A new generalized intuitionistic fuzzy set

A new generalized intuitionistic fuzzy set Ezzatallah Baloui Jamkhaneh Saralees Nadarajah Abstract A generalized intuitionistic fuzzy set GIFS B) is proposed. It is shown that Atanassov s intuitionistic fuzzy set, intuitionistic fuzzy sets of root type intuitionistic fuzzy sets of second type are special cases of this new one. Some important notions, basic algebraic properties of GIFS B, three operators their relationship are discussed. The algebraic properties include being closed under union, being closed under intersection, being closed under a necessity measure, being closed under a possibility measure de Morgan type identities. Keywords: Generalized intuitionistic fuzzy set, Intuitionistic fuzzy set, Intuitionistic fuzzy sets of root type, Intuitionistic fuzzy sets of second type. 2000 AMS Classification: 47S40. 1. Introduction The concept of fuzzy sets was introduced by Zadeh [22] whose basic component is only a degree of membership. Atanassov [2] generalized this idea to intuitionistic fuzzy sets IFS) using a degree of membership a degree of non-membership, under the constraint that the sum of the two degrees does not exceed one. A fuzzy set can be considered as IFS, since the sum of these grades is one. However, there are different situations when the sum of two degrees is smaller than one, which means that there is a certain ambiguity in the decision of membership or non-membership. For such cases the IFS is an appropriate tool. A generalized intuitionistic fuzzy set GIFS) were proposed by Mondal Samanta [14] under the constraint that the minimum of the two degrees does not exceed half. Following the definition of IFS, Atanassov [3] [4] Atanassov Gargov [5] introduced interval valued IFSs, IFSs of second type, temporal IFS. Srinivasan Palaniappan [19] introduced IFSs of root type. Some other extensions of the IFSs have also been introduced: IF soft sets due to Maji et al. [12]; IF rough sets due to Samanta Mondal [18]; rough IFSs due to Rizvi et al. [17]. Some recent applications of IFSs have been: sustainable energy planning in Malaysia Abdullah Najib [1]); image fusion Balasubramaniam Ananthi [6]); agricultural production planning from a small farm holder perspective Bharati Singh [7]); medical diagnosis Bora et al. [8]); pattern recognition Department of Statistics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran, Email: e baloui2008@yahoo.com School of Mathematics, University of Manchester, Manchester, UK, Email: mbbsssn2@manchester.ac.uk Corresponding Author.

2 Chu et al. [9]); reservoir flood control operation Hashemi et al. [10]); reliability optimization of complex system Mahapatra Roy [11]); fault diagnosis using dissolved gas analysis for power transformer Mani Jerome [13]); prioritizing the components of SWOT matrix in the Iranian insurance industry Nikjoo Saeedpoor [15]); prediction of the best quality of two-wheelers Pathinathan et al. [16]); study of the decision framework of wind farm project plan selection Wu et al. [21]). The aim of this paper is to introduce new generalized IFSs to derive their properties. The derived properties include: i) if A B are generalized IFSs then their union intersection are also generalized IFSs; ii) if A, B C are generalized IFSs, A is a subset of B B is a subset of C then A is a subset of C; iii) if A is a generalized IFS then its necessity possibility measures are also generalized IFSs; iv) if the degree of non-determinacy of an element of a generalized IFS is zero then that for the nth power of the set is also zero; v) if A is a generalized IFS then the necessity measure of the nth power of A is the same as the nth power of the necessity measure of A; vi) if A is a generalized IFS then the possibility measure of the nth power of A is the same as the nth power of the necessity measure of A; vii) if A is a generalized IFS m n then the mth power of A is a subset of the nth power of A; viii) if A is a generalized IFS m n then na is a subset of ma; ix) if A B are generalized IFSs A is a subset of B then na is a subset of nb; x) if A B are generalized IFSs A is a subset of B then the nth power of A is a subset of nth power of B; xi) if A B are generalized IFSs then the nth power of the union of A B is the same as the union of the nth powers of A B; xii) if A B are generalized IFSs then the nth power of the intersection of A B is the same as the intersection of the nth powers of A B; xiii) if A B are generalized IFSs then n times the union of A B is the same as the union of na nb; xiv) if A B are generalized IFSs then n times the intersection of A B is the same as the intersection of na nb. 2. Preliminaries In this section, we give some definitions of various types of IFS. We also define triangular norms triangular conorms. Let X denote a non-empty set. 1. Definition. Atanassov [2]). An IFS A in X is defined as an object of the form A x, µ A x), ν A x), where the functions µ A : X [0, 1] ν A : X [0, 1] denote, respectively, the degree of membership degree of non-membership functions of A, 0 µ A x) + ν A x) 1 for each x X. 2. Definition. Atanassov [3]). An intuitionistic fuzzy set of second type IFSST) A in X is defined as an object of the form A x, µ A x), ν A x), where the functions µ A : X [0, 1] ν A : X [0, 1] denote, respectively, the degree of membership degree of non-membership functions of A, 0 [µ A x)] 2 + [ν A x)] 2 1 for each x X. 3. Definition. Srinivasan Palaniappan [20]). An intuitionistic fuzzy set of root type IFSRT) A in X is defined as an object of the form A x, µ A x), ν A x),

