Generalized Entropy for Intuitionistic Fuzzy Sets

Malaysian Journal of Mathematical Sciences 0(): 090 (06) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal Generalized Entropy for Intuitionistic Fuzzy Sets Priti Gupta, H. D. Arora, and Pratiksha Tiwari 3 Department of Statistics, Maharshi Dayanand University, Rohtak, Haryana, India Department of Mathematics, Amity University, Noida, U.P., India 3 Department of Statistics, Maharshi Dayanand University, Rohtak, Haryana, India E-mail: hdarora@amity.edu ABSTRACT Information theory was introduced by Shannon in 948 and it includes the study of uncertainty measures. Analogous to information theory which is founded on probability theory, notion of fuzzy sets was established by Zadeh in 965 which is a mechanism to manage uncertainty. The pedestal for present study is information theory with intuitionistic fuzzy sets. Atanassov (986) devised Intuitionistic fuzzy sets which is an expedient devise to deal vagueness and uncertainty. In present study, we extended the fuzzy entropy proved by Gupta et al.(04) and then generalized it for intuitionistic fuzzy sets with axiomatic justication and presented importance of parameter α. Further, some properties of this measure are analyzed and the performance of proposed entropy measure at dierent values of α on the basis of linguistic variables is compared with the help of a numerical example. Keywords: Fuzzy entropy, intuitionistic fuzzy entropy, exponential intuitionistic fuzzy entropy, linguistic variables.

H. D. Arora. Introduction Shannon (948) dened measure of uncertainty for discrete and continuous probability distribution and proved various mathematical properties corresponding to both discrete and continuous probability distributions. Parallel to probability theory Zadeh discovered Fuzzy set theory in 965. Zadeh (965) introduced a measure of fuzzy information termed as fuzzy entropy which is based on Shannon entropy. In fuzzy set theory, non-membership value of a member is complement of its membership value from one, but practically it is not true, this is dealt by higher order Fuzzy set proposed by Atanassov (986) and is termed as intuitionistic fuzzy sets (IFSs). IFSs are advantageous in handling imprecision as well as hesitancy originated from inadequate information. It characterizes two characteristic functions for membership, µ A (x) and nonmembership, ν A (x) for an element x belonging to the universe of discourse. Later, Burillo and Bustince (996) dened the distance measure among IFSs. Also gave axiomatic denition of intuitionistic fuzzy entropy with its characterization. Thereafter, Szmidt and Kacprzyk (00, 005 a, b) extended the fuzzy entropy properties proposed by De Luca and Termini (97) and abridged the distance calculated using Hamming distance. Hung and Yang (006), Vlachos and Sergiadis (007), Chaira and Ray (008), Ye (00) and Verma and Sharma (03) developed entropies for IFS and discrimination measure between IFSs with corresponding properties and demonstrated the eciency in the framework of medical diagnosis, pattern recognition, edge detection, image segmentation and multi-criteria decision making. The article initially consist a brief introduction about IFSs followed by α exponential intuitionistic fuzzy entropy corresponding to α exponential fuzzy entropy. Further, to check the performance of proposed intuitionistic entropy measure on the basis of linguistic variables a numerical example is used.. Brief Introduction about Intuitionistic Fuzzy Sets. Atanassov (986) proposed a generalization of fuzzy sets characterized as Intuitionistic Fuzzy Sets (IFSs). It is has dealt with vagueness and hesitancy originated from inadequate information with expediency. It characterizes two characteristic functions for membership and non-membership µ A (x) and ν A (x) respectively for an element x belonging to the universe of discourse. An IFS is given as A = { x, µ A (x), ν A (x), x Ω} where µ A : Ω [0, ] and ν A : Ω [0, ] such that 0 µ A (x) + ν A (x) and µ A (x) and ν A (x) Ω denote degree of membership and non-membership of x A, respectively. For each IFS in Ω, 0 Malaysian Journal of Mathematical Sciences

