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Intuitionistic fuzzy rough set model based on conflict distance and applications
Applied Soft Computing 31 (2015) 266–273
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Intuitionistic fuzzy rough set model based on conflict distance and applications Yong Liu a,∗ , Yi Lin b a b
School of Business, Jiangnan University, Wuxi 214122, China Mathematics Department, Slippery Rock University, Slippery Rock, PA 16057, USA
a r t i c l e
i n f o
Article history: Received 10 January 2013 Received in revised form 25 February 2015 Accepted 28 February 2015 Available online 17 March 2015 Keywords: Intuitionistic fuzzy set Fuzzy rough set model Theories of uncertainty Conflict analysis
a b s t r a c t Due to the complexity and uncertainty of the objective world, as well as the limitation of human ability to understand, it is difficult for one to employ only a single type of uncertainty method to deal with the real-life problem of decision-making, especially problems involving conflicts. On the other hand, by incorporating the advantages of various theories of uncertainty, one is expected to develop a more powerful hybrid method for soft decision making and to solve such problems more effectively. In view of this, in this paper the thought and method of intuitionistic fuzzy set and rough set are used to construct a novel intuitionistic fuzzy rough set model. Corresponding to the fact that the decisionmaking information system of rough sets is of intuitionistic fuzzy information system, our method defines the conflict distance by using the idea of measuring intuitionistic fuzzy similarity so that it is introduced into the models of rough sets, leading to the development of our intuitionistic fuzzy rough set model. After that, we investigate the properties of the model, introduce a novel tool for conflict analysis based on our hybrid model, and employ this new tool to describe and resolve a real-life conflict problem. © 2015 Elsevier B.V. All rights reserved.
1. Introduction As a useful mathematical tool to deal with knowledge with inaccuracy, uncertainty, and fuzziness, rough set theory was proposed by Pawlak [1]. This theory has ever since been widely applied in many scientific disciplines, such as knowledge discovery, data mining, decision analysis, pattern recognition, etc. [2–4]. The classical rough set theory conducts data reasoning based on equivalence relationship. It turns out to be difficult for equivalence relationship to satisfy the harsh conditions in practical applications. For example, the particular binary relationship of the discussion is often not an equivalence relation. Instead, it tends to be either a fuzzy equivalence relation or a similarity relation. In view of this observation, based on the idea and method of the fuzzy set theory, Dubois and Prade put forward the fuzzy rough sets theory and method [5]. Because this theory can well describe the uncertainty of various types of knowledge, and reflect the real world with more objectivity, it soon became the research focus of rough set theory, and
∗ Corresponding author. Tel.: +86 15061889051. E-mail addresses: [email protected] (Y. Liu), [email protected] (Y. Lin). http://dx.doi.org/10.1016/j.asoc.2015.02.045 1568-4946/© 2015 Elsevier B.V. All rights reserved.
consequently developed into a well-established theory in a short period of a few years. As a result of simultaneously considering the positive, negative and hesitancy degrees for an object to belong to a set, intuitionistic fuzzy sets possess a stronger ability of expressing information and better describing and portraying delicate ambiguities of the uncertain nature of the objective world when compared to the theory of the traditional fuzzy sets [6,7]. Thus it was suggested to combine the concepts of intuitionistic fuzzy sets and rough sets, resulting in the introduction of intuitionistic fuzzy rough set model [8]. Due to the important value of theoretical research and practical application, the intuitionistic fuzzy rough set theory has been soon becoming a hot academic research issue. Currently, the related researches on intuitionistic fuzzy rough sets lie mainly in such aspects as constructing different models, exploring the properties of the models, and simplifying the attributes of the models. For the related studies on constructing different models and exploring their properties, Cornelis, Cock, Kerre, and others first revealed the relationship between intuitionistic fuzzy set theory and rough set theory and proved the fact that each fuzzy rough set is actually an intuitionistic L-fuzzy set, and then the concept of intuitionistic fuzzy sets is used to define approximation operators in the intuitionistic fuzzy approximation space. By employing the concept of cut sets of intuitionistic fuzzy sets, the upper and
Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273
lower approximation operators of intuitionistic fuzzy rough sets and the axiomatic method of the approximation operators based on the general binary intuitionistic fuzzy relationship are constructed [8–10], and the formulas of rough set are extended to the n dimension and the semantic and semantic reasonings of the formulas are given [11–14]. Jena and Ghosh prove that the upper and lower approximation sets of intuitionistic fuzzy rough sets are intuitionistic fuzzy set [15]. Based on intuitionistic fuzzy residual implication and intuitionistic fuzzy relationship, Xu, Lei, and Tan construct an intuitionistic fuzzy rough set model [16]. However, it is difficult for the model to deal with information systems with noise data. With the intuitionistic fuzzy triangle model T = min, intuitionistic fuzzy t-conforms S = max, and intuitionistic fuzzy inverse operator N, the approximation operators of the intuitionistic fuzzy rough sets are defined by Zhou et al., and then the (I; T) intuitionistic fuzzy rough set model is constructed based on the general logic operators of the intuitionistic fuzzy [17,18]. With the thought and method of intuitionistic fuzzy set and rough set, Yin, Lei and Lei put forward an improvement intuitionistic fuzzy rough set model based on hamming distance and reveal its properties, such as interval, symmetry, complete similarity and complete dissimilarity [19]. Lin and Yang establish the interval-valued intuitionistic fuzzy rough set based on the thought of implication, and then establish the related properties of the model from the angle of mathematics [20]. By combining interval intuitionistic fuzzy set and rough set, Z. M. Zhang, J. F. Tian and others construct the interval intuitionistic fuzzy rough set model based on the interval intuitionistic fuzzy relationship [21,22]. Based on interval-valued intuitionistic fuzzy compatibility relationship, Yang establishes the interval-valued intuitionistic fuzzy rough set model based on double universes and discusses its properties [23]. Zhang, Zhou and Li propose the general framework of intuitionistic fuzzy rough set, and discuss the intuitionistic fuzzy operator and the properties of the model with the dual domain case [24]. In order to deal with the incomplete and imprecise intuitionistic fuzzy information system, Z. T. Gong exploits the thought and method of intuitionistic fuzzy set and variable precision rough set to construct an extended intuitionistic fuzzy rough set model [25], while Abdullah, Aslam, Ullah and Amin establish bipolar fuzzy soft sets and generalized fuzzy soft expert set, and respectively utilize them to make decision problems and analysis of S-box image encryption [26,27]. For the related literature of attribute reduction, with the characteristics of intuitionistic fuzzy information systems, Lu, Lei and Hua put forward the genetic algorithm to reduce attributes [28]. Chen and Yang propose a new kind of attribute reduction algorithm of intuitionistic fuzzy rough set based on mutual information by combining information entropy theory and intuitionistic fuzzy rough set [29]. Esmail, Maryam, and Habibolla make research on the properties and structure of rough set model based on intuitionistic fuzzy information system, and give the method of attributes reduction and rules extraction [30]. By using the intuitionistic fuzzy distance formula, Huang, Li, Wei and et al. construct the intuitionistic fuzzy rough model, and design an attribute reduction method [31,32]. However, the influence and effects of the core, hesitancy degree and other factors are not considered in the model. According to the above discussion on literatures, it can be seen that most of publication lies in the theoretical study and construction of models, while there are few works that address how intuitionistic fuzzy rough sets could be used to deal with practical decision-making problems, especially conflict problems. In view of these, by taking advantage of the strengths of intuitionistic fuzzy set and rough set, and by making use of the thought of the intuitionistic fuzzy similarity measure, we will construct a novel intuitionistic fuzzy rough set model, and then transform it into the intuitionistic fuzzy conflict analysis model by combining the characteristics of the conflict information system.
267
The specific arrangement of this paper is as follows: the preliminary knowledge of intuitionistic fuzzy sets and intuitionistic fuzzy information systems is outlined in Section 2. The intuitionistic fuzzy rough set model based on conflict distance is established, and then its properties are studied in Section 3. The intuitionistic fuzzy conflict analysis model is proposed in Section 4. An example is used to verify the validity and rationality of the proposed method in Section 5. The paper is concluded in Section 6. 2. Preliminary knowledge Intuitionistic fuzzy sets, as proposed by Atanassov [6,7], are an expansion and further development of the traditional fuzzy sets. As a result of taking into account to the membership and nonmembership information by adding a new attribute parameters – the non-membership function in the intuitionistic fuzzy sets, intuitionistic fuzzy sets can provide additional options for describing the properties of things and possess a capability of dealing with uncertain information. That explains why these sets can well describe and portray the delicate natural ambiguity of the objective world. 2.1. Intuitionistic fuzzy sets Definition 1[6,7]. Suppose that X = {x1 , x2 , ..., xn } is a nonempty, finite set of objects, where, xi (i = 1, 2, ..., n) is the ith object. The set A = {< x, A (x), A (x) > |x ∈ X} of triples is called an intuitionistic fuzzy set, where A (x) and A (x) are respectively called the membership and non-membership for the object x to belong to A, that is, A (x) : X → [0, 1],
x ∈ X → A (x) ∈ [0, 1]
(1)
A (x) : X → [0, 1],
x ∈ X → A (x) ∈ [0, 1]
(2)
where, 0 ≤ A (x) + A (x) ≤ 1, x ∈ X while A (x) stands for the degree of hesitation or uncertainty for the object x to belong to A and A (x) = 1 − A (x) − A (x). So, the intuitionistic fuzzy number can be denoted as ˛ =< A (x), A (x) >.Definition 2[6,7]. For any two intuitionistic fuzzy numbers ˛1 =< 1 , 1 > and ˛2 =< 2 , 2 >, the rules of arithmetic operations are given as follows: (1) (2) (3) (4)
˛1 =< v1 , 1 >; ˛1 + ˛2 =< 1 + 2 − 1 2 , v1 , v2 >; ˛1 + ˛2 = ˛1 + ˛2 ; and ˛1 + ˛2 = (˛1 + ˛2 ), > 0.
Definition 3[33]. For any intuitionistic fuzzy number ˛ =< , >, the score function S(˛) of this number is defined as follows: S(˛) = − ,
S(˛) ∈ [−1, 1]
(3)
Number example 1: for intuitionistic fuzzy number ˛ =< 0.8, 0.1 >, the score function acquired is 0.7. Note: the larger the value of S(˛) is, the greater the intuitionistic fuzzy number ˛ =< , > is.Definition 4[33]. For any intuitionistic fuzzy number ˛ =< , >, the precision function H(˛) of this number is defined as follows: H(˛) = + ,
H(˛) ∈ [0, 1]
(4)
Number example 2: For intuitionistic fuzzy number ˛ =< 0.4, 0.5 >, the precision function acquired is 0.9. Note: The larger the value of H(˛) is, the higher the precision degree of the intuitionistic fuzzy number ˛ =< , > is.Definition 5[33]. For any two intuitionistic fuzzy numbers ˛1 =< 1 , 1 > and ˛2 =< 2 , 2 >, the following hold true:
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(1) if S(˛1 ) < S(˛2 ), then ˛1 < ˛2 , (2) when S(˛1 ) = S(˛2 ), if H(˛1 ) = H(˛2 ), then ˛1 = ˛2 ; if H(˛1 ) < H(˛2 ), then ˛1 < ˛2 .
