Content uploaded by Ali Fahmi

Author content

All content in this area was uploaded by Ali Fahmi on Feb 29, 2016

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 ܪௌ, and is an ordered finite

subset of the consecutive linguistic terms set ܵ=

൛ܵ,ܵଵ,…,ܵൟ.

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(ܪௌ)=[ܪௌష,ܪௌశ],ܪௌషܪௌశ (1)

ܪௌశ=max {ܵ}ܵאܪௌ

ܪௌష= min {ܵ}ܵאܪௌ

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(ܽଵ,ܽଶ,…,ܽ)=σݓܾ

ୀଵ (2)

where ܽଵ,ܽଶ,…,ܽis the aggregated set of arguments

and ܾis the jth largest argument of the aggregated set,

and the associated weighting vector

ܹ=(ݓଵ,ݓଶ,…,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:

ܾ=ܱܹܣௐೞ ( ܽெ

,ܽெ

ାଵ,…,ܽெ

) (4)

ܿ=ܱܹܣௐ ( ܽெ

,ܽெ

ାଵ,…,ܽெ

) (5)

where ܹ௦and ܹ௧represent 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:

ܹଵ=(ݓଵଵ,ݓଶଵ,…,ݓ

ଵ)்

ݓଵଵ=ߙ,ݓଶଵ=ߙ(1െߙ),ݓଷଵ=ߙ(1െߙ)ଶ,…,ݓିଵ

ଵ=

ߙ(1െߙ)ିଶ, (6)

ݓ

ଵ=(1െߙ)ିଵ

Also, the second type of weights of OWA operator is as

follows:

ܹଶ=(ݓଵଶ,ݓଶଶ,…,ݓଶ)்

ݓଵଶ=ߙ,ݓଶଶ=ߙ(1െߙ),ݓଷଶ=ߙ(1െߙ)ଶ,…,ݓିଵ

ଶ=

ߙ(1െߙ)ିଶ, (7)

ݓଶ=(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 ܣሚ=(ܽ,ܾ,ܿ,݀)is a

trapezoidal fuzzy number shown in Fig. 1, and its

membership function is given by:

ߤ(ݔ)=

ە

ۖ

۔

ۖ

ۓ

0, ݔ<ܽ,

௫ି

ି,ܽݔܾ,

1, ܾ<ݔ<ܿ,

ௗି௫

ௗି,ܿݔ݀,

0, ݔ>݀,

(8)

Fig. 1. Trapezoidal fuzzy number,

෩=(,,,)

Definition 7. Let A and B be two positive trapezoidal

fuzzy numbers, where ܣሚ=(ܽ,ܾ,ܿ,݀)and

ܤ෨=(,ݍ,ݎ,ݏ). The arithmetic operations between ܣሚ

and ܤ෨is as follows:

Addition operation:

ܣሚ۩ܤ෨=(ܽ+,ܾ+ݍ,ܿ+ݎ,݀+ݏ) (9)

Subtraction operation:

ܣሚٓܤ

෨=(ܽെݏ,ܾെݎ,ܿെݍ,݀െ) (10)

Multiplication operation:

ܣሚٔܤ

෨؆(ܽ,ܾݍ,ܿݎ,݀ݏ) (11)

Division operation:

ܣሚٕܤ

෨؆(

௦,

,

,ௗ

) (12)

Multiplication operation:

ݑ.ܣሚ=ܣሚ.ݑ=൜(ݑήܽ,ݑήܾ,ݑήܿ,ݑή݀)ifݑ0

(ݑή݀,ݑήܿ,ݑήܾ,ݑήܽ)ifݑ<0 (13)

Division operation:

෨

௨=ቐ(

௨,

௨,

௨,ௗ

௨)ifݑ0

(ௗ

௨,

௨,

௨,

௨)ifݑ0

(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 ܩு.

