Weighted Correlation Coefficient with a Trigonometric Function Entropy of Intuitionistic Fuzzy Set in Decision Making

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1 Weighted Correlatio Coefficiet with a Trigoometric Fuctio Etropy of Ituitioistic Fuzzy Set i Decisio Makig Wa Khadiah Wa Ismail, Lazim bdullah School of Iformatics ad pplied Mathematics, Uiversiti Malaysia Tereggau, 2100 Kuala Tereggau, Malaysia. Correspodig author. Tel.: ; lazim_m@umt.edu.my Mauscript submitted March 5, 2015; accepted Jue 9, doi: /iapm bstract: Weighted correlatio coefficiet is oe of the methods for ituitioistic fuzzy sets based multi-criteria decisio makig (IFS-MCDM). It uses a method of etropy to fid criteria weights for alteratives. Etropy weight that employed the arithmetic operatios of additio ad subtractio i IFS memberships is used for establishig the weighted correlatio coefficiet. However, the etropy obtaied from the use of the arithmetic operatios may deprive the sigificace of membership degree, o-membership degree ad hesitatio degree of IFS. This paper proposes a sie fuctio etropy weight for measurig weighted correlatio coefficiets of IFS-MCDM. example is give to illustrate the proposed method. The compariso results are also preseted to show the feasibility ad effectiveess of the proposed sie fuctio etropy weight. Key words: Etropy, correlatio coefficiet, ituitioistic fuzzy sets, sie fuctio. 1. Itroductio Ituitioistic fuzzy set (IFS) was itroduced by taassov [1] as the extesio ad geeralizatio of Zadeh s fuzzy set [2]. The cocept of dual memberships i IFS was well-received by may researchers thaks to the uique characteristics of memberships degree, o-membership degree ad hesitatio degrees. The IFS has successfully used i solvig various applicatios of multi-criteria decisio makig (MCDM) problems. Past researchers have studied the applicatios of IFS i a diverse array of MCDM (IFS-MCDM) disciplies such as medical diagostics [], eviromet performace idex [4], similarity measures [5], huma capital [6] ad patter recogitio [7]. I decisio makig process, sometimes the iformatio about criteria weights is completely ukow due to lack of kowledge or data ad limitatios of experts i providig iformatio about the problem domai [8], [9]. To address this issue, Ye [10] proposed a MCDM method based o weighted correlatio coefficiet usig etropy weights. This weighted correlatio coefficiet measures the stregth of correlatio betwee a alterative ad the ideal alterative i MCDM problem. Ye [10] used the etropy weight that was defied by Burillo & Bustice [11] based o the kowledge of complemetary membership degrees i IFS. The sum of membership degree, o-membership degree ad hesitatio degree must fulfill the uique maximum membership of 1. It seems that the etropy proposed by Burillo & Bustice [11] used a simple arithmetic operatio. The arithmetic operatio may deprive the importace of the three membership degrees, particularly i the situatio where 199 Volume 5, Number, July 2015