The larger the value of S(˛) is, the greater the intuitionistic fuzzy number ˛ =< [, ], [- , ] > is.Definition 10[33] Let ˛ =< [, ], [- , ] > be an interval intuitionistic fuzzy number, the precision function H(˛) of this number is defined as follows:
Number example 3: for two intuitionistic fuzzy numbers ˛1 =< 0.7, 0.2 > and ˛1 =< 0.7, 0.1 >, by the definition 3, we can get S(˛1 ) = 0.5 and S(˛2 ) = 0.6, therefore ˛1 < ˛2 . Number example 4: for two intuitionistic fuzzy numbers ˛1 =< 0.5, 0.2 > and ˛1 =< 0.4, 0.1 >, by the definition 3, we can get S(˛1 ) = 0.3 and S(˛2 ) = 0.3, however, we cannot make comparisons. According to the definition 4, H(˛1 ) = 0.7 and H(˛2 ) = 0.5, therefore, ˛1 > ˛2 .Definition 6[33]. For any intuitionistic fuzzy numbers ˛1 =< 1 , 1 > and ˛2 =< 2 , 2 >, the normalized Hamming distance of these numbers is defined as follows:
1 D(˛1 , ˛2 ) = (1 − 2 + + |1 − 2 |) (5) 2 Number example 5: for two intuitionistic fuzzy numbers ˛1 =< 0.8, 0.2 > and ˛1 =< 0.5, 0.1 >, by the definition 6, the normalized Hamming distance D(˛1 , ˛2 ) is 0.2. 2.2. Interval Intuitionistic fuzzy sets Due to the complexity and uncertainty of the objective world, as well as the limitation of the human understanding, it is often difficult to express the values of A (x) and A (x) accurately so that interval numbers should be employed. Therefore, Atanassov and others expand the concept of intuitionistic fuzz sets and put forward the concept of interval intuitionistic fuzzy sets [32,33].Definition 7[34,35] Let X = {x1 , x2 , . . ., xn } be a nonempty, finite set of objects, where xi (i = 1, 2, . . ., n) is the ith object. The interval intuitionistic fuzzy sets in X is defined as follows: A = {< x, A (x), A (x) > |x ∈ X}
(6)
where, A (x) = [A , A ] and A (x) = [ - A , A ] are respectively called the membership and non-membership for the object x to belong to A, A , A respectively the upper and lower extreme value of the membership A (x); A , - A respectively the upper and lower extreme value of the non-membership A (x), which satisfy 0 ≤ A + A ≤ 1, 0 ≤ A + - A ≤ 1, while A (x) stands for the degree of hesitation or uncertainty for the object x to belong to A and A (x) = 1 − A (x) − A (x). So the intuitionistic fuzzy number is denoted as ˛ =< A (x), A (x) >.Definition 8[34,35] For any interval intuitionistic fuzzy numbers ˛1 =< [1 , 1 ], [- 1 , 1 ] > and ˛2 =< [2 , 2 ], [- 2 , 2 ] >, the following hold true: (1) ˛1 + ˛2 =< [1 + 2 − 1 2 , 1 + 2 − - 1 2 ], [- 1 - 2 , 1 2 ] >; • (2) ˛1 ˛2 =< [1 2 , 1 2 ], [- 1 + - 2 − - 1 - 2 , 1 + 2 − 1 2 ] >; - and
(3) (˛1 ) =< 1 , 1 , [1 − (1 − - 1 ) , 1 − (1 − 1 ) ] >, -
> 0.
According to Definition 8, the operation algorithms can be obtained as follows: (1) (2) (3) (4) (5)
˛1 + ˛2 = ˛2 + ˛1 ; ˛1 • ˛2 = ˛2 • ˛1 ; (˛1 • ˛2 ) = ˛1 • ˛2 , > 0; ˛1 + ˛2 = (˛1 + ˛2 ), > 0; and ˛1 • ˛2 = (˛1 • ˛2 ) , > 0.
Definition 9[33] Let ˛ =< [, ], [- , ] > be an interval intuitionistic fuzzy number. Then the score function S(˛) of this number is defined as follows: −+− S(˛) = - , 2
S(˛) ∈ [−1, 1]
(7)
+++ H(˛) = - , 2
H(˛) ∈ [0, 1]
(8)
The larger the value of H(˛) is, the higher the precision degree of the intuitionistic fuzzy number ˛ =< , , [, ] >
is.Definition 11[33] Suppose ˛1 =< 1 , 1 , 1 , 1 > and ˛2 =< [2 , 2 ], [- 2 , 2 ] > are two interval intuitionistic fuzzy numbers. Then the following hold true: (1) if S(˛1 ) < S(˛2 ), then ˛1 < ˛2 ; and (2) when S(˛1 ) = S(˛2 ), if H(˛1 ) = H(˛2 ), then ˛1 = ˛2 ; if H(˛1 ) < H(˛2 ), then ˛1 < ˛2 . Definition 12[33] Suppose that ˛1 =< [1 1 ], [- 1 , 1 ] > and ˛2 =< [2 , 2 ], [- 2 , 2 ] > are two interval intuitionistic fuzzy numbers, the normalized Hamming distance of these numbers is defined as follows: D(˛1 , ˛2 ) =
1 4
− + 1 − 2 + 1 − 2 + 1 − 2 (9) 1 2 -
-
2.3. Intuitionistic fuzzy information system Definition 13Suppose that there exists the quadruple S = (U, A, V, f ), where, U = {U1 , U2 , . . ., Un } stands for the finite and nonempty universe set, and A = C ∪ D expresses the finite and nonempty attribute set such that C = {a1 , a2 , . . ., am } is the condition attribute set and D = {d1 , d2 , . . ., dp } the decision attribute set, while V = ∪Vib indicates the value range of U with respect to
the attribute set A such that Vib is the value of the object Ui with respect to the attribute b, and f : U × A → V stands for an information function. For ∀Ui ∈ U, ∀b ∈ A, Vib ∈ V , if Vib is an (interval)
intuitionistic fuzzy number ˛bi =< bi , ib >, then the quadruple S is known as the (interval) intuitionistic fuzzy information system, and denoted by IFS. Correspondingly, the rough sets defined on (interval) intuitionistic fuzzy information systems are known as (interval) intuitionistic fuzzy rough sets. 3. Intuitionistic fuzzy rough set model based on conflict distance
The upper and lower approximations of a rough set based on the (interval) intuitionistic fuzzy information system are often derived by using the algorithms of (interval) intuitionistic fuzzy numbers, while it is difficult for such rough sets to take full advantage of the information in the intuitionistic fuzzy information system in order to deal with decision-making problems of the real world, including problems of conflicts. Additionally, the intuitionistic fuzzy similarity measure only considers the impact of the non-membership degree and the membership degree [36–41]. For example, the similarity measure method that was put forward by Chen [36,37] to a certain extent does not agree with the real-life situation. The similarity measure, proposed by Hong and Kim [38] does not consider the effect of core values. Other similarity measure methods, such as those proposed by Zhang, Huang and Li [39,40], and Zhang and Dang [41], do not pay sufficient amount of attention to the particular backgrounds of concern when applied to deal with uncertain information. In view of this survey of the existing literature, this paper applies the thought and methodology of the similarity measure for (interval) intuitionistic fuzzy numbers to define the concept of conflict distance, on top of which a novel
Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273 Table 1 The intuitionistic fuzzy conflict information system on university-industry cooperation conflict.
U1 U2 U3
a1
a2
a3
<0.8,0.0> <0.0,0.6> <0.4,0.0>
<0.0,0.5> <0.8,0.0> <0.3,0.1>
<0.6,0.0> <0.0,0.7> <0.3,0.0>
variable precision intuitionistic fuzzy rough set model is put forward.Definition 14Suppose that IFS = (U, A, V, f ) is an (interval) intuitionistic fuzzy information system, for ∀Ui , Uk ∈ U, ∀b ∈ B ⊆ b of U , U with respect to the attribute b A, the conflict distance ik i k can be defined as follows: b ik
=
Sb (˛i ) − Sb (˛k ) + ıb (˛i ) − ıb (˛k ) + b (˛i ) − b (˛k ) i
k
i
k
i
k
4
(10)
where, B represents the attribute subset of A, while Vib , Vkb respectively stands for the values of the objects Ui , Uk with respect to the attribute b, and ˛bi =< bi , ib >, ˛bk =< bk , kb >, Sib (˛i ) = (bi − ib ), Skb (˛k ) = (bk − kb ), ib =
1 − bi − ib , kb = 1 − bk − kb , ıbi = bi + ib bi , ıbk =
bk + kb bk , ib = ib + ib ib , and kb = kb + kb kb .
