Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto Optiiztion Frewor for Rod Networ Directed by Unbloced Relibility for Given Networ Topology nd Inelstic Dend with Stochstic User Equilibriu SHUJUN HOU *, NAOKI MARUYAMA, MASAFUMI HIROTA nd SEIZO KATO Grdute School of Engineering Mie University 77 Kurichiy-cho, Tsu, -807 JAPAN * Corresponding uthor: shujunhou@yhoo.co Abstrct: - To provide ore relible service for drivers, it is necessry to estblish n optiiztion frewor in which relibility index is dopted s perfornce index to evlute rod networs. First, odified fourlevel odel of unbloced relibility (lin, pth, Origin-Destintion pir nd entire rod networ) is proposed, in which the unbloced relibilities of the Origin-Destintion pir nd the entire rod networ re forulted by the lw of totl probbility when the unbloced relibility of pth is conditionl probbility. Then, bi-level progr is estblished bsed on the new odel of unbloced relibility since plnners design the rod networ nd drivers respond the chnge in the rod networ. The objective function of the upper-level progr is the xiized blnce between the unbloced relibility of the entire rod networ nd the rod networ expnsion rtio which iplies cost of iproving the rod networ. The progr equivlent of Stochstic User Equilibriu is dopted s the lower-level odel so s to chieve consistency in route choices nd to odel the congestion effect in the rod networ. Next, set of lin cpcity expnsions is deterined s plnning schee by solving the bi-level progr. This plnning schee not only iproves the lin of lower relibility but lso tes into ccount the perfornce of the Origin-Destintion pir nd the entire rod networ. The proposed optiiztion frewor is cpble of iproving the rod networ to its highest possible relibility level with iniu scle of rod networ expnsion. Key-Words: - Rod Networ Plnning, Bi-level Progr, Unbloced Relibility, Stochstic User Equilibriu, Logit loding odel Introduction This pper studies n optiiztion ethod to iprove rod networ, in which the unbloced relibility is regrded s the perfornce index. Fro the viewpoint of the plnner, the highest perfornce nd the lowest cost re desired. With the incresing dend for better nd ore relible service, the rod networ syste hs incorported relibility nlysis s n integrl prt of its plnning, design nd opertion. Therefore, relibility index of rod networ syste perfornce is needed for rod networ plnning. The proble of optiizing rod networ for inelstic trffic dend nd given networ topology is nlyzed. There re severl relibility concepts in rod networ systes, such s the connectivity relibility [], trvel tie relibility [], cpcity relibility [], nd unbloced relibility. Unbloced relibility is the probbility of the rod unit or syste being ble to intin n unbloced stte t the pe hour t which the highest trffic volues re observed for one dy []. A four-level odel of rod networ unbloced relibility, coprising lin, pth, Origin- Destintion (OD) pir, nd entire rod networ, to ssess the opertion perfornce of the rod networ [], in which the OD pir unbloced relibility is regrd s prllel cobined structures of pths. In grph theory, networ is coprised of lins nd nodes. All lins referred to in this pper re directed, for exple, lin - nd lin - re different. A pth is sequence of nodes connected by directed lins so tht oveent is fesible fro the first node to the lst node in the sequence. The OD pir represents the source nd objective of trip. In this pper, the four-level odel is odified by the viewpoints of conditionl probbility nd totl probbility. ISSN: 09-9 9 Issue, Volue, June 009
Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto Plnners nd ngers in the upper level prt deterine the preters of rod networ, nd drivers in the lower level prt e choices with regrd to the route of their trvel in response to the chnge in the rod networ. Corresponding to this condition, bi-level progr odel is used to solve the proble of rod networ optiiztion. A bilevel progr hs hierrchicl structure in which upper-level nd lower-level decision ers select their strtegies so s to optiize their objective functions, respectively. The objective function of the upper-level progr is the xiized blnce between the unbloced relibility of the entire rod networ nd the rod networ expnsion rtio. The entire rod networ unbloced relibility is forulted by the lw of totl probbility when the unbloced relibility of pth is the conditionl probbility of route choice. The upper-level decision er should consider how the lower-level decision er would rect to given upper-level decision, lthough he cn not intervene in the lower-level decision of the decision er. The equivlent progr of Stochstic User Equilibriu (SUE) is dopted s lower-level odel so s to chieve consistency in route choices nd to odel the congestion effect in the networ. The SUE is stte in which no driver cn iprove his/her perceived trvel tie by unilterlly chnging routes nd the perceived trvel tie hs rndo error []. The lower-level progr is solved by the Method of Successive Averge [7] in conjunction with the logit loding odel [8] bsed on siple pth enuertion. A pth with no repeted nodes is clled siple pth. The Hooe- Jeeves (H-J) [9] lgorith is pplied to solve the proposed bi-level progr. Then, the proposed optiiztion frewor is tested by locl rod networ, which hs 7 nodes nd 8 lins. Noenclture A set of lins in networ C cpcity, pcu/h d control fctor of rod networ expnsion scle E expecttion vlue F objective function of upper-level progr f pth flow, pcu/h G nuber of lins in rod networ H function of unrelible probbility I set of ll origin nodes J set of ll destintion nodes L length of lin, M set of pths used n nuber of trffic zones p route choice proportion q trffic dend between Origin-Destintion pir, pcu/h R unbloced relibility index s initil step length of explortory ove for lin cpcity expnsion in Hoo-Jeeves lgorith, pcu/h T su of trvel tie of ech lin on studied pth, h t trvel tie of lin, h u iportnce fctor of n Origin-Destintion pir V trffic volue, pcu/h v step length of pttern ove in Hoo-Jeeves lgorith X uxiliry lin flow, pcu/h x lin flow, pcu/h w nuber of siple pths between Origin- Destintion pir y lin cpcity expnsion, pcu/h Z objective function of lower-level progr α proportion to iniize objective function of theticl progr β direction fctor in Hoo-Jeeves lgorith, β = for positive cpcity expnsion, nd β = for negtive cpcity expnsion δ indictor vrible, hs vlue of if lin is ε on pth fro i to j; 0 otherwise sll positive nuber s index of convergence test in Method of Successive Averge lgorith, pcu/h η convergence step length of explortory ove for lin cpcity expnsion in Hoo-Jeeves lgorith, pcu/h θ dispersion preter λ conversion fctor for physicl diension, h/( pcu) µ expnsion direction φ reduction fctor of step length of explortory ove in Hoo-Jeeves lgorith ω vrible of integrtion in objective function of Stochstic User Equilibriu proble Superscripts e free-flow stte counter for Method of Successive Averge lgorith h counter for pttern ove in Hoo-Jeeves lgorith used pth between Origin-Destintion pir, M ISSN: 09-9 9 Issue, Volue, June 009
Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto Subscripts lin in networ, A g counter for explortory ove in Hoo-Jeeves lgorith i origin node, i I j destintion node, j J Model of Unbloced Relibility. Lin Unbloced Relibility The vlue of the lin unbloced relibility equls the unbloced trips divided by the totl observtion trips t lin t the pe hour. A curve tht expresses the reltionship between relibility nd congestion cn be obtined by ens of trffic survey. The unrelible probbility of lin, H(x /C ), is defined s the following function [0]: x / C, 0 x / C H ( x / C ) = (), x / C > SUE. If pth-bsed ssignent of SUE (lowerlevel progr) is copleted, route choice pttern is obtined. By the lw of totl probbility, when sple spce is liited in n OD pir, the OD pir unbloced relibility, R, is expressed s follows: where