ArticlePDF Available

ELECTRE I Method Using Hesitant Linguistic Term Sets: An Application to Supplier Selection

Authors:

Abstract and Figures

Decision making is a common process in human activities. Every person or organization needs to make decisions besides dealing with uncertainty and vagueness associated with human cognition. The theory of fuzzy logic provides a mathematical base to model the uncertainities. Hesitant fuzzy linguistic term set (HFLTS) creates an appropriate method to deal with uncertainty in decision making. Managerial decision making generally implies that decision making process conducts multiple and conflicting criteria. Multi criteria decision analysis (MCDA) is a widely applied decision making method. Outranking methods are one type of MCDA methods which facilitate the decision making process through comparing binary relations in order to rank the alternatives. Elimination et Choix Traduisant la Réalité (ELECTRE), means elimination and choice that translates reality, is an outranking method. In this paper, an extended version of ELECTRE I method using HFLTS is proposed. Finally, a real case problem is provided to illustrate the HFLTS-ELECTRE I method.
Content may be subject to copyright.
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=tcis20
Download by: [Istanbul Technical University] Date: 29 February 2016, At: 03:38
International Journal of Computational Intelligence
Systems
ISSN: 1875-6891 (Print) 1875-6883 (Online) Journal homepage: http://www.tandfonline.com/loi/tcis20
ELECTRE I Method Using Hesitant Linguistic Term
Sets: An Application to Supplier Selection
Ali Fahmi, Cengiz Kahraman & Ümran Bilen
To cite this article: Ali Fahmi, Cengiz Kahraman & Ümran Bilen (2016) ELECTRE I Method Using
Hesitant Linguistic Term Sets: An Application to Supplier Selection, International Journal of
Computational Intelligence Systems, 9:1, 153-167, DOI: 10.1080/18756891.2016.1146532
To link to this article: http://dx.doi.org/10.1080/18756891.2016.1146532
Published online: 03 Feb 2016.
Submit your article to this journal
Article views: 24
View related articles
View Crossmark data
Received 7 September 2015
Accepted 18 December 2015
ELECTRE I Method Using Hesitant Linguistic Term Sets: An Application to Supplier
Selection
Ali Fahmi1, Cengiz Kahraman2, Ümran Bilen3
1Istanbul Technical University, Department of Management Engineering, Macka, 34367 Istanbul, Turkey
E-mail: fahmi@itu.edu.tr
2Istanbul Technical University, Department of Industrial Engineering, Macka, 34367 Istanbul, Turkey
E-mail: kahramanc@itu.edu.tr
3Istanbul Technical University, Naval Architecture and Marine Engineering, Maslak, 34469 Istanbul, Turkey
E-mail: umranbilen@gmail.com
Abstract
Decision making is a common process in human activities. Every person or organization needs to make decisions
besides dealing with uncertainty and vagueness associated with human cognition. The theory of fuzzy logic
provides a mathematical base to model the uncertainities. Hesitant fuzzy linguistic term set (HFLTS) creates an
appropriate method to deal with uncertainty in decision making. Managerial decision making generally implies that
decision making process conducts multiple and conflicting criteria. Multi criteria decision analysis (MCDA) is a
widely applied decision making method. Outranking methods are one type of MCDA methods which facilitate the
decision making process through comparing binary relations in order to rank the alternatives. Elimination et Choix
Traduisant la Réalité (ELECTRE), means elimination and choice that translates reality, is an outranking method. In
this paper, an extended version of ELECTRE I method using HFLTS is proposed. Finally, a real case problem is
provided to illustrate the HFLTS-ELECTRE I method.
Keywords: Multiple criteria decision analysis (MCDA), outranking methods, ELECTRE method, hesitant fuzzy
linguistic terms set (HFLTS), ordered weighted averaging (OWA), computing with words (CWW)
1. Introduction
Multiple criteria decision analysis (MCDA) denotes
to analyzing decision making situations that encounter
with multiple and conflicting criteria. It refers to
analyzing tools and systematic approaches that
empower the decision maker (DM). They help DM to
represent his/her preferences and consider all subjective
and objective conditions to assess the decision elements
[1, 2]. MCDA methods provide a quantitative
infrastructure to model the assessments of criteria and
alternatives [3]. MCDA methods are classified into
three major categories including multi-attribute value
theory (MAVT), multi-objective mathematical
programming, and outranking methods [4, 5, 6, 7, 8, 9].
(1) MAVT approach concerns hierarchical decision
making problems with the overall goal on the top level
and the criteria on the lowest level. MAVT involves in
the weighting the criteria and assessment of alternatives
with respect to criteria. Lastly, MAVT approach ranks
the criteria through calculations of criteria weights and
alternative assessments.
(2) Multi-objective mathematical programming
methods focus on finding the aspiration points. These
points can generate non-dominated solutions by
scalarizing functions through reaching down from ideal
solution [10, 11, 12].
International Journal of Computational Intelligence Systems, Vol. 9, No. 1 (2016) 153-167
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
153
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
(3) Outranking methods form two main steps:
building the outranking relations, and exploiting the
outranking relations [13]. At first, outranking methods
systematically compare criteria. The binary comparisons
construct the concordance and discordance sets which
are linguistic comparisons. Then, these comparisons
lead to numerical concordance and discordance indices.
There are different outranking methods including
ELECTRE family, PROMETHEE family, QUALIFLES
[14], ORESTE [15, 16], MELCHIOR [17], PRAGMA
[18], MAPPACC [18], and TACTIC [19]. ELECTRE
family includes ELECTRE I [20, 21], ELECTRE II [22,
23], ELECTRE III [24, 25], ELECTRE IV [26],
ELECTRE IS [27], ELECTRE TRI [28], and
ELECTREGKMS [29]. PROMETHEE family contains
PROMETHEE I [30], and PROMETHEE II [30].
The PROMETHEE family of outranking methods
includes the PROMETHEE I for partial ranking of the
alternatives and the PROMETHEE II for complete
ranking of the alternatives, the PROMETHEE III for
ranking based on interval, the PROMETHEE IV for
complete or partial ranking of the alternatives when the
set of viable solutions is continuous, the PROMETHEE
V for problems with segmentation constraints, the
PROMETHEE VI for the human brain representation
[31].
ELECTRE family methods are the most widely used
outranking approaches [29, 32]. ELECTRE I outranking
method is the first ELECTRE method which was
introduced by Benayoun, Roy, and Sussman [20] and
Roy [21]. Decision makers use ELECTRE I to construct
a partial prioritization and choose a set of promising
alternatives [4]. ELECTRE II is used for ranking the
alternatives based on the determination of concordance
and discordance matrices for each criterion and
alternative pair. In ELECTRE III, an outranking degree
is established, representing an outranking creditability
between two alternatives which makes this method
more sophisticated. ELECTRE III is based on the
principle of fuzzy logic and uses the preference and
indifference thresholds while determining the
concordance and discordance indices [13].
Outranking methods are extended to deal with
imprecision and uncertainty of decision making process.
Roy [33] and Siskos, Lochard, and Lombard [34]
developed fuzzy outranking method. This method uses
fuzzy concordance and discordance relation. Sevkli [35]
applied fuzzy ELECTRE method for supplier selection.
Ertay and Kahraman [36] evaluated the design
requirements by utilizing fuzzy outranking methods.
Afterwards, different extensions of fuzzy sets have
been applied to outranking methods. Interval type-2
fuzzy sets have been employed in the ELECTRE
method [37, 38]. Chen [39] implemented an ELECTRE
based group decision making method using interval
type-2 fuzzy sets. Devi and Yadav [40] developed an
ELECTRE group decision making based on
intuitionistic fuzzy sets for plant location selection. See
Vahdani et al. [41], Li, Lin, and Chen [42], and Wue
and Chen [43] developed ELECTRE method extension
in an intuitionistic fuzzy environment. Hatami-Marbini
and Tavana [4] applied fuzzy group decision making to
extend ELECTRE I methodology. See Hatami-Marbini
and Tavana [4] for more details.
Torra [44] proposed hesitant fuzzy sets (HFS) as a
generalization of fuzzy sets. HFS are quite suitable in
decision making when experts have to assess a set of
alternatives. HFS were introduced to the literature for
the common difficulty that often appears when the
membership degree of an element must be established
and the difficulty is not because of an error margin (as
in intuitionistic fuzzy sets) or due to some possibility
distribution (as in type-2 fuzzy sets), but rather because
there are some possible values that cause hesitancy
about which one would be the right one [45].
These new sets permit DM to represent his/her
hesitation in decision making process. The
mathematical background of hesitation is modeled by
different membership functions. HFS establishes a
relationship between the envelopes of fuzzy sets and
generates new fuzzy sets with combined membership
functions.
As decision making encounters linguistic
information, we need to use computing with words
(CWW) processes [46, 47, 48]. HFS provides an
appropriate infrastructure to develop the concept of
hesitant linguistic term set which is a useful CWW
process. The experts involved in decision problems
under uncertainty cannot easily provide a single term as
an expression of his/her knowledge such as poor, good,
very good, etc. because they might consider several
terms at the same time or looking for a more complex
linguistic term such as at least medium poor, at most
very high, etc. Thus, Rodriguez, Martinez, and Herrara
[49], and Liu and Rodriguez [50] proposed the hesitant
fuzzy linguistic term set (HFLTS) and its application in
decision making. This method could increase the ability
and flexibility of linguistic elicitation of DM. HFLTS
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
154
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
enables DM to represent the uncertainty in his/her
assessments of actions, criteria, alternatives, etc. [49].
