A Novel Solution Methodology for Transit Route Network Design Problem

Transcription

1 A Novel Soluton Methodology for Trnst Route Networ Desgn Proble Ghd Mouss, Moud Ows Abstrct Trnst route Networ Desgn Proble (TrNDP) s the ost portnt coponent n Trnst plnnng, n whch the overll cost of the publc trnsportton syste hghly depends on t. The n purpose of ths study s to develop novel soluton ethodology for the TrNDP, whch goes beyond pervous trdtonl sophstcted pproches. The novelty of the soluton ethodology, dopted n ths pper, stnds on the deternstc opertors whch re tcled to construct bus routes. The deternstc nner of the TrNDP soluton reles on usng lner nd nteger thetcl forultons tht cn be solved exctly wth ther stndrd solvers. The soluton ethodology hs been tested through Mndl s benchr networ proble. The test results showed tht the ethodology developed n ths reserch s ble to prove the gven networ soluton n ters of nuber of constructed routes, drect trnst servce coverge, trnsfer drectness nd soluton relblty. Although the set of routes resulted fro the ethodology would stnd lone s fnl effcent soluton for TrNDP, t could be used s n ntl soluton for et-heurstc procedures to pproch globl optl. Bsed on the presented ethodology, ore robust networ optzton tool would be produced for publc trnsportton plnnng purposes. Keywords Integer progrng, Trnst route desgn, Trnsportton, Urbn plnnng. I. INTRODUCTION ITH ncresng trffc on the rods, ore obltyrelted probles such s congeston, r polluton, nose W polluton, nd ccdents re creted. Publc trnsportton s very portnt ens to reduce trffc congestons, to prove urbn envronentl condtons nd consequently ffects peoples' socl lves. Therefore, the need for new publc trnsportton nfrstructures for servng new towns nd/or provng exstng trnsportton structures to cope wth such ncrese s n urge [1]. Plnnng, desgnng nd ngeent of publc trnsportton re the ey ssues for offerng copettve ode tht cn copete wth the prvte trnsportton [2]. These trnsportton plnnng, desgnng nd ngeent ssues re ddressed n the Trnst route Networ Desgn Proble (TrNDP) [3]. The TrNDP s to desgn set of bus routes nd nge ther operton n n effcent nner for both users nd opertors. Dfferent syste functons nd trgets, requred for ech group of prtcpnts, hve to be et through soluton G. Mouss, Ph.D., s wth Cvl Engneerng Deprtent, Assut Unversty, Assut 71515, Egypt, (phone: ; fx: ; e- l: ghd.ouss@gl.co). M. Ows s wth Cvl Engneerng Deprtent, Assut Unversty, Assut 71515, Egypt, (e-l: ows@yhoo.co). ethodology. TrNDP, stted sply, reltes to the deternton of set of routes defned over the street networ to del wth dend trps [4]. TrNDP s sorted s one of the ost dffcult probles to be solved n the feld of trnsportton. Ths ght be due to ts hgh degree of coplexty. There re fve n sources of coplexty tht often preclude fndng unque optl soluton for TrNDP [4]-[7]. 1. Proble Forulton: Forulton coplexty results fro the dffculty of defnng the decson vrbles nd thus expressng the coponents of the obectve functons. 2. Non-lnerty nd Non-convexty: Most TrNDP forultons exhbt non-lner decson vrbles nd constrnts. Non-convexty would be llustrted by the fct tht trnst desgner cn deploy ore resources n trnst networ (ncresng opertor s costs) nd stll obtn hgher totl trvel te (worse users costs). 3. Cobntorl Coplexty: Ths rses fro the dscrete nture of TrNDP. Dscrete vrbles re lwys nvolved n route networ desgn proble. Cobntorl optzton proble s specl cse of nteger probles. It es the coplexty of the proble grows exponentlly wth the sze of trnst networ. 