Multiattribute decision making models and methods using intuitionistic fuzzy sets

Size: px
Start display at page:

Download "Multiattribute decision making models and methods using intuitionistic fuzzy sets"

Transcription

1 Journal of Computer System Sciences 70 (2005) Multiattribute decision making models methods using intuitionistic fuzzy sets Deng-Feng Li Department Two, Dalian Naval Academy, Dalian , Liaoning, China Received 27 September 2003; received in revised form 28 June 2004 Available online 18 August 2004 Abstract The concept of intuitionistic fuzzy sets is the generalization of the concept of fuzzy sets. The theory of intuitionistic fuzzy sets is well suited to dealing with vagueness. Recently, intuitionistic fuzzy sets have been used to build soft decision making models that can accommodate imprecise information, two solution concepts about the intuitionistic fuzzy core the consensus winner for group decision-making have also been developed by other researchers using intuitionistic fuzzy sets. However, it seems that there is little investigation on multicriteria /or group decision making using intuitionistic fuzzy sets with multiple criteria being explicitly taken into account. In this paper, multiattribute decision making using intuitionistic fuzzy sets is investigated, in which multiple criteria are explicitly considered, several linear programming models are constructed to generate optimal weights for attributes, the corresponding decision-making methods have also been proposed. Feasibility effectiveness of the proposed method are illustrated using a numerical example Elsevier Inc. All rights reserved. Keywords: Fuzzy set; Intuitionistic fuzzy set; Multiattribute decision making; Linear programming model 1. Introduction The theory of fuzzy sets proposed by Zadeh [23] has attracted wide spread attentions in various fields, especially where conventional mathematical techniques are of limited effectiveness, including biological social sciences, linguistic, psychology, economics, more generally soft sciences. In such fields, variables are difficult to quantify dependencies among variables are so ill-defined addresses: lidengfeng65@hotmail.com, dengfengli@sina.com /$ - see front matter 2004 Elsevier Inc. All rights reserved. doi: /.css

2 74 D.-F. Li / Journal of Computer System Sciences 70 (2005) that precise characterization in terms of algebraic, difference or differential equations becomes almost impossible. Even in fields where dependencies between variables are well defined, it might be necessary or advantageous to employ fuzzy rather than crisp algorithms to arrive at a solution [18]. Out of several higher-order fuzzy sets, intuitionistic fuzzy sets introduced by Atanassov [1 3] have been found to be well suited to dealing with vagueness. The concept of an intuitionistic fuzzy set can be viewed as an alternative approach to define a fuzzy set in cases where available information is not sufficient for the definition of an imprecise concept by means of a conventional fuzzy set. In general, the theory of intuitionistic fuzzy sets is the generalization of fuzzy sets. Therefore, it is expected that intuitionistic fuzzy sets could be used to simulate human decision-making processes any activities requiring human expertise knowledge [12,13], which are inevitably imprecise or not totally reliable. Gau Buehrer [10] presented the concept of vague sets. Burillo Bustince [4] showed that the notion of vague sets coincides with that of intuitionistic fuzzy sets. Szmidt Kacprzyk [20] proposed a non-probabilistic type of entropy measure for intuitionistic fuzzy sets. De et al. [8] studied Sanchez s approach for medical diagnosis extended this concept with the notion of intuitionistic fuzzy set theory. Turanli Coker [22] introduced several types of fuzzy connectedness in intuitionistic fuzzy topological spaces. De et al. [7] defined some operations on intuitionistic fuzzy sets. Szmidt Kacprzyk [21] discussed distances between intuitionistic fuzzy sets. Bustince [5] presented different theorems for building intuitionistic fuzzy relations on a set with predetermined properties. Ciftcibasi Altunay [6] discussed different forms of fuzzy propositional expressions their relations. Li Cheng [15] studied similarity measures of intuitionistic fuzzy sets their application to pattern recognitions. Szmidt Kacprzyk [16 19] considered the use of intuitionistic fuzzy sets for building soft decision-making models with imprecise information, proposed two solution concepts about the intuitionistic fuzzy core the consensus winner for group decision making using intuitionistic fuzzy sets. A novel effective approach to deal with decision making in medical diagnosis using the composition of intuitionistic fuzzy relations was proposed in [9]. However, it seems that so far there has been little research on multicriteria or multiattribute in discrete decision situations /or group decision making using intuitionistic fuzzy sets, which is indeed one of the most important areas in decision analysis as most real world decision problems involve multiple criteria a group of decision makers [12,13]. In this paper, multiattribute decision making using intuitionistic fuzzy sets is investigated, in which attributes are explicitly considered, several corresponding linear programming models are constructed to generate optimal weights of attributes, the corresponding decision-making methods are also proposed. This paper is organized as follows. The definitions properties of intuitionistic fuzzy sets are briefly introduced in Section 2. Multiattribute decision-making models with intuitionistic fuzzy values are then proposed, the corresponding linear programming models methods are established in Section 3. A numerical example a short conclusion are given in Section 4 5, respectively. 2. Definitions properties of intuitionistic fuzzy sets Definition 1. (Atanassov [1 3]) Let X ={x 1,x 2,...,x n } be a finite universal set. An intuitionistic fuzzy set A in X is an obect having the following form: A ={<x, μ A (x ), υ A (x )> x X},

3 where the functions μ A : X [0, 1], x X μ A (x ) [0, 1] υ A : X [0, 1], x X υ A (x ) [0, 1] D.-F. Li / Journal of Computer System Sciences 70 (2005) define the degree of membership degree of non-membership of the element x X to the set A X, respectively, for every x X,0 μ A (x ) + υ A (x ) 1.We call π A (x ) = 1 μ A (x ) υ A (x ), the intuitionistic index of the element x in the set A. It is the degree of indeterminacy membership of the element x to the set A. It is obvious that for every x X 0 π A (x ) 1. Distance between intuitionistic fuzzy sets was first introduced by Atanassov [2]. A deeper discussion of the distance is given by Szmidt Kacprzyk [21]. Here, we introduce a normalized Hamming distance [14], which will be employed in Section 3. Let A B be two intuitionistic fuzzy sets in the set X. Namely, A ={<x, μ A (x ), υ A (x )> x X} B ={<x, μ B (x ), υ B (x )> x X}. The normalized Hamming distance between A B is defined as follows where D(A, B) = 1 2n n ( μ A (x ) μ B (x ) + υ A (x ) υ B (x ) + π A (x ) π B (x 10 ) ), (1) =1 π A (x ) = 1 μ A (x ) υ A (x ) π B (x ) = 1 μ B (x ) υ B (x ). Theorem 1. D defined by Eq. (1) is a metric. Proof. Evidently, D is symmetric D(A, A) = 0. Conversely, if D(A, B) = 0, according to Eq. (1), we must have μ A (x ) = μ B (x ), υ A (x ) = υ B (x ) π A (x ) = π B (x ) for all x X. Hence, it follows that A = B according to Definition 1. Thus D is positive definite.

