Transit route and frequency design: Bi-level modeling and hybrid artificial bee colony algorithm approach

Transcription

1 Titl Transit rout and frquncy dsign: Bi-lvl modling and hybrid artificial b colony algorithm approach Author(s) Szto, WY; Jiang, Y Citation Transportation Rsarch Part B: Mthodological, 2014, v. 67, p Issud Dat 2014 URL Rights NOTICE: this is th author s vrsion of a work that was accptd for publication in Transportation Rsarch Part B: Mthodological. Changs rsulting from th publishing procss, such as pr rviw, diting, corrctions, structural formatting, and othr quality control mchanisms may not b rflctd in this documnt. Changs may hav bn mad to this work sinc it was submittd for publication. A dfinitiv vrsion was subsquntly publishd in Transportation Rsarch Part B: Mthodological, 2014, v. 67, p DOI: /j.trb ; This work is licnsd undr a Crativ Commons Attribution-NonCommrcial-NoDrivativs 4.0 Intrnational Licns.

2 Transit Rout and Frquncy Dsign: Bi-lvl Modling and Hybrid Artificial B Colony Algorithm Approach W.Y. Szto *, Y. Jiang Dpartmnt of Civil Enginring, Th Univrsity of Hong Kong, Pokfulam Road, Hong Kong * mail: cszto@hku.hk Abstract This papr proposs a bi-lvl transit ntwork dsign problm whr th transit routs and frquncy sttings ar dtrmind simultanously. Th uppr-lvl problm is formulatd as a mixd intgr non-linar program with th objctiv of minimizing th numbr of passngr transfrs, and th lowr-lvl problm is th transit assignmnt problm with capacity constraints. A hybrid artificial b colony (ABC) algorithm is dvlopd to solv th bi-lvl problm. This algorithm rlis on th ABC algorithm to dsign rout structurs and a proposd dscnt dirction sarch mthod to dtrmin an optimal frquncy stting for a givn rout structur. Th dscnt dirction sarch mthod is dvlopd by analyzing th optimality condition of th lowr-lvl problm and using th rlationship btwn th lowrand uppr-lvl objctiv functions. Th stp siz for updating th frquncy stting is dtrmind by solving a linar intgr program. To fficintly rpair rout structurs, a nod insrtion and dltion stratgy is proposd basd on th avrag passngr dmand for th dirct srvics concrnd. To incras th computation spd, a lowr bound of th objctiv valu for ach rout dsign solution is drivd and usd in th fitnss valuation of th proposd algorithm. Various xprimnts ar st up to dmonstrat th prformanc of our proposd algorithm and th proprtis of th problm. Kywords: Transit rout and frquncy stting problm; Bus ntwork dsign; Bi-lvl programming; Artificial b colony algorithm; Mixd intgr program; Mathuristics 1. Introduction Transit ntwork dsign has rcivd considrabl attntion ovr th last two dcads du to its practical importanc. For xampl, in Hong Kong, ovr 90% of th 11 million daily trips that popl mak involv public transport. Hnc, a wll-dsignd transit ntwork is 1

3 important for mting passngr dmand. Guihair and Hao (2008) and Kpaptsoglou and Karlaftis (2009) providd comprhnsiv rviws in this ara. Prvious works on this topic focus on rout dsign (.g., Mandl, 1980; Murray, 2003; Wan and Lo, 2003; Li t al., 2011, 2012), frquncy stting (.g., Furth t al., 1982; LBlanc, 1988; Hadas and Shnaidrman, 2012), timtabling (.g., Wong t al., 2008; Flurnt t al., 2004), vhicl schduling (.g., Bunt t al., 2006), crw schduling (.g., Wrn and Roussau, 1993), far structur (.g., Li t al., 2009), flt siz dtrmination (.g., Li t al., 2008) and a combination of th abov (.g., Cdr and Wilson, 1986; L and Vuchic, 2005; Szto and Wu, 2010). Th majority of prvious studis hav considrd th optimization of transit rout structurs and srvic frquncis sparatly. For xampl, Frnandz and Marcott (1992), Constantin and Florian (1995), Zubita (1998), Gao t al. (2004), Uchida t al. (2005, 2007), and Liva t al. (2010) proposd modls for optimizing frquncis to achiv diffrnt objctivs within an xisting transit ntwork, whras Laport t al. (2010) and Yu t al. (2012) focusd xclusivly on dsigning rout structurs. Both transit rout structur and frquncy stting dtrmin th lvl of srvic (.g., in trms of in-vhicl congstion and waiting tim at bus stops); mor importantly thy dtrmin whthr th srvic has sufficint capacity to mt to passngr dmand. Thrfor, it is important to simultanously optimiz th transit rout structur and th frquncy stting. In transit ntwork dsign, it is ssntial to considr th in-vhicl congstion issu. In-vhicl congstion lads to incrasd waiting and travl tims, along with th comfort problm promptd by a lack of sats for passngrs. This comfort problm can b particularly srious if th trip tim is long or dmand is high. Gnrally, thr ar two approachs to addrssing th congstion issu: capacity constraint and th congstion cost function. Th capacity constraint approach (.g., Kurauchi t al., 2003; Li and Chn, 2004; Lam t al., 1999, 2002; Cpda t al., 2006; Sumal t al., 2009, 2011; Schmöckr t al., 2008, 2011; Szto t al., 2013; Cortés t al., 2013) incorporats capacity constraints in transit assignmnt modls that disallow flows on transit vhicls to b gratr than th corrsponding capacity. Th congstion cost function approach (.g., Spiss and Florian, 1989; d Ca and Frnándz, 1993; Lo t al., 2003; Li t al., 2008, 2009, 2011; Sun and Gao, 2007; Tklu, 2008; Szto t al., 2011a; Szto and Jiang, 2014) adopts an unboundd incrasing convx function to modl th ffct of in-vhicl congstion on waiting tim. Although both approachs hav bn usd in th litratur, practically spaking, th formr is mor ralistic bcaus th lattr can rsult in an unaccptabl lin flow that is far gratr than th corrsponding capacity. 2

4 In addition to th congstion issu, it is important to considr passngr transfrs btwn transit vhicls, as thy can gnrat passngr inconvninc. Th numbr of passngr transfrs is an important ntwork prformanc indicator, spcially in Hong Kong, for th following rasons. First, th total numbr of passngr transfrs rflcts th numbr of passngrs without dirct srvics to thir dstinations, which can indicat inconvninc. Scond, passngrs always complain whn thr ar no dirct srvics to thir dstinations (Szto and Jiang, 2012). Th total numbr of passngr transfrs also indirctly rflcts th numbr of complaints rgarding lack of dirct srvics. Optimizing th numbr of passngr transfrs can rduc th numbr of complaints implicitly. Howvr, vry fw studis hav considrd this numbr. Baaj t al. (1990) mbddd th transfr concpt into thir rout gnration procdurs, such that a rout with mor than two transfrs was abandond. Similarly, th numbr of passngr transfrs was modld implicitly in Zhao t al. (2005). Th travl cost calculatd in th objctiv function xcludd th travl costs of routs with mor than two transfrs, yt thy did not optimiz th total numbr of passngrs nding to transfr btwn transit vhicls. Guan t al. (2006) usd th total numbr of passngr transfrs as a surrogat of transfr and waiting tims in passngr lin assignmnt, which is th lowr lvl problm of thir transit ntwork dsign problm. Jara-Díaz t al. (2012) considrd th total numbr of passngr transfrs to invstigat th condition undr which a transit ntwork dsign with transfrs is prfrabl. Most of th xisting studis hav usd th total passngr travl tim as th objctiv function. Howvr, thr is no guarant that minimizing th total numbr of passngr transfrs also minimizs th total passngr travl tim. In som cass, thr can b a tradoff btwn th total numbr of passngr transfrs and total passngr travl tim (Szto and Wu, 2010). It is ssntial to xplicit captur th total numbr of passngr transfrs in th objctiv function. This papr proposs a bi-lvl modl for dsigning transit routs and thir frquncis that xplicitly minimizs th total numbr of passngr transfrs in th objctiv function of th uppr-lvl problm and incorporats strict capacity constraints to addrss th in-vhicl congstion in th lowr-lvl problm. This bi-lvl modl is formulatd as a mixd intgr non-linar program that is NP-hard and considrs th rout choic bhavior of passngrs through th lowr-lvl usr-quilibrium problm. Th modl also considrs th stop location choic of ach rout within ach zon of th study ara. This modl diffrs from th bi-lvl modls proposd by Constantin and Florian (1995), Gao t al. (2004), and Uchida t al. (2005, 2007) in th sns that thy only considrd frquncy stting, whras our modl furthr 3

5 considrs rout dsign and stop location choic. To solv transit ntwork dsign problms, xact mthods (.g., Wan and Lo, 2003) and mtahuristics such as gntic algorithms (GAs) (.g., van Ns t al., 1988; Billi t al., 2002; Chakroborty, 2002; Tom and Mohan, 2003; Ngamchai and Lovll, 2003; Shih t al., 1998; Fan and Machmhl, 2006a; Mazloumi t al., 2012) and simulatd annaling (.g., Fan and Machmhl, 2006b; Zhao and Zng, 2006) hav bn usd. A hybrid artificial b colony (ABC) algorithm a mathuristic that combins a mtahuristic and an xact algorithm is dvlopd for th transit ntwork dsign problm as an improvmnt to th original ABC algorithm, a mtahuristic proposd by Karaboga (2005) and motivatd by th foraging bhavior of hony bs. Compard with xisting volutionary algorithms such as GAs, th ABC algorithm has a bttr local sarch mchanism that improvs th solution quality. Mor rcntly, th ABC algorithm has bn applid to solv complx nginring optimization problms. For xampl, Kang t al. (2009) succssfully applid an ABC algorithm to th paramtr idntification of concrt dam-foundation systms. Karaboga (2009) proposd an ABC algorithm to solv a digital filtr dsign problm and obtaind good rsults. Karaboga and Ozturk (2009) usd an ABC algorithm to train nural ntworks for pattrn classification, and thir rsults on bnchmark instancs showd that such us was fficint. Szto t al. (2011b) improvd th ABC algorithm to solv a capacitatd vhicl routing problm. Szto and Jiang (2012) nhancd th ABC algorithm to solv a singl-lvl transit ntwork dsign problm without considring th in-vhicl congstion ffct. Long t al. (2014) improvd th ABC algorithm to solv a turn rstriction dsign problm. Szto and Jiang (2012) and Long t al. (2014) showd that thir proposd ABC algorithm is bttr than th GA for solving thir problms, but it has not yt bn improvd to solv bi-lvl transit ntwork dsign problms that considr in-vhicl congstion. This study nhancs th ABC algorithm to solv this problm. Th proposd algorithm rlis on th ABC algorithm to dsign rout structurs and a proposd dscnt dirction sarch mthod to dtrmin an optimal frquncy stting for a givn rout structur. A nod insrtion and dltion stratgy for rpairing th rout structurs is dvlopd basd on avrag-dirct-dmand, which is dfind as th avrag passngr dmand on th dirct srvics concrnd. Th dscnt dirction sarch mthod is dvlopd by analyzing th optimality condition of th lowr-lvl problm and using th rlationship btwn th lowr- and uppr-lvl objctiv functions. Th stp siz for updating th frquncy stting is dtrmind by solving a linar intgr program formd by th drivativ 4