This would be implemented by context-free grammars
based on the fuzzy linguistic tools to compare the
expressions.
In this study, the ELECTRE I method is extended
based on HFLTS. This new approach could be applied
to decision making problems and deal with the
uncertainty and imprecision of multiple criteria decision
making. This method contains two phases: HFLTS
phase and ELECTRE I phase. Firstly, HFLTS method
considers DM’s hesitation between several values to
evaluate the decision elements and manage the
uncertainty of decision making. These hesitant linguistic
terms are expressed by numerical fuzzy values. Next, in
ELECTRE phase, we apply ELECTRE I method to
analyze the decision making. This would be
accomplished through building the outranking relations,
and then exploiting the outranking relations. Finally, we
can rank the alternatives and make an appropriate
decision.
It is worthy to mention that ELECTRE method has
an important pitfall [4]. DM must initiate this method by
precise measurement of the performance ratings and the
weights of criteria [5]. On the other hand, DM may
prefer to express his/her judgment by linguistic
expressions [51, 52, 53]. This approach takes this
weakness into account and modifies it by utilizing
linguistic expressions in decision making process and
allows DM to evaluate the real world problem with
imprecise and vague ratings.
This paper is organized as follows: Section 2
includes needed definitions and arithmetic operations
about HFLTS and ELECTRE I method. In Section 3,
the step-by-step methodology of hesitant fuzzy
linguistic term set ELECTRE I (HFLTS-ELECTRE I) is
proposed. Next, a real case study is added in Section 4.
Lastly, the conclusions and future works are presented
in Section 5.
2. Preliminaries
In this section, the basic definitions and arithmetic
operations are given.
2.1. Comparative Linguistic Expressions
In many decision making situations, DM deals with
uncertain and imprecise information to express his/her
judgment. Zadeh [54] presented fuzzy sets and then
developed the concept of linguistic variables [51].
Based on these concepts, Rodriguez, Martinez, and
Herrera [49] developed the concepts of fuzzy hesitant
linguistic terms set (HFLTS).
Definition 1. [51] A linguistic variable has five
characteristics and is defined as (ܪ,ܶ(ܪ),ܷ,ܩ,ܯ),
where His the name of variable, T(H) is the term set of
H, i.e., the collection of its linguistic values, Uis the
universe of discourse, Gis a syntactic rule which
generates the terms in T(H), and Mis a semantic rule
which associates with each linguistic value X its
meaning, M(X) denotes a fuzzy subset of U.
Most of decision making situations deal with
comparative judgments. DM provides relative
assessments by comparing the alternatives. Therefore, in
order to apply the concept of HFLTS in decision
making problems, we should elicit the comparative
linguistic expressions [55].
Definition 2. [49, 50] A hesitant fuzzy linguistic term
set (HFLTS) is shown by ܪ, and is an ordered finite
subset of the consecutive linguistic terms set ܵ=
൛ܵ,ܵ,…,ܵ.
Definition 3. The envelope of an HFLTS is a linguistic
interval that its upper bound and lower bound determine
the limits of the envelope as follows:
݁݊ݒ(ܪ)=[ܪ,ܪ],ܪ൑ܪ (1)
ܪ=max {ܵ}׊ܵאܪ
ܪ= min {ܵ}׊ܵאܪ
As mentioned above, DM applies comparative
linguistic expressions to assess the criteria and
alternatives of a common decision making problem. It is
appropriate to represent the comparative expressions by
fuzzy membership functions [50]. Thus, trapezoidal
membership function is used and ordered weighted
averaging (OWA) operator is established [56] in order
to compute the elements of fuzzy membership function.
OWA weights would represent the decision maker’s
hesitation in linguistic terms. There are different
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
155
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
approaches to obtain the OWA weights, but in this study
Filev and Yager’s [56] approach is applied.
Definition 4. Let OWA operator maps from dimension
nas follows:
ܱܹܣ:ܴ՜ܴ
ܱܹܣ(ܽ,ܽ,…,ܽ)=σݓܾ
௝ୀଵ (2)
where ܽ,ܽ,…,ܽis the aggregated set of arguments
and ܾis the jth largest argument of the aggregated set,
and the associated weighting vector
ܹ=(ݓ,ݓ,…,ݓ)satisfies ݓא[0,1]׊݅=
1,2, … , ݊
σݓ=1
௜ୀଵ (3)
In order to calculate the parameters band c, OWA
operator is implemented as follows:
ܾ=ܱܹܣ ( ܽ
,ܽ
௜ାଵ,…,ܽ
) (4)
ܿ=ܱܹܣ ( ܽ
,ܽ
௜ାଵ,…,ܽ
) (5)
where ܹand ܹrepresent the form of OWA
weighting vectors for computing band c, respectively.
ݏ,ݐ=1,2and ݏ്ݐor ݏ=ݐ.
Definition 5. Let ߙbe the parameter in the unit interval
(0, 1). This parameter is used to calculate the first type
of weights of OWA operator as follows:
ܹ=(ݓ,ݓ,…,ݓ
)
ݓ=ߙ,ݓ=ߙ(1െߙ),ݓ=ߙ(1െߙ),…,ݓ௡ିଵ
=
ߙ(1െߙ)௡ିଶ, (6)
ݓ
=(1െߙ)௡ିଵ
Also, the second type of weights of OWA operator is as
follows:
ܹ=(ݓ,ݓ,…,ݓ)
ݓ=ߙ,ݓ=ߙ(1െߙ),ݓ=ߙ(1െߙ),…,ݓ௡ିଵ
=
ߙ(1െߙ)௡ିଶ, (7)
ݓ=(1െߙ)௡ିଵ
2.2. Fuzzy arithmetic operations
Based on the previous definitions, proposed method
employs trapezoidal fuzzy numbers and arithmetic
operations of trapezoidal fuzzy numbers should be
calculated. The required concepts are defined below.
Definition 6. A fuzzy number ܣ=(ܽ,ܾ,ܿ,݀)is a
trapezoidal fuzzy number shown in Fig. 1, and its
membership function is given by:
ߤ(ݔ)=
ە
ۖ
۔
ۖ
ۓ
0, ݔ<ܽ,
௫ି௔
௕ି௔,ܽ൑ݔ൑ܾ,
1, ܾ<ݔ<ܿ,
ௗି௫
ௗି௖,ܿ൑ݔ൑݀,
0, ݔ>݀,
(8)
Fig. 1. Trapezoidal fuzzy number,
=(,,,)
Definition 7. Let A and B be two positive trapezoidal
fuzzy numbers, where ܣ=(ܽ,ܾ,ܿ,݀)and
ܤ=(݌,ݍ,ݎ,ݏ). The arithmetic operations between ܣ
and ܤis as follows:
Addition operation:
ܣ۩ܤ=(ܽ+݌,ܾ+ݍ,ܿ+ݎ,݀+ݏ) (9)
Subtraction operation:
ܣٓܤ
=(ܽെݏ,ܾെݎ,ܿെݍ,݀െ݌) (10)
Multiplication operation:
ܣٔܤ
؆(ܽ݌,ܾݍ,ܿݎ,݀ݏ) (11)
Division operation:
ܣٕܤ
؆(
,
,
,
) (12)
Multiplication operation:
ݑ.ܣ=ܣ.ݑ=(ݑήܽ,ݑήܾ,ݑήܿ,ݑή݀)ifݑ൒0
(ݑή݀,ݑήܿ,ݑήܾ,ݑήܽ)ifݑ<0 (13)
Division operation:
=(
,
,
,
)ifݑ൒0
(
,
,
,
)ifݑ൑0
(14)
2.3 Basic definitions of ELECTRE I
As mentioned earlier, ELECTRE I is an outranking
method that is characterized due to analyzing binary
relations of the alternatives. Some definitions regarding
outranking relations are provided as follows [4, 5, 57,
58, 59]:
Definition 8. In ELECTRE I method, DM indicates the
preferences by binary outranking relations. Let xand y
be two alternatives. “x S y” means “xis at least as good
as y”.
a
bc
ߤ(ݔ)
1
dx
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
156
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
Definition 9. The concept of outranking relation is
based on two concepts including the concordance and
the discordance. The statement of “x S y” provides
insights into these concepts: The concordance concept:
For an outranking “x S y” to be validated, a sufficient
majority of the criteria should be in favor of this
assertion. On the other hand, when the concordance
condition holds, discordance concept implies that none
of the criteria in the minority should oppose too strongly
to the assertion “x S y”.
