# Evaluating Projects Based on Intuitionistic Fuzzy Group Decision Making

Authors: Babak Daneshvar Rouyendegh
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

There are various methods regarding project selection in different fields. This paper deals with an actual application of construction project selection, using two aggregation operators. First, the opinion of experts is used in a model of group decision making called intuitionistic fuzzy TOPSIS (IFT). Secondly, project evaluation is formulated by dynamic intuitionistic fuzzy weighted averaging (DIFWA). Intuitionistic fuzzy weighted averaging (IFWA) operator is utilized to aggregate individual opinions of decision makers (DMs) for rating the importance of criteria and alternatives. A numerical example for project selection is given to clarify the main developed result  in this paper.


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## Body

## 1. Introduction

Project selection and project evaluation involve decisions that are critical in terms of the profitability, growth, and survival of project management organizations in the increasingly competitive global scenario. Such decisions are often complex, because they require identification, consideration, and analysis of many tangible and intangible factors [1].There are various methods regarding project selection in different fields. Project selection problem has attracted great endeavor by practitioners and academicians in recent years. One of the major fields that have been applied to this problem is mathematical programming, especially Mix-Integer Programming (MIP), since the problems comprise selection of projects while other aspects are considered using real-value variables [2]. For instance, a MIP model is developed by [3] to conquer Research and Development (R&D) portfolio selection.Multicriteria decision making (MCDM) is a modeling and methodological tool for dealing with complex engineering problems [4]. Many mathematical programming models have been developed to address project-selection problems. However, in recent years, MCDM methods have gained considerable acceptance for judging different proposals. The objective of Mohanty’s [5] study was to integrate the multidimensional issues in an MCDM framework that may help decision makers to develop insights and make decisions. They computed weight of each criterion and then assessed the projects by doing technique for order preference by similarity to ideal solution algorithm (TOPSIS) [6]. To select R&D project, the application of the fuzzy analytical network process (ANP) and fuzzy cost analysis has been used by some researchers [7]. In their studies, triangular fuzzy numbers (TFNs) are used to prefer one criterion over another by using a pairwise comparison with the fuzzy set theory, where the weight of each criterion in the format of triangular fuzzy numbers is calculated [7]. The project selection problem was presented through a methodology which is based on the analytic hierarchy process (AHP) for quantitative and qualitative aspects of a problem [8]. It assists the measuring of the initial viability of industrial projects. The study shows that industrial investment company should concentrate its efforts in development of prefeasibility studies for a specific number of industrial projects which have a high likelihood of realization [8].Multiattribute decision making (MADM) is the other applied approach in which criteria are mostly defined in qualitative scale and the decision is made with respect to assigned weights using some methods, such as PROMETHEE [9, 10]. To have more comprehensive study on MADM methods in this field, readers are referred to [11–15].The rest of the paper is organized as follows. Section2 provides materials and methods, mainly fuzzy set theory (FST) and intuitionistic fuzzy set (IFS). The IFT and DIFWA are introduced in Section 3. How the proposed model is used in an actual example is explained in Section 4. Finally, the conclusions are provided in the final section.


## 2. Materials and Methods

### 2.1. FST

Zadeh (1965) introduced the fuzzy set theory (FST) to deal with the uncertainty due to imprecision and vagueness. A major contribution of this theory is capability of representing vague data; it also allows mathematical operators and programming to be applied to the fuzzy domain. An FS is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function, which assigns to each object a grade of membership ranging between zero and one [16, 17].A tilde “~” will be placed above a symbol if the symbol represents an FST. A TFN M̃ is shown in Figure 1. A TFN is denoted simply as (l/m,m/u) or (l,m,u). The parameters l,m and u (l≤m≤u), respectively, denote the smallest possible value, the most promising value, and the largest possible value that describe a fuzzy event. The membership function of TFN is as follows.Figure 1
A TFNM̃.Each TFN has linear representations on its left and right side, such that its membership function can be defined as(2.1)μ(xM̃)={0,x<l,x-lm-l,l≤x≤m,u-xu-m,m≤x≤u,0,x>u.A fuzzy number can always be given by its corresponding left and right representation of each degree of membership as in the following:(2.2)M̃=(Ml(y),Mr(y))=(l+(m-l)y,u+(m-u)y),y∈[0,1],
where l(y) and r(y) denote the left side representation and the right side representation of a fuzzy number (FN), respectively. Many ranking methods for FNs have been developed in the literature. These methods may provide different ranking results, and most of them are tedious in graphic manipulation requiring complex mathematical calculation [18].While there are various operations on TFNs, only the important operations used in this study are illustrated. If we define two positive TFNs(l1,m1,u1) and (l2,m2,u2), then(2.3)(l1,m1,u1)+(l2,m2,u2)=(l1+l2,m1+m2,u1+u2),(l1,m1,u1)*(l2,m2,u2)=(l1*l2,m1+m2,u1*u2),(l1,m1,u1)+k=(l1*km1*k,u1*k),wherek>0.


