fuzzy weighted averaging (IFWA) operator can then be defined as:
$I F W A ω ( I 1 , I 2 , … , I n ) = ⊕ j = 1 n ( ω j I j )$
where $ω = ( ω 1 , ω 2 , … , ω n ) T$ is the weight vector of $I j ( j = 1 , 2 , … , n )$ and $ω j > 0 , ∑ j = 1 n ω j = 1$.
From Definition 4, the following result can be obtained:
Theorem 1.
The aggregated value using an IFWA operator is also an IFN, where
$I F W A ω ( I 1 , I 2 , … , I n ) = ⊕ j = 1 n ( ω j I j ) = ( 1 − ∏ j = 1 n ( 1 − μ I j ) ω j , ∏ j = 1 n ( ν I j ) ω j )$
where$ω = ( ω 1 , ω 2 , … , ω n ) T$be the weight vector of$I j ( j = 1 , 2 , … , n )$and$ω j > 0 , ∑ j = 1 n ω j = 1$.
Definition 5
[30]. Let $I j ( j = 1 , 2 , ⋯ , n )$ be a collection of IFNs; the intuitionistic fuzzy weighted geometric (IFWG) operator can then be defined as:
$I F W G ω ( I 1 , I 2 , … , I n ) = ⊗ j = 1 n ( I j ) ω j$
where $ω = ( ω 1 , ω 2 , … , ω n ) T$ is the weight vector of $I j ( j = 1 , 2 , … , n )$ and $ω j > 0 , ∑ j = 1 n ω j = 1$.
Derived from Definition 5, the following result can be obtained:
Theorem 2.
The aggregated value using an IFWG operator is also an IFN, where
$I F W G ω ( I 1 , I 2 , … , I n ) = ⊗ j = 1 n ( I j ) ω j = ( ∏ j = 1 n ( μ I j ) ω j , 1 − ∏ j = 1 n ( 1 − ν I j ) ω j )$
where$ω = ( ω 1 , ω 2 , … , ω n ) T$is the weight vector of$I j ( j = 1 , 2 , … , n )$and$ω j > 0 , ∑ j = 1 n ω j = 1$.

