Applying linguistic information and intersection concept

to improve e®ectiveness of multi-criteria decision

analysis technology

Ping-Feng Pai

Department of Information Management, National Chi Nan University1, University Road, Nantou County 54561, Taiwan

Chen-Tung Chen

Department of Information Management, National United University

1, Lienda Road, Miaoli County 36003, Taiwanctchen@nuu.edu.tw

Wei-Zhan Hung

Department of International Business Studies, National Chi Nan University

1, University Road, Nantou County 54561, Taiwan

Published 21 January 2014

Multi-criteria decision-making (MCDM) is one of the most widely used decision methodologies.

Because every kind of MCDM approach has unique strengths and weaknesses, it is di±cult to

determine which kind of MCDM approach is best suited to a speci¯c problem. Therefore, a new

decision-making method is proposed herein, based on linguistic information and intersectionconcepts; it is called the linguistic intersection method (LIM). Notably, the linguistic variables

are more suited to expressing the opinion of each decision maker. There are four MCDM

methods: TOPSIS, ELECTRE, PROMETHEE and VIKOR which are included in the LIM.

First, each MCDM approach is used to determine the ranking order of all alternatives inaccordance with the linguistic evaluations of decision makers. Then, the intersection set is

determined with regard to the better alternatives of all methods. Third, the ¯nal ranking order

of alternatives in the intersection set can be determined by the proposed method. Lastly, anexample is given to describe the procedure of the proposed method. In order to verify the

e®ectiveness of the proposed method, a simulation test is provided to compare the LIM with the

linguistic MCDM method. According to the comparison results, the proposed method is more

stable in determining the ranking order of all decision alternatives.

Keywords: MCDM; linguistic variable; linguistic intersection method; simulation.

1. Introduction

Multi-criteria decision-making (MCDM) is one of the most widely used decision

methodologies in the sciences, business, governmental organizations and engineering

arenas. MCDM methods can help to improve the quality of decisions by allowing the

decision-making process to become more explicit, rational and e±cient.1,2 Currently,

International Journal of Information Technology & Decision Making

Vol. 13, No. 2 (2014) 291–315

°c World Scienti¯c Publishing CompanyDOI: 10.1142/S0219622014500436

291

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there are a variety of MCDM approaches, i.e., utility function,3 fuzzy integral,4

AHP,5 ANP,6 ELECTRE,7 TOPSIS,8 rough set9 and PROMETHEE.10 However,

every kind of MCDM approach has a unique scope and speci¯c limitations.

The technique for order performance by similarity to ideal solution (TOPSIS)

was ¯rst developed by Hwang and Yoon for solving MCDM problems.11 Speci¯cally,

TOPSIS involves choosing the best alternative according to the relative position,

which is the shortest distance from the positive ideal solution (PIS) and the farthest

from the negative ideal solution (NIS), from among all of the alternatives. TOPSIS

has been applied in tra±c police performance assessment,12 country tourism

industry competitiveness assessment,13 computer-integrated manufacturing tech-

nology selection,14 energy e±cient network selection,15 business failure prediction16

and multiclass classi¯er comparison.17 By using TOPSIS, a total ranking order of all

alternatives can be developed. But the drawback of TOPSIS is that it cannot make a

pairwise comparison and provide the degree of di®erence among the available

alternatives.

The ELECTRE method is a highly developed multi-criteria analysis model which

takes into account the uncertainty and vagueness in the decision-making process18; it

is based on the axiom of partial comparability, which is suitable for alternative

selection. There are various kinds of ELECTRE methods which have been developed,

such as ELECTRE I,19 ELECTRE II,20 ELECTRE III,21 ELECTRE IV,22 ELECTRE

GD,23 Electre-CBR-I24 and ELECTRE TRI.25 ELECTRE has been applied in

material selection,22 data mining,24 gas pipelines risk sorting,25 stock portfolio selec-

tion26 and bankruptcy prediction.27 However, it is not easy to obtain the total ranking

order of all alternatives in a real environment using the ELECTRE method for an

MCDM problem.

Preference Ranking Organization Method for Enrichment Evaluation (PRO-

METHEE) is amulti-criteria decision-makingmethod.28 It is well adapted to problems

where a ¯nite number of alternative actions are to be ranked by considering several,

sometimes con°icting, criteria.28 There are six basic types of preference functions in the

PROMETHEE method, so decision makers can establish °exible standards according

to the requirements of a particular decision-making problem with respect to each

criterion. PROMOTHEE can accommodate multiclass classi¯er comparison pro-

blems,17 portfolio selection,29 bankruptcy prediction30 and systems design.31 The

drawback of PROMETHEE is that it needs to use a pairwise comparison action based

on threshold values for determining a total order of considered alternatives. However,

it is not suitable for deciding the total order of a large range of decision alternatives.32

VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje/Multicriteria

Optimization and Compromise Solution) is a MCDM method developed by Opri-

covic.33 VIKOR determines a compromised solution that provides the group utility

with the maximum and minimum individual regret for the opponent34; thus, VIKOR

can ¯nd a compromised priority ranking order of alternatives according to the

selected criteria.35,36 VIKOR has been used in security risk assessment,37 material

selection,38 personnel selection39 and renewable energy project selection.40 However,

292 P.-F. Pai, C.-T. Chen & W.-Z. Hung

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it usually was unable to provide a total order of alternatives in accordance with the

ranking rules of the VIKOR method. Therefore, a sensitive analysis needs to be

executed in regard to the VIKOR method, in order to increase the robustness of a

ranking order for an MCDM problem.

Notably, e®ectiveness is more important than e±ciency in a MCDM problem.

Because every kind of MCDM approach has its own strengths and weaknesses, it is

di±cult to determine which kind of MCDM approach is best suited to a speci¯c

problem. Choosing an unsuitable MCDM approach to make decisions will reduce

the e®ectiveness and quality of the decisions. In order to avoid this problem, a

linguistic intersection method (LIM) is presented herein to handle MCDM problem

in a fuzzy environment. The LIM involves treating every kind of MCDM approach

as an expert and allowing them to determine the performance of each alternative

according to its calculative mechanism. There are four MCDM methods included in

the LIM: TOPSIS, ELECTRE, PROMETHEE and VIKOR. First, each MCDM

approach is used to determine the ranking order of all alternatives in accordance

with the linguistic evaluations by decision makers. Second, the intersection set is

determined with regard to the better alternatives of all methods. Third, the ¯nal

ranking order of alternatives in the intersection set can be determined by the pro-

posed method.

The information for choosing the best alternative in the decision-making process

comprises quantitative and qualitative information. Quantitative information is easy

to describe by its crisp values. However, qualitative information is di±cult to describe

by crisp values and is usually expressed by an expert's subjective opinion. Because an

expert's subjective opinion is subject to vagueness and imprecise relationships to

realworld situations, a more realistic approach may be to use linguistic assessments

instead of crisp values.41 The 2-tuple linguistic representation model is based on the

concept of symbolic translation.42,43 It is an e®ective method which reduces the

mistakes of information translation and avoid the information loss by computing with

words.44

This study is organized as follows. In Sec. 2, we discuss the evaluation information

generally used in MCDM problems. In Sec. 3, we describe the details of the proposed

method. In Sec. 4, an example is implemented to describe the procedure for the

proposed method. In Sec. 5, we compare the proposed method with individual

MCDM method. Finally, the conclusion and future research options are o®ered.

