A Method Based on Dial's Algorithm for Multi-time Dynamic Traffic Assignment

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1 Sensors & Transducers, Vol. 66, Issue 3, March 04, pp Sensors & Transducers 04 by IFSA Publshng, S. L. A Method Based on Dal's Algorthm for Mult-tme Dynamc Traffc Assgnment Rongje Kuang, Xaopng Guang, Yueyuan Xu, Yha Tan School of Traffc and Transportaton Engneerng, Lanzhou Jaotong Unversty, Lanzhou, , Chna Tel.: E-mal: @qq.com Receved: 7 December 03 /Accepted: 8 February 04 /Publshed: 3 March 04 Abstract: Due to statc traffc assgnment has poor performance n reflectng actual case and dynamc traffc assgnment may ncurs excessve compute cost, method of mult-tme dynamc traffc assgnment combnng statc and dynamc traffc assgnment balances factors of precson and cost effectvely. A method based on Dal's logt algorthm s proposed n the artcle to solve the dynamc stochastc user equlbrum problem n dynamc traffc assgnment. Before that, a fttng functon that can proxmately reflect overloaded traffc condton of lnk s proposed and used to gve correspondng model. Numercal example s gven to llustrate heurstc procedure of method and to compare results wth one of same example solved by other lterature's algorthm. Results show that method based on Dal's algorthm s preferable to algorthm from others. Copyrght 04 IFSA Publshng, S. L. Keywords: Dynamc traffc assgnment, Compute cost, Dal's logt algorthm, Dynamc stochastc user equlbrum, Heurstc procedure.. Introducton Statc user equlbrum s a state that acheves the user optmal. In ths condton, user optmal and user equlbrum are unfed, models based on User Equlbrum (UE) prncple thus are wdely proposed. Frst and second prncple of equlbrum s developed by Wardrop, whch emanate from concept of Nash Equlbrum n the game theory. Then Beckmann ntroduces mathematcal extreme value model of equlbrum theory. A hybrd model combnng traffc volume dstrbuton wth traffc assgnment s proposed n 97, and t solves assgnment problems through lnear programmng. Cesaro [] proposes combnaton model of traffc volume dstrbuton and traffc assgnment whch bases on the gravty model at the cost of gnorance of elements effectng traffc state. Basng on the equlbrum theory, Smth [] develops varatonal nequalty model that mproves the theory. Daganzon and Sheffc [3] unfes the concepton of stochastc and Wardrop equlbrum and proposes Stochastc User Equlbrum (SUE) model, whch can be effcently solved by the method ntroduced n Dal [4]. The statc traffc assgnment, however, can not reflect congeston of traffc on the lnk, ths to some extent prevent assgnment results from gettng more precse. Dynamc traffc assgnment models have sgnfcant employments n the rapdly developed advanced traveler nformaton systems and advanced traffc management systems. Generally, these models can be classfed nto two categores: the reactve assgnment model and the predctve one. The former 48 Artcle number P_935

2 Sensors & Transducers, Vol. 66, Issue 3, March 04, pp assumes that each traveler wll chose the shortest path to get to the destnaton accordng to current nstantaneous traffc stuaton. If dfferent paths have been chosen, travelers between the same Orgn- Destnaton (OD) par at the same tme thus may arrve at destnaton dfferently. Models proposed by We et al. [5] and Ran et al. [6] are belong to ths paradgm. The later consders the mpact of future traffc stuaton on path choce behavor, that s, the shortest path s usually determned by the actually experenced travel tme. Most of these models am at satsfyng the dynamc user-equlbrum condton, whch requres actual travel tme used by travelers s equal and mnmal. Models belongng to the later one are used n Fresz et al. [7] and Chen and Hsueh [8]. Dynamc traffc assgnment s regarded as a new approach to solve the traffc problems and t deals wth congeston, dynamc models, however, are more complcated. What s more, exstence and unqueness of the soluton are stll not absolutely guaranteed. But nvestgaton n Lo and Szeto [9] shows that statc traffc assgnment and dynamc one can even produce drectly opposte results, and ths llustrates the mportance of adoptng advantages from the dynamc traffc assgnment for plannng and polcy evaluaton. In the pont of vew on macro traffc polcy, traffc assgnment ams to predct dstrbuton of traffc flow on the route network and to get predctve results such as utlzaton of lnk and nfluence from new added OD par. The polcy thus can be made to mprove or remove correspondng nfrastructure or to control part of flows through traffc regulatons. The method n ths paper s to