A Network Signal Timing Design Bilevel Optimization Model with Traveler Trip-Chain Route Choice Behavior Consideration

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1 Journl of Trffi nd Trnsporttion Engineering 5 (2017) doi: / / D DAVID PUBLISHING A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion Chung-Yung Wng 1, Kuo-Chi Yen 1, Shou-Ren Hu 2, Chih-Peng Chu 3 nd Y-Tin Jhung 1 1. Deprtent of Logistis Mngeent, Ntionl Defense Univeity, Tipei City, 112, Tiwn, R.O.C.; 2. Deprtent of Trnsporttion nd Counition Mngeent Siene, Ntionl Cheng Kung Univeity, Tinn City, 701, Tiwn, R.O.C.; 3. Deprtent of Business Adinistrtion, Ntionl Dong Hw Univeity, Hulien City, 974, Tiwn, R.O.C. Abstrt: This study proposed bilevel optiiztion odel for the network signl tiing design proble by onsidering the link flows refleted by the trip-hin route hoie behvio of rod use. The bilevel progring odel is forulted bsed on the intertions between signl tiing ontrol nd trip-hin behvior, nd solution lgorith by obining vritionl inequlity sensitivity nlysis, the generlized invee trix ethod, nd grdient projetion pproh revised for trip-hin user equilibriu is developed for the trnsporttion design proble. The perforne of the developed odel frework ws verified through nueril nlysis under different test senrios. Key words: Signl tiing design, bilevel progring odel, trip-hin user equilibriu. Noenlture Link nuber Trvel osts of link 0 Free trvel osts of link Trvel ost derivtive of link Trvel osts of using route p between OD pir p (,) under user equilibriu priniple Trvel osts of using trip-hin route ˆp between OD pir (,) under user equilibriu priniple C Signl yle tie t inteetion CAP Cpity nd sturtion flow rte of rod link onneted to inteetion d Desent diretion d Vetor of desent diretion I g Effetive green tie in the Ith phse t inteetion whih onneted to link g Vetor of effetive green tie I g The iniu effetive green tie. p h Trffi flow on route p between OD pir (,) under user equilibriu priniple h Trffi flow on trip-hin route ˆp between OD pir (,) under user equilibriu priniple I l Loss tie in the Ith phse t inteetion onneted to link ni Ativity nodes pssed by the trip-hin routes between OD pir (,) N Set of tivity nodes pssed by the trip-hin routes between OD pir (,) p Pth vrible Route pssing tivity nodes on trip-hin ˆp ˆp P Set of trip-hin routes q Trip dend between OD pir (,) r Origin vrible s Destintion vrible S x z α ε ϕ Shortest route pssing tivity nodes on trip-hin Sturtion flow rte of link onneted to inteetion Trffi flow on link Objetive funtion Step size Perturbtion preter Convergene riterion The link/pth inditor vrible, it is zero-one δ ˆ p inditor vrible tht equls 1 when trip-hin γ p n i route psses link nd 0 when it does not. The tivity node/pth inditor vrible, it is zero-one inditor vrible. When route p between trip-hin OD pir ( s) r, psses the n th Ativity node on the trip-hin between the

2 204 υ A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion trip-hin OD pir, then γ n i p = 1 ; otherwise, γ n i p = 0. Step djustent preter 1. Introdution In rod network nlysis, the design of trffi signl tiing plns is highly dependent on the urte observtions of rod use route hoie behvio nd/or deisions. Signl tiing settings n hve effets on distributed trffi volue, whih influenes link trvel osts, in turn ffeting the routes tht rod use hoose. The ggregtion of user behvio is refleted in the trffi volue of vrious rod segents nd inteetions, nd the trffi volue t inteetions is the foundtion of signl tiing design. Fisk [1] estblished leder-follower reltionship between signl tiing designs nd the route hoies behvior of use, with the forer being the leder of rod network systes nd the ltter being the follower. Followe (the rod use) re influened by signl tiings in their seletion of their routes. In order to develop desirble signl tiing pln tht will potentilly result in the optil trnsporttion perforne in rod network, it is neessry to develop suitble odel frework for the behvio of followe. Therefore, Stkelberg equilibriu exists between signl ontrol nd user behvior. Mny previous studies explored the optiiztion of signl tiings using bilevel progring, with the upper level onsidering signl tiing optiiztion nd the lower level onsidering user equilibriu route hoies behvior. However, these reserhes only onsidered the route seletions between trip origins