2. Evaluation Information

In general, quantitative and qualitative information will be collected in the decision-

making process. Considering the personnel selection problem, quantitative infor-

mation is easy to describe by crisp values, i.e., working experience, the grade of

TOEIC test, the educational background and licenses.45–47 Qualitative information

is usually expressed by an expert's subjective opinion, i.e., working ability, the degree

of loyalty and honesty.

Applying linguistic intersection concept to MCDA technology 293

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Qualitative information can be expressed by a 2-tuple linguistic variable. The

membership function of a 2-tuple linguistic variable can be expressed as a triangle

fuzzy number.48 Notably, there are two types of 2-tuple linguistic variable applied in

this study (shown in Table 1). The membership functions of the two types of 2-tuple

linguistic variable are shown in Figs. 1 and 2.

Let S ¼ fs0; s1; s2; . . . ; sgg be a ¯nite and totally ordered linguistic term set. A 2-

tuple linguistic variable can be expressed as (si; �i), where si is the central value of

ith linguistic term in S and �i is a numerical value representing the di®erence

between calculated linguistic term and the closest index label in the initial linguistic

term set. The symbolic translation function � is presented to translate crisp value �

into a 2-tuple linguistic variable.48 Then, the symbolic translation process is applied

to translate � (� 2 ½0; 1�) into a 2-tuple linguistic variable as49:

� : ½0; 1� ! S � � 1

2g;1

2g

� �; ð2:1Þ

�ð�Þ ¼ ðsi; �iÞ; ð2:2Þwhere i ¼ roundð� � gÞ, �i ¼ � � i

g and �i 2 ½� 12g ;

12gÞ.

Fig. 1. Membership functions of linguistic variables at type 1.

Fig. 2. Membership functions of linguistic variables at type 2.

Table 1. Di®erent types of linguistic variables.

Linguistic variable

Type 1: Extremely poor ðs 50Þ, Poor ðs 51Þ, Fair ðs 52Þ, Good ðs 53Þ, Extremely good ðs 54Þ Fig. 1

Type 2: Extremely poor ðs 70Þ, Poor ðs 71Þ, Medium poor ðs 72Þ, Fair ðs 73Þ, Medium good ðs 74Þ,Good ðs 75Þ, Extremely good ðs 76Þ

Fig. 2

294 P.-F. Pai, C.-T. Chen & W.-Z. Hung

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A reverse function ��1 is de¯ned to return an equivalent numerical value � from

2-tuple linguistic information ðsi; �iÞ. It can be represented as follows49:

��1ðsi; �iÞ ¼i

gþ �i ¼ �: ð2:3Þ

Let x ¼ fðr1; �1Þ; ðr2; �2Þ; . . . ; ðrn; �nÞg be a 2-tuple linguistic variable set and

W ¼ fðw1; �w1Þ; ðw2; �w2Þ; . . . ; ðwn; �wnÞg be the set of linguistic weights of each

linguistic variable. The linguistic arithmetic mean �X is computed as50:

�X ¼ �1

n

Xni¼1

��1ðri; �iÞ !

¼ ðsm; �mÞ: ð2:4Þ

The linguistic weighted arithmetic mean is computed as50:

�Xw ¼ �1

n

Pni¼1ð��1ðri; �iÞ ���1ðwi; �wiÞÞPn

i¼1 ��1ðwi; �wiÞ

!¼ ðswm; �w

mÞ: ð2:5Þ

In general, decision makers would use di®erent kinds of 2-tuple linguistic variables

based on their knowledge or experiences in expressing their opinions.51 A trans-

formation function is needed to transfer these 2-tuple linguistic variables from

di®erent kinds of linguistic sets to a standard linguistic set at a unique domain. In

Ref. 52, the domain of the linguistic variables increased as the number of linguistic

variables increased. To overcome this drawback, a translation function is applied

here to transfer a crisp number or 2-tuple linguistic variable to a standard linguistic

term at the unique domain.49 Suppose that the interval [0, 1] is the unique domain.

The linguistic variable sets with di®erent types will be de¯ned by partitioning the

interval [0, 1]. Transforming a crisp number � (� 2 [0, 1]) into ith linguistic term

ðs nðtÞi ; �nðtÞi Þ of type t yields:

�tð�Þ ¼ ðs nðtÞi ; �nðtÞi Þ; ð2:6Þ

where i ¼ roundð� � gtÞ, �nðtÞi ¼ � � i

gtgt ¼ nðtÞ � 1, and nðtÞ is the number of lin-

guistic variable of type t.

Transforming ith linguistic term of type t into a crisp number � (� 2 [0, 1]) yields:

��1t ðs nðtÞi ; �

nðtÞi Þ ¼ i

gtþ �

nðtÞi ¼ �; ð2:7Þ

where gt ¼ nðtÞ � 1 and �nðtÞi 2 ½� 1

2gt; 12gt

Þ.Therefore, the transformation from ith linguistic term ðs nðtÞi ; �

nðtÞi Þ of type t

to k th linguistic term ðs nðtþ1Þk ; �

nðtþ1Þk Þ of type t þ 1 at interval [0, 1] can be

expressed as:

�tþ1ð��1t ðs nðtÞi ; �

nðtÞi ÞÞ ¼ ðsnðtþ1Þ

k ; �nðtþ1Þk Þ; ð2:8Þ

where gtþ1 ¼ nðt þ 1Þ � 1 and �nðtþ1Þk 2 ½� 1

2gtþ1; 12gtþ1

Þ.

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3. Linguistic Intersection Method

Traditionally, the quantitative and qualitative data are collected and considered

simultaneously to make decisions in a MCDM method. However, it is not easy to

explain why a special kind of MCDM method can e®ectively rank alternatives when

the criteria for making decision are usually con°icting, and alternatives can perform

better than each other with regard to di®erent criteria.

In reality, an e®ective manager will make decisions by simultaneously considering

the suggestions of multiple experts. We can consider di®erent kinds of MCDM

methods (such as linguistic TOPSIS, linguistic ELECTRE, linguistic PROMETHEE

and linguistic VIKOR) as di®erent experts who provide information pertaining to

each alternative.

In this study, we execute four kinds of MCDM methods. Then, the ranking order

of alternatives can be determined by each MCDM approach. The intersection set

(consensus alternative set) is picked up from the chosen set by each MCDM

approach. Lastly, the ¯nal ranking order of each alternative in the consensus

alternative set can be determined by considering the importance of each MCDM

approach from the viewpoint of the manager. A conceptualization of the LIM is

shown in Fig. 3.

Generally speaking, the contents of the decision-making process will be included

as follows:

(1) A set of alternatives is called A ¼ fA1;A2; . . . ;Amg.(2) A set of criteria is called C ¼ fC1;C2; . . . ;Cng. The quantitative criteria are

from C1 to CZ . The qualitative criteria are from CZþ1 to Cn.

MCDM 1 MCDM 2 MCDM m

Start

End

Collect quantitative and qualitative information

MCDM 3 ………

Determining the ranking order of alternatives in consensus set by LIM

Determine the consensus set

Fig. 3. The conceptualization of the LIM.

