Informatica 37 (2013) 399-409 399 Intuitionistic Fuzzy Jensen-Renyi Divergence: Applications to Multiple-Attribute Decision Making Rajkumar Verma and Bhu Dev Sharma Department of Mathematics Jaypee Institute of Information Technology (Deemed University) Noida-201307, U.P., India E-mail: rkver83@gmail.com, bhudev.sharma@jiit.ac.in Keywords: intuitionistic fuzzy set, Renyi entropy, Jensen-Shannon divergence, Jensen- Renyi divergence, MADM Received: May 17, 2013 Vagueness in the scientific studies presents a challenging dimension. Intuitionistic fuzzy set theory has emerged as a tool for its characterization. There is need to associate measures which can measure vagueness and differences in the underlying characterizing IFSs. In the present paper we introduce an information theoretic divergence measure, called intuitionistic fuzzy Jensen-Renyi divergence. It is a difference measure in the setting of intuitionistic fuzzy set theory, involving parameters that provide flexibility and choice. The strength of the new measure lies in its properties and applications. An approach to multiple-attribute decision making based on intuitionistic fuzzy Jensen-Renyi divergence is proposed. A numerical example illustrates the application of the new measure and the role of various parameters therein to multipleattribute decision making problem formulated in terms of intuitionistic fuzzy sets. Povzetek: Razvita je nova verzija intuitivne mehke logike za uporabo v procesu odločanja. 1 Introduction In probability theory and statistics, divergence measures are commonly used for measuring the differences between two probability distributions [13 and 22]. Kullback-Leibler [13] divergence is the well known such information theoretic divergence. Another important information theoretic divergence measure is the Jensen-Shannon divergence (JSD) [22] which has attracted quite some attention. It has been shown that the square root of JSD turns out to be a metric [9], satisfying (i) non-negativity (ii) (minimal) zero value only for identical distributions (iii) symmetric and (iv) satisfying triangular inequality, i.e. it is bounded from below and from above in terms of the norms of the distributions. However it may be mentioned that JSD itself is not a metric. It satisfies the first three axioms, and not the triangular inequality. These divergence measures have been applied in several disciplines like signal processing, pattern recognition, finance, economics etc. Some generalizations of Jensen-Shannon divergence measure have been studied in the last couple of years. For instance, He et al. [10] proposed a one parametric generalization of JSD based on Renyi's entropy function [21], called Jensen-Renyi divergence and used it in image registration. Other than probabilistic, there are vague/fuzzy phenomena. These are best characterized in terms of 'fuzzy sets', and their generalizations. The theory of fuzzy sets proposed by Zadeh [32] in 1965 addresses these situations and has found applications in various fields. In fuzzy set theory, the membership of an element is a single value lying between zero and one, where the degree of non-membership is just automatically equal to one minus the degree of membership. As a generalization of Zadeh's fuzzy sets, Atanassov [1, 2], introduced intuitionistic fuzzy sets. In their general setting, these involve three non-negative functions expressing the degree of membership, the degree of non-membership, and hesitancy, their sum being one. These considerations imbue IFSs with inbuilt structure to consider varieties of factors responsible of vagueness in the phenomena. IFSs