4. NP-hrd: TrNDP optzton s clssfed s NP-hrd, whch refers to the proble for whch the nuber of eleentry nuercl opertons s not lely to be expressed by functon of polynol for. 5. Mult Obectve Nture of TNDP: Mny pst pproches hve recognzed reducng users costs or opertor s costs s ther solo obectve. In prctce, users nd opertors costs re conflctng obectves. Over the lst fve decdes, the TrNDP hs been under study for ny reserchers, ost lely becuse the proble s prctclly portnt, theoretclly nterestng, hghly coplcted, nd ult-dscplnry s well. Ech resercher consdered dfferent obectve functon, proble s Mthetcl Progrng representton nd serch lgorth (heurstcs or Met heurstc) to be pleented n the soluton ethodology. Heurstcs s wdely ccepted by vst nuber of reserchers s the best possble pproch for TrNDP. Lpn nd Slns used seleton routes s ntl soluton for TrNDP. These seleton routes were expnded by nsertng other networ nodes under specfed condtons [8]. B nd Mhssn pleented the concept of nsertng node usng Lsp lnguge. They used dfferent strteges for nsertng nodes to seleton routes n ttept of trdng off between user 283

2 cost nd opertor cost [5], [9], [10].Shh nd Mhssn s extended B nd Mhssn wor ntroducng Trnst Centre (TC) noton [11]. Muttonw & Urquhrt used Pr Inserton Algorth (PIA) to prove B nd Mhssn wor. PIA sply uses pr node nserton nsted of sngle node nserton per te of nserton [12]. Recently, Met-heurstcs hve been consdered s one of the ost prctcl pproches to TrNDP soluton, becuse these ethods re generlly desgned for cobntorl optzton probles. They pleent effcent echnss to rech optl or ner optl solutons. Exples of Metheurstcs pproches re Genetc Algorth (GA), Sulted Annelng (SA), nd Swr Intellgence (SI). Pttnn et l. generted set of fesble bus routes through heurstc procedure, then GA ws clled to select optl (or ner optl) bus routes networ. Ther obectve ws to nze the totl syste cost for users nd opertors [13]. Zho defned neghbor serch spce round selected routes, clled ster pths, nd then pleented SA for the selecton of fnl soluton [14]. Generlly speng, SI s reltvely new pproch to TrNDP solvng tht tes nsprton fro the socl behvors of nsects n selectng ther routes between nest nd food. Trcng the behvor of nts n fndng pths ccordng to the densty of ther pheroone s found n [15]. Bee behvor s lso tcled n [16]. To the best of our nowledge, there s rre wor on deternstc technques for TrNDP. Ths y bc to coplexty sources entoned before, besdes the well-nown deternstc lgorth Trvel Slesn Proble (TSP) s devoted to the cse of one to ny trnst dend proble "unle our cse ny to ny". In ths study, we too the chllenge to propose sple deternstc soluton ethodology for TrNDP. We decoposed the proble nto three sub-probles. Ech subproble s fored n deternstc nner. Frst, n pproxte trnst networ s obtned fro the ordnry gven street (rod) networ through sple lner ssgnent odel. Second, nother lner odel s used to generte Crtcl Dend (llustrted n Secton II.D) on trnst networ lns fro the ssgned networ. Thrd, Crtcl Dend s used n the fnl step of constructng trnst routes to dentfy trnst routes over the trnst networ through thetcl forulton resebles the one used n TSP. The output of ethodology should reflect the obectves of desgn nd nvolte plnnng constrnts. The structure of ths pper s s follows. Secton II gves bsc trnst concepts to help the reder to get good grsp of the render of ths pper. Secton III descrbes the route desgn stge obectves nd constrnts. Secton IV gves the soluton ethodology. Secton V Mndl benchr trnst proble s used to evlute the proposed ethodology. Secton VI s the concluson. The nontons used n the pper re presented s follows: d the trnst dend fro () to () expressed s trps per unt te r t s t R C X nu n vehcle trvel te between () nd () for pssengers dend ( ) usng route (r), r R d trvel te between node () nd node () through the shortest pth R=(r 1, r 2,,r n ) set of bus routes the fxed cost on ln () (te or length) flow on ln () trps per unt te q flow on pth connectng