4 76 D.-F. Li / Journal of Computer System Sciences 70 (2005) For any intuitionistic fuzzy sets A, B C, where C ={<x, μ C (x ), υ C (x )> x X}. Using Eq. (1), we have i.e., D(A, B)= 1 2n 1 2n n ( μ A (x ) μ B (x ) + υ A (x ) υ B (x ) + π A (x ) π B (x ) ) =1 n ( μ A (x ) μ C (x ) + υ A (x ) υ C (x ) + π A (x ) π C (x ) ) = n n ( μ C (x ) μ B (x ) + υ C (x ) υ B (x ) + π C (x ) π B (x ) ) =1 = D(A, C) + D(C, B), D(A, B) D(A, C) + D(C, B). So D is triangular. Hence, we have completed the proof of Theorem 1. If A B are conventional fuzzy sets, i.e., A ={<x, μ A (x ), 1 μ A (x )> x X} B = {<x, μ B (x ), 1 μ B (x )> x X}, D(A, B) defined by Eq. (1) becomes D(A, B) = 1 n n μ A (x ) μ B (x ). =1 If A B are crisp sets, i.e., A ={<x, μ A (x ), 1 μ A (x )> x X} B ={<x, μ B (x ), 1 μ B (x )> x X}, where { 1 if x X, μ A (x ) = 0 otherwise { 1 if x X μ B (x ) = 0 otherwise. D(A, B) is the cardinality of the symmetric difference of A B, i.e., the set-theoretic difference between their union intersection. 3. Models methods for multiattribute decision makingusingintuitionistic fuzzy values 3.1. Presentation of multiattribute decision-making problems under intuitionistic fuzzy environment Suppose there exists an alternative set X ={x 1,x 2,...,x n } which consists of n non-inferior decisionmaking alternatives from which a most preferred alternative is to be selected. Each alternative is assessed on m attributes. Denote the set of all attributes A ={a 1,a 2,...,a m }. Assume that μ i υ i are the degree of membership the degree of non-membership of the alternative x X with respect to the

5 D.-F. Li / Journal of Computer System Sciences 70 (2005) attribute a i A to the fuzzy concept excellence, respectively, where 0 μ i 1, 0 υ i 1 0 μ i + υ i 1. In other words, the evaluation of the alternative x X with respect to the attribute a i A is an intuitionistic fuzzy set. Denote X i ={<x, μ i, υ i >}. The intuitionistic indices π i = 1 μ i υ i are such that the larger π i the higher a hesitation margin of the decision maker as to the excellence of the alternative x X with respect to the attribute a i A whose intensity is given by μ i. Intuitionistic indices allow us to calculate the best final result ( the worst one) we can expect in a process leading to a final optimal decision. During the process the decision maker can change his evaluations in the following way. He can increase his evaluation by adding the value of the intuitionistic index. So in fact his evaluation lies in the closed interval [μ l i, μu i ]=[μ i, μ i + π i ], where μ l i = μ i μ u i = μ i + π i = 1 υ i. Obviously, 0 μ l i μu i 1 for all x X a i A. Similarly, assume that ρ i τ i are the degree of membership the degree of non-membership of the attribute a i A to the fuzzy concept importance, respectively, where 0 ρ i 1, 0 τ i 1 0 ρ i + τ i 1. The intuitionistic indices η i = 1 ρ i τ i are such that the larger η i the higher a hesitation margin of decision maker as to the importance of the attribute a i A whose intensity is given by ρ i. Intuitionistic indices allow us to calculate the biggest weight ( the smallest one) we can expect in a process leading to a final decision. During the process the decision maker can change his evaluating weights in the following way. He can increase his evaluating weights by adding the value of the intuitionistic index. So in fact his weight lies in the closed interval [ω l i, ωu i ]=[ρ i, ρ i + η i ], where ω l i = ρ i ω u i = ρ i + η i = 1 τ i. Obviously, 0 ω l i ωu i 1 for each attribute a i A. In addition, in this paper assume that ω l i 1 m ω u i 1 in order to find weights ω i [0, 1] (i = 1, 2,...,m) satisfying ω l i ω i ω u i m ω i = Optimization model of multiattribute decision making under intuitionistic fuzzy environment For each alternative x X, its optimal comprehensive value can be computed via the following programming { max z = m } β i ω i μ l i β i μ u i (i = 1, 2,...,m; = 1, 2,...,n), ω s.t l i ω i ω u (2) i (i = 1, 2,...,m), ω i = 1 for each = 1, 2,...,n. To solve Eq. (2), we can solve the following two linear programmings min{z l = m μ l i ω i} ω l i ω i ω u i (i = 1, 2,...,m) ω i = 1 (3)

6 78 D.-F. Li / Journal of Computer System Sciences 70 (2005) m max{z u = μ u i ω i} ω l i ω i ω u i ω i = 1 (i = 1, 2,...,m), for each = 1, 2,...,n. Solving Eqs. (3) (4) by Simplex method, we can obtain their optimal solutions ω = ( ω 1, ω 2,..., ω m) T ω =( ω 1, ω 2,..., ω m )T ( =1, 2,...,n), respectively. In total, 2n linear programmings need to be solved since there are n alternatives in the set X. After generating the corresponding optimal weight vectors, the optimal comprehensive value of alternative x X can be computed as an interval [ z l, zu ], where m z l = m μ l i ω i = μ i ω i m z u = μ u i ω m i = 1 υ i ω i for each = 1, 2,...,n. That is, the optimal comprehensive value of the alternative x X is an intuitionistic fuzzy set Ā ={<x, z l, 1 zu {<x >}=, μ i ω m } i, υ i ω i >. (7) However, optimal solutions of Eqs. (3) (4) are different in general, i.e., the weight vectors ω = ω for all alternatives x X, or ω i = ω i for all i = 1, 2,...,m = 1, 2,...,n. Therefore, the comprehensive values of all n alternatives x X can not be compared. Since X is a non-inferior alternative set, there exists no evident preference on some alternatives. Hence, for each alternative x X, its obective function z l in Eq. (3) should be assigned a equal weight 1/n. Eq. (3) is then aggregated into the following linear programming: n μ l i ω i min z0 l = =1 n ω l i ω i ω u i (i = 1, 2,...,m), ω i = 1. (8) (4) (5) (6)

7 D.-F. Li / Journal of Computer System Sciences 70 (2005) In a similar way, Eq. (4) is aggregated into the following linear programming n μ u i ω i max z0 u = =1 n ω l i ω i ω u i (i = 1, 2,,m) ω i = 1 Solving Eqs. (8) (9) by Simplex method, we can obtain their optimal solutions (9) ω 0 = ( ω 0 1, ω0 2,..., ω0 m )T ω 0 = ( ω 0 1, ω 0 2,..., ω 0 m )T, respectively. After generating the corresponding optimal weight vectors, the optimal comprehensive value of the alternative x X can be computed as an interval [ z l, z u ], where z l m = m μ l i ω0 i = μ i ω 0 i z u m = μ u i ω 0 m i = 1 υ i ω 0 i (11) for each = 1, 2,...,n. That is, the optimal comprehensive value of the alternative x X is an intuitionistic fuzzy set given by { } Ā = {<x, z l, 1 z u m } > = <x, μ i ω 0 i, υ i ω 0 i >. (12) In generating the above-intuitionistic fuzzy set only two linear programmings (i.e. Eqs. (8) (9)) need to be solved. However, the optimal solutions of Eqs. (8) (9) are normally different, so ω 0 = ω 0 in general, or ω 0 i = ω 0 i for all i = 1, 2,...,m. Therefore, it is possible that z l > z u. If this is the case, it follows that the intuitionistic index is negative, or π = 1 z l (1 z u ) = z u z l < 0. However, this is not permitted by Definition 1. (10)

8 80 D.-F. Li / Journal of Computer System Sciences 70 (2005) Note that Eq. (8) is equivalent to the following linear programming n μ l i ω i max z 0 l = =1 n ω l i ω i ω u i (i = 1, 2,...,m), ω i = 1. Since Eqs. (9) (13) have the same constraints, they can be combined to formulate the following linear programming n (μ u i μl i )ω i =1 max z = n ω l i ω i ω u i (i = 1, 2,, m), (14) ω i = 1. Normally, Eqs. (9) (13) are not equivalent to Eq. (14). However, Some of solutions of Eqs. (9) (13) can be generated by solving Eq. (14). Eq. (14) can be rewritten as follows n π i ω i =1 max z = n ω l i ω i ω u i (i = 1, 2,, m), (15) ω i = 1. The optimal solution ω 0 = (ω 0 1, ω0 2,...,ω0 m )T can be obtained solving Eq. (14) or Eq. (15) by Simplex method. Then, the optimal comprehensive value of the alternative x X can be computed as an interval ], where [z 0l,z0u (13) z 0l m = m μ l i ω0 i = μ i ω 0 i z 0u = m μ u i ω0 i = 1 υ i ω 0 i (17) (16)