have been applied in many practically uncertain/vague situations, such as decision making [3, 4, 8, 14, 16-18, 20, 25, 27-30 and 33] medical diagnosis [5, 24] and pattern recognition [6, 11, 12, 19 and 24] etc. Atanassov [2] and Szmidt and Kacprzyk [26] suggested some methods for measuring distance/difference between two intuitionistic fuzzy sets. Their measures are generalizations of the well known Hamming and Euclidean distances. Dengfeng and Chutian [6] and Dengfeng [7] proposed some other similarity and dissimilarity measures for measuring differences between pairs of intuitionistic fuzzy sets. In addition, Yanhong et al. [31] undertook a comparative analysis of these similarity measures. Recently, Verma and Sharma [25] proposed a generalized intuitionistic fuzzy divergence and studied its applications to multi criteria decision making. In this paper, we extend the idea of Jensen-Renyi divergence to intuitionistic fuzzy sets and propose a new divergence measure, called intuitionistic fuzzy Jensen-Renyi divergence (IFJRD) to measure the difference between two IFSs. After studying its properties, we give 400 Informatica 37 (2013) 399-409 R. Verma et al. an example of its applications in multiple-attribute decision making based on intuitionistic fuzzy information. The paper is organized as follows: In Section 2 some basic definitions related to probability theory, fuzzy set theory and intuitionistic fuzzy set theory are briefly given. In Section 3, the intuitionistic fuzzy Jensen-Renyi divergence (IFJRD) between two intuitionistic fuzzy sets is proposed. Some of its basic properties are analysed there, along with the limiting case. In Section 4 some more properties of the proposed measure are studied. In Section 5 application of proposed intuitionistic fuzzy Jensen-Renyi divergence measure to multiple-attribute decision making are illustrated and our conclusions are also presented here. 2 Preliminaries We start with probabilistic background. We denote the set of n-complete (n > 2) probability distributions by G = {p = (A, P2,"•, P„): Pt > 0, ±pt = lj . (1) For a probability distribution P =(p„ P2 v*^ p„)e r„, the well known Shannon's entropy [23], is defined as H(P) = -£p, log P,. (2) Various generalized entropies have been introduced in the literature taking the Shannon entropy as basic and have found applications in various disciplines such as economics, statistics, information processing and computing etc. A generalizations of Shannon's entropy introduced by Rényi's [21], Rényi's entropy of order« , is given by 1 " Ha(P) = --log(£ p« ), a* 1, a> 0. (3) 1 a ,=i Forae (0,1), it is easy to see that Ha(P) is a concave function of P, and in the limiting case a ® 1, it tends to Shannon's entropy. It can also be easily verified that Ha(P) is a non-increasing function of ae (0,1) and thus H«(P)> H(P) " ae (0,1) (4) In sequel, we will restrict a e (0,1), unless otherwise specified and will use base 2 for the logarithm. Next, we mention Jensen-Shannon divergence [15]. Let 1,12 > 0,1+12 = 1 be the weights of two probability distributions P, Qe Tn, respectively. Then the Jensen-Shannon divergence, is defined as JSi(P, Q)= H(lP +12Q)-1 H(P)-1 H(Q). (5) Since H(P) is a concave function, according to Jensen's inequality, JSÀ(P, Q) is nonnegative and vanishes when P = Q. One of the major features of the Jensen-Shannon divergence is that we can assign different weights to the probability distributions involved (6) according to their importance. This is particularly useful in the study of decision problems. A generalization of the above concept is the Jensen-Renyi divergence proposed by