The Intuitionistic Fuzzy ELECTRE model

Abstract

The purpose of this research is to postulate and define a new model for Multi-Criteria Decision-Making (MCDM) problems utilizing the Intuitionistic Fuzzy ELimination Et Choix Traduisant la RÉalité (IFELECTRE) method, otherwise identified as the Intuitionistic Fuzzy Index of Hesitation Degree method. The Intuitionistic Fuzzy Sets (IFS) method offers certain advantages in using vagueness over a Fuzzy Set (FS): the IFELECTRE method is used to handle more complicated problems, whereas the Decision-Makers (DMs) have some vagueness in assigning option values to the objects considered. The processes of evaluating qualitative and quantitative scales are combined in this work and the proposed model enables different DMs to assess and use IFS. The original ELECTRE method cannot be operated effectively owing to a lack of precise information under different conditions.

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The Intuitionistic Fuzzy ELECTRE model
Babak Daneshvar Rouyendegh (Babek Erdebilli)
To cite this article: Babak Daneshvar Rouyendegh (Babek Erdebilli) (2017): The Intuitionistic
Fuzzy ELECTRE model, International Journal of Management Science and Engineering
Management, DOI: 10.1080/17509653.2017.1349625
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INTERNATIONAL JOURNAL OF MANAGEMENT SCIENCE AND ENGINEERING MANAGEMENT, 2017
https://doi.org/10.1080/17509653.2017.1349625
The Intuitionistic Fuzzy ELECTRE model
Babak DaneshvarRouyendegh (Babek Erdebilli)a,b
aDepartment of Industrial Engineering, Atilim University, Ankara, Turkey; bDepartment of Industrial Engineering, Ankara Yildirim Beyazit University,
Ankara, Turkey
ABSTRACT
The purpose of this research is to postulate and dene a new model for Multi-Criteria Decision-
Making (MCDM) problems utilizing the Intuitionistic Fuzzy ELimination Et Choix Traduisant la RÉalité
(IFELECTRE) method, otherwise identied as the Intuitionistic Fuzzy Index of Hesitation Degree
method. The Intuitionistic Fuzzy Sets (IFS) method oers certain advantages in using vagueness over
a Fuzzy Set (FS): the IFELECTRE method is used to handle more complicated problems, whereas the
Decision-Makers (DMs) have some vagueness in assigning option values to the objects considered. The
processes of evaluating qualitative and quantitative scales are combined in this work and the proposed
model enables dierent DMs to assess and use IFS. The original ELECTRE method cannot be operated
eectively owing to a lack of precise information under dierent conditions.
1. Introduction
Multi-Criteria Decision-Making (MCDM) is used to solve risk
assessment problems with multiple criteria, to help nd the best
alternatives. Solving complex problems using MCDM methods
with several criteria is one of the fastest developing elds in the
process of research. MCDM has also been described as the most
popular eld of decision-making processes (Triantaphyllou,
2000; Yang & Hung, 2007).
However, classical MCDM methods cannot work eectively
with uncertain and imprecise data in decision problems. Zadeh
(1965) proposed Fuzzy Set eory (FST), which is implemented
on MCDM problems. It is an eective tool to use for uncertain
and imprecision data and is more natural for humans to express
than mathematical rules. Also, in the actual world, knowledge
is oen more fuzzy than precise (Vahdani & Hadipour, 2011).
Using classical FST, it is very hard for DMs to quantify
team opinions precisely as a number between zero and one.
erefore, it would be more appropriate to show opinion
degree through Intuitionistic Fuzzy (IF) models (Atanassov,
1986, 1999).
e Fuzzy Set (FS) is an appropriate tool to deal with vague-
ness. Also, Intuitionistic Fuzzy Sets (IFS) is characterized by
three functions: (a) membership, (b) non-membership, and
(c) hesitancy.
In real world applications, when evaluating alternatives,
DMs may not be able to express their preferences completely
and exactly because of the fact that they may not have sucient
access to information regarding candidate alternatives (Xu &
Liao, 2014).
In recent years, the IFS has been applied in many elds
such as medical diagnosis (De, Biswas, & Roy, 2001; Szmidt &
Kacprzyk, 2000; Szmidt & Kacprzyk, 2001), decision-making
problems (Atanassov, Pasi, & Yager, 2005; Hong & Choi, 2000;
Hung & Yang, 2004; Liu & Wang, 2007; Szmidt & Kacprzyk,
2002, 2003; Wang, 2009; Xu, 2007a, 2007b, 2007c; Xu & Yager,