3. Methodology
As mentioned above, we propose a new fuzzy
ELECTRE I method based on HFLTS. This method
facilitates decision making process due to utilizing
linguistic terms and context-free grammars for
evaluating the importance weights of the criteria and
performance ratings. Since DMs compare the criteria
and alternatives to present relative assessments,
comparative linguistic expressions are needed for the
assessments of alternatives and criteria. Therefore,
HFLTS method contributes to determine fuzzy numbers
for hesitant judgments. Then, fuzzy ELECTRE I
outranking method is established to complete the
decision making process. Our methodology includes
two main phases: HFLTS operations and the application
of fuzzy ELECTRE I outranking method.
I. First phase: HFLTS operations
Step 1. Determine the semantics and syntax of linguistic
terms set and context-free grammar ܩ.
In this phase, DM should determine the decision
making conditions to apply CWW process. The first
step is to determine the semantic and syntax of
comparative linguistic terms set. Definition 1 also lets
us to determine the context-free grammar ܩ. An
example of context-free grammar is provided below
[50]:
ܩ={ܸ,ܸ,ܫ,ܲ}
where ܸshows non-terminal symbols and ܸshows
terminal symbols as follows:
ܸ=ۃ݌ݎ݅݉ܽݎݕݐ݁ݎ݉ۄ,ۃܾ݅݊ܽݎݕݎ݈݁ܽݐ݅݋݊ۄ,
ۃݑ݊ܽݎݕݐ݁ݎ݉ۄ,ۃܿ݋݆݊ݑ݊ܿݐ݅݋݊ۄ
ܸ=൛ܽݐ݈݁ܽݏݐ,ܽݐ݉݋ݏݐ,ܾ݁ݐݓ݁݁݊,ܽ݊݀,ܵ,ܵ,…,ܵ
ܫאܸ
Pindicates the production rules for the context-free
grammar ܩ. Bracket signs shows optimal elements
and the symbol “|” indicates alternative elements. In this
example, the production rules are as follows:
ܲ=ە
ۖ
۔
ۖ
ۓ
݌ݎ݅݉ܽݎݕݐ݁ݎ݉׸=ݏ|ݏหݏ,
ۃܾ݅݊ܽݎݕݎ݈݁ܽݐ݅݋݊ۄ׸=ܾ݁ݐݓ݁݁݊,
ۃݑ݊ܽݎݕݎ݈݁ܽݐ݅݋݊ۄ׸=ܽݐ݈݁ܽݏݐ|ܽݐ݉݋ݏݐ|݈݋ݓ݁ݎݐ݄ܽ݊|݃ݎ݁ܽݐ݁ݎݐ݄ܽ݊,
ۃܿ݋݆݊ݑ݊ܿݐ݅݋݊ۄ׸=ܽ݊݀ ۙ
ۖ
ۘ
ۖ
ۗ
This set specifies the fuzzy partition of the HFLTS
phase. This partitioning lets us to represent the fuzzy
envelopes of the assessments. Three values are
considered including left value (L), middle value (M),
and right value (R) for the partitioning. As you see in
Fig. 2, each triangular represents three values L, M, and
R. In the proposed approach, assessments are
represented by trapezoidal fuzzy numbers.
Fig. 2. Fuzzy partitioning.
Step 2. Gather the assessments of performance and
criteria.
In this step, DM evaluates the performance ratings
with respect to criteria in square matrices. Also, the
weights of criteria are calculated by DM’s assessments
of criteria with respect to goal.
Step 3. Transform the assessments into HFLTS by
transformation function ܧ.
Let ܵ=൛ݏ,ݏ,…,ݏbe the linguistic terms set.
DM evaluates the performance and criteria by
comparative linguistic terms based on the defined
context-free grammar ܩ. The transformation of the
evaluations into HFLTS will be carried out by
transformation function ܧas follows:
ܧ(ݏ)={ݏ|ݏאܵ}={ݏ},
ܧ൫ܾ݁ݐݓ݁݁݊ݏܽ݊݀ݏ=൛ݏหݏ൑ݏ൑ݏܽ݊݀ݏא
ܵൟ=൛ݏ,ݏ௜ା,…,ݏ,
ܧ(ܽݐ݈݁ܽݏݐݏ)=൛ݏหݏ൒ݏܽ݊݀ݏאܵ=
൛ݏ,ݏ௜ା,…,ݏ,
ܧ(ܽݐ݉݋ݏݐݏ)=൛ݏหݏ൑ݏܽ݊݀ݏאܵ
={ݏ,ݏ,…,ݏ}
ݏݏ
ݏݏ
ݏ௚ିଵ
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
157
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
Step 4 : Obtain an aggregate set for each assessment.
In this step, the fuzzy envelope is obtained through
calculating the elements of a trapezoidal fuzzy number.
We calculate these elements by aggregate set as follows:
Let Abe the aggregate set for ܧ൫ܾ݁ݐݓ݁݁݊ݏܽ݊݀ݏ.
ܣ௕௘௧௪௘௘௡ =
൛ܽ
,ܽ
,ܽ
௜ାଵ,ܽ
,ܽ
௜ାଵ,ܽ
௜ାଶ,ܽ
௜ାଵ,…,ܽ,ܽ
௝ିଵ,ܽ
,ܽ
(15)
ܣ௔௧௟௘௔௦௧ =
൛ܽ
,ܽ
,ܽ
௜ାଵ,ܽ
,ܽ
௜ାଵ,ܽ
௜ାଶ,ܽ
௜ାଵ,…,ܽ,ܽ
௚ିଵ,ܽ
,ܽ
(16)
ܣ௔௧௠௢௦௧ =
൛ܽ
,ܽ
,ܽ
,ܽ
,ܽ
,ܽ
,ܽ
,…,ܽ
,ܽ
௜ିଵ,ܽ
,ܽ
(17)
As you see in Fig. 2, ܽ
௞ାଵ=ܽ
=ܽ
௞ିଵ for ݇=
1,2, … , ݃െ1. So that, the Eqs. (15), (16), and (17) are
simplified as follows:
ܣ௕௘௧௪௘௘௡ =൛ܽ
,ܽ
,ܽ
௜ାଵ,…,ܽ
,ܽ
ܣ௔௧௟௘௔௦௧ =൛ܽ
,ܽ
,ܽ
௜ାଵ,…,ܽ
,ܽ
ܣ௔௧௠௢௦௧ =൛ܽ
,ܽ
,ܽ
,…,ܽ
,ܽ
Step 5. Obtain fuzzy envelope for each assessment.
ܶ௕௘௧௪௘௘௡ =(ܽ,ܾ,ܿ,݀),
ܽ=݉݅݊ܣ௕௘௧௪௘௘௡ =݉݅݊൛ܽ
,ܽ
,ܽ
௜ାଵ,…,ܽ
,ܽ
=
ܽ
,
݀=maxܣ௕௘௧௪௘௘௡ =max൛ܽ
,ܽ
,ܽ
௜ାଵ,…,ܽ
,ܽ
=
ܽ
,
ܾ=ܱܹܣ ( ܽ
,ܽ
௜ାଵ,…,ܽ
),
ܿ=ܱܹܣ ( ܽ
,ܽ
௜ାଵ,…,ܽ
).
As in Definition 5, we can compute the weights of
OWA operator to calculate element b, and cfor
“between” comparative linguistic terms set. The first
type of weights of OWA operator is used to calculate b
and the second type of weights of OWA operator to
calculate cas follows:
I) If i+j is odd, then
ܾ=ܱܹܣቆܽ
,ܽ
௜ାଵ,…,ܽ
೔శೕషభ
,
ܿ=ܱܹܣቆܽ
,ܽ
௝ିଵ,…,ܽ
೔శೕశభ
,
ܹ=൫ݓ,ݓ,…,ݓ௝ି௜ାଵ ଶ
Τ
and ߙ=௚ି(௝ି)
௚ିଵ
ݓ݄݁ݎ݁ݓ=ߙ௝ି௜ିଵ
,ݓ
=(1െߙ)ߙ௝ି௜ି
,…,ݓ௝ି௜ିଵ
=(1െߙ)ߙ,ݓ௝ି௜ାଵ
=(1െߙ)
ܾ=ݓ.ܽ
௜ା௝ିଵ
,ݓ.ܽ
௜ା௝ିଷ
,…,ݓ௝ି௜ିଵ
.ܽ
௜ାଵ,ݓ௝ି௜ାଵ
.ܽ
and ܹ=൫ݓ,ݓ,…,ݓ௝ି௜ାଵ ଶ
Τ
and ߙ=1െߙ=
(௝ି௜)ିଵ
௚ିଵ
where ݓ=ߙ,ݓ=ߙ(1െߙ),…,ݓೕష೔షభ
=ߙ(1െߙ)௝ି௜ିଷ
,ݓ௝ି௜ାଵ
=(1െߙ)௝ି௜ିଵ
ܿ=ݓ.ܽ
௜ା௝ାଵ
,ݓ.ܽ
௜ା௝ିଵ
,…,ݓ௝ି௜ିଵ
.ܽ
௜ାଵ,ݓ௝ି௜ା
.ܽ
II) If i+j is even, then
ܾ=ܱܹܣቆܽ
,ܽ
௜ାଵ,…,ܽ
೔శೕ
,
ܿ=ܱܹܣቆܽ
,ܽ
௝ିଵ,…,ܽ
೔శೕ
,
ܹ=൫ݓ,ݓ,…,ݓ௝ି௜ାଶ ଶ
Τ
where ݓ=ߙೕష೔
,ݓ=(1െߙ)ߙೕష೔షమ
,…,ݓೕష೔
ܾ=ݓ.ܽ
௜ା௝
,ݓ.ܽ
௜ା௝ିଶ
,…,ݓ௝ି௜
.ܽ
௜ାଵ,ݓ௝ି௜ାଶ
.ܽ
and ܹ=൫ݓ,ݓ,…,ݓ௝ି௜ାଶ ଶ
Τ
where ݓ=ߙ,ݓ=ߙ(1െߙ),…,ݓೕష೔
=ߙ(1െߙ)௝ି௜ି
,ݓ௝ି௜ାଶ
=(1െߙ)௝ି௜
ܿ=ݓ.ܽ
௜ା௝
,ݓ.ܽ
௜ା௝ିଶ
,…,ݓ௝ି௜
.ܽ
௝ିଵ,ݓ௝ି௜ାଶ
.ܽ
Finally, the HFLTS envelope will be formed as
ܶ௕௘௧௪௘௘௡ =(ܽ
,ܾ,ܿ,ܽ
). See Fig. 3.
