A hybrid cross entropy algorithm for solving dynamic transit network design problem

Transcription

1 A hybid coss entopy algoithm fo solving dynamic tansit netok design poblem TAI-YU MA Tanspot Economics Laboatoy Univesity Lyon 2 - CNRS Lyon, Fance This pape poposes a hybid multiagent leaning algoithm fo solving the dynamic simulation-based bilevel netok design poblem. The objective is to detemine the optimal fequency of a multimodal tansit netok, hich minimizes total uses tavel cost and opeation cost of tansit lines. The poblem is fomulated as a bilevel pogamming poblem ith equilibium constaints descibing non-coopeative Nash equilibium in a dynamic simulation-based tansit assignment context. A hybid algoithm combing the coss entopy multiagent leaning algoithm and Hooke-Jeeves algoithm is poposed. Computational esults ae povided on the Sioux Falls netok to illustate the pefomance of the poposed algoithm. Keyods: multiagent, leaning, netok design, tansit system, simulation 1. INTRODUCTION Tansit netok design poblem (TNDP) has been an impotant poblem in tanspotation science and idely studied in the past [1][2][3]. The objective is to detemine optimal tansit line fequencies in a tansit netok, hich minimizes total use cost and tansit opeation cost unde esouce and use equilibium flo constaints. The late esults fom the solution of taffic assignment poblem aiming to detemine the taffic flo at Nash equilibium (use optimal) states. The TNDP can be geneally fomulated as a bilevel pogamming poblem, hee the uppe-level is a constained minimization poblem fo the optimal tansit line fequency decision; the loe-level is a vaiational inequality (VI) poblem fo solving the use-optimal equilibium flo. As the evaluation of the objective function at the uppe-level equies a solution of the VI poblem at the loe-level, the poblem has been ell knon as a difficult poblem in mathematical pogamming and tanspotation science. In the past, the TNDP has been studied by many authos. Gao et al. [4] fomulated the TNDP as a bilevel pogamming model and poposed a solution pocedue based on sensitivity analysis. The uppe-level poblem is fomulated as a minimization poblem unde the equilibium tansit assignment constaints. The loe-level poblem is fomulated as the VI poblem ith diffeential cost functions. The poposed appoach equies the calculation of the deivatives of flo ith espect to the line fequency to obtain optimal solutions. Macotte [1] poposed a fomal desciption of the TNDP and povided seveal heuistic pocedues fo solving it. The use-optimal flo is obtained by solving the VI poblem in a static netok ith asymmetic link cost functions. LeBlanc poposed a seies of papes fo the TNDP [2][5]. In [2], a bilevel static multimodal tansit netok design model has been poposed. The autho fist solved a mode-split 1

2 assignment poblem to obtain a use-optimal equilibium flo by Fank-Wolfe algoithm and then applied Hooke-Jeeves algoithm to iteatively deive optimal fequencies in a static tansit netok. Othe solution techniques fo the bilevel pogamming poblem can be found in [6]. Hoeve, fo dynamic simulation-based tansit assignment, the above deivative-based methods cannot be applied since the functional fom of the deivatives is geneally unavailable. The simulation-based VI poblem is geneally difficult to solve in the dynamic tansit system. Fo this issue, Ma and Lebacque [7][8] poposed a coss entopy (CE) based solution algoithm to iteatively deive optimal tavel choice pobabilities toads use equilibium based on minimizing the Kullback-Lieble elative entopy (coss entopy) beteen to consecutive pobability distibutions. In this ok, a hybid algoithm is poposed by combing the multiagent coss entopy leaning algoithm and the Hooke-Jeeves algoithm fo solving the simulation-based tansit netok design poblem. The poposed algoithm is deivative-fee, convenient fo solving the simulation-based TNDP. Fo the tansit system simulation, a multiagent appoach is poposed to captue explicitly the tansit system dynamics. We popose a multi-laye netok to effectively epesent the tansit netok and simulate the movement of diffeent agents (passenges and vehicles). Passenge s aiting time at stop is explicitly calculated subject to the capacity constaint of the vehicle. The est of the pape is oganized as follos. Section 2 descibes the mathematical fomulation of the bilevel pogamming poblem fo the TNDP. It follos in Section 3 the dynamic tansit system desciption based on the multiagent appoach along ith the tansit netok model and tavel cost fomulation. Section 4 pesents the poposed solution algoithm by combining the Hooke-Jeeves algoithm and the CE multiagent appoach. A state-of-the-at algoithm based on the method of successive aveage (MSA) fo solving simulation-based dynamic taffic assignment poblem is poposed. Section 5 povides the computational esults of the CE multagent appoach and the MSA appaoch on the Sioux Falls netok [2] to validate the obtained loe-level use euilibium solution. Then e sho the optimal tansit fequency obtained by the hybid algoithm. Section 6 concludes the pape. 2. THE TRANSIT NETWORK DESIGN MODEL Notation l tansit line L set of tansit lines Y l uppe bound of the fequency of tansit line l Y l loe bound of the fequency of tansit line l y l fequency of tansit line l y vecto of the fequency of tansit lines θ l cost incease fo fequency in tansit line l m designation of a use C m genealized tavel cost of use m k oigin-destination pai