He [10], given by JRa(P, Q)= Ha1 P +I2Q) -lHa(P)~1 Ha(Q\ *e(0,l) where Ha(P) is Renyi's entropy, and 1 = (ll,12) is the weight vector, with 1,12 > 0 ,1+12 = 1, as before. Properties of Jensen-Renyi Divergence: Briefly we note some simple properties: i. JR1a(P, Q) is nonnegative and is equal to zero when P = Q. ii. Forae (0,l), JRXa(P, Q) is a convex function ofP and Q. iii. JR1a(P, Q), achieves its maximum value when P and Q are degenerate distributions. The Jensen-Shannon divergence (5) is a limiting case of JR1a(P, Q) when a® 1. Definition 1. Fuzzy Set [32]: A fuzzy set A in a finite universe of discourse X = {xl, x2,..., xa} is defined as A = {(x, m W)| xe X}, (7) where (x): X ® [0,1] is measure of belongingness or degree of membership of an element xe X to A . Thus, automatically the measure of non-belongingness of xe X to A is (1 -m~A (x)). Atanassov [1, 2] introduced following generalization of fuzzy sets, called intuitionistic fuzzy sets. Definition 2. Intuitionistic Fuzzy Set [1, 2]: An intuitionistic fuzzy set A in a finite universe of discourse X = {x1, x2,... , xn} is defined as A ={vB(x) " xe X; (ii) A = B iff A c B and B c A; (9) JRjA, B)= Ha(lA+1B) -1Ha(A)-1 Ha(B) where Ha(•) is Renyi's entropy for intuitionistic fuzzy set (•),ae (0,1), 1+12 = 1, 11,12 > 0 , and (Km a W+M* (xP V (x)+AVB(x) 1A+1b= That is JKa(A, B) log A 1PA (x)+K2PB (x) (iii) AC = {x, Va(x),mA(x)) | xe X}; (1 - a) (iv) AUB = ((e X}; [\ mn^A (x),vb (x)) / J (v) AnB = i(e X}. [\ maXVA (x),VB WW J Extending the idea from probabilistic to intuitionistic phenomena, in the neXt section, we propose a divergence measure called 'Intuitionistic Fuzzy Jensen-Renyi Divergence' (IFJRD) on intuitionistic fuzzy sets to quantify the difference between two intuitionistic fuzzy sets and discuss its limiting case. -1 log< -1 log (11Ma (x) + K2MB (x))" + (1Va (x)+1Vb (x))a + (l1PA (x)+K2PB (x))' ^ (Ma (x))a+V (x))a + (Pa (x))a_ (Mb (x))a+V (x))a + (p (x))a (10) 3 Intuitionistic Fuzzy Jensen-Renyi Divergence (IFJRD) Single element universe: First, let A and B be two intuitionistic fuzzy sets defined on a single element universal set X = {x} . Precisely speaking, we have: A =(Ma (x),va (x),pA(x)), and B = (mB(x\vB(x\nB(x)), where Ma (x) + V a (x)+pA (x) =1, and Mb (x)+VB (x)+PB (x) = 1, with 0 0, with equality if and only if A = B . ii. 0 < JR^A, B)< 1. iii. For three IFSs A, B, C in X and A c B c C, JR,a(A, B)< JR*jA, C), and JRjB, C)< JRjA, C). Proof: (i) The result directly follows from Jensen's inequality. (ii) Since JR*a(A, B) is convex for ae (0,1), refer Proposition 1 of He et al. [10], therefore, forae (0,1), JR*la(A, B) increases as || A - B ||1 increases, where || A - B ||1 = |Ma (x)-Mb (x) + VA (x) - VB w| + pA (x)-Pb (x) . (11) Thus, JR*a(A, B) " ae (0,1), attains its maximum for following degenerate cases: A =(1,0,0), B =(0,1,0) or A =(0,1,0), B =(1,0,0) or A = (0,0,1),B =(0,1,0). 0 < JRjA, B)< 1. 1 400 Informatica 37 (2013) 399-409 R. Verma et al. (iii) For A, B, C e IFS(X), || A - B ||, < || A - C | and ||B- C ^ < || A-C ||1, if A c B c C. Thus, A, B)< M,a(A, C) and JR1a(B, C)< JRjA C) V ae (0,1). (12) This proves the theorem. Limiting case: When a® 1 and 1=12 = 1, then measure (10) reduces to J-divergence on intuitionistic fuzzy sets proposed by Hung and Yang [11] as J (A, B) (( mA (x)+MbMl log( Ma (x)+Mb Wi 1 + 2 J I 2 ( (x)+nB (x) j log|nM±ni£) 'nA (x)+Pb (x) 1 ( nA (X)+Pb (x) 2 J I 2 (M(x)iogmA(x) 1 (mB(x)iogmB(x) i1 + + Va (x)logVA (x) y + PA (x)log pA (x)J + Vb (x)log^B (x) y +PB(x)log p b (x)j Ma (x) > Mb (x ) and ( (x) < ( (x). 