nd destintions, negleting the trip-hin behvior tht n rise fro inidentl soil or eonoi tivities. Consequently, the bilevel progring odels proposed in these studies did not onern the sitution of trip-hin behvior on signl tiing design. This is Corresponding uthor: Chung-Yung Wng, Ph.D., professor; reserh fields: trnsporttion plnning, trffi ssignent probles, network design proble, trip-hin network equilibriu odel nd solution lgorith. E-il: stellr@s35.url.o.tw. signifint reserh gp by inorporting trip-hin needs in the trvel behvior of urbn rod use into the signl tiing design proble. In order to tke the trip hin behvior of rod use into ount, we integrted the vritionl inequlity sensitivity nlysis pprohes proposed by Tobin [2] nd Tobin nd Friesz [3] nd the generlized invee trix ethod to derive the objetive desent diretion of the bilevel progring odel nd developed solution lgorith. For the lower level of the odel, we onstruted trip-hin user equilibriu trffi ssignent odel nd developed route-bsed lgorith to solve it. The reinder of this pper is orgnized s follows. Setion 2 nlyzes the reserh proble nd relevnt literture, while Setion 3 desribes the odeling proess for the bilevel signl tiing optiiztion odel tht onside trip-hin route hoie deisions. Setion 4 explins how to obine the sensitivity nlysis ethod nd the generlized invee trix ethod to solve the bilevel progring odel. A ethod for solving the lower-level trip-hin-bsed user equilibriu trffi ssignent odel is lso presented. Setion 5 presents nueril nlysis on the verifition of the proposed ethod using test network, nd Setion 6 surizes this reserh by onlusions nd suggestions. 2. Proble Stteent nd Literture Review A nuber of reserhe hve onsidered the route seletion behvior of rod use when optiizing signl tiing designs using bilevel progring, inluding Yng nd Ygr [4], Chen nd Hsueh [5], Chen nd Chou [6], Ciprini nd Fuso [7], Chiou [8], Sith [9], Ukkusuri et l. [10], nd Chiou [11]. However, their odels only onsidered the route hoies between O-D (origin-destintion) pi, nd negleted trip-hin behvior. In ft, there re lso soe needs to oplete ultiple tivities into single trip, whih results in trip-hin behvior. Hägetrnd [12] proposed n tivity-bsed odel to

3 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion 205 onsider trip-hin behvior. Mny epiril studies, suh s those onduted by Hensher nd Reyes [13], MGukin et l. [14], Moreny nd Vliquette [15], Currie nd Delbos [16], nd Zho et l. [17], showed the trip-hin behvior should be onsidered. Therefore, we onsider the trip-hin bsed route hoies of rod use to identify the tul trnsporttion dend within network before n pproprite trffi signing tiing design shee n be estblished. With regrd to the trip-hin behvior of rod use, Mruy nd Hrt [18] indited tht if the individul segents of rod user s trip-hin needs re onsidered seprtely, the results would be unble to reflet the reltionships ong the, thereby reduing the ury of preditions for subsequent trnsporttion dend. To overoe this proble, Mruy nd Hrt [19] estblished trip-hin bsed odel under the ssuption of stti network equilibriu to ondut ordon-bsed ongestion priing. Mruy nd Sulee [20] indited tht diret reltionship exists between ongestion priing nd the trip-hin behvior of rod use nd therefore used the trip-hin bsed network equilibriu odel proposed by Mruy nd Hrt [19] to opre the vlidity nd firness of ordon-bsed nd re-bsed priing shees. Higuhi et l. [21] developed novel obined trnsport ode nd trip-hin bsed route hoie network equilibriu odel with two stges bsed on vritionl inequlity probles nd dopted the relxtion ethod to solve it. However, ll of the bove studies fixed the order of tivities, thereby eliinting order s ftor influening route hoies. Trip-hin desriptions in the odels were lso overly siplified, erely inditing tht if route n ws on the trip-hin route between O-D pir (r, s), then trip-hin vrible η n =1. However, n urbn rod network is not diretly onsisted of routes, but rther routes oprising respetive links. Expressing trip-hin behvior in this nner rende it fro nlyzing the reltionships between trip-hin needs nd link flows in the rod network. Furtherore, these studies used link-bsed Frnk-Wolfe lgorith pprohes, whih exhibit poor oputtionl effiieny. Wng nd Chen [22] developed trip-hin user equilibriu trffi ssignent odel tht onsidered the effet of the orde in whih rod use