9 D.-F. Li / Journal of Computer System Sciences 70 (2005) for each = 1, 2,...,n. That is, the optimal comprehensive value of the alternative x X is an intuitionistic fuzzy set given by { m } A 0 ={<x,z 0l, 1 z0u >}= <x, μ i ω 0 i, υ i ω 0 i >. (18) Theorem 2. Suppose intuitionistic fuzzy sets Ā A 0 Then for each alternative x X [ z l, zu ] [z0l,z0u ]. are defined by Eqs. (7) (18), respectively. (19) Proof. Since ω 0 = (ω 0 1, ω0 2,...,ω0 m )T is an optimal solution of Eq. (14) or Eq. (15), it is a feasible solution of Eq. (3). Note that ω = ( ω 1, ω 2,..., ω m) T is an optimal solution of Eq. (3). Then, according to Eqs. (3) (5), we have m z l = m μ l i ω i μ l i ω0 i = z0l (20) for each alternative x X. In a similar way, ω = ( ω 1, ω 2,, ω m )T is an optimal solution of Eq. (4). Then, according to Eqs. (4) (6), we have m z u = μ u i ω m i μ u i ω0 i = z0u (21) for each alternative x X. Obviously, we have z 0u = m μ u i ω0 i μ l i ω0 i = z0l (22) for each alternative x X. Then, combining Eqs. (20) (21) with Eq. (22), we have z u z0u z0l zl. Therefore, it follows that for each alternative x X. [ z l, zu ] [z0l,z0u ] Multiattribute decision-making method under an intuitionistic fuzzy environment Using the above Eq. (14) or Eq. (15), n optimal comprehensive values A 0 of all alternatives x X( = 1, 2,...,n) can be obtained. Now, we are interested in how a final best compromise alternative or the final ranking order of the alternative set X can be generated.

10 82 D.-F. Li / Journal of Computer System Sciences 70 (2005) In a similar way to the TOPSIS method proposed by Hwang Yoon [11], we define the following index for each alternative x X D(A 0 ξ =,B) D(A 0,B)+ D(A0,G), (23) where A 0 ={<x,z 0l, 1 z0u >}={<x, m μ i ω 0 i, m υ i ω 0 i >} given by Eq. (18) is an intuitionistic fuzzy set corresponding to the optimal comprehensive value of the alternative x X. G ={<g, 1, 0 >} is an intuitionistic fuzzy set corresponding to the evaluation of the ideal alternative g. B ={<b,0, 1 >} is an intuitionistic fuzzy set corresponding to the evaluation of the negative ideal alternative b. Obviously, normally g/ X b/ X. D(A 0,B)is a distance measure between the intuitionistic fuzzy sets A0 B. D(A 0,G)is a distance measure between the intuitionistic fuzzy sets A0 G. There are several distance formulae between intuitionistic fuzzy sets [3]. In this paper, we choose the distance formula given by Eq. (1) in Section 2. Obviously, for each alternative x X,wehave 0 ξ 1 Furthermore, ξ = 0ifA 0 = B (or x is the negative ideal alternative b); ξ = 1if(A 0 = Gor x is the ideal alternative g). It is easy to see that the higher ξ the better the alternative x. According to Eq. (1), D(A 0,B) D(A0,G)are reduced into the following formulae D(A 0,B)= z0l z0u = z0l + z0u + (z 0u z 0l) 2 = z 0u z 0l 2 (1 z0u ) 0 (24) D(A 0,G)= z0l = z0u z 0l 2 (1 z0l ) + (1 z0u ) + (z0 z0l ) 2 = 1 z 0l. (1 z0u ) (25) Hence, Eq. (23) can be simply written as follows ξ = z 0u 1 + z 0u z 0l. (26)

11 D.-F. Li / Journal of Computer System Sciences 70 (2005) From Eqs. (16) (17), Eq. (26) can also be written as follows μ u i ω0 i ξ = 1 + m (μ u i μl i )ω0 i 1 m υ i ω 0 i = 1 + (1 m υ i ω 0 i ) m μ i ω 0 i 1 m υ i ω 0 i = 2 m. (μ i + υ i )ω 0 i Thus, the best alternative x X can be generated so that ξ = max{ξ x X} (27) (28) the alternatives are ranked according to the increasing order of ξ for all x X. 4. An numerical example Consider an air-condition system selection problem. Suppose there exist three air-condition systems x 1, x 2 x 3 Denote the alternative set by X ={x 1,x 2,x 3 }. Suppose three attributes a 1 (economical), a 2 (function) a 3 (being operative) are taken into consideration in the selection problem. Denote the set of all attributes by A ={a 1,a 2,a 3 }. Using statistical methods, the degrees μ i of membership the degrees υ i of non-membership for the alternative x X with respect to the attribute a i A to the fuzzy concept excellence can be obtained, respectively. Namely, x 1 x 2 x 3 ((μ i, υ i )) 3 3 = a 1 a 2 a 3 ((μ i, υ i )) 3 3 = a 1 a 2 a 3 (0.75, 0.10) (0.80, 0.15) (0.40, 0.45), (0.60, 0.25) (0.68, 0.20) (0.75, 0.05) (0.80, 0.20) (0.45, 0.50) (0.60, 0.30) x 1 x 2 x 3 [0.75, 0.90] [0.80, 0.85] [0.40, 0.55], [0.60, 0.75] [0.68, 0.80] [0.75, 0.95] [0.80, 0.80] [0.45, 0.50] [0.60, 0.70] In a similar way, the degrees ρ i of membership the degrees τ i of non-membership for the three attributes a i A to the fuzzy concept importance can be obtained, respectively. Namely, a 1 a 2 a 3 ((ρ i, τ i )) 1 3 =((0.25, 0.25) (0.35, 0.40) (0.30, 0.65)).

12 84 D.-F. Li / Journal of Computer System Sciences 70 (2005) Therefore, attribute weights lie in the closed interval as follows, a 1 a 2 a 3 ([ω l i, ωu i ]) 1 3 = ([0.25, 0.75] [0.35, 0.60] [0.30, 0.35]). According to Eq. (14) or Eq. (15), the following linear programming can be obtained { max z = 0.35ω } ω ω 3, ω , 0.35 ω , 0.30 ω , ω 1 + ω 2 + ω 3 = 1. Solving the above linear programming, its optimal solution can be obtained as follows ω 0 = (ω 0 1, ω0 2, ω0 3 )T = (0.25, 0.40, 0.35) T. Using Eqs. (16) (17), the optimal comprehensive value of the alternative x X can be computed as follows: z 0l 1 z 0u 1 = , z0l 2 = , z0u 2 = , z0l 3 = = , z0u 3 = Thus, the optimal comprehensive value of the alternative x X can be expressed as an intuitionistic fuzzy set A 0 1 ={<x 1, , >},A 0 2 ={<x 2, , >}, A 0 3 ={<x 3, 0.610, >}, respectively. For alternatives x 1, x 2 x 3, the following index for each alternative can be generated using Eq. (26): ξ 1 = ξ 2 = ξ 3 = z1 0u z1 0u = z0l = , 1 z2 0u z2 0u = z0l = , 2 z3 0u z3 0u = z0l = Then, the best alternative is x 1. The optimal ranking order of the alternatives is given by x 1 x 3 x 2. (29) 5. Conclusions In the above analysis, we have proposed several linear programming models methods for multiattribute decision making under intuitionistic fuzziness. In such decision situations, attributes are

