Analysis on Criteria based Emotive Music Composition ... · Analysis on Criteria based Emotive Music Composition Selection using a New Trapezoidal Fuzzy DEMATEL - TOPSIS Hybrid ...

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122

Analysis on Criteria based Emotive Music Composition Selection

using a New Trapezoidal Fuzzy DEMATEL - TOPSIS Hybrid

Technique

S. Aseervatham1*, A. Victor Devadoss1

(1) Department of Mathematics, Loyola College, Chennai, India.

Copyright 2015 © S. Aseervatham and A. Victor Devadoss. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided

the original work is properly cited.

Abstract

The decision data of human judgments with preferences are often vague in many real life cases. Human

judgments with preferences are often unclear and hard to estimate by exact numerical values. So that the

traditional ways of using crisp values are inadequate. The relationship among criteria and choosing and

rating alternatives based on criterion are often expressed in terms of linguistic terms by the experts. These

causal relationships among criterion have been investigated by the Decision making trial and evaluation

laboratory (DEMATEL) with the use of trapezoidal fuzzy numbers. Furthermore, fuzzy TOPSIS method is

used to express the rankings of alternatives based on criterion. In this paper the case study on choosing

emotional music composition is discussed based on musical features by the proposed hybrid technique of

Fuzzy TOPSIS and DEMATEL using trapezoidal fuzzy number.

Keywords: Linguistic Variable, Trapezoidal fuzzy number, DEMATEL, TOPSIS.

1 Introduction

Fuzzy set theory is useful when the situation is full of uncertainty and imprecision due to the human

judgments making the decision very complex and unstructured. Human judgments with preferences are often

unclear and hard to estimate by exact numerical values has created the need for fuzzy logic. Further, Use of

linguistic assessments instead of numerical values is more sensible approach, in which all assessments of

criteria in the problem are evaluated by means of linguistic variables. The Decision making trial and

evaluation laboratory (DEMATEL) method is a powerful method for capturing the causal relationship

between criteria. This method is originated from Geneva research center of the Battelle Memorial Institute.

In recent years, the DEMATEL has become very popular because it can visualize the structure of

complicated causal relationships. Then alternatives ranking based on criterion should be determined which

can assist the decision making. TOPSIS, ELECTRE and VIKOR such techniques are applied for the ranking

Journal of Fuzzy Set Valued Analysis 2015 No. 2 (2015) 122-133

Available online at www.ispacs.com/jfsva

Volume 2015, Issue 2, Year 2015 Article ID jfsva-00239, 12 Pages

doi:10.5899/2015/jfsva-00239

Research Article

Journal of Fuzzy Set Valued Analysis 2015 No. 2 (2015) 122-133 123

http://www.ispacs.com/journals/jfsva/2015/jfsva-00239/

International Scientific Publications and Consulting Services

process. Here the fuzzy TOPSIS (technique for order preference by similarity to an ideal solution) is

developed with fuzzy DEMATEL to rank all competing alternatives in terms of their overall performances.

This method was developed by Hwang & Yoon (1981). Two artificial alternatives are defined as positive-

ideal and negative-ideal solution. Maximization of the benefit of criteria is evaluated by the positive ideal

solution whereas negative ideal solution does vice-versa. TOPSIS selects the alternative which is the closest

to the positive ideal solution and farthest from negative ideal solution.

2 Preliminaries and notations

Definition 2.1. Linguistic Variable (LV):

A linguistic variable is a variable which represents word or sentence in a natural language but not a

number.

Definition 2.2. Trapezoidal Fuzzy Number (TzFN):

It is represented with four points as follows: (1) (2) (3) (4), , ,Z z z z z . Its membership function and

graphical representation defined as follows,

Figure 1: Membership and graphical diagram of TzFN

Definition 2.3. Basic Operations in TzFNs:

It is represent Let (1) (2) (3) (4)

1 1 1 1 1, , ,Z z z z z and (1) (2) (3) (4)

2 2 2 2 2, , ,Z z z z z be two hexagonal fuzzy numbers.

Then the addition and subtraction operations are defined by,

[i] (1) (1) (2) (2) (3) (3) (4) (4)

1 2 1 2 1 2 1 2 1 2, , ,Z Z z z z z z z z z

[ii] (1) (1) (2) (2) (3) (3) (4) (4)

1 2 1 2 1 2 1 2 1 2, , ,Z Z z z z z z z z z

Definition 2.3. Linguistic variables and its corresponding TzFNs:

Here some examples of using Trapezoidal fuzzy numbers for their corresponding linguistic variables are

given in the following tables.