prtiipted in tivities on their route hoie deisions nd used route-bsed GP (grdient projetion) ethod to develop n lgorith. The results showed tht onventionl trffi ssignent odels whih did not onsider trip-hin behvior were erely speil se of trip-hin-bsed trffi ssignent odels. Thus, the odel frework they developed ws ore generlized trip-hin-bsed trffi ssignent odel. To solve the bilevel progring odels for optil signl tiing plns, Allsop [23] nd Grtner [24] used IOA (itertive optiiztion-ssignent) lgoriths to updte signl tiing designs bsed on fixed network flow nd then solve network equilibriu flows bsed on fixed signl tiing designs. They fit fixed the deision vribles in the upper-level proble before solving the lower-level proble nd then fixed the deision vribles in the lower-level proble to solve the upper-level proble. This proess ws then repeted until the upper-level proble onverged. However, the IOA lgorith obtins Nsh solutions rther thn the true solutions for the bilevel optiiztion odels. Mrotte [25] expnded the optiiztion probles hidden within network design probles to for bilevel progring proble, inditing tht bilevel progring probles re tully type of NSG (non-oopertive Stkelberg ge). Fisk [1] lso stted tht Stkelberg equilibriu exists between the signl ontrol nd rod user s route hoie behvior proble. In the Stkelberg equilibriu solutions of bilevel progring odels, the deision vribles of the upper-level nd lower-level odels hve n ipliit funtionl reltionship, nd for this reson, the prtil

4 206 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion derivtives of the deision vribles nnot be lulted diretly. To overoe this proble, Tobin [2] proposed vritionl inequlities sensitivity nlysis ethod nd verified tht when vritionl inequlity hs unique solution nd the vribles nd funtions stisfy the ondition of strit opleentry slkness, preter perturbtion n be used to obtin the optil solution in the doin of the preter ε = 0. Tobin nd Friesz [3] lso dopted vritionl inequlities sensitivity nlysis pproh tht used the tre perturbtions in the deision vribles ner the equilibriu solution to effetively estite the derivtive funtion ner the equilibriu solution. The funtion provides desent diretion in the serh for the optil solution nd then returns the Stkelberg solution, whih is the optil solution for the bilevel progring odel. While investigting bilevel progring odels for inteetion signl designs, Wong nd Yng [26] used the sensitivity nlysis pproh proposed by Tobin [2] to deterine the reltionships ong signl tiing, link flow, nd user behvior, but they overlooked the ft tht route solutions y not be unique nd thus derived singulr solution. To ddress this issue, Cho [27] used the generlized invee trix ethod to onvert routes into unique links before perforing the sensitivity nlysis. This pproh eliinted the possibility of degenerte solutions ourring in the sensitivity nlysis. Wng [28] obined the vritionl inequlities sensitivity nlysis pproh with the generlized invee trix ethod nd suessfully solved tie-dependent bilevel progring odel for signl tiing optiiztion with link pity onstrints. Chen nd Chou [6] siilrly used the vritionl inequlities sensitivity nlysis ethod nd the generlized invee trix ethod to solve tie-dependent bilevel progring odel for signl tiing optiiztion. The reserh proble onsidered by this study n be surized s follows: Bilevel progring odels n be used to optiize trffi signl tiings. The upper-level odels re signl tiing optiiztion probles with iniizing totl trvel osts in the syste s the objetive, nd the trvele route hoie behvio n serve s onstrints in the lower-level odels. In previous reserhes with bilevel progring odels for signl tiing optiiztion, lower-level odels did not onsider trip-hin behvior nd thus ould not oprehensively reflet the tul trnsporttion dends in rod network. The solution of bilevel progring odel for signl tiing optiiztion is Stkelberg equilibriu solution. The deision vribles of the upper-level nd lower-level odels hve n ipliit funtionl reltionship nd do not for losed funtion. For this reson, the prtil derivtives of the deision vribles nnot be lulted diretly. Using the vritionl inequlities sensitivity nlysis ethod proposed by Tobin [2] nd the generlized invee trix ethod presented by Cho [27] n provide the derivtive of the ipliit funtion. 