p R = R p () M is the loding proportion on the pth between the OD pir. The contribution of ech pth to the OD pir unbloced relibility is given by Eq. ().. Unbloced Relibility of Entire Rod Networ In rel networ the iportnce of the OD pir should be considered. The iportnce fctor of n OD pir, u, tht is the rtio of n OD dend to totl dend, cn be clculted by the following function: Then, the lin unbloced relibility, R, is defined by the following eqution: R = H( x / C ) () u = n q n q i= j= j i (). Pth Unbloced Relibility The pth unbloced relibility, R, on pth connecting origin i nd destintion j, cn be clculted siilr to the lin unbloced relibility s follows []: V C, 0 V C = () 0, V C > R where V/C is the noinl degree of congestion on pth.. Origin-Destintion Pir Unbloced Relibility The physicl ening of the OD pir unbloced relibility is the rtio of the unbloced trip tie between n OD pir to the trffic dend. Fro the perspective of probbility theory, the unbloced relibility of pth is the conditionl probbility tht driver intins n unbloced stte on his pth when the totl drivers choose the route by rule in the rod networ. In this pper the rule is where q is the trffic dend between the OD pir nd n is the nuber of trffic zones. For the entire rod networ, the choice probbility of used pth, pˆ, cn be clculted by the following function: pˆ = u p () The su of p nd the su of pˆ re unitry for two lyers, such s n OD pir nd the totl networ, respectively. By the lw of totl probbility, when sple spce is the set of ll used pths, the unbloced relibility of entire rod networ R, is expressed s follows: R = n n i = j= M j i R pˆ (7) The se expression lso cn be deduced fro the physicl ening of the entire rod networ unbloced relibility, which is the rtio of the unbloced trip ties to the totl trip ties in the entire rod networ. ISSN: 09-9 9 Issue, Volue, June 009
Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto Upper-level Model of Bi-level Progr The upper-level odel reflects the expecttion of the plnner tht is xiu networ surplus obtined by subtrcting cost index ssocited with iproving the rod networ fro perfornce index. In this pper, the perfornce of the rod networ is expressed s n indictor of the unbloced relibility. The upper-level odel ust be ble to respond to the chnge of the networ property nd cost of iproveent. The upper-level odel is estblished s follows: ( ) x F( y) = R d λ L y (8) where y ( y, L,,, ) = y L y G is vector of the lin cpcity expnsions, G is the totl nuber of lins in rod networ, nd F(y) is the upper-level objective function, which is expected to be xiized when the lin flows re provisionlly fixed. R is the whole networ unbloced relibility, which is function of lin flows. The second ter on the right-hnd side of the upper-level odel is clled the penlty ter, which represents the cost ssocited with iproving the rod networ. Where d is control fctor of the rod networ expnsion scle, λ is the unit trnsfortion fctor, L is the length of lin nd y is the lin cpcity expnsion of lin. The control fctor, d, cn gnify the ipct of the cost of iproving the rod networ, nd ( y ) L is the rod networ expnsion scle. The conversion fctor for the physicl diension, λ, which is constnt in certin rod networ, is forulted s follows: λ = (9) ( L C ) where L nd C re the length nd cpcity of ech lin in n existing rod networ, respectively. Thus, L indictes the existing rod networ ( ) C λ L y expresses the rod networ scle, nd ( ) expnsion rtio which iplies cost of iproving the rod networ. Then, the penlty ter, d λ ( L y ), becoes diensionless fctor nd ensures the networ cn be expnded to resonble degree. Stochstic User Equilibriu Assignent s Lower-level Model of Bi-level Progr In this pper, the lower-level odel is the SUE ssignent. The SUE ssignent is to find the lin (or pth) flows on rod networ given trffic dend trix nd probbilistic route choice odel. Generlly, the driver perception of trvel tie is ssued to be rndo. The stochstic