Generalized Entropy for Intuitionistic Fuzzy Sets π A (x) = µ A (x) ν A (x) denotes index of hesitancy or intuitionistic fuzzy index. Obviously, 0 π A (x).the properties of fuzzy entropy proposed by De Luca and Termini(97) were extended by Szmidt and Kacprzyk (00) for an entropy measure in IFSs are as follows:. Intuitionistic fuzzy entropy is zero i set is crisp.. Intuitionistic fuzzy entropy is one i membership value is same as nonmembership values for every element. 3. Intuitionistic fuzzy entropy decreases as set get sharpened i.e if A is less fuzzy than B it implies that E(A) E(B). 4. Intuitionistic fuzzy entropy of a set is same as its complement. Li, Lu and Cai (003) suggested a method of converting intuitionistic fuzzy sets to fuzzy sets by allocating hesitation degree equally to membership and non-membership values. In subsequent section, a generalized entropy measure for intuitionistic fuzzy sets have been proposed which is based on α exponential fuzzy entropy given by Gupta et al. (04) as represented by equation () and validated some axioms for the same. H α (A) = 0 < α n( α e α ) [µ α A(x i )e µα A (xi) i= + ( µ A (x i )) α e ( µ A(x i)) α ], () 3. Extension of α Exponential Intuitionistic Fuzzy Entropy In this section, we have extended the result given by Gupta et al. (04) as represented by equation (). Let n H α (A) = n( α e α )) i= Hi α(a) where Hα(A) i = [µ α A (x i)e µα A (xi) + ( µ A (x i )) α e ( µ A(x i)) α ] Malaysian Journal of Mathematical Sciences

H. D. Arora Figure : H i α (A) (µ A (x i )) at µ A(x i)) = Hα(A) i = [α(α )µα (µ A (x i )) A (x i)e µα A (xi) + α ( 3α)µ α A (x i )e µα A (xi) At µ A (x i )) = + α( 3α)( µ A (x i )) α e ( µ A(x i)) α + α e ( µ A(x i)) α ( µ A (x i )) 3α + α(α )e ( µ A(x i)) α ( µ A (x i )) α + α µ A (x i ) 3α e µ A(x i) α ] equation () reduces to Hα(A) i (µ A (x i )) = α(.7465463α0.5 αe 0.5 3α +.7465463α0.5 α +.7465463α0.5 α.7465463α0.5 α 65.3876388α0.5 α ) And H i α (A) (µ A (x i)) < 0 at µ A (x i )) = when 0 < α as shown in Figure () Hence equation () holds for 0 < α. () 4. α Exponential Intuitionistic Fuzzy Entropy We dene α exponential intuitionistic fuzzy entropy corresponding to (), using method proposed by Li, Lu and Cai (003). According to this method, IFS A = { x, µ A (x), ν A (x), x Ω} is transformed in to fuzzy set (A ) having membership function µ A (x) = µ A (x) + (π A (x))/ = µ A(x)+ ν A (x).thus parallel to α exponential fuzzy entropy, we introduce α exponential intuitionistic Malaysian Journal of Mathematical Sciences

Generalized Entropy for Intuitionistic Fuzzy Sets fuzzy entropy dened as follows: K α (A) = n( α e α ) i= [( µ A(x i ) + ν A (x i ) + ( µ A(x i ) + ν A (x i ) ) α e ( µ A (x i )+ ν A (x i ) ) α ] ) α e ( µ A (x i )+ ν A (x i ) ) α for 0 < α. K α (A) = n( α e α ) i= [( µ A(x i ) + ν A (x i ) ) α e ( µ A (x i )+ν A (x i ) ) α + ( µ A(x i ) + ν A (x i ) ) α e ( µ A (x i )+ ν A (x i ) ) α ] (3) for 0 < α. In order to prove that (3) is a valid entropy measure for intuitionistic fuzzy sets, four properties given by Szmidt and Kacprzyk (00) has been checked as follows:. Property : Consider a crisp set A i.e. membership values of elements are either 0 or. Then K α (A) = 0. By putting µ A(x i)+ ν A (x i) = γ A (x i ), equation (3) reduces to K α (A) = n( α e α ) [γa(x α i )e γα A (xi) i= + ( γ A (x i )) α e ( γ A(x i)) α ] for 0 < α which is same as equation (). Therefore by the properties of fuzzy entropy, fuzzy entropy becomes zero if and only ifγ A (x i ) = 0 or for all i. µ A (x i)+ ν A (x i) = 0 µ A (x i ) ν A (x i ) = (4) for all i.again ν A (x i)+ µ A (x i) = 0 for all i. Also, ν A (x i ) µ A (x i ) = (5) µ A (x i ) + ν A (x i ) (6) Malaysian Journal of Mathematical Sciences 3