precsely have traffc assgnment wth low compute cost at the extent of macro traffc polcy. To make traffc assgnment results approach realstc condton, such a target has attracted an ncreasng number of study efforts. Basng on shortcomngs from the statc traffc assgnment and defectons of hgh compute cost n the dynamc traffc assgnment, new models between the statc traffc assgnment and the dynamc one are proposed n the past few years. The ntroducton of new models ams to overcome defect on the statc type's poor performance n reflectng actual traffc stuatons, whch may lead to error of polcy maker's decson, and on the bg complexty and huge compute cost n dynamc type. Lnk state s defned and then t dvdes traffc flows nto dfferent states to have traffc assgnment. Method proposed n Kong and L [0] has traffc assgnment through multpath ncrement loadng. And t also consders the nfluence of the traffc flow whch has been assgned n the network, usng an algorthm that fgures results of traffc assgnment accordng to varaton of mpedance n each lnk. Model based on traffc assgnment between statc and dynamc one s proposed n Tang [], and t s solved by Frank- Wolfe algorthm n the lterature. In ths paper, a method based on Dal's logt assgnment algorthm [4] s proposed n the artcle to solve the dynamc stochastc user equlbrum problem n mult-tme dynamc traffc assgnment. The artcle uses the Dal's algorthm n that t s a probablstc assgnment algorthm whch can reflect travelers' actual behavor better. Numercal example from [0] wll be fgured by method ntroduced n the paper and ts results wll be compared to results n [0], n order to show ts mprovement. What s note worthy s that method of Kong uses dfference between traffc volume of each perod bordered upon as ncrement of traffc volume, whch can be expressed wth q=q nm -q nm- [0]. If q 0, then traffc flow s loaded through all or nothng, else the longest dstance path n the network wll be searched and there wll be a q reducton n ts traffc volume. And new mpedance value then s updated by Bureau of Publc Roods (BPR) model []. In the method of Kong, all trps between a fxed Orgn-Destnaton par are assgned to the lnk of shortest length path. That approach used n the lterature of Kong s so-called all-or-nothng (AON) assgnment, whch assgns all traffc flows on the shortest path. AON assgnment s known to contradct actual trp behavor, and the lnk volumes these traffc assgnment models output are to some extent naccurate to the pont of compromsng the transportaton planner's desgn decsons, due to the effects of trp volumes on travel tme and the trp maker's non-determnstc choce functon on route selecton. In order to satsfy the realstc traffc demand and to reflect the traffc assgnment condtons n dfferent lnks at dfferent tme, the model dvdes tme nto several dfferent perods and then t does have traffc assgnment accordng to the regulaton of demand varaton and average demand n each perod. In the dynamc model, tme dvdng s usually based on the tme-traffc volume curve due to dvson of tme hnges largely on the regulaton of varaton of traffc volume. The varety regulaton of traffc flow n dfferent tme can be shown and graphed n Fg.. Stable varaton of traffc volume can be regarded as a perod, thus each perod of tme are not same all the tme. In the route networks, lnk travel tme s not always n proporton to dstance, ths s because traffc flow on each lnk plays an mportant role n mpactng lnk travel tme, the more t becomes large, the more lnk travel tme becomes huge. That s, lnk travel tme s to some extent n proporton to the traffc mpedance rather than the dstance. Due to the traffc demand volume s a varable, the traffc mpedance can thus be treated as varable as well. So, the problem can be generalzed as follows: If the mpedance s quantfed as length of path, wth the varaton of traffc volume between two gven node, the shortest path s changes too. Ths compels us to update the mpedance of lnk frequently so that large part of traffc flow can be assgned on the new shortest path. 49