3 3 K set of oigin-destination (o-d) pai k f (t) flo on path at time t f vecto of flos path index R k set of paths connecting o-d pai k d k (t) demand of oigin-destination pai at time t D k demand of oigin-destination pai t time index T the time of the last vehicle/use leaves the netok The TNDP is fomulated as a bilevel pogamming poblem. Fo the uppe-level poblem, a decision make aims to minimize the total cost of the tansit system unde feasible fequency constaints and dynamic use equilibium at the loe-level. Fo the loe-level poblem, each use aims to minimize his/he tavel cost, this is a poblem of noncoopeative Nash equilibium in a multiagent system. At the uppe-level, the poblem is fomulated as a constained minimization poblem subject to feasible line fequency and to the equilibium path flo detemined at the loe-level. At the loe-level, the poblem is fomulated as a VI poblem ith the line fequencies imposed at the uppe-level. (Uppe-level) (3) Min St. Z ( y, f ) = C ( t, f ) + y θ (1) Y l k m dk m l l l y Y, l L (2) l l f is the noncoopeative Nash equilibium path flo vecto detemined by solving the minimization poblem of (4)-(7). (Loe-level, VI poblem) Find use optimal equilibium path flo vecto f such that Min S( f ) = f [ C ( t, f ) C ( t, f )] (4) k, s R k T t St. f ( t ) = d ( t ), k K, t [ t, T ] R k T d t k k ( t) = Dk, f k s ( t),, t [ t, T ], (7) hee the function [ q ] + = max(, q). The objective function (1) minimizes total genealized tavel cost of uses and total opeation cost. The constaints (2) mean that the line fequencies ae bounded. The constaint (3) is the use optimal path flo vecto satisfying the equilibium condition + (5) (6)

4 solved by (4)-(7) [9]. At the loe-level, the use equilibium flo is stated as f [ C ( t, f ) Cs ( t, f )] + =,, s Rk, k, t [ t, T ] (8) can be obtianed by solving the minimization poblem of (4)-(7). The use equilibium flo states that fo uses of the same oigin and destination (OD) the genealized tavel cost esulting fom depatue time and oute choices is equal and no less than that of unused choice altenatives. The constaints (5)-(7) state the consevation of flo and non-negativity of path flo. Pevious studies [1][11] shoed that e can define a elative gap function to measue ho the genealized tavel cost is fa fom the idealized shotest path cost. The gap function is defiend as f h H k K R [ C C ] Gap( f ) =, (9) d C h H k K hee h :depatue time choice index h H ith H = {,1,2,..., n}. Given a selected depatue time inteval h, a andom depatue time ill be taken ithin [ t + h, t + ( h + 1) ) ith t the ealiest depatue time and a small discetized time inteval (e.g. 5 minuites). C : expeienced tavel cost ith espect to (h, k, ) d : time-dependent tavel demand ith espect to (h, k) C : minimum path genealized tavel cost ith espect to (h, k) The gap function epots the aveage gap toads to dynamic use equilibium. When the gap function conveges to a stable value and the obtained aveage tavel cost fo utilized depatue time intevals and paths ae no moe than that on unused altenatives, the appoximate of use equilibium is achieved. 3. MULTIAGENT-BASED TRANSIT SYSTEM To captue the dynamics of the movements of the uses and the effect of congestion at stations, the multiagent appoach is adopted. The multiagent appoach is vey convenient fo simulating the dynamics of opeations of tansit vehicles and use flo on the system [12]. Its advantage esides on its flexibility in captuing the inteactions beteen agents ith thei envionment. We utilize a multilaye netok stuctue to explicitly model complex connections ithin multimodal stations ith the pesence of diffeent tanspot modes and sevice lines. The detail of the tansit netok and multiagent simulation is descibed as follos. 3.1 Tansit netok and tansit paths

