Vol. 34 ( 2014 ) No. 4. J. of Math. (PRC) : A : (2014) Frank-Wolfe [7],. Frank-Wolfe, ( ).

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1 Vol. 4 ( 214 ) No. 4 J. of Math. (PRC) 1,2, 1 (1., 472) (2., 714) :.,.,,,..,. : ; ; ; MR(21) : 9B2 : : A : (214) ,,,,, [1 ].,, [4 6],, Frank-Wolfe, Frank-Wolfe [7],.,,.,,,., UE,, UE. O-D,,,,, O-D Logt, O-D,,. Frank-Wolfe,..,. : : (712171). : (198 ),,,, :.

2 76 Vol. 4 2 G = (N, A), N, A, R, S, R S ; A ; (r, s) r, s O-D ; P rs O-D (r, s).,, : Ω = x fp rs p P rs mn x Ω = q rs, f rs p xa t a (ω)dω r R s S qrs B(ω)dω, (2.1) }, x a = fp rs δap, rs x a C a, a, p, r, s. (2.2) r R s S p P rs : x a a, x = (, x a, ) T ; fp rs O-D (r, s) p, f = (, fp rs, ) T ; t a (x a ) a, x a, t(x) = (, t a (x a ), ) T ; C a a, C = (, C a, ) T ; q rs O-D (r, s) ; q rs = D rs (u rs ) O-D (r, s), (r, s) u rs, u rs, q = (, q rs, ) T ; u rs = B rs (q rs ) = D 1 (q rs ). O-D (r, s) p a, δap rs = 1,., (2.1) (2.2) (x, q ). K-K-T, (2.1) (2.2) ( t a (x a)δap rs u rs )fp rs =, t a (x a)δap rs u rs, fp rs, (2.) qrs(u rs B rs (qrs)) =, u rs B rs (qrs), qrs, (2.4) d a (x a C a ) =, x a C a, d a, (2.5) t a (x a ) = t a (x a ) + d a, Lagrange u rs, d a O-D (r, s) a,., d a, t a (x a ),,,,, x a > C a, d a d a, d a = d a d a >,,, UE.,, Lagrange d a [1]. Sheff UE [7], Frank-Wolfe. UE

3 No. 4 : 761 : (2.1) qrs B(ω)dω = Drs(u rs ) Drs(u rs ) B(ω)dω B(ω)dω, (.1) q rs (.1),, u rs (r, s). e rs = D rs (u rs) q rs, Drs(u rs ) ers B rs (ω)dω = B rs (D rs (u rs) ω)dω = W rs (ω)dω, q rs D rs(u rs ) qrs W rs (ω) = B rs (D rs (u rs) ω). O-D (r, s), e rs, W rs (e rs ), UE : mn (x,e) Ω xa t a (ω)dω + r R s S ers W rs (ω)dω, (.2) Ω = (x, e) fp rs p P rs + e rs = D rs (u rs), e rs, x Ω, r, s }. (.).1 N O-D,, O-D A ( O-D, = 1, 2,, N ), d a d a, t a (x a ) = t a (x a ) + d a, p ( = 1, 2,, N). p A, A ; A, p m p m p + 1. O-D, α 2,,,...2 α k O-D, Frank- Wolfe O-D, O-D. : k p k t p k A k, α k = max exp( θt p k ) m p exp( θt p ), ζ, (.4) p A k θ Logt, A k k, O-D ( p k ), t p p, m p,

4 762 Vol. 4, p. O-D p A k m p = k. (.4) ζ ε,,.. O-D, O-D N, O-D, O-D, ;, k = 1, O-D A = ; ε >, O-D, t k x k t k, d k a t a (C a ), x a > C a, a = d k a, a A;,, O-D A (A O-D, = 1, 2,, N), m p k = 1. 1 t k+1 a = t k a + d k a, O-D p k+1 ; p k+1 A, A := A p k+1, p k+1 m p k+1 = 1, A, m p k+1 = m p k + 1. O-D, f k, ẽ k. (.4) α k+1 O-D, A : f k+1 = α k+1 e k+1 = α k+1 ẽ k + (1 α k+1 )e k. 2 ;. x k+1, e k+1 t k+1, max xk+1 a x k a + max =1,2,,N ek+1 d k+1 a = a A, k := k + 1, 1..4 e k+1 d k a + d k a, x k+1 a C a,,, + max xk+1 a C a < ε, d k+1 a = max, d k a + 1 k + 1 (xk+1 a C a )},, f k + (1 α k+1 )f k, Ω D (u ) < ( = 1, 2,, N), ε >, ζ,.,, a, x k+1 a x k a < ε, ek+1. x k+1 a x k a < ε, k. O-D, e k < ε, max, xk+1 a C a } < ε, k, M = max p P exp( θt p)}, m = mn p P exp( θt p)},