4 Properties of intuitionistic fuzzy Jensen- Renyi divergence measure The measure JR1a(A, B) defined in (10) has the following properties: Theorem 2: For A, Be IFS(X), (i) JRia(AU B, AI B)= JRjA B), (ii) JRjAH B, AU B) = JRjB, A). Proof: We prove (i) only, (ii) can be proved analogously. (i) From definition in (10), we have: JRia(A U B, A n B) (13) 1 y •2(1 -a) tr 2 Definition 4: JR1a(A, B) on Finite Universe: Previously, we considered single element universe set. The idea can be extended to any finite universe set. If A and B are two IFSs defined in finite universe of discourse X = [xl, x2,..., x}, then, we define, the associated intuitionistic fuzzy Jensen-Renyi divergence by 1 a JRa( A, B) = - y JRlMx,), B(xt)) (14) ah where A(x) = {(x, Ma (x \ nA (x ),pA (x))} , and B(xt)= {(x,¡iB(x), VB(x),p(x) )}. In the next section, we study several properties of JR1a(A B). While proving these properties, we consider separation of X into two parts X1 and X2 , such that log 1 AuB (x )+12mA1B (x, ))a + (\VAnB (x, )+1(VAUB (x, )]i + | 1 (1 -MAUB (x, )-VA1B (x, )) + 1 (1 -MAm (x )-vaub (x,)) - A log -l2 log n (1 - a) y log (maUb (x ))a + (vanb (x))a + (1 -maUb (x )-va1b (x ))' (ma1b (x ))" + (vaW (x ))a + (1 -maIb (x )-(aUb (x))a (1Mb (x )+KMa (x))a + (( (x ) + 1Ya (x ))a + | 1(1 -Mb(x) ( B(x)) + 1 (1 -Ma (x )-(a (x )) Al |(Mb(x))a+(B(x))a A ogl+(1 -MB(x)-(b(x)) M (x )M(a (x ))a -l2 log + (1 -Ma (x )-(A (x))a X ={x | x e X, A(xt )c B(xt)}, X2 ={x | x e X, A(xt )= B(xt)}. Further it may be noted that for all x e X1, Ma (x) < Mb (x) and va (x) > (b (x), as also for V x e X,, (15) (16) +y x El, log (11Ma (x ) + aMb (x ))a + 1va (x )+1(Vb (x ))a + (1 -Ma (x )-Va (x )) ] + 1 (1 -Mb (x )-VB (x )),J -l1log A2 log M (x ))a + (vA (x ))a + (1 -Ma (x )-Va (x ))a (Mb (x ))a +((x ))a + (1 -Mb (x )-n (x ))a = JRjA, b) . + + 1 x eX Intuitionistic Fuzzy Jensen-Rényi Divergence... Informatica 37 (2013) 399-409 409 This proves the theorem. Theorem 3: For A, Be IFS(X), (i) JRia(A, A U B)+ JRjA, AI B) = JRjA, B), (ii) JRia(B, A U B)+ JRjB, AI B) = JR^B A). Proof: In the following, we prove only (i), (ii) can be proved analogously. (i) Using definition in (10), we first have JRjA, au B) 7(1 -a) i(l-a) E log - 1 log -12log (1«a (X )+12«aUb (x))a + 1(a (x )+1(a1b (x ))a + | l(l-Ma(x)-((x)) + + 12 (l-MaUB (x )-VA1B (x \«A (x ))a + K (x ))a ' + (l-«A (x )-( (x))' («AUB (x ^KlB (x ))a + (l -MaUB (x )-(A1B (x ))' n (l -a) E log (1l«A (x )+12«B(x ))a + (((x )+12vB (x ))a +1 1 (l-Ma (x)-(A (x)) + 1 (l-«b(x) (B(x)\ —1 log I2 log («A (x ))a + ( (x))a + (l-«A (x )-(A (x))£ («B (x))" + ((x ))a + (l-«B (x )-(B (xF + EE xeX2 log (ll«A (x )+12«A (x ))a + (1l( (x ) + 1((A (x ))a + | l(l ~«A (x )-(a (x )) + 1 (l -«A (x )~VA (x )\ — 1 log -12log («a (x ))a + (yA (x))a + (l -«A (x )-(A (x ))C («a (x )Y + (a (x ))a + (l-«A (x )( (x ))" EE log (1«a (x )+^2«B (x ))a + (1(a (x )+^2^B (x ))a + | 1 (l-«A (x )-(A (x )) + 1 (l-«b (x )-(b (x )). -1log 12 log («A (x ))a + ( (x ))a + (l-«A (x )-( (x ))a «(x ))a + ((B (x))a +(l-«b (x )-(b (x )y Next, again from definition in (l0), we have JRjA, An B) (1«A (x ) + 1«AnB (x ))a + (1((A (x )+1((AUB (x ))a iog] 1-«a(x)((x)) + 1 (l -«AIB (x )-(AUB (x l E n (l -a)~l («a (x))" + ((x ))a + (l -«A (x )-(a (x )) («AIB (x ))a + (aub (x )) -1llog + (l -«A1B (x )-(AUB (x ))a 7(l -a) E log (1«A (x )+12«A (x ))a + (A(a (x )+1((a (x ))a + 1(l-«A (x )-( (x)) I +1 (l -«a (x )-(a (x )) —1 log -12 log («A (x))a + ((A (x))a + (l -«A (x )-(A (x ))" (Ma (x ))a + ( (x ))a +(l-«a (x )-(a (x )y + E x.eX 2 log (l7) (1«A (x )+12«B (x ))a + (1l(A (x ) + l2(B (x ))a + 1(l -«a (x )-(a (x )) l +12 (l -«B (x )-(B (x )),l — 1 log 12 log ma (x ))a+( (x ))a ' + (l -«A (x )-(A (x ))a, («B (x ))"+(b (x ))" + (l-Mb (x )-(b (x ))a l (2 l l e X 400 Informatica 37 (2013) 399-409 R. Verma et al. 7(1 -a) z log (iMa (x )+Kmb (x))a + (lvA (x )+1vB (x ))a +!1 (1 -Ma (x )-( (x)) + 1 (1 -Mb (x )-(b (x)), -11log -llog (m (x ))a+(yA (x ))a +(1 -Ma (x )-( (x ))a (Mb (x))a + (B (x))a + (1 -Mb (x )~VB (x )) (18) Adding (17) and (18), we get the result. Theorem 4: For A, B, C e IFS(X), (i) JRia(AU B, C) < JRjA, C)+ JRjB, C); (ii) JRia(AI B, C) < JRia(A, C)+ JRia(B, C); Proof: We prove (i) only, (ii) can be proved analogously. (i) Let us consider the expression JRla(A, C)+ JRia(B, C)- JRia(AU B, C) (19) z 7(1 -a)tf log- Ima (x )+1Mc (x ))a + (1Va (x ) + 1(c (x ))a + | 1 (1 -Ma (x )-(a (x)) + 1 (1 -Mc (x )-vc (x)), -1 log--12log (m (x )T+((a (x )r + (1 -Ma (x )-( (x))< (mc (x )Y + (c (x ))a + ((1 -Mc (x )~VC (x) 1 z 7(1 -a)tf log (11Mb (x ) + AMc (x))a +1cb (x )+1(c (x ))a +! 1(1 -MB(x)-(B(x)) + 1 (1 -Mc (x )-(c (x)) - 1 log 12 log (Mb (x ))a + (B (x))a + (1 -Mb (x )-(b (x )) (mc (x ))a + (c (x ))a + ((1 -Mc (x )-(c (x))) —— z 7(1 -a) « log -11log I2 log (AMaub (x )+\Mc (x))a + (l1(A1B (x )+12(C (x ))a + ! 1 (1 -MaUb(x)-(A1B(x)) + 1 (1 -Mc (x )~vc (x )) (MaUB (x ))a + ((arb (x))a + (1 -MaUB (x ) (AIB (x (Mc (x ))a+(c (x ))a + ((1 -Mc (x )( (x ))) (1 -a) z log (1Mb (x )+1Mc (x))a +(((x )+Mc (x ))a +!1 (1 -Mb (x )-(B (x)) + 1 (1 -Mc (x )-(c (x )). -1 log -12log (Mb (x )) + (b (x T + (1 -Mb (x )-(B (x))a M (x )Y+((/c (x))a + (1 -Mc (x )-(c (x)) 7(1 -a) z x eX1 log {Km a (x )+12Mc (x))a + (l1(A (x )+12(C (x ))a + 1 (1 -Ma (x )-(a (x )) + K2(1 -Mc (x )-(0 (x)).J -Klog -1 log (Ma (x ))a+(a (x ))a + (1 -Ma (x )( (x ))a (Mc (x))"+(((x ))a + (1 -Mc (x )-Vc (x ))a > 0 This proves the theorem. Theorem 5: For A, B, C e IFS(X), JRjAU B, C)+JRjAH B, C) = jrJa, c )+jrJb, c ) • Proof: Using definition in (10), we first have: JRjAU B, C) a 1 a 1 a 1 + + Intuitionistic Fuzzy Jensen-Rényi Divergence... Informatica 37 (2013) 399-409 409 t 3(1 -a) £ log (1MaUb (x■ )+1Ma (x,))a + (1vaib (x )+1nc (x))a + l11 (1 M AUB (X )-VA1B (X ) +1 (l-mc (x )-vc (x)) -1log)(mAUB (X ))a + (vA1B (X ))a + (l-maUb (X )-VA1B (X ))a log (mc (x))a + (nc (x))a + (l-mc (x )-vc (x))a 7(1 -a) t X s X1 log (iMb (X )+KMc (x))a + (1(b (x )+1((c (x ))a +!1 (1 -Mb (X )-Vb (X )) + 1 (1 -Mc (X )-Vc (X)) -11log I2 log (Mb (x ))a + ((B (x ))a + (1 -Mb (X )-(b (X )) (Mc (x ))a + (c (x ))a + (1 -Mc (x )-(c (x))' +t xsX2 log (iMa (x )+12Mc (x ))a + (11VA (x )+12Vc (x ))a 1 (1 -Ma (x )-(a (x )) v + 1 (1 -Mc (x )-Vc (x)) -1 log* I2 log (Ma (x ))a + ((A (x ))a + (1 -Ma (x )-Va (x ))a \(Mc (x )Y+(c (x ))a [+(1 -Mc (x )( (x ))a (20) Next, again using definition in (10), we have JRia(A1 B, c) (1MaIb (x )+1Mc (x ))a + (AVaJb (x )+1((c (x ))a log-] + (1 -Maib (x )-(aUb (x ) + 1(1 -Mc (x )-vc (x)) t 7(1 -a) if -1log,(MAnB (x t + (nA ub (x ))a + (1 -Maib (x )-VaUb (x ))a -12log (Mc (x))a + (vc (x))a + (1 -Mc (x )-nc (x))a 7(1 -a) t log (1Ma (X )+1Mc (X ))a + (1(a (x )+1( (x ))a +! 1 (1 -Ma (X )~VA (X )) + 1 (1 -Mc (x )-Vc (x)). -11log 1"2 log (ma (X ))"+( (X ))" + (1 -ma (X )-(a (X )) (Mc (x )t + (vc (X ))a + (1 -Mc (x )-(c (x))' +t log (1Mb (x )+KMc (X ))a +(1(b (x )+1( (X ))a +. 