## 3. The EDAS Method with Intuitionistic Fuzzy Information

Integrating the EDAS method with IFSs, we built the IF-EDAS method in which the assessment values were given by IFNs. The calculating procedures of the developed method are described below. Let $Z = { Z 1 , Z 2 , … Z n }$ be the set of attributes, $z = { z 1 , z 2 , … z n }$ be the weight vector of attributes $Z j$, where $z j ∈ [ 0 , 1 ] , j = 1 , 2 , … , n , ∑ j = 1 n z j = 1$. Assume that $D = { D 1 , D 2 , … D l }$ is a set of decision makers that have a significant degree of $d = { d 1 , d 2 , … d l }$, where $d k ∈ [ 0 , 1 ] ,$ $k = 1 , 2 , … , l .$ $∑ k = 1 l d k = 1$. Let $Y = { Y 1 , Y 2 , … Y m }$ be a discrete collection of alternatives. $Q = ( q i j ) m × n$ is the overall intuitionistic fuzzy decision matrix, where $q i j$ means the value of alternative $Y i$ regarding the attribute $Z j$. The specific calculating procedures are presented below.
Step 1. Set up each decision maker’s intuitionistic fuzzy decision matrix $Q ( k ) = ( q i j k ) m × n$ and calculate the overall intuitionistic fuzzy decision matrix $Q = ( q i j ) m × n$.
$Q ( k ) = [ q i j k ] m × n = [ q 11 k q 12 k … q 1 n k q 21 k q 22 k … q 2 n k ⋮ ⋮ ⋮ ⋮ q m 1 k q m 2 k … q m n k ]$
$Q = [ q i j ] m × n = [ q 11 q 12 … q 1 n q 21 q 22 … q 2 n ⋮ ⋮ ⋮ ⋮ q m 1 q m 2 … q m n ]$
where $q i j k$ is the assessment value of the alternative $Y i ( i = 1 , 2 , … , m )$ on the basis of the attribute $Z j ( j = 1 , 2 , … , n )$ and the decision maker $D k ( k = 1 , 2 , … , l )$.
Step 2. Normalize the overall intuitionistic fuzzy decision matrix $Q = ( q i j ) m × n$ to $Q N = [ q i j N ] m × n$.
Step 3. Use the CRiteria Importance Through Intercriteria Correlation (CRITIC) method to determine the weighting matrix of attributes.
The CRITIC method was designed in this part to decide the attributes’ weights. The calculating procedures of this method are presented below.
(1)
Depending on the normalized overall intuitionistic fuzzy decision matrix $Q N = ( q i j N ) m × n$, the correlation coefficient between attributes can be calculated as:
$I C j t = ∑ i = 1 m ( S ( q i j N ) − S ( q j N ) ) ( S ( q i t N ) − S ( q t N ) ) ∑ i = 1 m ( S ( q i j N ) − S ( q j N ) ) 2 ∑ i = 1 m ( S ( q i t N ) − S ( q t N ) ) 2 , j , t = 1 , 2 , … , n$
where $q j N = 1 m ∑ i = 1 m S ( q i j N )$ and $q t N = 1 m ∑ i = 1 m S ( q i t N )$.
(2)
Calculate the attributes’ standard deviation.
$I S j = 1 m − 1 ∑ i = 1 m ( S ( q i j N ) − S ( q j N ) ) 2 , j = 1 , 2 , … , n$
where $q j N = 1 m ∑ i = 1 m S ( q i j N )$.
(3)
Calculate the attributes’ weights.
$z j = I S j ∑ t = 1 n ( 1 − I C j t ) ∑ j = 1 n ( I S j ∑ t = 1 n ( 1 − I C j t ) ) , j = 1 , 2 … , n$
where $z j ∈ [ 0 , 1 ]$ and $∑ j = 1 n z j = 1$.
Step 4. Calculate the value of average solution (AV) regarding all proposed attributes.
$A V = [ A V j ] 1 × n = [ ∑ i = 1 m q ^ i j N m ] 1 × n$
$[ A V j ] 1 × n = [ ∑ i = 1 m q ^ i j N m ] 1 × n = ( 1 − ∏ i = 1 m ( 1 − μ i j N ) 1 m , ∏ i = 1 m ( ν i j N ) 1 m ) 1 × n$
Step 5. Depending on the AV results, the positive distance from average (PDA) and negative distance from average (NDA) can be calculated as:
$P D A i j = [ P D A i j ] m × n = max ( 0 , ( s ( q i j N ) − s ( A V j ) ) ) s ( A V j )$
$N D A i j = [ N D A i j ] m × n = max ( 0 , ( s ( A V j ) − s ( q i j N ) ) ) s ( A V j )$
Step 6. Calculate the values of $S P i$ and $S N i$ which denote the weighted sum of PDA and NDA.
Step 7. Depending on the above calculated results, $S P i$ and $S N i$ can be normalized as:
Step 8. Calculate the values of the appraisal score ($A S$) regarding each alternative’s $N S P i$ and $N S N i$:
$A S i = 1 2 ( N S P i + N S N i )$
Step 9. In terms of the calculated results of $A S$, all the alternatives can be ranked. The higher the value of $A S$, the higher the value of the optimal alternative that is selected.