Table 1: Linguistic variables for factors relationship

Scale Linguistic Variable for factor relationship Trapezoidal Fuzzy Number

1 Very Low (VL) (0, 0, 0, 0.25)

2 Low (L) (0, 0.05, 0.15, 0.25)

3 Medium Low (ML) (0.15, 0.25, 0.35, 0.45)

4 Medium (M) (0.35, 0.45, 0.55, 0.65)

5 Medium High (MH) (0.55, 0.65, 0.75, 0.85)

6 High (H) (0.75, 0.85, 0.95, 1)

7 Very High (VH) (0.95, 1, 1, 1)




Table 2: Linguistic variables for alternatives ratings based on factors

Scale Linguistic Variable Trapezoidal Fuzzy Number

1 Very Poor (VP) (0, 0, 0, 2.5)

2 Poor (P) (0, 0.5, 1.5, 2.5)

3 Medium Poor (MP) (1.5, 2.5, 3.5, 4.5)

4 Medium (M) (3.5, 4.5, 5.5, 6.5)

5 Medium Good (MG) (5.5, 6.5, 7.5, 8.5)

6 Good (G) (7.5, 8.5, 9.5, 10)

7 Very Good (VG) (9.5, 10, 10, 10)

3 The Proposed Methodology

In this section, the fuzzy DEMATEL and fuzzy TOPSIS methods are combined to analyze the correlations

among factors and rating the alternatives for the corresponding criterion in an uncertain linguistic

environment. The working procedure of fuzzy DEMATEL for giving causal relationship between one factor

to another and then finding the ranking of alternatives for the factors are briefly explained as follows.

Method 3.1. Fuzzy DEMATEL

The correlation among factors in an uncertain linguistic environment is determined by using fuzzy

DEMATEL method. The set of attributes f1, f2, f3,…, fn are taken as the evaluation criterion. The correlation

among these criterion factors can be characterized by the link between one another. Particularly the link with

the direction represents the influential relationship of any factor fi on fj where the relationships between the

factors are expressed in appropriate linguistic terms by the group of expert’s opinion. These linguistic terms

are often converted by its corresponding fuzzy numbers. Here Trapezoidal fuzzy numbers are utilized to

convert the linguistic variables. Then the aggregation of fuzzy numbers is derived in following steps to create

a dynamical system.

Step 1: Collect the attributes from survey, theoretical studies, etc., which are related to the problem and sort

them as n-factors. Let F = {f1, f2, …, fn} be a finite set of factors and E = {E1, E2, …, En} be the finite

set of experts, where Ek denotes the kth expert. It is assumed that the experts have the identical

importance and their judgments on the intensities among factors are expressed in linguistic variables.

Step 2: Form the initial uncertain direct-relation matrix using linguistic variable terms responded by the kth-

expert as [ ]k k

ijU u where k =1,2,…, K. If there does not exist a correlation between fi and fj, then

denote ' 'k

iju . Particularly, there does not exist a correlation between fi itself. Now, the correlation

among the factors by Ekth expert’s opinion is,

f1 f2 . . . fn

12 11

2 21 2

1 2

. . .

. . .

. . . . . .

. . . . . .

. . . . . .

. . .

k k

n

k k

n

k

k kn

n n

f u u

f u u

U

fu u

and k =1,2,…, K




Step 3: Consider the trapezoidal fuzzy number TzFN zk = (z1(k),z2(k),z3(k),z4(k)) [where z1(k) ≤ z2(k) ≤ z3(k) ≤ z4(k)]

for the corresponding linguistic term responded by the decision maker Ek, where k = 1,2, …, K. Now

transform the matrix [ ]k k

ijU u into [ ]k k

ijU z ; where (1) (2) (3) (4), , ,k k k k k

ij ij ij ij ijz z z z z and k =1,2,…, K

and ' ' (0,0,0,0)k k

ij iju z .