1 (1 -Mb (X )~Vb (X )) + 1 (1 -Mc (x )-Vc (x )), -1 log -12log (Mb (x ))a + ((B (x ))a + (1 -Mb (x )-Vb (X ))£ (Mc (x ))" + (vc (x))a + (1-Mc (x )-Vc (x))' (21) Adding (20) and (21), we get the result. Theorem 6: For A, Bs IFS(X), (a) JRjA, B) = JRXa{Ac, Bc) (b) JRjA, B)= JRjAc, B); (c) JRjA, b)+JRjAc, B) = JRJAc , Bc )+JRJA, Bc ). where Ac and Bc represents the complement of intuitionistic fuzzy sets A and B respectively. Proof: (a) The proof simply follows from the relation of membership and non-membership functions of an element in a set and its complement. (b) Let us consider the expression JRjA, Bc )- JRXa(Ac, B) (22) (11ma (X ) + Mb (X ))a +(( (x )+1Mb (X ))a (1 -Ma (X )-VA (X )) + 1 (1 -Mb (X )-Vb (X )) (Ma (x ))a+(vA (x ))a + (1 -Ma (x )-VA (X ))a Vb (x ))"+(Mb (x ))a + (1 -vb (x )-Mb (x )) 7(1 -a) log -1log -12log a 1 x sX1 a 1 400 Informatica 37 (2013) 399-409 R. Verma et al. •3(1 -a) log (Ma (x )+kmB (x ))a + (1mA (x )+12nB(x))a +| 1(1 -Va(x)-Ma(x)) +1 (1 -mB (x )-vb (x)) -1ilog -l^log- Va (x ))a + (mA (x ))a + (1 -Va (x )-mA (x))£ (Mb (x ))a+(yB (x,))a + (1 -Mb (x )-yB (x)) = 0. (c) It immediately follows (a) and (b). This completes proof the theorem. In the next section, we suggest an application of the measure proposed to multiple-attribute decision making problem and give an illustrative example. 5 Applications of intuitionistic fuzzy Jensen-Rényi divergence to multiple-attribute decision making Vagueness is a fact of life and needs attention in matters of management. It can have several forms, for example, imperfectly defined facts, indirect data, or imprecise knowledge. For mathematical study, vague phenomena have got to be first suitably represented. IFSs are found to be suitable tools for this purpose. In this section, we present a method based on our proposed intuitionistic fuzzy Jensen-Rényi divergence defined over IFSs, to solve multiple-attribute decision making problems. It may be remarked that for a deterministic or probabilistic phenomenon where patterns show stability of the form, parameters have perhaps limited rule, but in vague phenomena, parameters provide a class of measures and choice for making appropriate selection by testing further. Intuitionistic fuzzy Jensen-Rényi divergence defined has parameters of two categories- the averaging parameters, 1 s, and an extraneous parameter a , each serving a different purpose. In the example below, we bring out their role in multiple-attribute decision making. Multiple-attribute decision making problems are defined on a set of alternatives, from which the decision maker has to select the best alternative according to some attributes. Suppose that there exists an alternative set A = {4, A2,..., Am} which consists of m alternatives, the decision maker will choose the best alternative from the set A according to a set of n attributes G ={G., G,,..., G }. Further let D =(d..) be 1 2 n J \ ij 'nxm the intuitionistic fuzzy decision matrix, where djj = (m¡j y i¡P) is an attribute value provided by the decision maker, such that m indicates the degree with which the alternative AV satisfies the attribute G¡, yjj indicates the degree with which the alternative A does not satisfies the attribute Gl, and p indicates the indeterminacy degree of alternative to the attribute Gl, such that: m e[o,i], n £ [0,1], m+yv =1, p„ = 1 - u„-v.. i = 1,2,...,n and j = 1,2,...,m. j t ij ij .j j J J To harmonize the data, first step is to look at the attributes. These, in general, can be of different types. If all the attributes G = {G1, G2,..., Gn} are of the same type, then the attribute values do not need harmonization. However if these involve different scales and/or units, there is need to convert them all to the same scale and/or unit. Just to make this point clear, let us consider two types of attributes, namely, (i) cost type and the (ii) benefit type. Considering their natures, a benefit attribute (the bigger the values better is it) and cost attribute (the smaller the values the better) are of rather opposite type. In such cases, we need to first transform the attribute values of cost