A Method for Optimizing the Phased Development of Rail Transit Lines

Transcription

1 Urban Ral Transt (2015) 1(4): DOI /s ORIGINAL RESEARCH PAPERS A Method for Optmzng the Phased Development of Ral Transt Lnes We-Chen Cheng 1 Paul Schonfeld 2 Receved: 11 November 2015 / Revsed: 23 December 2015 / Accepted: 29 December 2015 / Publshed onlne: 21 January 2016 The Author(s) Ths artcle s publshed wth open access at Sprngerlnk.com Abstract Ths paper develops a method for optmzng the constructon phases for ral transt lne extenson projects wth the objectve of maxmzng the net present worth and examnes the economc feasblty of such extenson projects under varous fnancal constrants (.e., unconstraned, revenue-constraned, and budget-constraned cases). A Smulated Annealng algorthm s used for solvng ths problem. Ral transt projects may be dvded nto several phases due to budget lmts or demand growth that justfes dfferent sectons at dfferent tmes. A mathematcal model s developed to optmze these phases for a smple, one-route ral transt system, runnng from a Central Busness Dstrct (CBD) to a suburban area. Some nterestng results ndcate that the economc feasblty of lnks wth low demand s affected by the completon tme of those lnks and ther demand growth rate after ther mplementaton. Senstvty analyss explores the effects of nterest rates on optmzed results (.e., constructon phases and objectve value). Wth further development, such a method should be useful to transportaton planners and decson-makers n optmzng constructon phases for ral transt lne extenson projects. & Paul Schonfeld pschon@umd.edu We-Chen Cheng rcheng@rkk.com 1 2 Rummel Klepper & Kahl, L.L.P., 81 Mosher St., Baltmore, MD 21217, USA Department of Cvl & Envronmental Engneerng, Unversty of Maryland, College Park, MD 20742, USA Edtor: Xuesong Zhou Keywords Ral transt Phased development Optmzaton Smulated annealng Net present worth 1 Introducton Projects for new or extended ral transt lnes may be subdvded nto phases based on demand growth consderatons and budget lmts over tme. Any addtons of statons or extensons of ral lnes affect many users and nvolve substantal nvestments. Consequences of addng statons may nclude ncreased moblty, hgher land values, ncreased employment opportuntes, envronmental mpacts, and reduced congeston. Therefore, such a project requres a comprehensve evaluaton of all drect and ndrect consequences, ncludng postve and negatve effects on dfferent affected groups [22]. No general gudelnes are yet avalable on how many phases are needed and when each phase should be mplemented. The phases and executon tme are usually based on avalable budgets, demand forecasts, and poltcal reasons (e.g., equty among regons). The scheduled phases may be far from optmal f sgnfcant effects of extensons are neglected, such as faster demand growth after servce qualty and accessblty mprovements (e.g., new statons). Schedulng decsons affect system performance over the entre analyss perod. Therefore, n order to overcome exstng analytc weaknesses, we propose a model that optmally subdvdes a predetermned ral transt lne nto sectons for phased development and optmzes the mplementaton tmes of those sectons over a plannng horzon. The evaluaton and schedulng of addtons to lnes (.e., lnks and statons) are performed jontly by ths model. Based on varous specfed evaluaton crtera, the model can optmze a phased development plan. Ths