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 ܩு. An

example of context-free grammar is provided below

[50]:

ܩு={ܸே,்ܸ,ܫ,ܲ}

where ܸேshows non-terminal symbols and ்ܸshows

terminal symbols as follows:

ܸே=൜ۃݎ݅݉ܽݎݕݐ݁ݎ݉ۄ,ۃܾ݅݊ܽݎݕݎ݈݁ܽݐ݅݊ۄ,

ۃݑ݊ܽݎݕݐ݁ݎ݉ۄ,ۃ݆ܿ݊ݑ݊ܿݐ݅݊ۄൠ

்ܸ=൛ܽݐ݈݁ܽݏݐ,ܽݐ݉ݏݐ,ܾ݁ݐݓ݁݁݊,ܽ݊݀,ܵ,ܵଵ,…,ܵൟ

ܫאܸே

Pindicates the production rules for the context-free

grammar ܩு. Bracket signs shows optimal elements

and the symbol “|” indicates alternative elements. In this

example, the production rules are as follows:

ܲ=ە

ۖ

۔

ۖ

ۓ

ݎ݅݉ܽݎݕݐ݁ݎ݉=ݏ|ݏଵห…หݏ,

ۃܾ݅݊ܽݎݕݎ݈݁ܽݐ݅݊ۄ=ܾ݁ݐݓ݁݁݊,

ۃݑ݊ܽݎݕݎ݈݁ܽݐ݅݊ۄ=ܽݐ݈݁ܽݏݐ|ܽݐ݉ݏݐ|݈ݓ݁ݎݐ݄ܽ݊|݃ݎ݁ܽݐ݁ݎݐ݄ܽ݊,

ۃ݆ܿ݊ݑ݊ܿݐ݅݊ۄ=ܽ݊݀ ۙ

ۖ

ۘ

ۖ

ۗ

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 ܧீಹ.

Let ܵ=൛ݏ,ݏଵ,…,ݏൟbe the linguistic terms set.

DM evaluates the performance and criteria by

comparative linguistic terms based on the defined

context-free grammar ܩு. The transformation of the

evaluations into HFLTS will be carried out by

transformation function ܧீಹas follows:

ܧீಹ(ݏ)={ݏ|ݏאܵ}={ݏ},

ܧீಹ൫ܾ݁ݐݓ݁݁݊ݏܽ݊݀ݏ൯=൛ݏหݏݏݏܽ݊݀ݏא

ܵൟ=൛ݏ,ݏାଵ,…,ݏൟ,

ܧீಹ(ܽݐ݈݁ܽݏݐݏ)=൛ݏหݏݏܽ݊݀ݏאܵൟ=

൛ݏ,ݏାଵ,…,ݏൟ,

ܧீಹ(ܽݐ݉ݏݐݏ)=൛ݏหݏݏܽ݊݀ݏאܵൟ

={ݏ,ݏଵ,…,ݏ}

ݏݏଶ

ݏଵݏ

ݏିଵ

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 ܧீಹ൫ܾ݁ݐݓ݁݁݊ݏܽ݊݀ݏ൯.

ܣ௧௪ =

൛ܽ

,ܽெ

,ܽ

ାଵ,ܽோ

,ܽெ

ାଵ,ܽ

ାଶ,ܽோ

ାଵ,…,ܽ,ܽோ

ିଵ,ܽெ

,ܽோ

ൟ

(15)

ܣ௧௦௧ =

൛ܽ

,ܽெ

,ܽ

ାଵ,ܽோ

,ܽெ

ାଵ,ܽ

ାଶ,ܽோ

ାଵ,…,ܽ,ܽோ

ିଵ,ܽெ

,ܽோ

ൟ

(16)

ܣ௧௦௧ =

൛ܽ

,ܽெ

,ܽ

ଵ,ܽோ

,ܽெ

ଵ,ܽ

ଶ,ܽோ

ଵ,…,ܽ

,ܽோ

ିଵ,ܽெ

,ܽோ

ൟ(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:

ܣ௧௪ =൛ܽ

,ܽெ

,ܽெ

ାଵ,…,ܽெ

,ܽோ

ൟ

ܣ௧௦௧ =൛ܽ

,ܽெ

,ܽெ

ାଵ,…,ܽெ

,ܽோ

ൟ

ܣ௧௦௧ =൛ܽ

,ܽெ

,ܽெ

ଵ,…,ܽெ

,ܽோ

ൟ

Step 5. Obtain fuzzy envelope for each assessment.