13 D.-F. Li / Journal of Computer System Sciences 70 (2005) explicitly considered are not compound, which differ from of the ways used by Szmidt Kaeprzyk [16,17,19]. Moreover, the evaluations of each alternative with respect to each attribute on a fuzzy concept excellence are given using intuitionistic fuzzy sets, the weights of each attribute are also given using intuitionistic fuzzy sets. This allows us to use flexible ways to simulate real decision situations, thereby building more realistic scenarios describing possible future events. In conclusion, multiattribute decision-making models using intuitionistic fuzzy sets can represent a wide spectrum of possibilities, which enables the explicit consideration of the best the worst results one can expect. Acknowledgements The author thank the valuable reviews also appreciate the constructive suggestions from the anonymous referees. References [1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Systems 20 (1986) [2] K.T. Atanassov, Intuitionistic Fuzzy Sets, Springer, Heidelberg, [3] K.T. Atanassov, Two theorems for intuitionistic fuzzy sets, Fuzzy Sets Systems 110 (2000) [4] P. Burillo, H. Bustince, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets Systems 79 (1996) [5] H. Bustince, Construction of intuitionistic fuzzy relations with predetermined properties, Fuzzy Sets Systems 109 (2000) [6] T. Ciftcibasi, D. Altunay, Two-sided (intuitionistic) fuzzy reasoning, IEEE Trans. Systems Man Cybernet A-28 (1998) [7] S.K. De, R. Biswas, A.R. Roy, Some operations on intuitionistic fuzzy sets, Fuzzy Sets Systems 114 (2000) [8] S.K. De, R. Biswas, A.R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets Systems 117 (2001) [9] G. Deschriver, E. E. Kerre, On the composition of intuitionistic Fuzzy Relations, Fuzzy Sets Systems, in press. [10] W.L. Gau, D.J. Buehrer, Vague sets IEEE Trans, Systems Man Cybernet 23 (1993) [11] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making: Methods Applications, A State of the Art Survey, Springer, Berlin, [12] L.I. Dengfeng, Fuzzy multiattribute decision making models methods with incomplete information. Fuzzy Sets Systems 106 (2) (1999) [13] L.I. Dengfeng, Fuzzy Multiobective Many-Person Decision Makings Games, National Defense Industry Press, Beiing, [14] L.I. Dengfeng, Some measures of dissimilarity in intuitionistic fuzzy structures, J. of Comput. System Sci. in press. [15] L.I. Dengfeng, Chuntian Cheng. New similarity measures of intuitionistic fuzzy sets application to pattern recognitions. Pattern Recognition Lett. 23 (1 3) (2002) [16] E. Szmidt, J. Kacprzyk, Intuitionistic fuzzy sets in group decision making, NIFS 2 (1) (1996) [17] E. Szmidt, J. Kacprzyk, Remarks on some applications of intuitionistic fuzzy sets in decision making, NIFS 2 (3) (1996) [18] E. Szmidt, J. Kacprzyk, Group decision making via intuitionistic fuzzy sets, FUBEST 96, Sofia, Bulgaria, October 9 11, 1996, pp [19] E. Szmidt, J. Kacprzyk, Intuitionistic fuzzy sets for more realistic group decision making, International Conference on Transition to Advanced Market Institutions Economies, Warsaw, June 18 21, 1997, pp [20] E. Szmidt, J. Kacprzyk, Entropy for intuitionistic fuzzy sets, Fuzzy Sets Systems 118 (2001) [21] E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets Systems 114 (2001) [22] N. Turanli, D. Coker, Fuzzy connectedness in intuitionistic fuzzy topological spaces, Fuzzy Sets Systems 116 (2000) [23] L.A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965)

On properties of four IFS operators

On properties of four IFS operators Fuzzy Sets Systems 15 (2005) 151 155 www.elsevier.com/locate/fss On properties of four IFS operators Deng-Feng Li a,b,c,, Feng Shan a, Chun-Tian Cheng c a Department of Sciences, Shenyang Institute of

More information

ATANASSOV S INTUITIONISTIC FUZZY SET THEORY APPLIED TO QUANTALES

ATANASSOV S INTUITIONISTIC FUZZY SET THEORY APPLIED TO QUANTALES Novi Sad J. Math. Vol. 47, No. 2, 2017, 47-61 ATANASSOV S INTUITIONISTIC FUZZY SET THEORY APPLIED TO QUANTALES Bijan Davvaz 1, Asghar Khan 23 Mohsin Khan 4 Abstract. The main goal of this paper is to study

More information

Entropy for intuitionistic fuzzy sets

Entropy for intuitionistic fuzzy sets Fuzzy Sets and Systems 118 (2001) 467 477 www.elsevier.com/locate/fss Entropy for intuitionistic fuzzy sets Eulalia Szmidt, Janusz Kacprzyk Systems Research Institute, Polish Academy of Sciences ul. Newelska

More information

Generalized Triangular Fuzzy Numbers In Intuitionistic Fuzzy Environment

Generalized Triangular Fuzzy Numbers In Intuitionistic Fuzzy Environment International Journal of Engineering Research Development e-issn: 2278-067X, p-issn : 2278-800X, www.ijerd.com Volume 5, Issue 1 (November 2012), PP. 08-13 Generalized Triangular Fuzzy Numbers In Intuitionistic

More information

A note on the Hausdorff distance between Atanassov s intuitionistic fuzzy sets

A note on the Hausdorff distance between Atanassov s intuitionistic fuzzy sets NIFS Vol. 15 (2009), No. 1, 1 12 A note on the Hausdorff distance between Atanassov s intuitionistic fuzzy sets Eulalia Szmidt and Janusz Kacprzyk Systems Research Institute, Polish Academy of Sciences

More information

Maejo International Journal of Science and Technology

Maejo International Journal of Science and Technology Full Paper Maejo International Journal of Science and Technology ISSN 1905-7873 Available online at www.mijst.mju.ac.th Similarity measures between temporal intuitionistic fuzzy sets Omar H. Khalil 1,

More information

Generalized Entropy for Intuitionistic Fuzzy Sets

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

More information

Cross-entropy measure on interval neutrosophic sets and its applications in Multicriteria decision making

Cross-entropy measure on interval neutrosophic sets and its applications in Multicriteria decision making Manuscript Click here to download Manuscript: Cross-entropy measure on interval neutrosophic sets and its application in MCDM.pdf 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 Cross-entropy measure on interval neutrosophic

More information

AN INTERVAL-VALUED PROGRAMMING APPROACH TO MATRIX GAMES WITH PAYOFFS OF TRIANGULAR INTUITIONISTIC FUZZY NUMBERS

AN INTERVAL-VALUED PROGRAMMING APPROACH TO MATRIX GAMES WITH PAYOFFS OF TRIANGULAR INTUITIONISTIC FUZZY NUMBERS Iranian Journal of Fuzzy Systems Vol. 11, No. 2, (2014) pp. 45-57 45 AN INTERVA-VAUED PROGRAMMING APPROACH TO MATRIX GAMES WITH PAYOFFS OF TRIANGUAR INTUITIONISTIC FUZZY NUMBERS D. F. I AND J. X. NAN Abstract.

More information

Soft Matrices. Sanjib Mondal, Madhumangal Pal

Soft Matrices. Sanjib Mondal, Madhumangal Pal Journal of Uncertain Systems Vol7, No4, pp254-264, 2013 Online at: wwwjusorguk Soft Matrices Sanjib Mondal, Madhumangal Pal Department of Applied Mathematics with Oceanology and Computer Programming Vidyasagar

More information

International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March ISSN

International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March ISSN International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 969 Soft Generalized Separation Axioms in Soft Generalized Topological Spaces Jyothis Thomas and Sunil Jacob John

More information

Group Decision Making Using Comparative Linguistic Expression Based on Hesitant Intuitionistic Fuzzy Sets

Group Decision Making Using Comparative Linguistic Expression Based on Hesitant Intuitionistic Fuzzy Sets Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 932-9466 Vol. 0, Issue 2 December 205), pp. 082 092 Applications and Applied Mathematics: An International Journal AAM) Group Decision Making Using

More information

Correlation Coefficient of Interval Neutrosophic Set

Correlation Coefficient of Interval Neutrosophic Set Applied Mechanics and Materials Online: 2013-10-31 ISSN: 1662-7482, Vol. 436, pp 511-517 doi:10.4028/www.scientific.net/amm.436.511 2013 Trans Tech Publications, Switzerland Correlation Coefficient of

More information

On some ways of determining membership and non-membership functions characterizing intuitionistic fuzzy sets

On some ways of determining membership and non-membership functions characterizing intuitionistic fuzzy sets Sixth International Workshop on IFSs Banska Bystrica, Slovakia, 10 Oct. 2010 NIFS 16 (2010), 4, 26-30 On some ways of determining membership and non-membership functions characterizing intuitionistic fuzzy

More information

Ranking of Intuitionistic Fuzzy Numbers by New Distance Measure

Ranking of Intuitionistic Fuzzy Numbers by New Distance Measure Ranking of Intuitionistic Fuzzy Numbers by New Distance Measure arxiv:141.7155v1 [math.gm] 7 Oct 14 Debaroti Das 1,P.K.De 1, Department of Mathematics NIT Silchar,7881, Assam, India Email: deboritadas1988@gmail.com