2008), pattern recognition (Li & Cheng, 2002; Liang & Shi,
2003; Vlachos & Sergiadis, 2007; Wang & Xin, 2005; Zhang &
Fu, 2006), supplier selection (Boran, 2011; Rouyendegh, 2015),
personnel selection (Boran, Genc, & Akay, 2011), facility loca-
tion selection (Boran, Boran, & Menlik, 2012), and evaluation
of renewable energy (Roy, 1968).
e ELECTRE which was originally devised by Roy
(Benayoun & Billsberry, 2007) is a decision-making process
widely used in operations management for determination
predilection and priorities within complex problems. e
ELECTRE method supports the DM covering problems with
qualitative or quantitative criteria.
e acronym ELECTRE stands for ELimination Et Choix
Traduisant la RÉalité (Andriosopoulos, Gaganis, Pasiouras,
& Zopounidis, 2012; Armaghan & Renaud, 2012; Hashemi,
Hajiagha, Zavadskas, & Mahdiraji, 2016; Roy, 1985; Roy &
Bertier, 1973) and was at rst cited ELECTRE for trading
reasons. is approach has evolved into dierent variants.
Now, dierent types are known as ELECTRE I, ELECTRE IS,
ELECTRE II (Roy, 1978), and ELECTRE III (Beccali, Cellura,
& Ardente, 1998; Chen, 2016).
e traditional ELECTRE model provides users with the
ability to take ordered numerical scales into consideration
without translating the original scales into esoteric rankings
with undetermined series. Furthermore, indierence and
preference thresholds, which were not possible to determine
in the previously mentioned models, could be accounted for
in computing awed information. e ELECTRE method is
classied in two steps; building the outranking relations, and
the outranking relations (Kahraman, Ruan, & Doğan, 2003).
As the IFS theory has expanded in both depth and scope,
it is natural to investigate ELECTRE within the context of IF.
Now we present IFELECTRE and develop an algorithm for
handling detailed MCDM problems. is algorithm takes
into consideration decisions provided by DMs and dierent
criteria based on ranking the alternatives fully. To do so, this
paper is organized as follows: Section 2 describes FST, and
© 2017 International Society of Management Science and Engineering Management
KEYWORDS
Multi-Criteria Decision-
Making (MCDM); Decision-
Makers (DMs); ELimination
Et Choix Traduisant la RÉalité
(ELECTRE); Intuitionistic
Fuzzy Set (IFS)
ARTICLE HISTORY
Received 13 July 2016
Accepted28 June 2017
CONTACT Babek Erdebilli babekd@atilim.edu.tr
JEL CLASSIFICATION
MCDM; DMs; ELECTRE; IFS
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2 B. ERDEBILLI
IFELECTRE. Some basic knowledge about the intuitionistic
preference relation is also given briey in the second section.
Section 3 presents a model that integrates IF and ELECTRE. An
illustrative example about New Plant (NP) is given in Section
4, and Section 5 concludes the paper.
2. Materials and methods
2.1. Fuzzy set theory (FST)
Zadeh (1965) introduced Fuzzy Set eory (FST) to deal with
uncertainty. A basic contribution of FST is the ability to repre-
sent uncertain and vague data. FST also allows mathematical
operators and programming to be performed within a fuzzy
environment.
Denition 1: A Fuzzy Set (FS) is a set of objects with a process
for assigning grades of membership. Such a set is character-
ized by membership functions, which assign to each object
a grade of membership between zero and one (Rouyendegh,
2012; Zadeh, 1965).
Denition 2: If a symbol represents an FST, a tilde (~) will be
placed over the symbol. A triangular fuzzy number (TFN)
̃
M
(tilde over M) is denoted simply as (x,y,z), in which x≤y≤z,
as shown in Figure 1. e parameters x, y, and z, respectively,
denote the smallest possible value x, the most promising value
y, and the largest possible value z that dene a fuzzy event
(Kahraman, Ruan, & Ethem, 2002). e α cut is the condence
level over the judgments.
e membership function of a TFN is as follows.
Denition 3: Each TFN has linear presentations on its le
and right sides, or lower bound and upper bound, of the TFN
such that its membership function can be described as (Boran,
2011):
e le and right representations of each degree of member-
ship are as follows:
where x(α) and y(α) denote the le and right side representa-
tions of a Fuzzy Number(FN), respectively. Many methods
for FNs have been developed in the literature. ese methods
may provide dierent ranking results (Boran, 2011; Kahraman
et al., 2002; Rouyendegh & Erol, 2012).
A={⟨r,
𝜇
A(r),
𝜐
A(r)⟩
⋮
r∈R}.
𝜇
(𝛼∕
⌣
M)
⎧
⎪
⎪
⎨
⎪
⎪
⎩
0, a<x
(a−x)∕(y−x),x≤=a≤=
y
(z−a)∕(z−y),y≤=a≤=z
0, a>z.
⌣
M
=xa,ya=

