- Email: [email protected]
Simulated annealing algorithm for the optimal translation sequence of the jth agent in rough communication
Journal of Systems Engineering and Electronics Vol. 19, No. 3, 2008, pp.507–512
Simulated annealing algorithm for the optimal translation sequence of the jth agent in rough communication Wang Hongkai1 , Guan Yanyong1 & Xue Peijun2 1. School of Sciences, Univ. of Jinan, Jinan 250022, P. R. China; 2. School of Mathematics and System Sciences, Shandong Univ., Jinan 250100, P. R. China (Received March 21, 2007)
Abstract: In rough communication, because each agent has a different language and cannot provide precise communication to each other, the concept translated among multi-agents will loss some information and this results in a less or rougher concept. With different translation sequences, the problem of information loss is varied. To get the translation sequence, in which the jth agent taking part in rough communication gets maximum information, a simulated annealing algorithm is used. Analysis and simulation of this algorithm demonstrate its effectiveness.
Keywords: rough sets, rough communication, translation sequence, optimal, simulated annealing algorithm.
1. Introduction Rough sets theory[1] , proposed by Z. Pawlak in 1982, is a novel mathematical tool to deal with vagueness and uncertainty. Rough sets theory has extensive applications in knowledge discovery, decision analysis, pattern recognition[2−6] , and so on. Rough communication[7] , a proper tool of dealing with several information sources, is presented as a novel extension of rough sets. When dealing with several sources of information where only one concept is present, each agent has a different language and cannot provide precise communication to others, so it is called rough communication. It is done by defining the approximation of a rough set from one approximation space to another. By doing this, we lose some information or precision of the concept and this results in a less precise or rougher concept. To solve the problem of information loss in rough communication, Ref. [8] makes some quantization analysis by defining the concepts of the degree of fidelity, the degree of distortion, and rough information flow. Ref. [9] proposes the concepts of lower and upper approximation information rough communication
using rough sets. It analyzes the conditions that cause information to remain invariable or cause information to be lost or gained, and obtains some important conclusions. Using the concepts of common knowledge and possible knowledge, Ref. [10] analyzes the concept translation result under different sequences of rough communication. It is highly significant for taking a decision when we do not know the exact translation sequence. A new definition of rough communication is proposed in Ref. [11], and the relation theorem between old-rough communication[7] and new-rough communication is obtained. Refs. [12–13] study the rough communication of a fuzzy concept. By using α−rough communication cut, the relation theorem between rough communication of fuzzy concept and that of classical concept is obtained. Refs. [14–15] study the rough communication of dynamic concept. There are n ! translation sequences in rough communication, where n agents take part. In fact, with different paths, the amount of information loss is varied. How to find the optimal translation sequence, in which the jth agent taking part in rough communication of a concept A gets maximum information, is a meaningful problem. In view of the specialty of this
* This project was supported by the Natural Science Foundation of Shandong Province (Y2006A12); the Scientific Research Development Project of Shandong Provincial Education Department (J06P01) and the Doctoral Foundation of University of Jinan (B0633).
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Wang Hongkai, Guan Yanyong & Xue Peijun
problem, a simulated annealing algorithm is used. Analysis and simulation of this algorithm demonstrate its effectiveness of the approach. The simulated annealing algorithm[16] proposed by Kirkpartrick in 1983 is a stochastic optimization algorithm, which simulates the process of energetics of metal with high temperature, which decreases during the annealing of solid matter similar with the process of ordinary combinatorial optimization. Simulated annealing algorithm is different from other local search algorithms, such as greedy algorithm in search strategy. To avoid getting into local optimization, in the process of search, it accepts not only the solution in neighborhood, which makes energy function fall, but also the solution making energy function increase by given probability. The simulated annealing algorithm is an effective global optimization method [17] , and it has extensive applications in optimization fields.
2. Rough communication[7] Let U be the universe set and R be an equivalence relation on U ; the family of all definable sets in the approximation space apr is denoted by Def (apr); A+ , A− ∈ Def (apr), A+ ∩ A− = φ, call (A+ , A− ) a rough set in approximation space (U, R), and +
A = RA −
A = U − RA
(1) (2)
Let (U, R1 ), (U, R2 ), . . . , (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , agent n, respectively. Therefore, according to Fig. 1, agent 1 has a direct translation of crisp concept A and the knowledge of agent 2 about concept A is received through agent 1 and so on.
− + − (A+ i , Ai ) = (Ri Ai−1 , Ri Ai−1 ),
2in
(3)
where the family of all rough sets in approximation space (U, Rk ) is denoted by Rough(aprk ). A family of n rough sets in which all are related to a same concept A, but by different agents, it is called a sequence on population and is denoted by SEQA − + − + − SEQA = {(A+ 1 , A1 ), (A2 , A2 ), . . . , (An , An )} − (A+ k , Ak ) ∈ Rough(aprk ),
k ∈ {1, 2, . . . , n}
(4)
− (A+ 1 , A1 ) = (R1 A, U − R1 A);
where
− + − (A+ i , Ai ) = (Ri Ai−1 , Ri Ai−1 ),
2in
+ − − Ri+1 A+ i ⊆ Ai , Ri+1 Ai ⊆ Ai , which denotes that − the information represented by (A+ i , Ai ) is less than − that of (A+ i−1 , Ai−1 ). Obviously, in the translation process, we lose some information or precision of the concept A, which results in a less precise or rougher concept.