node () nd (), δ duy vrble, 1 f flow q pss through ln, 0 otherwse A the set of networ rcs (lns) θ odel decson vrble equl to 1 f drected rc - s selected n bus route (r) under constructon, 0 otherwse X - flow vlue on ln - d(r) Trnst route networ drectness ndctor D o dend covered drectly by the set of routes R D o1 Dend covered wth one trnsfer by the set of routes R D tot totl pssengers dend wthn servce re to be covered by bus servce TD networ totl dend, orgn nd destnton nodes N K set of vlble pths between () nd () = 1,2,, n S of ll networ lns cobntons tht nclude t lest three nodes nd perfor closed crculr route, S = [s 1, s 2, s 3,... s n ] CD - Crtcl Dend on drected ln - N networ set of nodes t - rc trvel te or length T r bus route (r) te (or length) xu llowble bus route trvel te (or length) T r x II. BASIC TRANSIT CONCEPTS A. Trnst Networ Infrstructure Representton The ter "networ" n trnsportton plnnng s used to descrbe structure of streets nd ntersectons (nodes) to be used n Mthetcl Progrng. Street networ nd nodes re consdered the nfrstructure for TrNDP. Two types of grphs re presented n Fg. 1. Undrected grph whch rcs re b-drectonl s depcted n Fg. 1 (). A drected grph s grph n whch the rcs hve specfed drectons by rrow heds s shown n Fg. 1 (b). A drected grph representton ws dpted n the proposed soluton ethodology. For gven urbn networ, t cn be defned s G = (N, A), where (N) s the set of nodes ( N = n). The set of nodes s connected by the set of rcs (lns) ( A = ). 284

3 Fg. 1 Grphs nd drected grphs B. Street (Rod) Networ versus Trnst Networ Essentl speng, plnners should dffer between two ters, nely; street networ nd trnst networ. Rod networ refers to ll exstng streets nd ther ntersected nodes presented n the studed networ. Trnst networ, s entoned before, s to dentfy sets of bus routes over the rod networ chevng soe trnst obectves under soe constrnts. These sets of connected bus routes consttute trnst networ. C. Route Drectness (d(r)) It should be noted, prtculr trnst networ wll dffer fro the orgnl street networ fro whch t s derved, provded soe lns present n the street networ re bsent fro the trnst networ. As consequence, shortest pth dstnces for trvelers between the vrous node prs wll need to be reclculted for ech new route set tht s evluted, usng dstnce or te trx specfc to tht new trnst route networ. Route drectness (d(r)) s n ndctor to esure bus route devton fro the shortest pth ong n trnst nodes prs snce; d(r) = 1 ndctes tht ll bus users would te the shortest pth long ther trvel between orgn nd destnton. The vlue of d(r) whch exceeds one, t would ndcte the dely cused by the set of bus routes to ll users.at the networ level, we would evlute the overll networ drectness d(r) ccordng (1) tng nto ccount the weght of ech trnst route dend. d(r) = N N d TD t t D. Crtcl Dend (CD) In ths secton, we would defne new noton of Crtcl Dend (CD) for dend lod profle of the trnst route. Fg. 2 depcts two possble lod profles of the trnst route encopsses fve ln segents nd sx ternls (A to F). Cse () shows tht the sustnble dend (CD) on ll segents s (28 pssenger/hr) whch s the lower dend of ths trnst route. Cse (b) revels the portnce of CD n conservng route sustnblty. If CD decresed to pproch zero n ths segent, the route would be splt nto two routes. The vlue of the CD s n ndctor of ll route segents rdershp. r s (1) Fg. 2 Trnst route dend lod profle In ths pper, we would concern of clcultng CD for ll trnst networ segents whch buld ll trnst networ routes. It s consdered the crteron for selectng these segents n the constructng route process. For certn trnst route, CD s consdered s fngerprnt, whch s conserved long ll segents. We would forulte lner thetcl odel to obtn CD for trnst lod profle gven n cse (). It y pper too