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Procedia Engineering
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 15 (2011) 2037 – 2041 www.elsevier.com/locate/procedia
Advanced in Control Engineering and Information Science
A Generalized Interval Valued Intuitionistic Fuzzy Sets theory Zhang Zhenhua a, b Yang Jingyu a Ye Youpei a Zhang QianSheng b
a
School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094, China b
Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, China
Abstract In this paper, a novel generalized interval-valued intuitionistic fuzzy sets (GIVIFS) is presented, which is the generalization of conventional intuitionistic fuzzy sets (IFS) and interval-valued intuitionistic fuzzy sets (IVIFS). By analyzing the degree of hesitancy, this paper introduces generalized interval-valued intuitionistic fuzzy sets with parameters (GIVIFSP). And then, it is proved that GIVIFS is a closed algebraic system as IFS and IVIFS.
© 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and/or peer-review under responsibility of [CEIS 2011] keywords: intuitionistic fuzzy sets; interval-valued intuitionistic fuzzy sets; generalized interval-valued intuitionistic fuzzy sets; generalized interval-valued intuitionistic fuzzy sets with parameters
1. Introduction In 1986, Atanassov presented IFS. In 1989, Atanassov proposed IVIFS based on comparative analysis of interval-valued fuzzy sets (IVVS) and IFS. Hence, many scholars applied IFS and IVIFS to decision analysis and pattern recognition widely. In the research field of IVIFS, Yager, Yuan Xuehai, and Li Hongxing discussed the cut set characteristics of IVIFS in [5, 6, 7], and in [8, 12, 13] Xu Zeshui and Zhang Qiansheng applied it to pattern recognition based on [4]. Xu Zeshui and Li Dengfeng also applied it to decision-making analysis in [9, 10, 11], and Lei Yingjie and Zhang Qiansheng researched on interval-valued intuitionistic fuzzy reasoning in [14, 15]. By introducing the degree of membership MA(x), the degree of non-membership NA(x) and the degree of hesitancy HA(x), the IFS theory and the IVIFS theory are established, which generalizes Zadeh’s fuzzy Zhang Zhenhua. Tel.: 0086- 020- 39328032, 13660061726; fax:. E-mail address:
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1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2011.08.380
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sets (FS) ([1, 2, 3]). According to the IVIFS definition, MA(x), NA(x) and HA(x) are intervals, where MA(x) denotes the range of support party, NA(x) denotes the range of opposition party, and HA(x) denotes the range of absent party. Moreover, the inferior of MA(x) (INF(MA(x))) is the firm support party of event A, the inferior of NA(x) (INF(NA(x))) is the firm opposition party of event A, the inferior of HA(x) (INF(HA(x))) is the firm absent party of event A, the superior of HA(x) (SUP(HA(x))) is the maximum absent party of event A, and SUP(HA(x))-INF(HA(x)) denotes the convertible absent part, where INF(MA(x))+ INF(NA(x))+SUP(HA(x))=1. Atanassov has divided the convertible absent part into two parts: SUP (MA(x))-INF (MA(x)) being the absent party which can be converted into support party, and SUP (NA(x))-INF (NA(x)) being the absent party which can be converted into the opposition party, where SUP (MA(x))-INF (MA(x))+ SUP (NA(x))-INF (NA(x)) = SUP(HA(x))-INF(HA(x)). Thus, Atanassov’s IVIFS is based on point estimation, which means that these intervals can be regarded as the estimation result of an experiment. However, the proportions of the absent party converted to the support party and to the opposition party may not be constants. For example, SUP (MA(x))-INF (MA(x)) is a constant for one experiment, but it may be a different constant for any other case. Thus, according to interval estimation, we provide a novel GIVIFS model to meet