3. Mathematical model for the optimal translation sequence of the jth agent in rough communication Definition 1 Let (U, R1 ), (U, R2 ), . . . , and (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , and agent n respectively (n 2). Consider j − OP T − SEQA = {(A+(o) , A−(o) ), (A+(o) , A−(o) ), . . . , (A+(o) , A−(o) )} as the i1
i1
i2
i2
in
in
optimal translation sequence of the jth agent in rough communication (j ∈ {1, 2, . . . , n}), if |A+(o) | + |A−(o) | = max {|A+(m) | + |A−(m) |} ij
ij
(o)
(o)
ij
1mn!
ij
(5)
(o)
where i1 , i2 , . . . , in is a full arrangement of (m) (m) (m) 1, 2, . . . , n; i1 , i2 , . . . , in (m = 1, 2, . . . , n !) are n ! full arrangements of 1, 2, . . . , n. |A| is the cardinal number of the classical set A. Fig. 1
A sequence on a population
− Approximation of (A+ i−1 , Ai−1 ) to approximation space (U, Ri ) is a rough set in Rough (apri ) defined as follows
Definition 2 Let (U, R1 ), (U, R2 ), . . . , and (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , and agent n, respectively (n 2). Consider − + − + − SEQA = {(A+ , A ), (A , A ), . . . , (A , A in in )} to i i i i 1
1
2
2
Simulated annealing algorithm for the optimal translation sequence of the jth agent in rough communication be more optimal than SEQA − + , A− )}, if (A+ i i , Ai ), . . . , (Ai n n 2
=
− {(A+ i , Ai ), 1
1
2
− + − |A+ i | + |Ai | > |Ai | + |Ai | j
j
j
j
(6)
where i1 , i2 , . . . , in and i1 , i2 , . . . , in are two full arrangements of 1, 2, . . . , n. Suppose the feasible solution for the problem of optimal translation sequence of the jth agent in rough communication (j ∈ {1, 2, . . . , n}) is P = (pi1 , pi2 , . . . , pin ), where pik denotes the agent ik (k = 1, 2, . . . , n) taking part in rough communication. Each feasible solution corresponds to a full arrangement of 1, 2, . . . , n, and feasible region corresponds to all full arrangements of 1, 2, . . . , n. Obviously, there are n ! feasible solutions for the problem. Feasible solution P = (pi1 , pi2 , . . . , pin ) corresponds to rough communication translation sequence − + − + − SEQA = {(A+ i1 , Ai1 ), (Ai2 , Ai2 ), . . . , (Ain , Ain )} (as shown in Fig. 2).
also optimal solutions for the problem. Where i1 , . . . , im , . . . , iq , . . . , in is a full arrangement of 1, 2, . . . , n; i1 , . . . , iq , . . . , im , . . . , in is also a full arrangement of 1, 2, . . . , n, which trades im with iq in i1 , . . . , im , . . . , iq , . . . , in , and the other elements in i1 , . . . , im , . . . , iq , . . . , in are fixed. Proposition 2 Let (U, R1 ), (U, R2 ), . . . , and (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , and agent n, respectively (n 2). If the problem of rough communication translation sequence optimality of the jth agent (j ∈ {1, 2, . . . , n}) has the optimal solutions − P (O) = (pi1 , . . . , pij , . . . , pin ) and (A+ ij , Aij ) = (φ, φ), (Om ) then P = (pi(m) , pi(m) , . . . , pi(m) ), m = 1, 2, . . . , n ! 1
A sequence on a population
The mathematical model for the optimal translation sequence of the jth agent in rough communication (j ∈ {1, 2, . . . , n}) is as follows − max{|A+ ij | + |Aij |}
(7)
s.t.i1 , i2 , . . . , in is an arbitrary full arrangement of 1, 2, · · · , n. From Definitions 1 and 2, and the concept of rough communication, it is easy to get the following propositions. Proposition 1 Let (U, R1 ), (U, R2 ), . . . , and (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , and agent n, respectively (n 2). If the problem of rough communication translation sequence optimality of the jth agent (j ∈ {1, 2, . . . , n}) has the optimal solutions P (O1 ) = (pi1 , . . . , pim , . . . , piq , . . . , pin ) and Rim = Riq , then P (O2 ) = (pi1 , . . . , piq , . . . , pim , . . . , pin ) are
2
n
(o)
(o)
(o)
are also optimal solutions. Where i1 , i2 , . . . , in is (m) (m) (m) an arrangement of 1, 2, . . . , n; i1 , i2 , . . . , in is an arbitrary full arrangement of 1, 2, . . . , n.