## 4. The Empirical Example and Comparative Analysis

#### 4.1. An Empirical Example

The energy conservation of a building, considering industrial, construction, and transportation aspects, is one of three key energy-saving fields. Following the implementation of the Chinese energy consumption and pollution reduction policy, the emergence and development of green architecture construction practices have become the trend with respect to sustainable development. Chinese construction energy conservation presents not only a pressing situation but also a tremendous potential, but it has been overlooked because of the lack of consideration for green architecture energy-saving designs. People rarely consider the economic benefits of green architecture to be achieved from energy-saving designs. They have not formed a standardized evaluation method for the energy-saving design for economic benefits. Thus, choosing a green building energy-saving design program for economic assessment methods to conduct research has a certain theoretical guidance significance and application value. In this section, an empirical example for evaluating green building energy-saving design projects considered as complex MAGDM issues [32,33,34,35,36,37,38,39] is provided using the IF-EDAS method. Taking its own business development into consideration, a building company requires a green building energy-saving design project for a school. There are five potential green building energy-saving design projects $Y i ( i = 1 , 2 , 3 , 4 , 5 )$. In order to select the optimal green building energy-saving design project, the building company invites five experts $D = { D 1 , D 2 , D 3 , D 4 , D 5 }$ (expert’s weight $d = ( 0.20 , 0.20 , 0.20 , 0.20 , 0.20 )$) to assess these green building energy-saving design projects. All experts give their assessment information depending on the four following attributes: ① $Z 1$ is traffic convenience; ② $Z 2$ is product price; ③ $Z 3$ is green environmental protection ability; and ④ $Z 4$ is service quality. Evidently, $Z 2$ is the building cost attribute, while $Z 1$, $Z 3$, and $Z 4$ are the benefit attributes. To obtain the optimal green building energy-saving design project, the calculating procedures are as follows.
Step 1. Set up each decision maker’s intuitionistic fuzzy evaluation matrix $Q ( k ) = ( q i j k ) m × n$ $( i = 1 , 2 , … , m , j = 1 , 2 , … , n )$ as shown in Table 1, Table 2, Table 3, Table 4 and Table 5. From these tables and Equations (12)–(14), the overall intuitionistic fuzzy decision matrix can be calculated. The results are presented in Table 6.
Step 2. Normalize the evaluation matrix $Q = [ q i j ] m × n$ to $Q N = [ q i j N ] m × n$ (See Table 7).
Step 3. Decide the attribute weights $z j ( j = 1 , 2 , … , n )$ using the CRITIC method as presented in Table 8.
Step 4. Depending on the calculated results of Table 8, the value of the average solution (AV) can be obtained based on all proposed attributes using Equations (19) and (20) (see Table 9).
Step 5. Based on the results of AV, the positive distance from average (PDA) and negative distance from average (NDA) can be calculated using Equations (21) and (22) (see Table 10 and Table 11).
Step 6. Based on Equation (23) and the attributes’ weighting vector $ω = ( 0.1410 , 0.2263 , 0.3234 , 0.3093 )$, the values of $S P i$ and $S N i$ can be calculated:
$S N 1 = 0.1365 , S N 2 = 0.0324 , S N 3 = 0.0633 , S N 4 = 0.1095 , S N 5 = 0.0373$
Step 7. The results of Step 6 can be normalized using Equation (24):
$N S N 1 = 0.0000 , N S N 2 = 0.7626 , N S N 3 = 0.5367 , N S N 4 = 0.1979 , N S N 5 = 0.7269$
Step 8. Based on each alternative’s $N S P i$ and $N S N i$, the values of AS can be calculated:
$A S 1 = 0.0035 , A S 2 = 0.8813 , A S 3 = 0.3483 , A S 4 = 0.2512 , A S 5 = 0.3634$
Step 9. Based on the calculated results of AS, all the alternatives can be ranked; the higher the value of AS, the higher the optimal alternative that is selected. Evidently, the rank of all alternatives is $Y 2 > Y 5 > Y 3 > Y 4 > Y 1$ and $Y 2$ is the best green building energy-saving design project.

#### 4.2. Comparative Analysis

In this section, our developed method is compared with other methods to illustrate its superiority. First, our presented method was compared with IFWA and IFWG operators [30]. For the IFWA operator, the calculated result is $S ( Y 1 ) = 0.4263 ,$ $S ( Y 2 ) = 0.5703$, $S ( Y 3 ) = 0.4756 ,$ $S ( Y 4 ) = 0.4842$, and $S ( Y 5 ) = 0.4650$. Thus, the ranking order is $Y 2 > Y 4 > Y 3 > Y 5 > Y 1$. For the IFWG operator, the calculated result is $S ( Y 1 ) = 0.4096 ,$ $S ( Y 2 ) = 0.5607 ,$ $S ( Y 3 ) = 0.4659 ,$ $S ( Y 4 ) = 0.4572$, and $S ( Y 5 ) = 0.4631$. Therefore, the ranking order is $Y 2 > Y 3 > Y 5 > Y 4 > Y 1$.
Furthermore, our presented method was compared with the modified VIKOR method with IFSs [40]. Then, we obtained the following calculated results. The closest ideal score values were determined as follows: $C I ∗ ( Y 1 ) = 1.0000$, $C I ∗ ( Y 2 ) = 0.1803$, $C I ∗ ( Y 3 ) = 0.3301$, $C I ∗ ( Y 4 ) = 0.6522$, and $C I ∗ ( Y 5 ) = 0.3962$. The worst score values were determined as follows: $C I − ( Y 1 ) = 0.0000$, $C I − ( Y 2 ) = 0.5000$, $C I − ( Y 3 ) = 0.7144$, $C I − ( Y 4 ) = 0.1755$, and $C I − ( Y 5 ) = 0.1038$. Then, each alternative’s relative closeness was calculated as follows: $D R C 1 = 1.0000$, $D R C 2 = 0.2650$, $D R C 3 = 0.3161$, $D R C 4 = 0.7880$, and $D R C 5 = 0.7925$. Hence, the ranking order of alternatives is $Y 2 > Y 3 > Y 4 > Y 5 > Y 1$.
Finally, our presented method was compared with GRA (Grey Relational Analysis)-based intuitionistic fuzzy [41]. Then, we obtained the following calculated results. The grey relational grades of each alternative were calculated as follows: $γ 1 = 0.8297$, $γ 2 = 1.0000$, $γ 3 = 0.8206$, $γ 4 = 0.8717$, and $γ 5 = 0.8533$. Therefore, the ranking order of alternatives is $Y 2 > Y 4 > Y 5 > Y 1 > Y 3$. The results of dissimilar methods are recorded in Table 12.
From Table 12, it is evident that the optimal green building energy-saving design project is $Y 2$ in the mentioned methods, while the poorest choice is $Y 1$ in most situations. Therefore, these method ranking results are slightly different.