3. Model Forultion 3.1 The Model Before forulting the trip-hin bsed optiiztion network signl tiing design odel, we define the trip-hin nd its user equilibriu priniple. In this study, ll the lotions of the trip-hin tivities between the trip ends should be pssed. We ssued tht the trvele hve perfet infortion suh tht they ould ke orret deisions regrding trip-hin route hoie. The trip-hin bsed user equilibriu priniple n be defined s no trveler n iprove his trip-hin route trvel tie by unilterlly hnging trip-hin route. Tht is, the route hoie ust pss ll of the prtiulr interediry tivity lotions between their origin nd destintion nd inur the inil trvel osts. The objetive of this odel is to iniize the totl trvel tie. We ssued tht the trffi signls t ll of the inteetions were two-phse signls with fixed

5 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion 207 yle ties nd iniu green light tie. The loss of green light tie in eh phse nd the sturtion flow rtes t eh inteetion re known. The estblished odel is s follows: I I ( g ) x ( g ) in (1) subjet to the following onstrints: Cyle onservtion onstrint: I I ( g + l ) = C I, (2) Definitionl onstrint: I I g CAP = S C I,, (3) Boundry onstrint: I g I g I, (4), The trip-hin user equilibriu onstrint: I I I ( x, g )[ x ( g ) x ( g )] 0 ( ) x Ω g (5) Eq. (1) is the trget funtion for the signl tiing optiiztion of single rod network with green light tie s the deision vrible. Eq. (2) is the onstrint for yle onservtion in the signl tiings, where the yle tie of the signl t inteetion is C nd I equls the totl su of effetive green tie g nd I loss tie l, the effetive green tie nd loss tie in the Ith phse t inteetion whih onneted to link, respetively. Eq. (3) defines the reltionship I ong CAP, S, g, nd C, whih denote the pity nd sturtion flow rte of link onneted to inteetion, the effetive green tie t inteetion onneted to link, nd signl yle tie t inteetion. Eq. (4) is boundry onstrint stipulting tht the effetive green light tie ust be greter thn or equl to the iniu effetive green I tie g. Eq. (5) onstrins the trip-hin route seletion behvior of rod use; it oprises trip-hin bsed user equilibriu route seletion odel in the for of vritionl inequlity nd enopsses the following onstrints for flow onservtion, non-negtive route flow, definition, nd trip-hin definition: Flow onservtion onstrint: h = q, r, s, (6) Non-negtivity onstrints on route flows: Definitionl onstrints: x (7) 0 (8) = 0 r s h δ, (9) { 0,1 }, r, s,, δ = (10) Trip-hin definitionl onstrints: hp ˆ = hp γ n r s p ( r s) ( r s) i p,,,,,, (11) ni N { } γ = 0,1, n,, N, p (12) Eq. (6) is the onstrint for the onservtion of trip-hin flow, ening tht the totl route flows of the trip-hins between ny given O-D pir ( r, s) ust equl q, the trip-hin dend for sid O-D pir. Eq. (7) is the non-negtivity onstrint on trip-hin route flows, nd Eq. (8) is the non-negtivity onstrint on link flows. Eq. (9) presents the definitionl onstrint, whih indites the reltionship between the trffi flow on eh link in the network nd the flow on the routes in the trip-hin. Eq. (10) defines δ s known inditor vrible tht equls 1 if, trip-hin route tht psses the interedite tivity points between O-D r,, psses through link nd 0 otherwise. Eq. (11) defines the reltionship between h nd h p, the route flows of trip-hin route tht psses the interedite tivity points between O-D pir r, nd oon route; γ is n tivity pir ( s) ( s) h 0, r, s, x point/route djeny trix tht is known { 0,1 }

6 208 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion inditor vrible, nd N is set of tivity points (nodes) tht the trip-hin route ust pss between O-D pir ( r, s) ( N = { tivity 1, tivity 2,..., tivity } ). If route p psses the ith tivity point, n, then γ = 1; otherwise, γ = 0. If route p psses ll of the tivity points in the trip-hin between O-D pir ( r, s), then the produt of ll the γ vlues for this route ust equl 1; in other words, γ = 1. If n N route p does not pss one or ore of the tivity points in the trip-hin, then the produt of ll the γ vlues for this route ust equl 0; in other words, γ = 0. Thus, the onstrining onditions of n N Eq. (11) n restrit the flows of the trip-hin routes between O-D pir ( r, s) nd ensure tht ll of the tivity points in the trip-hin re pssed. Furtherore, γ = 1 does not restrit the order n N in whih the tivity points re pssed. Finlly, Eq. (12) is onstrint for the trip-hin definition, in whih γ is n tivity point/route