real need. First, we present the concept of GIVIFS, which is proved to be the generalization of IFS and IVIFS. And then we introduce the construction method of the generalized interval valued intuitionistic fuzzy sets with parameters (GIVIFSP), and define complement operation, intersection operation and union operation on GIVIFS. Finally, we prove that GIVIFS is a closed algebraic system for all these operations as fuzzy sets, IFS, and IVIFS. Therefore, this paper generalizes the IVIFS theory, and provides some valuable conclusions for the field of application research of IVIFS, and it is also useful to the generalization of interval-valued intuitionistic fuzzy reasoning. 2. Construction of GIVIFS Definition 1. An IFS A in universe X is given by (Atanassov [1, 2, 3]): (1) A = {< x, μA(x), νA(x) > |x � X} where μA : X → [0, 1],νA : X → [0, 1] with the condition 0≤μA(x) + νA(x) ≤1 for each x � X. The numbers μA(x), νA(x) � [0, 1] denote the degree of membership and the degree of non-membership of x to A, respectively. For each IFS in X, we call πA(x) = 1 − μA(x) − νA(x) the degree of hesitancy ([9]) of x to A, 0≤πA(x) ≤1 for each x � X. Definition 2. An IVIFS A in universe X is given by (Atanassov [2, 3]): (2) A = {< x, MA(x), NA(x) > |x � X} − + − + − + where M A (x)=[t A ( x ), t A ( x )], N A (x)=[ f A ( x ), f A ( x)], and HA(x) = [π A ( x ), π A ( x )] with the condition M A ( x) ⊆ [0,1] , N A ( x) ⊆ [0,1] , P A ( x) ⊆ [0,1] . The interval MA(x), NA(x), and HA(x) denote the degree of membership, the degree of non-membership, and the degree of hesitancy of x to A, respectively. For each IVIFS in X, we call π A− ( x ) = 1 − t A+ ( x ) − f A+ ( x ), π A+ ( x ) = 1 − t A− ( x ) − f A− ( x ) lower bound and upper
bound of hesitancy of x to A respectively, where 0 ≤ π A− ( x ) ≤ π A+ ( x ) ≤ 1, for each x � X. Theorem1. Suppose that A is an IVIFS as mentioned above, then t A+ ( x) − t A− ( x) + f A+ ( x) − f A− ( x) = π A+ ( x) − π A− ( x). Based on definition2, we have (3). From definition 2, let all samples be divided into three parts, −
(3)
t A− ( x) being firm support party of event
A, f A ( x ) representing firm opposition party of event A, and
π A+ ( x)
showing all absent party. In
Zhenhua al. / Procedia (2011) Engineering 2037 – 2041 00 (2011) 000–000 Zhang Zhenhua, Zhang Yang Jingyu, YeetYoupei , Zhang Engineering Qiansheng / 15 Procedia
π A− ( x)
is firm absent party, and π A+ ( x ) − π A− ( x ) is convertible absent party, in which each sample may become one of the support party and the opposition party. Suppose that there is t A+ ( x ) − t A− ( x ) absent party,
of the samples supporting event A and f A+ ( x ) − f A− ( x ) of the samples opposing event A, and then
0 ≤ t A+ ( x) − t A− ( x) ≤ π A+ ( x) − π A− ( x), 0 ≤ f A+ ( x) − f A− ( x) ≤ π A+ ( x) − π A− ( x). Let α A ( x) = t A+ ( x) − t A− ( x), β A ( x) = f A+ ( x) − f A− ( x), and we will get the GIVIFS definition as follows: Definition 3. Let X be a universe of discourse. A GIVIFS A in X is an object having the form: A={|x∈X} where the intervals MA(x), NA(x), HA(x) are the same as definition 2. Let t A+ ( x ) = t A− ( x ) + α A ( x ) and f A+ ( x ) = f A− ( x ) + β A ( x ) , then 0 ≤ max{α A ( x ), β A ( x )} ≤ π A+ ( x ) − π A− ( x ) . Obviously, if π A+ ( x ) = π A− ( x ) = 0, then α A ( x) = β A ( x) = 0 and GIVIFS is fuzzy sets; and if α A ( x) = β A ( x) = 0, then GIVIFS is IFS; and if
α A ( x) + β A ( x) = t A+ ( x) − t A− ( x) + f A+ ( x) − f A− ( x) = π A+ ( x) − π A− ( x) , then GIVIFS is IVIFS. Suppose that the proportion of absent party converted to the support party is
λA1 ( x) and that
converted to the opposition party is 1 − λ A1 ( x) . The model will become interval valued intuitionistic fuzzy sets with single parameter, where
α A ( x) = λA1 ( x)(π A+ ( x) − π A− ( x)), β A ( x) = (1 − λA1 ( x))(π A+ ( x) − π A− ( x)).