sple to need forulton, but ts usefulness would pper n trnst networ level, when trnst routes ren t constructed yet. Mxze Soluton; xze CD (2) s.t. CD ( n) D ( out) = 0 N (3) X - CD X - N, (4) CDA B + CDB + CDC + CDD E + CDE F CDA B = CDB ; CDB = CDC ; CDC = CDD E ; CDD E = CDE F CD 80; -58 CDB 58; -28 CDC 28; - CDE 52; ths would result n CDA B = CDB = CDC = CDD E = CDE F = 28 s.t. -80 A B 42 D E CD 42; -52 F CD = pssenger/hr Note: A negtve lower bound on lns flow s pertted durng the optzton process. Ths ens; f CD- tes negtve vlue, t s n opposte flow drecton on tht ln (.e. the drecton of flow s fro () to ()). III. TRNDP DESIGN OBJECTIVES AND CONSTRAINTS Trnst route desgn s ult-obectve proble whch consttutes or obstcle n defnng the globl optl soluton. We could dentfy three or obectves to be nzed n TrNDP, nely opertor cost, user cost nd socl cost. The opertor cost s esured n ters of totl trvel te (length) of trnst routes whch reflects totl operton cost nd requred fleet sze to ntn certn frequency (level of servce). User cost s esured by Trnst networ drectness, the ore drected networ, the less user cost. The socl cost s nzed by xzng drect 285

4 dend coverge (Do) for the gven trnst networ. Incresng Do s otvton for n ncrese n trnst rdershp, nd consequently sgnfcnt decrese n networ congeston occurs. The concerned obectve functon s stged n two levels, frst, n ssgnent stge coprose s de between opertor nd user cost. Pssengers re ssgned to shortest or th-shortest pths (user cost), whle nzng the overll networ cost (opertor cost). Second, n route constructon stge, we focused on nzng socl cost (xzng drect dend coverge) whch could be consdered the prry concern n the desgn. TrNDP constrnts refer usully to resource vlblty nd prctcl pplcblty. Trnst route xu length s or constrnt n route desgn stge. There s constrnt on xu trnst route length bsed on ether heurstc gudelnes, pst experence or coon prctce ccepted by trnst plnners. Longer bus routes y cuse bus drver ftgue nd consequently result n sfety hzrds. Mxu round trp shouldn t exceed 2 hours. IV. SOLUTION METHODOLOGY The proposed soluton ethodology doesn t to rech the globl optlty. It contrbutes to TrNDP by presentng sple nnovtve soluton ethodology whch s cpble of producng effcent trnst networ n systetc deternstc ethodology. The output networ cn stnd s n effcent soluton for TrNDP or be used s subroutne for other etheurstcs procedures. The chllenge for conservng the deternstc nner of ethodology ppered n route desgn stge. Three or probles hnder dentfyng trnst routes over the trnst networ through TSP forulton; these probles re tcled n Secton IV.C. 1. Hlton Grph: TSP stpultes tht the grph of networ under study contns t lest one Hlton crcut (HC). HC s pth tht vsts ech vertex exctly once, except the strt nd end re vsted twce [17]. 2. Sub-Tours: A ey prt of TSP s to e sure the tour s contnuous (the soluton s one tour), tht the rcs re lned fro the bse cty ll the wy to every cty vsted. Pervous forultons of TSP qute often wll get solutons contnng detched tours between nteredte nodes nd not connected to the bse cty. 3. One Crculr Route: As nown, TrNDP s desgn set of open bus routes chevng desgn gols. TSP results n one crculr route, so ts usge n trnst plnnng s lted to cses of CBD or rlwy stton (Bse cty) would be connected to other dend zones (one to ny) unle our cse of study (ny to ny). These probles re tcled n Secton IV-C. A. Trnst Pssenger Assgnent It could be defned s the query of pssenger flows on trnst networ segents. Pssenger ssgnent s process of predctng pssengers behvor n selectng bus routes ccordng to route te length nd bus frequency for ech bus route [18]. Trnst pssengers n ny cses hve to del wth overlppng bus routes wth soe routes shrng sectons nd coon stops. Ths proble