3 where the functions µ A : X [0, 1] ν A : X [0, 1] denote, respectively, the degree of membership degree of non-membership functions of A, 0 1 2 µa x) + 1 2 νa x) 1 for each x X. 4. Definition. Atanassov [4]). A temporal IFS A in X is defined as an object of the form AT) {x, t), µ A x, t), ν A x, t) : x, t) E T, where the functions µ A x, t) ν A x, t) denote, respectively, the degree of membership degree of non-membership functions of A of the element x X at the time-moment t T, A E is a fixed set 0 µ A x, t) + ν A x, t) 1 for each x, t) E T. 5. Definition. A triangular norm is a binary operation on [0, 1], i.e., an operator T : [0, 1] 2 [0, 1] such that for all x, y, z [0, 1] the following conditions are satisfied: i) Communicativity: Tx, y) Ty, x), ii) Associativity: T x, Ty, z)) T Tx, y), z), iii) Monotonicity: Tx, y) Tx, z) whenever y z, iv) Boundary condition: Tx, 1) x. 6. Definition. A triangular conorm is a binary operation on [0, 1], i.e., an operator S : [0, 1] 2 [0, 1] such that for all x, y, z [0, 1] the following conditions are satisfied: i) Communicativity: Sx, y) Sy, x), ii) Associativity: S x, Ty, z)) S Tx, y), z), iii) Monotonicity: Sx, y) Sx, z) whenever y z, iv) Boundary condition: Tx, 0) x. Generalized fuzzy intuitionistic metric spaces can be defined based on triangular norms triangular conorms. 3. New generalized intuitionistic fuzzy sets 7. Definition. Let X denote a non-empty set. Our generalized IFS A in X is defined as an object of the form A x, µ A x), ν A x), where the functions µ A : X [0, 1] ν A : X [0, 1] denote, respectively, the degree of membership degree of non-membership functions of A, 0 µ A x) + ν A x) 1 for each x X n or 1 n, n 1, 2,..., N. The collection of all of our generalized IFSs is denoted by GIFS B, X). One of the geometrical interpretations of the GIFS B, X) is shown in Figures 1 2. Let X denote a universal set F a subset in the Euclidean plane with cartesian coordinates. For a GIFS B A, a function f A from X to F can be constructed such that if x X then p ν A x), µ A x)) f A x) F, 0 µ A x), ν A x) 1. Let X be a set of ages of men over [0, 75]. Let A be a set of young men whose ages are between 20 30. Define the membership non membership functions

4 Figure 1. A geometrical interpretation of GIFS B with 1 2. Figure 2. A geometrical interpretation of GIFS B with 0.5. of A as µ A x) ν A x) ) 1/2 x 10, if 10 x 20, 10 1, if 20 x 30, ) 1/2 40 x, if 30 x 40, 10 0, otherwise, ) 1/2 20 x, if 5 x 20, 15 0, if 20 x 30, ) 1/2 x 30, if 30 x 45, 15 1, otherwise.