### 2.2. Basic Concept of IFS

The application of IFS method within the overall goal to select the best project has been described. IFSs introduced by Atanassov [19] are an extension of the classical FST, which is a suitable way to deal with vagueness. IFSs have been applied to many areas such as medical diagnosis [20–22], decision-making problems [23–46], pattern recognition [47–52], supplier selection [53, 54], enterprise partners selection [55], personnel selection [56], evaluation of renewable energy [57], facility location selection [58], web service selection [59], printed circuit board assembly [60], and management information system [61].The following briefly introduces some necessary introductory concepts of IFS. IFSA in a finite set X can be written as [19]
(2.4)A={〈x,μA(x),vA(x)〉∣x∈X},whereμA(x),VA(x):X⟶[0,1]
are membership function and nonmembership function, respectively, such that
(2.5)0≤μA(x)⨁VA(x)≤1.
A third parameter of IFS is πA(x), known as the intuitionistic fuzzy index or hesitation degree of whether x belongs to A or not:
(2.6)πA(x)=1-μA(x)-VA(x).
It is obviously seen that for every x∈X(2.7)0≤πA(x)≤1iftheπA(x).If it is small, knowledge aboutx is more certain. If πA(x) is great, knowledge about x is more uncertain. Obviously, when
(2.8)μA(x)=1-vA(x)μA(x)=1-v(x)
for all elements of the universe, the ordinary FST concept is recovered [60].LetA and B be IFSs of the set X, then multiplication operator is defined as follows [19]:
(2.9)A⨁B={μA(x)⋅μB(x),vA(x)+vB(x)-vA(x)⋅vB(x)∣x∈X}.


## 2.1. FST

Zadeh (1965) introduced the fuzzy set theory (FST) to deal with the uncertainty due to imprecision and vagueness. A major contribution of this theory is capability of representing vague data; it also allows mathematical operators and programming to be applied to the fuzzy domain. An FS is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function, which assigns to each object a grade of membership ranging between zero and one [16, 17].A tilde “~” will be placed above a symbol if the symbol represents an FST. A TFN M̃ is shown in Figure 1. A TFN is denoted simply as (l/m,m/u) or (l,m,u). The parameters l,m and u (l≤m≤u), respectively, denote the smallest possible value, the most promising value, and the largest possible value that describe a fuzzy event. The membership function of TFN is as follows.Figure 1
A TFNM̃.Each TFN has linear representations on its left and right side, such that its membership function can be defined as(2.1)μ(xM̃)={0,x<l,x-lm-l,l≤x≤m,u-xu-m,m≤x≤u,0,x>u.A fuzzy number can always be given by its corresponding left and right representation of each degree of membership as in the following:(2.2)M̃=(Ml(y),Mr(y))=(l+(m-l)y,u+(m-u)y),y∈[0,1],
where l(y) and r(y) denote the left side representation and the right side representation of a fuzzy number (FN), respectively. Many ranking methods for FNs have been developed in the literature. These methods may provide different ranking results, and most of them are tedious in graphic manipulation requiring complex mathematical calculation [18].While there are various operations on TFNs, only the important operations used in this study are illustrated. If we define two positive TFNs(l1,m1,u1) and (l2,m2,u2), then(2.3)(l1,m1,u1)+(l2,m2,u2)=(l1+l2,m1+m2,u1+u2),(l1,m1,u1)*(l2,m2,u2)=(l1*l2,m1+m2,u1*u2),(l1,m1,u1)+k=(l1*km1*k,u1*k),wherek>0.


## 2.2. Basic Concept of IFS

The application of IFS method within the overall goal to select the best project has been described. IFSs introduced by Atanassov [19] are an extension of the classical FST, which is a suitable way to deal with vagueness. IFSs have been applied to many areas such as medical diagnosis [20–22], decision-making problems [23–46], pattern recognition [47–52], supplier selection [53, 54], enterprise partners selection [55], personnel selection [56], evaluation of renewable energy [57], facility location selection [58], web service selection [59], printed circuit board assembly [60], and management information system [61].The following briefly introduces some necessary introductory concepts of IFS. IFSA in a finite set X can be written as [19]
(2.4)A={〈x,μA(x),vA(x)〉∣x∈X},whereμA(x),VA(x):X⟶[0,1]
are membership function and nonmembership function, respectively, such that
(2.5)0≤μA(x)⨁VA(x)≤1.
A third parameter of IFS is πA(x), known as the intuitionistic fuzzy index or hesitation degree of whether x belongs to A or not:
(2.6)πA(x)=1-μA(x)-VA(x).
It is obviously seen that for every x∈X(2.7)0≤πA(x)≤1iftheπA(x).If it is small, knowledge aboutx is more certain. If πA(x) is great, knowledge about x is more uncertain. Obviously, when
(2.8)μA(x)=1-vA(x)μA(x)=1-v(x)
for all elements of the universe, the ordinary FST concept is recovered [60].LetA and B be IFSs of the set X, then multiplication operator is defined as follows [19]:
(2.9)A⨁B={μA(x)⋅μB(x),vA(x)+vB(x)-vA(x)⋅vB(x)∣x∈X}.