networ loding pproch is clled logit-bsed loding odel when the perceived lin trvel ties follow logit odel. This odel cn be derived fro the concepts of rndo perceived trvel tie nd perceived trvel tie iniiztion by ssuing tht the rndo ters of ech perceived trvel tie function re independently nd identiclly distributed Gubel vrints. The ggregte shre function gives the totl nuber of individuls selecting prticulr trvel lterntive. The proportion of trffic dend between the OD pir choosing pth is given by: p = w exp( θt exp( θt = ) ) (0) where the dispersion preter, θ, which scles the error of the perceived trvel tie, is constnt for certin rod networ, the clculted trvel tie of pth, T, is the su of the trvel tie of ech lin on this pth nd w is the nuber of siple pths between n OD pir. The lin trvel tie, t (x ), which represents the reltionship between the flow nd the trvel tie for lin, is clculted using the Bureu of Public Rods function [], s shown below: e ( ) x x = t + 0. t () C The pth flows re expressed s function of the OD flow nd loding proportion s: f = q p () The lin flows re chieved by the indictor reltionship of pth-lin: = n n i= j= M j i, x f δ () ISSN: 09-9 9 Issue, Volue, June 009
Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto The stochstic networ loding proble y be solved either in the spce of the lin flows or in the spce of the pth flows. An iportnt disdvntge of pth-bsed solutions is tht they require explicit enuertion of the pth choice set, which y be huge nuber in lrge networs. However, this drwbc hs been relxed treendously in recent yers with the rpid developent of high-speed coputers nd lrge-cpcity storge, nd the proposed upper-level progr needs pth enuertion. The depth-first serch (DFS) [] is dopted to find the totl siple pth in the networ. DFS is grph trversl lgorith, which extends the current pth s fr s possible before bctrcing to the lst choice node nd trying the next lterntive pth. The grph trversl refers to the proble of visiting ll the nodes in grph. The DSF lgorith cn be specilized to find pth when given strt node nd end node. A stc is used to eep trc of the pth between the strt node nd the current node nd void cycles. A stc is n bstrct dt type nd dt structure bsed on the principle of Lst-In- First-Out. As n bstrct dt type, the stc is continer of nodes nd hs two bsic opertions, push nd pop. Push dds given node to the top of the stc leving previous nodes below. Pop reoves nd returns the current top node of the stc. The pproch of pth enuertion is illustrted by rod networ s shown in Fig., which consists of 7 nodes nd 8 lins. The networ cn be equivlently represented by n djcency list s shown in Tble. The DFS lgorith is oved long the djcency list s the digitl for of the networ. The OD pir (, ) is studied to generte siple pths. Node is the origin nd node is the destintion. The generted siple pths re shown in Fig.. This is nown s the DFS since n djcent node of sll code is serched first. Node hs two djcent nodes, node nd node 7, nd the current 7 Fig. Test rod networ Outline Node of trffic zone Node of intersection Lin 7 Tble Adjcency list Node Nuber of djcent nodes Adjcent nodes 7 7 7 7 Fig. Totl siple pths of Origin-Destintion pir (-) stc is -. Thus, the stc grows into ---. Node hs two djcent nodes, node nd node. Node hs been in the stc, so node is dded to the stc nd loop is voided. For node, the first djcent node (first serch tie), node, hs been in the stc, nd the second djcent node (second serch tie), node, is dded to the stc. The first siple pth, -----, is found since node is the destintion. When the serch reches the destintion, it bctrcs long the stc. Node becoes the current top node nd node is not in the stc while the pop opernd is executed. For the third serch tie of djcent nodes of node, node 7 is dded to the stc, nd the stc is -----7. For djcent nodes of node 7, either node or hs been in the stc, so node is dded to the stc. The second siple pth, -----7-, is found. When ll possible brnches hve been serched, six siple pths re found between the OD pir (, ). Thus, fter ll OD pirs hve been serched, siple pths re found in the rod networ. Once these pths re generted nd stored, the pth choice proportions nd the lin flows cn be coputed ccording to Eqs. (0) to (). Tht is, single logit-bsed loding hs been copleted. It is necessry to forulte iniiztion progr since the solution by single logit-bsed loding is 7 ISSN: 09-9 9 Issue, Volue, June 009
Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto not equilibriu. Consider the following unconstrined theticl progr []: n n in Z( x) = q E in i= j= j i + x t ( x, y ) x 0 { T } t ( ω, y ) dω T ( x, y) () where E in { T } T ( x, y) is the expected perceived trvel tie between ech OD pir. The objective function of the lower-level progr is given by Z(x). The lin flows x = ( x, L, x, L, x G ) is vector. The totl cpcity of ech lin is coposed of the lin cpcity expnsion nd the existing lin cpcity when trvel tie, t, is clculted by Eq. (). The itertion ethod is used to solve the progr of Eq. (). The iniiztion procedures cn be written s: x = x + α ( X x ) () + In this eqution, x is the vlue of the decision vrible vector t the th itertion, α is sclr representing the ove size, nd X is descending direction vector coputed t x, where x = ( x, L, x L, xg ) is vector vrible of the lin flows nd X = ( X, L, X L, X G ) is vector vrible of the uxiliry lin flows. Eqution () ens x + is equl to the weighted verge of x nd X. It is difficult, however, to optiize the ove size with the stndrd descent lgorith in the SUE objective function since the expected perceived trvel tie is difficult to clculte. Consequently, the MSA is pplied to solve the SUE becuse it is bsed on predeterined ove size long the descending direction. In other words, the sequence of ove sizes α, α,, is deterined priori. For solving SUE, ech ove size is the reciprocl of the itertion tie. The MSA lgorithic frewor in conjunction with previously entioned logit-bsed loding odel is given s follows: Step 0: Generte set of siple pths using the DFS lgorith. Step : Perfor logit-bsed networ loding with n epty rod networ to get lin flow x. Set =. Step : Clculte the lin trvel ties, t (x ), with respect to the current lin flows. Step : Perfor logit-bsed networ loding with the current set of lin trvel ties, t (x ), to yield uxiliry lin flows X, which is in descending direction. Step : Find the new lin flows for ll lins: x = x + ( X x + ) () Step : If convergence is ttined, stop. If not, set = + nd go to step. Convergence criterion is given: ( X x ) < ε (7) Solving Bi-level Progr The H-J lgorith is direct serch ethod which includes n explortory ove nd pttern ove. The gol of solving the proposed bi-level progr is to find set of lin cpcity expnsions to xiize the objective function of the upper-level progr, which responds to the SUE ssignent result. This ethod for given networ topology is coon study since it cn lso solve the proble of how to dd lins. For exple, set of lins which hve infinitesil cpcity re dded to n existing rod networ. Clculting this rod networ s given networ topology proble, the plnner cn deterine which lins re worthy of iproveent nd the resonble cpcities of these iproved lins. In the plnning schee, lin need not be built if it hs less thn certin threshold cpcity. There re three vectors used in the lgorith, the test lin cpcity expnsion, ŷ, benchr lin cpcity expnsion, y, nd the success lin cpcity h expnsion, y. The four experientil preters, s, φ, η nd v, re provided for the cse study, s shown in Tble. The step-by-step procedure of this schee is given below. Tble Preters of Hoo-Jeeves lgorith Mens Sybol Vlue Initil step length of explortory ove, pcu/h s,000 Reduction fctor of step length of explortory ove φ 0. Convergence step length of explortory ove, pcu/h η 00 Step length of pttern ove v ISSN: 09-9 97 Issue, Volue, June 009
Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto Step 0: Initiliztion. Given n initil solution of y 0, solve the lowerlevel progr with y 0. Tht is, the current cpcity is the su of the initil cpcity expnsion, y 0, nd the cpcity of the existing rod networ. Clculte the objective function of the upper-level to obtin F(y 0 ). Set F =. Give the step length of the explortory ove, s, the reduction fctor of the step length of the explortory ove, φ, the convergence step length of the explortory ove, η, nd the step length of the pttern ove, v. Set the 0 0 y = y nd ( y) F( y ) direction fctor, β =, counter, g = (fro the first lin), for the explortory ove nd counter, h =0, for the pttern ove. Step : Explortory oves. Step -: If g > G, ech lin hs been tested, so go to step. Let µ g be vector tht contins in the gth position nd 0 elsewhere. Step -: Set yˆ = y + β s μg. Solve the lowerlevel progr with ŷ. Clculte the objective function of the upper-level to obtin F(ŷ). Step -: If F ( yˆ ) > F( y) renew the benchr lin cpcity expnsions nd objective function of the upper-level, y = yˆ nd F( y) = F( yˆ ), respectively, set g = g + nd go to step -. Otherwise, perfor the next step. Step -: If β =, set β = - nd go to step -; otherwise g = g +, β = nd go to step -. Step : Pttern oves. h Step -: If ( y) F( y ) F >, renew the success solution nd objective function vlue of the upperlevel progr, y h + = y nd F( y ) h + = F( y), respectively. Perfor the pttern ove, h h+ h y = y + v( y y ). Solve the lower-level progr with y. Clculte the objective function of the upper-level to obtin F ( y). Set h = h+, g = nd go to step. Otherwise perfor the next step. Step -: If the convergence criterion, s < η, is et, stop (the current solution, y h, is the optil lin cpcity expnsion); otherwise, reduce the step length of the explortory ove, s = φ s, return to the success solution nd objective function vlue of h h the upper-level progr, y = y nd F ( y) = F( y ), respectively, set g = nd go to step. Cse Study of Locl Rod Networ. Condition of Existing Rod Networ nd Origin-Destintion Mtrix The rod networ shown in Fig. ws nlyzed to test the proposed optiiztion frewor. Tble shows the lin length of the existing rod networ. The st to th nodes express trffic zones regrded s the re of the origins nd destintions of totl trips. The th nd the 7th nodes express the intersection of the rods. The cpcity of ech lin is,00 pcu/h in the existing rod networ. The free flow speed is 0 /h. Tble shows the OD trix of the pe hour in the orning. An OD trix displys the trips coing nd rriving t respective trffic zones. Tble is the trffic dend for trffic ssignent in this cse study. Tble Lin length of existing rod networ Lin Length () Lin Length () - 8.8-9. - 9. - 9. - 7.8-7 7.0-7 9. - 9. - 8.8-7.8-7.8-9. - 7.8 7-8.8-9. 7-9.0-7 9.0 7-7.0 Tble Origin-Destintion trix Unit: pcu/h D O Su - 0 900 70 0,070 0-0 0 00,70 900 0-0 0,0 70 0 0-90,90 0 00 0 90 -,790 Su,070,70,0,90,790 9,0. Route Choice nd Loding Proportion There is dispersion preter, θ, which scles the perceived trvel tie in Eq. (0) when the loding proportion of trffic dend is clculted. If θ rises, the perception error will decrese. Drivers will tend to select the iniu esured trvel-tie pth ISSN: 09-9 98 Issue, Volue, June 009
Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto when θ is very lrge. If θ drops, the perception error will increse. In the liit of θ pproching zero, ll pths re used. The totl trvel tie of the entire rod networ with respect to the different vlues of θ is considered to deterine n pproxition vlue of the preter θ. The totl trvel tie of the entire rod networ in one hour equls the su-product of ech lin trvel tie nd trffic volue on it. The curve of the totl trvel tie with respect to different vlues of θ is shown in Fig.. In order to disply clerly, the curve is cut off t trvel tie of,00 hours. When θ = 0, 0., 0., the totl trvel ties re 8,7,,87 nd,707 hours, respectively. The curve in Fig. reches the lowest point, tht is,, hours, t bout θ = 0.9. In this cse study, the pproxition vlue of the preter θ is 0.9. Totl trvel tie (hours),00,90,80,70,0,0,0,0,0,0,00 0.0 0. 