5 Since 0 µ A x) 2 + ν A x) 2 1, x X, A x, µ A x), ν A x) is a GIFS B 2). Also, we can define the membership non membership functions of A as ) 2 x 10, if 10 x 20, µ A x) ν A x) 10 1, if 20 x 30, ) 2 40 x, if 30 x 40, 10 0, otherwise, ) 2 20 x, if 5 x 20, 15 0, if 20 x 30, ) 2 x 30, if 30 x 45, 15 1, otherwise. Since 0 µ A x) 0.5 + ν A x) 0.5 1, x X, A x, µ A x), ν A x) is a GIFS B 0.5). 3.1. Remark. It is obvious that for all real numbers α, β [0, 1], i) if 0 α + β 1 1 then we have 0 α + β 1. With this consideration if A IFS then A GIFS B. ii) if 0 α + β 1 1 then 0 α + β 1. With this consideration if A GIFS B then A IFS. iii) if 1 2 then α 2 α 1 β 2 β 1. It follows that GIFS B 1 ) GIFS B 2 ). 3.2. Remark. GIFS B 1) IFS, GIFS B 2) GIFSST, GIFS B 1 2) GIFSRT. 8. Definition. Let X denote a non-empty set. Let A B denote two GIFS B s such that A x, µ A x), ν A x) B x, µ B x), ν B x). Define the following relations operations on A B: i. A B if only if µ A x) µ B x) ν A x) ν A x), x X, ii. A B if only if µ A x) µ B x) ν A x) ν B x), x X, iii. A B x, max µ A x), µ B x)), min ν A x), ν B x)), iv. A B x, min µ A x), µ B x)), maxν A x), ν B x)), v. A + B x, µ A x) + µ B x) µ A x) µ B x), ν A x) ν B x), so 2A x, 1 1 µ A x) ) 2, νa x) 2 na x, 1 1 µ A x) ) n, νa x) n,

6 vi. A.B x, µ A x).µ B x), ν A x) + ν B x) ν A x) ν B x), so A 2 x, µ A x) 2, 1 1 ν A x) ) 2 A n x, µ A x) n, 1 1 ν A x) ) n, vii. A x, ν A x), µ A x). Proposition 3.1 For A, B, C GIFS B, we have i. A A, ii. A B, B C A C. Proof. The proof is obvious. Proposition 3.2 For A, B GIFS B, we have i. A B GIFS B, ii. A B GIFS B, iii. 1 A + B GIFS B, < 1 A + B IFS, iv. 1 A.B GIFS B, < 1 A.B IFS. Proof. i) Suppose maxµ A x), µ B x)) µ A x). Since min ν A x), ν B x)) ν A x), we have 0 µ A B x) + ν A B x) maxµ A x), µ B x))) + min ν A x), ν B x))) µ A x) + min ν A x), ν B x))) µ A x) + ν A x) 1. Suppose now maxµ A x), µ B x)) µ B x). Since min ν A x), ν B x)) ν B x), we have 0 maxµ A x), µ B x))) + min ν A x), ν B x))) The proof of i) is complete. ii) Proof of i) is similar. iii) Since µ B x) + min ν A x), ν B x))) µ B x) + ν B x) 1. A + B x, µ A x) + µ B x) µ A x) µ B x), ν A x) ν B x), we have µ A+B x) + ν A+B x) µ A x) + µ B x) µ A x) µ B x) ) + νa x) ν B x) ) µ A x) 1 µ B x) ) + µ B x) ) + νa x) ν B x) ) 0