More information

A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions

A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions Jian Wu a,b, Francisco Chiclana b a School of conomics

More information

ISSN: Received: Year: 2018, Number: 24, Pages: Novel Concept of Cubic Picture Fuzzy Sets

ISSN: Received: Year: 2018, Number: 24, Pages: Novel Concept of Cubic Picture Fuzzy Sets http://www.newtheory.org ISSN: 2149-1402 Received: 09.07.2018 Year: 2018, Number: 24, Pages: 59-72 Published: 22.09.2018 Original Article Novel Concept of Cubic Picture Fuzzy Sets Shahzaib Ashraf * Saleem

More information

NEUTROSOPHIC VAGUE SOFT EXPERT SET THEORY

NEUTROSOPHIC VAGUE SOFT EXPERT SET THEORY NEUTROSOPHIC VAGUE SOFT EXPERT SET THEORY Ashraf Al-Quran a Nasruddin Hassan a1 and Florentin Smarandache b a School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia

More information

Index Terms Vague Logic, Linguistic Variable, Approximate Reasoning (AR), GMP and GMT

Index Terms Vague Logic, Linguistic Variable, Approximate Reasoning (AR), GMP and GMT International Journal of Computer Science and Telecommunications [Volume 2, Issue 9, December 2011] 17 Vague Logic in Approximate Reasoning ISSN 2047-3338 Supriya Raheja, Reena Dadhich and Smita Rajpal

More information

Intuitionistic Fuzzy Sets - An Alternative Look

Intuitionistic Fuzzy Sets - An Alternative Look Intuitionistic Fuzzy Sets - An Alternative Look Anna Pankowska and Maciej Wygralak Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87, 61-614 Poznań, Poland e-mail: wygralak@math.amu.edu.pl

More information

PYTHAGOREAN FUZZY INDUCED GENERALIZED OWA OPERATOR AND ITS APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING

PYTHAGOREAN FUZZY INDUCED GENERALIZED OWA OPERATOR AND ITS APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING International Journal of Innovative Computing, Information and Control ICIC International c 2017 ISSN 1349-4198 Volume 13, Number 5, October 2017 pp. 1527 1536 PYTHAGOREAN FUZZY INDUCED GENERALIZED OWA

More information

Intuitionistic Fuzzy Sets: Spherical Representation and Distances

Intuitionistic Fuzzy Sets: Spherical Representation and Distances Intuitionistic Fuzzy Sets: Spherical Representation and Distances Y. Yang, F. Chiclana Centre for Computational Intelligence, School of Computing, De Montfort University, Leicester LE1 9BH, UK Most existing

More information

Intuitionistic Fuzzy Sets: Spherical Representation and Distances

Intuitionistic Fuzzy Sets: Spherical Representation and Distances Intuitionistic Fuzzy Sets: Spherical Representation and Distances Y. Yang, F. Chiclana Abstract Most existing distances between intuitionistic fuzzy sets are defined in linear plane representations in

More information

Some aspects on hesitant fuzzy soft set

Some aspects on hesitant fuzzy soft set Borah & Hazarika Cogent Mathematics (2016 3: 1223951 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Some aspects on hesitant fuzzy soft set Manash Jyoti Borah 1 and Bipan Hazarika 2 * Received:

More information

Intuitionistic random multi-criteria decision-making approach based on prospect theory with multiple reference intervals

Intuitionistic random multi-criteria decision-making approach based on prospect theory with multiple reference intervals Scientia Iranica E (014) 1(6), 347{359 Sharif University of Technology Scientia Iranica Transactions E: Industrial Engineering www.scientiairanica.com Intuitionistic random multi-criteria decision-making

More information

Adrian I. Ban and Delia A. Tuşe

Adrian I. Ban and Delia A. Tuşe 18 th Int. Conf. on IFSs, Sofia, 10 11 May 2014 Notes on Intuitionistic Fuzzy Sets ISSN 1310 4926 Vol. 20, 2014, No. 2, 43 51 Trapezoidal/triangular intuitionistic fuzzy numbers versus interval-valued

More information

New Similarity Measures for Intuitionistic Fuzzy Sets

New Similarity Measures for Intuitionistic Fuzzy Sets Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2239-2250 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43171 New Similarity Measures for Intuitionistic Fuzzy Sets Peerasak Intarapaiboon

More information

ON INTUITIONISTIC FUZZY SOFT TOPOLOGICAL SPACES. 1. Introduction

ON INTUITIONISTIC FUZZY SOFT TOPOLOGICAL SPACES. 1. Introduction TWMS J. Pure Appl. Math. V.5 N.1 2014 pp.66-79 ON INTUITIONISTIC FUZZY SOFT TOPOLOGICAL SPACES SADI BAYRAMOV 1 CIGDEM GUNDUZ ARAS) 2 Abstract. In this paper we introduce some important properties of intuitionistic

More information

A New Fuzzy Positive and Negative Ideal Solution for Fuzzy TOPSIS

A New Fuzzy Positive and Negative Ideal Solution for Fuzzy TOPSIS A New Fuzzy Positive and Negative Ideal Solution for Fuzzy TOPSIS MEHDI AMIRI-AREF, NIKBAKHSH JAVADIAN, MOHAMMAD KAZEMI Department of Industrial Engineering Mazandaran University of Science & Technology

More information

The weighted distance measure based method to neutrosophic multi-attribute group decision making

The weighted distance measure based method to neutrosophic multi-attribute group decision making The weighted distance measure based method to neutrosophic multi-attribute group decision making Chunfang Liu 1,2, YueSheng Luo 1,3 1 College of Science, Northeast Forestry University, 150040, Harbin,

More information

Intuitionistic Hesitant Fuzzy VIKOR method for Multi-Criteria Group Decision Making

Intuitionistic Hesitant Fuzzy VIKOR method for Multi-Criteria Group Decision Making Inter national Journal of Pure and Applied Mathematics Volume 113 No. 9 2017, 102 112 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Intuitionistic

More information

Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment

Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment International Journal of General Systems, 2013 Vol. 42, No. 4, 386 394, http://dx.doi.org/10.1080/03081079.2012.761609 Multicriteria decision-making method using the correlation coefficient under single-valued

More information

Credibilistic Bi-Matrix Game

Credibilistic Bi-Matrix Game Journal of Uncertain Systems Vol.6, No.1, pp.71-80, 2012 Online at: www.jus.org.uk Credibilistic Bi-Matrix Game Prasanta Mula 1, Sankar Kumar Roy 2, 1 ISRO Satellite Centre, Old Airport Road, Vimanapura

More information

WEIGHTED NEUTROSOPHIC SOFT SETS APPROACH IN A MULTI- CRITERIA DECISION MAKING PROBLEM

WEIGHTED NEUTROSOPHIC SOFT SETS APPROACH IN A MULTI- CRITERIA DECISION MAKING PROBLEM http://www.newtheory.org ISSN: 2149-1402 Received: 08.01.2015 Accepted: 12.05.2015 Year: 2015, Number: 5, Pages: 1-12 Original Article * WEIGHTED NEUTROSOPHIC SOFT SETS APPROACH IN A MULTI- CRITERIA DECISION

More information

NEUTROSOPHIC PARAMETRIZED SOFT SET THEORY AND ITS DECISION MAKING

NEUTROSOPHIC PARAMETRIZED SOFT SET THEORY AND ITS DECISION MAKING italian journal of pure and applied mathematics n. 32 2014 (503 514) 503 NEUTROSOPHIC PARAMETRIZED SOFT SET THEORY AND ITS DECISION MAING Said Broumi Faculty of Arts and Humanities Hay El Baraka Ben M

More information

A New Approach for Optimization of Real Life Transportation Problem in Neutrosophic Environment

A New Approach for Optimization of Real Life Transportation Problem in Neutrosophic Environment 1 A New Approach for Optimization of Real Life Transportation Problem in Neutrosophic Environment A.Thamaraiselvi 1, R.Santhi 2 Department of Mathematics, NGM College, Pollachi, Tamil Nadu-642001, India