s sub-proble of trnst pssenger ssgnent, clled coon-lnes proble. Vrous ssuptons nd studes re de n order to trc pssengers behvor towrds gven supply of trnst servce [19]-[21] Spess developed the concept of clever pssenger nto pssenger optl strteges. Pssengers would nze the su of wtng te nd n-vehcle te n ther bordng strteges. We would perfor ssgnent step on the street networ (nsted of the trnst networ) to obtn n pproxte soluton for the trnst networ. Spess forulton s nly xed nteger non-lner progrng whch s dffcult (possble) to be solved exctly for rel sze networs. We would use only the frst ter of hs odel (pssenger tendency to nze n-vehcle te) ssung tht bus frequences re very hgh on ll trnst networ segents. Ths ssupton would be vld n tht cse n whch the shortest trnst lne s turned to the ttrctve one. Assgnent odel would be esly wrtten s follows: Mnze s.t. A q = d K X = K N N C (5) δ X,, N (6) q A (7) X L.F f x V s (8) q 0,, (9) X 0 (10) K represents the set of shortest pths between node pr () nd (), t could be obtned by usng Dstr s Algorth nd Yen s K-shortest Pth [22], [23]. Assgnent odel would be consdered the optl (effcent) soluton for TrNDP, snce ll trps re ssgned to the shortest pth or th shortest pth. Shortest pth set -for even sll networ- s lrge set; therefore t s not prctcl soluton to consder ll these pths s bus routes. Our soluton ethodology depends on constructng hgh dend coverge routes fro the resulted trnst networ (ssgned networ). For ore llustrton, sll nstnce of street networ s gven n Fg. 3 () wth lns length n. nd trnst O/D trx s gven n Tble I. By pplyng ssgnent odel n (5)-(10), t would gve the ssgned trnst networ depcted n Fg. 3 (b). 286

5 TABLE I O/D MATRIX REPRODUCED FROM [24] OD B. Crtcl Dend Clculton Assgned trnst networ n Fg. 3 (b) denotes ggregte dend lod profle of ll trnst routes, whle trnst routes hven t been dentfed yet (sequence of nodes for ech trnst route). As entoned before, the CD s consdered n ndctor for Trnst route sustnblty. We would explot (2)-(4) to generte ggregte CD on the trnst networ lns. In the rel world, buses go n drected pth fro ts strt ternl trversng other nodes n ts route (pth) tll t reches ts end ternl. So, t s worth notng tht the proposed forulton of CD helps n fndng the best drectonl syste of trnst networ, see Fg. 3 (c) (Note: All networ lns n ths exple re b-drecton, so there s n opposte syste of drectons wth the se vlue of CD on lns). Ths would d t route constructon stge. Trnst route effectveness would be defned by the suton of CD vlues over ts lns. Hgher suton vlue s n ndctor of hgher route overll rder-shp. C. Route Constructon Procedure To conserve the deternstc nner of proposed soluton ethodology, we would use lner nteger forulton for trnst route constructon, whch cn be solved exctly up to resonble sze wth brnch nd bound lgorth. There re lso effcent stochstc technques for lrger szes. Integer progrng lgorth presented n (11) resebles TSP forulton, except t s to fnd the route wth the hghest Fg. 3 An exple networ reproduced fro [24] CD suton (the ost effcent trnst route). Mxze N N θ CD (11) s.t. θ t T r x (12) θ = θ + θ f + θ f θ 2 N (13) N (14) θ =2 (15) S= [s 1, s 2, s 3,... s n ] s n N, s 3 (16) s θ, n s n sn -1 n sn S (17) Inequlty (12) sserts tht bus round trp trvel te won t exceed ts xu llowble te or length, (13) provdes connected bus route [25].Inequlty (14) ssures tht every node wll be vsted once t ost. Equton (15) sserts tht bus route under-constructon ust vst node (f). Node (f) s 287