x+

y−x

a,z+

y−z

a

,a∈[0, 1]
,
Denition 4: Let
̃
M1
and
̃
M2
be two TFNs. While there are var-
ious operations on TFNs, only the important ones are utilized
in this work. TFNs are given as follows (Singh & Yadav, 2015):
2.2. Intuitionistic Fuzzy Set (IFS)
e IFS method was developed by Atanassov (1986) as an
appropriate way to deal with uncertainty. In this section, we
review and describe some necessary introductory concepts
related to IFS.
Denition 5: Let X be a nite set, and let A⊂ X be a xed set
(Atanassov, 1986, 1999). e inside of X can be described as
where
A membership function and a non-membership function are
explained by the following equation:
Denition 6: e third boundary of IFS is
𝜋x(s)
, meaning hesi-
tation degree, depending on X’s subordination to A (Atanassov,
1986, 1999; Büyüközkan & Güleryüz, 2016):
where
𝜋x(s)
is dened as the degree of uncertainty of x to A.
Let A and B be IFSs of the set S, then the multiplication
operators are, correspondingly (Atanassov, 1986):
3. The proposed IFELECTRE model
In this section, we present and oer an IFELECTRE model for
the evaluation of the units’ performance. An algorithm takes into
consideration the judgments provided by dierent DMs and, later,
contains the quantitative and the qualitative data. erefore, experts
are asked to compare each alternative with a criterion. (Figure 2)
Now we replace the algorithm with a nine-step procedure
as follows.
Denition 7: Symbols
IFDM: Intuitionistic Fuzzy Decision Matrix,
IFWA: Intuitionistic Fuzzy Weighted Averaging,
μij: Degree of membership,
νij: Degree of non-membership,
πij: Degree of hesitation,
R: An Intuitionistic Fuzzy Decision Matrix,
⌣
M1
=
(
x
1
,y
1
,z
1)
,
⌣
M
2
=
(
x
2
,y
2
,z
2)
⌣
M1
+
⌣
M
2
=
(
x
1
+x
2
,y
1
+y
2
,z
1
+z
2)
⌣
M1
−
⌣
M
2
=
(
x
1
−x
2
,y
1
−y
2
,z
1
−z
2)
⌣
M1
∗
⌣
M
2
=
(
x
1
∗x
2
,y
1
∗y
2
,z
1
∗z
2)
k
∗
⌣
M
1
=
(
kx
1
,ky
1
,kz
1)
, where k>
0
k
∗
⌣
M
1
=
(
kz
1
,ky
1
,kx
1)
, where k<
0.
X
=
{
s,𝜇
x
(s),v
x
(s)
|
s∈S
},
𝜇
x
(s):
𝜇x
(s)∈[0, 1],S
→
[0, 1]
vx
(s):v
x
(s)∈[0, 1],S→[0, 1]
.
0≤𝜇x(s)+vx(s)≤0∀s∈S,R
→
[0, 1].
𝜋x(s)=1−𝜇x(s)−vx(s),
X
+Y=
{
𝜇x(s)⋅𝜇y(s),vx(s)+vy(s)−vx(s)⋅vy(s)
|
s∈S
}.
Figure 1.A TFN
̃
M
.
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INTERNATIONAL JOURNAL OF MANAGEMENT SCIENCE AND ENGINEERING MANAGEMENT 3
Cxy: Concordance Index,
Dxy: Discordance Index,
E: Concordance Matrices,
F: Discordance Matrices,
L: Concordance Dominances Matrices,
K: Discordance Dominances Matrices,
T: Aggregate Dominances Matrix.
In this study, all criteria importance as well as criteria weight
are well-considered in the form of 1–9 scales (Saaty, 1996).
Also, the DM importance and alternatives weights are consid-
ered as linguistic variables. We identify the alternative, criteria,
and DMs. Let A be a set of alternatives, C a set of criteria, and
L represent DecisionMakers (DMs).
Step 1. Determine weight of criteria
Assuming equal weights of criteria, we cannot assume that
alternatives have equal importance. Let W = {w1, w2,...,wn} be
a set of weights designating a set of position ranking positions.
e decision group or the DMs are given the task of forming
individual pairwise comparisons by using a standard 1–9 scale
(Table 1). To ascertain W, all of the individual DMs’ estimations
for the weight of the individual criterion must be determined.
Both distances from each DM can be aggregated as the dis-
tances of the DMs by taking geometric mean:
D
i=