296 P.-F. Pai, C.-T. Chen & W.-Z. Hung

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(3) A set of decision makers is called D ¼ fE1;E2; . . . ;Ekg.(4) The ~wj (j ¼ 1; 2; . . . ; nÞ can be represented as the linguistic weight of jth

criterion.

(5) D ¼ ½xij �m�nði ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; nÞ is a decision making matrix. It can

be represented as:

D ¼ ½xij �mn ¼

C1 . . . Cz Czþ1 . . . Cn

A1

A2

. . .

Am

x11 . . . x1z ~x1zþ1 . . . ~x1nx21 . . . x2z ~x2zþ1 . . . ~x2n. . . . . . . . . . . . . . . . . .

xm1 . . . xmz ~xmzþ1 . . . ~xmn

2664

3775 : ð3:1Þ

For quantitative criteria, xij represents the performance of i-th alternative with

respect to jth criterion. We use crisp value CVij to represent xij .

For qualitative criteria,~xij represents the linguistic performance of ith alternative

with respect to jth criterion. Decision makers can use 2-tuple linguistic variables to

express their opinions about linguistic performances. Let F kj ðAiÞ ¼ ðS k

ij ; �kijÞ rep-

resent the linguistic performance of ith alternative with respect to jth criterion which

is expressed by kth decision maker.

Transferring crisp value CVij to a 2-tuple linguistic variable is computed as:

FjðAiÞ ¼ ��1CVij � miniðCVijÞ

maxiðCVijÞ � miniðCVijÞ� �

: ð3:2Þ

Aggregating the opinions of all decision makers about the linguistic performances of

ith alternative with respect to jth criterion is computed as:

~xij ¼ �1

K

XKk¼1

��1ðS kij ; �

kijÞ

!¼ ðSij ; �ijÞ: ð3:3Þ

Aggregating the opinions of all decision makers about the linguistic weight of jth

criterion is computed as:

~W j ¼ �1

K

XKk¼1

��1ðS wjk ; �

wjkÞ

!¼ ðS w

j ; �wj Þ: ð3:4Þ

There are four kinds of MCDM methods which are presented here based on linguistic

variables and the decision matrix.

3.1. Linguistic TOPSIS

The linguistic TOPSIS was developed by Chen and Cheng.53 Linguistic TOPSIS has

been applied in information system selection,53 stock portfolio selection,54 service

quality evaluation55 and factory cleaning system selection.56 According to~xij and ~wj ,

Applying linguistic intersection concept to MCDA technology 297

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the linguistic weighted matrix can be computed as:

~V ¼ ½~v ij �m�n; i ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; n; ð3:5Þ

where ~v ij ¼ �ð��1ð~xijÞ ���1ð ~wjÞÞ.Thus, the positive ideal solution and negative ideal solution can be represented as

A� ¼ ð~v �1; ~v

�2; . . . ; ~v

�nÞ and A� ¼ ð~v �

1 ; ~v�2 ; . . . ; ~v

�n Þ, where ~v �

j ¼ �ðmaxið��1ð~vijÞÞÞand ~v �

j ¼ �ðminið��1ð~vijÞÞÞ.The distance between alternative Ai and the positive ideal solution (A�) can be

calculated as:

d �i ¼ dðAi;A

�Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnj¼1

maxi

ð��1ð~v ijÞÞ ���1ð~v ijÞ� �

2

vuut : ð3:6Þ

The distance between alternative Ai and the negative ideal solution (A�) can be

calculated as:

d�i ¼ dðAi;A

�Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnj¼1

minið��1ð~v ijÞÞ ���1ð~v ijÞ

� �2

vuut : ð3:7Þ

Then, the closeness coe±cient of each alternative AiðCCiÞ can be computed as:

CCi ¼d�i

ðd �i þ d�

i Þ; i ¼ 1; 2; . . . ;m: ð3:8Þ

The ranking order of alternatives can be determined in accordance with the closeness

coe±cient. If CCi > CCj , then alternative Ai is a better than alternative Aj .

3.2. Linguistic ELECTRE

The linguistic ELECTRE is presented by Chen and Hung.57 Linguistic ELECTRE

has been applied in project selection57 and stock portfolio selection.26,54 According to

ELECTRE, three threshold values of each criterion (Cj) will be de¯ned: preference

threshold pj , indi®erence threshold qj and veto threshold vj .

The concordance index CjðAi;AlÞ represents the degree of alternative Ai being

better than Al with respect to jth criterion. It can be computed as:

CjðAi;AlÞ ¼

1; ��1ð~xijÞ � ��1ð~xljÞ � qj

��1ð~xijÞ ���1ð~xljÞ þ pjpj � qj

; ��1ð~xljÞ � qj � ��1ð~xijÞ � ��1ð~xljÞ � pj

0; ��1ð~xijÞ � ��1ð~xljÞ � pj

8>>>><>>>>:

:

ð3:9Þ

298 P.-F. Pai, C.-T. Chen & W.-Z. Hung

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The overall concordance index CðAi;AlÞ represents the total degree of alternative Ai

being better than Al . It can be computed as:

C ðAi;AlÞ ¼Xnj¼1

��1ð ~wjÞPnk¼1 �

�1ð ~wkÞ� �

CjðAi;AlÞ: ð3:10Þ

The discordance index DjðAi;AlÞ represents the degree of alternative Ai being not

better than Al with respect to jth criterion. It can be computed as

DjðAi;AlÞ ¼

1; ��1ð~xijÞ � ��1ð~xljÞ � vj

��1ð~xljÞ � pj ���1ð~xijÞvj � pj

; ��1ð~xljÞ � pj � ��1ð~xijÞ � ��1ð~xljÞ � vj

0; ��1ð~xijÞ � ��1ð~xljÞ � pj

8>>><>>>:

:

ð3:11ÞThe credibility matrix SðAi;AlÞ represents the preference degree of alternative Ai

being better than Al . It can be computed as:

SðAi;AlÞ ¼C ðAi;AlÞ; if DjðAi;AlÞ � CðAi ;AlÞ 8 j

C ðAi;AlÞQ

j 2JðAi ;AlÞ

1� DjðAi;AlÞ1� C ðAi;AlÞ

; otherwise

8><>: ;

ð3:12Þwhere JðAi;AlÞ which represents the set of criteria which satis¯ed the discordance

index of the criteria is larger than the overall concordance index for the alternatives

Ai and Al .

The degree value of alternative Ai is better than all of the other alternatives and

can be computed as:

�þe ðAiÞ ¼

XAl 2A

SðAi;AlÞ: ð3:13Þ

The degree value of all of the other alternatives is better than alternative Ai and

can be computed as:

��e ðAiÞ ¼

XAl 2A

SðAl ;AiÞ: ð3:14Þ

Calculate the net°ow of alternative Ai as:

�eðAiÞ ¼ �þe ðAiÞ � ��

e ðAiÞ: ð3:15ÞNormalize the net°ow of alternatives as:

OTIeðAiÞ ¼�eðAiÞ

m þ 1

2: ð3:16Þ

According to the OTIe, we can determine the ranking order of all alternatives to the

linguistic ELECTRE method.