Fig. 3. Fuzzy Envelope for comparative linguistic expression
"between"
ad
cb
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
158
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
ܶ௔௧௟௘௔௦௧ =(ܽ,ܾ,ܿ,݀),
ܽ=݉݅݊ܣ௔௧௟௘௔௦௧ =݉݅݊൛ܽ
,ܽ
,ܽ
௜ାଵ,…,ܽ
,ܽ
=ܽ
,
݀=maxܣ௔௧௟௘௔௦௧ =max൛ܽ
,ܽ
,ܽ
௜ାଵ,…,ܽ
,ܽ
=ܽ
ܾ=ܱܹܣ ( ܽ
,ܽ
௜ାଵ,…,ܽ
)
ܿ=ܱܹܣ ( ܽ
,ܽ
௜ାଵ,…,ܽ
)
As can be seen in Definition 5, the weights of OWA
operator to compute element bfor “at least”
comparative linguistic terms set are as follows:
݊=݃െ݅+1,
ܹ=(ݓ,ݓ,…,ݓ)and ߙ=݅݃
where ݓ=ߙ௚ି௜,ݓ=(1െߙ)ߙ௚ି௜ିଵ,…,ݓ௚ି௜
=
ߙ(1െߙ)
ݓ௚ି௜ାଵ
=1െߙ
ܾ=ݓ.ܽ
,ݓ.ܽ
௚ିଵ,…,ݓ௚ି௜
.ܽ
௜ାଵ,ݓ௚ି௜ାଵ
.ܽ
,
ܾ=ߙ௚ି௜.ܽ
,(1െߙ)ߙ௚ି௜ିଵ.ܽ
௚ିଵ,…,
ߙ(1െߙ).ܽ
௜ାଵ,(1െߙ).ܽ
ܿ=ܽ
Finally, the HFLTS envelope is formed as ܶ௔௧௟௘௔௦௧ =
(ܽ
,ܾ,ܽ
,ܽ). See Fig. 4.
Fig. 4. Fuzzy Envelope for comparative linguistic expression
"at least".
ܶ௔௧௠௢௦௧ =(ܽ,ܾ,ܿ,݀),
ܽ=݉݅݊ܣ௔௧௠௢௦௧ =݉݅݊൛ܽ
,ܽ
,ܽ
,…,ܽ
,ܽ
=ܽ
,
݀=maxܣ௔௧௠௢௦௧ =max൛ܽ
,ܽ
,ܽ
,…,ܽ
,ܽ
=ܽ
ܾ=ܱܹܣ ( ܽ
,ܽ
,…,ܽ
)
ܿ=ܱܹܣ ( ܽ
,ܽ
,…,ܽ
)
As you see in Definition 5, the weights of OWA
operator to compute element cfor “at most”
comparative linguistic terms set are as follows:
݊=݅+1,
ܹ=(ݓ,ݓ,…,ݓ)and ߙ=݅݃
where ݓ=ߙ,ݓ=ߙ(1െߙ),…,ݓ=ߙ(1
ߙ)௜ିଵ,ݓ௜ା
=(1െߙ)
ܿ=ݓ.ܽ
,ݓ.ܽ
௜ିଵ,…,ݓ.ܽ
,ݓ௜ାଵ
.ܽ
,
ܿ=ߙ.ܽ
,ߙ(1െߙ).ܽ
௜ିଵ,…,ߙ(1െߙ)௜ିଵ.ܽ
,(1െߙ).ܽ
ܾ=ܽ
Finally, the HFLTS envelope is formed as ܶ௔௧௠௢௦௧ =
(ܽ
,ܽ
,ܿ,ܽ
). See Fig. 5.
Fig. 5. Fuzzy Envelope for comparative linguistic expression
"at most"
II. Second phase: ELECTRE I outranking method
Step 6. Defuzzify and rank the alternatives for each
criterion.
In this step, the trapezoidal fuzzy values, which DM
evaluated the alternatives in previous steps, will be
defuzzified. The center of gravity defuzzification
method is applied to rank alternatives with respect to
each criterion.
Then, the utility matrix is built. By taking utility
matrix into account, assessment matrices will be formed
to compare alternatives with respect to each criterion.
The trapezoidal fuzzy elements of final utility matrix
form the performance ratings as in Eq. (18).
(18)
The decision matrix is also built to compare the
importance of criteria. Next, the weight of each criterion
could be obtained through computing the average of the
elements in each row using summation and division
rules of trapezoidal fuzzy numbers. See Definition 7 for
more details. Finally, the weight matrix will be obtained
as Eq. (20).
(19)
(20)
a,b cd
b
c,d
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
159
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
where , , and
Since the values in the matrices are between 0 and 1,
there is no need to obtain a normalized matrix. The
weight matrix and the utility matrix are
multiplied to construct weighted utility matrix as
follows:
where
,
, and
Step 7. Calculate the fuzzy concordance indices and
discordance indices.
The fuzzy concordance matrix is built as Eq. (21).
The elements of this matrix are measured by pairwise
comparison of alternatives. In this regard, two
alternatives and are considered. Using
Definition 8, the outranking relation will be
defined. The arguments in favor of this statement “
” is measured to obtain the fuzzy
concordance indices as follows:
(21)
where , ,
, and
and .
In addition, the discordance set is formed to
construct the discordance matrix. The concept of
discordance set is in contrast to the concept of
concordance set. If DM has doubt upon the statement “
” and DM is against the assertion “
is at least as good as ” , we can represent the
discordance matrix and the arguments as follows [60]:
(22)
where , if ,
, and , and
.
The discordance indices are obtained between
alternatives iand jwith respect to the criterion .
and are ranked by Yager’s centroid index [61].
Then, the defuzzification method, center of gravity will
be employed to calculate the discordance index.
Since this method calculates the outranking relations
without defining concordance and discordance
thresholds, a combination method to unite fuzzy
concordance and discordance matrices is proposed. A
modified version of Aouam and Chang’s [62] formula is
applied to calculate the elements of matrix . Hence,
is formed by subtracting each element of the
discordance matrix from 1. Then, the fuzzy global
matrix will be calculated through peer to peer
multiplication of the elements of the matrices and
(using Eq. (11)) as follows:
(23)
where
(24)
Step 8. Compare the alternatives by fuzzy Kernel
diagram.
In the final step, the outranking relations is exploited
to compare the alternatives by the elements of matrix .
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
160
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
In this regard, graph is defined, where Vis
the set of nodes and Jis the set of arcs. One node is
considered for each alternative and an arc between two
nodes of alternatives and . Here, we propose
fuzzy Kernel diagram. This diagram could depict
uncertainty in comparison of alternatives. We draw
solid arc if absolutely outranks , that means
is absolutely preferred to as shown in Fig. 6 a. We
proposed to draw long dashed arc if strongly
outranks as shown in Fig. 6 b, and dashed arc if
weakly outranks as shown in Fig. 6 c. If and
are incomparable, we consider no arc between them
as shown in Fig. 6 d. In addition, Fig. 6 eillustrates the
indifference between and .
Fig. 6. Outranking relations represented by fuzzy Kernel
diagram.
In order to determine the arcs, the elements of fuzzy
global matrix should be defuzzified. In this step,
center of gravity defuzzification method is applied and
represents the defuzzified values. The intervals of
the defuzzified values are defined for associated binary
preferences are presented below:
(25)
In summary, you can find the associated steps of our
combined method as follows:
I. First phase: HFLTS operations
Step 1. Determine the semantics and syntax of linguistic
terms set and context-free grammar .
Step 2. Gather the assessments of performance and
criteria.
Step 3. Transform the assessments into HFLTS by
transformation function
Step 4 : Obtain aggregate set for each assessment.
Step 5. Obtain fuzzy envelope for each assessment.
II. Second phase: ELECTRE I outranking method
Step 6. Rank alternatives for each criterion.
Step 7. Calculate fuzzy concordance and discordance
indices.
Step 8. Compare alternatives by fuzzy Kernel diagram.