Step 4: The group uncertain direct-relation fuzzy matrix is denoted by G

and defined as follows: 1 2

...K

ijG U U U g

. This is done by aggregating the individual uncertain direct-relation

matrices. (ie.,) If (1) (2) (3) (4), , ,ij ij ij ijijg g g g g , then (1) (2) (3), ,ij ij ijg g g and

(4)

ijg are calculated by

(1) (1)

1

1 Kk

ij ij

k

g zK

, (2) (2)

1

1 Kk

ij ij

k

g zK

, (3) (3)

1

1 Kk

ij ij

k

g zK

and(4) (4)

1

1 Kk

ij ij

k

g zK

where i,j =

1,2,3,…,n .

Step 5: Then the normalized uncertain group direct-relation matrix ijX x is determined as follows: If

(1) (2) (3) (4), , ,ij ij ij ij ijx x x x x , then (1) (2) (3), ,ij ij ijx x x and

(4)

ijx are calculated by

( )

( )

l

ijl

ij

gx

M where

(4)

11

max 0n

iji n

j

M g

and l = 1,2,3,4.

Step 6: To compute the total-relation uncertain matrix, we should have to establish and analyze this model

by ensuring the convergence of 1 2

lim ...l

lT X X X

, where these crisp value

matrices l

X ’s are taken from the decomposition of the normalized matrix X . This is done by

separating each trapezoidal entry from the matrix X . Then make them as four crisp value matrices.

(i.e.,)

( ) ( )

12 1

( ) ( )

21 2

( ) ( )

1 2

0 . . .

0 . . .

. . 0 . . .

. . . 0 . .

. . . . 0 .

. . . 0

l l

n

l l

n

l

l l

n n

x x

x x

X

x x

l = 1,2,3,4.

Step 7: Then construct the total-relation uncertain matrix asijT t , where (1) (2) (3) (4), , ,ij ij ij ij ijt t t t t and

1

( ) l ll

ijt X I X

l = 1,2,3,4 and i,j = 1,2,3,…,n .




Step 8: All direct and indirect influence of factor fi on all other factors is denoted by ir and defined as

(1) (2) (3) (4) (1) (2) (3) (4)

1 1 1 1

, , , , , ,ij ij ij ij

n n n n

i i i i i

j j j j

r r r r r t t t t

and ir is called as the degree of

influential impact. And, both direct and indirect impacts on fj is influenced by all other factors is

denoted by jc and defined as (1) (2) (3) (4) (1) (2) (3) (4)

1 1 1 1

, , , , , ,ij ij ij ij

n n n n

j j j j j

i i i i

c c c c c t t t t

and jc

is called as the degree of influenced impact.

Step 9: Aggregate the weight of factors from ir and jc values using following formula. This will be

considered as initial weighting of factors in TOPSIS method to obtain fuzzy rating of alternatives.

The initial weight of factor is denoted by ( )w i and defined as,

( ) ( )

1

( ) ( )

1 1

( ) ( ) ( ) 1,2,..., 4

nl l

i j

jl l

n nl l

i j

i j

r c

w i w i w i l

r c

i = 1,2,3,…,n .

Method 3.2. Fuzzy TOPSIS

Here the TOPSIS method is developed to construct the casual relationship between factors and factor based

rating of alternatives. The algorithm is given as follows.

The survey, theoretical studies and expert opinions have been taken in to an account for deciding the problem

criterion where they were used in DEMATEL. Their corresponding alternative ratings are rated by the group

of experts in terms of appropriate linguistic variables in an uncertain environment.

Step 1: Consider the set of attributes { f1, f2, f3,…, fn } as the factors and { a1, a2, a3,…, am } as the alternatives

based on fi ; i = 1,2,…,n. This is converted into the dynamical system N for the expert-k as,

k k

ijn m

N x

, where k = 1,2,…,K.

Step 2: Convert the appropriate linguistic ratings of the factors based alternatives into corresponding

trapezoidal fuzzy numbers for the kth - expert as, kkijN x , Where

(1) (2) (3) (4), , ,k k k k k k

ij ij ij ij ij ijx za za za za za is TzFN of appropriate linguistic rating of alternative.

Step 3: Obtain the aggregated fuzzy rating ijx of alternative aj under criteria fi evaluated by experts using

trapezoidal fuzzy numbers of each matrices and take the aggregated fuzzy weight ( )w i of factor fi

from DEMATEL model procedure for TOPSIS calculation. (ie.) 1 21

...K

ij ij ij ijx x x xK

and

( )w i .