6 obtaind by th Lagrang function of th lowr-lvl problm. Th Simplx mthod is usd to solv th lowr-lvl problm. To incras th computation spd, a lowr bound of th objctiv valu for ach rout dsign solution is drivd and usd in th fitnss valuation for th hybrid ABC algorithm. Various xprimnts ar conductd to dmonstrat th ffctivnss of our proposd algorithm. Thy illustrat th ffcts of various nod insrtion and dltion stratgis and th ffcts of diffrnt paramtr valus and forms of fitnss functions on th prformanc of th hybrid ABC algorithm. A ralistic cas study is conductd to show that undr dmand uncrtainty, th optimal solution obtaind from th hybrid ABC algorithm is bttr than th xisting bus ntwork dsign in trms of th avrag numbr of passngr transfrs, and is mor robust in trms of handling passngr dmand. W also us th Winnipg ntwork to dmonstrat that th prformanc of our proposd mthod is bttr than that of a GA to solv our problm. Th xprimnts illustrat th ffcts of diffrnt dsign paramtrs such as minimum frquncy, maximum flt siz, and th maximum numbrs of routs and intrmdiat stops on th objctiv valu. Th rsults show that a highr minimum frquncy can lad to a highr numbr of passngr transfrs, and multipl dsign solutions ar possibl. Th main contributions of this study ar as follows. 1) Proposing a bi-lvl modl to simultanously solv th transit rout dsign and frquncy stting problms whil considring th candidat transit stop location availabl in ach zon in th study ara and two inconvninc factors: transfrs btwn transit vhicls and in-vhicl congstion. 2) Dvloping a nw mathuristic th hybrid ABC algorithm to solv th modl. 3) Examining th proprtis of th bi-lvl problm and th prformanc of th algorithm. 4) Dmonstrating th applicability of th proposd modl and algorithm in ralistic situations. Th rmaindr of this papr is organizd as follows. Sction 2 introducs th bi-lvl modl. Th proposd hybrid ABC mthod is dscribd in Sction 3, and numrical xampls ar prsntd in Sction 4. Finally, th conclusions and futur rsarch dirctions ar givn in Sction 5. 5

7 2. Bi-lvl formulation of th problm Considr a study ara with a connctd (bus) transit ntwork rprsntd by a dirctd graph G with N nods, E links (or arcs), and on dummy nod (nod 0) introducd for th as of formulating th problm. Th study ara is sparatd into many zons, ach of which is rprsntd by a cntroid. Th cntroid is th origin nod aggrgating th travl dmand within th zon. Each cntroid is connctd to all of th candidat transit stops and trminals in that zon, in which a transit trminal for a bus srvic can b a candidat stop for anothr bus srvic. Each cntroid also gnrats N typs of travl dmand, ach of which is dsignatd to on cntroid (or dstination nod) outsid th study ara. Each of th N cntroids is connctd to bus trminals or bus stops in thir individual zon. Both th bus trminals and cntroids in ach of ths zons ar connctd to th transit ntwork in th study ara. Th following notations ar usd in this papr. Sts Z U = a st of nods in th uppr-lvl ntwork, xcluding th dpot; G s = a st of cntroids within th study ara; H m = a st of candidat stops conncting to cntroid m; U V G d C Z L R T A = a st of starting bus trminals insid th study ara; = a st of nding trminals outsid th study ara; = a st of cntroids/dstinations outsid th study ara; = a st of bus trminals and candidat stops within th study ara; = a st of nods (including cntroids) in th lowr-lvl ntwork; = a st of transfr links or arcs in th lowr-lvl ntwork; = a st of transit links in th lowr-lvl ntwork; A + i A - i = a st of transit links coming out from nod i; and = a st of transit links going into nod i. Indics i, j, m = indics of nods; = th indx of a cntroid/dstination outsid th study ara; = th indx of an nding bus trminal outsid th study ara; and r = th rout indx. 6

8 Paramtrs c ij = th in-vhicl travl tim on th shortst path btwn nods i and j; c a = th in-vhicl travl tim on link a; s t = th avrag tim for stopping at a nod; d m = th travl dmand from nod m to cntroid ; W k cap R max f min S max = th maximum bus flt siz allowd for th ntwork; = th capacity of a bus; = th maximum numbr of routs in th bus ntwork; = th minimum frquncy of a rout; = th maximum numbr of stops (including th bus trminal) within th study ara on a rout; T max = th maximum rout travl tim within th study ara; and p Dcision Variabls = a vry larg valu usd in th sub-tour limination constraint. Lowr-lvl dcisions v t v a ω i v = th numbr of passngr transfrs on transit link t to dstination ; = th flow on link a to dstination ; = th total waiting tim at nod i for all flows to dstination ; = v a ; w = ω i. Uppr-lvl dcisions qir = th nod potntial at nod i, which is ndd in th sub-tour limination constraint for bus rout r ; X ijr = 1 if rout r (r = 1 to R max ) passs through nod j immdiatly aftr nod i, and 0 X 0jr X i0r X 00r othrwis; = 1 if rout r starts at nod j, and 0 othrwis; = 1 if rout r nds at nod i, and 0 othrwis; = 1 if rout r is not availabl, and 0 othrwis; f r = th frquncy of rout r; 7 X = X ijr ; and

9 f =. Functions of dcision variabls T r [ ] f r = th trip tim of rout r from th starting trminal to th nding trminal; d r f a = 1 if rout r conncts th trminal that links to cntroid, and 0 othrwis; = th frquncy of link a; and ' d i = th travl dmand from nod i to bus trminal Uppr-lvl problm Th uppr-lvl problm is to dtrmin th frquncy of and a rout structur for ach transit lin within th study ara. Th numbr of transfrs within th study ara is unlimitd and thir possibl locations must rmain within th study ara. Th uppr-lvl problm is formulatd as follows. min z xfq,, 1 = v (1) R t T G d t subjct to j U {0} i V {0} X X 0 jr i0r = 1 = 1 for r = 1 to R max, (2) for r = 1 to R max, (3) X X jir = 0 for j ZU, r = 1 to R max, (4) ijr i ZU U i j i j { 0, } i Z { 0, } X ijr 1, for j ZU, r = 1 to R max, (5) { } i ZU 0, i j X ijr 1 for j ZU, r = 1 to R max, (6) { } j ZU 0, j i X = 0 for j ZU, r = 1 to R max, (7) jjr ( t) t for r = 1 to R max, (8) T = X c + s s Rmax r ijr ij i ZU j ZU, j i r= 1 ( ) 2fT 1 X W, (9) r r 00r 8

10 ( ) f X f min 1 00r r for r = 1 to R max, (10) Xijr Smax for r = 1 to R max, (11) i C j C, j i ( ij ) Xijr c + st st Tmax for r = 1 to R max, (12) i C j C, j i Rmax X ijr 1 for m Gs, (13) { } r= 1i Hm j ZU 0, j i Rmax fk r capd r dm r= 1 m Gs for Gd ' r i ZU \ V i' r, (14) δ = X for ' V, and (15) qir q jr + px ijr p 1 for i, j Z, i j, r = 1 to R max. (16) Objctiv (1) is to minimiz th sum of transfr passngrs. Constraint (2) nsurs that all of th bus srvic routs start from a bus trminal slctd from th availabl locations insid th study ara. Constraint (3) nsurs that ach of th srvic routs nds at a bus trminal slctd from th availabl locations outsid th study ara. It should b notd that th rth rout is not ndd to provid bus srvics whn X 00r = 1. Constraint (4) nsurs that with th xcption of dummy nods, any nod on a srvic rout has on prcding and on following nod. Constraints (5)-(7) nsur that ach nod can b visitd by a particular rout at most onc. Constraint (8) calculats th in-vhicl travl tim (including stop tim) of a srvic rout. Constraint (9) nsurs that th flt siz usd cannot xcd th availabl flt siz. Constraint (10) nsurs that th frquncy of ach srvic rout is not lss than th minimum allowabl frquncy. Constraints (11) and (12) rstrict th numbr of intrmdiat stops and th trip tim within th study ara, rspctivly. Constraint (13) is th zon covring constraint, and nsurs that at last on of th candidat stops in ach zon is srvd by at last on transit lin. Constraint (14) is th dmand constraint, which nsurs that thr is nough lin capacity to mt passngr dmand hading to ach dstination/cntroid outsid of th study ara. Constraint (15) dtrmins whthr rout r nds at trminal. Constraint (16) is th sub-tour limination constraint, which is xtndd from Millr t al. (1960). In this formulation, th dcisions ar th rout structurs and frquncis. Howvr, undr th prcding stting and constraint (13), th rout dsign automatically also considrs th stop location choic in a zon bcaus thr is mor than on candidat stop in ach zon in gnral, and a rout may not pass through all of thm. 9 U

11 2.2. Lowr-lvl problm Th lowr-lvl problm rquirs anothr ntwork rprsntation to dpict th passngr rout choic bhavior undr a givn st of transit routs dfind by th uppr-lvl problm. Th ntwork rprsntation for th lowr-lvl problm is xtndd from th on proposd by Nguyn and Pallottino (1988). Th ntwork is also rprsntd by nods and links (or arcs). Howvr, a nod may rprsnt a bus stop in a transit lin, a boarding nod, an alighting nod, or a cntroid. A link is usd to connct two adjacnt nods. Each link has thr attributs: travl tim, frquncy, and capacity. Figur 1 is a graph rprsntation of a cntroid conncting on gnral transit stop srvd by n transit lins. Similar to th graphical rprsntation proposd by Nguyn and Pallottino (1988), thr is a pair of boarding and alighting arcs conncting th bus stop of ach transit lin, s, i = 1,..., n, to th stop nod (as rprsntd by th nod dfind by th dashd lin in i Figur 1) that corrsponds to th nod in th uppr-lvl ntwork. To nsur that ths arcs ar only usd for connctivity purposs, th travl tim is st to zro and th capacity is st to a vry larg numbr. Th frquncy of th alighting arc is also st to a vry larg numbr, whras th frquncy of th boarding arc is qual to th frquncy with which th passngrs ar ntring th transit lin. Unlik th graphical rprsntation proposd by Nguyn and Pallottino (1988), w rplac th stop nod with two othr nods an alighting nod s a and a boarding nod s b a transfr arc to connct thm, a cntroid that corrsponds to th cntroid in th uppr-lvl ntwork, on accss arc, and on grss arc. Th boarding (alighting) nod is usd to snd (rciv) passngrs to (from) diffrnt transit lins and rciv (snd) passngrs from (to) th cntroid via th accss (grss) link. Th travl tims of th accss and grss links ar qual to th walking tims from th cntroid to th transit stops, whil th frquncis and capacitis associatd with accss and grss links ar vry larg (i.., infinity). Th transfr arc has a travl tim of M, a vry high frquncy, and a vry larg capacity. Intuitivly, M can b intrprtd as th inconvninc cost (xprssd as tim-quivalnt) or transfr pnalty gnratd by a transfr, and can b calibratd from survy data. Whn a dirct srvic is always prfrrd to a transfr srvic, M is st to b a larg numbr. Thr is no alighting arc, alighting nod, or grss arc for a starting trminal and no boarding arc, boarding nod, or accss arc for an nding trminal. Th conscutiv bus stops of a transit lin ar connctd by a travl arc, in which th travl tim is st to b qual to th in-vhicl travl tim plus th stop tim at th nxt stop, and th stop tim at ach trminal is 10