2 228 Urban Ral Transt (2015) 1(4): model s demonstrated here for one hypothetcal ral transt lne but s desgned to be applcable to any such transt lnes. Tavares [21] optmzes the schedule for a set of nterconnected ralway projects wth the purpose of maxmzng the net present worth (NPW), usng Dynamc Programmng. Ths model s applcable for schedulng large sets of expensve and nterconnected development projects under tght captal constrants and wth a margnal net present value. He notes that maxmzng the NPW of a project n terms of ts schedule under eventual restrctons concernng ts total duraton can be consdered as a dual perspectve of the problem of mnmzng makespan (defned as the total duraton of a project) wth resource constrants. The model presented n the paper does not consder demand reductons durng constructon. The tems consdered n NPW are only constructon expendtures and payments receved after completon of projects. Snce t s a renewal project, all the tems that are affected by the project should be taken nto account. Kolsch and Padman [9] summarze and classfy prevous studes on the resource-constraned project schedulng problem (RCPSP) by ther objectves and constrants: NPW maxmzaton and makespan mnmzaton, wth and wthout resource constrants. For the resource-unconstraned case, generally t s optmal to schedule jobs wth assocated postve cash flows as early as possble, and jobs wth net negatve cash flows as late as possble, subject to restrctons mposed by network structure. Matszw et al. [14] propose an optmzaton model to determne route extenson networks for bus transt systems. It s smlar to a routng problem that maxmzes coverng areas and mnmzes the extenson length under resource constrants. It s mportant to expand the exstng servce network to tap nto emergng areas of demand not beng served. Maxmzng network coverage can ncrease rdershp. Whle ncreasng ths potental rdershp s sgnfcant, t s necessary to keep any route extenson to a mnmal length. Extendng routes to low-demand areas could result n low servce utlzaton. In our present study, the NPW maxmzaton objectve determnes how far routes should be extended to low-densty suburbs. Wang and Schonfeld [23] develop a smulaton model to evaluate waterway system performance and optmze the mprovement project decsons wth demand model ncorporated. They argue that mnmzng total costs rather than maxmzng the NPW over the entre analyss perod s not vald n a system where demand s elastcally affected by the system mprovements beng optmzed. The results show how demand elastcty can be used n estmatng net benefts. Shayanfar et al. [16] compare the relatve merts of three metaheurstc algorthms, namely smulated annealng, tabu search, and a genetc algorthm, for selectng and schedulng mprovements n road networks. Numerous other researchers have developed related models for optmzng varous characterstcs of publc transportaton systems. These nclude Guan et al., [5], Fan and Machemehl [3], Zhou et al. [24], L et al. [11], Tsa et al. [20], DJoseph and Chen [2], Km and Schonfeld [8], and Markovc et al. [13]. Km et al. [7] optmzed vertcal algnments and speed profles for ral transt lnes. La and Schonfeld [10] optmzed the locaton of ral transt lnes and statons, based on GIS databases and usng a genetc algorthm, but wthout consderng phasng decsons. Lo and Szeto [12] and Szeto et al. [19] deal wth the tmng of mprovements n dscrete network desgn. Guhare and Hao [6] revew transt network desgn and schedulng approaches, whle Farahan et al. [4] revew urban transportaton network desgn more generally. The modelng approach used n our present study s partly based on a model of Chen and Schonfeld [1], except for the decson varables. They developed a model that jontly optmzed the characterstcs of a ral transt route and ts assocated feeder bus routes n order to mnmze total costs. Somewhat smlarly, Spasovc and Schonfeld [17] also optmze the transt servce coverage wth a mnmum total cost objectve. Ther analytc results showed that n order to mnmze total costs, the operator cost, user access cost, and user wat cost should be equalzed. They also noted that the most sgnfcant factor n determnng the ral lne length s the demand. Thus, no route completon constrant s consdered n our present model because t mght overextend routes nto dstant suburbs wth nsuffcent demand densty. Sun and Schonfeld [18] analyze a related phased development problem, but for arport facltes rather than ral transt lnes. Although the publshed studes we found do not deal wth the optmzed phased development of transt lnes, ths problem can be treated as an RCPSP wth unque characterstcs. Frst, the actvtes n ths project represent the statons to be added. Second, the precedence relatons n ths problem are much easer than those n the general project schedulng problem. Our transt lne can only be extended sequentally from one end (.e., CBD) to the other (We can stll treat a lne through a CBD as two end-to-end radal lnes). Thrd, constrants on both captal budget and revenue are consdered n ths study. For the captal budget constrant, subsdes are equally dstrbuted here wthn each gven tme nterval, although any dstrbuton may easly be specfed. The revenue constrant s used for balancng the operatonal expendture. It s mportant to note that the resource constrants vary over the entre tme horzon, snce these two constrants are affected by the operatonal stuaton and decson made n prevous years. Hence, ths problem s a dynamc RCPSP. NPW maxmzaton s our chosen objectve. All the quantfable tems

3 Urban Ral Transt (2015) 1(4): that would be affected by the extenson should be consdered n ths problem (e.g., user watng costs, n-vehcle costs, and operatng and mantenance costs), ncludng soco-economc effects f they can be quantfed and estmated correctly. Due to the complexty of the dynamc RCPSP, ncludng the pervasveness of local optma, we use a Smulated Annealng algorthm to solve ths problem. The model formulaton and desgn of the SA algorthm are presented below. 2 Model Formulaton Table 1 defnes the notaton used n the paper. The followng smplfyng assumptons are made here. A gven demand at the startng tme nterval (t = 0) s already consstent wth network equlbrum. 1. Transt routes and staton locatons are predetermned. Hence, user access costs are omtted from ths analyss. 2. Effects of development schedules of other transportaton system changes on the demand of our lne are neglected. 3. Statons can only be added sequentally from the CBD outward. Wth a double crossover track at every staton, any staton can be at least temporarly the lne s termnal staton. Hence, turnaround tme s omtted from ths analyss. 4. There are no bndng constructon tme constrants. 5. Potental demand for each O/D par ncreases at a hgher rate after the staton s completed. Table 1 Notaton Varables Descrptons Unts B Total beneft $ C Total cost $ C C Captal cost $ C I In-vehcle cost $ C M Mantenance cost $ C O Operatng cost $ C S Suppler cost $ C U User cost $ C W Watng cost $ d Staton spacng mle f Taxaton rato for coverng operatonal expendture % F T Fleet sze vehcle h Headway h The orgn n the O/D matrx j The destnaton n the O/D matrx k Captal cost for staton and ral lne $ m The row n the O/D matrx n C Number of cars needed per tran cars/vehcle P Demand functon NPW Net present worth of total benefts $ q j Ral passenger flows from orgn to destnaton j people r Demand growth rate R Round trp tme h s Interest rate t Tme nterval t d Dwell tme hour u I Unt cost of user n-vehcle tme $/passenger-h u L Mantenance unt cost $/passenger-mle u T Hourly operatng cost $/vehcle-h u W Unt cost of user watng tme $/passenger-h U B User beneft $ V Cruse speed mles/h y Decson varable