## 5. Conclusions

In this paper, an IF-EDAS method was developed to tackle the MAGDM issues based on the description of the EDAS method and some fundamental notions of IFSs. Initially, the fundamental information of IFSs was simply reviewed. Second, the IFWA and IFWG operators were used to integrate the intuitionistic fuzzy information. Subsequently, based on the CRITIC method, the attributes’ weights were decided. In addition, applying the EDAS method to the intuitionistic fuzzy environment, a novel method was designed, and the calculating procedures were briefly depicted. Finally, an application for evaluating a green building energy-saving design project was provided to confirm the superiority of this novel method, and a comparative analysis between an IF-EDAS method and other methods was also made to further verify the merits of this method. In our future work, an IF-EDAS method may be extended to interval-valued intuitionistic fuzzy sets and extensively applied to different uncertain situations [42,43,44,45] and ambiguous environments [46,47,48,49,50].

## Funding

This research received no external funding.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Wan, S.P.; Lin, L.L.; Dong, J.Y. MAGDM based on triangular Atanassov’s intuitionistic fuzzy information aggregation. Neural Comput. Appl. 2017, 28, 2687–2702. [Google Scholar] [CrossRef]
2. Wan, S.P.; Wang, F.; Dong, J.Y. Additive consistent interval-valued Atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making. Eur. J. Oper. Res. 2017, 263, 571–582. [Google Scholar] [CrossRef]
3. Wang, J.; Wang, P.; Wei, G.W.; Wei, C.; Wu, J. Some power Heronian mean operators in multiple attribute decision-making based on q-rung orthopair hesitant fuzzy environment. J. Exp. Theor. Artif. Intell. 2019. [Google Scholar] [CrossRef]
4. Wang, P.; Wang, J.; Wei, G.W.; Wei, C.; Wei, Y. The Multi-Attributive Border Approximation Area Comparison (MABAC) for Multiple Attribute Group Decision Making Under 2-Tuple Linguistic Neutrosophic Environment. Informatica 2019, 30, 799–818. [Google Scholar] [CrossRef]
5. Zadeh, L.A. Fuzzy Sets. Inf. Control 1965, 8, 338–356. [Google Scholar] [CrossRef] [Green Version]
6. Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
7. Gou, X.J.; Xu, Z.S.; Lei, Q. New operational laws and aggregation method of intuitionistic fuzzy information. J. Intell. Fuzzy Syst. 2016, 30, 129–141. [Google Scholar] [CrossRef]
8. Li, M.; Wu, C. A Distance Model of Intuitionistic Fuzzy Cross Entropy to Solve Preference Problem on Alternatives. Math. Probl. Eng. 2016. [Google Scholar] [CrossRef] [Green Version]
9. Khan, M.S.; Lohani, Q.M.D. A Similarity Measure For Atanassov Intuitionistic Fuzzy Sets and its Application to Clustering. In Proceedings of the 2016 International Workshop on Computational Intelligence (IWCI), Dhaka, Bangladesh, 12–13 December 2016; pp. 232–239. [Google Scholar]
10. Li, P.; Liu, J.; Liu, S.F.; Su, X.; Wu, J. Grey Target Method for Intuitionistic Fuzzy Decision Making Based on Grey Incidence Analysis. J. Grey Syst. 2016, 28, 96–109. [Google Scholar]
11. Lei, F.; Wei, G.W.; Lu, J.P.; Wei, C.; Wu, J. GRA method for probabilistic linguistic multiple attribute group decision making with incomplete weight information and its application to waste incineration plants location problem. Int. J. Comput. Intell. Syst. 2019, 12, 1547–1556. [Google Scholar] [CrossRef] [Green Version]