12 st to zro. Th frquncy of a travl arc is st to th frquncy of th transit srvic, whras its capacity is th frquncy of that arc multiplid by th bus capacity. All of th gnral transit stops and trminals ar connctd through travl arcs. Bcaus th dmand of ach origin-dstination (OD) pair is fixd and th flow on ach link cannot b gratr than that link s capacity, th total dmand btwn an OD pair may b largr than th availabl capacity providd by all transit lins srving this OD pair. Hnc, th lowr-lvl formulation may not provid a fasibl solution. To addrss this issu, a virtual link (corrsponding to a walking path) with a vry larg capacity, a vry high frquncy, and a vry long trip tim is cratd to connct ach OD pair. Th flow on ach virtual link at optimality is thn qual to th unsrvd dmand of th corrsponding OD pair. In th xtrm cas, whn th capacity btwn an OD pair is zro, thr is still a fasibl and optimal solution for that OD pair. Transit lin 1 s 1 Transit lin 2 Boarding arc s 2 : : Alighting arc Transit lin n s n s b Transfr arc s a Stop nod Accss arc Cntroid Egrss arc Figur 1 A graph rprsntation of a cntroid conncting on gnral transit stop Basd on this ntwork rprsntation, th transit assignmnt formulation proposd by Spiss and Florian (1989) can b xtndd to captur transfr pnalty and in-vhicl congstion as follows. 11

13 min : z = cv + ω vω, 2 a a i a A Gd i ZL Gd (17) subjct to v f ω a a i v = v + d a a i + a Ai a Ai + for a Ai, i ZL, Gd, (18) for i ZL, Gd, (19) va fk a cap for aî A, (20) Gd va 0 for a A, G, and (21) d ωi 0 for i ZL, Gd. (22) Th lowr-lvl objctiv (17) is to minimiz th sum of th total in-vhicl travl and stop tims (i.., th first trm of th objctiv function) and total waiting tim (i.., th scond trm of th objctiv function). Constraint (18), which rlats link flow, frquncy, and waiting tim, is a rlaxd constraint of distributing nod flows into th arcs manating from that nod. Constraint (19) is th flow consrvation condition for a nod. Constraint (20) nsurs that th flow on ach travl arc is not gratr than that arc s capacity, with th capacity constraint usd to modl th in-vhicl congstion cost and xtra dlay du to passngr ovrloading. Constraints (21) and (22) ar non-ngativity conditions. In th lowr-lvl problm, th following points rlatd to th proposd capacity constraint must b clarifid. First, th capacity constraint must b incorporatd into th lowr- rathr than th uppr-lvl problm. Th capacity constraint is usd to modl that, du to limitd vhicl capacity, som passngrs may not b abl to board th first bus that arrivs at a bus stop, and hnc may xprinc xtra dlays. Th capacity constraint is placd in th lowr-lvl problm to nsur that such dlays ar considrd in passngrs rout choics. Th xtra dlay of a passngr on a link is qual to th Lagrang multiplir associatd with th capacity constraint, which appars in th quilibrium condition drivd from th Karush Kuhn Tuckr condition of th lowr-lvl problm. If th capacity constraint wr placd in th uppr-lvl, it would b assumd that passngrs would not considr th dlay in dtrmining thir rout choic bcaus th lowr-lvl problm would b idntical to th transit assignmnt problm proposd by Florian and Spiss (1989). This bhavioral assumption is unralistic. Th scond point is rlatd to th flow distribution. If th lowr-lvl capacity constraint is not binding, th formulation rducs to th original stratgy formulation (Spiss and Florian, 12

14 1989), whr th rsultant lin flow is proportional to th lin frquncy, strictly following th assumptions that passngrs arriv randomly, hadway is xponntially distributd, and passngrs slct th first bus from a st of attractiv lins that arrivs at th bus stop. If th capacity constraint is binding, th flow distribution may not satisfy thos assumptions bcaus th passngrs cannot board th first bus that arrivs at th bus stop if it is full. In such cass, th rsults ar approximations that ar accptabl for stratgic planning purposs. Th third point is rlatd to th in-vhicl congstion cost. In th proposd modl, passngrs only prciv congstion costs if th capacity constraint is binding or buss ar fully occupid. Othrwis, th congstion cost is nglctd. Howvr, in rality, passngrs may still prciv in-vhicl congstion costs, such as th cost du to insufficint sat capacity or in-vhicl crowding, vn whn th capacity has not bn rachd. Th mor passngrs thr ar insid a bus, th highr th in-vhicl congstion cost. Hnc, th congstion cost should b a continuous and incrasing function. In th proposd capacity constraint mthod, th in-vhicl congstion cost is a picwis function, which can b addrssd by dvloping a continuous, non-linar, and incrasing in-vhicl cost function that can b linarizd to rduc th problm to a linar programming problm. Howvr, driving this function has bn lft for futur study. Th proposd formulation has thr advantags. First, th lowr-lvl problm is a linar programming problm and can b solvd fficintly by xisting algorithms. Scond, it is asy for us to idntify whthr a particular rout sction is ovrloadd (by chcking whthr th corrsponding Lagrang multiplir is positiv) and whthr th ovrall transit supply is sufficint (by chcking virtual links carry flow) all of which maks it asir to dsign appropriat improvmnt stratgis. Third, this linar problm allows us to dvlop an fficint mthod for solving th transit ntwork dsign problm. 3. Solution mthod Constraint (9) is non-linar, and th dcision variabls ar both discrt and continuous. Hnc, th bi-lvl problm is a mixd-intgr non-linar problm. It has bn notd that a gnral ntwork dsign problm is alrady NP-hard (Magnanti and Wong, 1984), and it is wll-known that th transit rout dsign problm is NP-hard (Zhao and Gan, 2003; Fan and Machmhl, 2004; Fan and Mumford, 2010). Our proposd problm includs th frquncy stting problm and a lowr-lvl problm that is mor complicatd than th gnral ntwork dsign and th transit routing problms. Thus, our problm is also NP-hard. Givn th 13

15 xtrm difficulty of solving NP-hard problms for xact solutions, a hybrid artificial b colony (ABC) algorithm is proposd to solv th bi-lvl problm. Th hybrid rlis on th original ABC algorithm to solv th rout dsign problm and incorporats a proposd itrativ procdur to dtrmin th numbr of passngr transfrs and th optimal frquncy stting. In th itrativ procdur, th linar transit assignmnt problms (17)-(22) ar solvd in ach itration via th Simplx mthod. Thn, a dscnt dirction is obtaind using th dual solutions to th transit assignmnt problm and usd to formulat a linar intgr program, which is solvd to giv a stp siz to updat th frquncy for th nxt itration. To allviat th computational burdn of solving many transit assignmnt problms, a scrning mthod basd on th lowr bound of th uppr-lvl objctiv function is also dvlopd. Only potntially good rout dsign solutions ar rquird to find th corrsponding optimal frquncy. Howvr, th othr solutions ar kpt for a nighborhood sarch Artificial b colony (ABC) algorithm Th ABC algorithm blongs to a class of volutionary algorithms inspird by th intllignt bhavior of hony bs finding nctar sourcs around th hiv. This class of mtahuristics has rcivd incrasing attntion rcntly, with variations of b algorithms proposd to solv combinatorial problms. Howvr, in all of thm, a common sarch stratgy is applid; that is, complt or partial solutions ar considrd as food sourcs and diffrnt groups of bs try to xploit th solution spac in th hop of finding good quality nctar, or high quality solutions, for th hiv. Thy thn communicat dirctly to inform othr bs about th sarch spac and th food sourcs. In th ABC algorithm, th colony of bs is dividd into mployd bs, onlookrs, and scouts. Employd bs ar rsponsibl for xploiting availabl food sourcs (solutions) and gathring rquird information. Ths bs also shar information with onlookrs, and ach onlookr slcts a food sourc nar th food sourc chosn by on mployd b. Whn th sourc is abandond, th mployd b bcoms a scout and starts to sarch for a nw sourc in th vicinity of th hiv. This abandonmnt happns whn th quality of th food sourc dos not improv for a prdtrmind numbr of itrations. Th ABC algorithm is itrativ, and starts by associating all mployd bs with randomly gnratd food sourcs (solutions). In vry itration, ach mployd b slcts a food sourc in th nighborhood of th currntly associatd food sourc using a nighborhood 14

16 oprator, and valuats its nctar amount (fitnss) aftrwards. If its nctar amount is bttr than that of th currntly associatd food sourc, thn th mployd b kps th nw food sourc and discards th old on; othrwis, th mployd b rtains th old food sourc. Whn all of th mployd bs hav finishd this procss, thy shar th nctar information for th food sourcs with th onlookrs. Each of th onlookrs thn slcts a food sourc according to a probability proportional to th nctar amount of that food sourc. In this study, w us th traditional roultt whl slction mthod (Haupt and Haupt, 2004). Clarly, with this schm, good food sourcs attract mor onlookrs than bad ons. Aftr all of th onlookrs hav chosn thir food sourcs, ach of thm slcts a food sourc in th nighborhood of thir chosn food sourcs (through nighborhood oprators) and computs its fitnss. Th bst food sourc among th particular food sourc of an mployd b and its nighboring food sourcs is th food sourc of th mployd b. If a solution rprsntd by a particular food sourc dos not improv for a prdtrmind numbr of itrations, thn th food sourc is abandond by its associatd mployd b and th b bcoms a scout. Th scout thn sarchs randomly for a nw food sourc. This is don by assigning a randomly gnratd food sourc (solution) to this scout. Aftr ach nw food sourc is dtrmind, anothr itration of th ABC algorithm bgins. Th whol procss is rpatd until th trmination condition is satisfid Ovrviw of th hybrid artificial b colony (ABC) algorithm Th xisting ABC algorithm cannot b usd dirctly to solv our problm bcaus our problm is bi-lvl and has many constraints. Hnc, a hybrid ABC algorithm is dvlopd. Th flow chart of th hybrid ABC algorithm is givn in Figur 2, which dpicts th main algorithm (ABC algorithm) and th sub-algorithm (th proposd frquncy dtrmination algorithm). ABC algorithm Th stps of th ABC algorithm can b dscribd as follows. 1. Initializ th paramtrs, including th colony siz N c, th numbr of mployd bs N, th numbr of onlookrs N o, th numbr of scouts N s, and th prdtrmind numbr of itrations limit; st I, which is th countr of itrations, to b qual to zro; st th maximum numbr of itrations, I max = 500; 15

17 2. Prform th initialization phas of mployd bs: Gnrat an initial solution for ach mployd b and st th limit countr for ach solution to b zro; 3. Incras th numbr of itrations by 1, i.., I = I + 1; 4. Prform th mployd b phas: Conduct a nighborhood sarch for ach solution found by an mployd b. Evaluat th fitnss of ach nighbor solution. Rplac th solution by its nighbor solution found by th sarch and st its limit countr to zro, if th lattr is bttr. Othrwis, kp th solution of th mployd b, and incras th limit countr by 1; 5. Prform th onlookr b phas: Prform th roultt whl slction to dtrmin which solution obtaind by an mployd b is slctd by an onlookr. Thn, conduct a nighborhood sarch for ach solution slctd by an onlookr. Evaluat th fitnss of ach nighbor solution. Rplac th solution by its nighbor solution, if th lattr is bttr. Othrwis, kp th solution of th mployd b, and incras its limit countr by 1; 6. Prform th scout b phas: Rplac ach solution that fails to improv within limit succssiv itrations by a nw solution gnratd randomly; 7. Chck th stopping critrion: If I < Imax, rturn to stp 3; 8. Trminat and output th bst solution. 16