More information

Intuitionistic Fuzzy Logic Control for Washing Machines

Intuitionistic Fuzzy Logic Control for Washing Machines Indian Journal of Science and Technology, Vol 7(5), 654 661, May 2014 ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645 Intuitionistic Fuzzy Logic Control for Washing Machines Muhammad Akram *, Shaista

More information

Intuitionistic Fuzzy Metric Groups

Intuitionistic Fuzzy Metric Groups 454 International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012 Intuitionistic Fuzzy Metric Groups Banu Pazar Varol and Halis Aygün Abstract 1 The aim of this paper is to introduce the structure

More information

Towards Decision Making under Interval Uncertainty

Towards Decision Making under Interval Uncertainty Journal of Uncertain Systems Vol.1, No.3, pp.00-07, 018 Online at: www.us.org.uk Towards Decision Making under Interval Uncertainty Andrze Pownuk, Vladik Kreinovich Computational Science Program, University

More information

THE notion of fuzzy groups defined by A. Rosenfeld[13]

THE notion of fuzzy groups defined by A. Rosenfeld[13] I-Vague Groups Zelalem Teshome Wale Abstract The notions of I-vague groups with membership and non-membership functions taking values in an involutary dually residuated lattice ordered semigroup are introduced

More information

ROUGH NEUTROSOPHIC SETS. Said Broumi. Florentin Smarandache. Mamoni Dhar. 1. Introduction

ROUGH NEUTROSOPHIC SETS. Said Broumi. Florentin Smarandache. Mamoni Dhar. 1. Introduction italian journal of pure and applied mathematics n. 32 2014 (493 502) 493 ROUGH NEUTROSOPHIC SETS Said Broumi Faculty of Arts and Humanities Hay El Baraka Ben M sik Casablanca B.P. 7951 Hassan II University

More information

International Journal of Mathematics Trends and Technology (IJMTT) Volume 51 Number 5 November 2017

International Journal of Mathematics Trends and Technology (IJMTT) Volume 51 Number 5 November 2017 A Case Study of Multi Attribute Decision Making Problem for Solving Intuitionistic Fuzzy Soft Matrix in Medical Diagnosis R.Rathika 1 *, S.Subramanian 2 1 Research scholar, Department of Mathematics, Prist

More information

Uncertain Logic with Multiple Predicates

Uncertain Logic with Multiple Predicates Uncertain Logic with Multiple Predicates Kai Yao, Zixiong Peng Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 100084, China yaok09@mails.tsinghua.edu.cn,

More information

New Results of Intuitionistic Fuzzy Soft Set

New Results of Intuitionistic Fuzzy Soft Set New Results of Intuitionistic Fuzzy Soft Set Said Broumi Florentin Smarandache Mamoni Dhar Pinaki Majumdar Abstract In this paper, three new operations are introduced on intuitionistic fuzzy soft sets.they

More information

Multi-attribute Group decision Making Based on Expected Value of Neutrosophic Trapezoidal Numbers

Multi-attribute Group decision Making Based on Expected Value of Neutrosophic Trapezoidal Numbers New Trends in Neutrosophic Theory and Applications. Volume II Multi-attribute Group decision Making Based on Expected Value of Neutrosophic Trapezoidal Numbers Pranab Biswas 1 Surapati Pramanik 2 Bibhas

More information

The underlying structure in Atanassov s IFS

The underlying structure in Atanassov s IFS The underlying structure in Atanassov s IFS J. Montero Facultad de Matemáticas Universidad Complutense Madrid 28040, Spain e-mail: monty@mat.ucm.es D. Gómez Escuela de Estadística Universidad Complutense

More information

On flexible database querying via extensions to fuzzy sets

On flexible database querying via extensions to fuzzy sets On flexible database querying via extensions to fuzzy sets Guy de Tré, Rita de Caluwe Computer Science Laboratory Ghent University Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium {guy.detre,rita.decaluwe}@ugent.be

More information

AN INTRODUCTION TO FUZZY SOFT TOPOLOGICAL SPACES

AN INTRODUCTION TO FUZZY SOFT TOPOLOGICAL SPACES Hacettepe Journal of Mathematics and Statistics Volume 43 (2) (2014), 193 204 AN INTRODUCTION TO FUZZY SOFT TOPOLOGICAL SPACES Abdülkadir Aygünoǧlu Vildan Çetkin Halis Aygün Abstract The aim of this study

More information

Comparison of Fuzzy Operators for IF-Inference Systems of Takagi-Sugeno Type in Ozone Prediction

Comparison of Fuzzy Operators for IF-Inference Systems of Takagi-Sugeno Type in Ozone Prediction Comparison of Fuzzy Operators for IF-Inference Systems of Takagi-Sugeno Type in Ozone Prediction Vladimír Olej and Petr Hájek Institute of System Engineering and Informatics, Faculty of Economics and Administration,

More information

Chapter 6. Intuitionistic Fuzzy PROMETHEE Technique. AIDS stands for acquired immunodeficiency syndrome. AIDS is the final stage. 6.

Chapter 6. Intuitionistic Fuzzy PROMETHEE Technique. AIDS stands for acquired immunodeficiency syndrome. AIDS is the final stage. 6. Chapter 6 Intuitionistic Fuzzy PROMETHEE Technique 6.1 Introduction AIDS stands for acquired immunodeficiency syndrome. AIDS is the final stage of HIV infection, and not everyone who has HIV advances to

More information

Group Decision-Making with Incomplete Fuzzy Linguistic Preference Relations

Group Decision-Making with Incomplete Fuzzy Linguistic Preference Relations Group Decision-Making with Incomplete Fuzzy Linguistic Preference Relations S. Alonso Department of Software Engineering University of Granada, 18071, Granada, Spain; salonso@decsai.ugr.es, F.J. Cabrerizo

More information

On Intuitionistic Fuzzy Entropy as Cost Function in Image Denoising

On Intuitionistic Fuzzy Entropy as Cost Function in Image Denoising International Journal of Applied Information Systems (IJAIS) ISSN : 49-0868 Volume 7 -. 5, July 014 - www.ijais.org On Intuitionistic Fuzzy Entropy as Cost Function in Image Denoising Rajeev Kaushik Research

More information

An Analysis on Consensus Measures in Group Decision Making

An Analysis on Consensus Measures in Group Decision Making An Analysis on Consensus Measures in Group Decision Making M. J. del Moral Statistics and Operational Research Email: delmoral@ugr.es F. Chiclana CCI Faculty of Technology De Montfort University Leicester

More information

Normalized Hamming Similarity Measure for Intuitionistic Fuzzy Multi Sets and Its Application in Medical diagnosis

Normalized Hamming Similarity Measure for Intuitionistic Fuzzy Multi Sets and Its Application in Medical diagnosis Normalized Hamming Similarity Measure for Intuitionistic Fuzzy Multi Sets and Its Application in Medical diagnosis *P. Rajarajeswari, **N. Uma * Department of Mathematics, Chikkanna Arts College, Tirupur,

More information

Intuitionistic fuzzy Choquet aggregation operator based on Einstein operation laws

Intuitionistic fuzzy Choquet aggregation operator based on Einstein operation laws Scientia Iranica E (3 (6, 9{ Sharif University of Technology Scientia Iranica Transactions E: Industrial Engineering www.scientiairanica.com Intuitionistic fuzzy Choquet aggregation operator based on Einstein

More information

The Cosine Measure of Single-Valued Neutrosophic Multisets for Multiple Attribute Decision-Making

The Cosine Measure of Single-Valued Neutrosophic Multisets for Multiple Attribute Decision-Making Article The Cosine Measure of Single-Valued Neutrosophic Multisets for Multiple Attribute Decision-Making Changxing Fan 1, *, En Fan 1 and Jun Ye 2 1 Department of Computer Science, Shaoxing University,

More information

Properties of intuitionistic fuzzy line graphs

Properties of intuitionistic fuzzy line graphs 16 th Int. Conf. on IFSs, Sofia, 9 10 Sept. 2012 Notes on Intuitionistic Fuzzy Sets Vol. 18, 2012, No. 3, 52 60 Properties of intuitionistic fuzzy line graphs M. Akram 1 and R. Parvathi 2 1 Punjab University

More information

Research Article A New Approach for Optimization of Real Life Transportation Problem in Neutrosophic Environment