b , the influence and effect of the core, For the conflict distance ik support, opposition, hesitancy degree of the (interval) intuitionistic fuzzy numbers are taken into consideration.Case 1Universityindustry cooperation is an effective way for innovation, while the conflict problem affects and restricts the cooperative performance, so that it is necessary to study the conflict problem of university-industry cooperation. The conflict of Universityindustry cooperation mainly results from the interest, which is mainly embodied with the three aspects of intellectual property rights (a1 ), administration authority (a2 ) and knowledge gains (a3 ). University-industry cooperation involves three parties, which is respectively the sources of knowledge (universities, or scientific research institutes, denoted as U1 ), the recipients of knowledge (enterprises, denoted as U2 ) and the coordinator (government, intermediary of science and technology, denoted as U3 ). In order to acquire the relevant data, we invite the 30 experts (10 experts from University, 10 University from enterprises, the other from Government, intermediary) to make interviews. By collecting and sorting, the intuitionistic fuzzy conflict information system on universityindustry cooperation conflict is obtained and shown in Table 1. For the attribute a1 , the conflict distance between U1 and U2 is 0.800; the conflict distance between U1 and U3 is 0.620, the conflict distance between U2 and U3 is 0.260. For the attribute a2 , the conflict distance between U1 and U2 is 0.753; the conflict distance between U1 and U3 is 0.310, the conflict distance between U2 and U3 is 0.210. For the attribute a1 , the conflict distance between U1 and U2 is 0.763; the conflict distance between U1 and U3 is 0.605, the conflict distance between U2 and U3 is 0.203. According to the above computing results, the conflict distance between the sources of knowledge and the recipients of knowledge with respect to intellectual property rights is the largest, which is consistent with the actual situation.Definition 15For a given (interval) intuitionistic fuzzy information system IFS = (U, AF, f ), a threshold 0 ≤ ≤ 1, B = (b1 , b2 , . . .bl ) ⊆ C, X ⊆ U, if the − B
b ≤ , ∀b ∈ B , then neighborhood bases of x is [x]B = y ∈ U yx − B -lower approximation and − B -upper approximation of X can be respectively defined by:
apr B (X) =
apr B (X) =
269
Definition 15 can be seen as an (interval) variable precision intuitionistic fuzzy Rough set model based on conflict distance. The − B -lower approximation apr B (X) of the set X, can also be called as the positive region of the (interval) intuitionistic fuzzy rough set model based on conflict distance. In other words, by a given threshold value , apr B (X) is such a set that the universe U can be definitely classified into all the element sets of the set X. The (, ˇ) − B -upper approximation apr B (X) of the set x reflects a given threshold value , and the universe U can be classified into all the element set of the set X probably. Obviously, the (interval) intuitionistic fuzzy rough set model based on conflict distance can identify the relationship of “belonging to” or “containing” to some extent, which makes it able to recognize the presence of strong non-deterministic relationships, for example, to derive the predictive rules with less-than-one probabilities. Accordingly, the upper and lower approximations of the (interval) intuitionistic fuzzy rough set model based on conflict distance can be used to define the − B boundary, positive region, approximation accuracy and classification quality of X.Definition 16For a given (interval) intuitionistic fuzzy information system IFS = (U, A, V, f ), 0 ≤ ≤ 1, the − B boundary, positive region, approximation accuracy and classification quality X are respectively defined as follows:
bndB (X) = apr B (X) − apr B (X) ˛(X) =
(13)
|apr B (X)|
(14)
|apr B (X)|
(B, D) =
|apr B (X)|
(15)
|U|
Property 1
apr B (X) ⊆ X ⊆ apr B (X) ProofLet x ∈ apr B (X). Then there exist [x]B ⊆ X and x ∈ [x]B such that
x ∈ X. Therefore apr B (X) ⊆ X. From x ∈ X, we know [x]B ∩ X = / so
that thus we obtain x ∈ apr B (X). Therefore, X ⊆ apr B (X). QEDTheorem 1For a given (interval) intuitionistic fuzzy information system IFS = (U, A, V, f ), and X ⊆ U, B1 ⊆ B2 ⊆ C, 0 ≤ 1 ≤ 2 ≤ 1, the following hold true: (1) apr B 2 (X) ⊆ apr B 1 (X); (2) apr
1 1 B1
(X) ⊆ apr
1 1 B2
(X);
apr B11 (X) ⊆ apr B12 (X); and apr B21 (X)
⊆ apr B11 (X)
Proof(1) For ∀x ∈ apr B 2 (X), there exists [x]B12 ⊆ X, due to 0 ≤ 1 ≤ 1
2 ≤ 1. So we have [x]B11 ⊆ [x]B12 such that x ∈ apr B 1 (X). There1
fore apr B 2 (X) ⊆ apr B 1 (X). And for ∀y ∈ apr B11 (X), there exists [y]B11 ∩ 1
1 2 B 12 B1 1 B1
/ such that [y] ∩ X = / , which leads to y ∈ apr B12 (X). ThereX= fore apr B11 (X) ⊆ apr (X). (2) For ∀x ∈ apr (X), there exists [x]B11 ⊆ X, due to B1 ⊆ B2 ⊆ A, we obtain [x]B21 ⊆ [x]B11 such that x ∈ apr B 1 (X). Therefore apr B 1 (X) ⊆
2
1
apr B 1 (X); and for ∀y ∈ apr B21 (X), there exists [y]B21 ∩ X = / . Because 2
/ such that y ∈ apr B11 (X). Therefore B1 ⊆ B2 ⊆ A, we have [y]B11 ∩ X = apr B21 (X) ⊆ apr B11 (X). QED 4. Conflict analysis model based on intuitionistic fuzzy rough set
{x ∈ U : [x]B ⊆ X}
(11)
4.1. Intuitionistic fuzzy conflict analysis model
{x ∈ U : [x]B ∩ X = / ϕ}
(12)
In order to describe and deal with conflict problem, Pawlak [42–44], put forward the conflict analysis model based on rough
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Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273
set (denoted as Pawlak conflict analysis model). In Pawlak conflict analysis model, S = (U, for the conflict information A, V, f ) stands
system, where, U = U1 , ..., Ui , ..., Un delegates for the players set, A expresses the conflict issue set, while V = ∪Via indicates the value range of U with respect to the attribute set A such that Via is the value of the object Ui with respect to the conflict issue a and restricted to three values {−1, 0, 1} meaning favorable, neutral and against, respectively, and f : U × A → V expresses the conflict information function. In the conflict analysis model based on rough set, as proposed by Pawlak [42–44], the players have only three attitudes with respect to the issues of concern, and the conflict information system based on the rough set is too stiff to describe and portray real-life conflict problems so that the model contains inconsistence with the problems of actual conflict. In view of this shortcoming, we will employ the thought and methodology of the intuitionistic fuzzy set to soften the players’ attitudes of the issues and construct the intuitionistic fuzzy conflict information system in accordance with the actual situation of concern and the relevant background. For example, the value of the player’s attitude of an issue is <0.7,0.2>, it could show that favorable degree, against degree are respectively 0.7 and 0.2. After that, we establish the intuitionistic fuzzy conflict analysis model to deal with the conflict problems of the objective world.Definition 17Suppose that IFS = (U, A, V, f ) is an (interval) intuitionistic fuzzy information system, if the condition attribute set C = {a1 , a2 , . . .am } stands for the conflict issue set, then IFS can be known as the (interval) intuitionistic fuzzy conflict information system, and denoted by CIFS, while the (interval) intuitionistic fuzzy rough set based on conflict distance is known as the (interval) intuitionistic fuzzy conflict analysis model. Note: In the intuitionistic fuzzy conflict analysis model, for ∀Ui ∈ U, ∀a ∈ A, Via ∈ V , if ai = 1, ia = 0, or ai = 0, ia = 1 or ai = 0.5, ia = 0.5, then Via = {−1, 0, 1}, that is the intuitionistic fuzzy conflict analysis model degrade into Pawlak conflict analysis model.Definition 18In the given (interval) intuitionistic fuzzy conflict information system CIFS, for ∀Ui , Uk ∈ U, ∀aj ∈ C, the con-
n
=
=
j=1
m
=
k=1,i = / k j=1
n m
=
m
wj j =
(19)
(n − 1) j
wj ik
j=1 i,k=1,i = / k
(20)
n(n − 1)
j=1
where wj is the weight of the players with the conflict issue aj (j = 1, 2, ..., m), satisfying
m
wj = 1.
j=1
For the case 1, based on the expressions (19) and (20), let the weight vector of the conflict issue is (0.37, 0.28, 0.35), then 1C = 0.773, 2C = 0.568, 3C = 0.248, = 0.503.Property 1 j
j
C C ik = ki .
ik = ki ,
ProofFor ∀Ui , Uk ∈ U, j
∀aj ∈ C, according to Definition 14 and Eq.
j
(10), ik = ki can be calculated by using ik =
m
j
wj ik . Therefore
j=1 C = C . QEDProperty 2 ik ki j
kk = 0,
C kk = 0.
Proof, for ∀Ui , Uk ∈ U, ∀aj ∈ C, according to Definition 14 and Eq. (10), symbolic computations indicate that j
=
j
ik
k=1,i = / k j=1
(17)
m(n − 1) j
j=1 i,k=1,i = / k
mn(n − 1)
=
ik =
from
m
j
wj ik ,
it
follows
C = 0. kk
that
j=1 j
10 ≤ ik ≤ 1,
QEDTheorem
j
follows that only when ˛i = ˛k , ik = 0, such that 0 ≤ ik . j i
j i
j i
j j ≤ 1, 0 ≤ ≤ 1, 0 ≤ + i ≤ 1, 0 ≤ k ≤ Because 0 ≤ j j j j j j j 1, 0 ≤ k ≤ 1, 0 ≤ k + k ≤ 1, Si (˛i ) = (i − i ), Sk (˛k ) = j j j j j j j j j j (k − k ), i = 1 − i − i , k = 1 − k − k , ıi = i + j j j j j j j j j j j j j j i i , ık = k + k k , i = i + i i , k = k + k k , we
obtain
n m
ik
And
j
(16)
n(n − 1)
n m
j
j
ik
k=1,i = / ki=1,i = / k
m
C ik
n−1
m
n
n
k=1,i = / k
=
j
ik
n
iC =
n−1
j
wj ik
j j j0 ≤ ikj ≤ 1.ProofAccording j j +i (˛i )−k (˛k ) S (˛ )−S (˛ ) + ı (˛ )−ı (˛ ) i k i k j i k i k to ik = 4 , and from j j j j j j 0 ≤ Si (˛i ) − Sk (˛k ), 0 ≤ ıi (˛i ) − ık (˛k ), 0 ≤ i (˛i ) − k (˛k ), it
n(n − 1)
k=1,i = / k
iC =
m n
C ik
j
i,k=1,i = / k
j =
n
kk = 0.
flict distance ik can be referred to as the conflict coefficient of Ui , Uk with respect to the conflict issue aj . And the conflict degree of the conflict issue aj , the player Ui with respect to the conflict system and the conflict issue set C can be respectively defined as follows: n
the previous definition of conflict degree is adjusted and redefined as follows:Definition 19Suppose that the weight vector of the conflict issue set C = {a1 , a2 , ..., am } is wj = {w1 , w2 , ..., wm }, for ∀Ui , Uk ∈ U, (i, k = 1, 2, ...n, i = / k), ∀aj ∈ C, the conflict degree of the player Ui and the conflict system with respect to the conflict issue set C can be respectively defined as follows:
n
j j j j j j Si (˛i ) − Sk (˛k ) + ıi (˛i ) − ık (˛k ) + i (˛i ) − k (˛k ) ≤ 4
j
ik
j=1 k=1,i = / ki=1,i = / k
(18)
mn(n − 1)
j
j
j
C and / k; j = 1, 2, . . ., m, ik respecwhere, i, k = 1, 2, . . .n, i = ik tively stand for the conflict degree of Ui and Uk with respect to the conflict issue set C and the conflict issue aj . Because they can estimate the conflict among the players, they can be used to measure the conflict degree of the conflict information system. For the case 1, based on the expressions (16), (17) and (18), we can acquire 1 = 0.56, 2 = 0.424, 3 = 0.523, 1C = 0.772, 2C = 0.517, 3C = 0.224, = 0.503. In any conflict information system, the degree of importance of each conflict issue varies from one decision-maker to another so that issues of conflict should have different weights. Accordingly,
j
and only when i k = 0, we have j
j
j
j
i + k = 1, i as j = 0, i + k = 1,
j j Si (˛i ) − Sk (˛k ) +
j j j j j ıi (˛i ) − ık (˛k ) + i (˛i ) − k (˛k ) = 4such that ik ≤ 1. Therefore j
0 ≤ ks ≤ 1. From ik =
m
j
wj ik ,
j
0 ≤ ks ≤ 1, 0 ≤ wj ≤ 1,
j=1
m j=1
follows that 0 ≤ ik ≤ 1. QEDTheorem 2 j
C ik = 0, ik = 0 ⇔ Ui = Uk ;
j
C ik = 1, ik = 1 ⇔ Ui = −Uk .