5 No. 4 : 76 P O-D, exp( θt p k ) m p exp( θt p ) p A k M p A m m k M m 1, k. k k, α k = ζ, ζ = ε N A q, A p, fp k+1 = α k+1 f p k + (1 α k+1 )fp k = fp k + α k+1 ( f p k fp k ), f k+1 p f k p = α k+1 f p k f k p q α k+1 = q ζ, k, q O-D. O-D, k A, a, x a = δ ap fp k+1, p A x k+1 a x k a δ ap fp k+1 fp k p A A q ζ ε, k. e k+1 e k+1 ε, k. e k O-D, e k+1 = α k+1 ẽ k + (1 α k+1 )e k, e k+1 e k = α k+1 ẽ k e k α k+1 q = q ζ ε, k. max, x k+1 a C a } < ε, k. k max, x k+1 a C a } > ε, a, k x k a C a > ε >, dk a, d k a d k a, t k a, k, a p f p k =, fp k+1 = α k+1 f p k + (1 α k+1 )fp k = (1 α k+1 )fp k = (1 ζ )fp k, k. x k+1 a = p P δ apfp k+1, k, x k+1 a C a > ε,.,,,.,,.,. Lagrange,,,. 4, : 4 O-D : , O-D d rs = D rs exp(.5(1 urs u rs )), u rs O-D (r, s) ; D 1,12 = 46, D9,4 = 4,

6 764 Vol. 4 1: D 12,1 = 44, D4,9 = 42; 4,,, BPR : t a (x a ) = t a ()(1 +.5( xa C a ) 4 ), C a a. O-D, W rs () = B rs (D rs (u rs)) = u rs. x a, 1 ( ε = 1, ). 1: a (1,2) (1,5) (2,1) (2,) (2,6) (,2) (,4) (,7) (4,) (4,8) (5,1) (5,6) (5,9) (6,2) (6,5) (6,7) (6,1) x a a (7,) (7,6) (7,8) (7,11) (8,4) (8,7) (8,12) (9,5) (9,1) (1,6) (1,9) (1,11) (11,7) (11,1) (11,12) (12,8) (12,11) x a ,,,,, (6,7) ,. e 1,12 = 59.86, e 9,4 = 59.99, e 12,1 = 55.9, e 4,9 = : : (1,2) 2,, 528, (1,2), 1. 5, Frank-Wolfe,.,. O-D, ( )

7 No. 4 : 765,.,. [8]. [1] Yang H, Huang H. Mathematca and economc theory of road prcng [M].Amsterdam: Elsever, 25. [2] Chou S W. A novel algorthm for area traffc capacty control wth elastc travel demands [J]. Appled Mathematcal Model, 211, 5: [] Frederc B, Jean P.V. An effcent method to compute traffc assgnment problems wth elastc demands [J]. Transportaton Scence, 28, 42(2): [4] Larsson T, Patrksson M. An augmented Lagrangan dual algorthm for lnk capacty sde constraned traffc assgnment problems [J]. Transportaton Research, 1995, 29: [5] Ne Y, Zhang H M. Models and algorthms for the traffc assgnment problem wth lnk capacty constrants [J]. Transportaton Research, 24, 8: [6] Xu Y D, L S J, Teo K L. Vector network equlbrum problems wth capacty constrants of arcs [J]. Transportaton Research Part E, 212, 48: [7] Sheff Y. Urban transportaton networks: equlbrum analyss wth mathematcal program methods [M]. New Jersey: Prentce-Hall, Inc, Englewood Clffs, [8],. [J]., 212, 2(1): APPROXIMATE ALGORITHM FOR ELASTIC DEMAND TRAFFIC EQUILIBRIUM ASSIGNMENT PROBLEM WITH LINK CAPACITY CONSTRAINTS LIU Bng-quan 1,2, HUANG Chong-chao 1 (1.School of Mathematcs and Statstcs, Wuhan Unversty, Wuhan 472, Chna) (2.School of Math. and Inform. Sc., Wenan Normal Unversty, Wenan 714, Chna) Abstract: Ths paper studes the elastc demand traffc equlbrum assgnment problem wth lnk capacty constrants. We analyze the dfferences of lnk travel tme and traffc demand between the lnk capacty constrants model and the one wth lnk capacty unconstrant. The elastc demand model s transformed as fxed one accordng to excess demand formulaton and we propose an approxmate algorthm for the model to crcumvent the costly soluton of the constraned assgnment problem. The algorthm ensures gradually the lnks flow lower than correspondng capacty and trends to generalzed elastc demand user equlbrum by approxmatng actual lnk travel tme and adaptvely regulatng the delay and error factors n each teraton. It s superor to stochastc equlbrum methods by avodng paths numeraton. Both the convergence result and the numercal example show the algorthms are effectve and effcent. Keywords: equlbrum traffc assgnment; elastc demand; capacty constrants; approxmate algorthm 21 MR Subject Classfcaton: 9B2