12. Chen, S.M.; Cheng, S.H.; Lan, T.C. Multicriteria decision making based on the TOPSIS method and similarity measures between intuitionistic fuzzy values. Inf. Sci. 2016, 367, 279–295. [Google Scholar] [CrossRef]
13. Gupta, P.; Mehlawat, M.K.; Grover, N.; Chen, W. Modified intuitionistic fuzzy SIR approach with an application to supplier selection. J. Intell. Fuzzy Syst. 2017, 32, 4431–4441. [Google Scholar] [CrossRef]
14. Lu, J.P.; Wei, C. TODIM method for performance appraisal on social-integration-based rural reconstruction with interval-valued intuitionistic fuzzy information. J. Intell. Fuzzy Syst. 2019, 37, 1731–1740. [Google Scholar] [CrossRef]
15. Wu, L.P.; Gao, H.; Wei, C. VIKOR method for financing risk assessment of rural tourism projects under interval-valued intuitionistic fuzzy environment. J. Intell. Fuzzy Syst. 2019, 37, 2001–2008. [Google Scholar] [CrossRef]
16. Wu, L.P.; Wei, G.W.; Wu, J.; Wei, C. Some Interval-Valued Intuitionistic Fuzzy Dombi Heronian Mean Operators and their Application for Evaluating the Ecological Value of Forest Ecological Tourism Demonstration Areas. Int. J. Environ. Res. Public Health 2020, 17, 829. [Google Scholar] [CrossRef] [Green Version]
17. Wu, L.P.; Wang, J.; Gao, H. Models for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators. J. Intell. Fuzzy Syst. 2019, 36, 5693–5709. [Google Scholar] [CrossRef]
18. Keshavarz Ghorabaee, M.; Zavadskas, E.K.; Olfat, L.; Turskis, Z. Multi-Criteria Inventory Classification Using a New Method of Evaluation Based on Distance from Average Solution (EDAS). Informatica 2015, 26, 435–451. [Google Scholar] [CrossRef]
19. Keshavarz Ghorabaee, M.; Zavadskas, E.K.; Amiri, M.; Turskis, Z. Extended EDAS Method for Fuzzy Multi-criteria Decision-making: An Application to Supplier Selection. Int. J. Comput. Commun. Control 2016, 11, 358–371. [Google Scholar] [CrossRef] [Green Version]
20. Zhang, S.Q.; Wei, G.W.; Gao, H.; Wei, C.; Wei, Y. EDAS method for multiple criteria group decision making with picture fuzzy information and its application to green suppliers selections. Technol. Econ. Dev. Econ. 2019, 26, 1123–1138. [Google Scholar] [CrossRef] [Green Version]
21. Peng, X.D.; Liu, C. Algorithms for neutrosophic soft decision making based on EDAS, new similarity measure and level soft set. J. Intell. Fuzzy Syst. 2017, 32, 955–968. [Google Scholar] [CrossRef] [Green Version]
22. Feng, X.Q.; Wei, C.P.; Liu, Q. EDAS Method for Extended Hesitant Fuzzy Linguistic Multi-criteria Decision Making. Int. J. Fuzzy Syst. 2018, 20, 2470–2483. [Google Scholar] [CrossRef]
23. He, Y.; Lei, F.; Wei, G.W.; Wang, R.; Wu, J.; Wei, C. EDAS method for multiple attribute group decision making with probabilistic uncertain linguistic information and its application to green supplier selection. Int. J. Comput. Intell. Syst. 2019, 12, 1361–1370. [Google Scholar] [CrossRef] [Green Version]
24. Karasan, A.; Kahraman, C. A novel interval-valued neutrosophic EDAS method: Prioritization of the United Nations national sustainable development goals. Soft Comput. 2018, 22, 4891–4906. [Google Scholar] [CrossRef]
25. Li, Z.X.; Wei, G.W.; Wang, R.; Wu, J.; Wei, C.; Wei, Y. EDAS method for multiple attribute group decision making under q-rung orthopair fuzzy environment. Technol. Econ. Dev. Econ. 2020, 26, 86–102. [Google Scholar] [CrossRef]
26. Wang, P.; Wang, J.; Wei, G.W. EDAS method for multiple criteria group decision making under 2-tuple linguistic neutrosophic environment. J. Intell. Fuzzy Syst. 2019, 37, 1597–1608. [Google Scholar] [CrossRef]