6 fcttous node connected to ll networ nodes wth gnry lns, s depcted n Fg. 4. CD vlues on these gnry lns equl to zero. The only rule for ths fcttous node f s to crete open bus routes. Fg.4 Trnst networ connected wth fcttous node (f) Equton (16) defnes S s the set of ll networ lns cobntons tht nclude t lest three nodes nd perfor closed crculr route [26]. Equton (17) prevents the possble exstence of ny group of crculr bus routes except the one pssng the fcttous node (f). Note tht S set s esly defned here snce the networ s drected (.e. nodes cobnton wth ther lns drectons should concde to perfor crculr route, see for nstnce route Fg. 3 (c), ths cobnton doesn't perfor route nd thus doesn't requre constrnt to be prevented. Another wy to defne the set S y be s follows; 1- Apply (11)-(17) before ddng the fcttous node to the networ. 2- Ths would result n crculr or cobnton of detched crculr routes. 3- The resultnt crculr (or crculrs) wll be defned to the set S nd be prevented by pplyng (22). 4- Apply gn (11)-(17), ths y result n nother crculr routes subset of S. 5-Ths process would be repeted tll ll S set re defned nd there sn't ny nternl closed crculr route cn be fored. Applyng route constructon odel on networ Fg. 3 (c) would result trnst route ( ) wth CD = After frst route constructon, we would updte CD for networ lns by subtrctng the sllest vlue of the CD on the constructed bus route (CD = 46.03, ln 1-4), fro ll route lns. Ths vlue s the fngerprnt of the route. We would obtn updted networ n Fg. 3 (d). Ths subtrcton s done to elnte the possblty of selectng ths route n the next route constructon. Equtons (11)-(17) re used to construct bus routes tertvely nd ths gves the opertor flexblty to dscrd ny unproftble route fro the soluton of TrNDP. By followng the se procedure, we could construct the second bus route. Ths process cn contnue tll nuber of routes re constructed stsfyng TrNDP obectves or no provng n route networ desgn preters s de. The trnst route enuerton process gves the opertor possblty to choose the nuber of bus routes whch stsfy hs/her plnnng requreents nd budgetry constrnts. A sury of bus routes s gven n Tble II. Although the route nuber (2) doesn't ncrese networ dend coverge, t proves route networ drectness (.e. route nuber 2 pprovl or excluson returns to the opertor). Route no. Route node sequence TABLE II BUS ROUTES ENUMERATION b Route length n CD CD -0- trnsfer trps (%) Totl dend covered (%) c In ths exple nodes (1,2,5,6) re bus route strt nd end ternls [24] b Route xu length constrnt equls 2.5 [24] c Suton of CD s clculted fro updtng networ seefg. 3 (c) D. TrNDP Soluton Methodology Structure The proposed soluton ethodology for TrNDP cn be concluded n these steps: Step1. Construct coded street networ. Step2. Identfy trvel te (or length) on lns. Step3. Assgn nodes trnst trps usng odel (5-10), rechng ssgned Trnst networ. Step4. Assue rbtrry drected grph rrows,.e. only one drecton s chosen rbtrrly for ech b-drectonl ln (N.B the lgorth dusts the drecton lter n the process of rechng CD vlues on lns). Step5. Optze the ssgned Trnst networ (2-4), rechng drected trnst networ wth CD vlues on lns. Step6. Construct frst bus route (11)-(17), coverng the hghest CD suton, set r = 1. Step7. Subtrct the lest vlue of CD on the constructed route fro the route lns long ts pth to get updted networ wth new CD vlues. Step8. Copute the totl dend stsfed by the set of Rth routes (usng -0- nd -1- trnsfer), f tht dend D n tot (the nu totl dend needed to be stsfed) ternte route constructon process nd output set R routes; otherwse, go to step 6. V. COMPUTATIONAL CASE STUDY A. Mndl s Benchr Trnst Networ [27] In order to deonstrte the effectveness of the soluton ethodology proposed n ths pper, populr benchr networ s solved (Mndl s Swss trnst networ). Mndl s Swss trnst networ s the ost populr trnst networ tht hs been utlzed by ny reserchers s benchr networ to copre ther results wth Mndl s soluton results [4], [6], [9]-[12], [14], [27]-[30]. Mndl s trnst networ conssts of 15 nodes connected by 21 lns wth totl dend of trps; see Fg. 5 [27]. B. Soluton Procedure for Mndl s Trnst Networ [27] The odels presented n ths pper re nly lner nd nteger progrng odels. So, they cn be solved wth ther stndrd solvers. We contnued to construct bus routes for Mndl s networs [27] untl no proveent n the networ drect dend coverge. 288