Research Article A New Approach for Optimization of Real Life Transportation Problem in Neutrosophic Environment Mathematical Problems in Engineering Volume 206 Article ID 5950747 9 pages http://dx.doi.org/0.55/206/5950747 Research Article A New Approach for Optimization of Real Life Transportation Problem in Neutrosophic

More information

Correlation Analysis of Intuitionistic Fuzzy Connectives

Correlation Analysis of Intuitionistic Fuzzy Connectives Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 5, N. 1, 017. Trabalho apresentado no CNMAC, Gramado - RS, 016. Proceeding Series of the Brazilian Society of Computational

More information

960 JOURNAL OF COMPUTERS, VOL. 8, NO. 4, APRIL 2013

960 JOURNAL OF COMPUTERS, VOL. 8, NO. 4, APRIL 2013 960 JORNAL OF COMPTERS, VOL 8, NO 4, APRIL 03 Study on Soft Groups Xia Yin School of Science, Jiangnan niversity, Wui, Jiangsu 4, PR China Email: yinia975@yahoocomcn Zuhua Liao School of Science, Jiangnan

More information

Uncertain Entailment and Modus Ponens in the Framework of Uncertain Logic

Uncertain Entailment and Modus Ponens in the Framework of Uncertain Logic Journal of Uncertain Systems Vol.3, No.4, pp.243-251, 2009 Online at: www.jus.org.uk Uncertain Entailment and Modus Ponens in the Framework of Uncertain Logic Baoding Liu Uncertainty Theory Laboratory

More information

A New Method for Complex Decision Making Based on TOPSIS for Complex Decision Making Problems with Fuzzy Data

A New Method for Complex Decision Making Based on TOPSIS for Complex Decision Making Problems with Fuzzy Data Applied Mathematical Sciences, Vol 1, 2007, no 60, 2981-2987 A New Method for Complex Decision Making Based on TOPSIS for Complex Decision Making Problems with Fuzzy Data F Hosseinzadeh Lotfi 1, T Allahviranloo,

More information

Neutrosophic Soft Multi-Set Theory and Its Decision Making

Neutrosophic Soft Multi-Set Theory and Its Decision Making Neutrosophic Sets and Systems, Vol. 5, 2014 65 Neutrosophic Soft Multi-Set Theory and Its Decision Making Irfan Deli 1, Said Broumi 2 and Mumtaz Ali 3 1 Muallim Rıfat Faculty of Education, Kilis 7 Aralık

More information

An Introduction to Fuzzy Soft Graph

An Introduction to Fuzzy Soft Graph Mathematica Moravica Vol. 19-2 (2015), 35 48 An Introduction to Fuzzy Soft Graph Sumit Mohinta and T.K. Samanta Abstract. The notions of fuzzy soft graph, union, intersection of two fuzzy soft graphs are

More information

TWO NEW OPERATOR DEFINED OVER INTERVAL VALUED INTUITIONISTIC FUZZY SETS

TWO NEW OPERATOR DEFINED OVER INTERVAL VALUED INTUITIONISTIC FUZZY SETS TWO NEW OPERATOR DEFINED OVER INTERVAL VALUED INTUITIONISTIC FUZZY SETS S. Sudharsan 1 2 and D. Ezhilmaran 3 1 Research Scholar Bharathiar University Coimbatore -641046 India. 2 Department of Mathematics

More information

Multi-period medical diagnosis method using a single valued. neutrosophic similarity measure based on tangent function

Multi-period medical diagnosis method using a single valued. neutrosophic similarity measure based on tangent function *Manuscript Click here to view linked References 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 Multi-period medical diagnosis method using a single valued neutrosophic similarity measure based on tangent function

More information

Group Decision Analysis Algorithms with EDAS for Interval Fuzzy Sets

Group Decision Analysis Algorithms with EDAS for Interval Fuzzy Sets BGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOOGIES Volume 18, No Sofia 018 Print ISSN: 1311-970; Online ISSN: 1314-4081 DOI: 10.478/cait-018-007 Group Decision Analysis Algorithms with

More information

Measure of Distance and Similarity for Single Valued Neutrosophic Sets with Application in Multi-attribute

Measure of Distance and Similarity for Single Valued Neutrosophic Sets with Application in Multi-attribute Measure of Distance and imilarity for ingle Valued Neutrosophic ets ith pplication in Multi-attribute Decision Making *Dr. Pratiksha Tiari * ssistant Professor, Delhi Institute of dvanced tudies, Delhi,

More information

Research on the stochastic hybrid multi-attribute decision making method based on prospect theory

Research on the stochastic hybrid multi-attribute decision making method based on prospect theory Scientia Iranica E (2014) 21(3), 1105{1119 Sharif University of Technology Scientia Iranica Transactions E: Industrial Engineering www.scientiairanica.com Research Note Research on the stochastic hybrid

More information

A fixed point theorem on soft G-metric spaces

A fixed point theorem on soft G-metric spaces Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 885 894 Research Article A fixed point theorem on soft G-metric spaces Aysegul Caksu Guler a,, Esra Dalan Yildirim b, Oya Bedre Ozbakir

More information

Rough Neutrosophic Sets

Rough Neutrosophic Sets Neutrosophic Sets and Systems, Vol. 3, 2014 60 Rough Neutrosophic Sets Said Broumi 1, Florentin Smarandache 2 and Mamoni Dhar 3 1 Faculty of Arts and Humanities, Hay El Baraka Ben M'sik Casablanca B.P.

More information

@FMI c Kyung Moon Sa Co.

@FMI c Kyung Moon Sa Co. Annals of Fuzzy Mathematics and Informatics Volume 5 No. 1 (January 013) pp. 157 168 ISSN: 093 9310 (print version) ISSN: 87 635 (electronic version) http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com

More information

A NEW APPROACH TO SEPARABILITY AND COMPACTNESS IN SOFT TOPOLOGICAL SPACES

A NEW APPROACH TO SEPARABILITY AND COMPACTNESS IN SOFT TOPOLOGICAL SPACES TWMS J. Pure Appl. Math. V.9, N.1, 2018, pp.82-93 A NEW APPROACH TO SEPARABILITY AND COMPACTNESS IN SOFT TOPOLOGICAL SPACES SADI BAYRAMOV 1, CIGDEM GUNDUZ ARAS 2 Abstract. The concept of soft topological

More information

Vague Set Theory Applied to BM-Algebras

Vague Set Theory Applied to BM-Algebras International Journal of Algebra, Vol. 5, 2011, no. 5, 207-222 Vague Set Theory Applied to BM-Algebras A. Borumand Saeid 1 and A. Zarandi 2 1 Dept. of Math., Shahid Bahonar University of Kerman Kerman,

More information

Some Measures of Picture Fuzzy Sets and Their Application in Multi-attribute Decision Making

Some Measures of Picture Fuzzy Sets and Their Application in Multi-attribute Decision Making I.J. Mathematical Sciences and Computing, 2018, 3, 23-41 Published Online July 2018 in MECS (http://www.mecs-press.net) DOI: 10.5815/ijmsc.2018.03.03 Available online at http://www.mecs-press.net/ijmsc

More information

First Order Non Homogeneous Ordinary Differential Equation with Initial Value as Triangular Intuitionistic Fuzzy Number

First Order Non Homogeneous Ordinary Differential Equation with Initial Value as Triangular Intuitionistic Fuzzy Number 27427427427427412 Journal of Uncertain Systems Vol.9, No.4, pp.274-285, 2015 Online at: www.jus.org.uk First Order Non Homogeneous Ordinary Differential Equation with Initial Value as Triangular Intuitionistic

More information

A study on fuzzy soft set and its operations. Abdul Rehman, Saleem Abdullah, Muhammad Aslam, Muhammad S. Kamran

A study on fuzzy soft set and its operations. Abdul Rehman, Saleem Abdullah, Muhammad Aslam, Muhammad S. Kamran Annals of Fuzzy Mathematics and Informatics Volume x, No x, (Month 201y), pp 1 xx ISSN: 2093 9310 (print version) ISSN: 2287 6235 (electronic version) http://wwwafmiorkr @FMI c Kyung Moon Sa Co http://wwwkyungmooncom

More information

Hesitant triangular intuitionistic fuzzy information and its application to multi-attribute decision making problem

Hesitant triangular intuitionistic fuzzy information and its application to multi-attribute decision making problem Available online at www.isr-publications.com/nsa J. Nonlinear Sci. Appl., 10 (2017), 1012 1029 Research Article Journal Homepage: www.tnsa.com - www.isr-publications.com/nsa Hesitant triangular intuitionistic

More information

A new generalized intuitionistic fuzzy set

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

More information

Chebyshev Type Inequalities for Sugeno Integrals with Respect to Intuitionistic Fuzzy Measures

Chebyshev Type Inequalities for Sugeno Integrals with Respect to Intuitionistic Fuzzy Measures BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 9, No 2 Sofia 2009 Chebyshev Type Inequalities for Sugeno Integrals with Respect to Intuitionistic Fuzzy Measures Adrian I.