27. Keshavarz Ghorabaee, M.; Amiri, M.; Zavadskas, E.K.; Turskis, Z.; Antucheviciene, J. Stochastic EDAS method for multi-criteria decision-making with normally distributed data. J. Intell. Fuzzy Syst. 2017, 33, 1627–1638. [Google Scholar] [CrossRef]
28. Zhang, S.Q.; Gao, H.; Wei, G.W.; Wei, Y.; Wei, C. Evaluation Based on Distance from Average Solution Method for Multiple Criteria Group Decision Making under Picture 2-Tuple Linguistic Environment. Mathematics 2019, 7, 14. [Google Scholar] [CrossRef] [Green Version]
29. Li, X.; Ju, Y.B.; Ju, D.W.; Zhang, W.K.; Dong, P.W.; Wang, A.H. Multi-Attribute Group Decision Making Method Based on EDAS Under Picture Fuzzy Environment. IEEE Access 2019, 7, 141179–141192. [Google Scholar] [CrossRef]
30. Xu, Z.S.; Yager, R.R. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst. 2006, 35, 417–433. [Google Scholar] [CrossRef]
31. Liu, H.W.; Wang, G.J. Multi-criteria decision-making methods based on intuitionistic fuzzy sets. Eur. J. Oper. Res. 2007, 179, 220–233. [Google Scholar] [CrossRef]
32. Deng, X.M.; Wang, J.; Wei, G.W. Some 2-tuple linguistic Pythagorean Heronian mean operators and their application to multiple attribute decision-making. J. Exp. Theor. Artif. Intell. 2019, 31, 555–574. [Google Scholar] [CrossRef]
33. Tang, X.Y.; Wei, G.W. Dual hesitant Pythagorean fuzzy Bonferroni mean operators in multi-attribute decision making. Arch. Control Sci. 2019, 29, 339–386. [Google Scholar] [CrossRef]
34. Wei, G.W.; Zhang, S.Q.; Lu, J.P.; Wu, J.; Wei, C. An extended bidirectional projection method for picture fuzzy MAGDM and its application to safety assessment of construction project. IEEE Access 2019, 7, 166138–166147. [Google Scholar] [CrossRef]
35. Deng, X.M.; Gao, H. TODIM method for multiple attribute decision making with 2-tuple linguistic Pythagorean fuzzy information. J. Intell. Fuzzy Syst. 2019, 37, 1769–1780. [Google Scholar] [CrossRef]
36. Gao, H.; Lu, M.; Wei, Y. Dual hesitant bipolar fuzzy hamacher aggregation operators and their applications to multiple attribute decision making. J. Intell. Fuzzy Syst. 2019, 37, 5755–5766. [Google Scholar] [CrossRef]
37. Li, Z.X.; Lu, M. Some novel similarity and distance and measures of Pythagorean fuzzy sets and their applications. J. Intell. Fuzzy Syst. 2019, 37, 1781–1799. [Google Scholar] [CrossRef]
38. Wang, J.; Gao, H.; Lu, M. Approaches to strategic supplier selection under interval neutrosophic environment. J. Intell. Fuzzy Syst. 2019, 37, 1707–1730. [Google Scholar] [CrossRef]
39. Wang, R. Research on the Application of the Financial Investment Risk Appraisal Models with Some Interval Number Muirhead Mean Operators. J. Intell. Fuzzy Syst. 2019, 37, 1741–1752. [Google Scholar] [CrossRef]
40. Zeng, S.Z.; Chen, S.M.; Kuo, L.W. Multiattribute decision making based on novel score function of intuitionistic fuzzy values and modified VIKOR method. Inf. Sci. 2019, 488, 76–92. [Google Scholar] [CrossRef]
41. Zhang, S.F.; Liu, S.Y. A GRA-based intuitionistic fuzzy multi-criteria group decision making method for personnel selection. Expert Syst. Appl. 2011, 38, 11401–11405. [Google Scholar] [CrossRef]
42. Zuo, C.; Pal, A.; Dey, A. New Concepts of Picture Fuzzy Graphs with Application. Mathematics 2019, 7, 470. [Google Scholar] [CrossRef] [Green Version]
43. Huang, L.S.; Hu, Y.; Li, Y.X.; Kumar, P.K.K.; Koley, D.; Dey, A. A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications. Mathematics 2019, 7, 551. [Google Scholar] [CrossRef] [Green Version]