7 Fg. 5 Coded networ wth trvel te on lns n nutes [27] C. Results nd Dscussons The Trnst route desgn resulted n sx bus routes. A sury of the constructed routes s gven n Tble III. Route No. Route node sequence TABLE III CONSTRUCTED ROUTES SUMMARY length % CD D o %Totl Dend stsfed n n n n n n Tble IV presents coprson between ths study nlyss results nd hghlghted prevous wor tht tcled Mndl s trnst networ [27] n ther soluton s benchr proble. The frst row ndctes the source of the solutons to the benchr proble. The second row gves the prevous wor yer. The thrd row dentfes the solutons serch tool ethods to the proble. Fourth to twelve rows denote the route networ chrcterstcs (plnnng preters). The coprson of Tble IV shows wth the proposed ethodology provdes sgnfcntly better soluton thn others, prtculrly n ters of percentge of drect dend coverge, whch es the trnst networ ore ttrctve to the users. Ths study proposes less nuber of routes thn other studes (except Mndl [27]) n the process of seeng ore drect routes. We consdered the nuber of routes generted -n regrd wth other preters of coprson- the ost portnt crter n udgng the effcency of ny of these serch ethods. Less nuber of routes deonstrtes the ethodology strength n sseblng cnddte lns n one contnuous bus route nd entls ccordngly less ount coputtonl effort ofthe bus nd drver schedulng prt. TbleV presents coprson of ths study result nd other studes whch produced the se nuber of routes for Mndl s networ [27]. TABLE IV COMPARISON OF APPROACHES FOR MANDL S BENCHMARK NETWORK PROBLEM [27] Proble Shh nd Bgloee & Mndl [27] source Mhssn [11] Ceder [30] Yer Serch ethod Mndl 1 Ths S&M 2 Ths GI& Ths study study AS 3 study Nuber of routes trnsfer trps % (D o ) One trnsfer trps% (D o1) Two trnsfer trps% (D o2 ) Totl Dend stsfed (D tot) Networ drectness d(r) RNR Trnsfer drectness RNR 1.05 Totl route te length (n) Drect trps / route length (pss./n) Mndl s ethod [27] 2 Shh nd Mhssn s [11] 3Grvty Index & Ant [30] 4Results not Reported 5Clculted s d(r) but for trnsfer trps only TABLE V COMPARISON OF APPROACHES WITH THE SAME NUMBER OF ROUTES Proble source B nd Mhssn [9] Chroborty [4] Zho [14] Fn &Muford [28] Yer Serch ethod B&M 1 Ths study GA 2 Ths study SAFD 3 Ths study SA 4 Ths study Nuber of routes trnsfer trps %(D o ) trnsfer trps% (D o1 ) Totl route length (n) RNR 205 RNR B nd Mhssn s [9] 2 Genetc Algorth [4] 3 Sulted nnelng nd FD [14] 4 Sulted Annelng [28] 289

8 VI. CONCLUSION Provdng trnsport county wth sple nd effectve trnst route desgn technque s the n purpose of ths pper. Soluton ethodology dependng only on lner nd nteger opertors ws presented to solve Trnst route Networ desgn proble. The proposed soluton ethodology goes beyond pervous trdtonl heurstcs nd Met heurstcs (pproxte) pproches. It s hghly depended on dend trx. It s generc ethod snce t confrs to severl networ routes confgurtons. It doesn't bs towrds ny exstng trnst networ. It s flexble; snce plnner cn clssfy generted bus routes ccordng to dend coverge, whch enbles the opertor to execute selected routes ccordng to vlble exstng resources. The ey ndctor of good trnst plnnng s to cheve xu drect dend coverge wth eepng the good vlue of other networ preters. Ths would encourge ore people to select publc trnsport nd consequently chevng one of the ost portnt gols of trnst plnnng. In future wor, etheurstc would be dopted to prove results to pproch globl optl soluton. REFERENCES [1] E. Cprn, S. Gor, nd