18 ABC algorithm 1. Initializ paramtrs; I = 0; 2. Initializ mployd bs 3. I = I Prform mployd b phas 5. Prform onlookr b phas 6. Prform scout b phas Ys Frquncy dtrmination Input solution g at itration I i. Initializ frquncis ii. Is solution g infasibl I or LBg min z1, g? No iii. Solv th transit assignmnt problm iv. Trminat? Ys No v. Dtrmin a dscnt dirction No 7. I < I max Ys 8. Stop and output th bst solution vii. Rturn frquncy stting vi. Find th stp siz and updat th frquncy for ach rout (a) ABC algorithm Figur 2 Flow chart of th hybrid ABC algorithm (b) Frquncy dtrmination algorithm For th ABC algorithm in our proposd mthod, th initialization phas gnrats a population of initial solutions by th mployd bs. Aftrwards, ach mployd b is associatd with on randomly gnratd solution. Th mployd and onlookr b phass ar quit similar, as shown in Figur 3. Th only diffrnc lis in th rul for slcting a candidat food sourc for a nighborhood sarch. In th mployd b phas, ach mployd b slcts its associatd solution for a nighborhood sarch. In th onlookr b phas, ach onlookr slcts a solution basd on th fitnss valu. Hnc, w xpct promising solution aras to b visitd and xplord mor frquntly. Both phass rquir th frquncy dtrmination algorithm to dtrmin th frquncy associatd 17

19 with ach rout and valuat th fitnss valu of ach solution. Thy both conduct a grdy slction aftr valuating th fitnss of th nighbor solution. If th nighbor solution is bttr than th food sourc, th lattr is rplacd by th formr and its limit countr is st to zro. Othrwis, th currnt solution is maintaind and th limit countr is incrasd by 1. Finally, if all of th mployd bs or onlookrs complt thir nighborhood sarchs, thn th mployd or onlookr b phas is trminatd. Employd b phas Onlookr b phas Slct on nighbor solution for ach mployd b Slct on nighbor solution basd on fitnss proportion for ach onlookr Gnrat a nighbor solution and rpair th solution if ncssary Dtrmin frquncis and valuat th fitnss of ach solution Rplac th solution by its nighbor, if th lattr is bttr Updat limit countrs Figur 3 Flow chart of th mployd b and onlookr b phass In th scout b phas, all of th food sourcs ar scannd and th sourc that fails to improv within limit succssiv itrations is abandond and rplacd by a nwly gnratd random solution. Frquncy dtrmination algorithm For ach solution (i.., rout structur) obtaind in th mployd or onlookr b phas of th ABC algorithm, th following procdur is usd to dtrmin th frquncy stting: i. Gnrat th initial frquncis; 18

20 ii. Dcid whthr to obtain an optimal frquncy for ach rout: If LB g (th lowr bound of givn solution g) is gratr than min z I 1, g (th minimum uppr-lvl objctiv valu until itration I) or th rout dsign is infasibl, thn rturn th initial frquncy of ach rout. Othrwis, procd to th nxt stp; iii. Solv th lowr-lvl transit assignmnt problm; iv. If th trmination critrion of th frquncy dtrmination algorithm is satisfid, thn stop and rturn th optimal frquncy stting. Othrwis, go to th nxt stp; v. Dtrmin th dscnt dirction of th lowr- and uppr-lvl objctiv valus with rspct to th frquncy; vi. Find th stp siz of th frquncy by solving a linar intgr program and updat th frquncy with th obtaind stp siz, thn go to stp iii Solution gnration and rpairing procdurs Solution rprsntation in th ABC algorithm To sarch all of th possibl rout structurs, th solution rprsntation in th ABC algorithm should b spcifically dsignd. Figur 4 illustrats th rprsntation schm usd in th ABC algorithm. On solution consists of 100 lmnts rprsnting 10 routs, with 10 lmnts for ach rout. For xampl, th first 10 lmnts rprsnt th first bus rout, which starts at nod 1, gos through nods 18, 15, 10, 12, and 7, and trminats at nod 25. Similarly, rout 10 starts at nod 16, gos through nod 11, and trminats at nod 27. Rout 1 Rout A total of 100 lmnts Figur 4 Solution rprsntation schm 19

21 Initialization procdurs In th ABC algorithm, nw solutions ar gnratd in th initialization and scout b phass, both of which adopt th sam procdurs to gnrat a random solution, as shown in Figur 5. To initializ th rout lmnts, th following procdurs ar carrid out squntially. For rout r, th first nod is dtrmind by randomly slcting from th availabl starting trminals in th study ara. Thn, th last nod is pickd from all of th availabl nding trminals ', th numbr of intrmdiat stops is gnratd, and a corrsponding numbr of nods is insrtd btwn th two trminals. Th probability of slcting an intrmdiat stop nod i is dtrmind basd on passngr dmand by p i = d ' i j ZU d ' j, whr rprsnts th probability of choosing nod i. If thr is mor than on stop in a zon, stop i in that zon is randomly pickd. Th last stp is to st th rst of th lmnts, if any, to zro. p i Gnrat a solution Initializ routs lmnts Rpair a solution Chck zon covring and insrt uncovrd nods Rpair th solution if ncssary Optimiz stop squnc Dtrmin th lowr bound Dlt and insrt intrmdiat stops Figur 5 Solution gnration and rpairing procdurs Rpairing procdurs Th solution gnratd maks it difficult to avoid infasibility du to th proposd random oprations. Although w can add a pnalty to th fitnss valu of an infasibl solution and lav th algorithm itslf to volv, according to our prliminary xprimnts, th solution quality in trms of th numbr of fasibl solutions and th objctiv valu in th final itration is lowr than that obtaind by th algorithm with th proposd rout rpairing procdurs. Thrfor, w propos th rout rpairing procdurs to provid bttr (initial) solutions. Th procdurs includ chcking zon covring, stop squnc optimization, and 20

22 dlting and insrting intrmdiat stops. Chcking zon covring Th zon covring procdur is dsignd to nsur that vry dmand zon is visitd by at last on rout. Bcaus th total numbr of lmnts in a solution (which quals th maximum numbr of stops multiplid by th maximum numbr of routs) is gratr than th numbr of zons in th ntwork, thr must b som zons that ar visitd by mor than on rout. Howvr, thr is no guarant that all zons ar srvd or covrd in th initialization procdur. If cntroid m is not srvd, thn nod i, which is on of th candidat stops conncting to cntroid m, is insrtd into th slctd rout with th numbr of stops lss than th maximum allowabl numbr of stops and th last travl tim incrmnt aftr insrting nod i. If no rout can srv this cntroid du to constraint (11) on th maximum numbr of stops, a zon srvd by at last two transit lins is randomly slctd and on of th stops in th zon passing by th lins is rplacd by nod i. Ths two stps guarant that th zon-covring constraint (13) is satisfid. Stop squnc optimization For vry gnratd ABC solution, a dscnt sarch huristic is usd to improv th squnc of stops on ach rout. Th purpos of this squnc-improving procss is to minimiz th trip tim of ach rout, as it dos not dpnd on frquncis and is rlativly asy to implmnt. Th outlin of th huristic is as follows: 21 For ach rout in th ABC solution St i' = 1 Whil i' th numbr of intrmdiat stops 1 j' = i' +1 Whil j' th numbr of intrmdiat stops, do Exchang th i' th and j'th stops Evaluat th trip tim of th rout If th trip tim is rducd, thn st j' = numbr of intrmdiat stops + 1, i' = 0 ls undo th xchang and j' = j' + 1

23 ndif ndwhil i' = i' +1 ndwhil Nxt rout in th ABC solution Dlting and insrting intrmdiat stops Th stop squnc optimization procdur ssntially rarrangs th squnc of intrmdiat stops to form th shortst path. Nvrthlss, som routs may still violat th maximum trip tim constraint (12). Thrfor, a stop-rmoval opration is conductd to liminat nods whil nsuring th solution to satisfy th zon covring constraint. Various critria can b usd in slcting which nods to dlt, such as trip tim rduction and th changs in total flow of th dirct srvics involvd aftr prforming th nod rmoval. Diffrnt critria hav diffrnt ffcts on th objctiv valu and th algorithm prformanc. W propos th following avrag-dirct-dmand i' ψ r to approximat th chang in th uppr-lvl objctiv valu that rsults from rmoving nod i from rout r, which conncts trminal ' dirctly: ψ d d + 1 ' i' i' i r r = i' d p p r for r = 1 to R max, i Z, ' V, U whr i' δ p is a binary indicator variabl that is qual to 1 if rout p passs both nods i and '. i' δ p calculats th numbr of transit lins that provid a dirct srvic btwn nod i p r and trminal ' aftr rmoving nod i from rout r. Adding 1 to that numbr allows us to considr th cas whn rout r hading to trminal ' originally passs nod i. Hnc, th dnominator givs th numbr of transit lins that provid a dirct srvic btwn nod i and trminal ' bfor rmoving nod i from rout r. Th dmand ' d i is obtaind from th lowr-lvl problm, and is th total flow on th boarding arcs nding at th transit stop corrsponding nod i in th uppr-lvl ntwork and hading to th transit stop corrsponding to trminal ' H. Ovrall, this avrag-dirct-dmand intnds to captur th incras in th numbr of passngr transfrs du to dlting nod i. This avrag approximats th flow of ach dirct srvic and is dtrmind by vnly splitting th dmand btwn nod i and 22

24 trminal ' to all of th routs providing dirct srvics for that pair of nods. This valu can b intrprtd as th avrag incrmnt in th numbr of transfr passngrs whn nod i is dltd from rout r. Thrfor, a nod with a smallr ratio is prfrrd for rmoval, bcaus a smallr ratio indicats a lowr avrag incrmnt in th numbr of passngrs who nd to mak a transfr. To compnsat for th ngativ ffcts of dlting nods, including rducing srvic covrag and incrasing th numbr of passngrs who mak a transfr, a rvrs opration calld nod insrtion is subsquntly conductd to insrt as many nods as possibl whil nsuring that th rsultant solution satisfis th maximum trip tim constraint. Th nod chosn for insrtion is also basd on th proposd avrag-dirct-dmand, and a largr valu is prfrrd Lowr bound dtrmination and fitnss valuation Fitnss is usd to rflct a solution s quality and slct candidat solutions for a nighborhood sarch. Although th rciprocal of th uppr-lvl objctiv valu can b usd as a fitnss masur for th proposd algorithm, it is cumbrsom to calculat th uppr-lvl objctiv for ach solution bcaus th corrsponding optimal frquncy must b found by th proposd frquncy stting mthod, which involvs solving th lowr-lvl problm many tims. Thus, a lowr bound is calculatd to dtrmin th minimum numbr of passngr transfrs for ach solution, and thn usd to rplac th uppr-lvl objctiv valu in th fitnss function. Such a bound can b obtaind much mor quickly than th rciprocal of th uppr-lvl objctiv valu. Givn a rout dsign solution, th lowr bound provids th minimum numbr of passngr transfrs, which is an optimistic stimation of th uppr-lvl objctiv function valu. Th calculation of th lowr bound is basd on th assumption that ach transit rout ' has unlimitd capacity. Undr this condition, th passngr dmand d i, i ZU, ' V, can b mt without nding to mak a transfr if thr is any rout conncting nods i and '. Bcaus summing up all of th srvd dmand provids th maximum total passngr dmand without making a transfr, th minimum numbr of passngr transfrs, or th lowr bound, can b obtaind by calculating th diffrnc btwn th total passngr dmand ' di and th sum of all passngrs not making a transfr undr th assumption statd i ZU \ V' V 23