4. An Illustrative Example
In this part, a simple MCDA problem is added to
provide a step by step solution in order to illustrate the
proposed method. The problem is the selection of best
supplier among three alternatives by considering three
attributes including price, quality, and delivery. The best
way of illustrating the relations between the criteria and
alternatives is to use a hierarchy often applied in
Analytic Hierarchy Process (AHP) [63]. The hierarchy
of decision making is provided below:
Fig. 7. The hierarchy of MCDA supplier selection problem.
(a) Absolute preference
(d) Incomparability
(b) Strong preference
(c) Weak preference
(e) Indifference
Selection of best
supplier
QualityPrice Delivery
Su
p
.1 Su
p
.3Su
p
.2
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
161
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
The steps of the method are presented below:
Step 1. At first, the semantic and syntax of linguistic
term set Sand context-free grammar should be
defined. We determine the linguistic term set Sas:
Step 2. In this step, the importance of criteria is
assessed with respect to the goal as Table 1. This will
obtain the weights of criteria in next steps. Then, the
performance of alternatives with respect to each
criterion will be evaluated as you see in Tables 2, 3, and
4.
Table 1
The linguistic assessment of criteria with respect to the goal.
Price Quality Delivery
Price - At most li At most mi
Quality At least hi - At least hi
Delivery At least mi At most li -
Table 2
The linguistic assessment of alternatives with respect to price.
Supplier 1 Supplier 2 Supplier 3
Supplier 1 - At most vli At most li
Supplier 2 At least vhi - At least hi
Supplier 3 At least hi At most li -
Table 3
The linguistic assessment of alternatives with respect to
quality
Supplier
1Supplier 2 Supplier 3
Supplier 1 - At most li At most li
Supplier 2 At least
hi -Between li and
mi
Supplier 3 At least
hi
Between mi and
hi -
Table 4
The linguistic assessment of alternatives with respect to
delivery.
Supplier 1 Supplier 2 Supplier 3
Supplier 1 - At most li At most mi
Supplier 2 At least hi - At least hi
Supplier 3 At least mi At most li -
Steps 3, 4, and 5. Firstly, each of comparative linguistic
expressions is transformed into corresponding HFLTS.
Then, the aggregate set is formed for each assessment.
This aggregate set lets us to obtain fuzzy envelopes. For
more details, follow the provided example below:
By using Eq. (17), simplified aggregate set is obtained:
We form fuzzy envelope similar to the presented fuzzy
partitioning in Fig. 5.
,
,
.
,
where
,
,,
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
162
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
Fig. 8.
Eventually, a trapezoidal fuzzy number will be
obtained for each of evaluations as shown in Tables 5,
6, 7, and 8.
Table 5
The numerical assessments of the criteria with respect to the
goal.
Price Quality Delivery
Price - (0, 0, 0.15, 0.5) (0, 0, 0.35, 0.67)
Quality (0.5, 0.85, 1, 1) - (0.5, 0.85, 1,1)
Delivery (0.33, 0.65, 1, 1) (0, 0, 0.15, 0.5) -
As mentioned before, the weights of criteria should be
calculated using assessment of criteria with respect to
goal as shown in Table 5. Similarly, the average
performance of three suppliers is obtained by
calculating the average amount of each row in Tables 6,
7, and 8. Thus, the utility matrix, which represents the
average performance ratings, and the weights of criteria
are shown in Table 9. The weighted utility matrix is
given in Table10.
Table 6
The numerical assessments of the alternatives with respect to
price.
Supplier 1 Supplier 2 Supplier 3
Supplier 1 - (0, 0, 0.028,
0.33)
(0, 0, 0.15,
0.5)
Supplier 2 (0.67, 0.972, 1,
1) -(0.5, 0.85, 1,
1)
Supplier 3 (0.5, 0.85, 1, 1) (0, 0, 0.15, 0.5) -
Table 7
The numerical assessments of the alternatives with respect to
quality.
Supplier 1 Supplier 2 Supplier 3
Supplier 1 - (0, 0, 0.15, 0.5) (0, 0, 0.35, 0.67)
Supplier 2 (0.5, 0.85, 1,
1) -(0.17, 0.33, 0.5,
0.67)
Supplier 3 (0.5, 0.85, 1,
1)
(0.33, 0.5, 0.67,
0.83) -
Table 8
The numerical assessments of the alternatives with respect to
delivery.
Supplier 1 Supplier 2 Supplier 3
Supplier 1 - (0, 0, 0.15, 0.5) (0, 0, 0.35, 0.67)
Supplier 2 (0.5, 0.85, 1, 1) - (0.5, 0.85, 1, 1)
Supplier 3 (0.33, 0.65, 1, 1) (0, 0, 0.15, 0.5) -
Table 9
The utility matrix (average performance ratings) and criteria
weights.
Price Quality Delivery
Weights (0, 0, 0.25,
0.585)
(0.5, 0.85, 1,
1)
(0.165, 0.325,
0.575, 0.75)
Supplier 1 (0, 0, 0.89,
0.415)
(0, 0, 0.15,
0.5)
(0, 0, 0.25,
0.585)
Supplier 2 (0.585, 0.911,
1, 1)
(0.335, 0.59,
0.75, 0.835)
(0.5, 0.85, 1,
1)
Supplier 3 (0.25, 0.425,
0.575, 0.75)
(0.415,0.675,0
.835,0.915)
(0.165,0.325,0
.575,0.75)
Table 10
The weighted utility matrix.
Price Quality Delivery
Supplier 1 (0, 0, 0.222,
0.242)
(0, 0, 0.15,
0.5)
(0, 0, 0.143,
0.438)
Supplier 2 (0, 0, 0.25,
0.585)
(0.167,0.501,0
.75,0.835)
(0.082,0.276,0
.575,0.75)
Supplier 3 (0, 0, 0.143,
0.438)
(0.207, 0.573,
0.835,0.915)
(0.027,0.105,0
.330,0.562)
Step 6. In this step, the alternatives are ranked with
respect to each criterion as presented in Tables 11, 12,
and 13. We apply center of gravity defuzzification
00.15 0.17 0.67 10.8
0.5
0.33
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
163
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
method to rank the fuzzy numbers. Considered fuzzy
numbers are average amount of row elements of
performance ratings shown in Tables. 6, 7, and 8. This
ranking will help us to obtain fuzzy concordance and
discordance indices in next step. For instance, center of
gravity defuzzifies (0, 0, 0.089, 0.415) as follows:
Table 11
Ranking of alternatives with respect to price.
Average Defuzzified Value
Supplier 1 (0, 0, 0.089, 0.415) 0.143
Supplier 2 (0.585, 0.911, 1, 1) 0.855
Supplier 3 (0.25, 0.425, 0.575, 0.75) 0.488
Ranking Supplier 2 Supplier 3 Supplier 1
Table 12
Ranking of alternatives with respect to quality.
Average Defuzzified Value
Supplier 1 (0, 0, 0.15, 0.5) 0.178
Supplier 2 (0.335, 0.59, 0.75, 0.835) 0.597
Supplier 3 (0.415, 0.675, 0.835, 0.915) 0.700
Ranking Supplier 3 Supplier 2 Supplier 1
Table 13
Ranking of alternatives with respect to delivery.
Average Defuzzified Value
Supplier 1 (0, 0, 0.25, 0.585) 0.220
Supplier 2 (0.5, 0.85, 1, 1) 0.820
Supplier 3 (0.165, 0.325, 0.575, 0.75) 0.480
Ranking Supplier 2 Supplier 3 Supplier 1
Step 7. In this step, fuzzy concordance and
discordance indices are computed to construct fuzzy
concordance and discordance matrices. By taking into
account the ranking of alternatives, the fuzzy
concordance indices are obtained through adding
criteria weights each of which the alternative ranking is
higher. Then, it will be divided by the total amount of
criteria weights. Fuzzy concordance table is presented in
Table 14:
Table 14
Fuzzy concordance table.
Supplie r 1 Suppl ier 2 Supp lier 3
Supplier
1-
Supplier
2-
Supplier
3-
From Table 14, we have
Based on the Eq. (22), the discordance matrix is
obtained using Table 10 as follows:
Based on Eq. (24), the supplement of discordance
matrix will be as follows:
Finally, the fuzzy global matrix is built through
multiplication of matrices and as follows:
After defuzzification, the preferences of the
alternatives will be obtained by using Eq. (25).
Therefore, supplier 2 is absolutely preferred to supplier
1, supplier 3 is absolutely preferred to supplier 1, and
supplier 2 is weakly preferred to supplier 3. Supplier 3
is also incomparable to supplier 2.
Step 8. Consequently, the fuzzy Kernel graph is
drawn based on the matrix as Fig. 8. According to
Fig. 6, solid arc is drawn from suppliers 2 and 3 to
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
164
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
supplier 1 to represent absolutely preferred nodes, and
dashed line from supplier 2 to supplier 3. Eventually,
the fuzzy Kernel graph in Fig. 9 depicts that supplier 2
outranks other two suppliers using HFLTS-ELECTRE I
method.