Step 4: Construct the fuzzy decision matrix D with the entries ijx as, ijD x and the weight of the

criteria is taken as, 1 2 3 4( ) ( ) , ( ) , ( ) , ( ) 1,2,...,w i w i w i w i w i i n .




Step 5: Transform this D into normalized fuzzy decision matrix which is denoted by R and defined as,

ijn m

R r

, where (1) (2) (3) (4)

* * * *, , ,

ij ij ij ijij ij

i i i i

za za za zar x

M M M M

and * (4)maxi ij

jM za .

Step 6: The weighted normalized fuzzy decision matrix is constructed from R to the different importance

of each criteria as, ijn m

V v

, where ( )ijijv r w i .

Step 7: Determine the fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS) from

V . This can be obtained by, 1 2, ,..., nFPIS v v v

and 1 2, ,..., nFNIS v v v

, where

, , ,iv m m m m and , , ,iv m m m m

where (4)

maxij

jm v and

(1)

minij

jm v , since iv is weighted normalized TzFN’s.

Step 8: Calculate the distance of each effect from FPIS d and FNIS d

. The distance formula is

used to find the distance between two trapezoidal fuzzy numbers (1) (2) (3) (4)

1 1 1 1 1, , ,z z z z z and

(1) (2) (3) (4)

2 2 2 2 2, , ,z z z z z as,

2 2 2 2

(1) (1) (2) (2) (3) (3) (4) (4)

1 2 1 2 1 2 1 2 1 2

1,

4d z z z z z z z z z z

and 1

,n

ijj i

i

d d v v

and 1

,n

ijj i

i

d d v v

for j = 1,2,…,m.

Step 9: Calculate the closeness coefficient (CCj) and rank the order of alternatives according to the

coefficient. This is calculated by 1,2,...,j

j

j j

dCC j m

d d

.

Based on the value of closeness coefficient of each alternative, the ranking order of all alternatives from the

highest closeness to the lowest is determined.

4 Emotional Music Composition: A case study

This section intends to suggest the best composition selection for evoking emotions through music. Based

on the literature reviews and experts’ opinions, the important attributes of musical features for composition

selection are collected. The factors related to Western musical terms are mode (f1), tempo (f2), loudness (f3),

harmonization (f4), rhythmic structure (f5) and timbre quality of vocals and instruments (f6). The relational

mapping is given below.




Emotional Music

Mode LoudTempo TimbreRhythmHarmony

A1 A2 A3 A4

Figure 2: Emotional Music Features Composition

The relation between each attribute with others for evoking emotions is given as follows with linguistic ratings

of two experts.

Table 3: Experts evaluation on factors

Experts Exp 1 Exp 2

Attribute f1 f2 f3 f4 f5 f6 f1 f2 f3 f4 f5 f6

f1 - MH ML VH L H - M L MH H MH

f2 M - L H VH M H - MH MH M H

f3 VL H - VH M VH L M - M H MH

f4 H L ML - H H H M ML - L M

f5 L M MH M - M M MH L VH - H

f6 H M MH VH M - VH H H MH L -

The group uncertain direct-relation fuzzy matrix is shown below with the aggregation of TzFNs’ for linguistic

variables responded by two expert opinions.

Table 4: Direct-relation fuzzy matrix

Fact

or f1 f2 f3 f4 f5 f6

f1 (0,0,0,0) (0.45,0.55,0.65

,0.75)

(0.075,0.15,0.2

5,0.35)

(0.75,0.825,0.87

5,0.925)

(0.375,0.45,0.55

,0.625)

(0.65,0.75,0.85,

0.925)

f2 (0.55,0.65,0.75

,0.825) (0,0,0,0)

(0.275,0.35,0.4

5,0.55)

(0.65,0.75,0.85,

0.925)

(0.65,0.725,0.77

5,0.825)

(0.55,0.65,0.75,

0.825)

f3 (0,0.025,0.075,

0.25)

(0.55,0.65,0.75

,0.825) (0,0,0,0)

(0.65,0.725,0.77

5,0.825)

(0.55,0.65,0.75,

0.825)

(0.75,0.825,0.87

5,0.925)

f4 (0.75,0.85,0.95

,1)

(0.175,0.25,0.3

5,0.45)

(0.15,0.25,0.35

,0.45) (0,0,0,0)

(0.375,0.45,0.55

,0.625)

(0.55,0.65,0.75,

0.825)

f5 (0.175,0.25,0.3

5,0.45)

(0.45,0.55,0.65

,0.75)

(0.275,0.35,0.4

5,0.55)

(0.65,0.725,0.77

5,0.825) (0,0,0,0)

(0.55,0.65,0.75,

0.825)

f6 (0.85,0.925,0.9

75,1)

(0.55,0.65,0.75

,0.825)

(0.65,0.75,0.85

,0.925)

(0.75,0.825,0.87

5,0.925)

(0.175,0.25,0.35

,0.45) (0,0,0,0)

The normalized uncertain group direct-relation fuzzy matrix is constructed as follows.