H. D. Arora From equation (4) and (6), we get that µ A (x i ) = 0, ν A (x i ) =. From equation (5) and (6), we get that µ A (x i ) =, ν A (x i ) = 0. Conversely, if µ A (x i ) = 0, ν A (x i ) = or µ A (x i ) =, ν A (x i ) = 0, K α (A) clearly reduces to zero. Hence K α (A) = 0 i either µ A (x i ) = 0, ν A (x i ) = or µ A (x i ) =, ν A (x i ) = 0 for all i.. Property : Let µ A (x i ) = ν A (x i ) for all i. Then equation (3) reduces to i.e. K α (A) =.Let K α (A) = n n i= F (γ A(x i )), 0 < α. WhereF (γ A (x i )) = [(γ A(x i)) α e (γ A (x i ))α +( γ A (x i)) α e ( γ A (x i ))α ] n( α e α ) Thus, if K α (A) =, n n i= F (γ A(x i )) = Dierentiating equation (7) w.r.t γ A (x i ) F (γ A (x i )) = i. (7) F (γ A (x i )) γ A (x i ) F (γ A (x i)) γ A (x i) = = 0, so [(αγ A (x i )) α e (γ A(x i)) α ( (γ A (x i )) α ) + α( γ A (x i )) α e ( γ A(x i)) α ( γ A (x i )) α ] n( α e α ) (8) [(αγ A (x i )) α e (γ A(x i)) α ( (γ A (x i )) α ) + α( γ A (x i )) α e ( γ A(x i)) α ( γ A (x i )) α ] = 0 (9) Equation (9) holds whenγ A (x i ) = Again dierentiating equation (8), we get F (γ A (x i )) (γ A (x i )) = [(αγ A (x i )) α e (γ A(x i)) α (α ) + α (γ A (x i )) α e (γ A(x i)) α ( 3α) + α ( γ A (x i )) 3α e ( γ A(x i)) α + α( 3α)( γ A (x i )) (α ) e ( γ A(x i)) α + α(α )( γ A (x i )) α e ( γ A(x i)) α + α (γ A (x i )) 3α e (γ A(x i)) α ] n( α e α ) For all i and 0 < α. F (γ A (x i)) (γ A (x i)) 0 atγ A (x i ) = as represented in gure ( where y = F (γ A (x i)) (γ A (x i)) and alpha = α) Hence F (γ A (x i )) is a concave function obtaining its maximum value at γ A (x i ) = i.e. γ A (x i ) = µ A(x i)+ ν A (x i). Thus K α (A) attains maxima at µ A (x i ) = ν A (x i ) for all i. 4 Malaysian Journal of Mathematical Sciences

Generalized Entropy for Intuitionistic Fuzzy Sets Figure : F (γ A (x i )) (γ A (x i )) Figure 3: df dt 3. Property 3: In order to prove that property 3 holds. It is enough to show that the function f(a, b) = [( a+ b ) α (a+ b) ( e ) α +( a+b ) α ( a+b) ( e ) α ] Where a, b [0, ], is increasing w.r.t a and decreasing for b.in order to determine critical point of f, equate f f a = 0 and b = 0 we get a=b. Again, letf(a, b) = [( a+ b ) α (a+ b) ( e ) α + ( a+b ) α ( a+b) ( e ) α ], where a > b, Put t = a b > 0, 0 < α then f(t) = [( t+ (+t) )α ( e ) α + ( t ( t) )α ( e ) α ] (+t) α α α α ( t) α α f t = [ α(+t)α ( (+t) α )e α( t)α ( ( t) α )e α ] < 0 α as shown in gure 3. and thus function f (t) is a decreasing when t = a b > 0. Similarly, it can proved that whena < b, t = b a > 0 then f(t) is decreasing function. Hence f f a 0, when a > b and a 0, when a < b.similarly, f f b 0,when a > b and b 0, when a < b.thus by monotonicity of function f(t), property () and containment property of IFSs, we get that K α (A) K α (B) when,a B. 4. Property 4 :Clearly it holds by description of complement of intuitionistic fuzzy sets and equation ().i.e. K α (A) = K α (Ā). Malaysian Journal of Mathematical Sciences 5

H. D. Arora Figure 4: Comparison of Intuitionistic Fuzzy Entropy at dierent values ofα Hence K α (A) is a valid entropy measure for intuitionistic fuzzy sets. It is observed that kurtosis of K α (A) varies as the value of generalization parameter α assumes dierent values within the specied range as shown in gure 4. Thus addition of a parameter makes distribution more exible and auent for practical purposes. It has also been supported by Andargie and Rao(03) so the suitable value of generalization parameter α assists practical modeling of data. The proposed measure has following properties: Theorem 4.. Consider two intuitionistic fuzzy sets A and B then K α (A B) + K α (A B) = K α (A) + K α (B) holdsfork α (A). Proof. Universe of discourse can be divided into two subsets as Ω = {x X µ A (x) µ B (x)} {x X µ A (x) µ B (x)}, which we will denote as Ω, Ω respectively. In Ω, µ A (x) µ B (x) ν A (x) ν B (x), then µ (A B) (x) = max{µ A (x), µ B (x)} = µ B (x), ν (A B) (x) = min{ν A (x), ν B (x)} = ν B (x) µ (A B) (x) = min{µ A (x), µ B (x)} = µ A (x), ν (A B) (x) = max{ν A (x), ν B (x)} = ν A (x). So A B = { x, µ (A B) (x), ν (A B) (x), x Ω }. From equation (), we have K α (A B) = n( α e α ) i= [( µ A B(x i ) + ν A B (x i ) ) α e ( µ A B (x i )+ν A B (x i ) ) α + ( µ A B(x i ) + ν A B (x i ) ) α e ( µ A B (x i )+ ν A B (x i ) ) α ] 6 Malaysian Journal of Mathematical Sciences