7 µ A+B x) + ν A+B x) µ A x) + µ B x) µ A x) µ B x) ) + νa x) ν B x) ) 1 ν A x) ) + 1 ν B x) ) 1 ν A x) ) 1 ν B x) )) + ν A x) ν B x) ) 1 ν A x) ν B x) ) + νa x) ν B x) ) 1 u) + u, where u ν A x) ν B x). If 1 then 1 u) + u 1, hence A + B GIFS B. If < 1 then 1 u) + u 1, if only if ν A x) 0 or ν B x) 0. But for any, we have µ A+B x) + ν A+B x) µ A x) + µ B x) µ A x) µ B x) + ν A x) ν B x) µ A x) + µ B x) µ A x) µ B x) + 1 µ A x) ) 1 µ B x) ) 1, hence A + B IFS. The proof of iii) is complete. iv). The proof of iii) is similar. 9. Definition. The degree of non-determinacy uncertainty) of an element x X to the GIFS B A is defined by π A x) 1 µ A x) ν A x) ) 1. 3.3. Remark. It can be easily shown that π A x) + µ A x) + ν A x) 1. 10. Definition. For every GIFS B A x, µ A x), ν A x), we define the modal logic operators, the necessity measure on A the possibility measure on A, as A x, µ A x), 1 µ A x) ) 1 respectively. A x, 1 ν A x) ) 1, ν A x), 11. Definition. Let X denote a non-empty finite set. For every GIFS B as A x, µ A x), ν A x), two analogues of the topological operators, closure C) intersection I), can be defined on GIFS B s as CA) x, K, L, IA) x, k, l, K max y X µ Ay), k min y X µ Ay), L min y X ν Ay) l max y X ν Ay).

8 It is obvious that both CA) IA) are GIFS B. These two operators transform a given GIFS B to a new GIFS B. 12. Definition. Let X denote a non-empty finite set let A denote a finite GIFS B. The normalization of A denoted by NORMA) is defined by µ A x) NORMA) x, sup µ A x), ν Ax) inf ν A x) 1 inf ν A x). Proposition 3.3 Let A, B GIFS B. We have i. A GIFS B, ii. A GIFS B, iii. π A x) 0 π A nx) 0. Proof. i) Follows by noting that µ A x) + ν A x) µ A x) + Proof of ii) is similar to that of i). iii) Since we have 1 µa x) ) 1) 1. π A x) 1 µ A x) ν A x) ) 1, π A x) 0 1 µ A x) ν A x) ) 1 0 µ A x) + ν A x) 1 µ A x) 1 ν A x). By using this result, we have A n x, µ A x) n, 1 1 ν A x) ) n It is now obvious that π A nx) 0. x, µ A x) n, 1 µ A x) n. 3.4. Proposition. Let A denote a GIFS B n any positive real number. Then, the following relations are true at the extreme values of µ A x) ν A x): i. A n A) n, ii. A n A) n. Proof. i) Since A n x, µ A x) n, 1 1 ν A x) ) n, we have Also since A n A x, µ A x) n, 1 µ A x) n2)1. x, µ A x), 1 µ A x) ) 1,

9 we have A) n x, µ A x) n, 1 1 1 µ A x) )) n x, µ A x) n, 1 µ A x) n. Assume A n A) n. Consequently, we must have 1 µ A x) n2)1 1 µ A x) n, 1 µ A x) n2) 1 µ A x) n), 1 u 1 u), u 1 µ A x) n). Hence, i) is true if only if µ A x) 0 or 1, x X. ii) We know that A n x, µ A x) n, 1 1 ν A x) ) n A n x, 1 1 1 ν A x) ) ) ) n 1, 1 1 νa x) ) n. Also so A) n A x, 1 ν A x) ) 1, ν A x),, 1 1 ν A x) ) n x, 1 ν A x) ) n x, 1 ν A x) ) n, 1 1 νa x) ) n. Assume A n A) n. Consequently, we must have 1 1 1 ν A x) ) ) ) n 1 1 νa x) ) n, 1 1 ν A x) ) n ) 1 1 νa x) ) n, 1 u) 1 u, u 1 ν A x) ) n. Hence, ii) is true if only if ν A x) 0 or 1, x X. 3.5. Proposition. For every GIFS B A, we have i. m n A m A n, ii. m n na ma, iii. A n na, where m n are both positive numbers. Proof. i) Since A n x, µ A x) n, 1 1 ν A x) ) n,