M. Petrell, "Trnst Networ Desgn: A Procedure nd n Applcton to Lrge Urbn Are," Trnsport Res C- Eer, 20, pp. vol.3, no.14, [2] A. Ibes, L. dell Olo, B. Alonso, nd O. Snz, "Optzng Bus Stop Spcng n Urbn Ares," Trnsport Res D-Tr E, 46, pp , [3] R. Z. Frhn,E. Mndobch, W. Szeto, nd H. Rshd, "A Revew of Urbn Trnsportton Networ Desgn Probles," Eur J Oper Res, vol.229, no.2, pp , [4] P. Chroborty, "Genetc Algorths for Optl Urbn Trnst Networ Desgn," Coput Aded Cvl & Infrstructure Eng. Blcwell Pub., MA 02148, USA, 18, pp , [5] M.B,The Trnst Networ Desgn Proble: An AI-Bsed Approch, Ph.D, [6] A. Gn, nd F. Zho, Optzton of Trnst Networ to Mnze Trnsfers. Fnl Report, Reserch Offce Flord Deprtent of Trnsportton 605 Suwnnee Street, MS 30 Tllhssee FL , [7] M. R. Grey, & D. S. Johnson, Coputers nd Intrctblty: A Gude to the Theory of np-copleteness,. W.H. Freen, , [8] W. Lpn, & P. Slns, "The Desgn of Routes, Servce Frequences nd Schedules for Muncpl Bus Undertng: A Cse Study," Oper Res, vol. 18, pp , [9] M. H. B, nd H. Mhssn "An AI-Bsed Approch for Trnst Route Syste Plnnng nd Desgn,"J. Adv Trnsport, vol. 25, no.2, pp , [10] M. H. B, nd H. Mhssn, "Hybrd Route Generton Heurstc Algorth for the Desgn of Trnst Networs," Trnsport Res C-Eer, vol. 3, no.1, pp.31-50, [11] M. C. Shh, nd H. Mhssn, A Desgn Methodology for Bus Trnst Networs wth Coordnted Operton. (Ph.D), the Unversty of Texs t Austn, Austn, Texs, [12] A. Muttonw, nd M. Urquhrt, "A Route Set Constructon Algorth for the Trnst Networ Desgn Proble," Coput nd Oper Res, vol 36, no.8, pp , [13] S.B. Pttn, S. Mohn, nd V. M. To, "Urbn Bus Trnst Route Networ Desgn Usng Genetc Algorth," J Trns Eng., vol.124, no.4, pp , [14] F. Zho, Lrge-Scle Trnst Networ Optzton by Mnzng Trnsfers nd User Cost. J Publc Trns, vol. 9, no.2, pp , [15] Yu, Bn, Yng, & Zhongzhen, "An Ant Colony Optzton Model: ThePerod Vehcle Routng Proble wth te Wndows," Trnsport Res E-Log, vol. 47, no. 2, pp , [16] M. Nolc, nd D. Teodorovc, "Trnst Networ Desgn by Bee Colony Optzton," Expert Syst Appl, vol.40, pp , [17] W. R. Hlton, MeornduRespectng New Syste of Roots of Unty," Phlos Mg, vol.12, no.4, pp. 446, [18] G. Desulners, nd M. Hcn, Hndboo n OR nd MS. Elsever, (Chpter 2), [19] C. Chrqu, nd P. Robllrd, "Coon Bus Lnes. Hutes Eludes Coercls, Montrél, Québec, Cnd, Trnsportton, vol. 9, no.2, pp , [20] P. Mrguer, nd A. Ceder, A. (1984). Pssenger Wtng Strteges for Overlppng Bus Routes. Msschusetts Insttute of Technology, Cbrdge, Msschusetts, Trnsportton Scence, vol. 18, no.3, pp , [21] H. Spess, On Optl Route Choce Strteges n Trnst Networs, Publcton 285, Centre de Recherche sur les Trnsports, Unversté de Montrél,1983. [22] Dstr, A Note on Two Probles n Connecton wth Grphs. Nuer Mth,1, pp , [23] J. Yen, Fndng the K Shortest Loopless Pths n Networ. Mng Sc, vol.7, no.11, pp , [24] B. Yu, Z. Yng, P. Jn, S. Wu, nd B. Yo, "Trnst Route Networ Desgn-Mxzng Drect nd Trnsfer Dend Densty," Trnsport Res C-Eer, vol.22, pp.58-75, [25] N. E. Lownes, & R. B. Mcheehl, "Exct nd Heurstc Methods for Publc Trnst Crcultor Desgn," Trnsport Res B-Meth, vol.44, no.2, pp , [26] A. Ceder, nd S Jerby, Optl Routng Desgn for Shuttle Bus Servce. Trnsportton Reserch Record: Journl of the Trnsportton Reserch Bord, pp.14-22, [27] Mndl, C. E. (1980). Evluton nd Optzton of Urbn Publc Trnsportton Networs. Europen Journl of Operton Reserch, 5(6), [28] L. Fn, & C. Muford, "A Metheurstc Approch to the Urbn Trnst Routng Proble,"J Heurstcs, vol. 16, pp , [29] F. Zho, nd X. Zeng, "Optzton of Trnst Route Networ, Vehcle Hedwys nd Tetbles for Lrge-Scle Trnst Networs," Europen Journl of Opertonl Reserch, 186, , [30] Bgloee, S., & Ceder, A. (2011). Trnst-Networ Desgn Methodology for Actul-Sze Rod Networs. Trnsportton Reserch Prt B: Methodologcl, 45(10),