3 Sensors & Transducers, Vol. 66, Issue 3, March 04, pp Traffc volume/pcu t/h Fg.. Varety regulaton of traffc flow n dfferent tme [0].. Method Based on Dal's Algorthm Accordng to forgong defecton of the Kong's method, method based manly on Dal's logt assgnment algorthm s proposed to deal wth such dynamc traffc assgnment problem. And then we wll gve a comparson between Kong's result and wrter's. Due to the usng of BPR model n the lterature of Kong, t s stll used as functon of mpedance n the artcle... The Fttng Functon In order to acheve a better performance n precson and low compute cost than before, pontqueue representaton s needed to be concerned. Three knds of functons are wdely used, they are: the outflow functon, the travel tme functon, and the mxed functon. The outflow functon regards outflow of lnk as a nondecreasng functon of the number of vehcles travelng n the lnk. And t s used to fgure cumulatve flows n the lnk so that travel tme can be obtaned. Outflow functon s adopted by Cary et al. [3]. The lnk travel tme s defned from the nflows and outflows. Whle the travel tme functon reverses the procedure, that s, t calculates lnk travel tmes before outflows. Yang and Meng [4] and Huang and Lam [5] adopt ths knd of functon n developng ther traffc flow models. In mxed functon, outflows are determned by ext-flow functon and travel tme s determned by travel tme functon. The functon s employed n Lam and Huang [6]. In all of these three knds of functons, outflow s needed weather before or after the calculaton of travel tme. Introducton of fttng functon n ths artcle makes value of outflow dspensable. We assume that value of assgnment traffc flow n each lnk s equal to ts outflow volume when assgnment value s lower than capacty. In the condton that assgnment value s equal to or larger than capacty of lnk, the wrter gves a fttng functon that can approxmately descrbe relatonshps between nflow and remanng traffc flow of lnk, and the fttng functon s expressed wth: x c v v =, () where x s the remanng traffc flow of lnk, v denotes total of lnk, and c denotes lnk capacty. In the course of traffc assgnment, congeston wll occur when value of traffc volume surpasses the capacty of lnk part of volume thus could not pass through the lnk nstantly. In ths stuaton, due to nteracton between vehcles on the lnk exsted, value of remans traffc volume s usually larger than margn between traffc nflow and capacty of lnk. () s just about the functon that fgures results of remanng flow n the lnk approachng actual value. Fg. shows graph of () whch reflects relatonshp between total flow and remanng traffc flow of lnk n a condton that value of capacty s gven 70. Fg.. Image of functon between remanng flow of lnk and total flow of lnk (the range from 70 to 800)... The Model In order to gve expresson of conservaton of dynamc flow and to gve mathematcal descrpton for BPR model, here we state our man assumptons: ) All vehcles passes through the lnk clearly before the next perod n a condton that traffc flow n each lnk s lower than ts capacty. 50

4 Sensors & Transducers, Vol. 66, Issue 3, March 04, pp ) No overtakng and drawback wll occur n each lnk. Because there s no formulaton of outflow for BPR, and wth ntroducton of fttng functon, expresson of conservaton of dynamc flow mentoned before can thus be substtuted wth: ( ) = ra ( ( )) + xa ( ( )), () where S a () denotes total flow of lnk n current perod; r a - (S a (-)) denotes remanng flow n the former perod, whch s calculated wth S a () n former perod; x a (S a (-)) s traffc assgnment volume of lnk fgured accordng to mpedance determned by S a (-) (.e. nput flow of lnk a at perod t ). Basng on Logt SUE model [7], the dynamc model of BPR model thus can be depcted as follows: mnf ( h, x ) = ( t ) hk ( t)lnhk ( t) + ta( w) dw (3) θ ( t ) k K a A k xa ( t ) = hk ( t ) δ a k K ( ) = ra ( ( )) + xa ( S hk ( t ) = q( t ) k K c ra = ( ) ( ) a ( )), (4) where A(k) s the set of all lnks orgnated from node k, B(k) s the set of all lnks termnated at node k, q rs (t ) s the flow orgnated from node r and termnated at node s at perod t, t a (q) s the tme needed for flow q to pass through lnk a, x a s the nput flow of lnk a at perod t, S a (t ) s the total flow of lnk at perod t, c s the capacty of lnk, h k (t ) s the flux of path k at perod t, q(t ) s the traffc k demand n the network at perod t, δ a s the weather yes lnk a on the path k, k δ a = { and r a s the 0 else remanng flow n the lnk..3. The Algorthm Accordng to the Dal's theory, t stochastcally dverts trps to alternate paths, but trps are not explctly assgned to routs. Instead, they are probablstcally dverted at each node encountered to the competng lnks enterng the node. In multstep dynamc traffc assgnment, due to tme has been dvded nto several perods, traffc demand s dvded nto several groups correspondngly as well. The method ntroduced n the artcle uses Dal's algorthm to solve traffc assgnment n each step and updates traffc mpedance to prepare for the assgnment n the next step. We need to know the ntal mpedance n order to get the frst step assgnment result. Then the remanng flow can be