44. Dey, A.; Son, L.H.; Pal, A.; Long, H.V. Fuzzy minimum spanning tree with interval type 2 fuzzy arc length: Formulation and a new genetic algorithm. Soft Comput. 2020, 24, 3963–3974. [Google Scholar] [CrossRef]
45. Dey, A.; Pradhan, R.; Pal, A.; Pal, T. A genetic algorithm for solving fuzzy shortest path problems with interval type-2 fuzzy arc lengths. Malays. J. Comput. Sci. 2018, 31, 255–270. [Google Scholar] [CrossRef] [Green Version]
46. Liu, P.D.; Liu, X. Multiattribute Group Decision Making Methods Based on Linguistic Intuitionistic Fuzzy Power Bonferroni Mean Operators. Complexity 2017. [Google Scholar] [CrossRef]
47. Mu, Z.M.; Zeng, S.Z.; Liu, Q.B. Some Interval-Valued Intuitionistic Fuzzy Zhenyuan Aggregation Operators and Their Application to Multi-Attribute Decision Making. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 2018, 26, 633–653. [Google Scholar] [CrossRef]
48. Dey, A.; Pradhan, R.; Pal, A.; Pal, T. The fuzzy robust graph coloring problem. In Proceedings of the 3rd International Conference on Frontiers of Intelligent Computing: Theory and Applications (FICTA) 2014; Springer: Cham, Switzerland, 2015. [Google Scholar]
49. Lu, J.P.; He, T.T.; Wei, G.W.; Wu, J.; Wei, C. Cumulative Prospect Theory: Performance Evaluation of Government Purchases of Home-Based Elderly-Care Services Using the Pythagorean 2-tuple Linguistic TODIM Method. Int. J. Environ. Res. Public Health 2020, 17, 1939. [Google Scholar] [CrossRef] [Green Version]
50. Dey, A.; Broumi, S.; Son, L.H.; Bakali, A.; Talea, M.; Smarandache, F. A new algorithm for finding minimum spanning trees with undirected neutrosophic graphs. Granul. Comput. 2019, 4, 63–69. [Google Scholar] [CrossRef] [Green Version]
Table 1. Intuitionistic fuzzy evaluation information by $D 1$.
Table 1. Intuitionistic fuzzy evaluation information by $D 1$.
$Z 1$$Z 2$$Z 3$$Z 4$
$Y 1$(0.35,0.65)(0.52,0.48)(0.24,0.76)(0.43,0.57)
$Y 2$(0.39,0.61)(0.66,0.34)(0.75,0.25)(0.61,0.39)
$Y 3$(0.40,0.60)(0.33,0.67)(0.56,0.44)(0.28,0.72)
$Y 4$(0.67,0.33)(0.58,0.42)(0.41,0.59)(0.47,0.53)
$Y 5$(0.26,0.74)(0.42,0.58)(0.52,0.48)(0.62,0.38)
Table 2. Intuitionistic fuzzy evaluation information by $D 2$.
Table 2. Intuitionistic fuzzy evaluation information by $D 2$.
$Z 1$$Z 2$$Z 3$$Z 4$
$Y 1$(0.38,0.62)(0.43,0.57)(0.29,0.71)(0.55,0.45)
$Y 2$(0.63,0.37)(0.34,0.66)(0.48,0.52)(0.52,0.48)
$Y 3$(0.50,0.50)(0.27,0.73)(0.41,0.59)(0.16,0.84)
$Y 4$(0.46,0.54)(0.62,0.38)(0.57,0.43)(0.29,0.71)
$Y 5$(0.60,0.40)(0.46,0.54)(0.42,0.58)(0.33,0.67)
Table 3. Intuitionistic fuzzy evaluation information by $D 3$.
Table 3. Intuitionistic fuzzy evaluation information by $D 3$.
$Z 1$$Z 2$$Z 3$$Z 4$
$Y 1$(0.44,0.56)(0.58,0.42)(0.31,0.69)(0.40,0.60)
$Y 2$(0.58,0.42)(0.65,0.35)(0.42,0.58)(0.74,0.26)
$Y 3$(0.35,0.65)(0.48,0.52)(0.18,0.82)(0.62,0.38)
$Y 4$(0.27,0.73)(0.26,0.74)(0.62,0.38)(0.31,0.69)
$Y 5$(0.46,0.54)(0.44,0.56)(0.34,0.66)(0.65,0.35)
Table 4. Intuitionistic fuzzy evaluation information by $D 4$.
Table 4. Intuitionistic fuzzy evaluation information by $D 4$.
$Z 1$$Z 2$$Z 3$$Z 4$
$Y 1$(0.52,0.48)(0.37,0.63)(0.25,0.75)(0.22,0.78)
$Y 2$(0.51,0.49)(0.64,0.36)(0.77,0.23)(0.42,0.58)
$Y 3$(0.43,0.57)(0.58,0.42)(0.41,0.59)(0.66,0.34)
$Y 4$(0.68,0.32)(0.32,0.68)(0.64,0.36)(0.15,0.85)
$Y 5$(0.37,0.63)(0.63,0.37)(0.52,0.48)(0.27,0.73)
Table 5. Intuitionistic fuzzy evaluation information by $D 5$.