## 3. Intuitionistic Fuzzy TOPSIS (IFT) and Dynamic Intuitionistic Fuzzy Weighted Averaging (DIFWA) Methods

### 3.1. IFT

It should be mentioned here that the presented approach mainly utilizes the IFT method presented in [53, 56, 57] to handle a project selection problem with six projects and six criteria. In the current paper we validate the method in an actual context and show this method applicability with an extensive set of selection criteria. The IFT method is a suitable way to deal with MCDM problem in intuitionistic fuzzy environment (IFE). Let A={A1,A2,…,Am} be a set of alternatives and let X={X1,X2,…,Xn} be a set of criteria, the procedure for IFT method has been conducted in eight steps presented as follows.Step 1.
Determine the weights of importance of DMs.
In the first step, we assume that decision group containsl={l1,l2,…,ln} DMs. The importances of the DMs are considered as linguistic terms. These linguistic terms were assigned to IFN. Let Dk=[μk,vk,πk] be an intuitionistic fuzzy number for rating of kth DM.Then the weight of kth DM can be calculated as
(3.1)λk=(μk+πk(μk/(μk+vk)))∑k=1l(μk+πk(μk/(μk+vk))),whereλk∈[0,1],∑k=1lλk=1.Step 2.
Determine intuitionistic fuzzy decision matrix (IFDM).
Based on the weight of DMs, the aggregated intuitionistic fuzzy decision matrix (AIFDM) was calculated by applying intuitionistic fuzzy weighted averaging (IFWA) operator Xu [62]. In group decision-making process, all the individual decision opinions need to be fused into a group opinion to construct AIFDM.
LetR(k)=(rij(k))m×n be an IFDM of each DM. λ={λ1,λ2,λ3,…,λl} is the weight of DM. Consider
(3.2)R=(rij)m×n,
where
(3.3)rij=IFWAλ(rij(1),rij(2),…,rij(l))=λ1rij(1)⨁λ2rij(2)⨁λ3rij(3)⨁⋯⨁λlrij(l)=[1-∏k=1l(1-μij(k))λk,∏k=1l(vij(k))λk,∏k=1l(1-μij(k))λk-∏k=1l(vij(k))λk].Step 3.
Determine the weights of the selection criteria.
In this step, all criteria may not be assumed to be of equal importance.W represents a set of grades of importance. In order to obtain W, all the individual DM opinions for the importance of each criteria need to be fused. Let wj(k)=(μj(k),vj(k),πj(k)) be an IFN assigned to criterion Xj by the kth DM.
The weights of the criteria can be calculated as follows:(3.4)wj=IFWAλ(wj(1),wj(2),…,wj(l))=λ1wj(1)⨁λ2wj(2)⨁λ3wj(3)⨁⋯⨁λlwj(l)=[1-∏k=1l(1-μj(k))λk,∏k=1l(vj(k))λk,∏k=1l(1-μj(k))λk-∏k=1l(vj(k))λk].
Thus, a vector of criteria weight is obtained:W=[w1,w2,w3,…,wj], where wj=(μj,vj,πj)(j=1,2,…,n).Step 4.
Construct the aggregated weighted IFDM.
In Step4, the weights of criteria (W) and the aggregated IFDM are determined to the aggregated weighted IFDM which is constructed according to the following definition [19]:
(3.5)R′=R⨁W=(μij′,vij′)={〈x,μij⋅μj,vij+vj-vij⋅vj〉},πij′=1-vij-vj-μij⋅μj+vij⋅vj.R′ is a matrix composed with elements IFNs, rıj′=(μij′,vij′,πij′)(i=1,2,…,m;j=1,2,…,n).Step 5.
Determine intuitionistic fuzzy positive and negative ideal solution.
In this step, the intuitionistic fuzzy positive ideal solution (IFPIS) and intuitionistic fuzzy negative ideal solution (IFNIS) have to be determined. LetJ1 and J2 be benefit criteria and cost criteria, respectively. A* is IFPIS and A- is IFNIS. Then A* and A- are equal to
(3.6)A*=(r1′*,r2′*,…,rn′*),rj′*=(μj′*,vj′*,πj′*),j=1,2,…,n,A-=(r1′-,r2′-,…,rn′-),rj′-=(μj′-,vj′-,πj′-),j=1,2,…,n,
where
(3.7)μj′*={(max⁡i⁡{μij′}j∈J1),(min⁡i⁡{μij′}j∈J2)},vj*={(min⁡i⁡{vij′}j∈J1),(max⁡i⁡{vij′}j∈J2)},πj′*={(1-max⁡i⁡{μij′}-min⁡i⁡{vij′}j∈J1),(1-min⁡i⁡{μij′}-max⁡i⁡{vij′}j∈J2)},μj′-={(min⁡i⁡{μij′}j∈J1),(max⁡i⁡{μij′}j∈J2)},vj′-={(max⁡i⁡{vij′}j∈J1),(min⁡i⁡{vij′}j∈J2)},πj′-={(1-min⁡i⁡{μij′}-max⁡i⁡{vij′}j∈J1),(1-max⁡i⁡{μij′}-min⁡i⁡{vij′}j∈J2)}.Step 6.
Determine the separation measures between the alternative.
Separation between alternatives on IFS, distance measures proposed by Atanassov [63], Szmidt and Kacprzyk [64], and Grzegorzewski [65] including the generalizations of Hamming distance, Euclidean distance and their normalized distance measures can be used. After selecting the distance measure, the separation measures, Si* and Si-, of each alternative from IFPIS and IFNIS, are calculated:
(3.8)Si*=12∑j=1n[|μij′-μj′*|+|vij′-vj′*|+|πij′-πj′*|],Si-=12∑j=1n[|μij′-μj′-|+|vij′-vj′-|+|πij′-πj′-|].Step 7.
Determine the final ranking.
In the final step, the relative closeness coefficient of an alternativeAi with respect to the IFPIS A* is defined as follows:
(3.9)Ci*=Si-Si*+Si-,where0≤Ci*≤1.
The alternatives were ranked according to descending order of Ci*’s score.