Applying linguistic intersection concept to MCDA technology 299

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3.3. Linguistic PROMETHEE

The linguistic PROMETHEE was developed by Chen et al.58 Linguistic PRO-

METHEE has been applied to personnel selection,58 investment portfolio selection59

and logistic supplier selection.60 In the PROMETHEE method, the preference

functions and thresholds of all criteria should be determined initially. Then, calculate

the individual preference value of each pair of alternatives with respect to each

criterion. In this paper, two kinds of preference function are used to calculate the

preference value.

(1) Level criterion with a linear preference function can be shown as:

Hð~xrj ; ~xsjÞ ¼1; ��1ð~xrjÞ ���1ð~xsjÞ > p

1

2; q � ��1ð~xrjÞ ���1ð~xsjÞ � p

0; q � ��1ð~xrjÞ ���1ð~xsjÞ

8>>><>>>:

: ð3:17Þ

(2) Criterion with a linear preference and indi®erence function can be shown as:

Hð~xrj ; ~xsjÞ ¼

1; p < ��1ð~xrjÞ ���1ð~xsjÞ��1ð~xrjÞ � q

p� q; q � ��1ð~xrjÞ ���1ð~xsjÞ � p

0; ��1ð~xrjÞ ���1ð~xsjÞ < q

8>>><>>>:

: ð3:18Þ

For each pair of alternatives, Ar and As, the overall preference value of

alternative Ar is better than As and can be computed as:

�ðAr ;AsÞ ¼Xnj¼1

��1ð ~wjÞ

,Xnk¼1

��1ð ~wkÞ!

� Hjð~x rj ; ~x sjÞ: ð3:19Þ

The preference degree of alternative Ar is better than all of the other alternatives and

can be computed as:

�þp ðArÞ ¼

Xb2A

�ðAr ; bÞ: ð3:20Þ

The preference degree of all of the other alternatives is better than alternative Ar and

can be computed as:

��p ðArÞ ¼

Xb2A

�ðb;ArÞ: ð3:21Þ

Then, the net°ow of alternative Ar can be calculated as:

�pðArÞ ¼ �þp ðArÞ � ��

p ðArÞ: ð3:22ÞFinally, normalizing the net°ow of alternative Ar :

OTIpðArÞ ¼�pðAr Þ

m þ 1

2: ð3:23Þ

300 P.-F. Pai, C.-T. Chen & W.-Z. Hung

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According to the OTIp, we can determine the ranking order of all alternatives using

the linguistic PROMETHEE method.

3.4. Linguistic VIKOR

The linguistic VIKOR was proposed by Chen et al.61 Linguistic VIKOR has been

applied in personnel selection61 and emergency alternative selection.62 According to

the linguistic decision matrix, the linguistic positive-ideal solution ( ~F�j ) of each

general criterion can be calculated as:

~F�j ¼ � Max

i��1ð~xijÞ

� �; i ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; n: ð3:24Þ

The linguistic negative-ideal solution (F �j ) of eachgeneral criterion can be calculated as:

~F�j ¼ � Min

i��1ð~xijÞ

� �; i ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; n: ð3:25Þ

The group utility for the majority (Si) of each alternative can be calculated as:

Si ¼Xnj¼1

��1ð ~wjÞ ���1ð ~F �

j Þ ���1ð~x ijÞ��1ð ~F �

j Þ ���1ð ~F �j Þ

; 8 i: ð3:26Þ

The individual regret rating for the opponentRi of each alternative canbe calculatedas:

Ri ¼ Maxj

��1ð ~wjÞ ���1ð ~F �

j Þ ���1ð~xijÞ��1ð ~F �

j Þ ���1ð ~F �j Þ

!; 8 i; ð3:27Þ

whereRi represents themaximum regret by choosing the ith alternative as the solution

according to selecting the worst performance in general criteria.

The aggregated value (Qi) of each alternative can be calculated as:

Qi ¼ v � Si � S�

S� � S� þ ð1� vÞ � Ri � R�

R� � R� ; 8 i; ð3:28Þ

where S� ¼ MiniSi, S� ¼ MaxiSi , R

� ¼ MaxiRi and R� ¼ MiniRi .

Here, v represents the decision-making coe±cient; v is between 0 and 1. When v is

close to 1, it represents that decision maker choosing the alternative that mainly

considers maximizing group utility for the majority. On the other hand, it represents

a decision maker who chooses the alternative that mainly considers minimizing

individual regret for the opponent when v is close to 0.

In order to judge a condition, the best alternative must be good enough to outrank

the other alternatives. The two conditions are illustrated as follows.

Condition 1: Qða2Þ �Qða1Þ � DQ, where Qða1Þ is the aggregated value of

best alternative a1, Qða2Þ, is the aggregated value of the alternative a2 with the

second position in the ranking list. DQ ¼ 1=ðm � 1Þ, where m is the number of

alternatives.

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Condition 2: Alternative a1 should be the best ranked by comparing the values in S

or/and R.

However, if one of the conditions is not satis¯ed, a set of compromised solutions is

recommended. Under this situation, the ranking results are illustrated as follows:

(1) Alternatives a1 and a2 belong to the same class if only Condition 2 is not satis¯ed.

(2) Alternatives a1; a2; . . . ; a� belong to the same class if Condition 1 is not satis¯ed

and Qða�Þ �Qða1Þ < DQ, where a� is the alternative with � position in the

ranking list and Qða�þ1Þ �Qða1Þ � DQ.

3.5. Linguistic intersection method

In general, these decision problems are usually solved by multiple decision makers;

these are known as group decision-making problems. As such, e®ective managers will

make a decision by simultaneously considering the suggestions of multiple experts.

Therefore, we can consider the linguistic TOPSIS, linguistic ELECTRE, linguistic

PROMETHEE and linguistic VIKOR methods as four experts who provide the

ranking order information of each alternative. Then, LIM is deployed.

According to the ranking order of alternatives by each linguistic MCDMmethod,

we are able to choose the better alternatives. Suppose that the NB represents the

number of alternatives that a manager considers regarding the maximum volume of

alternatives where the alternative that better exists in each linguistic MCDM

approach. Let RankðCCiÞ, RankðOTIeðAiÞÞ, RankðOTIpðAiÞÞ and RankðQiÞ rep-

resent the alternative ranking order of each linguistic MCDM approach. There-

fore, we can determine the chosen set as �t ¼ fAi jRankðCCiÞ � NBg, �e ¼fAi jRankðOTIeðAiÞÞ � NBg, �p ¼ fAi jRankðOTIpðAiÞÞ � NBg and �v ¼ fAi jRankðQiÞ � NBg. The �t , �e, �p and �v are the chosen set of the linguistic

TOPSIS, ELECTRE, PROMETHEE and VIKOR methods. Then, the intersection

from four sets can be computed as �LIM ¼ �t \ �e \ �p \ �v. The �LIM represents

the common set which is agreeable by four experts (linguistic MCDM methods) and

can be de¯ned as a consensus alternative set.