In absent party
π A+ ( x),
if firm absent party is π A− ( x ) = (1 − λ A 0 ( x ))π A+ ( x ), then variable absent party
is π A+ ( x ) − π A− ( x ) = λ A 0 ( x )π A+ ( x ) , and then we can define GIVIFSP as follows: Definition 4. Let X be a universe of discourse. A GIVIFSP A in X is an object having the form: A={|x∈X} where MA(x), NA(x), and HA(x) are the same as definition3. Let t A+ ( x) = t A− ( x ) + λ A0 ( x)λ A1 ( x )π A+ ( x ), f A+ ( x ) = f A− ( x ) + λ A 0 ( x)λ A 2 ( x)π A+ ( x), π A− ( x) = (1 − λ A0 ( x ))π A+ ( x ),
where 0 ≤ λ Ai ( x) ≤ 1, i = 0,1, 2. MA(x), NA(x), And HA(x) represent the degree range of membership, the degree range of non-membership, and the degree range of hesitancy of x to A, respectively. It is clear that we will get the following conclusions: If π A+ ( x ) = π A− ( x) = 0, then λA0 ( x) = 0, and then GIVIFS is fuzzy sets; and if λA0 ( x) = 0, then GIVIFS is IFS; and if λ A0 ( x) > 0 and λ A1 ( x) + λA 2 ( x) = 1 ,
then GIVIFS is IVIFS. Furthermore, if λ Ai ( x) = λi , i = 0,1, 2, and λi is constant, then GIVIFSP is an interval valued intuitionistic fuzzy sets with fixed parameters, otherwise it is a variable model. 3. Algebraic properties of GIVIFS
We present the basic operations of containment relation, equal relation, intersection, union, and complement of GIVIFS as follows: Definition 5. Let X be a universe of discourse. A and B are two GIVIFSPs in X. A={|x∈X}, B={|x∈X},where MA(x), NA(x), and HA(x) are the same as definition 3. (1).A ⊆ B ⇔ M A ( x ) ⊆ M B ( x ), N B ( x ) ⊆ N A ( x ) ⇔ t A− ( x ) ≤ t B− ( x ), t A+ ( x ) ≤ t B+ ( x ), f A− ( x ) ≥ f B− ( x ), f A+ ( x ) ≥ f B+ ( x );
(2).A = B ⇔ A ⊆ B, B ⊆ A ⇔ M A ( x) = M B ( x ), N B ( x ) = N A ( x ) ⇔ t A− ( x) = t B− ( x), t A+ ( x) = t B+ ( x), f A− ( x) = f B− ( x), f A+ ( x ) = f B+ ( x);
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(3). A I B = {< x, M A ( x ), N A ( x) >| x ∈ X } I {< x, M B ( x ), N B ( x) >| x ∈ X } = {< x,[t A− ( x), t A+ ( x)],[ f A− ( x), f A+ ( x)] >| x ∈ X } I {< x,[t B− ( x), t B+ ( x)],[ f B− ( x ), f B+ ( x)] >| x ∈ X } = {< x,[t A− ( x) ∧ t B− ( x), t A+ ( x) ∧ t B+ ( x)],[ f A− ( x) ∨ f B− ( x), f A+ ( x) ∨ f B+ ( x)] >| x ∈ X }; (4). A U B = {< x, M A ( x), N A ( x) >| x ∈ X } U {< x, M B ( x), N B ( x ) >| x ∈ X } = {< x,[t A− ( x), t A+ ( x)],[ f A− ( x), f A+ ( x)] >| x ∈ X } U {< x,[t B− ( x), t B+ ( x)],[ f B− ( x), f B+ ( x)] >| x ∈ X } = {< x,[t A− ( x) ∨ t B− ( x), t A+ ( x) ∨ t B+ ( x)],[ f A− ( x) ∧ f B− ( x), f A+ ( x) ∧ f B+ ( x)] >| x ∈ X }; (5). AC = {< x, M A ( x), N A ( x ) >| x ∈ X }C = {< x, N A ( x), M A ( x ) >| x ∈ X } = {< x,[ f A− ( x ), f A+ ( x )],[t A− ( x ), t A+ ( x )] >| x ∈ X }.