### 3.2. DIFWA

The DIFWA method, proposed by Xu and Yager [33], is a suitable way to deal with problem in IFE. The procedure for DIFWA method has been given as follows.Step 1.
Utilize the DIFWA operator(3.10)rij=DIFWAλ(t)(rij(t1),rij(t2),…,rij(tp))=(1-∏k=1p(1-μrij(tk))λ(tk),∏k=1pvrij(tk)λ(tk),∏k=1p(1-μrij(tk))λ(tk)-∏k=1pvrij(tk)λ(tk)).
to aggregate all the intuitionistic fuzzy matrix R(tk)=(rij(tk))m×n(k=1,2,…,p) into a complex IFDM:
(3.11)R=(rij)m×n,whererij=(μij,vij,πij),μij=1-∏k=1p(1-μrij(tk))λ(tk),vij=∏k=1pvrij(tk)λ(tk),πij=∏k=1p(1-μrij(tk))λ(tk)-∏k=1pvrij(tk)λ(tk),i=1,2,…,n,j=1,2,…,m.Step 2.
Defineα+=(α1+,α2+,…,αm+)Tandα-=(α1-,α2-,…,αm-)T as the IFPIS and the IFNIS, respectively, where α+=(1,0,0)(i=1,2,…,m) are the m largest IFNs and α-=(0,1,0)(i=1,2,…,m) are the m smallest IFNs. Furthermore, for convenience of depiction, we denote the alternative xi(i=1,2,…,n) by xi=(ri1,ri2,…,rim)T,i=1,2,…,n.Step 3.
Calculate the distance between the alternativexi and the IFIS and the distance between the largest native xi and the IFNIS, respectively:
(3.12)d(xi,α+)=∑j=1mwjd(rij,αj+)=12∑j=1mwj(|μij-1|+|vij-0|+|πij-0|)=12∑j=1mwj(1-μij+vij+πij)=12∑j=1mwj(1-μij+vij+1-μij-vij)=∑j=1mwj(1-μij),d(xi,α-)=∑j=1mwjd(rij,αj-)=12∑j=1mwj(|μij-0|+|vij-1|+|πij-0|)=12∑j=1mwj(1-μij-vij+πij)=12∑j=1mwj(1+μij-vij+1-μij-vij)=∑j=1mwj(1-vij),
where rij=(μij,vij,πij),i=1,2,…,n,j=1,2,…,m.Step 4.
Calculate the closeness coefficient of each alternative:(3.13)c(xi)=d(xi,α-)d(xi,α+)+d(xi,α-),i=1,2,…,n.
Since
(3.14)d(xi,α+)+d(xi,α-)=∑j=1mwj(1-μij)+∑j=1mwj(1-vij)=∑j=1mwj(2-μij-vij)=∑j=1mwj(1+πij),
then (3.13) can be rewritten as
(3.15)c(xi)=∑j=1mwj(1-vij)∑j=1mwj(1-πij),i=1,2,…,n.Step 5.
Rank all the alternativesxi(1,2,…,n) according to the closeness coefficients c(xi)(1,2,…,n), the greater the value c(xi), the better the alternative xi.