Table 5: Normalized direct-relation fuzzy matrix

F f1 f2 f3 f4 f5 f6

f1 (0,0,0,0) (0.109,0.133,0.

158,0.182)

(0.018,0.04,0.0

61,0.085)

(0.182,0.2,0.21

2,0.224)

(0.091,0.109,0.1

33,0.152)

(0.158,0.182,

0.206,0.224)

f2 (0.133,0.157,

0.182,0.2) (0,0,0,0)

(0.067,0.08,0.1

09,0.133)

(0.158,0.182,0.

206,0.224)

(0.158,0.176,0.1

88,0.2)

(0.133,0.158,

0.182,0.2)

f3 (0,0.006,0.01

8,0.061)

(0.133,0.158,0.

182,0.2) (0,0,0,0)

(0.158,0.176,0.

188,0.2)

(0.133,0.158,0.1

82,0.2)

(0.182,0.2,0.2

12,0.224)

f4 (0.182,0.206,

0.230,0.242)

(0.042,0.061,0.

085,0.109)

(0.036,0.06,0.0

85,0.109) (0,0,0,0)

(0.091,0.109,0.1

33,0.152)

(0.133,0.158,

0.182,0.2)

f5 (0.042,0.061,

0.085,0.109)

(0.109,0.109,0.

133,0.158)

(0.182,0.067,0.

08,0.109)

(0.158,0.176,0.

188,0.2) (0,0,0,0)

(0.133,0.158,

0.182,0.2)

f6 (0.206,0.224,

0.236,0.242)

(0.133,0.158,0.

1820.2)

(0.2,0.158,0.18

,0.206)

(0.182,0.2,0.21

2,0.224)

(0.042,0.061,0.0

85,0.109) (0,0,0,0)

The total-relation uncertain group direct-relation fuzzy matrix is calculated in the following table.

Table 6: Total-relation uncertain group direct-relation fuzzy matrix

F f1 f2 f3 f4 f5 f6

f1 (0.165,0.285,0

.548,1.243)

(0.219,0.352,0.

632,1.338)

(0.110,0.219,0.

459,1.065)

(0.351,0.509,0.

822,1.618)

(0.2,0.32,0.585,1

.238)

(0.306,0.464,0

.787,1.581)

f2 (0.293,0.444,0

.748,1.506)

(0.140,0.265,0.

547,1.284)

(0.162,0.278,0.

534,1.182)

(0.358,0.535,0.

881,1.736)

(0.274,0.402,0.6

76,1.367)

(0.312,0.483,0

.832,1.679)

f3 (0.178,0.307,0

.576,1.323)

(0.254,0.389,0.

662,1.377)

(0.103,0.197,0.

410,1.007)

(0.348,0.508,0.

813,1.627)

(0.25,0.376,0.63

4,1.299)

(0.342,0.496,0

.801,1.607)

f4 (0.297,0.427,0

.697,1.365)

(0.153,0.277,0.

548,1.220)

(0.113,0.219,0.

449,1.025)

(0.174,0.31,0.6

04,1.352)

(0.184,0.298,0.5

53,1.174)

(0.269,0.419,0

.730,1.485)

f5 (0.189,0.319,0

.592,1.285)

(0.211,0.339,0.

606,1.289)

(0.146,0.249,0.

476,1.061)

(0.316,0.467,0.

767,1.539)

(0.11,0.21,0.444,

1.062)

(0.275,0.426,0

.735,1.505)

f6 (0.369,0.519,0

.819,1.588)

(0.274,0.425,0.

735,1.504)

(0.245,0.369,0.

632,1.289)

(0.402,0.581,0.

927,1.799)

(0.198,0.337,0.6

32,1.353)

(0.219,0.379,0

.719,1.577)

The degree of influential impact and the degree of influenced impact of factors are shown with the final

weightings of attributes as follows.