Fig. 9. Fuzzy Kernel graph
5. Conclusion
In this study, ELECTRE I outranking method using
hesitant fuzzy linguistic term set is proposed. Since
decision making involves uncertainty and imprecision,
DMs prefer to render linguistic judgments about
decision alternatives and criteria. Our proposed method
facilitates decision making process due to using
comparative linguistic expressions to evaluate the
alternatives and criteria and it could be a DM-friendly
MCDA method. Fuzzy sets theory and subsequent
developments empowers us to numerically interpret the
uncertainty of MCDA problems. Ultimately, ELECTRE
I method is established to investigate binary relations of
alternatives and criteria in order to compare them by
comparative linguistic terms.
In this proposed HFLTS-ELECTRE I method, the
combination matrix of concordance and discordance
remains fuzzy. Our method offers an evolution in
outranking relations through elimination of concordance
and discordance thresholds form ELECTRE I method.
Hence, we propose fuzzy Kernel diagram to represent
the preferences of actions. This new diagram uses
dashed lines to depict decision maker’s uncertainty
toward actions. Since fuzzy Kernel diagram is
implemented, defuzzification of the fuzzy concordance
matrix is not needed and we continue the calculations
by fuzzy values until the final step. Thus, the output
clearly illustrates decision maker’s assessments and
his/her preferences. Future works could focus on
extensions of HFLTS for other members of ELECTRE
family or outranking methods. In addition, extension of
fuzzy Kernel diagram could represent decision maker’s
preferences in detail.
References
[1]
I. N. Durbach and T. Stewart, Using expected values to
simplify decision making under uncertainty, Omega, vol.
37, nº 2, p. 312–330, 2009.
[2]
X. Wang and E. Triantaphyllou, Ranking irregularities
when evaluating alter- natives by using some ELECTRE
methods, Omega, vol. 36, p. 45–63, 2008.
[3]
R. T. Clemen, Making hard decisions: an introduction to
decision analysis, 2nd ed., Belmont: Duxbury Press at
Wadsworth Publishing Company, 1996.
[4]
A. Hatami-Marbini and M. Tavana, An extension of the
Electre I method for group decision-making under a
fuzzy environment, Omega, vol. 39 , p. 373–386, 2011.
[5]
J. Figueira, S. Greco and M. Ehrgott, Multiple criteria
decision analysis: state of the art surveys, New York:
Springer, 2005.
[6]
T. J. Stewart and F. B. Losa, Towards reconciling
outranking and value measurement practice, European
Journal of Operational Research, vol. 145, nº 3, p. 645–
659, 2002.
[7]
C. Zopounidis and M. Doumpos, Multicriteria
classification and sorting methods: a literature review,
European Journal of Operational Research, vol. 138, nº
2, p. 229–246, 2002.
[8]
B. Roy and D. Vanderpooten, An overview on The
European School of MCDA: emergence, basic features
and current work, European Journal of Operational
Research, vol. 99, nº 1, pp. 26-27, 1997.
[9]
M. Doumpos, Y. Marinakisa, M. Marinakia and C.
Zopounidis, An evolutionary approach to construction o
f
outranking models for multicriteria classifica- tion: the
case of the ELECTRE TRI method, European Journal o
f
Operational Research, vol. 199, nº 2, p. 496–505, 2009.
[10]
F. J. Andre, I. Herrero and L. Riesgo, Amodified DEA
model to estimate the importance of objectives with an
application to agricultural economics, Omega, vol. 38, nº
5, p. 371–382, 2010.
[11]
M. A. Hinojosa and A. M. Marmol, Axial solutions for
multiple objective linear problems: an application to
target setting in DEA models with preferences, Omega,
vol. 39, nº 2, p. 159–167, 2011.
[12]
R. E. Steuer, J. Silverman and A. J. Whisman, A
combined Tchebycheff/aspiration criterion vector
interactive multi objective programming procedure,
Management Science, vol. 39, nº 10, p. 1255–1260, 1993.
[13]
B. Roy, The outranking approach and the foundations o
f
ELECTRE methods, Theory and Decision, vol. 31, nº 1,
p. 49–73, 1991.
Sup. 2
Sup. 1
Sup. 3
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
165
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
[14] J. Paelinck, Qualiflex: a flexible multiple-criteria method,
Economic letters, vol. 1, nº 3, p. 193–197, 1978.
[15] M. Roubens, Preference relations on actions and criteria
in multi-criteria decision making, European Journal o
f
Operational Research, vol. 10, nº 1, p. 51–55, 1982.
[16] H. Pastijn and J. Leysen, Constructing an outranking
relation with oreste, Mathematical and Computer
Modelling, vol. 12, nº 10–11, p. 1255–1268, 1989.
[17] J. P. Leclercq, Propositions d’extension de la notion de
dominance en presence de relations d’ordre sur les
pseudo-criteres: MELCHIOR, Revue Belge de Recherche
Operationelle de Statistique et d’hformatique, vol. 24, nº
1, p. 32–46, 1984.
[18] B. Matarazzo, Multicriterion analysis of preferences by
means of pairwise actions and criterion comparisons
(MAPPAC), Applied Mathematics and Computation, vol.
18, nº 2, p. 119–141, 1986.
[19] J. C. Vansnick, On the problem of weights in multiple
criteria decision making: the non-compensatory
approach, European Journal of Operational Research,
vol. 24, p. 288–294, 1986.
[20] R. Benayoun, B. Roy and B. Sussman, ELECTRE: Une
méthode pour guider le choix en présence de points de
vue multiples, Note de travail 49, SEMA-METRA
international, direction scientifique, Paris, 1966.
[21] B. Roy, Classement et choix en presence de points de vue
multiples (la methode ELECTRE), Revue Francaise
d'Informatique et de Recherche Operationnelle, vol. 8, p.
57–75, 1968.
[22] B. Roy and P. Bertier, La Methode ELECTRE II: Use
Methode de Classement en Presence de Criteres
Multiples., Note de Travail No. 142, Direction
Scientifique, Group Metra, Paris, 1971.
[23] B. Roy and P. Bertier, La methode ELECTRE II: Une
application au mediaplanning, de OR 72, M. Ross, Ed.,
Amsterdam, North Holland, 1973, pp. 291-302.
[24] B. Roy, Electre III Un algorithme de classements fonde
sur une representation floue en presence de criteres
multiples, Cahairs du CERO, vol. 20, nº 1, p. 3–24, 1978.
[25] M. Rogers and M. Bruen, A new system for weighting
environmental criteria for use within ELECTRE III,
European Journal of Operational Research , vol. 107, nº
3, p. 552–563, 1998.
[26] B. Roy and J. C. Hugonnard, Ranking of suburban line
extension alternatives on the Paris metro system by a
multicriteria method, Transportation Research, vol. 16A,
nº 4, p. 301–312, 1982.
[27] B. Roy and J. C. Hugonnard, Classement des
prolongements de lignes de metro en banlieue parisienne
(presentation d’une methode multicritere originale),
Cahiers du CERO, vol. 24, nº 2,3,4, p. 153–171, 1982.
[28] W. Yu, ELECTRE TRI: Aspects methodologiques et
manuel d’utilisation, Document du LAMSADE 74,
Universite Paris-Dauphine, 1992.
[29]
S. Greco, M. Kadzinski, V. Mousseau and 56áRZLQVNL
ELECTREGKMS: robust ordinal regression for
outranking methods, European Journal of Operational
Research , vol. 214, nº 1, p. 118–135, 2011.
[30]
J. P. Brans and P. Vincke, A preference ranking
organization method: the PROMETHEE method for
MCDM, Management Science, vol. 31, nº 6, p. 647–656,
1985.
[31]
M. Behzadian, R. B. Kazemzadeh, A. Albadvi and M.
Aghdas, PROMETHEE: A comprehensive literature
review on methodologies and applications, European
Journal of Operational Research, vol. 200, nº 1, pp. 198-
215, 2010.
[32]
B. Roy and D. Bouyssou, Aide Multicritere a la Decision:
Methodes et Cas, Paris: Economica, 1993.
[33]
B. Roy, Partial preference analysis and decision-aid: the
fuzzy outranking relation concept, de Conflicting
objectives and decisions, New York, Wiley, 1977, pp. 40-
75.
[34]
J. L. Siskos, J. Lochard and J. Lombardo, A multicriteria
decision-making methodology under fuzziness:
application to the evaluation of radiological protection in
nuclear power plants, TIMS Studies in the Management
Sciences, vol. 20, p. 261–283, 1984.
[35]
M. Sevkli, An application of the fuzzy ELECTRE
method for supplier selection, International Journal o
f
Production Research, vol. 48, nº 12, pp. 3393-3405,
2010.
[36]
T. Ertay and C. Kahraman, Evaluation of design
requirements using fuzzy outranking methods,
International Journal of Intelligent Systems, vol. 22, p.
1229–1250, 2007.
[37]
B. Vahdani, A. Jabbari, V. Roshanaei and M. Zandieh,
Extension of the ELECTRE method for decision-making
problems with interval weights and data, International
Journal of Advanced Manufacturing Technology, vol. 50
, nº 5-8, p. 793–800, 2010.
[38]
B. Vahdani and H. Hadipour, Extension of the ELECTRE
method based on interval-valued fuzzy sets, Soft
Computing, vol. 15, p. 569–579, 2011.