## 3.1. IFT

It should be mentioned here that the presented approach mainly utilizes the IFT method presented in [53, 56, 57] to handle a project selection problem with six projects and six criteria. In the current paper we validate the method in an actual context and show this method applicability with an extensive set of selection criteria. The IFT method is a suitable way to deal with MCDM problem in intuitionistic fuzzy environment (IFE). Let A={A1,A2,…,Am} be a set of alternatives and let X={X1,X2,…,Xn} be a set of criteria, the procedure for IFT method has been conducted in eight steps presented as follows.Step 1.
Determine the weights of importance of DMs.
In the first step, we assume that decision group containsl={l1,l2,…,ln} DMs. The importances of the DMs are considered as linguistic terms. These linguistic terms were assigned to IFN. Let Dk=[μk,vk,πk] be an intuitionistic fuzzy number for rating of kth DM.Then the weight of kth DM can be calculated as
(3.1)λk=(μk+πk(μk/(μk+vk)))∑k=1l(μk+πk(μk/(μk+vk))),whereλk∈[0,1],∑k=1lλk=1.Step 2.
Determine intuitionistic fuzzy decision matrix (IFDM).
Based on the weight of DMs, the aggregated intuitionistic fuzzy decision matrix (AIFDM) was calculated by applying intuitionistic fuzzy weighted averaging (IFWA) operator Xu [62]. In group decision-making process, all the individual decision opinions need to be fused into a group opinion to construct AIFDM.
LetR(k)=(rij(k))m×n be an IFDM of each DM. λ={λ1,λ2,λ3,…,λl} is the weight of DM. Consider
(3.2)R=(rij)m×n,
where
(3.3)rij=IFWAλ(rij(1),rij(2),…,rij(l))=λ1rij(1)⨁λ2rij(2)⨁λ3rij(3)⨁⋯⨁λlrij(l)=[1-∏k=1l(1-μij(k))λk,∏k=1l(vij(k))λk,∏k=1l(1-μij(k))λk-∏k=1l(vij(k))λk].Step 3.
Determine the weights of the selection criteria.
In this step, all criteria may not be assumed to be of equal importance.W represents a set of grades of importance. In order to obtain W, all the individual DM opinions for the importance of each criteria need to be fused. Let wj(k)=(μj(k),vj(k),πj(k)) be an IFN assigned to criterion Xj by the kth DM.
The weights of the criteria can be calculated as follows:(3.4)wj=IFWAλ(wj(1),wj(2),…,wj(l))=λ1wj(1)⨁λ2wj(2)⨁λ3wj(3)⨁⋯⨁λlwj(l)=[1-∏k=1l(1-μj(k))λk,∏k=1l(vj(k))λk,∏k=1l(1-μj(k))λk-∏k=1l(vj(k))λk].
Thus, a vector of criteria weight is obtained:W=[w1,w2,w3,…,wj], where wj=(μj,vj,πj)(j=1,2,…,n).Step 4.
Construct the aggregated weighted IFDM.
In Step4, the weights of criteria (W) and the aggregated IFDM are determined to the aggregated weighted IFDM which is constructed according to the following definition [19]:
(3.5)R′=R⨁W=(μij′,vij′)={〈x,μij⋅μj,vij+vj-vij⋅vj〉},πij′=1-vij-vj-μij⋅μj+vij⋅vj.R′ is a matrix composed with elements IFNs, rıj′=(μij′,vij′,πij′)(i=1,2,…,m;j=1,2,…,n).Step 5.
Determine intuitionistic fuzzy positive and negative ideal solution.
In this step, the intuitionistic fuzzy positive ideal solution (IFPIS) and intuitionistic fuzzy negative ideal solution (IFNIS) have to be determined. LetJ1 and J2 be benefit criteria and cost criteria, respectively. A* is IFPIS and A- is IFNIS. Then A* and A- are equal to
(3.6)A*=(r1′*,r2′*,…,rn′*),rj′*=(μj′*,vj′*,πj′*),j=1,2,…,n,A-=(r1′-,r2′-,…,rn′-),rj′-=(μj′-,vj′-,πj′-),j=1,2,…,n,
where
(3.7)μj′*={(max⁡i⁡{μij′}j∈J1),(min⁡i⁡{μij′}j∈J2)},vj*={(min⁡i⁡{vij′}j∈J1),(max⁡i⁡{vij′}j∈J2)},πj′*={(1-max⁡i⁡{μij′}-min⁡i⁡{vij′}j∈J1),(1-min⁡i⁡{μij′}-max⁡i⁡{vij′}j∈J2)},μj′-={(min⁡i⁡{μij′}j∈J1),(max⁡i⁡{μij′}j∈J2)},vj′-={(max⁡i⁡{vij′}j∈J1),(min⁡i⁡{vij′}j∈J2)},πj′-={(1-min⁡i⁡{μij′}-max⁡i⁡{vij′}j∈J1),(1-max⁡i⁡{μij′}-min⁡i⁡{vij′}j∈J2)}.Step 6.
Determine the separation measures between the alternative.
Separation between alternatives on IFS, distance measures proposed by Atanassov [63], Szmidt and Kacprzyk [64], and Grzegorzewski [65] including the generalizations of Hamming distance, Euclidean distance and their normalized distance measures can be used. After selecting the distance measure, the separation measures, Si* and Si-, of each alternative from IFPIS and IFNIS, are calculated:
(3.8)Si*=12∑j=1n[|μij′-μj′*|+|vij′-vj′*|+|πij′-πj′*|],Si-=12∑j=1n[|μij′-μj′-|+|vij′-vj′-|+|πij′-πj′-|].Step 7.
Determine the final ranking.
In the final step, the relative closeness coefficient of an alternativeAi with respect to the IFPIS A* is defined as follows:
(3.9)Ci*=Si-Si*+Si-,where0≤Ci*≤1.
The alternatives were ranked according to descending order of Ci*’s score.