ܶ෨௧௪ =(ܽ,ܾ,ܿ,݀),

ܽ=݉݅݊ܣ௧௪ =݉݅݊൛ܽ

,ܽெ

,ܽெ

ାଵ,…,ܽெ

,ܽோ

ൟ=

ܽ

,

݀=maxܣ௧௪ =max൛ܽ

,ܽெ

,ܽெ

ାଵ,…,ܽெ

,ܽோ

ൟ=

ܽோ

,

ܾ=ܱܹܣௐమ ( ܽெ

,ܽெ

ାଵ,…,ܽெ

),

ܿ=ܱܹܣௐభ ( ܽெ

,ܽெ

ା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

ܾ=ܱܹܣௐమቆܽெ

,ܽெ

ାଵ,…,ܽெ

శೕషభ

మቇ,

ܿ=ܱܹܣௐభቆܽெ

,ܽெ

ିଵ,…,ܽெ

శೕశభ

మቇ,

ܹଶ=൫ݓଵଶ,ݓଶଶ,…,ݓିାଵ ଶ

Τ

ଶ൯்and ߙଵ=ି(ି)

ିଵ

ݓ݄݁ݎ݁ݓଵଶ=ߙଵିିଵ

ଶ,ݓଶଶ

=(1െߙଵ)ߙଵିିଷ

ଶ,…,ݓିିଵ

ଶ

ଶ

=(1െߙଵ)ߙଵ,ݓିାଵ

ଶ

ଶ=(1െߙଵ)