calculated accordng to fttng functon mentoned before so that teratve procedures could carry on. Method based on Dal's algorthm n solvng dynamc traffc assgnment problem can be llustrated as procedures as follows: Step 0 (Intalzaton): Dvde tme nto several perods accordng to stablty of varaton of traffc volume and then get the traffc demand q(t) n each perod. The ntal mpedance of network s gven. Step : Use Dal's algorthm to get the result of assgnment n perod t accordng to traffc demand and mpedance n Step 0 and calculate remanng traffc flow and total flow of each lnk at perod t. Step : Accordng to the precedng step, the traffc mpedance can be fgured through BPR model, and the use Dal's algorthm to assgn traffc volume. If lnk n the network s defned as effcency lnk, then mark t wth. And f all perods' assgnment have been fnshed, the dynamc traffc assgnment s complete, else turn to Step 3. Step 3: Take the result of assgnment nto equaton () and take ts fguraton result nto fttng functon () to calculate total flow of lnk, turn to Step. 3. Numercal Example 3.. Example and ts Results Solved by Kong's Method Network, as Fg. 3 shows, gves values of travelng tme and capacty n each lnk Fg. 3. Network and travelng tme [0]. Table gves traffc flow at dfferent tme. All travelers n each perod, accordng to traffc condton before, wll select travelng path the content wth from orgn node to destnaton 9. All values of capacty are gven 70, and traffc volume to be assgned nto network I has been dvded nto 0 groups. Results of traffc flow and transportaton mpedance n each lnk at dfferent tme fgured by Kong's method s shown as Table

5 Sensors & Transducers, Vol. 66, Issue 3, March 04, pp Table. The traffc flow of dfferent tme [00]. Perod Traffc volume (PCU) Perod Traffc volume (PCU) Table. Kong's results of traffc flow and transportaton mpedance n each lnk at dfferent tme (Part ) [0]. Perod Note: I denotes traffc mpedance, F denotes traffc flow n the lnk. Table 3. Kong's Results of Traffc Flow and Transportaton Impedance n each Lnk at Dfferent Tme (Part ) [0]. Perod Note: I denotes traffc mpedance, F denotes traffc flow n the lnk. 3.. Optmzaton Strategy Here, heurstc procedure based on Dal's algorthm wll be used to solve example n lterature of Kong. Before that, some of varables n heurstc procedure s needed to be llustrated. Illustratons of varable can be shown n Fg. 4. r M s N t a j A, B, C R, K, T Fg. 4. Network and travelng tme. H r s Mj Nj t a denotes the tme needed to pass through lnk a, hence t can be used as mpedance of lnk; H denotes capacty of lnk; M and N denote the shortest dstance from orgn node respectvely n forward pass and backward pass; A B and C denote lnk weght, probablty of selecton and assgnment volume respectvely; R denotes remanng flow of lnk durng the former perod; K denotes remanng flow of lnk durng current perod; and T denotes total flow n the lnk durng current. In the act of calculaton at each stage, varable K plays an mportant role, whch s used to fgure T of current stage and next stage. The mpedance of lnk bases on total flow of lnk. If lnk s defned as effcent lnk then mark t wth to avod mstakes durng the assgnment. Let parameter θ n Dal's algorthm equal to, and the results of traffc flow and mpedance n each lnk at dfferent tme by method based on Dal's method s shown n Table 4 and Table 5. 5

6 Sensors & Transducers, Vol. 66, Issue 3, March 04, pp Table 4. Results of method based on Dal's algorthm n traffc flow and transportaton mpedance n each lnk at dfferent tme (Part ). Perod Note: I denotes traffc mpedance, F denotes traffc flow n the lnk. Table 5. Results of method based on Dal's algorthm n traffc flow and transportaton mpedance n each lnk at dfferent tme (Part ). Perod Note: I denotes traffc mpedance, F denotes traffc flow n the lnk Results In comparson wth Table and Table 3, the advance of utlty of lnk can be found n Table 4 and Table 5, whch reflects traveler's behavor effcency. That s, not all travelers selectng the shortest path n the realstc condton. Besdes, results of Table 4 and Table 5 also take congeston effect nto consderaton. Ths makes method results close to actual case. In Table 4 and Table 5, every traffc flow of lnk -, -4, 4-5, 6-9 and 8-9 are larger than other lnks', whch means lnks wth hgher traffc flow may be overloaded and those lnks wll thus be defned as crtcal lnks. Crtcal lnks can be compared to bottleneck of network, and through capacty of network hnges largely on crtcal lnks. Hence t s of overrdng sgnfcance to mprove capacty of crtcal lnks. 4. Concluson Method based on