0. 0. 0. 0. 0. 0.7 0.8 0.9.0.......7.8 Dispersion preter Fig. Totl trvel ties with respect to the different vlues of dispersion preter When the DFS lgorith is copleted, siple pths re found. The trffic ssignent is perfored by the MAS lgorith in conjunction with the logit loding odel nd only 8 pths re used. Due to spce liittions, only the OD pir (, ) is used s n exple to show the pths used nd the loding proportions, s shown in tble. This cse study confirs tht the lower-level progr cn deterine the pths used in ccord with the SUE criterion. Of course, the lin flows hve lso been clculted. The unbloced relibility in the upper-level progr cn then be clculted by the proposed odel. Tble Pths used nd loding proportion of OD pir (, ) Order Loding proportion in existing rod networ. Optiiztion Result The control fctor of rod networ expnsion scle, d, significntly influences the objective function of the upper-level progr. The proposed bi-level progr cn coply with vrious liits of environentl nd finncil resources by setting the control fctor to different degrees. When d increses, the results of the bi-level progr show the decresing objective function vlue of the upperlevel progr, lower unbloced relibility nd networ expnsion scle. The contribution of ore networ expnsions to rod networ perfornce iproveent decreses with decresing d. In this pper, the specil constrints of the environentl nd finncil resource re not considered, nd the controllble fctor d =. The H-J lgorith is pplied, nd the result is set of lin cpcity expnsions s shown in tble. Only lins tht will be iproved re shown, nd the other lin cpcities re not chnged. The networ expnsion rtio, λ ( L y ), reches 0.% in Loding proportion in iproved rod networ Pssing nodes.% 7.%.8%.% 7 the existing rod networ scle by ppliction of the H-J lgorith to solve the proposed bi-level progr when the objective function of the upperlevel progr reches xiu. In tble, set of syetricl lin cpcity expnsions cn be obtined when there is syetry between the OD trix nd perfornce of lins in existing networ. The syetricl feture ens the opposing lins hve the se cpcity nd opposing OD pirs hve se trffic dend. Tble Lin cpcity expnsion Lin Expnsion cpcity (pcu/h) -,000-7 7 -,000 - - 7-7 ISSN: 09-9 99 Issue, Volue, June 009
Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto When the plnning schee of Tble is copleted, the unbloced relibility of ech level, such s, the lin, pth, OD pir nd the entire networ in rod networ, is iproved. The grde of lins ccording to unbloced relibility re shown in Fig.. The red, brown nd green rrows unbloced relibilities of the existing rod networ nd the iproved rod networ re shown in Fig.. The bsciss is the used pth tht is denoted by series of nodes. The ordinte is the unbloced relibility of the pth. The lower the pth relibility is, the ore the pth cn be prooted. A driver focuses his ttention on the perfornce of n OD pir. The desire line of the unbloced relibility is shown in Fig., which sybolizes the feture of 7 () Existing 7 (b) Iproveent Outline Node of trffic zone Node of intersection Lin unbloced relibility < % % Lin unbloced relibility < 0% 0% Lin unbloced relibility < 7% () Existing (b) Iproveent Outline Node of trffic zone OD pir unbloced relibility < % % OD pir unbloced relibility < 0% 0 % OD pir unbloced relibility < 7% Fig. Grding lins by unbloced relibility express the grde of relibility fro low to high, corresponding to R < %, % R < 0%, 0% R < 7%, respectively. There is no lin in which the unbloced relibility is greter thn 7%. Since the cpcities of the lins re iproved, the color of five lins chnges fro red to brown nd the unbloced relibility of lin -7 is lso incresed fro 9.