ProofThe results follow naturally from Theorem 1. QED.
wj = 1, it
Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273
According to Theorem 2, it is known that for situations involving two players, only when they have the same attitude toward the conflict issue, there will be no conflict; when one player completely rejects the whole issue, while the other completely support, their conflict value is 1. That is, these players conflict each other totally. If these two players conflict each other completely, then one of them rejects the whole issue, while the other support the issue completely. According to the definition and analysis of the conflict analysis model in this paper, the proposed model not only describes some real conflict problems such as the election problems involved in Pawlak [42–44], but also meticulously portrays the players’ attitudes of the issues or options by means of conflict attribute values, so that players can adjust the evolution of conflict by adopting some options to control the conflict degrees of the players. For example, when the conflict degree of the conflict information system is much bigger, players can use different options with respect to different conflict issue to reduce the conflict degree, so that the system could be realize the comparative equilibrium. 4.2. Coalitions based on intuitionistic fuzzy conflict analysis model Definition 20For the conflict issue set C, X ⊆ U, ∀x, y ∈ X, ˇ ∈ [0, 1], the coalition Xˇ based on the threshold ˇ is defined as follows: Xˇ =
C x ∈ X xy ≤ ˇ, ∀x, y ∈ X
(21)
C > ˇ, no coalition among the players can be formed. If xy For the case 1, for ˇ ≤ 0.63, the two coalition between the sources of knowledge and the coordinator, and the recipients of knowledge and the coordinator with respect to the conflict issues of intellectual property rights, administration authority and knowledge gains can be formed, while the coalition between the sources of knowledge and the recipients of knowledge is not generated, which is consistent with the actual condition.Property 3For ˇ ∈ [0, 1], X ˇ ⊆ X. And only when ˇ = 1, X ˇ = X.ProofAccording to Definition 20, the results can be readily shown. All the details are omitted. QED
5. Case analysis In China, economic development and social stabilization are for the local government much more important assignments. In order to develop the economy of seven counties (Pingyu, Runan, Shangcai, Queshan, Xincai, Xiping, and Suiping), and control the contradiction among counties, Zhumadian city (Henan province, China) has to take into consideration into many issues (the construction of roads, factories, entertainment, educational institutions, railway) to understand and deal with some emerging conflict problems. For the conveniences, the seven counties (Pingyu, Runan, Shangcai, Queshan, Xincai, Xiping, and Suiping) can respectively be seen as the players and written as U1 , U2 , U3 , U4 , U5 , U6 , U7 , and the issues (the construction of roads, factories, entertainment, educational institutions, railway) are denoted as a1 , a2 , a3 , a4 , a5 , while the conflict problem can be a conflict information system. For the practical decision-making conflict information system, due to the complexity and uncertainty of the objective world, as well as the limitation of human capability to understand, the conflict values of the players with respect to the conflict issues are often given in the form of the intuitionistic fuzzy numbers. According to the circumstances of seven counties with respect to the five conflict issues considered and invested by Zhumadian in 2013, Zhumadian city government invite 70 delegates (every 10 are from the same county) to express their requirements and attitudes over the five
271
Table 2A The given intuitionistic fuzzy conflict information system.
U1 U2 U3 U4 U5 U6 U7
a1
a2
a3
a4
a5
d
<0.6,0.3> <0.7,0.3> <0.7,0.3> <0.8,0.2> <0.7,0.3> <0.6,0.2> <0.6,0.3>
<0.8,0.1> <0.8,0.2> <0.5,0.2> <0.8,0.2> <0.6,0.2> <0.5,0.1> <0.7,0.1>
<0.7,0.3> <0.6,0.3> <0.6,0.3> <0.7,0.3> <0.7,0.3> <0.6,0.3> <0.5,0.3>
<0.9,0.1> <0.7,0.3> <0.6,0.3> <0.8,0.2> <0.6,0.3> <0.7,0.3> <0.7,0.1>
<0.8,0.2> <0.8,0.2> <0.8,0.2> <0.8,0.1> <0.8,0.2> <0.6,0.2> <0.6,0.2>
1 1 2 1 1 2 2
conflict issues, so that an intuitionistic fuzzy conflict information system CIFS = (U, A, V, f ) can be acquired, whose details are given in Table 2, where U = {U1 , U2 , U3 , U4 , U5 , U6 , U7 } is the set of players, C = {a1 , a2 , a3 , a4 , a5 } the set of conflict issues, and V the range of the conflict values of the players with respect to the conflict issues. In the Table 2, d is the decision attribute, and 1 and 2 respectively express the whole acceptance and opposition of the counties; for the value <0.6,0.3> of U1 with respect to the conflict issue a1 , it indicates that 60% delegates from Pingyu clearly favorable option to demand Zhumadian city to construct the roads, while 30% delegates are going to carry out the object option, and 10% delegates are neutral. When = 0.1, based on conflict distance, the universe on the attribute set C can be divided into: condition U/C = X10.1 , X20.1 , X30.1 , X40.1 , where, X10.1 = {U1 }, X20.1 = {U2 , U4 }, X30.1 = {U3 , U5 , U6 }, X40.1 = {U7 } According to the set D of decision attributes, the universe can be divided into: U/D = {Y1 , Y2 }, where, Y1 = {U1 , U2 , U4 , U5 }, Y2 = {U3 , U6 , U7 } According to the intuitionistic fuzzy rough set based on conflict distance, the lower approximation and upper approximation of the decision-making classes Y1 and Y2 can be acquired as follows:
apr B (Y1 ) = {U1 , U2 , U4 }, apr B (Y1 ) = {U1 , U2 , U3 , U4 , U5 , U6 }
apr B (Y2 ) = {U7 }, apr B (Y2 ) = {U3 , U5 , U6 , U7 } n
For n
the
conflict
n
issue
aj ,
from
j =
i,k=1,i = / k n(n−1)
j ik
=
j ik
k=1,i = / ki=1,i = / k n(n−1)
, it follows that the conflict degrees can respectively be computed and given as follows: 1 = 0.0532, 2 = 0.0870, 3 = 0.0390, 4 = 0.1148, 5 = 0.0567. For the current conflict system, based on the conflict degrees of the conflict issue aj , it is well known that educational institutions are the most important for the seven counties, while the construction of factories is much more important. According to the attribute reduction algorithm, for details see (Liu, Jian and Liu, 2013) [45], based on the attribute dependency, the weight with respect to the conflict issue aj is respectively given as follows: wj = (w1 , w2 , w3 , w4 , w5 ) = (0.1516, 0.2481, 0.1112, 0.3274, 0.1617)
For the player set U, from ik =
m
j
wj ik , it follows that the con-
j=1
flict degrees among the players can be respectively calculated and shown in Table 2.
272
Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273
Table 2B The conflict degrees among the players.
1 2 3 4 5 6 7
1
2
3
4
5
6
7
0 0.0934 0.1389 0.0820 0.1224 0.1226 0.0874
0.0934 0 0.0455 0.0691 0.0385 0.0367 0.0920
0.1389 0.0455 0 0.1091 0.0165 0.0450 0.0929
0.0820 0.0691 0.1091 0 0.0926 0.0958 0.0901
0.1224 0.0385 0.0165 0.0926 0 0.0908 0.0908
0.1226 0.0367 0.0450 0.0958 0.0908 0 0.0672
0.0874 0.0920 0.0929 0.0901 0.0908 0.0672 0
Table 3 The coalitions based on the different threshold values. Threshold Coalitions 0.1 0.09 0.08 0.06 0.04 0.037 0.03 0.01
{U2 , U3 , U5 , U6 , U7 }, {U1 , U2 , U4 , U5 , U6 , U7 } {U1 , U3 } {U1 , U4 } {U1 , U7 } {U2 , U3 , U5 } {U2 , U4 } {U2 , U3 , U6 } {U6 , U7 } {U1 } {U2 , U4 } {U2 , U3 , U5 } {U2 , U3 , U6 } {U2 , U4 } {U6 , U7 } {U1 } {U2 , U3 , U5 } {U2 , U3 , U6 ,} {U4 } {U7 } {U1 } {U2 , U5 } {U2 , U6 } {U3 , U5 } {U4 } {U7 } {U1 } {U2 , U6 } {U3 , U5 } {U4 } {U7 } {U1 } {U2 } {U3 , U5 } {U4 } {U6 } {U7 } {U1 } {U2 } {U3 } {U4 } {U5 } {U6 } {U7 }
n
m n
C
k=1,i = / k
j ik
wj
ik
k=1,i = / k j=1
For player (country) Ui , from iC = = , n−1 (n−1) it follows that this player’s conflict degree with the conflict issues set C can be computed as follows: 1C = 0.1078, 2C = 0.0625, 3C = 0.0747, 4C = 0.0898, 5C = 0.0753, 6C = 0.0764, 7C = 0.0868. For the conflict information system, from =
m
wj j , it follows
j=1
that the situation of the conflict information system can be obtained as = 0.2610, which indicates the degree of seven countries. In order to reduce further the conflict degree of the conflict information system, Zhumadian city can take some seduced options to make seven countries change their conflict options or attitudes, so that the conflict information system could reach the equilibrium. In order to realize the cooperation and development, the countries often form the coalitions. In the case, seven countries have to cooperate for the sake of obtaining much more capital from Zhumadian city. Based on the calculations and analysis of the conflict degrees above, by adjusting the threshold , the different coalitions of the players with respect to the conflict issues can be determined, Table 3. According to Tables 2 and 3, Shangcai and Xincai are the simplest to produce the coalition, while Runan and Xiping are much simpler. Based on Tables 2 and 3, it follows that due to the differences among the conflict degrees of the seven countries, the different coalitions can be formed by setting different thresholds. As the threshold decreases, the number of elements in the resultant coalition becomes smaller. When the threshold value is less than 0.0165, there will not be any coalition among the players, because the conflict degree goes beyond the endurance range of the players and the conflict information system cannot satisfy the established threshold. By means of the analysis of Tables 2 and 3, it is well known that Zhumadian city can take some countermeasures to impel or deduce seven countries change their conflict options or attitudes, so that the conflict information system could produce much more coalitions. Based on the above analysis and results, it is seen that the key factor of determining coalitions is the definition of the conflict function and the setting of the threshold value, while the conflict analysis model based on the intuitionistic fuzzy rough set, as proposed in this paper, can well avoid the
shortcomings of the previously proposed conflict analysis model based on the rough set. At the same time, this new model not only simplifies the calculation process but also makes it possible to select different strategies of forming coalitions in order to meet different purposes. 6. Conclusions As a social phenomenon, conflict widely exists in human social lives. As the current trend of globalization continues, how to effectively describe and resolve real-world conflict decision problems is becoming an urgent issue. However, due to the complexity and uncertainty of the objective world, as well as the limitation of the human being, it has been difficult to employ a single soft computing methodology to satisfactorily deal with real-life conflict problems. Because of this recognition, we realize the need to incorporate the advantages of various methodologies of uncertainty in order to develop a truly powerful hybrid method of soft computing. To this end, in this paper, the thoughts and methods of the intuitionistic fuzzy set and rough set are used to construct the intuitionistic fuzzy rough set model based on conflict distance. Our model can to some extent soften the configuration of the conflict information system of conflict analysis model, as proposed by Pawlak, and more effectively describe and depict the real conflict decision problem. For the decision-making problems that involve either a great deal of noise or preference information in the conflict information system, the next step of research will be to construct the dominance intuitionistic fuzzy variable precision rough set model in order to resolve these problems effectively and satisfactorily. Acknowledgement This work is partially funded by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme (Grant No. FP7-PIIF-GA-2013-629051); National Natural Science Foundation of China (71301064), the key project of the National Social Science Fund (12AZD111); the Ministry of education of Humanities and Social Science Youth Fund Project (13YJC630120); Jiangsu Province University Philosophy and Social Sciences for Key Research Program (2012ZDIXM030;2013ZDIXM019); Jiangsu Province Social Science Fund Project(14GLC008); The research base of Chinese IOT development strategy (133930), the Fundamental Research Funds for the Central Universities (JUSRP11583; JUSRP1507ZD). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Contents lists available at ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Intuitionistic fuzzy rough set model based on conflict distance and applications Yong Liu a,∗ , Yi Lin b a b
School of Business, Jiangnan University, Wuxi 214122, China Mathematics Department, Slippery Rock University, Slippery Rock, PA 16057, USA
a r t i c l e
i n f o
Article history: Received 10 January 2013 Received in revised form 25 February 2015 Accepted 28 February 2015 Available online 17 March 2015 Keywords: Intuitionistic fuzzy set Fuzzy rough set model Theories of uncertainty Conflict analysis
a b s t r a c t Due to the complexity and uncertainty of the objective world, as well as the limitation of human ability to understand, it is difficult for one to employ only a single type of uncertainty method to deal with the real-life problem of decision-making, especially problems involving conflicts. On the other hand, by incorporating the advantages of various theories of uncertainty, one is expected to develop a more powerful hybrid method for soft decision making and to solve such problems more effectively. In view of this, in this paper the thought and method of intuitionistic fuzzy set and rough set are used to construct a novel intuitionistic fuzzy rough set model. Corresponding to the fact that the decisionmaking information system of rough sets is of intuitionistic fuzzy information system, our method defines the conflict distance by using the idea of measuring intuitionistic fuzzy similarity so that it is introduced into the models of rough sets, leading to the development of our intuitionistic fuzzy rough set model. After that, we investigate the properties of the model, introduce a novel tool for conflict analysis based on our hybrid model, and employ this new tool to describe and resolve a real-life conflict problem. © 2015 Elsevier B.V. All rights reserved.
