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1 * (, ) ( ; ),.,., ; ;.,,. :,, PACS: Ox, Cw, a, TP 1 ().,.,,,, s d. [1]., [2,3],,,.,,.,,,.,,,., [4 6].,, 1. A S E N(42,2 2 ) D N(63,3 2 ) B:70 1 8:00 S E D,. A [40 110], B 70, E 9:00 9:30 7, 0, (E, D) 9:00 N(42, 2 2 ), 9:00 N(63, 3 2 ). * ( : 2007AA12Z238) ( : B). zsj south@sohu.com c 2012 Chinese Physical Society

2 A 75, S E B,, B, S D. A D 135, B D 140. Bellman [4],,,.,, Bellman. Erdös Rényi [7] Frank [8],,. Hall [9],,. Hall,. Fu Rilett [10] Hall,,,. Miller-Hooks Mahmassani [11,12],. Sigal [13] Kamburowski [14],,. Wellman [15],,,,. [16,17] s d., [18], Alberto [19]. [20 25]. Thangiah [20] ; Inagaki [21], Chang Ramakrishna [22],,, ; Nanayakkara [23].,,. Davies Lingras [24],. Yang [25].,.,.,,. Bellman,,.,,.,,. : 2,,, ; 3,, ; 4,, ; 5,

3 Acta Phys. Sin. Vol. 61, No. 16 (2012) G = {V, E, W } ( 2 ),, V, i, V = {1, 2,, n}; E v v ( E = m); W ( ), W i, e ij = (i, j) E W ij,. G. 1 2, s d r (s = i, e ij, j, e jk, k,, n = d),. r 1 (s, d), r 2 (s, d),, r k (s, d) s d k, R (s, d) s d. r (s, d), l r, l r = w ij + w i + (i,j) r wj, W, l r ; L (R s,d ) R (s, d), L (R s,d )., : ( l min = min wij x ij + w i x ij + ) w j x ij, i,j B x ij B, 2 B n 2, B {1, 2,, n}, (1) (2) 0 (i, j) / r x i,j = (i, j = 1, 2,, n, i j), 1 (i, j) r (3), (1), (2) (3) ; x i,j (i, j); (1) ; (2), B, B B ; (3) x i,j = 1 (i, j), x i,j = 0 (i, j). l min.,.. G, s,, d,,.,,.,, ( ) ().,. 2 G = {V, E, W e (t), W i (t)}, V, E, W e (t),, W i (t),, G.,, {W e (t), t T }, W e (t) t e e, T. t t [t 0, t m ], [t 0, t m ], t 0, t m, W e (t) =, t > t m. t, W e (t), f we (w e, t)., P i,,, W i (t),, W i (t) 0.,

4 , ;,,,,. (): V = {W e1 (t),, W em (t), W 1 (t),, W n (t)}, f {w e1 (t),, w em (t), w 1 (t),, w n (t)}.,. :,, l r, s d l min = min {L (R s,d )}. s d r (s = i, e ij, j, e jk, k,, n = d), f r = f (we1, w em, w 1,, w n ) dw eq dw v, e q, v., r i s d u p i (l i l 1,, l i l k,, l i l u ), k i,. 3 p r = max p i(l i i=1,2,,u l 1,, l i l k,, l i l u ), k i r.,,,. p i = f (l 1,, l u ) dl, G(l i l 1,,l i l u ) f (l 1,, l u ) r 1,, r u., p r,,. [26]. 1 E (X) Y X, Y, X E (Y ) P (X Y ) 0.5, E (X) E (Y ) P (X Y ) 0.5. E (X) E (Y ), P (X Y ) 0.5,, X Y, 1/2, 6 < x < 8 f X (x) =, 0, 1/8, 4 < y < 12 f Y (y) =, 0, (X, Y ) f (x, y) =f X (x) f Y (y) 1/16, 6 < x < 8, 4 < y < 12 =, 0, E(X) = 7 < E(Y ) = 8, P (X Y ) = f (x, y) dxdy = > 0.5 ( Ω 6 < Ω x < 8, 4 < y < 12, x y ), P (X Y ) 0.5. E (X) E (Y ) P (X Y ) 0.5, P (X Y ) 0.5 E (X) E (Y ), E (X) E (Y ) P (X Y ) 0.5. X E (X) Y E (Y ), X Y,.,,.,, NP-hard (, NP-hard ),,.,,,,.. 3, [22]. (genetic algorithm) Michigan Holand [27]

5 Acta Phys. Sin. Vol. 61, No. 16 (2012) ,,.,,,,. 3. N.,,.,,.,, (, ), 5. 4 () 3,,.,,. 3.1.,,,,.,. 2 N, N., 5 3.2,.. ( 2),,.,,,,,,.,,,, : 1,

6 Acta Phys. Sin. Vol. 61, No. 16 (2012) ; 2,, 1, ( ), ; 3 2 (),, ; 4, ; , [28].. (µ + λ)... s d, k,. 4, 1 5, (b), {10, 3}, 2, 3 9, , 2. f(k) = E (W i (t)) + E (W e (t)). (4) Holland,. 3.4,..., ();.,,. 4, 6(a).,, ,,., : 1) ; 2),.,

, 1 8 10 7 4 6 5 4, 4, 7 6, 7. 3.6 7 : 1) W i (t) W e (t); 2) ; 3) eval(k); 4) ; 5) ; 6) ; 7),, 3). 4,,.,., ;,.