Table 7: Weightings of the factors

i Influence of factor fi on

others j

Impacts of fj influenced by

others Wgt.

Final weight of the

factor

r1 (1.351,2.148,3.832,8.082) c1 (1.492,2.302,3.98,8.308) w(1) (0.1627,0.1633,0.1645,0.1

648)

r2 (1.539,2.408,4.218,8.753) c2 (1.252,2.046,3.73,8.014) w(2) (0.1737,0.173,0.1727,0.17

16)

r3 (1.475,2.271,3.896,8.241) c3 (0.879,1.532,2.96,6.628) w(3) (0.17,0.1679,0.1659,0.166

4)

r4 (1.190,1.951,3.581,7.621) c4 (1.948,2.909,4.81,9.673) w(4) (0.1533,0.1559,0.1592,0.1

602)

r5 (1.248,2.010,3.620,7.743) c5 (1.215,1.943,3.52,7.494) w(5) (0.1566,0.1581,0.16,0.161

4)

r6 (1.707,2.611,4.464,9.111) c6 (1.723,2.667,4.61,9.434) w(6) (0.1836,0.1805,0.1779,0.1

752)

Then construct the decision making matrix for choosing alternative with respect to factors as follows.




Table 8: Alternative rating of Experts

Experts Exp 1 Exp 2

Attribute a1 a2 a3 a4 a1 a2 a3 a4

f1 G MP G MG M G M P

f2 M P VG MP MG G VG G

f3 MP MG M G G M G MG

f4 VG M VG MP VG VG G G

f5 M G M P MP P MP MG

f6 G M G VG G G M VG

Substituting trapezoidal fuzzy numbers for linguistic variables present in the above table then construct the

normalized fuzzy decision matrix as follows and weight of the criterion given in the left side.

Table 9: Normalized fuzzy relation matrix with factor weights

F Wgt. Final weight of the

factor a1 a2 a3 a4

f1 w(1) (0.1627,0.1633,0.1645

,0.1648)

(0.667,0.788,

0.909,1)

(0.545,0.667,0.

788,0.879)

(0.667,0.79,0.90

9,1)

(0.333,0.424,

0.545,0.667)

f2 w(2) (0.1737,0.173,0.1727,

0.1716)

(0.45,0.55,0.6

5,0.75)

(0.375,0.45,0.5

5,0.625) (0.95,1,1,1)

(0.45,0.55,0.6

5,0.725)

f3 w(3) (0.17,0.1679,0.1659,0.

1664)

(0.486,0.595,

0.703,0.784)

(0.486,0.595,0.

703,0.811)

(0.595,0.7,0.811

,0.892)

(0.703,0.811,

0.919,1)

f4 w(4) (0.1533,0.1559,0.1592

,0.1602) (0.95,1,1,1)

(0.65,0.725,0.7

75,0.825)

(0.85,0.93,0.975

,1)

(0.45,0.55,0.6

5,0.725)

f5 w(5) (0.1566,0.1581,0.16,0.

1614)

(0.4,0.56,0.72

,0.88)

(0.6,0.72,0.88,

1)

(0.4,0.56,0.72,0.

88)

(0.44,0.56,0.7

2,0.88)

f6 w(6) (0.1836,0.1805,0.1779

,0.1752)

(0.75,0.85,0.9

5,1)

(0.55,0.65,0.75

,0.825)

(0.55,0.65,0.75,

0.825) (0.95,1,1,1)

The weighted normalized fuzzy decision matrix is formulated in the below table.

Table 10: Weighted normalized fuzzy decision matrix

F a1 a2 a3 a4

f1 (0.11,0.13,0.15,0.16) (0.09,0.11,0.13,0.14) (0.11,0.13,0.15,0.16) (0.05,0.07,0.09,0.11)

f2 (0.08,0.10,0.11,0.13) (0.07,0.08,0.09,0.11) (0.17,0.17,0.17,0.17) (0.08,0.10,0.11,0.12)

f3 (0.08,0.10,0.12,0.13) (0.08,0.10,0.12,0.13) (0.10,0.12,0.13,0.15) (0.12,0.14,0.15,0.17)

f4 (0.15,0.16,0.16,0.16) (0.10,0.11,0.12,0.13) (0.13,0.14,0.16,0.16) (0.07,0.09,0.10,0.12)

f5 (0.06,0.09,0.12,0.14) (0.09,0.11,0.14,0.16) (0.06,0.09,0.12,0.14) (0.07,0.09,0.12,0.14)

f6 (0.14,0.15,0.17,0.18) (0.10,0.12,0.13,0.14) (0.10,0.12,0.13,0.14) (0.17,0.18,0.18,0.18)

The fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS) are given as follows.