k

j=1
Dij
1∕k
i=1, …,
m
where W = {w1,w2,...,wn}
is an element of an Aggregated Intuitionistic Fuzzy Decision
Matrix (AIFDM).
Step 2. Calculate the weights of the DMs
e dierent inputs given by DMs within a group do not
necessarily carry equal weight or value and are referred to as
linguistic terms in IFNs in Table 2. Next, the collected opinions
of the experts are summed up using the following equations:
where
𝜆l∈[0, 1]
and
∑k
l=1
𝜆
l
=
1.
Step 3. Determine an AIFDM based on the DMs’ opinions
Using a DM’s weight as the base equation, the AIFDM is
determined by employing an IFWA operator (Grzegorzewski,
2004; Szmidt & Kacprzyk, 2000). In order to reach a precise
conclusion, each individual opinion obtained from a group
of DMs should be merged into a single opinion to build the
AIFDM model. In this case, each candidate receives a rating
given by the criterion found in the linguistic terms from Table 3.
Let
R(l)
=(r
l
ij
)
m∗n
be the IFDM of each DM.
λ={λ1,λ2,λ3,…,λk} is the weight of the DM.
where
erefore, the decision problem is expressed in matrix format
and should be identied ass
Rij = (μij,vij,πij) (i=1,2,...,m; j=1,2,...,n) is an element of
an AIFDM.
𝜆
l=