djeny trix tht is known { 0,1 } inditor vrible. As n be seen, the odel is bilevel progring odel. The upper-level odel optiizes the rod network syste, nd the lower level odel is trip-hin bsed user equilibriu trffi ssignent odel enopssing the onstrints of the upper-level odel. With regrd to trip-hin bsed user equilibriu onditions, Wng nd Chen [22] stted tht when use ust pss ertin interedite tivity points, the route tht they will selet is the one tht psses ll of the interedite tivity points between their origin nd destintion with inil trvel osts. In view of this, we pled the trip-hin bsed user equilibriu trffi ssignent odel in the lower level of our bilevel odel for signl tiings to better reflet the reltionship between signl tiings design nd rod user behvior. 4. Solution Algorith 4.1 Sensitivity Anlysis nd Generlized Invee Mtrix Method The objetive funtion of the bilevel progring odel for signl tiing optiiztion in this study puues iniu network osts nd is the su of the produts of link ost funtion (g) nd link flow x. Link osts re funtions of green light tie g, nd differenes in green light tie re refleted on link pity, whih in turn influene trvel osts nd the route hoie behvior of rod use. Thus, link flow x vries with green light tie g nd n ipliit funtionl reltionship exists between the two, without losed funtion to express it. Consequently, the derivtive of link flow x with regrd to green light tie g nnot be lulted diretly, whih kes the signl tiing optiiztion odel of this study diffiult to solve. Generlly, the solution of bilevel progring odel is lso referred to s Stkelberg equilibriu solution. To ddress the proble in whih prtil derivtives nnot be obtined diretly fro the ipliit funtionl reltionship between the deision vribles of the upper nd lower levels, Tobin [2] nd Tobin nd Friesz [3] proposed sensitivity nlysis ethods to derive the desent diretion of the objetive funtion in the upper level nd obtin the lol optiu solution of the odel. We lso dopted this pproh to solve the odel in this study. Aording to the onlusions de by Tobin [2] nd Tobin nd Friesz [3], n ipliit funtionl reltionship exists between link flow x nd effetive green light tie g; n sensitivity nlysis pproh n be used to derive the prtil derivtive of the ipliit funtion nd the desent diretion to serh for the optil solution of g, the deision vrible of the upper-level odel, s shown in the eqution below:

7 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion 1 ( 0) = J ( 0) [ J ( 0) ] ε x x ε (13) However, the sensitivity nlysis ethod requires route solution, whih in generl y not be unique for ny O-D pir, nd this retes soe diffiulty in obtining the solution. To enhne the generlity of the sensitivity nlysis ethod in user equilibriu probles, we dopted the generlized invee trix pproh proposed by Cho [27] to ondut the sensitivity nlysis. The generlized invee trix pproh for sensitivity nlysis in user equilibriu probles utilizes the ft tht link solutions re unique. This uniqueness is used to onvert the reltionships ong the link vribles into n O-D pir/route djeny trix nd liner independent link/route djeny trix tht express the route solution under the restritions of flow onservtion nd link/route definitions. We onduted nueril siultion to verify whether the proposed pproh n urtely obtin the derivtive of the ipliit funtion in trip-hin bsed network equilibriu probles. Fig. 1 displys the test network oprising six nodes nd twelve links. Nodes 1 nd 2 re the origins, nd Nodes 5 nd 6 re the destintions. Node 3 is n interedite tivity point tht ust be pssed between Origin 2 nd Destintion 5. We dopted n FHWA (Federl 209 Highwy Adinistrtion) ost funtion for the link osts, shown in Eq. (14). The free-flow trvel tie for eh link ws set s 1, nd Tbles 1 nd 2 exhibit the trnsporttion dends for eh O-D pir in the trip-hins nd the dt regrding the rod links in the network nd the signl tiings of the orresponding inteetions. Tbles 3 nd 4 present the flow dt of the vrious routes nd rod links when the network rehes the onditions of trip-hin user equilibriu: x Cp ( x ) = , 0 Fig. 1 Test Network 1. Tble 1 Trnsporttion dend between trip-hin O-D pi in Test Network 1. No. O-D pir Ativity node Trvel dend (pu/hr) Tble 2 Link nd inteetion signl tiing dt. Miniu Sturtion Cyle tie Initil green tie Lost tie green tie flow rte 60 s 27 s 7 s 50 pu/hr 3 s Tble 3 Trip-hin route flows when Test Network 1 rehes trip-hin user equilibriu. O-D pir Ativity node No. of pth Trip-hin pth Flow Trvel tie (14)