[39]
T. Chen, An ELECTRE-based outranking method for
multiple criteria group decision making using interval
type-2 fuzzy sets, Information Sciences, vol. 263 , p. 1–
21, 2014.
[40]
K. Devi and P. Yadav S., A multicriteria intuitionistic
fuzzy group decision making for plant location selection
with ELECTRE method, The International Journal o
f
Advanced Manufacturing Technology, vol. 66, p. 1219–
1229, 2013.
[41]
B. Vahdani, S. M. Mousavi, R. Tavakkoli-Moghaddam
and H. Hashemi, A new design of the elimination and
choice translating reality method for multi-criteria group
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
166
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
A. Fahmi et al. / Hesitant Linguistic ELECTRE I Method
decision-making in an intuitionistic fuzzy environment,
Applied Mathematical Modelling, vol. 37, nº 4, pp. 1781-
1799, 2013.
[42] J. Li, M. Lin and J. Chen, ELECTRE Method Based on
Interval-valued Intuitionistic Fuzzy Number, Applie
d
Mechanics and Materials, Vols. %1 de %2220-223, pp.
2308-2312, 2012.
[43] M. C. Wue and T. Y. Chen, The ELECTRE multicriteria
analysis approach based on Atanassov’s intuitionistic
fuzzy sets, Expert Systems with Applications, vol. 38, p.
12318–12327, 2011.
[44] V. Torra, Hesitant fuzzy sets, International Journal o
f
Intelligent Systems, vol. 25, nº 6, p. 529–539, 2010.
[45] R. M. Rodríguez, L. Martínez, V. Torra, Z. S. Xu and F.
Herrera, Hesitant fuzzy sets: state of the art and future
directions, International Journal of Intelligent Systems,
vol. 29, nº 6, p. 495–524, 2014.
[46] L. Zadeh, Fuzzy logic = computing with words, IEEE
Transactions on Fuzzy Systems, vol. 4, nº 2, pp. 103-111,
1996.
[47] F. Herrera, S. Alonso, F. Chiclana and E. Herrera-
Viedma, Computing with words in decision making:
foundations, trends and prospects, Fuzzy Optimization
and Decision Making, vol. 8, nº 4, p. 337–364, 2009.
[48] L. Martínez, D. Ruan and F. Herrera, Computing with
words in decision support systems: an overview on
models and applications, International Journal o
f
Computational Intelligence Systems, vol. 3, nº 4, p. 382–
395, 2010.
[49] R. M. Rodriguez, L. Martinez and F. Herrera, Hesitant
fuzzy linguistic term sets for decision making, IEEE
Transactions on Fuzzy Systems, vol. 20, nº 1, pp. 109-
119, 2012.
[50] H. Liu and R. M. Rodriguez, A fuzzy envelope for
hesitant fuzzy linguistic term set and its application to
multicriteria decision making, Information Sciences, vol.
258, p. 220–238, 2014.
[51] L. Zadeh, The concept of a linguistic variable and its
applications to approximate reasoning, Information
Sciences, 8, 9, pp. 199–249 (I). 301–357 (II), 43–80 (III),
1975.
[52] C. T. Chen, Extensions of the TOPSIS for group
decision-making under fuzzy environment, Fuzzy Sets
and Systems, vol. 114, nº 1, p. 1–9, 2000.
[53] S. H. Tsaur, T. Y. Chang and C. H. Yen, The evaluation
of airline service quality by fuzzy MCDM, Tourism
Management, vol. 23 , p. 107–115, 2002.
[54] L. Zadeh, Fuzzy sets, Information and Control, vol. 8, nº
3, p. 338–353, 1965.
[55] R. M. B. B. Rodríguez, H. Bustince, Y. C. Dong, B.
Farhadinia, C. Kahraman, L. Martínez, V. Torra, Y. J.
Xu, Z. S. Xu and H. F, A Position and Perspective
Analysis of Hesitant Fuzzy Sets on Information Fusion in
Decision Making. Towards High Quality Progress,
Information Fusion, vol. 29, pp. 89-97, 2016.
[56]
D. Filev and R. Yager, On the issue of obtaining OWA
operator weights, Fuzzy Sets and Systems, vol. 94, p.
157–169, 1998.
[57]
H. J. Zimmermann, Fuzzy set theory and its applications,
2nd, Ed., Boston: Kluwer Academic Publishers, 1991.
[58]
D. J. Dubois, Fuzzy sets and systems: theory and
applications, New York: Academic Press, 1980.
[59]
G. J. Klir and B. Yuan, Fuzzy sets and fuzzy logic: theory
and applications, New York: Prentice-Hall, 1995.
[60]
K. P. Yoon and C. L. Hwang, Multiple Attribute
Decision Making, An Introduction. Sage University
Paper series on Quantitative Applications in the Social
Sciences, Sage, Thousand Oaks, 1995.
[61]
S. J. Chen and C. L. Hwang, Fuzzy Multiple Attribute
Decision Making, Methods and Applications, Lecture
Notes in Economics and Mathematical Systems, vol. 375,
Heidelberg: Springer, 1992.
[62]
T. Aouam, S. I. Chang and E. S. Lee, Fuzzy MADM: An
outranking method, European Journal of Operational
Research, vol. 145 , p. 317–328, 2003.
[63]
T. L. Saaty, The Analytic Hierarchy Process, New York:
McGraw Hill, 1980.
Co-published by Atlantis Press and Taylor & Francis
Copyright: the authors
167
Downloaded by [Istanbul Technical University] at 03:38 29 February 2016
... The MCDA is sensitive to expert judgments, causing difficulty in evaluating weights when the experts use natural language such as "better," "somewhat worse," or "so much better" to express a kind of general preferences (Hafezalkotob and Hafezalkotob 2017;Doukas 2013;Fahmi et al. 2016). In mathematics, these natural languages are categorized as crisp sets. ...
Article
Flood is one of the major problems in the Sad Ekbatan Watershed, northern Hamadan province, Iran. This problem imposes severe damage related to economic issues. Therefore, prioritizing the study area based on the flooding degree can be considered for identifying hotspot flooded areas for performing soil and water conservation practices. In this study, to prioritize sub-watersheds from the viewpoint of flooding degree, five flood-related criteria were considered: entropy of drainage network (EN), index of connectivity (IC), Stream Power Index (SPI), curvature (C), and curve number (CN). Then, the fuzzy-based best worse multi-criteria decision-making (F-BWM) method was used to assign weights to these criteria and their combination to determine the flooding degree for each sub-watershed. The prioritization of sub-watersheds indicated that the sub-watersheds #14 and #21 are the most and least susceptible areas to flooding, respectively.
... They identified 11 MCDM methods that have been widely applied, highlighting the need for an efficient MCDM method. In the literature, many researchers have used MCDM methods in the supplier selection process, such as Multi-Attribute Utility Theory (Shaik and Abdul-Kader, 2011), Analytic Hierarchy Process (AHP) (Yadav and Sharma, 2016), fuzzy set theory (Chen et al., 2006), fuzzy AHP (Chan et al., 2008), case-based reasoning (Zhao and Yu, 2011), data envelopment analysis (DEA) (Garfamy, 2006), Simple Multi-Attribute Rating Technique (Ng, 2008), Goal Programming (Choudhary and Shankar, 2014), ELECTRE method (Fahmi et al., 2016), Simple Additive Weighing (Kaur and Kumar, 2013), and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) (Zouggari and Benyoucef, 2012). Other researchers prefer to integrate two methods and techniques to yield more robust decisions: fuzzy AHP and TOPSIS (Jain et al., 2018); AHP and Delphi (Su and Zhan, 2020); AHP and Monte Carlo Method Approach (Kristy and Zagloel, 2020); goal programming and AHP (Khorramshahgol, 2012); AHP and VIKOR (Büyüközkan et al., 2019); Analytical Network Process (ANP) and VIKOR (Abdel-Baset et al., 2019); fuzzy TOPSIS and MCGP (Liao and Kao, 2011); ELECTRE and fuzzy clustering (Azadnia et al., 2011); AHP and ELECTRE II (Wan et al., 2017); utility function and ELECTRE (de Almeida, 2007); fuzzy AHP and fuzzy multiobjective linear programming (Shaw et al., 2012); ANP and DPA (Kuo and Lin, 2012); and ANP and linear programming (Ghodsypour and O'Brien, 1998). ...
Full-text available
Article
Supplier selection is one of the most critical processes in supply chain management (SCM). Most small and medium enterprises (SMEs) face difficulties choosing the best supplier using conventional methods. A hybrid multi-criteria decision-making (MCDM) approach is proposed in supplier selection. This proposed framework integrates the Delphi technique as a data-gathering tool and Analytic Hierarchy Process (AHP) as the MCDM methodology for data analysis; both were used to select an effective supplier. This project applies the Delphi technique, allows experts to select the main criteria, and compares the trade-offs between the available alternatives depending on the main criteria. The criteria selected were price, delivery time, online ranking, rejection rate, and flexibility. Using the AHP approach, the criteria's weights were then assigned. The highest was for the price (43.84%), followed by the rejection rate (21.81%), online ranking (19.27%), delivery time (9.44%), and flexibility (5.64%). Lastly, a new framework was suggested using the weighted criteria collection for supplier selection.