FPIS = ( [0.16,0.16,0.16,0.16], [0.17,0.17,0.17,0.17], [0.17,0.17,0.17,0.17], [0.16,0.16,0.16,0.16],

[0.16,0.16,0.16,0.16], [0.18,0.18,0.18,0.18] )

FNIS = ( [0.05,0.05,0.05,0.05], [0.07,0.07,0.07,0.07], [0.08,0.08,0.08,0.08], [0.07,0.07,0.07,0.07],

[0.06,0.06,0.06,0.06], [0.1,0.1,0.1,0.1] )

The distance of each alternative from FPIS and FNIS with respect to each factors are shown in the following

table.




Table 11: Distances of alternatives from FPIS and FNIS

S Distance from FPIS Distance from FNIS

F a1 a2 a3 a4 a1 a2 a3 a4

f1 0.0307 0.04702 0.0307 0.082 0.0904 0.0713 0.0904 0.0373

f2 0.0691 0.08522 0.0033 0.0697 0.0385 0.0229 0.1006 0.0369

f3 0.0651 0.06447 0.0479 0.0317 0.0327 0.0345 0.0488 0.066

f4 0.0075 0.04462 0.017 0.0688 0.0854 0.0486 0.0783 0.0296

f5 0.065 0.0414 0.065 0.0627 0.0515 0.0723 0.0515 0.0517

f6 0.0257 0.0583 0.0583 0.0038 0.0606 0.0292 0.0292 0.0771

Then the d+, d- and the closeness coefficient are obtained for rank the order of alternatives as follows.

Table 12: Ranking of alternatives

Alternative d+ d- CCj Rank

a1 0.2631 0.3592 0.5773 2

a2 0.341 0.279 0.4497 4

a3 0.222 0.3989 0.6423 1

a4 0.3187 0.299 0.4837 3

The automatic compositional techniques of music can be developed with these ratings for better emotional

outcomes. The third alternative combination of musical features with necessary importance has a better

ability to promote emotions rather than others.

5 Conclusion

In this paper the causal relationships among criterion and its importance through weightings have been

discussed. The Modified DEMATEL technique is utilized for making causal relationship among factors with

the use of expert’s opinion. Then the combination of fuzzy TOPSIS technique with DEMATEL is utilized

as more appropriate tool for evaluating alternatives ranking based on factors. This method is useful when

the relationship among factors and choosing the alternatives relations with factors are expressed in an

uncertain linguistic environment. Moreover the case study on emotive music compositional techniques can

be observed for better compositional selection. The result can provide a suggestion to the music technicians

to decide the suitable ratings of musical features in compositions.

Acknowledgements

This research work is supported by UGC scheme MANF. Award Letter No.: F1-17.1/2011-12/MANF-

CHR-TAM-7467/(SA-III/Website).

References

[1] Abhinav Bansal, Trapezoidal Fuzzy Numbers (a,b,c,d): Arithmetic Behavior, International Journal of

Physical and Mathematical Sciences, (2011) 39-44.

[2] Akbar Alam-Tabriz, Neda Rajabani, Mojtaba Farrokh, An Integrated Fuzzy DEMATEL-ANP-TOPSIS

Methodology for Supplier Selection Problem, Global Journal of Management Studies and Researches, 1

(2) (2014) 85-99.




[3] D. R. Babak, T. E. Saputro, Supplier selection using integrated fuzzy TOPSIS and MCGP: a case study,

Procedia-Social and Behavioral Sciences, 116 (2014) 3957-3970.

http://dx.doi.org/10.1016/j.sbspro.2014.01.874

[4] J. K. Chen, I. S. Chen, Using a novel conjunctive MCDM approach based on DEMATEL, fuzzy ANP, and

TOPSIS as an innovation support system for Taiwanese higher education, Expert Systems with

Applications, 37 (3) (2010) 1981-1990.

http://dx.doi.org/10.1016/j.eswa.2009.06.079

[5] E. Fontela, A. Gabus, The DEMATEL Observer, DEMATEL 1976 Report, Battelle Geneva Research

Centre,Geneva, (1976).