1. Introduction As a useful mathematical tool to deal with knowledge with inaccuracy, uncertainty, and fuzziness, rough set theory was proposed by Pawlak [1]. This theory has ever since been widely applied in many scientific disciplines, such as knowledge discovery, data mining, decision analysis, pattern recognition, etc. [2–4]. The classical rough set theory conducts data reasoning based on equivalence relationship. It turns out to be difficult for equivalence relationship to satisfy the harsh conditions in practical applications. For example, the particular binary relationship of the discussion is often not an equivalence relation. Instead, it tends to be either a fuzzy equivalence relation or a similarity relation. In view of this observation, based on the idea and method of the fuzzy set theory, Dubois and Prade put forward the fuzzy rough sets theory and method [5]. Because this theory can well describe the uncertainty of various types of knowledge, and reflect the real world with more objectivity, it soon became the research focus of rough set theory, and
∗ Corresponding author. Tel.: +86 15061889051. E-mail addresses: [email protected] (Y. Liu), [email protected] (Y. Lin). http://dx.doi.org/10.1016/j.asoc.2015.02.045 1568-4946/© 2015 Elsevier B.V. All rights reserved.
consequently developed into a well-established theory in a short period of a few years. As a result of simultaneously considering the positive, negative and hesitancy degrees for an object to belong to a set, intuitionistic fuzzy sets possess a stronger ability of expressing information and better describing and portraying delicate ambiguities of the uncertain nature of the objective world when compared to the theory of the traditional fuzzy sets [6,7]. Thus it was suggested to combine the concepts of intuitionistic fuzzy sets and rough sets, resulting in the introduction of intuitionistic fuzzy rough set model [8]. Due to the important value of theoretical research and practical application, the intuitionistic fuzzy rough set theory has been soon becoming a hot academic research issue. Currently, the related researches on intuitionistic fuzzy rough sets lie mainly in such aspects as constructing different models, exploring the properties of the models, and simplifying the attributes of the models. For the related studies on constructing different models and exploring their properties, Cornelis, Cock, Kerre, and others first revealed the relationship between intuitionistic fuzzy set theory and rough set theory and proved the fact that each fuzzy rough set is actually an intuitionistic L-fuzzy set, and then the concept of intuitionistic fuzzy sets is used to define approximation operators in the intuitionistic fuzzy approximation space. By employing the concept of cut sets of intuitionistic fuzzy sets, the upper and
Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273
lower approximation operators of intuitionistic fuzzy rough sets and the axiomatic method of the approximation operators based on the general binary intuitionistic fuzzy relationship are constructed [8–10], and the formulas of rough set are extended to the n dimension and the semantic and semantic reasonings of the formulas are given [11–14]. Jena and Ghosh prove that the upper and lower approximation sets of intuitionistic fuzzy rough sets are intuitionistic fuzzy set [15]. Based on intuitionistic fuzzy residual implication and intuitionistic fuzzy relationship, Xu, Lei, and Tan construct an intuitionistic fuzzy rough set model [16]. However, it is difficult for the model to deal with information systems with noise data. With the intuitionistic fuzzy triangle model T = min, intuitionistic fuzzy t-conforms S = max, and intuitionistic fuzzy inverse operator N, the approximation operators of the intuitionistic fuzzy rough sets are defined by Zhou et al., and then the (I; T) intuitionistic fuzzy rough set model is constructed based on the general logic operators of the intuitionistic fuzzy [17,18]. With the thought and method of intuitionistic fuzzy set and rough set, Yin, Lei and Lei put forward an improvement intuitionistic fuzzy rough set model based on hamming distance and reveal its properties, such as interval, symmetry, complete similarity and complete dissimilarity [19]. Lin and Yang establish the interval-valued intuitionistic fuzzy rough set based on the thought of implication, and then establish the related properties of the model from the angle of mathematics [20]. By combining interval intuitionistic fuzzy set and rough set, Z. M. Zhang, J. F. Tian and others construct the interval intuitionistic fuzzy rough set model based on the interval intuitionistic fuzzy relationship [21,22]. Based on interval-valued intuitionistic fuzzy compatibility relationship, Yang establishes the interval-valued intuitionistic fuzzy rough set model based on double universes and discusses its properties [23]. Zhang, Zhou and Li propose the general framework of intuitionistic fuzzy rough set, and discuss the intuitionistic fuzzy operator and the properties of the model with the dual domain case [24]. In order to deal with the incomplete and imprecise intuitionistic fuzzy information system, Z. T. Gong exploits the thought and method of intuitionistic fuzzy set and variable precision rough set to construct an extended intuitionistic fuzzy rough set model [25], while Abdullah, Aslam, Ullah and Amin establish bipolar fuzzy soft sets and generalized fuzzy soft expert set, and respectively utilize them to make decision problems and analysis of S-box image encryption [26,27]. For the related literature of attribute reduction, with the characteristics of intuitionistic fuzzy information systems, Lu, Lei and Hua put forward the genetic algorithm to reduce attributes [28]. Chen and Yang propose a new kind of attribute reduction algorithm of intuitionistic fuzzy rough set based on mutual information by combining information entropy theory and intuitionistic fuzzy rough set [29]. Esmail, Maryam, and Habibolla make research on the properties and structure of rough set model based on intuitionistic fuzzy information system, and give the method of attributes reduction and rules extraction [30]. By using the intuitionistic fuzzy distance formula, Huang, Li, Wei and et al. construct the intuitionistic fuzzy rough model, and design an attribute reduction method [31,32]. However, the influence and effects of the core, hesitancy degree and other factors are not considered in the model. According to the above discussion on literatures, it can be seen that most of publication lies in the theoretical study and construction of models, while there are few works that address how intuitionistic fuzzy rough sets could be used to deal with practical decision-making problems, especially conflict problems. In view of these, by taking advantage of the strengths of intuitionistic fuzzy set and rough set, and by making use of the thought of the intuitionistic fuzzy similarity measure, we will construct a novel intuitionistic fuzzy rough set model, and then transform it into the intuitionistic fuzzy conflict analysis model by combining the characteristics of the conflict information system.
267
The specific arrangement of this paper is as follows: the preliminary knowledge of intuitionistic fuzzy sets and intuitionistic fuzzy information systems is outlined in Section 2. The intuitionistic fuzzy rough set model based on conflict distance is established, and then its properties are studied in Section 3. The intuitionistic fuzzy conflict analysis model is proposed in Section 4. An example is used to verify the validity and rationality of the proposed method in Section 5. The paper is concluded in Section 6. 2. Preliminary knowledge Intuitionistic fuzzy sets, as proposed by Atanassov [6,7], are an expansion and further development of the traditional fuzzy sets. As a result of taking into account to the membership and nonmembership information by adding a new attribute parameters – the non-membership function in the intuitionistic fuzzy sets, intuitionistic fuzzy sets can provide additional options for describing the properties of things and possess a capability of dealing with uncertain information. That explains why these sets can well describe and portray the delicate natural ambiguity of the objective world. 2.1. Intuitionistic fuzzy sets Definition 1[6,7]. Suppose that X = {x1 , x2 , ..., xn } is a nonempty, finite set of objects, where, xi (i = 1, 2, ..., n) is the ith object. The set A = {< x, A (x), A (x) > |x ∈ X} of triples is called an intuitionistic fuzzy set, where A (x) and A (x) are respectively called the membership and non-membership for the object x to belong to A, that is, A (x) : X → [0, 1],
x ∈ X → A (x) ∈ [0, 1]
(1)
A (x) : X → [0, 1],
x ∈ X → A (x) ∈ [0, 1]
(2)
where, 0 ≤ A (x) + A (x) ≤ 1, x ∈ X while A (x) stands for the degree of hesitation or uncertainty for the object x to belong to A and A (x) = 1 − A (x) − A (x). So, the intuitionistic fuzzy number can be denoted as ˛ =< A (x), A (x) >.Definition 2[6,7]. For any two intuitionistic fuzzy numbers ˛1 =< 1 , 1 > and ˛2 =< 2 , 2 >, the rules of arithmetic operations are given as follows: (1) (2) (3) (4)
˛1 =< v1 , 1 >; ˛1 + ˛2 =< 1 + 2 − 1 2 , v1 , v2 >; ˛1 + ˛2 = ˛1 + ˛2 ; and ˛1 + ˛2 = (˛1 + ˛2 ), > 0.
Definition 3[33]. For any intuitionistic fuzzy number ˛ =< , >, the score function S(˛) of this number is defined as follows: S(˛) = − ,
S(˛) ∈ [−1, 1]
(3)
Number example 1: for intuitionistic fuzzy number ˛ =< 0.8, 0.1 >, the score function acquired is 0.7. Note: the larger the value of S(˛) is, the greater the intuitionistic fuzzy number ˛ =< , > is.Definition 4[33]. For any intuitionistic fuzzy number ˛ =< , >, the precision function H(˛) of this number is defined as follows: H(˛) = + ,
H(˛) ∈ [0, 1]
(4)
Number example 2: For intuitionistic fuzzy number ˛ =< 0.4, 0.5 >, the precision function acquired is 0.9. Note: The larger the value of H(˛) is, the higher the precision degree of the intuitionistic fuzzy number ˛ =< , > is.Definition 5[33]. For any two intuitionistic fuzzy numbers ˛1 =< 1 , 1 > and ˛2 =< 2 , 2 >, the following hold true:
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(1) if S(˛1 ) < S(˛2 ), then ˛1 < ˛2 , (2) when S(˛1 ) = S(˛2 ), if H(˛1 ) = H(˛2 ), then ˛1 = ˛2 ; if H(˛1 ) < H(˛2 ), then ˛1 < ˛2 .