4. Simulated annealing algorithm We use the simulated annealing algorithm to resolve mathematical model (7) through iterative strategy of feasible solution, control of temperature parameter, and termination condition of algorithm. 4.1
Fig. 2
509
Iterative strategy of solution
There are two contents in the iterative strategy of solution: one is the decision of neighborhood of feasible solution, the other is the iterative method of solution in neighborhood. Neighborhood of solution: an arbitrary full arrangement corresponds to a feasible solution P = (pi1 , pi2 , . . . , pin ) of this question, so adopt 2 − Opt mapping to product neighborhood. The number of feasible solution is |N (P )| = Cn2 + 1. Iterative method of feasible solution: generation of new feasible solution in the neighborhood of current feasible solution, and its generation probability is subjected to uniform distribution, that is, ⎧ ⎨ 1/|N (P )|, P ∈ N (P ) GP P (t) = ⎩ / N (P ) 0, P ∈ 4.2
Control of temperature parameter
In this section, we adopt non-homogeneous simulated annealing algorithm, so the control of temperature pa-
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Wang Hongkai, Guan Yanyong & Xue Peijun
rameter includes: (1) choice of the original temperature; (2) method of temperature reduction. The original temperature of simulated annealing algorithm should be chosen as high enough, such that it cannot fall into the trap of local optimal solution immediately; but the original temperature of simulated annealing algorithm should not too high to lessen redundant iteration. We use the method in Refs. [18–19] to get the original temperature t0 t0 =
Fmin − Fmax ln P0
where P0 ∈ (0, 1) is the original control probability. Stochastically generate L original feasible solutions P1 , P2 , . . . , PL , and calculate the maximum value Fmax and the minimum value Fmin in terms of object − function {|A+ ij | + |Aij |}. Temperature reduction adopts the method of Lundy and Mess tr+1 = tr (1 + β tr )−1 t0 − tf , tf is a given relative small positive M t 0 tf integer, and M is the maximum steps of temperature reduction.
where β =
4.3
Termination condition of algorithm
There are two conditions synchronously to control the algorithm to stop. (1) Control method of all the iterative steps: if the number of all the iterative steps is beyond M , the algorithm stops. (2) Control method of non-mending rules: if the current optimal solution does not change in continuous Q steps of temperature reduction, the algorithm stops. If the algorithm is subjected to one of the above conditions, then it can stop. 4.4
Calculating process of simulated annealing algorithm
(1) If R1 = R2 = . . . = Rn , then the optimal solution is P = (pi1 , pi2 , . . . , pin ), where i1 , i2 , . . . , in is an arbitrary full arrangement of 1, 2, . . . , n, and the algorithm stops; otherwise, go to step (2). (2) Stochastically generate L original feasible solutions P1 , P2 , . . . , PL , and get the original temperature
t0 . In terms of the object function, get optimal solution P ∗ within P1 , P2 , . . . , PL and make P ∗ be the current solution P , and let k = 0. (3) By the iterative strategy of solution, get new solution P and calculate its object function value FP . (4) If exp[(FP − FP )/tr ] > random(0, 1), or FP FP , then substitute P for P , and P is the new current solution, and if FP > FP ∗ , then substitute P for P ∗ , and P is the new optimal solution; otherwise, do not change. (5) If current optimal solution does not change in continuous Q steps of temperature reduction, the algorithm stops; otherwise, go to step (6). (6) Temperature reduction: tr+1 = tr (1 + β tr )−1 , and let k = k + 1. (7) If k = M , then algorithm stops; otherwise, let k = 0 and go to step (3). Where random(0, 1) is a stochastically generated real number, which is greater than 0 and less than 1.
5. Simulation There are five agents who take part in a rough communication. The universe set consists of sixteen toys, that is, U = {t1 , t2 , . . . , t16 }. Agents 1, 2, 3, 4, and 5 classify the toys according to their color, shape, size, price, and producing area, respectively: Agent 1: {t1 , t2 , t3 }, {t4 }, {t5 , t6 , t7 }, {t8 }, {t9 }, {t10 , t11 , t12 }, {t13 , t14 }, {t15 , t16 }; {t2 , t4 }, {t5 , t6 }, {t7 , t9 }, Agent 2: {t1 , t3 }, {t8 , t10 , t12 }, {t11 , t13 }, {t14 , t15 , t16 }; Agent 3: {t1 , t3 }, {t2 , t4 , t8 }, {t5 , t6 }, {t7 , t9 }, {t10 , t11 , t12 , t13 , t14 }, {t15 , t16 }; Agent 4: {t1 , t3 }, {t2 , t4 }, {t5 }, {t6 , t7 }, {t8 , t9 , t10 }, {t11 , t12 , t13 , t14 }, {t15 }, {t16 }; Agent 5: {t1 , t2 , t4 }, {t5 , t6 }, {t3 , t7 , t9 }, {t8 }, {t10 , t11 , t12 }, {t13 , t14 }, {t15 , t16 }. Suppose A = {t1 , t3 , t5 , t6 , t7 , t9 , t10 , t12 , t15 , t16 } and j = 3. Using the simulated annealing algorithm given in Section 4, simulation is done, with L = 6, P0 = 0.1, tf = 8, Q = 50, M = 5 000. The results of simulation of this question are given in Table 1. From the simulation results, we can find that this simulated annealing is effective, and it can get the global optimization (approximate) solution.