According to the consensus alternative set, normalize the closeness coe±cient

value as:

PtðAiÞ ¼CCiP

Ai 2�LIMCCi

; Ai 2 �LIM

0; Ai 62 �LIM

8><>: : ð3:29Þ

According to the consensus alternative set, normalize the OTIe as:

PeðAiÞ ¼OTIeðAiÞPAi 2�LIM

OTIeðAiÞ; Ai 2 �LIM

0; Ai 62 �LIM

8><>: : ð3:30Þ

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According to the consensus alternative set, normalize the OTIp as:

PpðAiÞ ¼OTIpðAiÞP

Ai 2�LIMOTIpðAiÞ

; Ai 2 �LIM

0; Ai 62 �LIM

8><>: : ð3:31Þ

According to the consensus alternative set, normalize the Qi, as:

PvðAiÞ ¼ð1�QiÞP

Ai 2�LIMð1�QiÞ

; Ai 2 �LIM

0; Ai 62 �LIM

8><>: : ð3:32Þ

Let PLIMðAiÞ be the comprehensive performance of each alternative in the con-

sensus alternative set by the LIM. It can be integrated from four kinds of linguistic

MCDM methods as:

PLIMðAiÞ ¼ vt � PtðAiÞ þ ve � PeðAiÞ þ vp � PpðAiÞ þ vv � PvðAiÞ; ð3:33Þwhere vt þ ve þ vp þ vv ¼ 1. The vt , ve, vp and vv represent the importance of lin-

guistic TOPSIS, linguistic ELECTRE, linguistic PROMETHEE and linguistic

VIKOR from the viewpoint of the manager.

4. Numerical Example

Suppose a notable gift manufacturer wants to sell a popular/special gift. The man-

ufacturer wants to enter the Chinese market, so the gift manufacturer has to choose a

sales channel company for outsourcing retail business. There are seven criteria to be

considered: the market share rate of the sales channel company (C1), gross pro¯t

margin (C2), inventory turnover ratio (C3), current ratio (C4), brand image (C5),

retail place (C6) and the enterprise culture (C7). The market share rate, gross pro¯t

margin, inventory turnover ratio and price-earnings ratio are quantitative criteria.

Brand image, retail place and enterprise culture are qualitative criteria. The gift

manufacturer employs four enterprise consultants (Ek ; k ¼ 1; 2; 3; 4) to evaluate the

12 sales channel companies (Ai; i ¼ 1; 2; . . . ; 12).

The computation process of the LIM is as follows:

Step 1: Collect quantitative information of 12 sales channel companies, as in

Table 2.

Step 2: Transform quantitative information of 12 sales channel companies into ¯ve

scale linguistic variables.

Step 3: The four enterprise consultants choose the linguistic variable for °exibly

expressing their opinions. Enterprise consultants E1 and E2 choose ¯ve

scale linguistic variables to express their opinions. Enterprise consultants

E3 and E4 use seven scale linguistic variables to express their opinions. The

linguistic weight of each criterion by each consultant is shown in Table 3.

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The linguistic performance of each sales channel company, with respect to

each qualitative criterion by each consultant, is shown in Table 4.

Step 4: Transform the linguistic performance of sales channel companies with

respect to each criterion into ¯ve scale linguistic variables and aggregate the

linguistic performance.

Step 5: Transform the linguistic weight of each criterion into ¯ve scale linguistic

variables and aggregate the linguistic weight of each criterion as in Table 5.

Table 2. The quantitative information.

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

C1 0.7% 4% 3% 4.2% 1.6% 2.7% 1.2% 2.1% 0.5% 1.6% 4% 3%C2 9.8% 11.2% 8% 7.2% 12% 6.6% 6.4% 8% 7.5% 5.6% 10.2% 7%

C3 6.8 7 8 4.3 7.5 4.2 6.5 6 5.6 4.5 7 8

C4 1.2 2 2.2 1.4 2 1.8 1.3 1.5 1.8 2.1 1.5 1.4

Table 3. The linguistic weight of each criterion by each consultant.

C1 C2 C3 C4 C5 C6 C7

E1 ðs 74 ; 0Þ ðs 74 ; 0Þ ðs 76; 0Þ ðs 76 ; 0Þ ðs 75 ; 0Þ ðs 75 ; 0Þ ðs 75 ; 0ÞE2 ðs 75 ; 0Þ ðs 75 ; 0Þ ðs 74; 0Þ ðs 76 ; 0Þ ðs 74 ; 0Þ ðs 76 ; 0Þ ðs 75 ; 0ÞE3 ðs 53 ; 0Þ ðs 51 ; 0Þ ðs 52; 0Þ ðs 54 ; 0Þ ðs 53 ; 0Þ ðs 54 ; 0Þ ðs 51 ; 0ÞE4 ðs 50 ; 0Þ ðs 54 ; 0Þ ðs 53; 0Þ ðs 52 ; 0Þ ðs 50 ; 0Þ ðs 51 ; 0Þ ðs 54 ; 0Þ

Table 4. The linguistic ratings.

Criteria E1 E2 E3 E4 E1 E2 E3 E4

C5 A1 ðs 52; 0Þ ðs 53; 0Þ ðs 74 ; 0Þ ðs 72 ; 0Þ A7 ðs 50 ; 0Þ ðs 54; 0Þ ðs 76; 0Þ ðs 76 ; 0ÞA2 ðs 53; 0Þ ðs 54; 0Þ ðs 75 ; 0Þ ðs 70 ; 0Þ A8 ðs 52 ; 0Þ ðs 54; 0Þ ðs 70; 0Þ ðs 75 ; 0ÞA3 ðs 51; 0Þ ðs 53; 0Þ ðs 73 ; 0Þ ðs 75 ; 0Þ A9 ðs 54 ; 0Þ ðs 54; 0Þ ðs 76; 0Þ ðs 70 ; 0ÞA4 ðs 51; 0Þ ðs 53; 0Þ ðs 75 ; 0Þ ðs 76 ; 0Þ A10 ðs 52 ; 0Þ ðs 54; 0Þ ðs 73; 0Þ ðs 73 ; 0ÞA5 ðs 54; 0Þ ðs 53; 0Þ ðs 70 ; 0Þ ðs 76 ; 0Þ A11 ðs 53 ; 0Þ ðs 52; 0Þ ðs 75; 0Þ ðs 76 ; 0ÞA6 ðs 54; 0Þ ðs 53; 0Þ ðs 74 ; 0Þ ðs 75 ; 0Þ A12 ðs 53 ; 0Þ ðs 51; 0Þ ðs 72; 0Þ ðs 72 ; 0Þ

C6 A1 ðs 52; 0Þ ðs 53; 0Þ ðs 76 ; 0Þ ðs 72 ; 0Þ A7 ðs 54 ; 0Þ ðs 54; 0Þ ðs 74; 0Þ ðs 73 ; 0ÞA2 ðs 51; 0Þ ðs 54; 0Þ ðs 70 ; 0Þ ðs 72 ; 0Þ A8 ðs 50 ; 0Þ ðs 54; 0Þ ðs 74; 0Þ ðs 73 ; 0ÞA3 ðs 50; 0Þ ðs 54; 0Þ ðs 74 ; 0Þ ðs 74 ; 0Þ A9 ðs 52 ; 0Þ ðs 52; 0Þ ðs 73; 0Þ ðs 73 ; 0ÞA4 ðs 53; 0Þ ðs 52; 0Þ ðs 76 ; 0Þ ðs 72 ; 0Þ A10 ðs 51 ; 0Þ ðs 50; 0Þ ðs 74; 0Þ ðs 73 ; 0ÞA5 ðs 50; 0Þ ðs 53; 0Þ ðs 76 ; 0Þ ðs 74 ; 0Þ A11 ðs 50 ; 0Þ ðs 53; 0Þ ðs 73; 0Þ ðs 74 ; 0ÞA6 ðs 54; 0Þ ðs 52; 0Þ ðs 75 ; 0Þ ðs 75 ; 0Þ A12 ðs 50 ; 0Þ ðs 52; 0Þ ðs 75; 0Þ ðs 73 ; 0Þ