The larger the value of S(˛) is, the greater the intuitionistic fuzzy number ˛ =< [, ], [- , ] > is.Definition 10[33] Let ˛ =< [, ], [- , ] > be an interval intuitionistic fuzzy number, the precision function H(˛) of this number is defined as follows:
Number example 3: for two intuitionistic fuzzy numbers ˛1 =< 0.7, 0.2 > and ˛1 =< 0.7, 0.1 >, by the definition 3, we can get S(˛1 ) = 0.5 and S(˛2 ) = 0.6, therefore ˛1 < ˛2 . Number example 4: for two intuitionistic fuzzy numbers ˛1 =< 0.5, 0.2 > and ˛1 =< 0.4, 0.1 >, by the definition 3, we can get S(˛1 ) = 0.3 and S(˛2 ) = 0.3, however, we cannot make comparisons. According to the definition 4, H(˛1 ) = 0.7 and H(˛2 ) = 0.5, therefore, ˛1 > ˛2 .Definition 6[33]. For any intuitionistic fuzzy numbers ˛1 =< 1 , 1 > and ˛2 =< 2 , 2 >, the normalized Hamming distance of these numbers is defined as follows:
1 D(˛1 , ˛2 ) = (1 − 2 + + |1 − 2 |) (5) 2 Number example 5: for two intuitionistic fuzzy numbers ˛1 =< 0.8, 0.2 > and ˛1 =< 0.5, 0.1 >, by the definition 6, the normalized Hamming distance D(˛1 , ˛2 ) is 0.2. 2.2. Interval Intuitionistic fuzzy sets Due to the complexity and uncertainty of the objective world, as well as the limitation of the human understanding, it is often difficult to express the values of A (x) and A (x) accurately so that interval numbers should be employed. Therefore, Atanassov and others expand the concept of intuitionistic fuzz sets and put forward the concept of interval intuitionistic fuzzy sets [32,33].Definition 7[34,35] Let X = {x1 , x2 , . . ., xn } be a nonempty, finite set of objects, where xi (i = 1, 2, . . ., n) is the ith object. The interval intuitionistic fuzzy sets in X is defined as follows: A = {< x, A (x), A (x) > |x ∈ X}
(6)
where, A (x) = [A , A ] and A (x) = [ - A , A ] are respectively called the membership and non-membership for the object x to belong to A, A , A respectively the upper and lower extreme value of the membership A (x); A , - A respectively the upper and lower extreme value of the non-membership A (x), which satisfy 0 ≤ A + A ≤ 1, 0 ≤ A + - A ≤ 1, while A (x) stands for the degree of hesitation or uncertainty for the object x to belong to A and A (x) = 1 − A (x) − A (x). So the intuitionistic fuzzy number is denoted as ˛ =< A (x), A (x) >.Definition 8[34,35] For any interval intuitionistic fuzzy numbers ˛1 =< [1 , 1 ], [- 1 , 1 ] > and ˛2 =< [2 , 2 ], [- 2 , 2 ] >, the following hold true: (1) ˛1 + ˛2 =< [1 + 2 − 1 2 , 1 + 2 − - 1 2 ], [- 1 - 2 , 1 2 ] >; • (2) ˛1 ˛2 =< [1 2 , 1 2 ], [- 1 + - 2 − - 1 - 2 , 1 + 2 − 1 2 ] >; - and
(3) (˛1 ) =< 1 , 1 , [1 − (1 − - 1 ) , 1 − (1 − 1 ) ] >, -
> 0.
According to Definition 8, the operation algorithms can be obtained as follows: (1) (2) (3) (4) (5)
˛1 + ˛2 = ˛2 + ˛1 ; ˛1 • ˛2 = ˛2 • ˛1 ; (˛1 • ˛2 ) = ˛1 • ˛2 , > 0; ˛1 + ˛2 = (˛1 + ˛2 ), > 0; and ˛1 • ˛2 = (˛1 • ˛2 ) , > 0.
Definition 9[33] Let ˛ =< [, ], [- , ] > be an interval intuitionistic fuzzy number. Then the score function S(˛) of this number is defined as follows: −+− S(˛) = - , 2
S(˛) ∈ [−1, 1]
(7)
+++ H(˛) = - , 2
H(˛) ∈ [0, 1]
(8)
The larger the value of H(˛) is, the higher the precision degree of the intuitionistic fuzzy number ˛ =< , , [, ] >
is.Definition 11[33] Suppose ˛1 =< 1 , 1 , 1 , 1 > and ˛2 =< [2 , 2 ], [- 2 , 2 ] > are two interval intuitionistic fuzzy numbers. Then the following hold true: (1) if S(˛1 ) < S(˛2 ), then ˛1 < ˛2 ; and (2) when S(˛1 ) = S(˛2 ), if H(˛1 ) = H(˛2 ), then ˛1 = ˛2 ; if H(˛1 ) < H(˛2 ), then ˛1 < ˛2 . Definition 12[33] Suppose that ˛1 =< [1 1 ], [- 1 , 1 ] > and ˛2 =< [2 , 2 ], [- 2 , 2 ] > are two interval intuitionistic fuzzy numbers, the normalized Hamming distance of these numbers is defined as follows: D(˛1 , ˛2 ) =
1 4
− + 1 − 2 + 1 − 2 + 1 − 2 (9) 1 2 -
-
2.3. Intuitionistic fuzzy information system Definition 13Suppose that there exists the quadruple S = (U, A, V, f ), where, U = {U1 , U2 , . . ., Un } stands for the finite and nonempty universe set, and A = C ∪ D expresses the finite and nonempty attribute set such that C = {a1 , a2 , . . ., am } is the condition attribute set and D = {d1 , d2 , . . ., dp } the decision attribute set, while V = ∪Vib indicates the value range of U with respect to
the attribute set A such that Vib is the value of the object Ui with respect to the attribute b, and f : U × A → V stands for an information function. For ∀Ui ∈ U, ∀b ∈ A, Vib ∈ V , if Vib is an (interval)
intuitionistic fuzzy number ˛bi =< bi , ib >, then the quadruple S is known as the (interval) intuitionistic fuzzy information system, and denoted by IFS. Correspondingly, the rough sets defined on (interval) intuitionistic fuzzy information systems are known as (interval) intuitionistic fuzzy rough sets. 3. Intuitionistic fuzzy rough set model based on conflict distance
The upper and lower approximations of a rough set based on the (interval) intuitionistic fuzzy information system are often derived by using the algorithms of (interval) intuitionistic fuzzy numbers, while it is difficult for such rough sets to take full advantage of the information in the intuitionistic fuzzy information system in order to deal with decision-making problems of the real world, including problems of conflicts. Additionally, the intuitionistic fuzzy similarity measure only considers the impact of the non-membership degree and the membership degree [36–41]. For example, the similarity measure method that was put forward by Chen [36,37] to a certain extent does not agree with the real-life situation. The similarity measure, proposed by Hong and Kim [38] does not consider the effect of core values. Other similarity measure methods, such as those proposed by Zhang, Huang and Li [39,40], and Zhang and Dang [41], do not pay sufficient amount of attention to the particular backgrounds of concern when applied to deal with uncertain information. In view of this survey of the existing literature, this paper applies the thought and methodology of the similarity measure for (interval) intuitionistic fuzzy numbers to define the concept of conflict distance, on top of which a novel
Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273 Table 1 The intuitionistic fuzzy conflict information system on university-industry cooperation conflict.
U1 U2 U3
a1
a2
a3
<0.8,0.0> <0.0,0.6> <0.4,0.0>
<0.0,0.5> <0.8,0.0> <0.3,0.1>
<0.6,0.0> <0.0,0.7> <0.3,0.0>
variable precision intuitionistic fuzzy rough set model is put forward.Definition 14Suppose that IFS = (U, A, V, f ) is an (interval) intuitionistic fuzzy information system, for ∀Ui , Uk ∈ U, ∀b ∈ B ⊆ b of U , U with respect to the attribute b A, the conflict distance ik i k can be defined as follows: b ik
=
Sb (˛i ) − Sb (˛k ) + ıb (˛i ) − ıb (˛k ) + b (˛i ) − b (˛k ) i
k
i
k
i
k
4
(10)
where, B represents the attribute subset of A, while Vib , Vkb respectively stands for the values of the objects Ui , Uk with respect to the attribute b, and ˛bi =< bi , ib >, ˛bk =< bk , kb >, Sib (˛i ) = (bi − ib ), Skb (˛k ) = (bk − kb ), ib =
1 − bi − ib , kb = 1 − bk − kb , ıbi = bi + ib bi , ıbk =
bk + kb bk , ib = ib + ib ib , and kb = kb + kb kb .