𝜇l+𝜋l

𝜇l
𝜇l+vl


k
l=1

𝜇l+𝜋l

𝜇l
𝜇
l+
v
l,
R=(rij)m
�
∗n
�
,
r
ij =IFWAr𝜆
(
r(1)
ij ,r(2)
ij ,…,r(k)
ij
)
=𝜆1r(1)
ij +𝜆2r(2)
ij ,…,𝜆kr(k
)
ij
=[
1−
k
∏
l
=1
(1−𝜇(l)
ij )𝜆l,
k
∏
l
=1
(v(l)
ij )𝜆l,
k
∏
l
=1
(1−𝜇(l)
ij )𝜆l−
k
∏
l
=1
(v(l)
ij )𝜆l
].
R
ij =
⎡⎢⎢⎢⎣
r11 ⋯r1j
⋮⋱⋮
ri1⋯rij
⎤⎥⎥⎥⎦
.
Figure 2.The proposed classification IFELECTRE procedure.
Table 1.The fundamental scale of numbers 1–9 (Saaty, 1996).
Importance Denition
1 Same significance
3 Avarage significance of one over another
5 Powerful significance of one over another
7 Very powerful significance of one over another
9 Extreme significance of one over another
2, 4, 6, 8 Intermediate values
Table 2.The importance weight of the criteria (DM1).
DM1C1C2C3C4weight
C11 2 3 4 0.463
C21/2 1 2 2 0.255
C31/3 1/2 1 2 0.177
C41/4 1/2 1/2 1 0.104
Table 3.The importance weight of the criteria (DM2).
DM2C1C2C3C4weight
C11 3 2 4 0.405
C21/3 1 3 3 0.330
C31/2 1/3 1 2 0.172
C41/4 1/3 1/2 1 0.094
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4 B. ERDEBILLI
Step 8. Calculate matrix T
Matrix (T), where
Step 9. At the last step the overall weight is determined, and
the best alternatives are selected:
4. Application of the case study
In recent decades, chosing new sites has been an important
and critical decision to be made by companies. Many potential
criteria, such as distance to market, general investment cost,
community considerations, maximizing enlargement possibil-
ity, human resources, availability of necessary material, climate,
etc. should be considered in selecting a new site. Linguistic
terms are used to assess the quality of the material neeeded
to build these plants, and these terms represent mathematical
criteria and their weights.
In the following section, we will document how to apply
an IFELECTRE algorithm model to a pilot study. A company
plans to choose the best site for its new location. Five locations
are considered for evaluation. An experienced group from the
decision team decided the evaluate criteria to be considered
in new plant performance. In our research, we used four cri-
teria and attributes (C1 C2,..., C4) for evaluating new plants
NPi (i=1,2,3,4,5). e performance of NP is evaluated by
the committee (i=1,2,3,4,5) according to the attributes Cj
(j=1,2,…,4). Five, NP1, NP2, NP3, NP4, and NP5, remain for
further evaluation.
Four members comprised the team of experts, and each
member was coded as DM1, DM2, DM3, DM4, correspond-
ingly. Each DM showed his/her assesment based on the 1–9
scale standard as shown in Tables 4, 5, 6, and 7, respectively.
Figure3 illustrates an hierarcial model example. We used four
criteria, which were the most suitable for assessing the selection
of a new location, all of which had been identied as the most
crucial obstacles by the team of experts.
ese criteria include: C1, distance to market; C2, gen-
eral cost; C3, community considerations; and C4, maximum
enlargement possibility.
e opinions of the DMs on the criteria are shown in Tables
2, 3, 4, and 5.
Kij =F∙−Fij, where F∙=the maximum value of Fij.
T
ij =
K
ij
L
ij
+Kij
.
w
−=
∑m
j=
1,
i=j
Tij X∗=Max
−
w
.
Step 4. Determine an aggregated weighted IFDM
Find the summation of the weight of the criteria (Step1)
and calculate the ratings given to the alternatives (Step 2). e
aggregated weighted intuitionistic fuzzy decision matrix is
established. W and R are two IFSs of the set X. e following
example gives the denition (Atanassov, 1986):
Step 5. Determine the concordance and discordance indexes
e concordance index Cxy shows the degree of con-
dence in the pairwise comparison of the x and y alternatives
(
Xx
→
Xy,x,y=1, 2, …,m;x≠y
) and is constructed as
follows:
Dxy
shows the degree of disagreement in (
Xx
→
Xy
) and is con-
structed as follows:
Step 6. Calculate the E and F matrices using the following
formula
e concordance index
Cxy
between Xx and Xy for the pro-
posed model uses the concept of IFS and is dened as
e discordance index
Dxy
for the proposed model uses the
concept of IFS and is dened as
e distance between Xil and Xlj is shown as
Step 7. Calculate the Lij and Kij matrices
e Lij and Kij matrices are measured as follows:
R�
ij
=w∗R
ij
;R
�
ij
=(𝜇
�
ij
,𝜈
�
ij
,𝜋
�
ij
)
is an element of an AIFDM.
C
1
xy =
{
l
|
𝜇xl ≥𝜇ly,vxl ≥vly , and 𝜋xl ≥𝜋ly
}
C
2
xy =
{
l
|
𝜇xl ≥𝜇ly,vxl <vly , and 𝜋xl ≥𝜋ly
}
C
3
xy =
{
l
|
𝜇xl ≥𝜇ly,vxl ≥vly
}
D
1
xy =
{
l
|
𝜇xl <𝜇
ly,vxl ≥vly , and 𝜋xl ≥𝜋ly
}
D
2
xy =
{
l
|
𝜇xl <𝜇
ly,vxl ≥vly and 𝜋xl ≥𝜋ly
}
D
3
xy =
{
l
|
𝜇xl <𝜇
ly,vxl <vly
}.
C
xy =w1
c∗
∑
l∈C1
xy
wl+w2
c∗
∑
l∈C2
xy
wl+w3
c∗
∑
l∈C3
xy
wl
.
D
xy =
max
j∈Dij
W
D
×dis(x
il
,x
lj
)
max
j∈L
dis (x
il
,x
lj
)
dis
(x
il
,x
lj
)=