Generalized Entropy for Intuitionistic Fuzzy Sets For 0 < α. In Ω K α (A B) = n( α e α ) i= [( µ B(x i ) + ν B (x i ) + ( µ B(x i ) + ν B (x i ) ) α e ( µ B (x i )+ ν B (x i ) ) α ] For 0 < α. Again from equation (), we have ) α e ( µ B (x i )+ν B (x i ) ) α K α (A B) = n( α e α ) i= [( µ A B(x i ) + ν A B (x i ) ) α e ( µ A B (x i )+ν A B (x i ) ) α + ( µ A B(x i ) + ν A B (x i ) ) α e ( µ A B (x i )+ ν A B (x i ) ) α ] For 0 < α In Ω K α (A B) = n( α e α ) i= [( µ A(x i ) + ν A (x i ) + ( µ A(x i ) + ν A (x i ) ) α e ( µ A (x i )+ ν A (x i ) ) α ] ) α e ( µ A (x i )+ν A (x i ) ) α For 0 < α. From equation (0) and (), we get K α (A B) + K α (A B) = K α (A) + K α (B) in Ω.. Similarly the result holds in Ω. In the next section, we have checked the performance of proposed intuitionistic entropy measure on the basis of language variable given by De et al (000). 5. Numerical Example Let A = { x, µ A (x), ν A (x), x Ω} be IFS for any i R then A n = { x i, [µ A (x i )] n, [ ν A (x i )] n x i Ω} dened by De et al. (000) Malaysian Journal of Mathematical Sciences 7

H. D. Arora for IFS. According to De et al. (000) the classication of linguistic variables is given as, if A is considered as "LARGE" in Ω then A, A, A 3 and A 4 may be treated as "More or less LARGE", "Very LARGE", "Quite very LARGE" and "Very very LARGE" respectively. We use this to check performance of the proposed measure of intuitionistic fuzzy entropy. Mathematically a good intuitionistic fuzzy entropy measure should follow the given requirement by Li, Lu & Cai (003): E(A ) > E(A) > E(A ) > E(A) 3 > E(A 4 ) Consider the IFS A = { 6, 0., 0.8, 7, 0.3, 0.5, 8, 0.5, 0.4, 9, 0.9, 0, 0,, 0 }. For 0 α 0.5 the proposed intuitionistic fuzzy entropy measure satises K α (A) > K α (A ) > K α (A ) > K α (A) 3 ) > K α (A 4 ). But for 0.5 < α the proposed intuitionistic fuzzy entropy measure satises K α (A ) > K α (A) > K α (A ) > K α (A) 3 ) > K α (A 4 ) as given in table. Table : Comparison of proposed entropy at dierent values of α K α (A i ) K α (A ) K α (A) K α (A ) K α (A 3 ) K α (A 4 ) α = 0. 0.7880 0.790307 0.7846 0.77757 0.74948 α = 0.5 0.6588 0.6645 0.57354 0.48398 0.44756 α = 0.6 0.630348 0.63765 0.53605 0.4864 0.36033 α = 0.9 0.56836 0.55396 0.406565 0.304 0.443 α =.5 0.499986 0.469539 0.79405 0.7569 0.3089 α =.7 0.48569 0.397586 0.5493 0.46648 0.096 α =.0 0.4603 0.49463 0.637 0.08377 0.0894 According to table, we can conclude that proposed measure of intuitionistic fuzzy entropy is consistent with the structured linguistic variables if value of α lies between 0.5 and. 6. Conclusion In the present communication we have anticipated a α exponential intuitionistic fuzzy entropy for intuitionistic fuzzy sets corresponding to the entropy measure for fuzzy sets given by Gupta et al. (04) with some properties. Further, the performance of proposed entropy measure at dierent values of α is compared with the help of an example. Although the maximum and minimum value of entropy is independent of values of α but the entropy measure is consistent with the structured linguistic variables if value of α lies between 0.5 and 8 Malaysian Journal of Mathematical Sciences

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