Simulated annealing algorithm for the optimal translation sequence of the jth agent in rough communication Table 1
Results of simulation
511
tion, 2003, 36: 3015–3018. [7] Mousavi A, Jabedar-Maralani P. Double-faced rough sets
Simulation
1
2
Solution
(p3 , p2 , p4 , p5 , p1 )
(p5 , p3 , p1 , p2 , p4 )
and rough communication. Information Sciences, 2002,
Value
7
6
Simulation
3
4
Solution
(p3 , p5 , p1 , p2 , p4 )
(p3 , p2 , p4 , p1 , p5 )
Measure in Rough Communication. International Journal
Value
7
7
of Advances in Systems Science and Applications, 2005,
Simulation
5
6
Solution
(p3 , p5 , p1 , p4 , p2 )
(p3 , p2 , p4 , p1 , p5 )
Value
7
7
148: 41–53. [8] Wang Hongkai, Li Xiuhong, Shi Kaiquan.
Information
5(4): 638–643. [9] Liu Jiqin.
Information rough communication based on
rough sets. Journal of Systems Engineering and Electronics, 2007, 29(3): 437–442.
6. Conclusions
[10] Guan Yanyong, Wang Hongkai, Yao Bingxue. Analysis
During rough communication[7] , we lose some information or precision of the concept and this results in a less precise or rougher concept. In fact, with different paths, the amount of the missed knowledge is varied. The concept of optimal translation sequence of the jth agent in rough communication (j ∈ {1, 2, . . . , n}) is proposed and its corresponding optimality mathematical model is also constructed. On the basis of the specialty of this problem, simulated annealing algorithm is used. Analysis and simulation of this algorithm demonstrate its effectiveness.
of the concept translation result in rough communication. Proceedings of International Conference on Artificial Intelligence, 2006: 381–384. [11] Wang Hongkai, Zhao Shuli. A new definition form of rough communication. Computer science, 2006, 33(9): 189–190. (in Chinese) [12] Wang Hongkai, Xue Peijun, Shi Kaiquan. The problem of rough communication of fuzzy concept. The 11th Joint International Computer Conference, 2005: 674–677. [13] Wang Hongkai, Guan Yanyong, Yao Bingxue, et al. Fuzzy rough communication and optimization problem of its translation sequence.
References
tional Conference on Machine Learning and Cybernetics,
[1] Pawlak Z. Rough sets. International Journal of Computer
2006: 2268–2273. [14] Wang Hongkai, Liu Baocang, Guan Yanyong. One direc-
and Information Science, 1982, 11: 341–356. [2] Tsai Ying-Chieh, Cheng Ching-Hsue, Chang Jing-Rong. Entropy-based fuzzy rough classification approach for extracting classification rules. Expert Systems with Applica-
tion expansion rough communication of dynamic concept based on α−generation of assistant sets.
International
Journal of Advances in Systems Science and Applications, 2007, 7(1): 93–96.
tions, 2006, 31: 436–443. [3] Shyng Jhieh-Yu, Wang Fang-Kuo, Tzeng Gwo-Hshiung, et al. Rough set theory in analyzing the attributes of combination values for the insurance market. Expert Systems
[15] Wang Hongkai, Yao Jinjiang, Xue Peijun, et al. Rough communication of dynamic concept. Advances in soft computing. Proc. of the Second International Conference of Fuzzy Information and Engineering, 2007: 781–787.
with Applications, 2007, 32: 56–64. [4] Leung Yee, Wu Wei-Zhi, Zhang Wen-Xiu.
IEEE Proc. of the fifth Interna-
Knowledge
acquisition in incomplete information systems: a rough set approach. European Journal of Operational Research, 2006, 168: 164–180. [5] Pawlak Z, Slowinski R. Rough set approach to multiattribute decision analysis. European Journal of Operational Research, 1994, 72: 443–459. [6] Asharaf S, Murty M N. An adaptive rough fuzzy single pass algorithm for clustering large data sets. Pattern Recogni-
[16] Kirkpatrick S, Gelatt Jr C D, Vecchi M P. Optimization by simulated annealing. Science, 1983, 220: 671–680. [17] Kang Lishan, Xie Yun, You Shiyong, et al. Non-numerical parallel computation (Book One) SA. Beijing : Science Press, 1998. [18] Wang Ling, Zheng Dazhong. A class of simulated annealing approach for multi-objective optimization. Computer Engineering and Applications, 2002, 38(8): 4–5. (in Chinese)
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[19] Feng TseLin, Cheng YanKao, Ching ChiHsu. Applying the
Wang Hongkai was born in 1978. He is a lecturer of
genetic approach to simulated annealing in solving some NP-hard problems. IEEE Trans. on SMC, 1993, 6: 1752–
University of Jinan, Ph. D. of Shandong University. His main research fields include rough sets theory and
1767.