[6] A. Gabus, E. Fontela, Perceptions of the world problematique: communication procedure, communicating

with those bearing collective responsibility (DEMATEL Report no.1), Battelle Geneva Research Centre,

Geneva, (1973).

[7] A. Kaufman, M. M. Gupta, Introduction to fuzzy arithmetic: theory and applications, Van Nostrand

Reinhold, New York, (1985).

[8] Kwang H. Lee, First Course on Fuzzy Theory and Applications, Advances in Soft Computing, Springer,

(2006) 145-146.

[9] C. J. Lin, W. W. Wu, A Fuzzy extension of the DEMATEL method for Group Decision Making, European

Journal of Operational Research, 156 (2004) 445-455.

[10] C. J. Lin, W. W. Wu, A Causal Analytical Method for Decision-Making under Fuzzy Environment, Expert

Systems with Applications, 34 (1) (2008) 205-213.


[11] Mahdi Mahmoodi, S. F. Gelayol, A New Fuzzy DEMATEL-TODIM Hybrid Method for evaluation

criteria of Knowledge management in supply chain, International Journal of Managing Value and Supply

Chains, 5 (2) (2014) 29-42.

http://dx.doi.org/10.5121/ijmvsc.2014.5204

[12] Marcel Zentner, Didier Grandjean, Klaus R. Scherer, Emotions Evoked by the Sound of Music:

Characterization, Classification and Measurement, Emotion, 8 (4) (2008) 494-521.

http://dx.doi.org/10.1037/1528-3542.8.4.494

[13] S. K. Patil, R. Kant, A hybrid approach based on fuzzy DEMATEL and FMCDM to predict success of

knowledge management adoption in supply chain, Applied Soft Computing, 18 (2014) 126-135.

http://dx.doi.org/10.1016/j.asoc.2014.01.027

[14] J. D. Richard, K. R. Scherer, H. Hill Goldsmith, Handbook of Affective Sciences, Series in Affective

Sciences, Oxford University Press, (2003).

[15] Sylvain Le Groux, Paul F. M. J. Verschure, Subjective Emotional Responses to Musical Structure,

Expression and Timbre Features: A Synthetic Approach, 9th ISCMMR, (2012) 160-176.

http://dx.doi.org/10.1016/j.sbspro.2014.01.874



http://dx.doi.org/10.5121/ijmvsc.2014.5204

http://dx.doi.org/10.1037/1528-3542.8.4.494

http://dx.doi.org/10.1016/j.asoc.2014.01.027




[16] G. H. Tzeng, W. H. Chen, R. C. Yu, M. L. Shih, Fuzzy decision maps: a generalization of the DEMATEL

methods, Soft Computing, 14 (11) (2010) 1141-1150.

http://dx.doi.org/10.1007/s00500-009-0507-0

[17] A. Victor Devadoss, A. Felix, A Fuzzy DEMATEL approach to study cause and effect relationship of

youth violence, International Journal of Computing Algorithm, 02 (2013) 363-372.

[18] G. W. Wei, Uncertain Linguistic hybrid geometric mean operator and its application to group decision

making under uncertain linguistic environment, International Journal of Uncertainty, Fuzziness and

Knowledge-Based Systems, 17 (02) (2009) 251-267.

http://dx.doi.org/10.1142/S021848850900584X

[19] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning (Part II),

Information Sciences, 8 (1975) 301-357.

http://dx.doi.org/10.1016/0020-0255(75)90046-8

[20] L. A. Zadeh, Fuzzy sets, Information Control, 8 (3) (1965) 338-353.

http://dx.doi.org/10.1016/S0019-9958(65)90241-X

[21] Xu. Zeshui, Uncertain linguistic aggregation operators based approach to Multiple Attribute Group

Decision Making under uncertain linguistic environment, Information Sciences, 168 (1–4) (2004) 171-

184.

http://dx.doi.org/10.1016/j.ins.2004.02.003

http://dx.doi.org/10.1007/s00500-009-0507-0

http://dx.doi.org/10.1142/S021848850900584X

http://dx.doi.org/10.1016/0020-0255(75)90046-8

http://dx.doi.org/10.1016/S0019-9958(65)90241-X

http://dx.doi.org/10.1016/j.ins.2004.02.003

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