5 5 The tansit netok is epesented by a diected gaph G(N, A), hee N is the set of nodes and A the set of acs. The nodes ae classified into thee types: oigin/destination, station, and line node [13]. The netok stuctue is illustated in Figue 1. As shon in Fig. 1, the oigin/destination nodes ae connected ith elated seviced station nodes by alking acs. The station nodes ae connected ith its sevice tansit lines, ith othe station nodes ithin the same multimodal station and ith oigin/destination nodes. Each ac is chaacteized by its tavel time calculated as its length divided by alking o constant mode-specific vehicle speed. A tansit path (called path heeafte) is an acyclic path connecting an oigin-destination (OD) pai in the multilaye tansit netok. The tavel time of a path compises alking time accessing to O/D, tansit line nodes by boading/alighting acs, aiting time at line nodes, and tansfe time beteen stations ithin the same multimodal station. The tavel time on the boading acs epesents aveage alking time fom the station cente to the boading point of the vehicle. Hence, the genealized tavel cost function fo the path consists of the folloing pats: (a) alking time π ; (b) in-vehicle time v s π ; (c) aiting time π ( t, f ) ; (d) mode tansfe penalty λ ; (e) ealy/late aival penalty ρ ( t, f ) ; (f) fae Γ. By assuming the Fist-In-Fist-Out pinciple fo boading a vehicle [14], the aiting time depends on the supply (vehicle capacity and sevice fequency) and the demand at each tansit line node. Hence, the aiting time π i (t) fo a use aiving at line node i at time t is calculated as 1 πi ( t) = Di ( Si ( t)) t, hee S i (t) is cumulative aivals at line node i by time t, D 1 i ( t) is the invese function of cumulative depatue fom line node i by time t. The genealized tavel cost of path hen depating fom oigin at time t is then evaluated as: v s C ( t, f ) = α( π + π + π ( t, f )) + n λ + ρ ( t, f ) + Γ, (1) hee α is the unitay monetay value of tavel time obtained by tavel suvey data; n is the numbe of mode change hich can be diectly calculated by the used multimodal path; λ is the unitay penalty pe change obtained by tavel suvey data ; Γ is fae of path. Based on the expeimental study of Small [15], the ealy/late aival penalty hen aiving at destination at time t is defined a as: a a ρ ( t, f ) = µ a max(, τ ϖ t ) + µ b max(, t τ ϖ), (11) hee µ a and µ b ae unitay penalty associated ith ealy and late aival, espectively. The value of unitay penalty can be geneally obtained by tavel behavio suvey. τ is the desied aival time at destination hich is set as identical fo simplicity; ϖ is the half of toleable schedule delay inteval ithout penalty, geneally set as 2-5 minutes. 3.2 Multiagent-based tansit system The system is composed of to classes of agents, i.e. tansit vehicles and uses. Fo the vehicle agent, it epesents a mode-specific vehicle such as tamay/meto/tain opeating on espective tansit lines ith pedefined fequency and capacity constaints. Fo simplicity, the vehicle agents move ith constant speed neglecting accidents o delayed