C7 A1 ðs 53; 0Þ ðs 50; 0Þ ðs 76 ; 0Þ ðs 76 ; 0Þ A7 ðs 50 ; 0Þ ðs 54; 0Þ ðs 73; 0Þ ðs 73 ; 0ÞA2 ðs 52; 0Þ ðs 54; 0Þ ðs 70 ; 0Þ ðs 70 ; 0Þ A8 ðs 54 ; 0Þ ðs 51; 0Þ ðs 74; 0Þ ðs 72 ; 0ÞA3 ðs 52; 0Þ ðs 53; 0Þ ðs 73 ; 0Þ ðs 73 ; 0Þ A9 ðs 51 ; 0Þ ðs 50; 0Þ ðs 74; 0Þ ðs 73 ; 0ÞA4 ðs 50; 0Þ ðs 50; 0Þ ðs 75 ; 0Þ ðs 75 ; 0Þ A10 ðs 53 ; 0Þ ðs 50; 0Þ ðs 73; 0Þ ðs 76 ; 0ÞA5 ðs 52; 0Þ ðs 52; 0Þ ðs 76 ; 0Þ ðs 75 ; 0Þ A11 ðs 53 ; 0Þ ðs 51; 0Þ ðs 73; 0Þ ðs 74 ; 0ÞA6 ðs 52; 0Þ ðs 52; 0Þ ðs 73 ; 0Þ ðs 75 ; 0Þ A12 ðs 54 ; 0Þ ðs 51; 0Þ ðs 75; 0Þ ðs 76 ; 0Þ

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Step 6: Construct the linguistic decision matrix as in Table 6.

Step 7: Calculate linguistic weighted decision-making matrix.

Step 8: Calculate the positive ideal solution and the negative ideal solution.

Step 9: Calculate the distance to the positive ideal solution and negative ideal

solution and the closeness coe±cient of each sales channel company as in

Table 7.

Step 10: Determine the preference threshold pj , indi®erence threshold qj and vote

threshold vj of each criterion in the ELECTRE method as in Table 8.

Step 11: Calculate concordance index, discordance index and the credibility matrix

in ELECTRE method.

Step 12: Calculate in°ow, out°ow, net°ow and OTIeðAiÞ in the ELECTRE method

as in Table 9.

Table 5. Aggregated linguistic weight of each criterion.

C1 C2 C3 C4 C5 C6 C7

Aggregatedweight

ðs 52 ; 0:063Þ ðs 53 ;�0:063Þ ðs 53 ;�0:021Þ ðs 54;�0:125Þ ðs 52 ; 0:0625Þ ðs 53 ; 0:021Þ ðs 53 ;�0:021Þ

Table 7. The PIS, NIS and closeness coe±cient.

Alternative A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

Distance

to PIS0.7280 1.2935 1.3686 0.9020 1.3252 1.0195 0.9027 0.8067 0.8589 0.9510 0.7197 1.0061

Distance

to NIS0.7280 1.2006 1.2473 0.6746 1.2044 0.7559 0.6113 0.5925 0.6612 0.8202 0.9695 0.8752

Closeness

coe±cient0.5000 0.4814 0.4768 0.4279 0.4761 0.4258 0.4038 0.4235 0.4350 0.4631 0.5739 0.4652

Table 6. The linguistic decision matrix.

C1 C2 C3 C4 C5 C6 C7

A1 ðs 52 ; 0:032Þ ðs 52 ;�0:006Þ ðs 52; 0:037Þ ðs 51 ; 0:012Þ ðs 52 ;�0:067Þ ðs 51 ; 0:119Þ ðs 51 ;�0:025ÞA2 ðs 50 ; 0:030Þ ðs 52 ;�0:049Þ ðs 52;�0:002Þ ðs 50 ; 0Þ ðs 51 ; 0:066Þ ðs 52 ;�0:003Þ ðs 51 ; 0:036ÞA3 ðs 52 ;�0:120Þ ðs 51 ;�0:100Þ ðs 53;�0:021Þ ðs 51 ;�0:075Þ ðs 51 ;�0:016Þ ðs 51 ; 0:103Þ ðs 51 ; 0:071ÞA4 ðs 51 ; 0:084Þ ðs 50 ; 0:107Þ ðs 50 ; 0Þ ðs 52 ; 0:025Þ ðs 52 ;�0:043Þ ðs 52 ; 0:110Þ ðs 51 ;�0:007ÞA5 ðs 51 ;�0:083Þ ðs 53 ;�0:063Þ ðs 53;�0:117Þ ðs 53 ;�0:050Þ ðs 51 ;�0:004Þ ðs 52 ;�0:035Þ ðs 51 ; 0:045ÞA6 ðs 51 ;�0:007Þ ðs 51 ; 0:008Þ ðs 51; 0:095Þ ðs 51 ; 0:013Þ ðs 51 ; 0:078Þ ðs 52 ;�0:083Þ ðs 51 ;�0:016ÞA7 ðs 52 ; 0:062Þ ðs 51 ;�0:079Þ ðs 50; 0:019Þ ðs 51 ;�0:075Þ ðs 52 ;�0:102Þ ðs 52 ;�0:003Þ ðs 51 ;�0:077ÞA8 ðs 50 ; 0:106Þ ðs 50 ; 0:086Þ ðs 52;�0:059Þ ðs 50 ; 0:088Þ ðs 52 ;�0:079Þ ðs 52 ; 0:110Þ ðs 51 ;�0:042ÞA9 ðs 52 ; 0:032Þ ðs 52 ; 0:100Þ ðs 52; 0:037Þ ðs 53 ;�0:050Þ ðs 51 ; 0:113Þ ðs 51 ; 0:055Þ ðs 51 ;�0:094ÞA10 ðs 51 ;�0:083Þ ðs 50; 0Þ ðs 50; 0:057Þ ðs 53 ; 0:037Þ ðs 51 ; 0101Þ ðs 51 ; 0:023Þ ðs 51 ;�0:016ÞA11 ðs 52 ;�0:120Þ ðs 51 ; 0:007Þ ðs 53;�0:021Þ ðs 54 ;�0:125Þ ðs 51 ; 0:078Þ ðs 52 ;�0:051Þ ðs 51 ;�0:016ÞA12 ðs 50 ; 0Þ ðs 51 ;�0:096Þ ðs 51; 0:018Þ ðs 52 ; 0:025Þ ðs 52 ;�0:079Þ ðs 52 ;�0:115Þ ðs 51 ;�0:103Þ

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Step 13: Determine the preference function and threshold of PROMETHEE as in

Table 10.

Step 14: Calculate the preference function of each criterion and then calculate the

overall preference value.