25 ' abov, ( 1 NRi ) i ZU \ V' V d ' i, whr ' NR i quals 1 if thr is no rout conncting i and ', and 0 othrwis. If th subscript g is usd to dnot a rout dsign solution, thn th lowr bound of solution g, LB g, can b mathmatically xprssd as whr ' ' ' 槫槫 ( 1 ), (23) LB = d - - NR d NR g i i i i恄 ZU\ V' V i恄 ZU\ V' V ' i Rmax r= 1 ( 1 RTi ' r ) = for i Z \ V U, ' V, (24) RT = X + X RT for i Z \ V U, w Z \ V U, w i, r = 1 to R max, (25) iwr iwr ijr jwr j ZU \ V, j i, j w RT ijr = l if rout r passs through nod i and nod j, and 0 othrwis. With th lowr bound of rout dsign solution g, calculatd via 1 Fg =. LB + P g g P g is a pnalty trm for solution g and is givn by LB g, th fitnss of solution g, F g, is (26) R R Pg = a Trg T + Vrg W r= 1 r= 1 max max max (, max,0) β max,,0, (27) whr V rg, is th flt siz for rout r in solution g; T rg, is th trip tim on rout r in solution g, and α and β ar th pnalty paramtrs rlatd to th maximum trip tim constraint and th flt siz constraint, rspctivly. Th pnalty mthod dals with infasibl solutions that cannot b rpaird by any of th rpairing oprators. Th first trm in (27) pnalizs th violation of constraint (12) whil th scond trm pnalizs th violation of constraint (9). Infasibl solutions ar kpt for nighborhood sarchs bcaus it is possibl for th global minima to b locatd clos to infasibl solutions. Nvrthlss, by varying th pnalty paramtr valus, it is asy to adjust th probability of sarching an infasibl solution rgion. Whn th pnalty valu is larg, th probability is small and vic vrsa Frquncy dtrmination procdur Th following subsctions dscrib th dtails of ach stp in th frquncy stting procdur 24

26 dpictd in Figur 2. Stp i: Frquncy initialization This stp is conductd to obtain th initial frquncy for ach rout. Th initial frquncy for rout r is calculatd from f rg, V rg, =, (28) 2T rg, whr frg, is th frquncy of rout r of solution g. Givn trip tim T, rg, th total flt should b allocatd carfully to mt th minimum frquncy and dmand consrvation rquirmnts. Thus, w propos th following procdur for allocating buss to dtrmin th initial frquncy of ach rout. 1. Assign buss according to minimum frquncy constraint (10). Givn th trip tim on a rout, th minimum numbr of buss rquird to mt th minimum frquncy constraint can b dtrmind by quation (28); 2. Assign buss according to dmand rquirmnt constraint (14). This procdur nsurs that th srvic capacity providd for ach dstination is not lss than th dmand nding at dstination, undr th assumption that th total srvic capacity for th study ara is not lss than th total dmand. In th bginning, all of th transit lins with th sam nding trminals ar groupd, and thn two frquncy valus ar calculatd and compard for ach group. On is th assignd group frquncy, which is th sum of th frquncis obtaind in stp 1. Th othr is th rquird frquncy, which is th minimum frquncy rquird to mt th total dmand for ach dstination group calculatd by m Gs d m k cap, G. For ach group, if th rquird frquncy is largr than th assignd frquncy, thn th group frquncy is insufficint and th diffrnc btwn th rquird and assignd frquncis is addd to th frquncy of th rout with th last trip tim. Th rout with th last trip tim is chosn bcaus whn on mor bus is assignd to that rout, it producs th highst incras in th frquncy and th lin capacity compard with othr routs. 3. Round up th flt siz and rcalculat frquncis. Aftr th forgoing two stps, th numbr of buss allocatd to ach rout is calculatd and roundd up to th narst intgr. Th frquncy of ach rout is thn rcalculatd, which usually nds with a slightly highr valu than th prvious rsult. 25 d

27 This procdur handls th frquncy-rlatd constraints by dtrmining th flt siz of ach rout. Aftrwards, if th sum of th flt siz of ach rout is lss than or qual to th total flt siz dfind by constraint (9), thn th rout structur simultanously satisfis constraints (9), (10), and (14); othrwis, th rout structur is infasibl and th fitnss of th infasibl solution is pnalizd. If thr ar rsidual buss that hav not bn allocatd to any rout, thy ar addd to th rout with th last trip tim bcaus at global optimality all buss must b usd. Stp ii: Lowr bound scrning Aftr stp i, w can idntify whthr a rout structur is fasibl. For infasibl solutions, it is not ncssary to sarch for optimal frquncy. For fasibl solutions, only potntially good solutions procd to obtain optimal frquncy. Candidat solutions ar idntifid by comparing th lowr bound with th currnt bst objctiv valu. If th lowr bound of a nw solution is largr than th currnt bst objctiv valu found by th hybrid ABC algorithm, thn it is impossibl to dtrmin th uppr-lvl objctiv valu of a nw solution that is smallr than th currnt bst objctiv valu by adjusting its frquncy and not changing th rout dsign. In this cas, th rout solution cannot b globally optimal and it is rdundant to carry out th frquncy stting procdur. Howvr, if th lowr bound is lss than th currnt bst objctiv valu, thn obtaining a bttr objctiv valu by sarching optimal frquncy sttings is possibl, and hnc th solution is potntially good. Stp iii: Solving th transit assignmnt problm With an initial frquncy and a fasibl rout structur, th lowr-lvl transit assignmnt problm is solvd by th Simplx mthod. Aftrwards, both th primal and dual solutions ar rcordd. Th primal solution indicats th numbr of transfr passngrs and th dual solution is usd, if ncssary, to dtrmin th dscnt dirction and stp siz for updating frquncy in latr stps. Stp iv: Trmination critria chcking Th following stopping critria ar usd: k Critrion (1) z1, g LBg ε1 and Critrion (2) z + z ε, k 1 k 1, g 1, g 2 26

28 k whr ε1 and ε 2 ar prdfind maximum accptabl rrors and z 1, g is th uppr-lvl objctiv valu of solution g aftr th k th itration. Both critria ar drivd basd on th k dfinition of th lowr bound, which stats that th uppr-lvl objctiv valu z 1, g cannot b rducd to a valu that is smallr than th lowr bound for rout structur g. Critrion (1) is usd as a stopping critrion whn th frquncy is optimal or narly k optimal. If z 1, g and LB g ar qual, thn th frquncy is optimal. If th diffrnc is small, thn th frquncy is probably optimal, and is at last narly optimal. Critrion (2) is usd whn two succssiv objctiv valus ar clos nough, which implis that th two succssiv solutions ar probably clos nough, and th latst solution is probably optimal. Stp v: Dtrmination of th dscnt dirction In this stp, th dscnt dirction of th uppr-lvl problm with rspct to frquncy is dtrmind. This dscnt dirction is also th dscnt dirction for th lowr-lvl problm undr th condition that th pnalty paramtr for transfrs (i.., M) is larg nough. Hnc, w can rly on th dscnt dirction of th lowr-lvl problm, which is drivd as follows. Dscnt dirction of th lowr-lvl problm For th as of prsntation, w rwrit th lowr-lvl formulation as a function of th frquncis f in th following vctor form and omit th solution subscript g. ( v( f) ) ( w( f) ) min : z = h + h v,w ( ( )) ( ) ( ) ( ) subjct to: g1 v f k f m w f 0, (30) v( f ) ( ) ( ( )) (29) g2 v f d= 0, (31) ( ( )) ( ) g3 vf cf 0, (32) ( ) v f 0, (33) ( ) w f 0, (34) whr and w f rprsnt th vctors v a and ω i, rspctivly, which ar ( v( f )) a a 2 ( ( )) functions of frquncis. h cv and h w f ω. For constraints 27 1 = a = i i

29 (30) to (32), 1 = v a, = f a ; m w f = ω i ; g 2 ( v( f) ) ( ) = ; d d i ; g3 v( f ) va and c( f ) fk a cap =. Th dimnsions of ths matrics ar not fixd, but vary with th solutions of th uppr-lvl problm. Th dscnt dirction is drivd basd on th ncssary Karush Kuhn Tuckr (KKT) conditions. At global optimality, th following conditions hold: ( ( )) ( ) ( ( )) ( ( )) w ( ) ( ) ( ( )) ( ) h * * v 1 v f vg1 v f * * vg2 v f vg3 v f T T T + π + φ + μ = 0 * * h 2 ( ) ( ) w w f k f m w f 0 0 * * whr v ( f ) and w ( ) π, = π ia = v a + a Ai a Ai = ϕ i ( ) * ( ( )) ( ) * * ( ( )) ( ) ( ) 3 (35) π g 1 v f k f m w f = 0, (36) μ g v f c f * v w π 0, (37) μ ( f) ( f) * 0, (38) f stand for th optimal solutions of th lowr-lvl problm and φ, and μ = [ ] ar, rspctivly, th optimal multiplirs for quations µ a ( ) * * (30), (31), and (32). Th sufficint conditions of global optimality at ( ), ( ) v f w f ar also satisfid bcaus th lowr-lvl problm is a linar programming problm with a convx solution st (i.., a convx problm). To obtain th dscnt dirction of th objctiv function, w form th Lagrang function L, * * diffrntiat th Lagrang function with rspct to f, and substitut ( ( ), ( ),,, ) th drivativ to gt g ( v( f )) k( f ) [ ] ( ( )) v a = v f w f π φ μ to 28

30 * * * * L v ( ( )) w f = vh1 v f + wh2( w ( f) ) f f * * T v * k( f) * w * + π vg1( v ( f) ) m( w ( f) ) k( f) wm w f f f f * * T v * T v * cf + φ vg2( v ( f) ) + μ vg ( v ( f) ) f f f Rarranging quation (39), w hav ( ) ( ( )) ( ) 3. * T * T * T * { h ( v ( f) ) π g ( v ( f) ) φ g ( v ( f) ) μ g ( v ( f) )} * L v f = v + v + v + v f * w * T * { wh2 ( w ( f) ) + π ( k( f) ) wm( w ( f) )} f T kf ( ) * T cf ( ) π m( w ( f) ) μ. f f (40) Substituting quation (35) into (40), w obtain ( ) T kf * f L = π m( w ( f) ) μ f T ( ) cf f (39). (41) Equation (41) provids th stpst ascnt dirction of th Lagrang function at th currnt v f w f π φ μ. Accordingly, f L is th stpst dscnt dirction at that * * solution ( ( ), ( ),,, ) point. Bcaus L= z2 implis that v f w f π φ μ, fl= f z2 at that point. This * * f f at ( ( ), ( ),,, ) f L provids a dscnt dirction of th lowr-lvl objctiv function with * * rspct to f at ( ( ), ( ),,, ) v f w f π φ μ. Dscnt dirction of th uppr-lvl problm For th proposd formulation, th lowr-lvl objctiv function can b dcomposd into two positiv and linar trms, whr th scond trm has th cofficint M. That is, T z2 M vt R t T G d = Cx +. (42) vt is th total flow on all th transfr links, and is idntical to th objctiv function R t T G d of th uppr-lvl problm; x is th vctor for th rst of th dcision variabls of th lowr-lvl problm; and C is th vctor of th cofficints of x. 29