6 situation. The vehicle capacity is assumed fixed and tanspot mode-specific. When the Bus netok Bus lines Non-oad mode tansit netok Tansit lines Meto/bus line node Meto/bus station node O/D Boading/Aligting ac Walking ac Tansit line ac Figue 1 Tansit t netok stuctue vehicle agent aives to a line node, a constant stop time makes the uses boad the vehicle. If the vehicle capacity is achieved, not seviced uses need to ait fo the next vehicle at the same line node. Fo the use agent, each one iteatively adapts his/he depatue time and path choice in ode to minimize his/he genealized tavel cost. The use behavio is based on the bounded-ationality assumption [16], assuming that the uses have no complete infomation about the tavel choice decision of the othe uses, neithe the eal-time congestion infomation of the tansit netok. The depatue time and path choices ae adjusted in a day-to-day basis based on the pefomance of choice altenatives on the pevious day. The leaning pocess is simila to the einfocement leaning pocess hee uses shift thei choices to moe attactive altenatives based on past expeiences. The difficulty is ho to update the choice pobability toads the use equilibium in a multiagent system. 4. SOLUTION ALGORITHM In this section, a hybid solution algoithm is poposed fo solving the dynamic simulation-based TNDP. As use s expeienced tavel cost depends on tansit netok supply (vehicle capacity and line fequency) and tavel demand dynamics (uses of the same OD competing the same esouces), the deivative-based methods cannot be applied to solve the poposed bilevel pogamming poblem. The algoithm is composed of to main steps. Fist, the coss entopy leaning algoithm is applied fo solving the loe-level VI poblem. The solution obtained at this stage is then utilized fo the computation of the value of the objective function at the uppe-level. Secondly, to optimize the line fequencies, the Hooke-Jeeves algoithm is applied, hich iteatively finds the optimal configuation of line fequency by moving each line fequency toads to bette solutions [17][2]. The solution algoithms ae descibed as follos.

7 7 4.1 Hooke-Jeeves algoithm Step 1: Initialize the vecto y of line fequency, set exploatoy seach step size as =. Set iteation index i=. Step 2: Fo each l L, set tial fequency vecto y ' by inceasing y l in y i by, if y l + > Y l set y l = Y l. Calculate the value of the objective function in (1) by summing total genealized tavel cost of uses (obtained by the coss entopy leaning algoithm descibed belo) and opeation cost. If Z ( y', f ) < Z( yi, f ), set y i = y' and Z( y i, f ) = Z( y', f ) ; otheise set tial fequency vecto y ' by deceasing y l in y i by. If the esulting y l < Y l, then set y l = Y l. Compute Z ( y', f ), if Z ( y', f ) < Z( yi, f ), set y i = y' and Z( y i, f ) = Z( y', f ). Set i : = i + 1. Step 3: If no impovement found, set := /2. If the esulting step size smalle than a small positive value, i.e., < ξ, stop the algoithm; otheise goto Step Coss entopy leaning algoithm The coss entopy leaning algoithm is designed fo solving dynamic multimodal use equilibium (UE) poblems. The algoithm consides the use equilibium is a ae event to be leaned. Based on an iteative pocedue, the poposed algoithm adaptively leans optimal tavel choice pobability by minimizing the Kullback-Lieble elative entopy beteen to consecutive pobability distibutions. The esulting pobability updates shift the uses to cheape choice altenatives toads the UE. The eade is efeed to [7][8] fo moe detailed desciption. In cuent application, the use s decision choice concens only the depatue time choice and path choice in the dynamic capacitated tansit netok. Conside the uses ae located at oigins aiming to aive to espective destinations ithin the desied aival time τ, assumed the same fo all taveles fo simplicity. Based on the bounded-ationality assumption, each tavele is assumed to choose a depatue time and path folloing elated choice pobability distibutions p h and p. Based on the expeienced genealized tavel cost C ( t, f ), the optimal choice pobabilities toads the UE ae iteatively deived. The detail of the coss entopy leaning algoithm is descibed as follos. Step 1: Initialize unifom pobability distibutions fo depatue time choice and path choice. Set p h = 1 / H, h H and p = 1 / Rk, Rk. R k is the path set of OD pai k. Step 2: Dynamic tansit system simulation and the tavel cost calculation. The use agents move into the netok accoding to his/he depatue time and path choice. When aiving at his/he destination, compute the expeienced genealized tavel cost by (1)-(11).