8 210 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion Tble 4 Link flows when Test Network 1 rehes trip-hin user equilibriu. No. Link Flow No. Link Flow Tble 5 Coprison of tul equilibriu solutions nd fit-order pproxite solutions. No. Link ε = 0 ε = 0.1 ε = 0.5 Atul sol. Estited sol. Atul sol. Estited sol ε = 0.1 is sll perturbtion preter. Next, bsed on the solution of trip-hin user equilibriu, we put sll perturbtion on the green tie. And opre the tul solution with the estited solution tht is lulted by vritionl inequlity sensitivity nlysis nd generlized invee pproh. The results re surized in Tble 5. When we put sll perturbtion to green tie, the estited solution is lost se s the tul solution. If the perturbtion ws too big, the differene between estited solution nd tul solution would be inresed. The nueril test deonstrtes tht the ipliit differene ould be lulted by vritionl inequlity sensitivity nlysis nd generlized invee pproh. Then the extly desent serh diretion ould be found. Therefore, with the optil step sizes, the Stkelberg solution of bilevel progring odel would be esily solved. 4.2 Solution Algorith Nueril siultions deonstrte tht the sensitivity nlysis pproh nd generlized invee trix ethods n produe the prtil derivtive of the ipliit funtion nd n be pplied to the bilevel progring odel of this study to provide diretion in the serh for the objetive funtion of the upper-level odel. Below, we desribe the steps of the solution lgorith in this study: Step 1: Set the initil green light ties g n for eh inteetion nd let n = 1 ; Step 2: Clulte the pities of eh link using the forul below: I g n I CAP n I = S I,, (15) C Step 3: Eploy the solution lgorith developed by Wng nd Chen [22] bsed on grdient projetion to solve the trip-hin bsed user equilibriu trffi ssignent odel below: in z = S.t. Flow onservtion onstrint: n ( ω, g ) Non-negtivity onstrint on route flows: x 0 hp ˆ = q r, s dω (16) (17)

9 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion 211 Definitionl onstrint: The trip-hin definitionl onstrint: h ˆ = hp γ, r, s, p r, s, (18) (19) (20) (21) (22) (23) Step 4: Use the foreentioned sensitivity nlysis nd generlized invee trix ethods to derive ε x ( 0) ; x δ h p ˆ 0 r, s = Step 5: Clulte the desent diretion for green light tie: n d = ε ( x( x, 0) ) (24) Using the FHWA ost funtion, for exple, the desent diretion is n 4 4 Sg ( ) ( ) 0 εx x ε x 0 C (25) 4 ( ) 5 n 5 S 0.60x g C Step 6: Updte the green tie to n+ 1 n 1 n g = g + d ; n + 1 Step 7: Adjust the green ties t eh inteetion until the onstrints below re stisfied: (26) (27) Step 8: Let n = n + 1 for the onvergene test: stop if g n g n+1 ; otherwise, return to Step 2. In the lgorith steps bove, the proedure for the grdient projetion ethod to solve the trip-hin h = δ = δ r, s, { 0,1 } r, s,, ( ) ( r s) p, n N γ { 0, 1}, r, s, n N p =, g I d n ( x( x ))= =,0 g ε I n + 1 I, I ( g l ) n 1 I I + = C I + bsed user equilibriu trffi ssignent odel in Step 3 is s follows: Step 0: Algorith initiliztion; Step 0.1: Let n = 0, set free-flow trvel tie { 0 } s the strting solution for trvel tie of the links in the network, nd lulte the shortest route tht psses ll of the tivity points between O-D pir ( r, s) ; Step 0.2: Bsed on the strting solution, generte set of trip-hin routes nd define the flow h = q, r,s for the trip-hin tht psses the tivity nodes between O-D pir ( r, s) s ( n+ 1) { ˆ } ; h p Step 1: Clultions for the ster proble; Step 1.1: Let n = n + 1, lulte the link flows ( n) ( ) bsed on { h p ˆ }, nd updte trvel tie { n ( x)} for eh link in the network; Step 1.2: Clulte the shortest route tht psses ll the tivity points between O-D pir (r,s), p ˆ, nd list it s the fit in the set of fesible trip-hin routes, ( n) { p ˆ } ; Step 2: Clultions for restrited ster proble; Step 2.1: Use Eqs. (28)-(30) to updte trip-hin ( n+ 1) ( +1) route flow { h nd link flow x n : h h ( n+ 1) } ( n) ( n) ( n) { 0, ( h α d )} ( n + 1) p ˆ = x +, x r R, s S, p = q h ( n + 1 ) ( n+ 1) n ) h ( 1 = + δ (28) (29) (30) Step 2.2: Convergene test: onvergene is hieved if the perentge differene between two rounds in link flow is less thn ertin onvergene riterion ϕ, s shown in Eq. (31); otherwise, return to Step 1: ( n+ 1) ( n) x x x ϕ (31) ( n) x Let the shortest route tht psses ll of the tivity, ˆ r R, s S,