ܾ=ݓଵଶ.ܽெ

ାିଵ

ଶ,ݓଶଶ.ܽெ

ାିଷ

ଶ,…,ݓିିଵ

ଶ

ଶ.ܽெ

ାଵ,ݓିାଵ

ଶ

ଶ.ܽெ

and ܹଵ=൫ݓଵଵ,ݓଶଵ,…,ݓିାଵ ଶ

Τ

ଵ൯்and ߙଶ=1െߙଵ=

(ି)ିଵ

ିଵ

where ݓଵଵ=ߙଶ,ݓଶଵ=ߙଶ(1െߙଶ),…,ݓೕషషభ

మ

ଶ

=ߙଶ(1െߙଶ)ିିଷ

ଶ,ݓିାଵ

ଶ

ଶ=(1െߙଶ)ିିଵ

ଶ

ܿ=ݓଵଵ.ܽெ

ାାଵ

ଶ,ݓଶଵ.ܽெ

ାିଵ

ଶ,…,ݓିିଵ

ଶ

ଵ.ܽெ

ାଵ,ݓିାଵ

ଶ

ଵ.ܽெ

II) If i+j is even, then

ܾ=ܱܹܣௐమቆܽெ

,ܽெ

ାଵ,…,ܽெ

శೕ

మቇ,

ܿ=ܱܹܣௐభቆܽெ

,ܽெ

ିଵ,…,ܽெ

శೕ

మቇ,

ܹଶ=൫ݓଵଶ,ݓଶଶ,…,ݓିାଶ ଶ

Τ

ଶ൯்

where ݓଵଶ=ߙଵೕష

మ,ݓଶଶ=(1െߙଵ)ߙଵೕషషమ

మ,…,ݓೕష

మ

ଶ

ܾ=ݓଵଶ.ܽெ

ା

ଶ,ݓଶଶ.ܽெ

ାିଶ

ଶ,…,ݓି

ଶ

ଶ.ܽெ

ାଵ,ݓିାଶ

ଶ

ଶ.ܽெ

and ܹଵ=൫ݓଵଵ,ݓଶଵ,…,ݓିାଶ ଶ

Τ

ଵ൯்

where ݓଵଵ=ߙଶ,ݓଶଵ=ߙଶ(1െߙଶ),…,ݓೕష

మ

ଵ

=ߙଶ(1െߙଶ)ିିଶ

ଶ,ݓିାଶ

ଶ

ଵ=(1െߙଶ)ି

ଶ

ܿ=ݓଵଵ.ܽெ

ା

ଶ,ݓଶଵ.ܽெ

ାିଶ

ଶ,…,ݓି

ଶ

ଵ.ܽெ

ିଵ,ݓିାଶ

ଶ

ଵ.ܽெ

Finally, the HFLTS envelope will be formed as

ܶ෨௧௪ =(ܽ

,ܾ,ܿ,ܽோ

). 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

ܶ෨௧௦௧ =(ܽ,ܾ,ܿ,݀),

ܽ=݉݅݊ܣ௧௦௧ =݉݅݊൛ܽ

,ܽெ

,ܽெ

ାଵ,…,ܽெ

,ܽோ

ൟ=ܽ

,

݀=maxܣ௧௦௧ =max൛ܽ

,ܽெ

,ܽெ

ାଵ,…,ܽெ

,ܽோ

ൟ=ܽோ

ܾ=ܱܹܣௐమ ( ܽெ

,ܽெ

ାଵ,…,ܽெ

)

ܿ=ܱܹܣௐమ ( ܽெ

,ܽெ

ା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,

ܹଶ=(ݓଵଶ,ݓଶଶ,…,ݓଶ)்and ߙ=݅݃

ൗ

where ݓଵଶ=ߙି,ݓଶଶ=(1െߙ)ߙିିଵ,…,ݓି

ଶ=

ߙ(1െߙ)

ݓିାଵ

ଶ=1െߙ

ܾ=ݓଵଶ.ܽெ

,ݓଶଶ.ܽெ

ିଵ,…,ݓି

ଶ.ܽெ

ାଵ,ݓିାଵ

ଶ.ܽெ

,

ܾ=ߙି.ܽெ

,(1െߙ)ߙିିଵ.ܽெ

ିଵ,…,

ߙ(1െߙ).ܽெ

ାଵ,(1െߙ).ܽெ

ܿ=ܽெ

Finally, the HFLTS envelope is formed as ܶ෨௧௦௧ =

(ܽ

,ܾ,ܽெ

,ܽ). See Fig. 4.

Fig. 4. Fuzzy Envelope for comparative linguistic expression

"at least".

ܶ෨௧௦௧ =(ܽ,ܾ,ܿ,݀),

ܽ=݉݅݊ܣ௧௦௧ =݉݅݊൛ܽ

,ܽெ

,ܽெ

ଵ,…,ܽெ

,ܽோ

ൟ=ܽ

,

݀=maxܣ௧௦௧ =max൛ܽ

,ܽெ

,ܽெ

ଵ,…,ܽெ

,ܽோ

ൟ=ܽோ

ܾ=ܱܹܣௐభ ( ܽெ

,ܽெ

ଵ,…,ܽெ

)

ܿ=ܱܹܣௐభ ( ܽெ

,ܽெ

ଵ,…,ܽெ

)

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,

ܹଵ=(ݓଵଶ,ݓଶଶ,…,ݓଶ)்and ߙ=݅݃

ൗ

where ݓଵଵ=ߙ,ݓଶଵ=ߙ(1െߙ),…,ݓଵ=ߙ(1െ

ߙ)ିଵ,ݓାଵ

ଵ=(1െߙ)

ܿ=ݓଵଵ.ܽெ

,ݓଶଵ.ܽெ

ିଵ,…,ݓଵ.ܽெ

ଵ,ݓାଵ

ଵ.ܽெ

,

ܿ=ߙ.ܽெ

,ߙ(1െߙ).ܽெ

ିଵ,…,ߙ(1െߙ)ିଵ.ܽெ

ଵ,(1െߙ).ܽெ

ܾ=ܽெ

Finally, the HFLTS envelope is formed as ܶ෨௧௦௧ =

(ܽ

,ܽெ

,ܿ,ܽோ

). 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

a

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 56áRZLQVNL

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