fuzzy sets theory. E-mail: ss [email protected]
Simulated annealing algorithm for the optimal translation sequence of the jth agent in rough communication Wang Hongkai1 , Guan Yanyong1 & Xue Peijun2 1. School of Sciences, Univ. of Jinan, Jinan 250022, P. R. China; 2. School of Mathematics and System Sciences, Shandong Univ., Jinan 250100, P. R. China (Received March 21, 2007)
Abstract: In rough communication, because each agent has a different language and cannot provide precise communication to each other, the concept translated among multi-agents will loss some information and this results in a less or rougher concept. With different translation sequences, the problem of information loss is varied. To get the translation sequence, in which the jth agent taking part in rough communication gets maximum information, a simulated annealing algorithm is used. Analysis and simulation of this algorithm demonstrate its effectiveness.
Keywords: rough sets, rough communication, translation sequence, optimal, simulated annealing algorithm.
1. Introduction Rough sets theory[1] , proposed by Z. Pawlak in 1982, is a novel mathematical tool to deal with vagueness and uncertainty. Rough sets theory has extensive applications in knowledge discovery, decision analysis, pattern recognition[2−6] , and so on. Rough communication[7] , a proper tool of dealing with several information sources, is presented as a novel extension of rough sets. When dealing with several sources of information where only one concept is present, each agent has a different language and cannot provide precise communication to others, so it is called rough communication. It is done by defining the approximation of a rough set from one approximation space to another. By doing this, we lose some information or precision of the concept and this results in a less precise or rougher concept. To solve the problem of information loss in rough communication, Ref. [8] makes some quantization analysis by defining the concepts of the degree of fidelity, the degree of distortion, and rough information flow. Ref. [9] proposes the concepts of lower and upper approximation information rough communication
using rough sets. It analyzes the conditions that cause information to remain invariable or cause information to be lost or gained, and obtains some important conclusions. Using the concepts of common knowledge and possible knowledge, Ref. [10] analyzes the concept translation result under different sequences of rough communication. It is highly significant for taking a decision when we do not know the exact translation sequence. A new definition of rough communication is proposed in Ref. [11], and the relation theorem between old-rough communication[7] and new-rough communication is obtained. Refs. [12–13] study the rough communication of a fuzzy concept. By using α−rough communication cut, the relation theorem between rough communication of fuzzy concept and that of classical concept is obtained. Refs. [14–15] study the rough communication of dynamic concept. There are n ! translation sequences in rough communication, where n agents take part. In fact, with different paths, the amount of information loss is varied. How to find the optimal translation sequence, in which the jth agent taking part in rough communication of a concept A gets maximum information, is a meaningful problem. In view of the specialty of this
* This project was supported by the Natural Science Foundation of Shandong Province (Y2006A12); the Scientific Research Development Project of Shandong Provincial Education Department (J06P01) and the Doctoral Foundation of University of Jinan (B0633).
508
Wang Hongkai, Guan Yanyong & Xue Peijun
problem, a simulated annealing algorithm is used. Analysis and simulation of this algorithm demonstrate its effectiveness of the approach. The simulated annealing algorithm[16] proposed by Kirkpartrick in 1983 is a stochastic optimization algorithm, which simulates the process of energetics of metal with high temperature, which decreases during the annealing of solid matter similar with the process of ordinary combinatorial optimization. Simulated annealing algorithm is different from other local search algorithms, such as greedy algorithm in search strategy. To avoid getting into local optimization, in the process of search, it accepts not only the solution in neighborhood, which makes energy function fall, but also the solution making energy function increase by given probability. The simulated annealing algorithm is an effective global optimization method [17] , and it has extensive applications in optimization fields.
2. Rough communication[7] Let U be the universe set and R be an equivalence relation on U ; the family of all definable sets in the approximation space apr is denoted by Def (apr); A+ , A− ∈ Def (apr), A+ ∩ A− = φ, call (A+ , A− ) a rough set in approximation space (U, R), and +
A = RA −
A = U − RA
(1) (2)
Let (U, R1 ), (U, R2 ), . . . , (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , agent n, respectively. Therefore, according to Fig. 1, agent 1 has a direct translation of crisp concept A and the knowledge of agent 2 about concept A is received through agent 1 and so on.
− + − (A+ i , Ai ) = (Ri Ai−1 , Ri Ai−1 ),
2in
(3)
where the family of all rough sets in approximation space (U, Rk ) is denoted by Rough(aprk ). A family of n rough sets in which all are related to a same concept A, but by different agents, it is called a sequence on population and is denoted by SEQA − + − + − SEQA = {(A+ 1 , A1 ), (A2 , A2 ), . . . , (An , An )} − (A+ k , Ak ) ∈ Rough(aprk ),
k ∈ {1, 2, . . . , n}
(4)
− (A+ 1 , A1 ) = (R1 A, U − R1 A);
where
− + − (A+ i , Ai ) = (Ri Ai−1 , Ri Ai−1 ),
2in
+ − − Ri+1 A+ i ⊆ Ai , Ri+1 Ai ⊆ Ai , which denotes that − the information represented by (A+ i , Ai ) is less than − that of (A+ i−1 , Ai−1 ). Obviously, in the translation process, we lose some information or precision of the concept A, which results in a less precise or rougher concept.