## 3.2. DIFWA

The DIFWA method, proposed by Xu and Yager [33], is a suitable way to deal with problem in IFE. The procedure for DIFWA method has been given as follows.Step 1.
Utilize the DIFWA operator(3.10)rij=DIFWAλ(t)(rij(t1),rij(t2),…,rij(tp))=(1-∏k=1p(1-μrij(tk))λ(tk),∏k=1pvrij(tk)λ(tk),∏k=1p(1-μrij(tk))λ(tk)-∏k=1pvrij(tk)λ(tk)).
to aggregate all the intuitionistic fuzzy matrix R(tk)=(rij(tk))m×n(k=1,2,…,p) into a complex IFDM:
(3.11)R=(rij)m×n,whererij=(μij,vij,πij),μij=1-∏k=1p(1-μrij(tk))λ(tk),vij=∏k=1pvrij(tk)λ(tk),πij=∏k=1p(1-μrij(tk))λ(tk)-∏k=1pvrij(tk)λ(tk),i=1,2,…,n,j=1,2,…,m.Step 2.
Defineα+=(α1+,α2+,…,αm+)Tandα-=(α1-,α2-,…,αm-)T as the IFPIS and the IFNIS, respectively, where α+=(1,0,0)(i=1,2,…,m) are the m largest IFNs and α-=(0,1,0)(i=1,2,…,m) are the m smallest IFNs. Furthermore, for convenience of depiction, we denote the alternative xi(i=1,2,…,n) by xi=(ri1,ri2,…,rim)T,i=1,2,…,n.Step 3.
Calculate the distance between the alternativexi and the IFIS and the distance between the largest native xi and the IFNIS, respectively:
(3.12)d(xi,α+)=∑j=1mwjd(rij,αj+)=12∑j=1mwj(|μij-1|+|vij-0|+|πij-0|)=12∑j=1mwj(1-μij+vij+πij)=12∑j=1mwj(1-μij+vij+1-μij-vij)=∑j=1mwj(1-μij),d(xi,α-)=∑j=1mwjd(rij,αj-)=12∑j=1mwj(|μij-0|+|vij-1|+|πij-0|)=12∑j=1mwj(1-μij-vij+πij)=12∑j=1mwj(1+μij-vij+1-μij-vij)=∑j=1mwj(1-vij),
where rij=(μij,vij,πij),i=1,2,…,n,j=1,2,…,m.Step 4.
Calculate the closeness coefficient of each alternative:(3.13)c(xi)=d(xi,α-)d(xi,α+)+d(xi,α-),i=1,2,…,n.
Since
(3.14)d(xi,α+)+d(xi,α-)=∑j=1mwj(1-μij)+∑j=1mwj(1-vij)=∑j=1mwj(2-μij-vij)=∑j=1mwj(1+πij),
then (3.13) can be rewritten as
(3.15)c(xi)=∑j=1mwj(1-vij)∑j=1mwj(1-πij),i=1,2,…,n.Step 5.
Rank all the alternativesxi(1,2,…,n) according to the closeness coefficients c(xi)(1,2,…,n), the greater the value c(xi), the better the alternative xi.