b , the influence and effect of the core, For the conflict distance ik support, opposition, hesitancy degree of the (interval) intuitionistic fuzzy numbers are taken into consideration.Case 1Universityindustry cooperation is an effective way for innovation, while the conflict problem affects and restricts the cooperative performance, so that it is necessary to study the conflict problem of university-industry cooperation. The conflict of Universityindustry cooperation mainly results from the interest, which is mainly embodied with the three aspects of intellectual property rights (a1 ), administration authority (a2 ) and knowledge gains (a3 ). University-industry cooperation involves three parties, which is respectively the sources of knowledge (universities, or scientific research institutes, denoted as U1 ), the recipients of knowledge (enterprises, denoted as U2 ) and the coordinator (government, intermediary of science and technology, denoted as U3 ). In order to acquire the relevant data, we invite the 30 experts (10 experts from University, 10 University from enterprises, the other from Government, intermediary) to make interviews. By collecting and sorting, the intuitionistic fuzzy conflict information system on universityindustry cooperation conflict is obtained and shown in Table 1. For the attribute a1 , the conflict distance between U1 and U2 is 0.800; the conflict distance between U1 and U3 is 0.620, the conflict distance between U2 and U3 is 0.260. For the attribute a2 , the conflict distance between U1 and U2 is 0.753; the conflict distance between U1 and U3 is 0.310, the conflict distance between U2 and U3 is 0.210. For the attribute a1 , the conflict distance between U1 and U2 is 0.763; the conflict distance between U1 and U3 is 0.605, the conflict distance between U2 and U3 is 0.203. According to the above computing results, the conflict distance between the sources of knowledge and the recipients of knowledge with respect to intellectual property rights is the largest, which is consistent with the actual situation.Definition 15For a given (interval) intuitionistic fuzzy information system IFS = (U, AF, f ), a threshold 0 ≤ ≤ 1, B = (b1 , b2 , . . .bl ) ⊆ C, X ⊆ U, if the − B
b ≤ , ∀b ∈ B , then neighborhood bases of x is [x]B = y ∈ U yx − B -lower approximation and − B -upper approximation of X can be respectively defined by:
apr B (X) =
apr B (X) =
269
Definition 15 can be seen as an (interval) variable precision intuitionistic fuzzy Rough set model based on conflict distance. The − B -lower approximation apr B (X) of the set X, can also be called as the positive region of the (interval) intuitionistic fuzzy rough set model based on conflict distance. In other words, by a given threshold value , apr B (X) is such a set that the universe U can be definitely classified into all the element sets of the set X. The (, ˇ) − B -upper approximation apr B (X) of the set x reflects a given threshold value , and the universe U can be classified into all the element set of the set X probably. Obviously, the (interval) intuitionistic fuzzy rough set model based on conflict distance can identify the relationship of “belonging to” or “containing” to some extent, which makes it able to recognize the presence of strong non-deterministic relationships, for example, to derive the predictive rules with less-than-one probabilities. Accordingly, the upper and lower approximations of the (interval) intuitionistic fuzzy rough set model based on conflict distance can be used to define the − B boundary, positive region, approximation accuracy and classification quality of X.Definition 16For a given (interval) intuitionistic fuzzy information system IFS = (U, A, V, f ), 0 ≤ ≤ 1, the − B boundary, positive region, approximation accuracy and classification quality X are respectively defined as follows:
bndB (X) = apr B (X) − apr B (X) ˛(X) =
(13)
|apr B (X)|
(14)
|apr B (X)|
(B, D) =
|apr B (X)|
(15)
|U|
Property 1
apr B (X) ⊆ X ⊆ apr B (X) ProofLet x ∈ apr B (X). Then there exist [x]B ⊆ X and x ∈ [x]B such that
x ∈ X. Therefore apr B (X) ⊆ X. From x ∈ X, we know [x]B ∩ X = / so
that thus we obtain x ∈ apr B (X). Therefore, X ⊆ apr B (X). QEDTheorem 1For a given (interval) intuitionistic fuzzy information system IFS = (U, A, V, f ), and X ⊆ U, B1 ⊆ B2 ⊆ C, 0 ≤ 1 ≤ 2 ≤ 1, the following hold true: (1) apr B 2 (X) ⊆ apr B 1 (X); (2) apr
1 1 B1
(X) ⊆ apr
1 1 B2
(X);
apr B11 (X) ⊆ apr B12 (X); and apr B21 (X)
⊆ apr B11 (X)
Proof(1) For ∀x ∈ apr B 2 (X), there exists [x]B12 ⊆ X, due to 0 ≤ 1 ≤ 1
2 ≤ 1. So we have [x]B11 ⊆ [x]B12 such that x ∈ apr B 1 (X). There1
fore apr B 2 (X) ⊆ apr B 1 (X). And for ∀y ∈ apr B11 (X), there exists [y]B11 ∩ 1
1 2 B 12 B1 1 B1
/ such that [y] ∩ X = / , which leads to y ∈ apr B12 (X). ThereX= fore apr B11 (X) ⊆ apr (X). (2) For ∀x ∈ apr (X), there exists [x]B11 ⊆ X, due to B1 ⊆ B2 ⊆ A, we obtain [x]B21 ⊆ [x]B11 such that x ∈ apr B 1 (X). Therefore apr B 1 (X) ⊆
2
1
apr B 1 (X); and for ∀y ∈ apr B21 (X), there exists [y]B21 ∩ X = / . Because 2
/ such that y ∈ apr B11 (X). Therefore B1 ⊆ B2 ⊆ A, we have [y]B11 ∩ X = apr B21 (X) ⊆ apr B11 (X). QED 4. Conflict analysis model based on intuitionistic fuzzy rough set
{x ∈ U : [x]B ⊆ X}
(11)
4.1. Intuitionistic fuzzy conflict analysis model
{x ∈ U : [x]B ∩ X = / ϕ}
(12)
In order to describe and deal with conflict problem, Pawlak [42–44], put forward the conflict analysis model based on rough
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Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273
set (denoted as Pawlak conflict analysis model). In Pawlak conflict analysis model, S = (U, for the conflict information A, V, f ) stands
system, where, U = U1 , ..., Ui , ..., Un delegates for the players set, A expresses the conflict issue set, while V = ∪Via indicates the value range of U with respect to the attribute set A such that Via is the value of the object Ui with respect to the conflict issue a and restricted to three values {−1, 0, 1} meaning favorable, neutral and against, respectively, and f : U × A → V expresses the conflict information function. In the conflict analysis model based on rough set, as proposed by Pawlak [42–44], the players have only three attitudes with respect to the issues of concern, and the conflict information system based on the rough set is too stiff to describe and portray real-life conflict problems so that the model contains inconsistence with the problems of actual conflict. In view of this shortcoming, we will employ the thought and methodology of the intuitionistic fuzzy set to soften the players’ attitudes of the issues and construct the intuitionistic fuzzy conflict information system in accordance with the actual situation of concern and the relevant background. For example, the value of the player’s attitude of an issue is <0.7,0.2>, it could show that favorable degree, against degree are respectively 0.7 and 0.2. After that, we establish the intuitionistic fuzzy conflict analysis model to deal with the conflict problems of the objective world.Definition 17Suppose that IFS = (U, A, V, f ) is an (interval) intuitionistic fuzzy information system, if the condition attribute set C = {a1 , a2 , . . .am } stands for the conflict issue set, then IFS can be known as the (interval) intuitionistic fuzzy conflict information system, and denoted by CIFS, while the (interval) intuitionistic fuzzy rough set based on conflict distance is known as the (interval) intuitionistic fuzzy conflict analysis model. Note: In the intuitionistic fuzzy conflict analysis model, for ∀Ui ∈ U, ∀a ∈ A, Via ∈ V , if ai = 1, ia = 0, or ai = 0, ia = 1 or ai = 0.5, ia = 0.5, then Via = {−1, 0, 1}, that is the intuitionistic fuzzy conflict analysis model degrade into Pawlak conflict analysis model.Definition 18In the given (interval) intuitionistic fuzzy conflict information system CIFS, for ∀Ui , Uk ∈ U, ∀aj ∈ C, the con-
n
=
=
j=1
m
=
k=1,i = / k j=1
n m
=
m
wj j =
(19)
(n − 1) j
wj ik
j=1 i,k=1,i = / k
(20)
n(n − 1)
j=1
where wj is the weight of the players with the conflict issue aj (j = 1, 2, ..., m), satisfying
m
wj = 1.
j=1
For the case 1, based on the expressions (19) and (20), let the weight vector of the conflict issue is (0.37, 0.28, 0.35), then 1C = 0.773, 2C = 0.568, 3C = 0.248, = 0.503.Property 1 j
j
C C ik = ki .
ik = ki ,
ProofFor ∀Ui , Uk ∈ U, j
∀aj ∈ C, according to Definition 14 and Eq.
j
(10), ik = ki can be calculated by using ik =
m
j
wj ik . Therefore
j=1 C = C . QEDProperty 2 ik ki j
kk = 0,
C kk = 0.
Proof, for ∀Ui , Uk ∈ U, ∀aj ∈ C, according to Definition 14 and Eq. (10), symbolic computations indicate that j
=
j
ik
k=1,i = / k j=1
(17)
m(n − 1) j
j=1 i,k=1,i = / k
mn(n − 1)
=
ik =
from
m
j
wj ik ,
it
follows
C = 0. kk
that
j=1 j
10 ≤ ik ≤ 1,
QEDTheorem
j
follows that only when ˛i = ˛k , ik = 0, such that 0 ≤ ik . j i
j i
j i
j j ≤ 1, 0 ≤ ≤ 1, 0 ≤ + i ≤ 1, 0 ≤ k ≤ Because 0 ≤ j j j j j j j 1, 0 ≤ k ≤ 1, 0 ≤ k + k ≤ 1, Si (˛i ) = (i − i ), Sk (˛k ) = j j j j j j j j j j (k − k ), i = 1 − i − i , k = 1 − k − k , ıi = i + j j j j j j j j j j j j j j i i , ık = k + k k , i = i + i i , k = k + k k , we
obtain
n m
ik
And
j
(16)
n(n − 1)
n m
j
j
ik
k=1,i = / ki=1,i = / k
m
C ik
n−1
m
n
n
k=1,i = / k
=
j
ik
n
iC =
n−1
j
wj ik
j j j0 ≤ ikj ≤ 1.ProofAccording j j +i (˛i )−k (˛k ) S (˛ )−S (˛ ) + ı (˛ )−ı (˛ ) i k i k j i k i k to ik = 4 , and from j j j j j j 0 ≤ Si (˛i ) − Sk (˛k ), 0 ≤ ıi (˛i ) − ık (˛k ), 0 ≤ i (˛i ) − k (˛k ), it
n(n − 1)
k=1,i = / k
iC =
m n
C ik
j
i,k=1,i = / k
j =
n
kk = 0.
flict distance ik can be referred to as the conflict coefficient of Ui , Uk with respect to the conflict issue aj . And the conflict degree of the conflict issue aj , the player Ui with respect to the conflict system and the conflict issue set C can be respectively defined as follows: n
the previous definition of conflict degree is adjusted and redefined as follows:Definition 19Suppose that the weight vector of the conflict issue set C = {a1 , a2 , ..., am } is wj = {w1 , w2 , ..., wm }, for ∀Ui , Uk ∈ U, (i, k = 1, 2, ...n, i = / k), ∀aj ∈ C, the conflict degree of the player Ui and the conflict system with respect to the conflict issue set C can be respectively defined as follows:
n
j j j j j j Si (˛i ) − Sk (˛k ) + ıi (˛i ) − ık (˛k ) + i (˛i ) − k (˛k ) ≤ 4
j
ik
j=1 k=1,i = / ki=1,i = / k
(18)
mn(n − 1)
j
j
j
C and / k; j = 1, 2, . . ., m, ik respecwhere, i, k = 1, 2, . . .n, i = ik tively stand for the conflict degree of Ui and Uk with respect to the conflict issue set C and the conflict issue aj . Because they can estimate the conflict among the players, they can be used to measure the conflict degree of the conflict information system. For the case 1, based on the expressions (16), (17) and (18), we can acquire 1 = 0.56, 2 = 0.424, 3 = 0.523, 1C = 0.772, 2C = 0.517, 3C = 0.224, = 0.503. In any conflict information system, the degree of importance of each conflict issue varies from one decision-maker to another so that issues of conflict should have different weights. Accordingly,
j
and only when i k = 0, we have j
j
j
j
i + k = 1, i as j = 0, i + k = 1,
j j Si (˛i ) − Sk (˛k ) +
j j j j j ıi (˛i ) − ık (˛k ) + i (˛i ) − k (˛k ) = 4such that ik ≤ 1. Therefore j
0 ≤ ks ≤ 1. From ik =
m
j
wj ik ,
j
0 ≤ ks ≤ 1, 0 ≤ wj ≤ 1,
j=1
m j=1
follows that 0 ≤ ik ≤ 1. QEDTheorem 2 j
C ik = 0, ik = 0 ⇔ Ui = Uk ;
j
C ik = 1, ik = 1 ⇔ Ui = −Uk .