8 Step 3: Update the depatue time choice pobability by p + 1 h = p h e h' H C h / γ p C h' / γ h' e, h H, hee C h is the aveage genealized tavel cost ith espect to the uses choosing the depatue time inteval h at iteation. γ is the contol paamete esulting fom the solution of the folloing minimization poblem: + 1 Min γ subject to ph ph α (13) h H, hee α = κ / is a numeical divegent seies such that the flo adjustment conveges. κ is a positive constant. is an iteation index. Step 4: Update the path choice pobability accoding to the aveage pefomance of path choice samples by applying the fomulas (12)-(13). Set : = + 1. (12) Step 5: When goto Step 2. max = o the esulting pobability updates stabilize, stop; otheise 4.3. Method of Successive Aveages (MSA) The method of successive aveages appoach has been idely applied fo solving the simulation-based dynamic taffic assignment poblem [11][18][19]. The MSA method is a geneal iteative path flo adjustment scheme fo solving fixed point poblems. The adjustment pocess consists of shifting iteatively taveles to cheape outes. Given knon OD demand, the taveles ae initially loaded on the time-dependent shotest paths based on fee flo tavel time. The shotest paths ae then iteatively updated based on taveles expeienced tavel cost. By shifting taveles to cuent found shotest paths, use equilibium can be appoximately achieved by an iteative adjustment pocess. The algoithm is teminated hen the gap function conveges to a small value o the maximum iteation is achieved. Existing applications of the MSA method teat only the path choice poblem, given knon time-dependent demand [11][19]. As e aim at solving the dynamic use equilibium poblem ith espect to depatue time and path choice, a MSA-based solution scheme is poposed as follos. Step 1: Initialization. Compute the time-dependent shotest paths fo each OD pai and assign OD demand on depatue time intevals. Set iteation index =. a) Geneate the shotest path u fo OD pai k and depatue time inteval h as v u = ag min[ α( π + π ) + n R + Γ ], h, k (14)

9 9 Initialize the shotest path set { } U = u fo all h and k. b) Depatue time assignment of tavel demand D k. Estimate the genealized tavel cost C on the shotest path u by C u v u = α( π + π ) + n λ + ρ ( t, f ) + Γ, h, k, (15) u u hee ρ u ( t, f ) is the ealy/late aival penalty on the shotest path u hen depating at time t = t + h. Fo each OD pai k, soting the depatue time index set H in a ascending ay ith espect to C. The obtained ascending depatue time choice set ' fo OD pai k is denoted as H k. Assign unifomly the tavel demand D k on the depatue time intevals 1,2,..., s in H k, denoted as H ' ~ k, such that sq D < ( s +1) Q, hee Q is the maximum alloed passenge flo (i.e. u k u u numbe of uses tanspoted pe depatue time inteval fo a given OD) on the path fo one depatue time inteval. The obtained time-dependent demand fo h and k is Dk ~ d, = h Hk, and otheise. This assignment makes uses utilize the loest s tavel cost depatue time intevals unde path flo capacity constaints. Step 2: Dynamic netok loading. Load all passenges on the netok based on thei depatue time and path choice and un the simulation until all passenges aive at thei destination. Compute the genealized tavel cost fo all passenges and the value of the gap function. Step 3: If the gap function value is stabilized o the maximum iteation is achieved then stop; otheise, goto Step 4. Step 4: Update time-dependent link tavel time fo all used links and compute ne time-dependent shotest paths u based on Dikjsta s algoithm fo each h and k Step 5: Update time-dependent demand d. Compute aveage genealized cost fo all ~ depatue time intevals. Find the least aveage cost depatue time inteval h +1 k. If ~ + ~ h 1 k H, then updated time-dependent demand is detemined by k ~ d, if h H k + 1 d = + 1 (16) 1 ~ + 1 Dk, if h = + 1 ~ + ~ Hoeve, if h 1 k H, update time-dependent demand by k u ~ + 1 d, if h + 1 d = + 1 (17) 1 ~ + 1 d + Dk, if h = u