2 the degree of hesitatio is small. I order to boost the sigificace of etropy weight i IFS-MCDM ad to mie the importace of hesitatio degree, a similar fuctio with maximum value of 1 could be proposed. Parkash et al., [12] provide some clues over this matter by itroducig weighted measures of fuzzy etropy based o the maximum etropy priciple. However, the multiplicatio of weights to cyclic or trigoometric fuctio i Parkash et al., [12] would exaggerate the cotributio of weight to etropy. O the other had, etropy is basically a measure of weight i perceptio based o theory of probability. s a result, Ye [10] proposed trigoometric fuctio etropy specifically for IFS based o the fuzzy etropy of Parkash et al., [12]. The emergece of trigoometric or oe uit cycle fuctio of IFS etropy ad the issue of maximum operatios i complemetary ature of IFS may shed some light o the improvemet of weighted correlatio coefficiet i IFS-MCDM. Motivated by these works, the preset paper proposes the weighted correlatio coefficiet usig a trigoometric fuctio of IFS etropy for solvig IFS-MCDM. The proposed weighted correlatio coefficiet uses the sie fuctio of IFS etropy istead of arithmetic operatios of IFS etropy. I short, the obective of this paper is to itroduce sie fuctio based etropy weight of IFS i measurig weighted correlatio coefficiet. 2. Prelimiaries This sectio itroduces the basic defiitios relatig to fuzzy set theory, IFS, ituitioistic fuzzy etropy ad weighted correlatio coefficiet. Out of several higher order fuzzy sets, IFS [1] is a primary extesio of the covetioal fuzzy sets theory. Both are alleviate some drawbacks of Zadeh s fuzzy set ad have bee foud to be highly useful to deal with vagueess. The IFS allocates both membership ad o-membership to each elemet of the uiverse. Defiitio 1: Ituitioistic fuzzy sets [1] IFS i X is a expressio is defied by = { < x, μ ( x,ν ) ( x ) > x ÎX }, (1) where μ ( x): X [0, 1] ad v ( x): X [0, 1] with the coditio 0 μ ( x) ν ( x) 1. The umbers μ () x ad v () x represet respectively the membership degree ad o-membership degree of the elemet x to the set. For each IFS i X: for all X. π( x ) = 1- μ( x-ν ) ( x, ) (2) x The π ( x ) is called the ituitioistic idex or hesitacy degree of the elemet x i the set. It ca be see that 0 π ( x) 1, x X. Defiitio 2: Correlatio coefficiet of IFS [1] Let ad B be two IFSs i the uiverse of discourse X { x1, x2,..., x }. The correlatio coefficiet of ad B is give by C(,B) k(,b ) =, T ( ).T ( B) () where the correlatio of two IFSs ad B is give by C(,B ) = ( μ( xi ) μb( x i) +ν( xi ) νb( xi )) ad i=1 200 Volume 5, Number, July 2015

3 2 2 i=1 i i the iformatioal ituitioistic eergies of two IFSs ad B are give by T( ) = ( μ ( x ) +ν ( x )) 2 2 ad T( B ) = ( μb( x i) +νb( x i)), respectively. i=1 Defiitio : Weighted correlatio coefficiet [12]. The correlatio coefficiet betwee a alterative i ad the ideal alterative with etropy weights for criteria defied as follows: ( ) Ci(, ) w 1 μ C i = Wi(, i) = =. T ( ) T ( ) 2 2 i i w ( μ ( C ) ( )) =1 i +ν C i (4) whe 1-H w =, - H = 1 (5) where 1 w [0, 1], w 1, ( ) 1 H E C ad 0H 1 m for ( 1, 2,..., ). ll the defiitios provide the fudametal kowledge for decisio makig usig weighted correlatio coefficiet.. Weighted Correlatio Coefficiet with Sie Fuctio Etropy Ye [10] developed weighted correlatio coefficiet with the aim to solve a decisio problem. The method deals with the icomplete iformatio where the criteria weights are ukow ad the memberships of criteria may take i form of ituitioistic fuzzy sets. ew improvemet of fidig etropy is itroduced i the proposed method. The basic etropy based o the defiitio of complemetary i IFS is replaced with etropy based o trigoometric or cyclic fuctio of sie. The maximum value of oe i IFS is retaied with the itroductio of etropy weight sie fuctio. The basic steps of the proposed weighted correlatio coefficiet are orgaized as follows. Step 1: Costruct a fuzzy decisio matrix. Evaluatios of alteratives are represeted by IFS. The ituitioistic fuzzy valued decisio matrix D = d i m where d i = ( μ i,νi ) is the evaluatio of alterative i with respect to criteria C. Step 2: Fid the etropy weight of each criteria, C. The trigoometric sie fuctio of IFS etropy is proposed as μ x -ν x μ x+ν x 1 siπ 1 ( ) ( ) siπ 1 ( ) ( ) 1 E1 ( ) (6) Step : Fid the weight of each criteria, C. Sice the weight of criteria is completely ukow, the weights of criteria ca be obtaied usig etropy weight computed usig Eq. (5). Step 4: Fid the weighted correlatio coefficiet. 201 Volume 5, Number, July 2015