type into the attribute values of benefit type. So, we transform the intuitionistic fuzzy decision matrix D =(d..) into the normalized intuitionistic fuzzy decision matrix R =(rv) by the method given by Xu and Hu [30], where , , id.., for benefit attribute G r. = (msv.,ps)=j( j )c , (23) I (d)C, for cost attribute Gt i= 1,2,...,n; j = 1,2,...,m where (d . )C is the complement of d , such that (dj)C={vj m ,pj). With attributes harmonized, using the measure defined in (10), we now stipulate following steps to solve our multiple-attribute intuitionistic fuzzy decision making problem: Step 1: Based on the matrix R =(rj) , specify the options A} (j = 1,2,...,m) by the characteristic sets: A = { G m V P | G £ g} . j = 1,2,...,m and i= 1,2,...,n Step 2: Find the ideal solution A*, given by: [(m1.,v1.,p1.),(m2.,v2.,p2.) ' A* = - ..-( m., y n* ,pJ) where, for each i=1,2,...,n, f max m„ ,min yV 1 - max m„ - min y „ (24) (25) Step 3: Calculate Jr. s(a. , A*) using the following expression for it: Intuitionistic Fuzzy Jensen-Rényi Divergence... Informatica 37 (2013) 399-409 409 1 y n(l-a) tf log -1 log - 22 log- (1mAl (x, )+22mAxi \ + (ivAj (x Mn-U). IPa, (x)+2paxi)a k (x )M( (x))" + p , (x). 'Mx ))a + Mx))a + pA'(X ))a (26) where l ,1 e [0,1], and l + 1 = 1 " j = 1,2,. A A2 A A4 4 G1 (0.5,0.4, (0.4,0.3, (0.5,0.2, (0.4,0.2, (0.6,0.4, 0.1) 0.3) 0.3) 0.4) 0.0) G 2 (0.7,0.2, (0.8,0.2, (0.9,0.1, (0.8,0.0, (0.5,0.2, 0.1) 0.0) 0.0) 0.2) 0.3) G3 (0.4,0.3, (0.5,0.2, (0.6,0.1, (0.7,0.3, (0.8,0.1, 0.3) 0.3) 0.3) 0.0) 0.1) G4 (0.6,0.2, (0.6,0.3, (0.8,0.1, (0.9,0.1, (0.4,0.2, 0.2) 0.1) 0.1) 0.0) 0.4) G (0.4,0.5, (0.6,0.4, (0.3,0.5, (0.5,0.3, (0.9,0.0, 5 0.1) 0.0) 0.2) 0.2) 0.1) G6 (0.3,0.1, (0.7,0.1, (0.6,0.2, (0.6,0.1, (0.4,0.3, 6 0.6) 0.2) 0.2) 0.3) 0.3) First, we transform the attribute values of cost type (G3) into the attribute values of benefit type (G3) by using Eq. (23): C = (G f = {(0-3,0.4,0.3), (0.2,0.5,0.3), (0.1,0.6,0.3),] 3 ( 3) {(0.3,0.7,0.0), (0.1,0.8,0.1) J , and then D =(d..) is transformed into R = (r.) , we V j /6x5 j ^6x5 ' get the following table: Table II: Normalized intuitionistic fuzzy decision matrix R A A2 A. A4 4 G (0.5,0.4, (0.4,0.3, (0.5,0.2, (0.4,0.2, (0.6,0.4, 0.1) 0.3) 0.3) 0.4) 0.0) G2 2 (0.7,0.2, (0.8,0.2, (0.9,0.1, (0.8,0.0, (0.5,0.2, 0.1) 0.0) 0.0) 0.2) 0.3) G3 (0.3,0.4, (0.2,0.5, (0.1,0.6, (0.3,0.7, (0.1,0.8, 0.3) 0.3) 0.3) 0.0) 0.1) G4 (0.6,0.2, (0.6,0.3, (0.8,0.1, (0.9,0.1, (0.4,0.2, 0.2) 0.1) 0.1) 0.0) 0.4) G5 (0.4,0.5, (0.6,0.4, (0.3,0.5, (0.5,0.3, (0.9,0.0, 0.1) 0.0) 0.2) 0.2) 0.1) G6 (0.3,0.1, (0.7,0.1 , (0.6,0.2, (0.6,0.1, (0.4,0.3, 6 0 .6) 0 . 2) 0.2) 0.3) 0.3) , m, in Step 4: Rank the alternatives Aj, j = 1,2, accordance with the values JR21 a (Aj, A*), j = 1,2,..., m, and select the best one alternative, denoted by At with smallest JRX a(Aj, A*). Then At is the best choice. In order to demonstrate the application of the above proposed method to a real multiple attribute decision making, we consider below a numerical example. Example: Consider a customer who wants to buy a car. Let five types of cars (alternatives) Aj (j = 1,2,3,4,5) be available. The customer takes into account six attributes to decide which car to buy: (1) G1: fuel economy, (2) G2: aerodynamic degree, (3) G3: price, (4) G4: comfort, (5) G5: design and (6) G6: safety. We note that G3 is a cost attribute while other five are benefit attributes. Next let us assume that the characteristics of the alternatives Aj (j = 1,2,3,4,5) are represented by the intuitionistic fuzzy decision matrix D = (dj) shown in the following table: _Table I: Intuitionistic fuzzy decision matrix D The step-wise