Step 15: Calculate the in°ow, out°ow, net°ow and OTIpðAiÞ of PROMETHEE as in

Table 11.

Step 16: Determine the linguistic positive-ideal solution and the linguistic negative-

ideal solution of each general criterion.

Step 17: Compute the group utility for the majority Si, the individual regret rating

for the opponent Ri and Qi; set decision-making coe±cient v ¼ 0:5 as in

Table 12.

Table 8. The preference threshold, indiference

threshold and vote threshold of each criterion.

C1 C2 C3 C4 C5 C6 C7

p 3 4% 3 1 1/6 1/6 1/6

q 0.5 1% 0.2 0.1 1/12 1/12 1/12

v 5 8% 5 2 1/2 1/2 1/2

Table 9. In°ow, out°ow, net°ow and OTIeðAiÞ.

Alternative A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

In°ow 9.256 9.575 10.470 8.869 10.104 9.801 9.550 9.306 8.506 7.029 10.322 8.662

Out°ow 9.249 8.691 8.584 9.886 7.800 8.116 8.9042 10.684 10.498 10.586 8.784 9.669

OTIeðAiÞ 0.500 0.536 0.578 0.457 0.596 0.570 0.526 0.442 0.417 0.351 0.564 0.458

Table 10. The preference functions of the criteria.

Criterion Preference function Threshold

C1 Criterion with linear preference and indi®erence area P ¼ 3, q ¼ 0:5

C2 Criterion with linear preference and indi®erence area P ¼ 4%, q ¼ 1%

C3 Criterion with linear preference and indi®erence area P ¼ 3, q ¼ 0:2C4 Criterion with linear preference and indi®erence area P ¼ 1, q ¼ 0:1

C5 Level criterion with linear preference P ¼ 1=6, q ¼ 1=12

C6 Criterion with linear preference and indi®erence area P ¼ 1=6, q ¼ 1=12

C7 Level criterion with linear preference P ¼ 1=6, q ¼ 1=12

Table 11. The in°ow, out°ow, net°ow and OTIpðAiÞ.

Alternative A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

In°ow 7.9949 8.4529 9.0201 7.4471 9.2407 8.5346 7.9715 7.8562 7.4947 6.1783 8.9338 8.2356

Out°ow 8.0943 7.3521 7.1041 8.3515 7.1084 7.3685 7.7742 9.2033 9.4178 9.0752 7.4032 9.1079

OTIpðAiÞ 0.4959 0.5459 0.5798 0.4623 0.5888 0.5486 0.5082 0.4439 0.4199 0.3793 0.5638 0.4637

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Step 18: The manager determines the NB in each linguistic MCDM approach. For

fairness, we set the NB in each MCDM the same and equal to six (50% of

the number of alternatives). Then, the selected alternative set by the

TOPSIS is �t ¼ fA1;A2;A3;A5;A11;A12g, the selected alternative set by

the ELECTRE is �e ¼ fA2;A3;A5;A6;A7;A11g, the selected alternative

set by the PROMETHEE method is �p ¼ fA2;A3;A5;A6;A7;A11g and the

selected alternative set by the VIKOR is �v ¼ fA2;A3;A5;A6;A8;A11g.Finally, we calculate the intersection set as �LIM ¼ �t \ �e \ �p \ �v ¼fA2;A3;A5;A11g.

Step 19: If the manager considers the importance of each linguistic MCDMmethod is

equal, then vt ¼ 14, ve ¼ 1

4, vp ¼ 14 and vv ¼ 1

4. And then, calculate the rela-

tive evaluation value of TOPSIS (Pt), ELECTRE (Pe), PROMETHEE

(Pp), VIKOR (Pv) and the overall performance of each alternative PLIM (see

Table 13). Finally, the ranking order of alternatives is A3 > A5 > A11 > A2.

5. The Comparison Result

In order to justify the e®ectiveness of the proposed method, this paper compares the

ranking result of the LIM with linguistic TOPSIS, linguistic ELECTRE, linguistic

PROMETHEE and linguistic VIKOR, respectively. The ranking order of alterna-

tives according to ¯ve methods is shown as Table 14. It is found that the intersection

alternatives are A2, A3, A5 and A11. The ranking order of these four alternatives

leaves them at the top ¯ve positions of any linguistic MCDM methods. Therefore, it

is robust enough for the ranking result of alternatives to use the LIM method. On the

other hand, the alternatives which are not in the intersection set do not have a

consistent ranking order amongst linguistic MCDM methods. If we consider the

Table 12. The value of Si , Ri and Qi .

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

Si 2.5692 1.8133 1.5132 2.6847 1.5920 2.0748 2.4926 2.7197 2.8758 3.0875 1.8671 2.5693

Ri 0.8750 0.6974 0.4297 0.7100 0.5329 0.7292 0.7875 0.6125 0.5625 0.7708 0.6125 0.7000

Qi 0.8354 0.3959 0.0000 0.6868 0.1409 0.5146 0.7128 0.5884 0.5819 0.8830 0.3177 0.6389

Table 13. The values of Pt , Pe, Pp, Pv and

PLIM.

A2 A3 A5 A11

Pt 0.2459 0.2435 0.2175 0.2931

Pe 0.2359 0.2543 0.2619 0.2479Pp 0.2396 0.2545 0.2585 0.2475

Pv 0.1920 0.3179 0.2731 0.2169PLIM 0.2284 0.2676 0.2527 0.2514

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sum of rank (SOR) of each alternative in each linguistic MCDM method, the ¯nal

ranking order of alternatives is the same as with the LIM method (see Table 15).

In order to verify the ranking order of alternatives, the proposed numerical ex-

ample is simulated by random data. In the simulation processes, the threshold value

of each criterion with respect to each MCDM approach is not changed. The random

range of data of each criterion is shown in Table 16. The random data are generated

by following a uniform distribution. We use random variables to generate new data

regarding the performance of each alternative with respect to each criterion. The

simulation runs 1,000,000 times. First, we compute the average ranking result of each

MCDM method when the alternative is the best according to other MCDM

approaches. The computation results are shown in Table 17. According to Table 17,

the average rankings with regard to LIM for TOPSIS, ELECTRE, PROMETEE,

VIKOR and SOR are 4.4231, 2.4820, 2.4897, 2.3171 and 1.6090, respectively. If we

use the SOR method to select the best alternative, the average ranking order with

LIM method is better than other linguistic MCDM methods. Therefore, the ranking

Table 14. The ranking results of ¯ve methods.

Rank TOPSIS ELECTRE PROMETHEE VIKOR LIM

1 A11 A5 A5 A3 A3

2 A1 A3 A3 A5 A5

3 A2 A6 A11 A2 A11

4 A3 A11 A6 A11 A2

5 A5 A2 A2 A6

6 A12 A7 A7 A8

7 A10 A1 A1 A9

8 A9 A12 A12 A12

9 A4 A4 A4 A7

10 A6 A8 A8 A4

11 A8 A10 A9 A1

12 A7 A9 A10 A10

Table 15. The ranking order information.