3. Mathematical model for the optimal translation sequence of the jth agent in rough communication Definition 1 Let (U, R1 ), (U, R2 ), . . . , and (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , and agent n respectively (n 2). Consider j − OP T − SEQA = {(A+(o) , A−(o) ), (A+(o) , A−(o) ), . . . , (A+(o) , A−(o) )} as the i1
i1
i2
i2
in
in
optimal translation sequence of the jth agent in rough communication (j ∈ {1, 2, . . . , n}), if |A+(o) | + |A−(o) | = max {|A+(m) | + |A−(m) |} ij
ij
(o)
(o)
ij
1mn!
ij
(5)
(o)
where i1 , i2 , . . . , in is a full arrangement of (m) (m) (m) 1, 2, . . . , n; i1 , i2 , . . . , in (m = 1, 2, . . . , n !) are n ! full arrangements of 1, 2, . . . , n. |A| is the cardinal number of the classical set A. Fig. 1
A sequence on a population
− Approximation of (A+ i−1 , Ai−1 ) to approximation space (U, Ri ) is a rough set in Rough (apri ) defined as follows
Definition 2 Let (U, R1 ), (U, R2 ), . . . , and (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , and agent n, respectively (n 2). Consider − + − + − SEQA = {(A+ , A ), (A , A ), . . . , (A , A in in )} to i i i i 1
1
2
2
Simulated annealing algorithm for the optimal translation sequence of the jth agent in rough communication be more optimal than SEQA − + , A− )}, if (A+ i i , Ai ), . . . , (Ai n n 2
=
− {(A+ i , Ai ), 1
1
2
− + − |A+ i | + |Ai | > |Ai | + |Ai | j
j
j
j
(6)
where i1 , i2 , . . . , in and i1 , i2 , . . . , in are two full arrangements of 1, 2, . . . , n. Suppose the feasible solution for the problem of optimal translation sequence of the jth agent in rough communication (j ∈ {1, 2, . . . , n}) is P = (pi1 , pi2 , . . . , pin ), where pik denotes the agent ik (k = 1, 2, . . . , n) taking part in rough communication. Each feasible solution corresponds to a full arrangement of 1, 2, . . . , n, and feasible region corresponds to all full arrangements of 1, 2, . . . , n. Obviously, there are n ! feasible solutions for the problem. Feasible solution P = (pi1 , pi2 , . . . , pin ) corresponds to rough communication translation sequence − + − + − SEQA = {(A+ i1 , Ai1 ), (Ai2 , Ai2 ), . . . , (Ain , Ain )} (as shown in Fig. 2).
also optimal solutions for the problem. Where i1 , . . . , im , . . . , iq , . . . , in is a full arrangement of 1, 2, . . . , n; i1 , . . . , iq , . . . , im , . . . , in is also a full arrangement of 1, 2, . . . , n, which trades im with iq in i1 , . . . , im , . . . , iq , . . . , in , and the other elements in i1 , . . . , im , . . . , iq , . . . , in are fixed. Proposition 2 Let (U, R1 ), (U, R2 ), . . . , and (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , and agent n, respectively (n 2). If the problem of rough communication translation sequence optimality of the jth agent (j ∈ {1, 2, . . . , n}) has the optimal solutions − P (O) = (pi1 , . . . , pij , . . . , pin ) and (A+ ij , Aij ) = (φ, φ), (Om ) then P = (pi(m) , pi(m) , . . . , pi(m) ), m = 1, 2, . . . , n ! 1
A sequence on a population
The mathematical model for the optimal translation sequence of the jth agent in rough communication (j ∈ {1, 2, . . . , n}) is as follows − max{|A+ ij | + |Aij |}
(7)
s.t.i1 , i2 , . . . , in is an arbitrary full arrangement of 1, 2, · · · , n. From Definitions 1 and 2, and the concept of rough communication, it is easy to get the following propositions. Proposition 1 Let (U, R1 ), (U, R2 ), . . . , and (U, Rn ) be n approximation spaces, which represent the knowledge of agent 1, agent 2, . . . , and agent n, respectively (n 2). If the problem of rough communication translation sequence optimality of the jth agent (j ∈ {1, 2, . . . , n}) has the optimal solutions P (O1 ) = (pi1 , . . . , pim , . . . , piq , . . . , pin ) and Rim = Riq , then P (O2 ) = (pi1 , . . . , piq , . . . , pim , . . . , pin ) are
2
n
(o)
(o)
(o)
are also optimal solutions. Where i1 , i2 , . . . , in is (m) (m) (m) an arrangement of 1, 2, . . . , n; i1 , i2 , . . . , in is an arbitrary full arrangement of 1, 2, . . . , n.