31 Proposition 1: Whn M is gratr than th largst lmnt of C, th gradint - f L at th currnt frquncy solution is also a dscnt dirction of th uppr-lvl objctiv function. Proof: Without loss of gnrality, ach of th currnt frquncis is lss than infinity. Morovr, th waiting tim for ach transit lin can only tnd to zro whn th frquncy tnds to infinity. Thrfor, on can always rduc th total waiting tim and, hnc, th objctiv valu of th lowr-lvl problm by incrasing th frquncy of at last on transit lin. Thrfor, ach of th lmnts of - f L is always ngativ for th currnt solution, maning that th objctiv valu of th lowr-lvl problm for th currnt frquncy solution can always b rducd by incrasing th frquncy of at last on transit lin. This implis that w can always find a dscnt dirction, including th stpst dscnt dirction, for th currnt frquncy solution. Bcaus w only considr dscnt dirctions, w do not nd to considr th constraints for th lowr-lvl problm. Without considring th constraints of th lowr-lvl problm, th objctiv valu of th lowr-lvl problm can b rducd aftr th currnt solution movs slightly along th stpst dscnt dirction. As M is gratr than th largst lmnt of C, it is mor fficint to rduc th valu of th scond trm than that of th first trm along th dscnt dirction. Hnc, th valu of th scond trm must b rducd along this dirction. Thrfor, f L is a dscnt dirction to th uppr-lvl problm. This complts th proof. Proposition 1 implis that it is possibl to rduc th valu of th uppr-lvl objctiv function by rducing th valu of th objctiv function of th lowr-lvl problm. Stp vi: Stp siz dtrmination and frquncy updating In addition to th dscnt dirction givn by (41), a stp siz must b dtrmind to updat f. Hnc, w invstigat th individual componnt of th gradint of th Lagrang function to dtrmin a good mathmatical proprty that simplifis th procdur for dtrmining th stp siz. Th gradint of th Lagrang function is f fk L * a * * a cap = pia ω i µ a fr i a fr a fr, (43) 30

32 whr L f r rprsnts th gradint with rspct to th frquncy of lin r and * rprsnts th solution at optimality. Th first trm on th right sid is dfind by th dual solution of th rlaxd nod-flow distribution constraint (30). Th scond trm is dfind by th dual solution of th capacity constraint (32). Not that if link a is a boarding arc of transit lin r, thn f f a r fa = 1; othrwis = 0. Similarly, f r fk a cap = f r k cap, if link a is a travl arc of transit lin r and fk a cap = f r 0 othrwis. Hnc, L f r is a function that is only influncd by th frquncy of transit lin r. Such sparabl proprty prmits us to adjust th frquncy of ach transit lin or to dtrmin th stp siz fr for ach lin r, sparatly. Aftr obtaining th dscnt dirction for ach rout, th following intgr linar program is proposd to dtrmin th stp siz: subjct to whr [ ] = V r min : z V L 3 = r r fr f (44) Vr fr =, for r = 1 to Rmax, (45) 2T r f + f f, for r = 1 to R, (46) r r min ( fr + fr) kcap dr dm, Gd, (47) r m Gs max Vr = 0, (48) r V and Vr is th chang in th flt siz of rout r. fr is th stp siz of th frquncy of rout r. Vr and fr ar rlatd through quation (45), which is drivd from (28). Th objctiv of th intgr program is to minimiz th incras in th valu of th Lagrang function by adjusting th flt siz of ach rout, which is quivalnt to minimizing th objctiv valu of th lowr-lvl problm by dtrmining th optimal flt allocation. Constraints (46) and (47) ar drivd from th uppr-lvl constraints (10) and (14). Constraint (48) rprsnts th flt siz consrvation constraint. Bcaus th flt siz V r is an intgr dcision variabl, th problm bcoms a linar intgr programming 31

33 problm. Although it is possibl to adopt [ f r ] as a vctor of continuous dcision variabls instad of using V as a vctor of intgr dcision variabls, th problm of transforming optimal continuous solutions into optimal intgral solutions is vn mor complx and non-trivial. Hnc, w adopt V as a vctor of dcision variabls. Proposition 2: z 3 is always a non-positiv numbr at optimality. Proof: It is asy to obsrv that th solution at th origin, V = 0, is fasibl, as it must satisfy all of th constraints of th intgr program. Morovr, whn V = 0, z 3 is qual to 0. As th intgr programming problm is a minimization typ, th objctiv valu of an optimal solution must not b gratr than that of any fasibl solution, including V = 0. Hnc, z 3 is always non-positiv at optimality. This complts th proof. Th implication of proposition 2 is that th optimal allocation dtrmind by th intgr program must rduc th valu of th Lagrang function and hnc th valu of th uppr objctiv function of th uppr-lvl problm, if z 3 is ngativ at optimality. Aftr solving th intgr program, th itrativ procdur rturns to stp iii with th updatd frquncy obtaind by first obtaining k D f r from k D V r using (45) and f = f +D f for r= 1 to Rmax. (49) k k- 1 k r r r Hr, an additional suprscript k is introducd, rprsnting th k th itration of th frquncy found in th itrativ frquncy dtrmination procdur. Violation of assumptions To us th dscnt dirction information drivd from (43), w assum: that (i) th optimal basis rmains optimal (ii) M is gratr than th largst cofficint in C. Th first assumption rquirs that ach chang in frquncy is within a crtain allowabl rang; othrwis swapping btwn basic and non-basic variabls occurs. Howvr, in th intgr program, th xact allowabl rang is not usd. Instad, w us th fasibl rgion dfind by (10) and (14) in th uppr-lvl problm to approximat it. As a rsult, th approximation crats a problm whn th fasibl rgion of th uppr-lvl problm dos not li within th allowabl rang. Consquntly, an inappropriat stp siz is found and th nw 32

34 frquncy falls out of th allowabl rang. Th objctiv valu may subsquntly incras, such that it taks xtra itrations to rduc th objctiv valu of th linar intgr program. Thr ar two mthods for addrssing this issu. First, an additional constraint is addd to limit th maximum chang for ach of k D V r. Howvr, if th constraint is too tight, it lads to mor itrations to obtain an optimal allocation. Thrfor, a balanc dcision should b carfully mad. Trial and rror tsting is mor likly to provid hints on whr to st th maximum chang for k D V r. Scond, w us th rsults of our snsitivity analysis in th linar programming to driv additional constraints, which nsur that th basis rmains unchangd. Without loss of gnrality, w rwrit th lowr-lvl problm in th following compact form: min z = cx (50) 2 Ax = b, (51) x 0, (52) whr A is a matrix, b and c ar column vctors, and x is a vctor of dcision variabls in which th lmnts involv all th lmnts of th auxiliary variabls v and w. According to th fundamntal rsults of th snsitivity analysis, at optimality, all cofficints in row 0 of th final tablau ar non-positiv (for th minimization problm) and all of th right sids ar non-ngativ; that is, for th cofficints of th variabls v and w in row 0 of th final tablau, w hav: 1 cb B A c 0, (53) whr 1 B is th invrs of B and B is th squar matrix, which contains th columns from [ A I ] that corrspond to th st of basic variabls (in ordr) and c B is th vctor of lmnts in c that corrsponds to th basis variabl. For th cofficints of auxiliary variabls in row 0 of th final tablau, w hav 1 cb B 0. (54) For th right sids, w hav 1 B b 0. (55) Givn that only th cofficints in (18) and th right sid in (20) involv lin frquncy, only A and b involv frquncy in som of th lmnts, and thus w only nd to 33

35 considr conditions (53) and (55). Aftr rvising th lmnts involving lin frquncy by adding fr to thm, w hav a rvisd matrix A ' and a rvisd column vctor b ', which ar linar functions of [ f r ]. W thn hav th following two sts of additional linar constraints for th intgr program: 1 cb B A c 0 and (56) 1 B b 0. (57) Not that both B 1 and c B ar known at optimality and c is obtaind from th lowr-lvl problm. Th scond assumption is that M is gratr than th largst cofficint in C. Whn this assumption is mt, - f L must b a dscnt dirction of th uppr-lvl objctiv function. Othrwis, thr is no guarant that - f L is also a dscnt dirction of th uppr-lvl objctiv function. Whn - f L is not a dscnt dirction of th uppr-lvl objctiv function, w may nd to switch to using th frquncy stting huristic proposd by Szto and Wu (2010) to dtrmin th frquncy. This huristic is tim-consuming, as it rlis on solving th lowr-lvl problm many tims to dtrmin th dscnt dirction of ach lin and th optimal frquncy. Th dtails of th frquncy stting huristic ar not rportd hr but radrs can rfr to Szto and Wu (2010) for th dtails. Altrnativly, th mthod proposd in this papr can b usd as a huristic to dtrmin frquncis Nighbor solution gnration Du to th complxity of th problm, spcific nighborhood sarch oprators ar dvlopd to gnrat nighbor solutions. As ach rout compriss thr parts starting trminal, intrmdiat stops, and nding trminal ths nighborhood sarch oprators intnd to mutat all of ths parts. Four oprators ar proposd to achiv this purpos: a) starting trminal swap, b) nding trminal swap, c) intrmdiat stop swap, and d) intrmdiat stop insrtion (Figur 6). Th oprations ar conductd randomly in th nighborhood sarch phas. 34

36 Bfor Aftr Bfor Aftr Bfor Aftr Bfor Aftr (a) Starting Trminal Swap (b) Ending Trminal Swap (c) Intrmdiat Stop Swap (d) Intrmdiat Stop Insrtion Figur 6 Nighborhood sarch oprations Th starting and nding trminal swap oprators ar trivial. Thy randomly slct two routs and xchang th two starting and nding trminals, rspctivly. Bfor swapping thir starting and nding trminals, th nding trminals of two candidat routs ar chckd to nsur that thy ar diffrnt to gnrat diffrnt solutions. For th intrmdiat stop swap and insrtion oprations, th nods slctd to prturb ar basd on th proposd avrag-dirct-dmand valu. For instanc, th nod to b transfrrd is th on that inducs th minimum avrag-dirct-dmand incrmnt. Onc a candidat nod is found, a scanning procdur is carrid out to chck whthr th slctd nod is in th rciving rout. If so, th nxt bst nod is slctd. Not that an intrmdiat stop dltion oprator is not usd hr bcaus it dos not gnrat a bttr solution by itslf. It is only usd whn th rout is too long; that is, w rpair th rout structur bcaus it violats th travl tim constraint or th constraint on th numbr of stops. 4. Exprimnts To invstigat som of th problm s proprtis and th prformanc of th proposd solution mthod, a small ntwork was cratd and tstd, aftr which th proposd mthod was applid to solv a ralistic bus ntwork problm in Tin Shui Wai (TSW), Hong Kong. Th prformanc of th proposd algorithm was dmonstratd by comparing it with that of a GA on th Winnipg ntwork. For th small and TSW ntworks, th cntroid and stop wr 35