Alternative Rank of

TOPSIS

Rank of

ELECTRE

Rank of

PROMETHEE

Rank of

VIKOR

Sum of

rank (SOR)

Overall

rank

A3 4 2 2 1 9 1

A5 5 1 1 2 9 1

A11 1 4 3 4 12 3A2 3 5 5 3 16 4

A6 10 3 4 5 22 5

A1 2 7 7 11 27 6A12 6 8 8 8 30 7

A7 12 6 6 9 33 8

A8 11 10 10 6 37 9

A4 9 9 9 10 37 9A9 8 12 11 7 38 11

A10 7 11 10 12 40 12

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order of alternatives from using the LIM method is more stable than other linguistic

MCDM methods (shown in Table 17).

In order to demonstrate that the LIM is a stable method, we show the average

ranking of each MCDM approach when the ¯rst six alternatives are determined by

LIM. According to the simulation results, we can observe the trends such that when

the alternative rank in LIM is lower, the average ranking in most of the MCDM

approaches is lower (see Fig. 4 and Table 18). Therefore, the LIM method is a

relatively robust decision-making method. Suppose that the random data are

Table 16. The random range of data for each criterion.

Criterion Description Data type Random range

C1 Market share rate Quantitative data 1–10%C2 Gross pro¯t margin Quantitative data 1–20%

C3 Inventory turnover ratio Quantitative data 5–15

C4 Current ratio Quantitative data 1–3

C5 Brand image Qualitative data Expert E1 and E2

ðs 50 ; 0Þ � ðs 54 ; 0ÞExpert E3 and E4

ðs 70 ; 0Þ � ðs 76 ; 0ÞC6 Retail place Qualitative data

C7 Enterprise culture Qualitative data

Table 17. The average ranking of each MCDM approach when the alternative is the best and the random

data follows uniform distribution.

Average ranking order in other methods for 1,000,000 simulations

TOPSIS ELECTRE PROMETHEE VIKOR SOR Averageranking

Best

alternative

TOPSIS ��� 8.4114 8.7016 8.3833 7.0463 8.1357

ELECTRE 6.9207 ��� 2.5803 2.0418 1.8536 3.3491PROMETHEE 6.9796 2.6716 ��� 1.9418 1.8745 3.3669

VIKOR 6.8982 2.0929 1.9416 ��� 1.6284 3.1403

SOR 5.3304 1.8365 1.8000 1.6310 ��� 2.6495

LIM 4.4231 2.4820 2.4897 2.3171 1.6090 2.6642

Fig. 4. The variation of average ranking of MCDM approaches when the alternative order is determined

by LIM and the random data follows uniform distribution.

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generated from a normal distribution. The computation results are shown in

Table 19. The simulation result is similar to Table 17. The average ranking order of

the best alternative in the LIM method is better than TOPSIS, ELECTRE, PRO-

METHEE and VIKOR. In addition, the trends regarding the average ranking in

most of the MCDM approaches is lower when the alternative rank in LIM is lower

(see Fig. 5 and Table 20).

In this research, the results of the simulation whose random data was generated

based on uniform distribution is similar to those whose random data was generated

Table 18. The average ranking of each MCDM approach when the ranking

order of alternative is determined by LIM and the random data followsuniform distribution.

LIM 1 2 3 4 5 6

TOPSIS 4.4231 3.9419 3.6657 3.4821 3.3923 3.6087

ELECTRE 2.4820 3.5416 4.1938 4.6605 5.0169 5.4457

PROMETHEE 2.4897 3.6135 4.2572 4.6792 5.0127 5.0435VIKOR 2.3171 3.4343 4.1241 4.6088 5.0072 5.3043

SOR 1.6090 2.6859 3.5393 4.2570 4.9094 5.5435

Table 19. The average ranking of each MCDM approach when the alternative is the best and the random

data follows normal distribution.

Average ranking in 1,000,000 simulations

TOPSIS ELECTRE PROMETHEE VIKOR SOR Average

ranking

Bestalternative

TOPSIS ��� 8.8054 9.2024 8.9689 7.6129 8.6474ELECTRE 7.7839 ��� 2.4192 2.0607 1.9557 3.5549

PROMETHEE 7.6424 2.2542 ��� 2.0636 1.8504 3.4527

VIKOR 7.5842 1.8105 1.9971 ��� 1.6248 3.2541

SOR 5.9248 1.7838 1.7595 1.6361 ��� 2.7760LIM 4.5323 2.7468 2.6984 2.5782 1.7805 2.8672

Fig. 5. The variation of average ranking of MCDM approaches when the alternative order is determined

by LIM and the random data follows normal distribution.

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based on normal distribution. This justi¯es the notion that the LIMmethod is a stable

method with regard to di®erent data sources. In this investigation, the average ranking

order of the best alternatives using LIM is lower than the best alternative using

TOPSIS, ELECTRE, PROMETHEE andVIKOR. From the simulation, we found the

trend was such that the average ranking order in most of the MCDM approaches is

lower when the alternative rank using LIM is lower. This means that the alternatives

are the best using the LIMmethod. Therefore, the LIMmethod is an e®ective decision-

making tool. Although the trend of ranking results of the TOPSIS method is di®erent

from other MCDMmethods, it shows that the ranking order of the TOPSIS method is

relatively unstable in comparison to ELECTRE, PROMETHEE, VIKOR and LIM. A

possible reason may be that the ranking order of alternatives is determined based on

the relative position of the PIS and the NIS.When the distance of PIS andNIS is small,

the alternative is not easily distinguished by the TOPSIS method.

6. Conclusion and Future Research

In this paper, a LIM is presented to handle MCDM problems in a fuzzy environment.

The advantage of the LIM is illustrated as follows:

(1) The picked alternative is executed by the LIM in accordance with the agreement

of other linguistic MCDM approaches. It can promote e®ective decision making.

(2) Although the LIM is somewhat complex in execution, the process is easy to

calculate and can be executed by computer program.

(3) In this paper, the linguistic TOPSIS, linguistic ELECTRE, linguistic PRO-

METHEE and linguistic VIKOR are used to determine the performance of each

alternative. In reality, the LIM is an extendable method which can be extended

by adding other MCDM approaches to enhance the quality of the decision

making.

(4) The LIM can e®ectively deal with MCDM problems based on quantitative and

qualitative information simultaneously. In addition, the LIM is a more stable

decision-making method, as can be seen by the simulation results. In the future,

the LIM will be compared with di®erent kinds of fuzzy MCDM approaches and

an interactive program will be designed based on an algorithm of the LIM to

enhance the power of decision making for managers.

Table 20. The average ranking of each MCDM approach when the ranking

order of alternative is determined by LIM and the random data followsnormal distribution.

LIM 1 2 3 4 5 6

TOPSIS 4.5323 4.0414 3.7451 3.5077 3.2790 3.0000

ELECTRE 2.7468 3.8109 4.4232 4.8420 5.1625 5.9231

PROMETHEE 2.6984 3.8638 4.5006 4.8886 5.1675 5.4615VIKOR 2.5782 3.6893 4.3319 4.7647 5.0915 5.4231

SOR 1.7805 2.9234 3.7570 4.4141 5.0058 5.7692

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Acknowledgments

The authors thank four anonymous referees for suggestions and comments on this

study. This work is partially supported by the National Science Council of Taiwan

under grants No. NSC 101-2410-H-239-004-MY2 and NSC 101-2410-H-260-005-MY2.

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