1∕2

(𝜇
il
−𝜇
lj
)2+(v
il
−v
lj
)2+(𝜋
il
−𝜋
lj
)2
.
C
xy ≥
−
C, where
−
C=
∑m
x=1,x≠y
∑m
y=1,y≠xC
xy
m∗(m−1)
D
xy ≥
−
D, where
−
D=
∑m
x=1,x≠y
∑m
y=1,y≠xDxy
m∗(m−1).
Lij =E∙−Eij, where E∙=the maximum value of Eij
Table 4.The importance weight of the criteria (DM3).
DM3C1C2C3C4weight
C11 3 4 5 0.486
C21/3 1 3 3 0.274
C31/4 1/3 1 3 0.171
C41/5 1/3 1/3 1 0.069
Table 5.The importance weight of the criteria (DM4).
DM4C1C2C3C4weight
C11 4 3 5 0.440
C21/4 1 4 4 0.313
C31/3 1/4 1 4 0.189
C41/5 1/4 1/4 1 0.058
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INTERNATIONAL JOURNAL OF MANAGEMENT SCIENCE AND ENGINEERING MANAGEMENT 5
e concordance and discordance sets are as follows:
C
1
kl =
⎡
⎢⎢⎢⎢⎢⎣
−
−
2, 3, 4
3, 4
1, 3, 4
−
−
1, 2, 3
3, 4
1, 4
−
−
−
3, 4
1, 3, 4
1, 2
−
2
−
1
−
−
−
3
−
⎤
⎥⎥⎥⎥⎥⎦
,
C
2
kl =⎡⎢⎢⎢⎢⎢⎣
−
2
−
−
2
1, 3
−
3
−
2, 3
1
−
−
−
2
−
2
−
−
2
−
−
−
−
−
⎤⎥⎥⎥⎥⎥⎦
,
C
3
kl =⎡⎢⎢⎢⎢⎢
⎣
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
1
−
4
−
−
−
4
−
⎤⎥⎥⎥⎥⎥
⎦
e opinions of each DM regarding each criterion were
accumulated to ascertain the weight of each criterion:
e decision-makers also give the relative weight as follows:
Each DM presents her/his assessment for ratings of DM’s rating
performance by a linguistic term as shown in Table 2, and the
importance of the DM’s and their weight is shown in Table 8.
e degree of the DMs on the group decision is shown in
Table 6, and the linguistic terms used for the ratings of the DMs
are shown in Table 7, respectively.
Establish the aggregated IFDM based on the opinions of
DMs; the linguistic terms are shown in Table 8.
e ratings of the NPs were evaluated by the four DMs using
the linguistic terms dened in Table 9. For example, DM1 and
DM2 assigned a Very Good (VG) rating to NP1 based on the
System C
1
. However, DM
3
and DM
4
assigned a Good (G) rating
to NP1 based on the C1.
e aggregated IFDM based on aggregation of DMs’ opin-
ions were constructed as follows:
Wi=[0.449, 0.293, 0.177, 0.081].
W�
=[W
c
,W
�
c
,W
��
c
,W
D
,W
�
D
,W
��
d
]=[1, 2∕3, 1∕3, 1, 2∕3, 1∕3]
.
R
54 =⎡⎢⎢
⎢⎢⎢⎣
DM1
(0.805, 0.084, 0.111)
(0.699, 0.194, 0.107)
(0.784, 0.129, 0.087)
(0.762, 0.085, 0.153)
(
1.000, 0.000, 0.000
)
DM2
(0.713, 0.176, 0.111)
(0.727, 0.162, 0.111)
(0.784, 0.129, 0.087)
(0.698, 0.194, 0.108)
(
0.804, 0.085, 0.111
)
DM3
(0.759, 0.100, 0.141)
(0.699, 0.194, 0.107)
(0.794, 0.094, 0.112)
(1.000, 0.000, 0.000)
(
0.828, 0.065, 0.107
)
DM4
(0.805, 0.084, 0.111)
(0.727, 1.162, 0.111)
(0.851, 0.05, 0.099)
(1.000, 0.000, 0.000)
(
1.000, 0.000, 0.000
)
⎤⎥⎥⎥⎥⎥⎦
Table 6.Linguistic terms for rating DMs.
Linguistic terms IFNs
Very 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 7.The importance of DMs and their weights.
DM1DM2DM3DM4
Linguistic terms Very Important Medium Important Impor tant
Weight 0.342 0.274 0.192 0.192
Figure 3.Hierarchy for new plant selection.
Calculate the E and F matrices,
Calculate the Lij and Kij dominances matrices,
D
1
kl =
⎡
⎢⎢⎢⎢⎢⎣
−
4
−
1
−
2
−
−
−
−
2, 3, 4
1, 2, 4
−
2
−
3, 4
3, 4
3, 4
−
3
1, 2, 3, 4
1, 2, 3, 4
1, 3, 4
1
−
⎤
⎥⎥⎥⎥⎥⎦
,
D
2
kl =⎡⎢⎢⎢⎢⎢⎣
−
1, 3
1
2
−
−
−
−
2
−
−
3
−
−
−
−
1
−
−
−
−
−
2
2
−
⎤⎥⎥⎥⎥⎥⎦
,
D