## 4. Case Study

In this section, we will describe how an IFT method was applied via an example of  selection  of  the most appropriate projects. Criteria to be considered in the selection of projects are determined by the expert team from a construction group. In our study, we employ six evaluation criteria. The attributes which are considered here in assessment ofPi(i=1,2,…,6) are (1)C1 is benefit and (2)C2,…,C6 are cost. The committee evaluates the performance of projects Pi(i=1,2,…,6) according to the attributes Cj(j=1,2,…,6), respectively. Criteria are mainly considered as follows(i)
net present value(C1),(ii)
quality(C2),(iii)
duration(C3),(iv)
contractor’s rank(C4),(v)
contractor’s technology(C5),(vi)
contractor’s economic status(C6).Therefore, one cost criterion,C1 and five benefit criteria, C2,…,C6 are considered. After preliminary screening, six projects P1,P2,P3,P4,P5, and P6 remain for further evaluation. A team of four DMs such as DM1, DM2, DM3, and DM4 has been formed to select the most suitable project.Now utilizing the proposed IFT to prioritize these construction projects, the following steps were taken.Degree of the DMs on group decision, shown in Table1, and linguistic terms used for the ratings of the DMs and criteria, as Table 2, respectively.Table 1
Linguistic term for rating DMs.
Linguistic termsIFNsVery important(0.80, 0.10)Important(0.50, 0.20)Medium(0.50, 0.50)Bad(0.3, 0.50)Very Bad(0.20, 0.70)Table 2
The importance of DMs and their weights.
DM1DM2DM3DM4Linguistic termsVery importantMediumImportantImportantWeight0.3420.2740.1920.192Construct the aggregated IFDM based on the opinions of DMs and the linguistic terms shown in Table3.Table 3
Linguistic terms for rating the alternatives.
Linguistic termsIFNsExtremely good (EG)[1.00; 0.00; 0.00]Very good (VG)[0.85; 0.05; 0.10]Good (G)[0.70; 0.20; 0.10]Medium bad (MB)[0.50; 0.50; 0.00]Bad (B)[0.40; 0.50; 0.10]Very bad (VB)[0.25; 0.60; 0.15]Extremely bad (EB)[0.00, 0.90,0.10]The ratings given by the DMs to six projects were shown in Table4.Table 4
The ratings of the projects.
AlternativeCriteriaDM1DM2DM3DM4P1C1VGVGGGC2GVGMBMBC3VGGBVGC4VGVGGGC5VGVGMBGC6GVGMBMBP2C1GVGMBBC2VGVGGMBC3VGVGBBC4VGVGMBGC5GGGGC6VGVGMBBP3C1VGVGGVGC2VGGGVGC3VGGVGGC4VGVGVGVGC5VGVGVGVGC6VGVGVGVGP4C1MBGMBVGC2GVGGGC3MBVGGGC4VGGVGVGC5VGVGGVGC6VGVGVGVGP5C1GMBMBVBC2MBBBVBC3BGMBVGC4MBBVBVGC5BMBVBVGC6MBVBVBMBP6C1VGVGMBMBC2GVGBMBC3VGVGBMBC4VGVGBMBC5VGVGVBVGC6VGVGGMBThe aggregated IFDM based on aggregation of DMs’ opinions was constructed as follows:(4.1)C1C2C3R=A1A2A3A4A5A6[(0.80,0.08,0.12)(0.69,0.20,0.11)(0.76,0.12,0.12)(0.68,0.20,0.12)(0.78,0.11,0.11)(0.74,0.13,0.13)(0.82,0.07,0.11)(0.79,0.10,0.11)(0.79,0.10,0.11)(0.83,0.16,0.1)(0.75,0.14,0.11)(0.70,0.19,0.11)(0.55,0.38,0.07)(0.42,0.52,0.06)(0.64,0.40,0.06)(0.75,0.13,0.12)(0.69,0.19,0.12)(0.75,0.13,0.12)]C4C5C6×A1A2A3A4A5A6[(0.80,0.09,0.11)(0.78,0.11,0.11)(0.69,0.20,0.11)(0.78,0.11,0.11)(0.69,0.21,0.10)(0.75,0.13,0.12)(0.84,0.05,0.11)(0.84,0.05,0.11)(0.84,0.05,0.11)(0.81,0.08,0.11)(0.82,0.07,0.11)(0.85,0.05,0.10)(0.55,0.33,0.12)(0.54,0.33,0.13)(0.40,0.54,0.06)(0.75,0.13,0.12)(0.85,0.05,0.10)(0.78,0.11,0.11)].The linguistic terms shown in Table5 were used to rate each criterion. The importance of the criteria represented as linguistic terms was shown in Table 6.Table 5
The linguistic terms for the importance of the criteria.
Linguistic termsIFNsVery good (VG)[0.80; 0.10]Good (G)[0.50; 0.20]Medium good (G)[0.50; 0.50]Very bad (VB)[0.30; 0.50]Bad (B)[0.20; 0.60]Table 6
The importance weight of the criteria.
CriteriaDM1DM2DM3DM4C1GVGVGMBC2VGVGVGVGC3MBGVGMBC4GGVGGC5GVGMBGC6MBGMBVGThe opinions of DMs on criteria were aggregated to determine the weight of each criterion:(4.2)W{X1,X2,X3,X4,X5,X6}=[(0,71,0.19,0.10)(0,90,0.00,0.10)(0,65,0.27,0.80)(0,78,0.11,0.11)(0,80,0.10,0.10)(0,67,0.24,0.9)]T.
After the weights of the criteria and the rating of the projects were determined, the aggregated weighted IFDM was constructed as follows:
(4.3)C1C2C3R′=A1A2A3A4A5A6[(0.57,0.26,0.18)(0.62,0.20,0.18)(0.49,0.36,0.19)(0.48,0.35,0.17)(0.70,0.11,0.19)(0.48,0.37,0.15)(0.58,0.25,0.17)(0.70,0.10,0.19)(0.51,0.34,0.14)(0.59,0.32,0.09)(0.70,0.14,0.19)(0.45,0.41,0.14)(0.39,0.50,0.11)(0.38,0.52,0.10)(0.42,0.56,0.02)(0.53,0.30,0.17)(0.62,0.19,0.19)(0.49,0.36,0.15)]C4C5C6×A1A2A3A4A5A6[(0.62,0.19,0.19)(0.62,0.20,0.18)(0.46,0.39,0.15)(0.61,0.21,0.18)(0.55,0.29,0.16)(0.50,0.34,0.16)(0.66,0.16,0.19)(0.67,0.15,0.18)(0.56,0.28,0.16)(0.63,0.18,0.19)(0.57,0.16,0.18)(0.57,0.28,0.15)(0.43,0.40,0.17)(0.43,0.40,0.17)(0.27,0.65,0.08)(0.59,0.23,0.19)(0.70,0.14,0.18)(0.52,0.33,0.15)].The net present value is cost criteriaj1={X1}, and quality, duration, contractor’s rank, contractor’s technology, and contractor’s economic status are benefit criteria j1={X2,X3,X4,X5}.Then IFPIS and IFNIS were provided as follows:(4.4)A*={(0.59,0.25,0.16),(0.71,0.10,0.19),(0.51,0.34,0.15),(0.66,0.15,0.18),(0.68,0.14,0.18),(0.57,0.28,0.15)},A-={(0.39,0.5,0.11),(0.38,0.5,0.12),(0.42,0.56,0.02),(0.43,0.4,0.17),(0.43,0.4,0.17),(0.27,0.65,0.08)}.
Negative and positive separation measures based on normalized Euclidean distance for each project and the relative closeness coefficient were calculated in Table 7.Table 7
Separation measures and the relative closeness coefficient of each project.
AlternativesS*S-Ci*P10.361.380.79P20.421.350.77P30.041.730.98P40.231.540.87P50.180.020.01P60.31.460.83Six projects were ranked according to descending order ofCi*’s. The result score is always the bigger the better. As visible in Table 6, project 3 has the largest score, and project 5 has the smallest score of the six projects which is ranked in the last pace. The projects were ranked as P3 >P4 >P6 >P1> P2 > P5. Project 3 was selected as appropriate project among the alternatives.In the second part, we utilize the proposed DIFWA to prioritize these construction projects, and the following steps were taken.First, utilize the DIFWA to aggregate all the IFDMR(tk) into a complex IFDM R:
(4.5)C1C2C3R′=A1A2A3A4A5A6[(0.57,0.26,0.18)(0.62,0.20,0.18)(0.49,0.36,0.19)(0.48,0.35,0.17)(0.70,0.11,0.19)(0.48,0.37,0.15)(0.58,0.25,0.17)(0.70,0.10,0.19)(0.51,0.34,0.14)(0.59,0.32,0.09)(0.70,0.14,0.19)(0.45,0.41,0.14)(0.39,0.50,0.11)(0.38,0.52,0.10)(0.42,0.56,0.02)(0.53,0.30,0.17)(0.62,0.19,0.19)(0.49,0.36,0.15)]C4C5C6×A1A2A3A4A5A6[(0.62,0.19,0.19)(0.62,0.20,0.18)(0.46,0.39,0.15)(0.61,0.21,0.18)(0.55,0.29,0.16)(0.50,0.34,0.16)(0.66,0.16,0.19)(0.67,0.15,0.18)(0.56,0.28,0.16)(0.63,0.18,0.19)(0.57,0.16,0.18)(0.57,0.28,0.15)(0.43,0.40,0.17)(0.43,0.40,0.17)(0.27,0.65,0.08)(0.59,0.23,0.19)(0.70,0.14,0.18)(0.52,0.33,0.15)].
Denote the IFIS, IFNIS, and the alternatives by
(4.6)α+=((1,0,0),(1,0,0),(1,0,0))T,α-=((0,1,0),(0,1,0),(0,1,0))T,
and calculate the closeness coefficient of each alternative:
(4.7)C(P1)=0.622,C(P2)=0.618,C(P3)=0.671,C(P4)=0.650,C(P5)=0.447,C(P6)=0.633.
Rank all the projects according to the closeness coefficients.The projects were ranked asC(P3)>C(P4)>C(P6)>C(P1)>C(P2)>C(P5). The greater value of C(Xi), the better alternative; thus the best alternative is also project 3.


## 5. Conclusion

The IFT and DIFWA have been emphasized in this paper which occurs in construction projects evaluation. In the evaluation process, the ratings of each project, given with intuitionistic fuzzy information, were represented as IFNs. The IFWA operator was used to aggregate the rating of DM. In project selection problem the project’s information and performance are usually uncertain. Therefore, the decision makers are unable to express their judgment on the project with crisp value, and the evaluations are very often expressed in linguistic terms. IFT and DIFWA are suitable ways to deal with MCDM because it contains a vague perception of DMs’ opinions. An actual life example in construction sector was illustrated, and finally the result is as follows Among 6 construction projects with respect to 6 criteria, after using these two methods, the best one is project 3 and project 4, project 6, project 1, project 2, project 5 will follow it, respectively. The presented approach not only validates the methods, as it was originally defined in Boran and Xu in a new application field that was the evaluation of construction projects, but also considers a more extensive list of benefit and cost-oriented criteria, suitable for construction project selection. Finally, the IFT and DIFWA methods have capability to deal with similar types of the same situations with uncertainty in MCDM problems such as ERP software selection and many other areas.



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*Source: 824265-2012-07-03.xml*