10 we have A m x, µ A x) m, 1 1 ν A x) ) m. Since m n, we have µ A x) n µ A x) m, so µ A x) n µ A x) m µ A nx) µ A mx). Also since ν A x) 1, we have 1 ν A x) ) m 1 νa x) ) n, so 1 1 ν A x) ) n 1 1 νa x) ) m νa nx) ν A mx), completing the proof. The proof of ii) is similar to that of i). The proof of iii) is immediate. 3.6. Proposition. Let A, B GIFS B. We have i. A B na nb, ii. A B A n B n, iii. A B) n A n B n, iv. A B) n A n B n, v. n A B) na nb, vi. n A B) na nb. Proof. i) Since A B, we have µ A x) µ B x) µ A x) µ B x) 1 µ B x) 1 µ A x) 1 µ B x) ) n 1 µa x) ) n, so 1 1 µ A x) ) n 1 1 µb x) ) n µna x) µ mb x). Also since A B, we have ν B x) ν A x) completing the proof. ii) follows since iii) follows since ν B x) n ν A x) n ν nb x) ν na x), A B B A nb na na nb A n B n. A B x, max µ A x), µ B x)), min ν A x), ν B x)) A B) n x, max µ A x), µ B x))) n, 1 1 minν A x), ν B x)) ) n x, max µ A x) n, µ B x) n),1 1 min ν A x), ν B x) )) n x, max µ A x) n, µ B x) n),1 max 1 ν A x), 1 ν B x) )) n x, max µ A x) n, µ B x) n),1 max 1 ν A x) ) n, 1 ν B x) ) n ) x, max µ A x) n, µ B x) n),min 1 1 ν A x) ) n,1 1 ν B x) ) n ) A n B n. iv) follows since A B x, min µ A x), µ B x)), maxν A x), ν B x))

11 A B) n x, minµ A x), µ B x))) n, 1 1 maxν A x), ν B x)) ) n x, min µ A x) n, µ B x) n), 1 1 max ν A x), ν B x) )) n x, min µ A x) n, µ B x) n), 1 min 1 ν A x), 1 ν B x) )) n x, min µ A x) n, µ B x) n), 1 min 1 ν A x) ) n, 1 ν B x) ) n ) x, min µ A x) n, µ B x) n), max 1 1 ν A x) ) n,1 1 ν B x) ) n ) A n B n. v) follows since n A B) x, 1 1 max µ A x), µ B x)) ) n, minνa x), ν B x)) n x, 1 1 max µ A x), µ B x) )) n,min ν A x) n, ν B x) n) x, 1 min 1 µ A x),1 µ B x) )) n, min ν A x) n, ν B x) n) x, 1 min 1 µ A x) ) n, 1 µ B x) ) n ), min ν A x) n, ν B x) n) x, max 1 1 µ A x) ) n,1 1 µ B x) ) n ), min ν A x) n, ν B x) n) na nb. The proof of vi) is similar to that of v). 4. The operators D α A), F α,β A) G α,β A) Let A x, µ A x), ν A x) denote a GIFS B. 13. Definition. Let α [0, 1] A GIFS B. We define the operator of D α A) as D α A) x, µ A x) + απ A x) ) 1, ν A x) + 1 α)π A x) ) 1. Clearly, D α A) is a GIFS B. Theorem 4.1 For every GIFS B A for every α, β [0, 1], we have i. α β D α A) D β A), ii. D 0 A) A, iii. D 1 A) A. Proof. The proof of i) is immediate. ii) We have D 0 A) x, µ A x) + 0 π A x) ) 1, ν A x) + 1 0)π A x) ) 1 x, µ A x), ν A x) + π A x) ) 1 x, µ A x), 1 µ A x) ) 1 A, where the penultimate equality follows since π A x) 1 µ A x) ν A x). So, ii) follows.