4 The correlatio coefficiet betwee a alterative i ad the ideal alterative with etropy weights for criteria is measured usig Eq. (4). Step 5: Rak the alterative ( i 1, 2,,..., m) ad select the best oe(s) i accordace with i weighted correlatio coefficiet. The highest rakig or the best alterative is chose based o the highest weighted correlatio coefficiet. This paper proposes a modificatio to etropy weight, specifically i Step 2, without missig the geerality of weighted correlatio coefficiet of Ye [10]. 4. Illustrative Example I order to demostrate the feasibility of the weighted correlatio coefficiet, the authors cosider the decisio makig problem discussed i [14]. Example. air-coditio system selectio problem [14]. Suppose that there exist three air-coditios systems; 1, 2 ad which form the set of all alteratives 1 2,,. Suppose four criteria C 1(good quality), C 2 (easy to operate), C (ecoomical) ad C 4(good service after sellig) are take ito accout i this problem. Now it is a eed to choose a air-coditio system which satisfiesc 1, C 2, C ad C 4. The selectio process is executed usig the proposed method. Step 1: Costruct a fuzzy decisio matrix. D C C C C (0.85, 0.10) (0.75, 0.10) (0.70, 0.15) (0.70, 0.10) , 0.10) (0.80, 0.15) (0.85, 0.10) (0.70, 0.15) (0.80, 0.10) (0.80, 0.10) (0.75, 0.15) (0.70, 0.15) Step 2: Fid the etropy of each criteria, C,( 1, 2,, 4) usig Eq. (6). siπ siπ EC ( 1) The etropy for other criteria E( C 2 ),E( C ) ad EC ( 4 ) are computed with the similar fashio. It is obtaied as E( C 2 ) , E( C ) ad EC ( 4 ) Step : Fid the weights of each criteria, C,( 1,2,, 4) usig Eq. (5). w where H1 (4.797) ad 4 1 H With the similar calculatio, weights for the rest of criteria are w , w ad w Volume 5, Number, July 2015

5 Step 4: Fid the weighted correlatio coefficiet. The correlatio coefficiet betwee a alterative i ad the ideal alterative with etropy weights for criteria is measured usig Eq. (4). For the alterative 1, 1, the weighted correlatio coefficiet is give as 0.251(0.85) (0.70) W1(, 1) ( ) ( ) With the similar fashio, the weighted correlatio coefficiets for other alteratives are W (, ) ad W (, ) Step 5: Rak the alterative i i( 1, 2, ) accordig the value of weighted correlatio coefficiet. The highest weight is the best alterative Therefore, the air-coditio system 1 is chose as a ideal air-coditio system which satisfies C 1, C 2, C ad C 4. Thus, the rakig result is cosistet with the result obtaied by Liu & Wag [14]. The problem of Liu & Wag [14] was tested usig the score fuctio. The same problem was also tested by Wu & Zhag [15] usig ituitioistic fuzzy weighted etropy. The results from these two tests were cosistet with the result usig the proposed method. It is also worth to ote here that Ye [10] used the complemetary arithmetic operatios of IFS i fidig etropy weight. The comparative results of the problem with differet approaches are show i Table 1. Method Table 1. Ideal ir-coditio System: Comparative Result The selected air-coditio system Score fuctio [10] Weighted correlatio coefficiet with arithmetic operatios [14] Ituitioistic fuzzy weighted etropy [15] The proposed weighted correlatio coefficiet with Sie fuctio The result shows that the alterative 1 is cosistetly selected as a ideal air-coditio system albeit differet methods used. 5. Coclusio The paper preseted a ew approach for determiig the etropy weight i evaluatig the weighted correlatio coefficiet. To determie the criteria weights, a sie fuctio etropy weight proposed by Ye [10] is used to replace the etropy proposed by Burillo & Bustice [11] i which the latter used the arithmetic operatio i the relatioship of IFS memberships i proposig etropy. The etropy weights were adapted ito the weighted correlatio coefficiet. The alteratives could be raked accordigly ad the most desirable oe(s) ca be selected accordig to the weighted correlatio coefficiet. Fially, a illustrative example was implemeted to demostrate the feasibility of the proposed method. Validatio with the other methods was also provided as a comparative aalysis. ckowledgmet This work is part of the proect FRGS Thaks are due to Malaysia Miistry of Educatio ad 20 Volume 5, Number, July 2015