Theorem2. GIVIFS is closed for complement operation, containment operation, and union operation. Proof: Support that A and B are GIVIFSs. We will prove that AC , A I B , and A U B are also GIVIFSs. AC = {< x, M A ( x ), N A ( x) >| x ∈ X }C = {< x, N A ( x), M A ( x) >| x ∈ X } = {< x,[ f A− ( x), f A+ ( x)],[t A− ( x), t A+ ( x)] >| x ∈ X } = {< x,[ f A− ( x ), f A− ( x) + β A ( x)],[t A− ( x), t A+ ( x) + α A ( x)] >| x ∈ X }, 0 ≤ max{α A ( x), β A ( x)} ≤ π A+ ( x) − π A− ( x).
Thus, AC is also GIVIFS. According to definition 3, π A+ I B ( x ), π A− I B ( x ), π A+ U B ( x ), π A−U B ( x ) can be defined as π A+ ( x ), π A− ( x ) . Obviously, 0 ≤ t A+ ( x ) − t A− ( x ) ≤ π A+ ( x ) − π A− ( x ), 0 ≤ f A+ ( x ) − f A− ( x ) ≤ π A+ ( x ) − π A− ( x ), 0 ≤ t B+ ( x ) − t B− ( x ) ≤ π B+ ( x ) − π B− ( x ),
0 ≤ f B+ ( x ) − f B− ( x ) ≤ π B+ ( x) − π B− ( x), π A+I B ( x) = 1 − t A−I B ( x) − f A−I B ( x), π A−I B ( x) = 1 − t A+I B ( x) − f A+I B ( x), Table 1. A ∩ B & A∪B +
No t AI B 01. t A+ 02. 03. 04. 05. 06. 07. 08. 09. 10. 11. 12. 13. 14. 15. 16.
t A+ t A+ t A+ t A+ t A+ t A+ t A+ t B+ t B+ t B+ t B+ t B+ t B+ t B+ t B+
t A−I B t A− t A− t A− t A− t B− t B− t B− t B− t A− t A− t A− t A− t B− t B− t B− t B−
f A+I B f A+ f A+ f B+ f B+ f A+ f A+ f B+ f B+ f A+ f A+ f B+ f B+ f A+ f A+ f B+ f B+
f A−I B f A− f B− f A− f B− f A− f B− f A− f B− f A− f B− f A− f B− f A− f B− f A− f B−
t A+I B − t A−I B t A+ − t A− t A+ − t A− t A+ − t A− t A+ − t A− t A+ − t B− t A+ − t B− t A+ − t B− t A+ − t B− t B+ − t A− t B+ − t A− t B+ − t A− t B+ − t A− t B+ − t B− t B+ − t B− t B+ − t B− t B+ − t B−
f A+I B − f A−I B f A+ − f A− f A+ − f B− f B+ − f A− f B+ − f B− f A+ − f A− f A+ − f B− f B+ − f A− f B+ − f B− f A+ − f A− f A+ − f B− f B+ − f A− f B+ − f B− f A+ − f A− f A+ − f B− f B+ − f A− f B+ − f B−
π A+I B
1 − t A− − 1 − t A− − 1 − t A− − 1 − t A− − 1 − t B− − 1 − t B− − 1 − t B− − 1 − t B− − 1 − t A− − 1 − t A− − 1 − t A− − 1 − t A− − 1 − t B− − 1 − t B− − 1 − t B− − 1 − t B− −
f A− f B− f A− f B− f A− f B− f A− f B− f A− f B− f A− f B− f A− f B− f A− f B−
π A−I B
1 − t A+ − 1 − t A+ − 1 − t A+ − 1 − t A+ − 1 − t A+ − 1 − t A+ − 1 − t A+ − 1 − t A+ − 1 − t B+ − 1 − t B+ − 1 − t B+ − 1 − t B+ − 1 − t B+ − 1 − t B+ − 1 − t B+ − 1 − t B+ −
f A+ f A+ f B+ f B+ f A+ f A+ f B+ f B+ f A+ f A+ f B+ f B+ f A+ f A+ f B+ f B+
According to each condition mentioned in Table1, the following inequality is obtained for each x ∈ X:
0 ≤ max{t A+ I B ( x ) − t A−I B ( x ), f A+I B ( x ) − f A−I B ( x )} ≤ π A+I B ( x ) − π A−I B ( x ) = t A+ I B ( x ) − t A− I B ( x ) + f A+I B ( x ) − f A−I B ( x ) .