10 Step 6: Update path flo assignment fo h, k, and (17). Set := +1, and goto Step f based on simila fomula of (16) 5. NUMERICAL STUDY In this section, e pesent and validate the solutions obtained fo the loe-level poblem by the CE leaning algoithm and the MSA algoithm. Then e epot the obtained solution of the bilevel poblem based on the hybid algoithm. The simulation of the tansit system is based on the discete event simulation technique implemented in C++ on a Dell Latitude E64 ith 2.53GHz and 3.48G memoy. The poposed algoithm is tested on the multilaye Sioux Falls tansit netoks (146 nodes and 446 acs in a multilevel diected gaph) by extending LeBlanc s netok in [2] (24 nodes and 76 links) (Fig. 2). Thee ae thee tamay lines (1, 2, 3) and to meto lines (A and B) ith sevice uns in both diections. The length of acs is shon in the squae backets of Fig. 2. The global paamete setting fo the expeiment is descibed as follos. The tansit modes contain only tamay and meto ith stict capacity constaints. The capacity pe vehicle fo (tamay, meto) ae set as (3, 6) and (25, 3) fo lo and high congestion scenaios. The speed fo tamay and meto is set espectively as 5. and 12.5 m/sec. The stop time at meto and tamay line nodes is set as 2 seconds fo all the vehicles. The alking speed is set as 1.4 m/sec. The length of the boading, alighting and tansfe acs (beteen to diffeent stations) is 1 m. The tansfe ac fom the O/D to connected station node is 3m. Fo simplicity, the desied aival time to destination is unifomly set as 9:. The depatue time choice ange is set beteen 7: and 9: ith discetized time inteval of 5 minutes. Fo simplification, Γ and λ ae set as. The detail of the paamete settings is listed in Table 1. Table 1 The paamete settings of the expeiments Eq. Paamete Value Eq. Paamete Value (2) Y l 1 1 (1) ϖ 3 sec. (2) Y l 2 Belo (12) κ 1.6 (1) θ l, tamay 1 (1) θ l, meto 4 (9) α 7 (9) Γ, λ (1) µ a 4 (1) µ b 15 y {1,1,1,1,1} (1) τ 9: Remak: 1. vehicle/hou 5.1. Results fo the CE leaning algoithm and the MSA appoach Befoe solving the TNDP, e illustate the pefomance of the CE leaning algoithm and the MSA algoithm fo the loe-level poblem. The algoithms ae tested on diffeent demand level (16, 4 and 8 passenges), vehicle capacity (lo and

11 11 high) and sevice fequency (2 minutes and 1 minutes). Table 2 shos the computational esults of the algoithms. It indicates that at highe congestion level (demand=8, fequency=1 minutes), the gap function conveges to a highe level. This is due to insufficient tansit sevice hich makes taveles leave thei home quite ealy and geneate highe ealy aival penalty. As fo the pefomance of the algoithms, the gap function quickly conveges to a stable value afte 1 iteations fo the CE algoithm. When compaed ith the MSA method, the CE leaning algoithm has simila pefomance in tems of solution quality and computational times. The validation of obtained use equilibium solution fo the loe-level poblem is epoted in Table 3, hee fo each 5 minute depatue time inteval, the numbe of uses and the aveage genealized tavel cost on the k-shotest (k=5) paths ae pesented. The esult indicates that the genealized tavel costs on all used paths ae no moe than that on all unused paths except fe exceptions. Note that the genealized tavel cost fo unused paths on some depatue time intevals is estimated by summing in-vehicle tavel time, aveage aiting time (half headay beteen vehicles) and aival penalty hen depating at the middle point of the depatue time inteval. Figue 2 The Sioux Falls netok ith tansit lines

12 Table 2 Compaative study of CE and MSA methods fo loe-level poblem Demand 16 (4 pe OD) 4 (1 pe OD) 8 (2 pe OD) CE MSA CE MSA CE MSA Value of the gap function Total genealized cost Computational time (sec.) Value of the gap function Total genealized cost Computational time (sec.) Remaks: 1. Fequency = 2 minutes, capacity fo meto and tamay: 6 passenges/veh and 5 passenges/veh, espectively. 2. Fequency = 6 minutes, capacity fo meto and tamay: 3 passenges/veh and 25 passenges/veh, espectively. 3 The untime fo 2 iteations. Table 3 Validation of obtained solution based on the CE method (total demand = 8 (2 fo each OD pai), ith fequency = 2 minutes fo meto and tamay, capacity fo meto and tamay : 6 passenges/veh and 5 passenges/veh, espectively) OD = (1,13) Numbe of passenges on each path Aveage genealized tavel cost on each path Depatue time inteval :25-7: :3-7: :35-7: :4-7: :45-7: :5-7: :55-8: :-8: :5-8:

13 13 Numbe of passenges on each path OD = (1,2) Aveage genealized tavel cost on each path Depatue time inteval :15-7: :2-7: :25-7: :3-7: :35-7: :4-7: :45-7: :5-7: OD = (2,13) Numbe of passenges on each path Aveage genealized tavel cost on each path Depatue time inteval :4-7: :45-7: :5-7: :55-8: :-8: :5-8: :1-8: :15-8: :2-8:

14 Numbe of passenges on each path OD = (2,2) Aveage genealized tavel cost on each path Depatue time inteval :3-7: :35-7: :4-7: :45-7: :5-7: :55-8: :-8: :5-8: Results fo the hybid algoithm The pefomance of the Hooke-Jeeves algoithm fo solving the uppe-level poblem is shon in Table 4. To scenaios ith espect to diffeent levels of demand ae set as 16 and 8 uses. As can be seen in Table 4, the Hooke-Jeeves algoithm conveges efficiently to nea-optimal line fequency. Note that e utilize 1 iteations to obtain nea use-optimal flo by the CE leaning algoithm fo solving the loe-level poblem in ode to educe the computational time. 6. CONCLUSION In this ok, a hybid multiagent leaning algoithm is poposed to solve the dynamic simulation-based tansit netok design poblem. The poblem is fomulated as a bilevel pogamming poblem hee the uppe-level is a constained minimization poblem fo the optimal tansit line fequency decision, and the loe-level is a vaiational inequality (VI) poblem fo solving use-optimal equilibium flo poblem. The poposed hybid algoithm is composed of to main steps to iteatively solve the bilevel poblem. In the fist step, the coss entopy leaning algoithm is poposed to solve the loe-level poblem. Then the Hooke-Jeeves algoithm is applied to iteatively find optimal line fequencies fo the uppe-level poblem. Computational esults on the extended Sioux Falls netok illustate that the poposed method can find nea-optimal solution unde dynamic use equilibium constaints. We compae the coss entopy leaning algoithm ith the method of successive aveage fo solving dynamic tansit assignment poblem. The esults sho that the pefomance of the to appoaches is simila fo the test netok. Futue extensions include applying the queuing theoy fo

15 15 modeling passenge flo at stations, and modeling the heteogeneity of passenge s oute choice behavio. Table 4 Summay of Hooke-Jeeves iteations Total demand= 16 (4 pe OD) f 1 f 2 f 3 f A f B Z f1 f2 f3 Total demand = 8 (2 pe OD) Remak: 1. f 1 : tamay line 1; f 2 tamay line 2, f 3 tamay line 3; f A meto line A, f B meto line B. 2. The vehicle capacity setting is 6 passenges and 3 passenges fo meto and tamay, espectively. f A f B Z REFERENCES 1. P. Macotte, Netok design poblem ith congestion effects: a case of bilevel pogamming. Mathematical Pogamming, vol. 34, 1986, pp L.J. LeBlanc, Tansit system netok design, Tanspotation. Reseach pat B, Vol. 22, 1988, pp I. Constantin and M. Floian, Optimizing fequencies in a tansit netok: a nonlinea bi-level pogamming appoach, Intenational Tansactions in Opeational Reseach, Vol. 2, 1995, pp Z. Gao, H. Sun and L.L. Shan, A continuous equilibium netok design model and algoithm fo tansit systems. Tanspotation Reseach Pat B, Vol. 38, 24, pp M. Abdulaal and L.J. LeBlanc Continuous equilibium netok design models. Tanspotation Resech, Vol. 138, 1979, pp B. Colson, P. Macotte and G. Savad, An ovevie of bilevel optimization, Annals of Opeations Reseach, Vol. 153, 27, pp T.Y. Ma and J.P. Lebacque, A coss entopy based multi-agent appoach to taffic assignment poblems. In: Poceedings of the Taffic and Ganula Flo 7, Spinge Belin Heidelbeg, 27, pp

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