12 iii) We have D 1 A) x, µ A x) + 1 π A x) ) 1, ν A x) + 1 1)π A x) ) 1 x, µ A x) + π A x) ) 1, ν A x) completing the proof. x, 1 ν A x) ) 1, ν A x) A, 14. Definition. Let α.β [0, 1], where α + β 1. Let A GIFS B. We define the operator of F α,β A) as F α,β A) x, µ A x) + απ A x) ) 1, ν A x) + βπ A x) ) 1. Theorem 4.2 For every GIFS B A for any α, β [0, 1], where α + β 1, we have i. F α,β A) GIFS B, ii. 0 γ α F γ,β A) F α,β A), iii. 0 γ β F α,β A) F α,γ A), iv. D α A) F α,1 α A), v. A F 0,1 A), vi. A F 1,0 A), vii. F α,β A F β,α A). Proof. i) follows since µ Fα,β A)x) + ν Fα,β A)x) [ µa x) + απ A x) ) 1 ] [ νa + x) + βπ A x) ) 1 ] µ A x) + ν A x) + π A x) α + β) µ A x) + ν A x) + π A x) 1. The proofs of ii) iii) are immediate. iv) follows since F α,1 α A) x, µ A x) + απ A x) ) 1, ν A x) + 1 α)π A x) ) 1 D α A). v) follows by Theorem 4.1 after noting that D 0 A) F 0,1 A) D 1 A) F 1,0 A) from iv). vi) follows by Theorem 4.1 after noting that D 0 A) F 0,1 A) D 1 A) F 1,0 A) from iv). vii) since F β,α A) x, µ A x) + βπ A x) ) 1, ν A x) + απ A x) ) 1 ) F α,β A x, ν A x) + απ A x) ) 1, µ A x) + βπ A x) ) 1,

13 we have ) F α,β A x, µ A x) + βπ A x) ) 1, ν A x) + απ A x) ) 1 F α,β A ) Fβ,α A). 15. Definition. Let α, β [0, 1] A GIFS B. We define the operator of G α,β A) as G α,β A) x, α 1 µa x), β 1 νa x). Theorem 4.3 For every GIFS B A, for any real numbers α, β, γ [0, 1], we have i. G α,β A) GIFS B, ii. α γ G α,β A) G γ,β A), iii. β γ G α,β A) G α,γ A), iv. τ [0, 1] G α,β G γ,τ A)) G αγ,βτ A) G γ, G α,β A)), v. G α,β CA)) C G α,β A)), vi. G α,β IA)) I G α,β A)), vii. G α,β A ) Gβ,α A). Proof. i) follows since ii) We have G α,β A) µ Gα,β A)x) + ν Gα,β A)x) G α,β A) x, α 1 µa x), β 1 νa x) ) ) α 1 µa x) + β 1 νa x) αµ A x) + βν A x) µ A x) + ν A x) 1. x, α 1 µa x), β 1 νa x) G γ,β A) x, γ 1 µa x), β 1 νa x). Since α γ, we have α 1 γ 1 so α 1 µ A x) γ 1 µ A x), completing the proof of ii). The proof of iii) is similar to that of ii). iv) We have G γ,τ A) x, γ 1 µa x), τ 1 νa x), G α,β G γ,τ A)) x, α 1 1 γ µa x), β 1 1 τ νa x) x, αγ) 1 µa x), βτ) 1 νa x) G αγ,βτ A)