7 , , 4, 7 6, : 1) W i (t) W e (t); 2) ; 3) eval(k); 4) ; 5) ; 6) ; 7),, 3). 4,,.,., ;,., [29]. 7:30 8:30, 0.1, 10 min, 1 min; N(5, 3 2 )., ( 8 ). ( 6:30 8:30).,, 12 min (a) ( ); (b) ( )

8 GIS ArcGIS10, VS. NET C#. ArcGIS10, ( 8)., W e (t); p (p 0),, W i (t). VS. NET ArrayList,. : pop size = 50, p c = 0.85, p m = 0.02, Maxgen = , - 3, 1.,. 1 3 s,. 1 - /min /s ,,,,.,.. 1),,,., Bellman,,. 2), [16],,.,,,,. 3), [24,25],.,. 4),,,,.,,,,

9 [1] Peer S K, Dinesh K S 2007 Comput. Math. Appl [2] Li S B, Wu J J, Gao Z Y, Lin Y, Fu B B 2011 Acta Phys. Sin (in Chinese) [,,,, ] [3] Wang K, Zhou S Y, Zhang Y F, Pei W J, Liu Q 2011 Acta Phys. Sin (in Chinese) [,,,, ] [4] Bellman E 1958 Quart. Appl. Math [5] Dijkstra E W 1959 Numer. Math [6] Dreyfus S 1969 Operat. Res [7] Erdös P, Rényi A 1960 Publ. Math. Inst. Hung. Acad. Sci [8] Frank H 1969 Operat. Res [9] Hall R W 1986 Transport. Sci [10] Fu L, Rilett L R 1998 Transport. Res. B [11] Miller-Hooks E D, Mahmassani H S 1998 Comput. Operat. Res [12] Miller-Hooks E D, Mahmassani H S 2000 Transport. Sci [13] Sigal C E, Pritsker A A B, Solberg J J 1980 Operat. Res [14] Kamburowski J 1985 Operat. Res [15] Wellman M P 1995 Proceedings of the 11th Conference on Uncertainty in Artificial Intelligence Montreal, Quebec, Canada, August 18 20, 1995 p18 [16] Jaillet P 1992 Networks [17] Fan Y Y, Kalaba R E, Moore J E 2005 Comput. Math. Appl [18] Dong J Y, Zhang J Y, Chen Z 2007 Acta Phys. Sin (in Chinese) [,, ] [19] Alberto D V, Roberto M, Norman C, Andrea E R, Gambardella L M 2008 Eur. J. Operat. Res [20] Thangiah S R, Nygard K, Juell P 1991 Proceedings of the 7th IEEE Conference on Artificial Intelligence Applications Miami, USA, February 24 28, 1991 p422 [21] Inagaki J, Haseyama M, Kitajima H 1999 Proc. IEEE Int. Symp. Circuits and Systems Orlando, USA, May 30 June 2, 1999 p137 [22] Chang W A, Ramakrishna R S 2002 IEEE Trans. Evol. Comput [23] Nanayakkara S, Srinivasan D, Lup L, Xavier G, Elizabeth T, Ong S H 2007 IEEE Congress on Evolutionary Computation Singapore, September 25 28, 2007 p4469 [24] Davies C, Lingras P 2003 Eur. J. Operat. Res [25] Yang S, Cheng H, Wang F 2010 IEEE Trans. Syst. Man Cybernet. C [26] Charles J S 1996 A Course in Probability and Statistics (California: Duxbury Press) p81 [27] Holland J H 1975 Adaptation in Natural and Artificial Systems (Michigan: University of Michigan Press) p22 [28] Thomas B W, White C C 2007 Eur. J. Operat. Res [29] Bell M G H, Iida Y 1997 Transportation Network Analysis (Chichester: John Willey and Sons) p

10 Dynamic stochastic shortest path algorithm Zhang Shui-Jian Liu Xue-Jun Yang Yang ( Key Laboratory of Virtual Geographic Environment of Ministry of Education, Nanjing Normal University, Nanjing , China ) ( Received 30 December 2011; revised manuscript received 27 April 2012 ) Abstract The static shortest path problem has been solved well. However, in reality, more networks are dynamic and stochastic. The states and costs of network arcs and nodes are not only uncertain but also correlated with each other, and the costs of the arcs and nodes are subject to a certain probability distribution. Therefore, it is more general to model the shortest path problem as a dynamic and stochastic optimization problem. In this paper, the dynamic and stochastic characteristics of network nodes and arcs and the correlation between the nodes and arcs are analyzed. The dynamic stochastic shortest path is determined. The dynamic stochastic optimization model of shortest path is provided, and a shortest path genetic algorithm is proposed to solve dynamic and stochastic shortest path problem. The effective and reasonable genetic operators are designed according to the topological characteristics of the network. The experimental results show that this algorithm can be used to effectively solve the dynamic stochastic shortest path problem. The proposed model and algorithm can be applied to the network flow optimization problem in transportation, communication networks, etc. Keywords: shortest path problem, genetic algorithm, dynamic and stochastic network PACS: Ox, Cw, a, TP * Project supported by the National High Technology Research and Development Program of China (Grant No. 2007AA12Z238) and the Postdoctoral Science Foundation of Jiangsu Province, China (Grant No B). zsj south@sohu.com

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