Therefore, A I B is GIVIFS. Obviously, A U B is also GIVIFS. Theorem 2 show that GIVIFS is a closed algebraic system for complement operation, intersection operation and union operation, and it is the same as FS, IFS, and IVIFS on this algebraic property.
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4. Concluding remarks
We proposed the definition of GIVIFS, and prove that GIVIFS is the generalization of IFS and IVIFS. Hence, we introduce GIVIFSP and construct a type of GIVIFSP model. Finally, we prove that GIVIFS is a closed algebraic system for complement operation, intersection operation and union operation as fuzzy sets, IFS, and IVIFS. Acknowledgements
This paper is supported by the National Natural Science Foundation of China (Grant No.61070061, Grant No.70801020), the Foundation for Young Scholars of Guangdong University of Foreign Studies (Grant No.GW20052013), the "Eleventh Five-Year" Philosophy and Social Sciences project of Guangdong Province Plan (396-Z1320084), Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No.396-GK100018). References [1] Atanassov k. Intuitionistic fuzzy sets. Fuzzy Sets and Systems 1986; 20(1):87-96. [2] Atanassov K, Gargov G. Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems 1989; 31:343-349. [3] Atanassov K. Intuitionistic Fuzzy Sets Theory and Applications. Heidelberg, New York: Physica-verl; 1999. [4] Eulalia Szmidt, Janusz Kacprzyk. Dilemmas with distances between intuitionstic fuzzy sets: straightforward approaches may not work. Studies in Computational Intelligence 2008; 109:415-430. [5] Yuan Xuehai, Li Hongxing. Cut sets on interval-valued intuitionistic fuzzy sets. IEEE Sixth International Conference on Fuzzy Systems and Knowledge Discovery, FSDK 2009; 6:167-171. [6] Yuan Xuehai, Li Hongxing, Sun Kaibiao. Theory based on interval-valued level cut sets of Zadeh fuzzy sets. Fuzzy Information and Engineering 2009; 2: 501-510. [7] Ronald R. Yager. Some aspects of intuitionistic fuzzy sets. Fuzzy Optim Decis Making 2009; 8:67–90. [8] Xu Zeishui. On similarity measures of interval-valued intuitionistic fuzzy sets and their application to pattern recognitions. Journal of Southeast University (English Edition) 2007; 23(1):139 – 143. [9] Xu Zeshui, Ronald R. Yager. Intuitionistic and interval-valued intuitionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group. Fuzzy Optim Decis Making 2009; 8:123-139. [10] Li Dengfeng. Topsis-Based Nonlinear-Programming methodology for multiattribute decision making with Interval-valued intuitionistic fuzzy sets. IEEE Transactions on Fuzzy Systems 2010; 18(2):299-311. [11] Li Dengfeng. Mathematical-Programming approach to matrix games with payoffs represented by Atanassov’s Intervalvalued intuitionistic fuzzy sets. IEEE Transactions on Fuzzy Systems 2010; 18(6):1112-1128. [12] Zhang Qiansheng, Yao Haixiang, Zhang zhenhua. Some similarity measures of interval-Valued intuitionistic fuzzy sets and application to pattern recognition. International Journal of Applied Mechanics and Materials 2011; 44-47: 3888-3892. [13] Zhang Qiansheng, Jiang Shengyi, Jia Baoguo, Luo Shihua. Some information measures for interval-valued intuitionistic fuzzy sets. Information Sciences 2010; 180(12):5130-5145 [14] Zhang Qiansheng, Yao Haixiang, Zhang zhenhua. An interval-valued fuzzy reasoning approach based on weighted similarity measure. Advanced Materials Research 2011; 143-144: 161-165. [15] Shen Xiaoyong, Lei Yingjie, Hua Jixue, Shi Zhaohui. Description and reasoning method of uncertain temporal knowledge based on IFTPN. Control and Decision 2010; 25(10):1457-1462.
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