14 G γ,τ G α,β A)) x, γ 1 1 α µa x), τ 1 1 β νa x) x, γα) 1 µa x), τβ) 1 νa x) x, αγ) 1 µa x), βτ) 1 νa x) G αγ,βτ A), so v) follows since G α,β G γ,τ A)) G αγ,βτ A) G γ,τ G α,β A)). CA) x, max µ Ay), min ν Ay) y X y X G α,β CA)) x, α 1 max µ Ay), β 1 min y X x, max α 1 µa y), min β 1 νa y) y X y X C G α,β A)). ν Ay) y X vi) follows since IA) x, min µ Ay), max ν Ay) y X y X G α,β IA)) x, α 1 min µ Ay), β 1 max y X x, min α 1 µa y), max β 1 νa y) y X y X I G α,β A)), ν Ay) y X where α, β [0, 1]. vii) Let A x, µ A x), ν A x) denote a GIFS B. Then, so G α,β A ) Gβ,α A). A x, ν A x), µ A x), G β,α A) x, β 1 µa x), α 1 νa x), ) G α,β A x, α 1 νa x), β 1 µa x), ) G α,β A x, β 1 µa x), α 1 νa x),

5. Conclusions We have introduced a new generalized IFS GIFS B ) as an extension to the IFS. The basic algebraic properties of GIFS B have been presented. Some operators on GIFS B are defined their relationship have been proved. A list of open problems is as follows: i) define the generalized fuzzy intuitionistic number, norms, distances, metrics, metric spaces, etc for the generalized IFS study of their properties; ii) develop statistical probabilistic tools for the generalized IFS; iii) construct an axiomatic system for the generalized IFS; iv) develop efficient algorithms computer software for the construction of degrees of membership nonmembership of a given generalized IFS; v) define study the properties of generalized IF boolean algebras; vi) develop information entropy measures corresponding to generalized IFSs; vii) develop preference theory utility theory for the generalized IFS; viii) compare with other generalizations of the IFS. Acknowledgments The authors thank the Editor the two referees for carefully reading for comments which greatly improved the paper. References [1] Abdullah, L. Najib, L. Sustainable energy planning decision using the intuitionistic fuzzy analytic hierarchy process: Choosing energy technology in Malaysia, International Journal of Sustainable Energy, doi: 10.1080/14786451.2014.907292, 2014. [2] Atanassov, K.T. Intuitionistic fuzzy sets, Fuzzy Sets Systems 20, 87-96, 1986. [3] Atanassov, K.T. Intuitionistic Fuzzy Set, Theory Applications, Springer Verlag, New York, 1999. [4] Atanassov, K.T. On the temporal intuitionistic fuzzy sets, In: Proceedings of the Ninth International Conference IPMU 2002, Volume III, Annecy, France, July 1-5, pp. 1833-1837, 2002. [5] Atanassov, K.T. Gargov, G. Interval valued intuitionistic fuzzy sets, Fuzzy Sets Systems 31, 343-349, 1989. [6] Balasubramaniam, P. Ananthi, V.P. Image fusion using intuitionistic fuzzy sets, Information Fusion 20, 21-30, 2014. [7] Bharati, K. Singh, R. Intuitionistic fuzzy optimization technique in agricultural production planning: A small farm holder perspective, International Journal of Computer Applications 89, 25-31, 2014. [8] Bora, M., Bora, B., Neog, T.J. Sut, D.K. Intuitionistic fuzzy soft matrix theory its application in medical diagnosis, Annals of Fuzzy Mathematics Informatics, 2014. [9] Chu, C.-H., Hung, K.-C. Julian, P. A complete pattern recognition approach under Atanassov s intuitionistic fuzzy sets, Knowledge-Based Systems 66, 36-45, 2014. [10] Hashemi, H., Bazargan, J., Meysam Mousavi, S. Vahdani, B. An extended compromise ratio model with an application to reservoir flood control operation under an interval-valued intuitionistic fuzzy environment, Applied Mathematical Modelling 38, 3495-3511, 2014. [11] Mahapatra, G.S. Roy, T.K. Reliability optimisation of complex system using intuitionistic fuzzy optimisation technique, International Journal of Industrial Systems Engineering 16, 279-295, 2014. [12] Maji, P.K., Biswas, R. Roy, A.R. Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics 9, 677-692, 2001. [13] Mani, G. Jerome, J. Intuitionistic fuzzy expert system based fault diagnosis using dissolved gas analysis for power transformer, Journal of Electrical Engineering Technology, 2014. 15

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