10 212 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion points between O-D pir (r, s) in the route set be ˆp, the trip-hin link flow of whih n be expressed s below: h = q h p ˆ r, s (32) ( n ) The desent diretion d in Step 2.1 n be obtined fro the fit-order derivtive of the trip-hin route vrible h in objetive eqution Eq. (16), s shown below: z( x) h = ( n ) The step size α in Step 2.1 n be obtined fro the reiprol of the seond-order prtil derivtive of the trip-hin route vrible h in objetive eqution Eq. (16), s shown below: Thus, 2 z(xx ) 2 ( ) h p ˆ α = p ˆ, δ + = δ + r R, s S, υ δ δ 2 2, (33) (34) for signl tiing optiiztion. Below, we used the testt network in Fig. 2 to ondut nueril nlysiss on the verifition of the proposed odel frework. In Fig. 3, Nodes 1, 5, 9, nd 13 re the origins nd destintions; Node 3 is the interedite tivity point thtt ust be pssed between O-D pir Nodes 1 nd 13, nd Node 11 is the interedite tivity point thtt ust be pssed between O-D pir Nodes 5 nd 9. We dopted the FHWA ost funtion in Eq. (13) to lulte link osts. Tble 6 displys the trnsporttionn dends nd the interedite nodes tht ust be pssed between the vrious O-D pi. The inteetions hve two-phse signls, nd the free-flow trvel tie for lll the links ws set s 1. Tble 7 presentss the yle tie, iniu green tie, sturtion flow rte, nd tie loss. r R, s S, (35) where preter υ is onstnt between 0 nd 1 tht indites fster solution speed loser to 1 nd higher solution preision loser to Nueril Anlysis The lgorith presented bove n find lol optiu solution for the bilevel progring odel Fig. 2 Test Network 2. Tble 6 Trnsporttion dend between trip-hin O-D pi in Test Network 2. O-D pir Ativity node Trvel dendd Tble 7 Inteetion dt for Test Network 2 (units in s). Cyle tie 60 Initil green tie 27 Miniu green tie 5 Sturtion flow rte 60 Lost tie 3

11 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion 213 2,000 1,950 1,900 1,850 1,800 1,750 1,700 1,650 1,600 1,550 1, Fig. 3 Convergene of objetion vlues for Stkerlberg solutions in Test Network 2. Tble 8 Link flow nd trvel tie in Test Network 2. Trip-hin pth Flow Trvel ost Obj. fun. vlues 1, We nlyzed the rod network nd O-D pir dt bove using the sensitivity nlysis nd generlized invee trix pproh developed in Setion 4.2. The oputing environent is t the PC under Pentiu 4 3.4Gb Hz pltfor, nd the solution lgoriths were ipleented by pplying Borlnd C ++ Veion 5.02 lnguge. The results re shown in Fig. 3 nd Tbles 8 nd 9. Fig. 3 displys the onvergene of the objetion funtion vlues during the solving proess of the Stkelberg solution to the upper-level odel. Tble 8 presents the trip-hin route deisions nd user equilibriu in the lower-level of the upper-levell odel hs odel when the Stkelberg solution been obtined. Tble 9 shows the trvel tie, flow, pity, nd green tie for the rod links onneting eh inteetionn when the Stkelberg solution of the upper-level odel hs been obtined. The results of the nueril nlysis on Test Network 2 indite the following:

12 214 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion Tble 9 Stkelberg solutions of Test Network 2. No. Link Free flow trvel tie Trvel tie Flow Cpity Green tie rtio The results in Fig. 4 show tht in the solution lgorith, the objetive funtion vlue of eh round dereses until onvergene. This indites tht the obintion of the vritionl inequlities sensitivity nlysis pproh nd the generlized invee trix ethod n effetively obtin the derivtive of the ipliit funtion of the deision vribles in the upper nd lower level odels in the signl tiing optiiztion odel with trip-hin route hoies behvior in this study. Furtherore, the proposed pproh n produe the desent diretion for the objetive funtion in the upper-level odel, whih filittes the serh for the Stkerlberg solution of the odel. The results in Tble 8 revel tht the used trip-hin pths between eh O-D pir hve the se trvel tie nd stisfy the trip-hin user equilibriu onditions when the Stkerlberg solution of the upper-level odel hs been obtined. This study optiizes the design of signl tiings t inteetions while iniizing totl osts in the rod network, s shown in Tble 9. This is hieved through solution of the signl tiing optiiztion odel with trip-hin route seletion behvior. The proposed odel is therefore pioneer oponent worthy of dding to the trnsporttion network siene literture.

