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

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.


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 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 positiveideal 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.

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: , , , Z z z z z  .Its membership function and graphical representation defined as follows, , , , Z z z z z  be two hexagonal fuzzy numbers.
Then the addition and subtraction operations are defined by,

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

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 f 1 , f 2 , f 3 ,…, f n 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 f i on f j 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 = {f 1 , f 2 , …, f n } be a finite set of factors and E = {E 1 , E 2 , …, E n } be the finite set of experts, where E k denotes the k th 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 k thexpert as where k =1,2,…, K.If there does not exist a correlation between f i and f j , then denote '' k ij u .Particularly, there does not exist a correlation between f i itself.Now, the correlation among the factors by E k th expert's opinion is, Step 3: Consider the trapezoidal fuzzy number TzFN z k = (z 1(k) ,z 2(k) ,z 3(k) ,z 4(k) ) [where z 1(k) ≤ z 2(k) ≤ z 3(k) ≤ z 4(k) ] for the corresponding linguistic term responded by the decision maker , , , Step 4: The group uncertain direct-relation fuzzy matrix is denoted by G and defined as follows: . This is done by aggregating the individual uncertain direct-relation matrices.(ie.,) , , , ,, ij ij ij g g g and ( 4) ij g are calculated by (1) (1) (2) (3) Step 5: Then the normalized uncertain group direct-relation matrix , , , ,, x x x and ( 4) Step 6: To compute the total-relation uncertain matrix, we should have to establish and analyze this model by ensuring the convergence of , 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.,) Step 7: Then construct the total-relation uncertain matrix as , , , Step 8: All direct and indirect influence of factor f i on all other factors is denoted by i r and defined as and i r is called as the degree of influential impact.And, both direct and indirect impacts on f j is influenced by all other factors is denoted by j c and defined as   (3) (4) is called as the degree of influenced impact.
Step 9: Aggregate the weight of factors from i r and j c 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 () wi and defined as,

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 { f 1 , f 2 , f 3 ,…, f n } as the factors and { a 1 , a 2 , a 3 ,…, a m } as the alternatives based on f i ; i = 1,2,…,n.This is converted into the dynamical system N for the expert-k as, , where k = 1,2,…,K.
Step 2: Convert the appropriate linguistic ratings of the factors based alternatives into corresponding trapezoidal fuzzy numbers for the k th -expert as, , , , x za za za za za  is TzFN of appropriate linguistic rating of alternative.
Step 3: Obtain the aggregated fuzzy rating ij x of alternative a j under criteria f i evaluated by experts using trapezoidal fuzzy numbers of each matrices and take the aggregated fuzzy weight () Step 9: Calculate the closeness coefficient (CC j ) and rank the order of alternatives according to the coefficient.This is calculated by 1, 2,..., 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.

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 (f  The relation between each attribute with others for evoking emotions is given as follows with linguistic ratings of two experts.
The group uncertain direct-relation fuzzy matrix is shown below with the aggregation of TzFNs' for linguistic variables responded by two expert opinions.The degree of influential impact and the degree of influenced impact of factors are shown with the final weightings of attributes as follows.The weighted normalized fuzzy decision matrix is formulated in the below table.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.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.

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.

Figure 1 :
Figure 1: Membership and graphical diagram of TzFN

Figure 2 :
Figure 2: Emotional Music Features Composition

Table 2 :
Linguistic variables for alternatives ratings based on factors Transform this D into normalized fuzzy decision matrix which is denoted by R and defined as, Step 4: Construct the fuzzy decision matrix D with the entries ij x as,   ij Dx  and the weight of the criteria is taken as,   ( ) ( ) , ( ) , ( ) , ( ) 1, 2,..., w i w i w i w i w i i n    .http://www.ispacs.com/journals/jfsva/2015/jfsva-00239/InternationalScientificPublicationsand Consulting ServicesStep 5:i v is weighted normalized TzFN's.FNIS d  .The distance formula is used to find the distance between two trapezoidal fuzzy numbers ://www.ispacs.com/journals/jfsva/2015/jfsva-00239/International Scientific Publications and Consulting Services http

Table 3 :
Experts evaluation on factors

Table 4 :
Direct-relation fuzzy matrix

Table 5 :
Normalized direct-relation fuzzy matrixThe total-relation uncertain group direct-relation fuzzy matrix is calculated in the following table.

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

Table 7 :
Weightings of the factorsThen construct the decision making matrix for choosing alternative with respect to factors as follows.http://www.ispacs.com/journals/jfsva/2015/jfsva-00239/ International Scientific Publications and Consulting Services

Table 8 :
Alternative rating of Experts 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

Table 10 :
Weighted normalized fuzzy decision matrix

Table 11 :
Distances of alternatives from FPIS and FNISThen the d + , d -and the closeness coefficient are obtained for rank the order of alternatives as follows.