37 assumd to b idntical. Th paramtrs, unlss spcifid, wr st as follows: M = 2000 ; N c = 100 ; N = 50 ; N o = 50 ; limit = 50; th maximum numbr of itrations was 500; a 9 = b = 10 ; and ε1 ε = =. Th proposd ABC mthod was codd in C++ and complid with Visual Studio 2008, and th lowr-lvl and linar intgr programs wr solvd by CPLEX For all of th tsts, 20 runs with diffrnt initial sds wr conductd and th avrag prformancs wr rportd Small ntwork xprimnts Figur 7 shows th small ntwork that w cratd. Th starting bus trminals wr nods 1 and 2. Nods 7-11 wr nding trminals. Th study ara consistd of nods 1-6. Each zon in this xampl had only on stop. Hnc, th passngr dmand btwn stops quald that btwn th corrsponding cntroids. Th passngr dmand is givn in Appndix I. Th maximum trip tim was st to 23 minuts, th total flt siz was 60, th maximum numbr of intrmdiat stops was 3, and th maximum numbr of routs was T Figur 7 Small ntwork Bnchmark rsults with multipl solutions To calculat th bnchmark rsult, a brut forc mthod (that numratd all possibl solutions) was applid. Th frquncy dtrmination procdur and th lowr bound scrning mthod wr incorporatd to spd up th brut forc mthod. Th optimal objctiv valu obtaind was 411. From th proposd ABC algorithm, th avrag objctiv valu of ach run and th lowst objctiv valu both quald 411, implying that all of th runs succssfully found th optimal objctiv valu. Tabl 1 provids th two optimal solutions, which possss diffrnt rout structurs and 36

38 hadways that ar th rciprocal of th corrsponding frquncis. Th two structurs ar quit similar. Th main diffrncs ar th frquncy stting and th stop squnc of th fourth rout. From th oprator s prspctiv thy may b significantly diffrnt in trms of othr critria such as ful consumption, missions, and opration cost. For illustrativ purposs, th ful cost was stimatd by multiplying th frquncy by th corrsponding trip tim. Th ful cost was HK$ and HK$621.7 pr hour for solutions 1 and 2, rspctivly. If th first dsign was chosn, a 3% gratr ful cost was spnt than if th scond dsign was chosn, implying that th oprator had to slct th dsign wisly. From th passngrs prspctiv, diffrnt frquncis and stop squncs indicatd diffrnt waiting tims and opportunitis to find a sat. Ths two factors also affct passngrs rout choics in rality, and can b considrd in slcting on out of all optimal solutions. Although th proposd ABC algorithm only kpt th bst solution ovr itrations, it was possibl to crat a solution pool that containd all of th optimal solutions sarchd. Hnc, it was not difficult to slct an optimal solution that gav th bst prformanc in th othr masurs. Tabl 1 Optimal solutions of th small ntwork Optimal solution 1 Optimal solution 2 Stop squnc Trip tim Hadway Trip tim Hadway Stop squnc (min) (min) (min) (min) 1, 3, 2, T, , 3, 2, T, , 3, 2, T, , 3, 2, T, , 3, 2, T, , 3, 2, T, , 5, 6, T, , 6, 5, T, , 4, T, , 4, T, Effctivnss of th lowr bound scrning mthod and hybrid ABC algorithm To tst th ffctivnss of th lowr bound scrning mthod, an ABC vrsion that did not us th scrning mthod was dvlopd. Th othr procdurs wr idntical, with th xcption of th lowr bound scrning mthod. For both vrsions, th infasibl solutions wr idntifid and wr not usd to dtrmin optimal frquncy. Although th lowr bound scrning mthod was not usd, th lowr bounds wr still calculatd and adoptd in th fitnss function. Th computation tims ar shown in Tabl 2. By comparing both mthods, w found that th lowr bound scrning mthod rducd th computation tim significantly. Th computational advantag may b du to th following rasons. On is that for th vrsion 37

39 with th lowr bound scrning mthod, only potntially good solutions wr rquird to dtrmin optimal frquncy, whras for th vrsion without, all of th fasibl solutions had to carry out th itrativ optimal frquncy dtrmination procdur. Accordingly, th total numbr of solutions rquird to dtrmin th optimal frquncis was rducd by th lowr bound scrning mthod. Th othr rason is th trmination critria of th dscnt frquncy sarch. For th vrsion with th lowr bound scrning mthod, th trmination critria rlid on both critria (1) and (2), whras for th vrsion without th mthod, th trmination critrion was only dfind by critrion (2), which was th diffrnc btwn th uppr-lvl objctiv valus of two succssiv itrations. Thus, th numbr of itrations rquird to dtrmin optimal frquncy was also rducd by th lowr bound scrning mthod. Th computational advantag is likly to b vn mor significant in larg ntworks with mor fasibl solutions and intgr variabls. Tabl 2 also shows that th hybrid ABC algorithm with th lowr bound mthod obtaind optimal solutions much mor quickly than th brut forc mthod (which was xact), illustrating th computational fficincy of th proposd algorithm. Tabl 2 Comparison of computation tim With lowr bound scrning Avrag computation tim (sconds) Brut forc mthod Hybrid ABC algorithm No Ys No Ys Effcts of dsign paramtrs Figurs 8-11 dmonstrat th ffcts of various dsign paramtrs on minimizing th numbr of passngr transfrs, including th minimum frquncy f min, th maximum numbr of intrmdiat stops S max, th maximum flt siz W, and th maximum numbr of routs R max. Without furthr spcification, th dfault paramtrs wr st as f min = 4.8 buss/h, W = 60, S max = 3, R max = 5, and T max = 26 minuts. Effct of minimum frquncy Minimum frquncy was usd to maintain a crtain lvl of srvic with rspct to waiting 38

40 tim. A highr frquncy mant a highr capacity and a shortr waiting tim. Howvr, undr th flt siz constraint, incrasing th minimum frquncy may hav rsultd in th dtrioration of anothr typ of srvic lvl, such as th total numbr of passngr transfrs, as illustratd in this xampl. Figur 8 dpicts th ffct of th minimum frquncy stting, showing that a tightr minimum frquncy constraint rsults in a highr numbr of passngr transfrs. Th frquncy rquirmnt was satisfid by cutting th trip tim and rducing th numbr of stops visitd, bcaus th total flt siz must b fixd. Thus, som passngrs rcivd th bnfit of a rducd waiting tim whil othrs bor an additional transfr cost. This finding raiss an intrsting quity rsarch dirction for futur rsarch. In th xtrm cas, whn th minimum frquncy was gratr than 5.4 buss/h, thr was no fasibl solution to satisfy th flt siz constraint. Numbr of Passngr Transfrs No fasibl solutions whn Minimum Frquncy (buss/h) Figur 8 Effct of minimum frquncy Effct of th maximum numbr of intrmdiat stops Mor intrmdiat stops may hav rducd th numbr of transfrs, at th cost of incrasing th rout travl tim and rducing th frquncy. Figur 9 shows that th total numbr of transfrs continud to dcras from 661 to 275 whn S max incrasd from 2 to 4. Mor stops could b addd to th xisting routs by incrasing th maximum numbr of intrmdiat stops, such that th xisting srvics could covr mor dmand locations and provid mor dirct srvics. Howvr, a furthr incras in 39 S max did not rduc th numbr

41 of transfrs bcaus no mor stops could b addd to th xisting routs in this rang of th maximum numbr of intrmdiat stops allowd. Eithr th maximum travl tim constraint (i.., constraint (12)) or th minimum frquncy constraint (i.., constraint (10)) instad of th constraint on th maximum numbr of intrmdiat stops (i.., constraint (11)) was binding at optimality. Visiting mor stops xtndd th round trip and total stop tims. Morovr, th prolongd travl tim rducd th frquncy if th flt siz for th rout rmaind unchangd. Hnc, th numbr of stops could not b addd to ach rout without limit. Th implication of this rsult is that practitionrs must idntify which constraints ar critical in improving th srvics bcaus th binding constraints ar diffrnt undr diffrnt conditions. Numbr of Passngr Transfrs Maximum Numbr of Intrmdiat Stops Figur 9 Effct of th maximum numbr of intrmdiat stops Effct of th maximum bus flt siz Th ffct of th maximum bus flt siz is shown in Figur 10. According to this figur, thr was no fasibl solution whn 0 < W < 55, bcaus th minimum frquncy rquirmnt could not b satisfid, implying that th flt siz was inadquat to provid th minimum accptabl lvl of srvic. Whn 55 W < 59, th numbr of transfrs dcrasd with th incras in th flt siz, bcaus mor buss wr allocatd to th xisting routs to srv th dmand btwn any pair of stops on th sam bus rout, and lss dmand btwn thm rquird a transfr du to th insufficint capacity of th dirct srvics. Whn W 59, th ffct of incrasing th flt siz on rducing th numbr of transfrs vanishd bcaus all of th dmands btwn any pair of stops on th sam bus rout wr mt. Rout capacity was 40

42 no longr th ky factor in rducing th numbr of transfrs and th maximum flt siz constraint was no longr binding. Furthr rducing th numbr of transfrs rquird adding mor dirct srvics. Numbr of Passngr Transfrs No fasibl solutions whn Maximum Bus Flt Siz Effct of th maximum numbr of routs Figur 10 Effct of th maximum bus flt siz 400 Numbr of Passngr Transfrs Maximum Numbr of Routs Figur 11 Effct of th maximum numbr of routs To illustrat th ffct of th maximum numbr of routs, R max was incrasd from 5 to 9. Th flt siz was adjustd to 120 in this tst, bcaus according to prliminary tsts no fasibl solution could b found if R max was gratr than 6 undr th dfault stting. Th rsults ar plottd in Figur 11. As xpctd, th numbr of transfrs dcrasd with an 41

43 incras in R max. Mor importantly, th numbr of transfrs was succssfully liminatd whn R max was qual to or gratr than 8, bcaus mor routs providd mor dirct srvics and covrd mor nods TSW ntwork Th main study ara was locatd in Tin Shui Wai (TSW), Hong Kong (Figur 12a). All of th routs lav TSW through th Tai Lam Tunnl (TLT), locatd on th south astrn sid of th ara, and thn continu via th highway, which is connctd to urban dstinations. Passngrs can transfr ithr at th TLT station or at othr nods outsid th TSW ara. Howvr, du to a lack of systmatic dsign, th xisting bus ntwork oprats in an infficint mannr, gnrating many transfrs. Som of th bus srvics ar routd using a low occupancy rat, which wasts rsourcs and inconvnincs passngrs. Howvr, from th passngrs prspctiv, idally, thr should b as many routs as possibl to provid dirct point-to-point srvics, but this is infasibl du to th rlativly fixd oprating cost of th oprator. As with othr bus oprators, th oprating cost was shown to b roughly proportional to th numbr of oprating vhicls. If a singl rout zigzags too much and has too many stops, th travl tim is long. Th bus oprators main concrn is thn how to rstructur th bus routs in TSW to rduc th numbr of passngr transfrs without incrasing th flt siz. Figur 12b shows th TSW ntwork. Th squar nods rprsnt th bus trminals insid TSW, th circl nods rprsnt th currnt bus stop locations, and T rprsnts th TLT bus intrchang. Th in-vhicl travl tims (in minuts) btwn nods ar shown nxt to th corrsponding links. As Figur 12b rvals, th TSW ara was dividd into 23 zons, ach of which had on stop or bus trminal. Th stops and trminals in this ara ar rprsntd by nods 1-23 and th svn bus trminals ar rprsntd by squard nods (Figur 12b). All of th bus routs originating from ths trminals trminatd at on of th fiv nding trminals, nods Th dmand matrix stimatd from th availabl data is givn in Appndix II. Effcts of diffrnt forms of fitnss functions and pnalty paramtr valus Bcaus a lowr bound is usd to approximat th uppr-lvl objctiv valu, th solution quality may not b rflctd accuratly. Thrfor, th following thr diffrnt forms of 42