Dal's algorthm has superorty, comparng wth Kong's method, n approachng the actual case. Such a dfference between two methods stems from shortcomngs of AON assgnment that s used n Kong's method. The frst s nstablty, dfferent out puts may be derved sgnfcantly by nsgnfcant change n nput. As a result, t wll be dffcult to predct the precse estmates of lnk tmes, that s, the nstablty serously hnders utlty of model. The second season s that the AON assgnment to some extent can not reflect actual behavor precsely. Although method ntroduced n Kong amelorates AON assgnment creatvely by overcomng ts defect of no capacty restrant, t s stll unable to allow for realstc random varaton of route selecton among ndvdual trp makers. Whle defecton of capacty restrant n Dal's algorthm s not reflected due to ts usage n the dynamc paradgm. The thrd one, due to all trps are assumed to use the shortest path accordng to AON assgnment, total vehcle hours are based. Thus planner may always overestmate the value of hs desgn because AON measure s often used as a macro-evaluator for planner. Dfferng from method of sngle statc traffc assgnment whch loads traffc flow nto network just for once, the artcle uses technque lke ncrement load to dvde tme nto several perods accordng to 53

7 Sensors & Transducers, Vol. 66, Issue 3, March 04, pp varaton of travelng volume, and t takes capacty restrant and traffc mpedance nto account to make assgnment results approach the realstc case as far as possble. For the property of hybrd method, when the tme s more precsely dvded, dynamc performance of hybrd method wll approach pure dynamc one. What s more, the proposton of usng Dal's algorthm to solve traffc assgnment for dynamc paradgm sgnfcantly decreases compute cost, especally n route enumeraton. Hence method proposed n the artcle s nstrumental n structurng advanced traveler nformaton systems and advanced traffc management systems. References []. F. J. Cesaro, A combned trp generaton and dstrbuton model, Transportaton Scence, Vol. 9, Issue 3, 975, pp. -3. []. M. J. Smth, The exstence, unqueness and stablty of traffc equlbra, Transportaton Research Part B, Vol. 3, Issue 4, 979, pp [3]. C. F. Daganzo, Y. Sheff, On stochastc models of traffc assgnment, Transportaton Scence, Vol., Issue 3, 977, pp [4]. R. B. Dal, A probablstc multpath traffc assgnment model whch obvates the need for path enumeraton, Transportaton Research Part B, Vol. 5, Issue, 97, pp [5]. B. W. We, T. L. Fresz, R. L. Tobn, Dynamc user optmal traffc assgnment on congested multdestnaton networks, Transportaton Research Part B, Vol. 4, Issue 3, 975, pp. -3. [6]. B. Ran, D. E. Boyce, L. J. Leblanc, A new class of nstantaneous dynamc user-optmal traffc assgnment models, Operaton Research, Vol. 4, Issue, 993, pp [7]. T. L. Fresz, D. H. Bernsten, T. E. Smth, R. L. Tobn, B. W. We, A varatonal nequalty formulaton of the dynamc network user equlbrum problem, Operaton Research, Vol. 4, Issue, 993, pp [8]. H.-K. Chen, C.-F. Hsueh, A model and an algorthm for the dynamc user-optmal route choce problem, Transportaton Research Part B, Vol. 3, Issue 3, 998, pp [9]. W. Y. Szeto, H. Lo, A cell-based smultaneous route and departure tme choce model wth elastc demand, Transportaton Research Part B, Vol. 38, Issue 7, 004, pp [0]. H. H. Kong, Y. Z. L, Mult-tme dynamc traffc assgnment and algorthm, Journal of Lanzhou Jaotong Unversty (Natural Scence), Vol. 3, Issue 4, 004, pp. -5. []. Q. L. Tang, Y. L, Research on dynamc mult-tme traffc based on assgnment model, Technology & Economy n Areas of Communcatons, Vol. 57, 00, pp []. Transportaton research board busness offce, Hghway capacty manual 000, Washngton, D.C: Natonal Research Councl, 000. [3]. M. Carey, A. Srmvasan, Externaltes, average and margnal cost, and tools on congested networks wth tme-varyng flows, Operatons Research, Vol. 4, Issue, 993, pp [4]. H. Yang, Q. Meng, Departure tme, route choce and congeston toll n a queung network wth elastc demand, Transportaton Research Part B, Vol. 3, Issue 4, 998, pp [5]. H. J. Huang, W. H. K. Lam, Modelng and solvng dynamc user equlbrum rout and departure tme choce problem n network wth queues, Transportaton Research Part B, Vol. 36, 00, pp [6]. W. H. K. Lam, H. J. Huang, Dynamc user optmal traffc assgnment model for many to one travel demand, Transportaton Research Part B, Vol. 9, Issue 4, 995, pp [7]. C. Fsk, Some developments n equlbrum traffc assgnment methodology, Transportaton Research Part B, Vol. 4, 980, pp Copyrght, Internatonal Frequency Sensor Assocaton (IFSA) Publshng, S. L. All rghts reserved. ( 54