4. Simulated annealing algorithm We use the simulated annealing algorithm to resolve mathematical model (7) through iterative strategy of feasible solution, control of temperature parameter, and termination condition of algorithm. 4.1
Fig. 2
509
Iterative strategy of solution
There are two contents in the iterative strategy of solution: one is the decision of neighborhood of feasible solution, the other is the iterative method of solution in neighborhood. Neighborhood of solution: an arbitrary full arrangement corresponds to a feasible solution P = (pi1 , pi2 , . . . , pin ) of this question, so adopt 2 − Opt mapping to product neighborhood. The number of feasible solution is |N (P )| = Cn2 + 1. Iterative method of feasible solution: generation of new feasible solution in the neighborhood of current feasible solution, and its generation probability is subjected to uniform distribution, that is, ⎧ ⎨ 1/|N (P )|, P ∈ N (P ) GP P (t) = ⎩ / N (P ) 0, P ∈ 4.2
Control of temperature parameter
In this section, we adopt non-homogeneous simulated annealing algorithm, so the control of temperature pa-
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Wang Hongkai, Guan Yanyong & Xue Peijun
rameter includes: (1) choice of the original temperature; (2) method of temperature reduction. The original temperature of simulated annealing algorithm should be chosen as high enough, such that it cannot fall into the trap of local optimal solution immediately; but the original temperature of simulated annealing algorithm should not too high to lessen redundant iteration. We use the method in Refs. [18–19] to get the original temperature t0 t0 =
Fmin − Fmax ln P0
where P0 ∈ (0, 1) is the original control probability. Stochastically generate L original feasible solutions P1 , P2 , . . . , PL , and calculate the maximum value Fmax and the minimum value Fmin in terms of object − function {|A+ ij | + |Aij |}. Temperature reduction adopts the method of Lundy and Mess tr+1 = tr (1 + β tr )−1 t0 − tf , tf is a given relative small positive M t 0 tf integer, and M is the maximum steps of temperature reduction.
where β =
4.3
Termination condition of algorithm
There are two conditions synchronously to control the algorithm to stop. (1) Control method of all the iterative steps: if the number of all the iterative steps is beyond M , the algorithm stops. (2) Control method of non-mending rules: if the current optimal solution does not change in continuous Q steps of temperature reduction, the algorithm stops. If the algorithm is subjected to one of the above conditions, then it can stop. 4.4
Calculating process of simulated annealing algorithm
(1) If R1 = R2 = . . . = Rn , then the optimal solution is P = (pi1 , pi2 , . . . , pin ), where i1 , i2 , . . . , in is an arbitrary full arrangement of 1, 2, . . . , n, and the algorithm stops; otherwise, go to step (2). (2) Stochastically generate L original feasible solutions P1 , P2 , . . . , PL , and get the original temperature
t0 . In terms of the object function, get optimal solution P ∗ within P1 , P2 , . . . , PL and make P ∗ be the current solution P , and let k = 0. (3) By the iterative strategy of solution, get new solution P and calculate its object function value FP . (4) If exp[(FP − FP )/tr ] > random(0, 1), or FP FP , then substitute P for P , and P is the new current solution, and if FP > FP ∗ , then substitute P for P ∗ , and P is the new optimal solution; otherwise, do not change. (5) If current optimal solution does not change in continuous Q steps of temperature reduction, the algorithm stops; otherwise, go to step (6). (6) Temperature reduction: tr+1 = tr (1 + β tr )−1 , and let k = k + 1. (7) If k = M , then algorithm stops; otherwise, let k = 0 and go to step (3). Where random(0, 1) is a stochastically generated real number, which is greater than 0 and less than 1.
5. Simulation There are five agents who take part in a rough communication. The universe set consists of sixteen toys, that is, U = {t1 , t2 , . . . , t16 }. Agents 1, 2, 3, 4, and 5 classify the toys according to their color, shape, size, price, and producing area, respectively: Agent 1: {t1 , t2 , t3 }, {t4 }, {t5 , t6 , t7 }, {t8 }, {t9 }, {t10 , t11 , t12 }, {t13 , t14 }, {t15 , t16 }; {t2 , t4 }, {t5 , t6 }, {t7 , t9 }, Agent 2: {t1 , t3 }, {t8 , t10 , t12 }, {t11 , t13 }, {t14 , t15 , t16 }; Agent 3: {t1 , t3 }, {t2 , t4 , t8 }, {t5 , t6 }, {t7 , t9 }, {t10 , t11 , t12 , t13 , t14 }, {t15 , t16 }; Agent 4: {t1 , t3 }, {t2 , t4 }, {t5 }, {t6 , t7 }, {t8 , t9 , t10 }, {t11 , t12 , t13 , t14 }, {t15 }, {t16 }; Agent 5: {t1 , t2 , t4 }, {t5 , t6 }, {t3 , t7 , t9 }, {t8 }, {t10 , t11 , t12 }, {t13 , t14 }, {t15 , t16 }. Suppose A = {t1 , t3 , t5 , t6 , t7 , t9 , t10 , t12 , t15 , t16 } and j = 3. Using the simulated annealing algorithm given in Section 4, simulation is done, with L = 6, P0 = 0.1, tf = 8, Q = 50, M = 5 000. The results of simulation of this question are given in Table 1. From the simulation results, we can find that this simulated annealing is effective, and it can get the global optimization (approximate) solution.