6 Uiversiti Malaysia Tereggau. Refereces [1] taassov, K. (1986). Ituitioistic fuzzy sets. Fuzzy Sets ad Systems, 20(1), [2] Zadeh, L.. (1965). Fuzzy sets. Iformatio ad Cotrol, 8(), 8-5. [] De, S. K., Biswas, R., & Roy,. R. (2001). applicatio of ituitioistic fuzzy set i medical diagosis. Fuzzy Sets Systems, 117(2), [4] bdullah, L., & Wa, I. W. K. (201). ew rakig of evirometal performace idex usig weighted correlatio coefficiet i ituitioistic fuzzy sets: case of SEN coutries. Moder pplied Sciece, 7(6), [5] Yusoff, B., Taib, I., bdullah, L., & Wahab,. F. (2011). ew similarity measure o ituitioistic fuzzy sets. Iteratioal Joural of Computatioal ad Mathematical Scieces, 5(2), [6] bdullah, L., Jaafar S., & Taib, I. (201). Ituitioistic fuzzy aalytic hierarchy process approach irakig of huma capital idicators. Joural of pplied Scieces, 1(), [7] Degfeg, L., & Chutia, C. (2002). New similarity measure of ituitioistic fuzzy sets ad applicatio to patter recogitios. Patter Recogitio Letter, 2(1-), [8] Chou, S. Y., Chag Y. H., & She, C. Y. (2008). fuzzy simple additive weightig system uder group decisio-makig for facility locatio selectio with obective/subective attributes. Europea Joural of Operatio Research, 189(1), [9] Yeh, C. H., & Chag, Y. H. (2009). Modellig subective evaluatio for fuzzy group multi-criteria decisio makig. Europea Joural of Operatio Research, 179, [10] Ye, J. (2010). Fuzzy decisio-makig method based o the weighted correlatio coefficiet uder ituitioistic fuzzy eviromet. Europea Joural of Operatio Research, 205, [11] Burillo, P., & Bustice, H. (1996). Etropy o ituitioistic fuzzy sets ad o iterval-valued fuzzysets. Fuzzy Sets ad Systems, 19, [12] Parkash, O., Sharma P. K., & Mahaa, R. (2008). New measures of weighted fuzzy etropy ad their applicatios for the study of maximum weighted fuzzy etropy priciple. Iformatio Scieces, 178(11), [1] Gersterkor, T., & Mako, J. (1991). Correlatio of ituitioistic fuzzy sets. Fuzzy Sets ad Systems, 44, 9-4. [14] Liu, H. W., & Wag, G. J. (2007). Multi-criteria decisio-makig methods based o ituitioistic fuzzy sets. Europea Joural of Operatio Research, 179, [15] Wu, J. Z., & Zhag, Q. (2011). Multi-criteria decisio makig method based o ituitioistic fuzzy weighted etropy. Expert Systems with pplicatios, 8, Wa Khadiah Wa Ismail is a research studet at the School of Iformatics ad pplied Mathematics, Uiversiti Malaysia Tereggau. Her research iterests iclude fuzzy sets ad similarity measures. 204 Volume 5, Number, July 2015

7 Lazim bdullah is a professor at the School of Iformatics ad pplied Mathematics, Uiversiti Malaysia Tereggau. He holds a B.Sc (Hos) i mathematics from the Uiversity of Malaya, Kuala Lumpur i Jue 1984 ad the M.Ed i mathematics educatio from Uiversity Sais Malaysia, Peag i He received his Ph.D. from the Uiversiti Malaysia Tereggau, (iformatio techology developmet) i His research focuses o the mathematical theory of fuzzy sets ad its applicatios i social ecology, evirometal scieces, health scieces, ad maufacturig egieerig. He is iterested i the measuremet of social idicators, idices of health related quality of life ad evirometal idex usig computatioal itelligece ad statistical approaches. His research fidigs have bee published i over two hudred publicatios icludig refereed ourals, coferece proceedigs, ad chapters i book, ad research books. Curretly he is a member of editorial boards to several iteratioal ourals related with computig ad iformatio techology. Besides, he is also a reviewer for umber of local ad iteratioal ourals, member of scietific committees of several symposia ad cofereces at atioal ad iteratioal levels. Dr. Lazim is a associate member of IEEE Computatioal Itelligece Society, a member of Malaysia Mathematical Society ad a member of the Iteratioal Society o Multiple Criteria Decisio Makig. 205 Volume 5, Number, July 2015