44 fitnss functions ar proposd and tstd: (a) Map of Tin Shui Wai Figur 12 Th study ntwork (b) Th Tin Shui Wai bus ntwork F1 = 10 - LB - P, 15 g g g F2 g = C I - LB g - P g, and 1 = LB + P F3 g g g. F 1 g and F 2 g adopt a similar functional form in th sns that both us a constant minus th lowr bound and th pnalty trm P g. Howvr, an arbitrarily slctd larg constant (i.., ) is adoptd in F 1 g, whil an adaptiv valu, C I, is usd in adding a small constant (i.., 1.0) to th maximum valu of ( LBg Pg) F 2 g. C I is dtrmind by + among all of th solutions obtaind in itration I. In addition to th forms of th fitnss functions, th pnalty paramtr valus affct th probability of sarching infasibl solutions. Th combind ffcts 43

45 of th form of a fitnss function and th pnalty paramtr valus ar plottd in Figur 13, assuming that th valu of th pnalty paramtr a is qual to th valu of th pnalty paramtr β. Th y-axis rprsnts th avrag uppr-lvl objctiv valu and th x-axis is th log pnalty paramtr valu, dmonstrating that by modifying th form of th fitnss function, th prformanc of our algorithm significantly improvs. Although th pnalty valus affct th prformanc, th ffct sms to b lss than that of th form of th fitnss function. Th bst avrag objctiv valu is givn by adoptd in th following xprimnts. F 3 g at a 10 8 = b =. This stting was Numbr of Passngr Transfrs F1 F2 F Log Pnalty Paramtr Valu Figur 13 Effcts of various forms of fitnss functions and pnalty paramtr valus Effct of limit Th prdfind numbr limit is usd to dtrmin whn an mployd b bcoms a scout. This valu may b roughly intrprtd as th sampling frquncy within a solution spac. A highr valu mans that mor nighbor solutions ar found and compard. In this tst, limit was incrasd from 0 to 500. Th avrag uppr-lvl objctiv valus ar plottd in Figur 14. Whn limit quals 0, it rprsnts th scnario that all food sourcs ar abandond and rgnratd in ach itration. Th avrag objctiv valu dcrasd initially and thn arrivd at th minimum point, whn limit quals 150. Aftrwards, th avrag objctiv valu slightly incrasd and bcam varid, indicating that th avrag algorithm prformanc worsnd if limit was too larg or small. On xplanation is that whn limit was small, th promising ara in th solution spac could not b wll xploitd, whras whn limit was too larg, many 44

46 sarch fforts wr trappd in th aras with low solution quality. Numbr of Passngr Transfrs Limit Figur 14 Effct of limit Effct of colony composition Th composition of mployd bs and onlookrs also affcts th prformanc of th hybrid ABC algorithm. Givn that th total numbr of itrations and th colony siz ar fixd, diffrnt prcntags of mployd bs rsult in diffrnt numbrs of initial solutions and diffrnt numbrs of onlookrs influnc th intnsity of nighbor solution sarching. Manwhil, th numbr of mployd bs also rflcts th numbr of scouts, which controls th maximum numbr of nw solutions gnratd in ach itration. Figur 15 shows that th composition has a significant ffct on th algorithm prformanc. In this xampl, whn 50% of th colony is mployd bs, th algorithm achivs th bst avrag objctiv valu. Eithr a highr or lowr prcntag prvnts th furthr improvmnt of th objctiv valu, probably bcaus whn th prcntag of mployd bs is low, only a fw promising food sourcs ar gnratd for onlookrs to xploit. In contrast, if th prcntag of mployd bs is high, only a fw onlookrs conduct nighborhood sarchs to xploit th solution spac nar promising food sourcs. Effct of diffrnt nod insrtion and dltion stratgis Th insrtion and dltion of nods, which is important in improving rout structurs, is includd in th rpairing procdurs and th nighborhood sarch. W propos an avrag-dirct-dmand valu, dfind as th avrag passngr dmand on th dirct srvics, 45

47 to slct a nod to insrt or dlt. To show th bnfit of using th proposd masur in slcting nods, thr diffrnt stratgis wr compard and th rsults ar prsntd in Tabl 3. Numbr of Passngr Transfrs % 35% 50% 65% 80% Prcntag of Employd Bs Figur 15 Effct of colony combinations S1: Insrting and dlting nods basd on th proposd avrag-dirct-dmand incrmnt and dcrmnt, rspctivly. S2: Insrting and dlting nods basd on th cost incrmnt and dcrmnt, rspctivly. S3: Insrting and dlting nods basd on th total dirct dmand incrmnt and dcrmnt, rspctivly. Tabl 3 Comparison of th diffrnt insrtion and dltion stratgis 50% dmand 100% dmand S1 S2 S (+107.3%) (+93.4%) (+117.3%) (+99.7%) Two scnarios low and normal dmand wr tstd. Th avrag uppr-lvl objctiv valus of 20 runs ar rportd in Tabl 3. Th numbr in ach pair of bracs is th incrmnt prcntag with rspct to stratgy S1 in th sam row and shows that th proposd stratgy S1 outprformd th othrs in both scnarios. Th highr th dmand, th mor notabl th advantag, bcaus th avrag-dirct-dmand took th chang in th uppr-lvl objctiv 46

48 valu du to insrting a nod into considration. In contrast, if only th total dirct dmand or cost chang was considrd, th nods with a highr dmand or shortr distanc wr visitd mor frquntly, making thm mor likly to provid xcssiv srvics for ths nods and induc mor transfrs for othr nods. Robustnss of th obtaind solution Th travl dmands of th ntwork wr stimatd, but th ral dmands may vary from day to day. To illustrat th robustnss of th solution obtaind by th proposd algorithm, 1,000 dmand matrics wr gnratd by prturbing th stimatd dmand matrix and usd for th valuation. For ach prturbd dmand matrix, its lmnt th prturbd dmand from nod m to dstination was randomly gnratd from a uniform distribution [0.8 d, 1.2 d ]. Tabl 4 compars th bst solutions obtaind by th hybrid ABC algorithm for th studid situation using th prturbd dmand matrics. Std and no. stand for standard dviation and numbr, rspctivly. According to this tabl, th dsign solution obtaind by th hybrid ABC algorithm was significantly bttr than th xisting dsign, with th formr rducing th numbr of transfrs by 340% on avrag and succssfully liminating th unsrvd dmand in all cass. Th high unsrvd dmand rsultd from a lack of srvic capacity to mt th total dmand of som dstinations and th dmand at som stops, whr passngrs faild to board any lin. Ths two issus wr addrssd by th hybrid ABC algorithm. Th dtaild solution obtaind by th hybrid ABC algorithm is shown in Tabl 6 and th xisting dsign is shown in Tabl 5. Intuitivly, thr ar fwr stops in th proposd dsign, rflcting a mor fficint mting of dmand. In addition, th hadways of th routs ar mor vnly distributd for th obtaind solution. Compard with th xisting dsign, th standard dviation of hadways for th bst dsign dcrasd from 2.9 to 2.4 minuts, indicating that th diffrnc in th lvl of srvic, in trms of frquncy, among all of th routs, dcrasd. m m Tabl 4 Comparison of th solutions undr random dmand Avrag no. of transfrs Std of no. of transfrs Avrag unsrvd dmand Std of unsrvd dmand Hybrid ABC algorithm Currnt

49 Tabl 5 Existing rout structurs and hadways Routs Stop squnc Numbr of Hadway buss (minut) 1 20, 19, T, , 17, 18, 23, 22, 21, T, , 17, 18, 23, 22, 21, T, , 6, 9, 10, 12, 13, 19, 21,T, , 6, 5, 4, 11,12, 13, 19, T, , 15, 8, 9, 10, 12, 13, 19, T, , 6, 8, 16, 17, 18, 23, 22,21, T, , 13, 12, 10, 8, 16, 17, 18, 23, 22, T, , 6, 1, 2, 3, 4, 11, 12, 13, 19, T, , 10, 11, 5, 6, 8, 16, 17, 18, 23, 22, T, Tabl 6 Bst solution obtaind by th proposd ABC algorithm Routs Stop squnc Numbr of Hadway buss (minut) 1 14, 13, T, , 1, 11, 13, 21, T, , 23, 18, 16, 15, 14, T, , 5, 1, 6, 9, 8, T, , 5, 6, 8, 16, 17, 22, T, , 1, 2, 3, 4, 12, 13,T, , 8, 18, 20, 22, 23, 13, 12, T, , 18, 16, 15, 10, 12, 13, 19, T, , 2, 3, 4, 5, 7, 9, 14, 19, T, , 23, 6, 10, 11, 15, 17, 21, 19, T, Winnipg ntwork To valuat th prformanc of th proposd hybrid ABC algorithm, it was compard with that of a GA using th Winnipg ntwork obtaind from Emm 3.4. Th ntwork is shown in Figur 16 (a) and compriss 154 zons, 1067 nods, and 2995 links. Th ntwork is furthr dividd into svn districts including on cntral ara. In ach district, 30 lins ar dsignd for commutrs travling from thir hom district to th cntral ara. Each lin is allowd to visit 15 stops at most within 45 minuts. Th total flt siz is 1,200 buss with a capacity of 60 passngrs pr bus. For a fair comparison, th GA uss th sam solution rprsntation, initialization procdur, and frquncy dtrmination procdur as th hybrid ABC algorithm. Th 48

50 population siz is st to b qual th colony siz. Unlik th hybrid ABC algorithm, th GA rquirs crossovr and mutation oprators to gnrat nw solutions. Nvrthlss, th standard crossovr and mutation oprators cannot b applid dirctly bcaus th proposd solution rprsntation is diffrnt from th solution rprsntation in th traditional GA (Haupt and Haupt, 2004). Thus, th stop crossovr oprator dvlopd by Szto and Wu (2010) was usd and th proposd nighborhood oprators wr adoptd as mutation oprators. (a) Winnipg Ntwork (b) Districts of Winnipg Ntwork Figur 16 Winnipg ntwork Both algorithms ran 20 tims. Th computation tim of ach run was 300 sconds. Th computation prformancs of th two algorithms is summarizd in Tabl 7. Th numbr within ach pair of brackts in th third column is th prcntag improvmnt of th hybrid ABC algorithm with rspct to th GA in th corrsponding masur. Th scond row shows that th avrag numbr of passngr transfrs obtaind by th hybrid ABC algorithm was lowr than that of th GA. A t-tst was also conductd to xamin whthr th diffrnc in thir avrag numbrs of passngr transfrs is statistically significant. Th t-valu obtaind is and hnc w can conclud that th diffrnc is significant at th 5% lvl and th avrag prformanc of th hybrid ABC algorithm is bttr at this significanc lvl. By comparing th standard dviations of th numbr of transfrs in 20 runs in th third row in Tabl 7, it was concludd that th solution quality obtaind by th hybrid ABC algorithm was much mor stabl. Mor importantly, as rflctd in th last row, th bst solution found by th hybrid ABC algorithm was suprior to that obtaind by th GA. Th advantag of th hybrid ABC algorithm may b attributd to a bttr local sarch stratgy, with a solution spac xplord by both mployd bs and onlookrs. 49