Simulated annealing algorithm for the optimal translation sequence of the jth agent in rough communication Table 1
Results of simulation
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tion, 2003, 36: 3015–3018. [7] Mousavi A, Jabedar-Maralani P. Double-faced rough sets
Simulation
1
2
Solution
(p3 , p2 , p4 , p5 , p1 )
(p5 , p3 , p1 , p2 , p4 )
and rough communication. Information Sciences, 2002,
Value
7
6
Simulation
3
4
Solution
(p3 , p5 , p1 , p2 , p4 )
(p3 , p2 , p4 , p1 , p5 )
Measure in Rough Communication. International Journal
Value
7
7
of Advances in Systems Science and Applications, 2005,
Simulation
5
6
Solution
(p3 , p5 , p1 , p4 , p2 )
(p3 , p2 , p4 , p1 , p5 )
Value
7
7
148: 41–53. [8] Wang Hongkai, Li Xiuhong, Shi Kaiquan.
Information
5(4): 638–643. [9] Liu Jiqin.
Information rough communication based on
rough sets. Journal of Systems Engineering and Electronics, 2007, 29(3): 437–442.
6. Conclusions
[10] Guan Yanyong, Wang Hongkai, Yao Bingxue. Analysis
During rough communication[7] , we lose some information or precision of the concept and this results in a less precise or rougher concept. In fact, with different paths, the amount of the missed knowledge is varied. The concept of optimal translation sequence of the jth agent in rough communication (j ∈ {1, 2, . . . , n}) is proposed and its corresponding optimality mathematical model is also constructed. On the basis of the specialty of this problem, simulated annealing algorithm is used. Analysis and simulation of this algorithm demonstrate its effectiveness.
of the concept translation result in rough communication. Proceedings of International Conference on Artificial Intelligence, 2006: 381–384. [11] Wang Hongkai, Zhao Shuli. A new definition form of rough communication. Computer science, 2006, 33(9): 189–190. (in Chinese) [12] Wang Hongkai, Xue Peijun, Shi Kaiquan. The problem of rough communication of fuzzy concept. The 11th Joint International Computer Conference, 2005: 674–677. [13] Wang Hongkai, Guan Yanyong, Yao Bingxue, et al. Fuzzy rough communication and optimization problem of its translation sequence.
References
tional Conference on Machine Learning and Cybernetics,
[1] Pawlak Z. Rough sets. International Journal of Computer
2006: 2268–2273. [14] Wang Hongkai, Liu Baocang, Guan Yanyong. One direc-
and Information Science, 1982, 11: 341–356. [2] Tsai Ying-Chieh, Cheng Ching-Hsue, Chang Jing-Rong. Entropy-based fuzzy rough classification approach for extracting classification rules. Expert Systems with Applica-
tion expansion rough communication of dynamic concept based on α−generation of assistant sets.
International
Journal of Advances in Systems Science and Applications, 2007, 7(1): 93–96.
tions, 2006, 31: 436–443. [3] Shyng Jhieh-Yu, Wang Fang-Kuo, Tzeng Gwo-Hshiung, et al. Rough set theory in analyzing the attributes of combination values for the insurance market. Expert Systems
[15] Wang Hongkai, Yao Jinjiang, Xue Peijun, et al. Rough communication of dynamic concept. Advances in soft computing. Proc. of the Second International Conference of Fuzzy Information and Engineering, 2007: 781–787.
with Applications, 2007, 32: 56–64. [4] Leung Yee, Wu Wei-Zhi, Zhang Wen-Xiu.
IEEE Proc. of the fifth Interna-
Knowledge
acquisition in incomplete information systems: a rough set approach. European Journal of Operational Research, 2006, 168: 164–180. [5] Pawlak Z, Slowinski R. Rough set approach to multiattribute decision analysis. European Journal of Operational Research, 1994, 72: 443–459. [6] Asharaf S, Murty M N. An adaptive rough fuzzy single pass algorithm for clustering large data sets. Pattern Recogni-
[16] Kirkpatrick S, Gelatt Jr C D, Vecchi M P. Optimization by simulated annealing. Science, 1983, 220: 671–680. [17] Kang Lishan, Xie Yun, You Shiyong, et al. Non-numerical parallel computation (Book One) SA. Beijing : Science Press, 1998. [18] Wang Ling, Zheng Dazhong. A class of simulated annealing approach for multi-objective optimization. Computer Engineering and Applications, 2002, 38(8): 4–5. (in Chinese)
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[19] Feng TseLin, Cheng YanKao, Ching ChiHsu. Applying the
Wang Hongkai was born in 1978. He is a lecturer of
genetic approach to simulated annealing in solving some NP-hard problems. IEEE Trans. on SMC, 1993, 6: 1752–
University of Jinan, Ph. D. of Shandong University. His main research fields include rough sets theory and
1767.
fuzzy sets theory. E-mail: ss [email protected]