... Based on the results of research between the Creative Economy Agency (Bekraf) and the Central Statistics Agency (BPS) in 2016, it was recorded that the creative economy contributed to national economic growth of 922.59 billion rupiah or 7.44 percent of the national Gross Domestic Product (GDP). Much empirical evidence supports that the creative industry has an impact on GDP through the creative industry sub-sector and creates a new form of governance of the cultural industry (Fahmi et al. 2016;Daubaraite and Startiene 2015). The creative industry has several problems related to its development in Indonesia. ...
Full-text available
Article
This study aims to determine the concept of creative economy development in Indonesia after the COVID-19 pandemic based on management strategies, policies, and the role of other economic actors. This research is a survey research based on ethnography. It is called ethnography because researchers will conduct survey activities in the field by taking several creative economic actors in Indonesia as samples. Based on the research results, it turns out that there are several strategies and policies that can be taken by several parties, both local governments, economic actors, economic activists/observers, and the general public. Thus, the creative economy in Indonesia will continue to survive and be developed to maintain the integrity of the welfare of the Indonesian people after the COVID-19 pandemic. Because it is the source of community life.
... In the area of the American School, for example, Analytic Hierarchy Process (AHP) [10,11] Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [12][13][14], Analytic Network Process (ANP) [15][16][17], VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) [18,19] or Decision-Making Trial and Evaluation Laboratory (DEMATEL) [20,21] methods demonstrated its effectiveness in supplier selection and evaluation problems. Moreover, the European School methods are extensively used in this subject and, for example, include methods such Elimination and Choice Expressing Reality (ELECTRE I) [22,23], ELECTRE II [24,25], ELECTRE III [26,27], PROMETHEE II [28][29][30] or Complex Proportional Assessment (CO-PRAS) [31,32]. There are numerous works that have used a kind of mixed-mode and fuzzy expansions of MCDM methods related with this issue, presented in [33,34] which have recognized their value in the supplier evaluation and selection problem. ...
Full-text available
Article
The supplier selection process is considered one of the most relevant decisions in supply chain management due to its effect on the product quality and on buyer performance. Supplier selection is often unstructured, and is generally based on the lowest-price proposal. However, this type of selection involves a high risk, sometimes resulting in project delays, poor quality of acquired goods, and large financial losses. Price is undoubtedly an important criterion when choosing a supplier; however, other equally important criteria must be considered. Therefore, supplier selection should be formulated as a multi-criteria decision-making (MCDM) problem. This study uses the PROMETHEE-GAIA (Preference Ranking Organization Method for Enrichment of Evaluations—Geometrical Analysis for Interactive Assistance) method to classify and select suppliers in an agrifood company. One of the advantages of this method is that it allows decision-makers to set their preferences considering all the relevant criteria simultaneously, and their relative importance. The case study demonstrates that PROMETHEE constitutes a flexible MCDM tool for supplier evaluation and selection, rank the different alternatives, and provide valuable insights. The results show that the supplier selection process has a strong point related to the existence of two groups of suppliers, one focused on economic criteria and other related to the innovative capacity. However, a flaw emerges, as little relevance is associated to the environmental criterion.
... The fuzzy set theory is an efficient tool to cope with the uncertainty in the evaluation process. Therefore, this study proposes a novel method to collect expert evaluations by HFLTS which have recently gained popularity in the literature (Fahmi et al., 2016;Yu et al., 2016;Basar, 2017). ...
Article
The construction project delivery system may be traditional or non-traditional, dependent upon the project-specific requirements and construction industry climate. Many factors are to be considered when making a decision, making it a multi-criteria decision. At the first stage of a project, most selection factors are vague. The project delivery system selection influences all project stages. In the next decade for Libya, there is a tremendous need to adopt a Projects Delivery System (PDS), which avoids the adverse objectives and conflicts that have characterized the Libyan construction industry for too long. This paper aims to assist decision-makers in understanding these concepts better. By utilizing the Fuzzy model, based on proposed criteria assigned to delivery systems evaluation for Libyan construction projects and the opinion of experts, PDSs were evaluated. The study outcome is twofold. Firstly, it selects the proper criteria and explores a Libyan construction project delivery decision-making system based on the Fuzzy theory perspective. Then has ranked the project delivery systems for the Libyan construction industry as follows: 1st rank construction manager at risk, 2nd rank construction manager as an agency, 3rd rank design-build, and 4th rank design-bid-build system. This study can be a quick and effective approach for construction owners.
Full-text available
Article
Occupational health and safety (OHS) risk assessment studies have gained importance recently as a result of increasing occupational accidents and occupational diseases. The health sector has a greater risk than many sectors for occupational accidents and occupational diseases. Although the health sector is one of the priority sectors in Turkey, OHS practices have not been fully implemented in this field. For this reason, this study adopts a two-stage approach to assess the OHS risks in the health sector by combining the Fine-Kinney and multi-criteria hesitant fuzzy linguistic term set (HFLTS) methods. The proposed method was applied to the OHS risks in the operating room of a public hospital in Turkey. As a solution to the problem, first, the potential hazards and related risks in the operating room were determined by the experts. In this first stage, 44 hazards were determined from the opinions of experts and records of past incidents. Parameter weights were then determined using the multi-criteria HFLTS method. The multi-criteria HFLTS method was used to evaluate seven hazards to be categorized as substantial-risk or higher according to the Fine-Kinney method, taking into account parameter weights. Sensitivity analysis was then carried out. Finally, actions were taken to mitigate the risks.
Full-text available
Article
Digital supply chains (DSCs) are collaborative digital systems designed to quickly and efficiently move information, products, and services through global supply chains. The physical flow of products in traditional supply chains is replaced by the digital flow of information in DSCs. This digitalization has changed the conventional supplier selection processes. We propose an integrated and comprehensive fuzzy multicriteria model for supplier selection in DSCs. The proposed model integrates the fuzzy best-worst method (BWM) with the fuzzy multi-objective optimization based on ratio analysis plus full multiplicative form (MULTIMOORA), fuzzy complex proportional assessment of alternatives (COPRAS), and fuzzy technique for order preference by similarity to ideal solution (TOPSIS). The fuzzy BWM approach is used to measure the importance weights of the digital criteria. The fuzzy MULTIMOORA, fuzzy COPRAS, and fuzzy TOPSIS methods are used as prioritization methods to rank the suppliers. The maximize agreement heuristic (MAH) is used to aggregate the supplier rankings obtained from the prioritization methods into a consensus ranking. We present a real-world case study in a manufacturing company to demonstrate the applicability of the proposed method.
Article
Supplier selection and evaluation is approached in the literature as a multi-criteria decision problem in which usually more than one decision-maker has to judge the importance of criteria and the performance of suppliers. Fuzzy techniques are commonly applied to deal with the uncertainty in the evaluation process. Intuitionistic and hesitant fuzzy representations have been applied to group decision problems. However, none of the studies in the literature presents a comparison of these two fuzzy representations when applied to multi-criteria group decision-making (MCGDM) problems. Thus, this paper presents the results of a comparative study of the intuitionist fuzzy and the hesitant fuzzy representations applied to supplier selection problem. The techniques were implemented and tested in a pilot application to a textile manufacturing company. The comparison was based on congruency of results, adequacy to group decision, data collection effort and flexibility of judgement, computational complexity and modelling of uncertainty.
Article
In the first part of this paper, we describe the main features of real-world problems for which the outranking approach is appropriate and we present the concept of outranking relations. The second part is devoted to basic ideas and concepts used for building outranking relations. The definition of such outranking relations is given for the main ELECTRE methods in Part 3. The final part of the paper is devoted to some practical considerations.
Chapter
As its name suggests, computing with words, CW, is a methodology in which words are used in place of numbers for computing and reasoning. The point of this note is that fuzzy logic plays a pivotal role in CW and vice-versa. Thus, as an approximation, fuzzy logic may be equated to CW. There are two major imperatives for computing with words. First, computing with words is a necessity when the available information is too imprecise to justify the use of numbers. And second, when there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost and better rapport with reality. Exploitation of the tolerance for imprecision is an issue of central importance in CW. In CW, a words is viewed as a label of a granule, that is, a fuzzy set of points drawn together by similarity, with the fuzzy set playing the role of a fuzzy constraint on a variable. The premises are assumed to be expressed as propositions in a natural language. For purposes of computation, the propositions are expressed as canonical forms which serve to place in evidence the fuzzy constraints that are implicit in the premises. Then, the rules of inference in fuzzy logic are employed to propagate the constraints from premises to conclusions. At this juncture, the techniques of computing with words underlie -- in one way or another -- almost all applications of fuzzy logic. In coming years, computing with words is likely to evolve into a basic methodology in its own right with wide-ranging ramifications on both basic and applied levels.
Chapter
There are some classical decision rules such as dominance, maximin and maximum which are still fit for the MADM environment. They do not require the DM’s preference information, and accordingly yield the objective (vs. subjective) solution. However, the right selection of these methods for the right situation is important. (See Table 1.3 for references).