3
kl =⎡⎢⎢⎢⎢⎢
⎣
−
−
−
−
−
4
−
−
−
−
−
−
−
1
−
−
−
−
−
−
−
−
−
−
−
⎤⎥⎥⎥⎥⎥
⎦
.
E
=
⎡
⎢⎢⎢⎢⎢⎣
−
0.195
0.550
0.258
0.902
0.417
−
1.037
0.557
0.843
0.299
0
−
0.258
0.902
0.688
0.195
0.443
−
0.671
0
0
0
0.204
−
⎤
⎥⎥⎥⎥⎥⎦
,
F
=⎡⎢⎢⎢⎢⎢
⎣
−
1
0.270
0.209
0
0.167
−
0
0.048
0
0.533
1
−
0.076
0
0.533
1
1
−
0.353
1
1
1
0.209
−
⎤⎥⎥⎥⎥⎥
⎦
.
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6 B. ERDEBILLI
the selection process, we determined the rank of each proxy by
IFNs. e ratings of the DMs was aggregated using an IFELECTRE
model. DMs are oen preferred over a single DM to lower partiality
in the decision process. In this pilot study, four criteria were selected
– C1, distance to market; C2, general cost; C3, community consid-
erations; and C
4
, maximize enlargement possibility – as the funda-
mental criteria for evaluating the alternatives expressed by DMs.
e proposed IFELECTRE algorithm is an eective model since
the involvement of fuzzy theory can adequately resolve the natural
ambiguity associated with the mapping of a DM’s perception of
precise numbers. Consequently, the postulated hybrid model not
only veries the methods used, but it also entertains a considerably
larger list of dierent criteria. e algorithm has the capacity to
handle similar situations and problems, and the integration of the
IF and ELECTRE methods can address the weaknesses found in
the MCDM model owing to its nature, and is able to assess deci-
sions made with uncertainty and vagueness. IFELECTRE allows
a more suitable description of the decision-making procedure,
making it more advantageous than the classical ELECTRE method.
e present study not only conrms the model, but also oers
a much more thorough list of dierent criteria acceptable for
new NP assessments. Furthermore, it is observed that the group
decision-making process is a useful practice for NP choice. e
IFELECTRE model has the ability to handle other similar situ-
ations that contain a degree of uncertainty in MCDM problems.
For further research, other MCDM methods can be applied and
their results can be compared with that of the proposed model.
Disclosure statement
No potential conict of interest was reported by the author.
ORCID
Babak Daneshvar Rouyendegh http://orcid.org/0000-0001-8860-3903
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Calculate the aggregate dominances matrices,
Aer using this model, numerical example results show
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55 =
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1
0.665
0.838
0.388
0
−
0.543
0.881
0.572
0
0
−
0.639
0
0
0
0.487
−
⎤
⎥⎥
⎥⎥⎥⎦
.
Table 8.Linguistic terms for rating the alternatives.
Linguistic terms IFNs
Extremely 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]
Table 9.The ratings of the alternatives.
Alternative Criteria DM1DM2DM3DM4
NP1C1VG VG G G
C2VG G MB MB
C3VG G MB G
C4VG VG G G
NP2C1G VG MB MB
C2G VG G MB
C3G VG MB MB
C4G VG MB G
NP3C1G VG VG G
C2G VG G VG
C3VG G G VG
C4VG VG VG VG
NP4C1VG VG MB MB
C2G VG MB MB
C3VG EG G MB
C4EG EG G MG
NP4C1EG EG G G
C2VG VG G G
C3VG VG G VG
C4VG EG G EG
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