More information

A NOVEL TRIANGULAR INTERVAL TYPE-2 INTUITIONISTIC FUZZY SETS AND THEIR AGGREGATION OPERATORS

A NOVEL TRIANGULAR INTERVAL TYPE-2 INTUITIONISTIC FUZZY SETS AND THEIR AGGREGATION OPERATORS 1 A NOVEL TRIANGULAR INTERVAL TYPE-2 INTUITIONISTIC FUZZY SETS AND THEIR AGGREGATION OPERATORS HARISH GARG SUKHVEER SINGH Abstract. The obective of this work is to present a triangular interval type- 2

More information

Fuzzy Order Statistics based on α pessimistic

Fuzzy Order Statistics based on α pessimistic Journal of Uncertain Systems Vol.10, No.4, pp.282-291, 2016 Online at: www.jus.org.uk Fuzzy Order Statistics based on α pessimistic M. GH. Akbari, H. Alizadeh Noughabi Department of Statistics, University

More information

Extension of TOPSIS for Group Decision-Making Based on the Type-2 Fuzzy Positive and Negative Ideal Solutions

Extension of TOPSIS for Group Decision-Making Based on the Type-2 Fuzzy Positive and Negative Ideal Solutions Available online at http://ijim.srbiau.ac.ir Int. J. Industrial Mathematics Vol. 2, No. 3 (2010) 199-213 Extension of TOPSIS for Group Decision-Making Based on the Type-2 Fuzzy Positive and Negative Ideal

More information

AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC

AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC Iranian Journal of Fuzzy Systems Vol. 9, No. 6, (2012) pp. 31-41 31 AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC E. ESLAMI Abstract. In this paper we extend the notion of degrees of membership

More information

@FMI c Kyung Moon Sa Co.

@FMI c Kyung Moon Sa Co. Annals of Fuzzy Mathematics and Informatics Volume 1, No. 1, (January 2011), pp. 97-105 ISSN 2093-9310 http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com Positive implicative vague

More information

Favoring Consensus and Penalizing Disagreement in Group Decision Making

Favoring Consensus and Penalizing Disagreement in Group Decision Making Favoring Consensus and Penalizing Disagreement in Group Decision Making Paper: jc*-**-**-**** Favoring Consensus and Penalizing Disagreement in Group Decision Making José Luis García-Lapresta PRESAD Research

More information

An OWA-TOPSIS method for multiple criteria decision analysis

An OWA-TOPSIS method for multiple criteria decision analysis University of Windsor Scholarship at UWindsor Odette School of Business Publications Odette School of Business Spring 2011 An OWA-TOPSIS method for multiple criteria decision analysis Ye Chen Kevin W.

More information

Friedman s test with missing observations

Friedman s test with missing observations Friedman s test with missing observations Edyta Mrówka and Przemys law Grzegorzewski Systems Research Institute, Polish Academy of Sciences Newelska 6, 01-447 Warsaw, Poland e-mail: mrowka@ibspan.waw.pl,

More information

Uninorm trust propagation and aggregation methods for group decision making in social network with four tuple information

Uninorm trust propagation and aggregation methods for group decision making in social network with four tuple information Uninorm trust propagation and aggregation methods for group decision making in social network with four tuple information Jian Wu a,b, Ruoyun Xiong a, Francisco Chiclana b a School of Economics and Management,

More information

Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets

Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets Xiaolu Zhang, 1 Zeshui Xu, 1 School of Economics and Management, Southeast University, Nanjing 11189, People s Republic

More information

Divergence measure of intuitionistic fuzzy sets

Divergence measure of intuitionistic fuzzy sets Divergence measure of intuitionistic fuzzy sets Fuyuan Xiao a, a School of Computer and Information Science, Southwest University, Chongqing, 400715, China Abstract As a generation of fuzzy sets, the intuitionistic

More information

Choose Best Criteria for Decision Making Via Fuzzy Topsis Method

Choose Best Criteria for Decision Making Via Fuzzy Topsis Method Mathematics and Computer Science 2017; 2(6: 11-119 http://www.sciencepublishinggroup.com/j/mcs doi: 10.1168/j.mcs.20170206.1 ISSN: 2575-606 (rint; ISSN: 2575-6028 (Online Choose Best Criteria for Decision

More information

Exponential operations and aggregation operators of interval neutrosophic sets and their decision making methods

Exponential operations and aggregation operators of interval neutrosophic sets and their decision making methods Ye SpringerPlus 2016)5:1488 DOI 10.1186/s40064-016-3143-z RESEARCH Open Access Exponential operations and aggregation operators of interval neutrosophic sets and their decision making methods Jun Ye *

More information

An Evaluation of the Reliability of Complex Systems Using Shadowed Sets and Fuzzy Lifetime Data

An Evaluation of the Reliability of Complex Systems Using Shadowed Sets and Fuzzy Lifetime Data International Journal of Automation and Computing 2 (2006) 145-150 An Evaluation of the Reliability of Complex Systems Using Shadowed Sets and Fuzzy Lifetime Data Olgierd Hryniewicz Systems Research Institute

More information

GIFIHIA operator and its application to the selection of cold chain logistics enterprises

GIFIHIA operator and its application to the selection of cold chain logistics enterprises Granul. Comput. 2017 2:187 197 DOI 10.1007/s41066-017-0038-5 ORIGINAL PAPER GIFIHIA operator and its application to the selection of cold chain logistics enterprises Shanshan Meng 1 Nan Liu 1 Yingdong

More information

Chi-square goodness-of-fit test for vague data

Chi-square goodness-of-fit test for vague data Chi-square goodness-of-fit test for vague data Przemys law Grzegorzewski Systems Research Institute Polish Academy of Sciences Newelska 6, 01-447 Warsaw, Poland and Faculty of Math. and Inform. Sci., Warsaw

More information

Decomposition and Intersection of Two Fuzzy Numbers for Fuzzy Preference Relations

Decomposition and Intersection of Two Fuzzy Numbers for Fuzzy Preference Relations S S symmetry Article Decomposition and Intersection of Two Fuzzy Numbers for Fuzzy Preference Relations Hui-Chin Tang ID Department of Industrial Engineering and Management, National Kaohsiung University

More information

A Grey-Based Approach to Suppliers Selection Problem

A Grey-Based Approach to Suppliers Selection Problem A Grey-Based Approach to Suppliers Selection Problem Guo-Dong Li Graduate School of Science and Engineer Teikyo University Utsunomiya City, JAPAN Masatake Nagai Faculty of Engineering Kanagawa University

More information

Solving fuzzy matrix games through a ranking value function method

Solving fuzzy matrix games through a ranking value function method Available online at wwwisr-publicationscom/jmcs J Math Computer Sci, 18 (218), 175 183 Research Article Journal Homepage: wwwtjmcscom - wwwisr-publicationscom/jmcs Solving fuzzy matrix games through a

More information

An Appliaction of Generalized Fuzzy Soft Matrices in Decision Making Problem

An Appliaction of Generalized Fuzzy Soft Matrices in Decision Making Problem IOSR Journal of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:9-765x Volume 0, Issue Ver. II. (Feb. 04), PP -4 www.iosrournals.org n ppliaction of Generalized Fuzzy Soft Matrices in Decision Making Problem

More information