% to.7%. Lin - is prooted to green due to the decresing flow on this lin, lthough its cpcity does not chnge. Twenty-eight used-pth Fig. Desire line of OD pir unbloced relibility trip between n OD pir, not necessrily the ctul route followed. The OD pir unbloced relibility is grded by the se colors s Fig.. Totl red desire lines re prooted to brown desire lines, tht is, ost unrelible OD pirs dispper. There is distinct iproveent in the unbloced relibility of the entire rod networ fro.8% of the existing rod networ to 0.% of the iproved rod networ. Unbloced relibility 0.9 0.8 0.7 0. 0. 0. 0. 0. 0. 0 Existing rod netwro Iproved rod networ -- - -- -- -- --- -7-- -7- -- -7- - --- --7- - -- --7- -- -7- - - -7- -- -- -7- -- -7-- - -7- Used pth Fig. Unbloced relibility of pth ISSN: 09-9 00 Issue, Volue, June 009
Shujun Hou, Noi Mruy, Msfui Hirot, Seizo Kto 7 Conclusion To optiize rod networs, new odel of bilevel progr is estblished in which the upperlevel progr xiizes the objective function by subtrcting the networ expnsion fro the entire rod networ unbloced relibility, nd the SUE ssignent is regrded s the lower-level progr. A plnning schee represented s set of lin cpcities is obtined by solving this bi-level progr. The proposed unbloced relibility odel tes full dvntge of the route choice behvior of drivers since the loding proportion of the pths, which is clculted by the SUE ssignent, enters both the OD pir unbloced relibility nd the entire rod networ unbloced relibility. The iproveent of unbloced relibility coes fro the lin cpcity expnsion nd the fct tht the OD trix is newly nd resonbly ssigned. The lower unbloced relibility units of the rod networ, respectively corresponding to lins, pths nd OD pirs, re iproved. The proposed optiiztion frewor is cpble of iproving the entire rod networ to chieve its highest possible relibility level with inil scle of rod networ expnsion. The estblished bi-level progr is n effective nd dvntgeous tool for optiizing rod networ. References [] Iid, Y. nd Wbyshi, H., An Approxition Method of Terinl Relibility of Rod Networ Using Prtil Minil Pth nd Cut Set, Proceedings of the Fifth World Conference on Trnsport Reserch, Vol., 989, pp. 7-80. [] Iid, Y., Bsic Concepts nd Future Directions of Rod Networ Relibility Anlysis, Journl of Advnced Trnsporttion, Vol., No., 999, pp. -. [] Chen, A., Yng, H., Lo, H. K., nd Tng, W., A Cpcity Relted Relibility for Trnsporttion Networs, Journl of Advnced Trnsporttion, Vol., No., 999, pp. 8-00. [] Chen, Y., Ling, Y., nd Du, H., The Appliction of Relibility in the Rod Networ Perfornce Evlution (in Chinese), Chin Civil Engineering Journl, Vol., No., 00, pp. -0. [] Hou, S., Mruy, N., nd Kto, S., A Study on Unbloced Relibility Assessing Rod Networ Opertion Perfornce for Plnning, Interntionl Syposiu on EcoTopi Science 007, No. 9, 007, pp. 7-79. [] Dgnzo, C.F., Sheffi, Y., On Stochstic Models of Trffic Assignent, Trnsporttion Science, Vol., 977, pp. -7. [7] Dil, R. B., A Probbilistic Multi-pth Trffic Assignent Algorith which Obvites Pth Enuertion, Trnsporttion Reserch, Vol., 97, pp. 8-. [8] Powell, W. B. nd Sheffi, Y., The Convergence of Equilibriu Algoriths with Predeterined Step Sizes, Trnsporttion Science, Vol., 98, pp. -. [9] Abdull, M., nd LeBlnc, L., Continuous Equilibriu Networ Design Models, Trnsporttion Reserch Prt B, Vol., 979, pp. 9-. [0] Zhu, S., Wng, W., Deng, W., Tng, Y., nd Wng, B., Reserch on Trffic Networ Relibility nd Access Rod Algorith (in Chinese), Chin Journl of Highwy nd Trnsport, Vol., No., 000, pp. 9-9. [] Hou, S., Mruy, N., nd Kto, S., New Methodology to Clculte Unbloced Relibility to Assess Rod Networ Opertion Perfornce, Journl of Mechnicl Systes for Trnsporttion nd Logistics, Vol., No., 008, pp. 0-. [] Bureu of Public Rods, Trffic Assignent Mnul, U. S. Deprtent of Coerce, Urbn Plnning Division, 9. [] Coren, T. H., Leiserson, C. E., Rivest, R. L. nd Stein, C., Introduction to Algoriths, Second Edition, MIT Press, 00. [] Sheffi, Y., Urbn Trnsporttion Networs: Equilibriu Anlysis with Mtheticl Progring Methods, Prentice-Hll, 98. ISSN: 09-9 0 Issue, Volue, June 009