Table 5. Intuitionistic fuzzy evaluation information by $D 5$.
$Z 1$$Z 2$$Z 3$$Z 4$
$Y 1$(0.63,0.37)(0.45,0.55)(0.39,0.61)(0.53,0.47)
$Y 2$(0.53,0.47)(0.37,0.63)(0.60,0.40)(0.59,0.41)
$Y 3$(0.47,0.53)(0.29,0.71)(0.52,0.48)(0.27,0.73)
$Y 4$(0.41,0.59)(0.53,0.47)(0.56,0.44)(0.19,0.81)
$Y 5$(0.33,0.67)(0.48,0.52)(0.54,0.46)(0.21,0.79)
Table 6. Overall intuitionistic fuzzy evaluation information.
Table 6. Overall intuitionistic fuzzy evaluation information.
$Z 1$$Z 2$$Z 3$$Z 4$
$Y 1$(0.4745,0.5255)(0.4752,0.5248)(0.2981,0.7019)(0.4373,0.5627)
$Y 2$(0.5346,0.4654)(0.5532,0.4468)(0.6300,0.3700)(0.5901,0.4099)
$Y 3$(0.4324,0.5676)(0.4030,0.5970)(0.4298,0.5702)(0.4361,0.5639)
$Y 4$(0.5235,0.4765)(0.4808,0.5192)(0.5667,0.4333)(0.2913,0.7087)
$Y 5$(0.4168,0.5832)(0.4923,0.5077)(0.4732,0.5268)(0.4477,0.5523)
Table 7. The normalized intuitionistic fuzzy evaluation information.
Table 7. The normalized intuitionistic fuzzy evaluation information.
$Z 1$$Z 2$$Z 3$$Z 4$
$Y 1$(0.4745,0.5255)(0.5248,0.4752)(0.2981,0.7019)(0.4373,0.5627)
$Y 2$(0.5346,0.4654)(0.4468,0.5532)(0.6300,0.3700)(0.5901,0.4099)
$Y 3$(0.4324,0.5676)(0.5970,0.4030)(0.4298,0.5702)(0.4361,0.5639)
$Y 4$(0.5235,0.4765)(0.5192,0.4808)(0.5667,0.4333)(0.2913,0.7087)
$Y 5$(0.4168,0.5832)(0.5077,0.4923)(0.4732,0.5268)(0.4477,0.5523)
Table 8. The attributes weights $z j$.
Table 8. The attributes weights $z j$.
$Z 1$$Z 2$$Z 3$$Z 4$
$z j$$0.1410$$0.2263$$0.3234$$0.3093$
Table 9. The value of the average solution.
Table 9. The value of the average solution.
Average Solution
$Z 1$$( 0.4785 , 0.5215 )$
$Z 2$$( 0.5216 , 0.4784 )$
$Z 3$$( 0.4921 , 0.5079 )$
$Z 4$$( 0.4487 , 0.5513 )$
Table 10. The results of $P D A i j$.
Table 10. The results of $P D A i j$.
$Z 1$$Z 2$$Z 3$$Z 4$
$Y 1$0.00000.00630.00000.0000
$Y 2$0.11730.00000.28020.3150
$Y 3$0.00000.14470.00000.0000
$Y 4$0.09410.00000.15170.0000
$Y 5$0.00000.00000.00000.0000
Table 11. The results of $N D A i j$.
Table 11. The results of $N D A i j$.
$Z 1$$Z 2$$Z 3$$Z 4$
$Y 1$0.00840.00000.39420.0255
$Y 2$0.00000.14330.00000.0000
$Y 3$0.09630.00000.12660.0282
$Y 4$0.00000.00440.00000.3508
$Y 5$0.12890.02650.03840.0023
Table 12. Evaluation results of dissimilar methods.
Table 12. Evaluation results of dissimilar methods.
MethodsRanking OrderThe Optimal AlternativeThe Worst Alternative
IFWA$Y 2 > Y 4 > Y 3 > Y 5 > Y 1$$Y 2$$Y 1$
IFWG$Y 2 > Y 3 > Y 5 > Y 4 > Y 1$$Y 2$$Y 1$
The modified VIKOR $Y 2 > Y 3 > Y 4 > Y 5 > Y 1$$Y 2$$Y 1$
The GRA method$Y 2 > Y 4 > Y 5 > Y 1 > Y 3$$Y 2$$Y 3$
The developed method$Y 2 > Y 5 > Y 3 > Y 4 > Y 1$$Y 2$$Y 1$

## Share and Cite

MDPI and ACS Style

Liang, Y. An EDAS Method for Multiple Attribute Group Decision-Making under Intuitionistic Fuzzy Environment and Its Application for Evaluating Green Building Energy-Saving Design Projects. Symmetry 2020, 12, 484. https://doi.org/10.3390/sym12030484

AMA Style

Liang Y. An EDAS Method for Multiple Attribute Group Decision-Making under Intuitionistic Fuzzy Environment and Its Application for Evaluating Green Building Energy-Saving Design Projects. Symmetry. 2020; 12(3):484. https://doi.org/10.3390/sym12030484

Chicago/Turabian Style

Liang, Yuan. 2020. "An EDAS Method for Multiple Attribute Group Decision-Making under Intuitionistic Fuzzy Environment and Its Application for Evaluating Green Building Energy-Saving Design Projects" Symmetry 12, no. 3: 484. https://doi.org/10.3390/sym12030484

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.