13 A Network Signl Tiing Design Bilevel Optiiztion Model with Trveler Trip-Chin Route Choie Behvior Considertion Conlusions nd Suggestions Previous studies hve pointed out tht the fto influening route hoie behvior inlude inidentl tivities during trvel s well s the lotions of origins nd destintions. We therefore developed trip-hin bsed bilevel progring odel for signl tiing optiiztion tht kes ore resonble ssuptions of user route hoie behvior. The lower-level of the odel developed in this study ould be used to lulte the trip dends with nd without tivities nodes between OD pi bsed on trips s well s bsed on trip-hins. Therefore, this inreses the generlizbility of the odel s well s the flexibility of proble nlysis. To solve the proposed bilevel optiiztion odel, we dopted n pproh obining sensitivity nlysis for vritionl inequlities, the generlized invee trix ethod, nd grdient projetion ethod developed for trip-hin user equilibriu behvio. The nueril tests on Test Network 1 deonstrted tht the results derived using the obined ethod pproxite the tul trip-hin user equilibriu flow solutions. This pproh n therefore effetively estite the hnges in link flow nd provide the desent diretion for the objetive funtion in the upper-level odel. The nueril nlysis results bsed on Test Network 2 showed tht the solution lgorith developed in this study n produe n pproprite desent diretion t the end of eh round during the solution of the bilevel progring odel s well s onverging trget vlue. The finl optiized signl tiing results n fulfill the onstrints on trip-hin route hoie behvio. To further inrese the lignent of signl tiing designs with tul rod network trffi dends, we suggest the following diretions for future reserh: The odel in this study ws estblished in light of losed rod network syste with norl two-phse signl settings. For the ske of onveniene in the odel verifition, the developed odel nd dopted prete did not tke the hnges in yle tie. This requires iproveent in the future. The inlusion of tie dependene nd dyni network designs in the trip-hin-bsed bilevel progring odel for signl tiing optiiztion would inrese the ury of predition of network dend so tht the trnsporttion dend n be stisfied in rel tie. This study did not onsider ultiple vehiles types trip-hin route seletion behvio or the influene of link pity on signl tiing designs. Giving onsidertion to ultiple vehiles types, suh s urbn bus nd sooter tht re oon in Tiwn, nd pity liits would enble the odel to ore urtely reflet tul trip-hin route seletion behvio. Nueril nlysis shows tht the fit-order prtil derivtive of link flow with regrd to the perturbtion preter of green tie derived using the sensitivity nlysis ethod for vritionl inequlities produes results tht pproxite the tul equilibriu flows. This pproh n therefore effetively estite hnges in link flows. Nevertheless, the error inreses with the perturbtion preter, nd thus the influene of the extent of perturbtion on the solution proess wrrnts further investigtion. Aknowledgents This reserh is prtilly funded by Ntionl Siene Counil, Tiwn, R.O.C. (projet no.: NSC H ). Referenes [1] Fisk, C. S Ge Theory nd Trnsporttion Syste Modeling. Trnsporttion Reserh Prt B 14 (3): [2] Tobin, R. L Sensitivity Anlysis for Vritionl Inequlities. Journl of Optiiztion Theory nd Applitions 48 (1): [3] Tobin, R. L., nd Friesz, T Sensitivity Anlysis for Equilibriu Network Flow. Trnsporttion Siene 22 (4): [4] Yng, H., nd Ygr, S Trffi Assignent nd Signl Control in Sturted Rod Networks.

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