procedure now goes as follows. Step 1: Based on R =(rj) , we have characteristic sets of the alternatives A (j = 1,2,..., 5) by A1 = A2 = A3 = A4 = A5 = A* = {(0.5,0.4,0.1), (0.7,0.2,0.1), (0.3,0.4,0.3),] {(0.6,0.2,0.2),(0.4,0.5,0.1),(0.3,0.1,0.6) J , {(0.4,0.3,0.3), (0.8,0.2,0.0), (0.2,0.5,0.3),] {(0.6,0.3,0.1), (0.6,0.4,0.0), (0.7,0.1,0.2) J , {(0.5,0.2,0.3), (0.9,0.1,0.0), (0.1,0.6,0.3),] {(0.8,0.1,0.1),(0.3,0.5,0.2),(0.6,0.2,0.2)J , {(0.4,0.2,0.4), (0.8,0.0,0.2), (0.3,0.7,0.0),] {(0.9,0.1,0.0),(0.5,0.3,0.2),(0.6,0.1,0.3) J , {(0.6,0.4,0.0), (0.5,0.2,0.3), (0.1,0.8,0.1),] {(0.4,0.2,0.4), (0.9,0.0,0.1), (0.4,0.3,0.3) J . Step 2: Using (24) and (25), we obtain A*: {(0.6,0.2,0.2), (0.9,0.0,0.1), (0.3,0.4,0.3),] {(0.9,0.1,0.0),(0.9,0.0,0.1),(0.7,0.1,0.2) J . Step3: We use formula (26) to measure JRX1 a(Aj, A*), choosing the various values of parameter. First we take 1 = 1 = 0.5 " j = 1,2,...,5; anda = 0.3, a = 0.5 12 ^ and a= 0.7 respectively, we get the following table: Table III: Values of JR11a(A , A*) for a = 0.3,0.5,0.7 a = 0.3 a = 0.5 a = 0.7 JRia 4 A*) 0.1453 0.1409 0.1345 JRia 4 A*) 0.1908 0.1584 0.1299 JRia 4 A*) 0.1617 0.1400 0.1214 JRia 4 A*) 0.0946 0.0905 0.0849 JRia 4 A*) 0.1483 0.1467 0.1424 Based on the calculated values of JR1j,a(Aj , A* ) in table III, we get the following orderings of ranks of the alternatives Aj (j = 1,2,3,4,5): Fora = 0.3, A, f A f A, f A f A2. m 400 Informatica 37 (2013) 399-409 R. Verma et al. For« = 0.5, A4 f 4 f A f A f a ■ Fora = 0.7, A4 f A f A2 f A f A5. Since JR. J(A4, A*) is smallest among the values of JRjjA, A') {] = 1,2,...,5} fora = 0.3, a = 0.5 and a = 0.7 , so A4 is the most preferable alternative. Thus here we find that variation in values of a brings about change in ranking, but leaves the best choice unchanged. Change in Consideration: In the above consideration, same values of were taken. But in a realistic situation these can also be different for different alternatives. The value of lj may then depend on an un-explicit (like past experience or pressures) on the decision maker. Let us next consider intuitionistic fuzzy Jensen-Renyi divergence measures JR. «(a., A*), taking different values of l]: We take ] = 0.5, l = 0.5 ; ] = 0.4, ] = 0.6 ; ] = 0.8, ] = 0.2 ; ] = 0.5, ] = 0.5 ; l = 0.3, l = 0.7 and a = 0.5. Calculating JR., a(Aj, A'), we get the following table: Table IV: Values of JR., (A., A*) for a = 0.5 JRla 4, a*) 0.0965 JRla A2, A*) 0.1644 JR« 4, A*) 0.0856 JRla 4, A*) 0.1178 JR« A5, A*) 0.1479 The resulting order of rankings then is A f 4f a4 f A, f a,. Thus A3 is the most preferable alternative. If we take l1 = 0.5, l = 0.5 ; = 0.7, = 0.3 ; l = 0.3, l = 0.7 ; l = 0.4, l4 = 0.6 ; l = 0.8, l5 = 0.2 and a = 0.5, calculating jr, «(a. , A*), we get the following table: Table V: Values of JR.^Aj, A') for a = 0.5 JR1a 4, A*) 0.1409 JR2a 4, A*) 0.1296 JR3a 4, A*) 0.1493 JR4a 4, A*) 0.1268 JR5a 4, A*) 0.0965 The resulting order of rankings then is A f A4 f A2 f A1 f A3. Resulting in A, as the most preferable option. Thus for a given value of parameter«, averaging parameters ls can effect the choice. The numerical example shows that change in order of the rankings results by change in parameters l & a establishing the significance of these parameters in multiattribute sensitive decision making problems. 6 Conclusions The paper provides a measure and application in multiple-attribute decision making problem under intuitionistic fuzzy environment. This study can lead to symmetric measure and resulting other insight into studying IFSs. References [1] Atanassov, K.T. 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