ProofThe results follow naturally from Theorem 1. QED.
wj = 1, it
Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273
According to Theorem 2, it is known that for situations involving two players, only when they have the same attitude toward the conflict issue, there will be no conflict; when one player completely rejects the whole issue, while the other completely support, their conflict value is 1. That is, these players conflict each other totally. If these two players conflict each other completely, then one of them rejects the whole issue, while the other support the issue completely. According to the definition and analysis of the conflict analysis model in this paper, the proposed model not only describes some real conflict problems such as the election problems involved in Pawlak [42–44], but also meticulously portrays the players’ attitudes of the issues or options by means of conflict attribute values, so that players can adjust the evolution of conflict by adopting some options to control the conflict degrees of the players. For example, when the conflict degree of the conflict information system is much bigger, players can use different options with respect to different conflict issue to reduce the conflict degree, so that the system could be realize the comparative equilibrium. 4.2. Coalitions based on intuitionistic fuzzy conflict analysis model Definition 20For the conflict issue set C, X ⊆ U, ∀x, y ∈ X, ˇ ∈ [0, 1], the coalition Xˇ based on the threshold ˇ is defined as follows: Xˇ =
C x ∈ X xy ≤ ˇ, ∀x, y ∈ X
(21)
C > ˇ, no coalition among the players can be formed. If xy For the case 1, for ˇ ≤ 0.63, the two coalition between the sources of knowledge and the coordinator, and the recipients of knowledge and the coordinator with respect to the conflict issues of intellectual property rights, administration authority and knowledge gains can be formed, while the coalition between the sources of knowledge and the recipients of knowledge is not generated, which is consistent with the actual condition.Property 3For ˇ ∈ [0, 1], X ˇ ⊆ X. And only when ˇ = 1, X ˇ = X.ProofAccording to Definition 20, the results can be readily shown. All the details are omitted. QED
5. Case analysis In China, economic development and social stabilization are for the local government much more important assignments. In order to develop the economy of seven counties (Pingyu, Runan, Shangcai, Queshan, Xincai, Xiping, and Suiping), and control the contradiction among counties, Zhumadian city (Henan province, China) has to take into consideration into many issues (the construction of roads, factories, entertainment, educational institutions, railway) to understand and deal with some emerging conflict problems. For the conveniences, the seven counties (Pingyu, Runan, Shangcai, Queshan, Xincai, Xiping, and Suiping) can respectively be seen as the players and written as U1 , U2 , U3 , U4 , U5 , U6 , U7 , and the issues (the construction of roads, factories, entertainment, educational institutions, railway) are denoted as a1 , a2 , a3 , a4 , a5 , while the conflict problem can be a conflict information system. For the practical decision-making conflict information system, due to the complexity and uncertainty of the objective world, as well as the limitation of human capability to understand, the conflict values of the players with respect to the conflict issues are often given in the form of the intuitionistic fuzzy numbers. According to the circumstances of seven counties with respect to the five conflict issues considered and invested by Zhumadian in 2013, Zhumadian city government invite 70 delegates (every 10 are from the same county) to express their requirements and attitudes over the five
271
Table 2A The given intuitionistic fuzzy conflict information system.
U1 U2 U3 U4 U5 U6 U7
a1
a2
a3
a4
a5
d
<0.6,0.3> <0.7,0.3> <0.7,0.3> <0.8,0.2> <0.7,0.3> <0.6,0.2> <0.6,0.3>
<0.8,0.1> <0.8,0.2> <0.5,0.2> <0.8,0.2> <0.6,0.2> <0.5,0.1> <0.7,0.1>
<0.7,0.3> <0.6,0.3> <0.6,0.3> <0.7,0.3> <0.7,0.3> <0.6,0.3> <0.5,0.3>
<0.9,0.1> <0.7,0.3> <0.6,0.3> <0.8,0.2> <0.6,0.3> <0.7,0.3> <0.7,0.1>
<0.8,0.2> <0.8,0.2> <0.8,0.2> <0.8,0.1> <0.8,0.2> <0.6,0.2> <0.6,0.2>
1 1 2 1 1 2 2
conflict issues, so that an intuitionistic fuzzy conflict information system CIFS = (U, A, V, f ) can be acquired, whose details are given in Table 2, where U = {U1 , U2 , U3 , U4 , U5 , U6 , U7 } is the set of players, C = {a1 , a2 , a3 , a4 , a5 } the set of conflict issues, and V the range of the conflict values of the players with respect to the conflict issues. In the Table 2, d is the decision attribute, and 1 and 2 respectively express the whole acceptance and opposition of the counties; for the value <0.6,0.3> of U1 with respect to the conflict issue a1 , it indicates that 60% delegates from Pingyu clearly favorable option to demand Zhumadian city to construct the roads, while 30% delegates are going to carry out the object option, and 10% delegates are neutral. When = 0.1, based on conflict distance, the universe on the attribute set C can be divided into: condition U/C = X10.1 , X20.1 , X30.1 , X40.1 , where, X10.1 = {U1 }, X20.1 = {U2 , U4 }, X30.1 = {U3 , U5 , U6 }, X40.1 = {U7 } According to the set D of decision attributes, the universe can be divided into: U/D = {Y1 , Y2 }, where, Y1 = {U1 , U2 , U4 , U5 }, Y2 = {U3 , U6 , U7 } According to the intuitionistic fuzzy rough set based on conflict distance, the lower approximation and upper approximation of the decision-making classes Y1 and Y2 can be acquired as follows:
apr B (Y1 ) = {U1 , U2 , U4 }, apr B (Y1 ) = {U1 , U2 , U3 , U4 , U5 , U6 }
apr B (Y2 ) = {U7 }, apr B (Y2 ) = {U3 , U5 , U6 , U7 } n
For n
the
conflict
n
issue
aj ,
from
j =
i,k=1,i = / k n(n−1)
j ik
=
j ik
k=1,i = / ki=1,i = / k n(n−1)
, it follows that the conflict degrees can respectively be computed and given as follows: 1 = 0.0532, 2 = 0.0870, 3 = 0.0390, 4 = 0.1148, 5 = 0.0567. For the current conflict system, based on the conflict degrees of the conflict issue aj , it is well known that educational institutions are the most important for the seven counties, while the construction of factories is much more important. According to the attribute reduction algorithm, for details see (Liu, Jian and Liu, 2013) [45], based on the attribute dependency, the weight with respect to the conflict issue aj is respectively given as follows: wj = (w1 , w2 , w3 , w4 , w5 ) = (0.1516, 0.2481, 0.1112, 0.3274, 0.1617)
For the player set U, from ik =
m
j
wj ik , it follows that the con-
j=1
flict degrees among the players can be respectively calculated and shown in Table 2.
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Y. Liu, Y. Lin / Applied Soft Computing 31 (2015) 266–273
Table 2B The conflict degrees among the players.
1 2 3 4 5 6 7
1
2
3
4
5
6
7
0 0.0934 0.1389 0.0820 0.1224 0.1226 0.0874
0.0934 0 0.0455 0.0691 0.0385 0.0367 0.0920
0.1389 0.0455 0 0.1091 0.0165 0.0450 0.0929
0.0820 0.0691 0.1091 0 0.0926 0.0958 0.0901
0.1224 0.0385 0.0165 0.0926 0 0.0908 0.0908
0.1226 0.0367 0.0450 0.0958 0.0908 0 0.0672
0.0874 0.0920 0.0929 0.0901 0.0908 0.0672 0
Table 3 The coalitions based on the different threshold values. Threshold Coalitions 0.1 0.09 0.08 0.06 0.04 0.037 0.03 0.01
{U2 , U3 , U5 , U6 , U7 }, {U1 , U2 , U4 , U5 , U6 , U7 } {U1 , U3 } {U1 , U4 } {U1 , U7 } {U2 , U3 , U5 } {U2 , U4 } {U2 , U3 , U6 } {U6 , U7 } {U1 } {U2 , U4 } {U2 , U3 , U5 } {U2 , U3 , U6 } {U2 , U4 } {U6 , U7 } {U1 } {U2 , U3 , U5 } {U2 , U3 , U6 ,} {U4 } {U7 } {U1 } {U2 , U5 } {U2 , U6 } {U3 , U5 } {U4 } {U7 } {U1 } {U2 , U6 } {U3 , U5 } {U4 } {U7 } {U1 } {U2 } {U3 , U5 } {U4 } {U6 } {U7 } {U1 } {U2 } {U3 } {U4 } {U5 } {U6 } {U7 }
n
m n
C
k=1,i = / k
j ik
wj
ik
k=1,i = / k j=1
For player (country) Ui , from iC = = , n−1 (n−1) it follows that this player’s conflict degree with the conflict issues set C can be computed as follows: 1C = 0.1078, 2C = 0.0625, 3C = 0.0747, 4C = 0.0898, 5C = 0.0753, 6C = 0.0764, 7C = 0.0868. For the conflict information system, from =
m
wj j , it follows
j=1
that the situation of the conflict information system can be obtained as = 0.2610, which indicates the degree of seven countries. In order to reduce further the conflict degree of the conflict information system, Zhumadian city can take some seduced options to make seven countries change their conflict options or attitudes, so that the conflict information system could reach the equilibrium. In order to realize the cooperation and development, the countries often form the coalitions. In the case, seven countries have to cooperate for the sake of obtaining much more capital from Zhumadian city. Based on the calculations and analysis of the conflict degrees above, by adjusting the threshold , the different coalitions of the players with respect to the conflict issues can be determined, Table 3. According to Tables 2 and 3, Shangcai and Xincai are the simplest to produce the coalition, while Runan and Xiping are much simpler. Based on Tables 2 and 3, it follows that due to the differences among the conflict degrees of the seven countries, the different coalitions can be formed by setting different thresholds. As the threshold decreases, the number of elements in the resultant coalition becomes smaller. When the threshold value is less than 0.0165, there will not be any coalition among the players, because the conflict degree goes beyond the endurance range of the players and the conflict information system cannot satisfy the established threshold. By means of the analysis of Tables 2 and 3, it is well known that Zhumadian city can take some countermeasures to impel or deduce seven countries change their conflict options or attitudes, so that the conflict information system could produce much more coalitions. Based on the above analysis and results, it is seen that the key factor of determining coalitions is the definition of the conflict function and the setting of the threshold value, while the conflict analysis model based on the intuitionistic fuzzy rough set, as proposed in this paper, can well avoid the
shortcomings of the previously proposed conflict analysis model based on the rough set. At the same time, this new model not only simplifies the calculation process but also makes it possible to select different strategies of forming coalitions in order to meet different purposes. 6. Conclusions As a social phenomenon, conflict widely exists in human social lives. As the current trend of globalization continues, how to effectively describe and resolve real-world conflict decision problems is becoming an urgent issue. However, due to the complexity and uncertainty of the objective world, as well as the limitation of the human being, it has been difficult to employ a single soft computing methodology to satisfactorily deal with real-life conflict problems. Because of this recognition, we realize the need to incorporate the advantages of various methodologies of uncertainty in order to develop a truly powerful hybrid method of soft computing. To this end, in this paper, the thoughts and methods of the intuitionistic fuzzy set and rough set are used to construct the intuitionistic fuzzy rough set model based on conflict distance. Our model can to some extent soften the configuration of the conflict information system of conflict analysis model, as proposed by Pawlak, and more effectively describe and depict the real conflict decision problem. For the decision-making problems that involve either a great deal of noise or preference information in the conflict information system, the next step of research will be to construct the dominance intuitionistic fuzzy variable precision rough set model in order to resolve these problems effectively and satisfactorily. Acknowledgement This work is partially funded by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme (Grant No. FP7-PIIF-GA-2013-629051); National Natural Science Foundation of China (71301064), the key project of the National Social Science Fund (12AZD111); the Ministry of education of Humanities and Social Science Youth Fund Project (13YJC630120); Jiangsu Province University Philosophy and Social Sciences for Key Research Program (2012ZDIXM030;2013ZDIXM019); Jiangsu Province Social Science Fund Project(14GLC008); The research base of Chinese IOT development strategy (133930), the Fundamental Research Funds for the Central Universities (JUSRP11583; JUSRP1507ZD). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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