4 230 Urban Ral Transt (2015) 1(4): Captal costs are reduced f multple statons and ther lnks are bult together. 7. The nterest rates are effectve rates whch already consder nflaton. Fgure 1 shows the proposed example ral transt lne, whch s 54.4 mles long and has 30 statons. Currently, only 4 statons are completed and n servce. The study s tme horzon s 30 years. Our bnary decson varable y (t) = 1 f lnk and ts staton already exst n tme perod t; y (t) = 0flnk s yet unbult n tme perod t. Here lnk s defned as the secton between statons - 1and, andlnk ncludes staton (t). The frst tme y changes from 0 to 1 whch ndcates that lnk saddednyeart. In the long term, the traffc ncrease may occur due to demographc and economc growth. Demand growth s consdered here by multplyng the demand relaton for the ntal perod (t = 0) wth a compound growth rate (1? r) t, where r s the growth rate per tme nterval (e.g., per week, month, or year) and t represents ntervals of growth (Fg. 2). The baselne demand functon for each orgn/ destnaton par s a lnear demand functon (.e., Q = a-b*p). As dscussed above, the orgn/destnaton (O/D) matrx values can contnuously ncrease at a specfc annual growth rate based on traffc demand forecasts. q (t) j = - q (0) j 9 (1? r) t, V, j, where q j denotes ral passenger flows from orgn to destnaton j. For our numercal study, the O/D matrx s symmetrc, wth q j = q j. There are 4 statons n servce n tme nterval zero. The O/D matrx s 2 3ðtÞ y 2 q 12 y 3 q 13 y 4 q 14 y 5 q 15 y 6 q y 2 q 21 y 3 q 23 y 4 q 24 y 5 q 25 y 6 q y 3 q 31 y 3 q 32 y 4 q 34 y 5 q 35 y 6 q D ðtþ ¼ y 4 q 41 y 4 q 42 y 4 q 43 y 5 q 45 y 6 q ; y 5 q 51 y 5 q 52 y 5 q 53 y 5 q 54 y 6 q y 6 q y 7 q where at t = 0, y 1 = y 2 = y 3 = y 4 = 1, y 5 = y 6 = = Beneft Functon User beneft (U B ), n any tme nterval t, s defned as the area under the demand (=margnal user beneft = P) curve for that nterval, ntegrated from 0 to q (t) j, where q (t) j s the traffc flow from to j n the tth smulaton nterval (Fg. 2). Snce q j may fluctuate n dfferent ntervals, the overall user beneft for the entre analyss perod s U B ¼ X30 X 30 X 30 Z! ðtþ q j P dq ; 6¼ j: ð1þ t 2.2 Cost Functon j 0 The user cost (C U ) conssts of three components: n-vehcle cost, watng cost, and access cost. Access cost s the total demand multpled the access tme.because we assume that staton locatons are predetermned, the access cost mght be omtted. The watng cost, C W, s the total demand multpled by the watng tme (whch s approxmated as half of the headway, h), and the unt cost of user watng tme, u W ($/passenger-h): C ðtþ W ¼ DðtÞ h 2 u W: ð2þ In-vehcle cost, C I, s the through flow multpled by the n-vehcle tme whch ncludes the rdng and dwell tme and the unt cost of n-vehcle tme, u I ($/passenger-hour). Through flow s equal to nflow mnus outflow at each staton, and t can be determned from the O/D matrx: Through flow ¼ 2 X30 X m m¼1 ¼1 X 30 j¼þ1 y j q j X j¼1 y q j!; ð3þ where m s the row n the O/D matrx, s the orgn n the O/D matrx, j s the destnaton n the O/D matrx.! C I ¼ 2 X30 X m X 30 y j q j X y q j m¼1 ¼1 d mþ1 V þ t d j¼þ1 y mþ1 u I ; j¼1 ð4þ Fg. 1 Proposed route

5 Urban Ral Transt (2015) 1(4): Fg. 2 User benefts where d m?1 represents the staton spacng between staton m? 1 and m, V s the transt speed, and t d s the lost tme at each staton. The factor t d accounts for the tme lost through deceleraton and acceleraton as well as for dwell tme at a staton. No out-of-pocket costs are ncluded n the user cost. Transt fares are not part of the user cost snce they are merely transfer payments from users to operators. Thus, the user cost s equal to the watng cost plus n-vehcle cost: C U ¼ C W þ C I : ð5þ The suppler cost (C S ) conssts of three components as shown n Eq. (6): C S ¼ C C þ C O þ C M : ð6þ These are captal cost (C C ), operatng cost (C O ), and mantenance cost (C M ).Captalcost(C C ) ncludes land acquston, desgn, and constructon, and ral-track layng costs: C C ¼ X30 t X 30 y ðt1þ k ; ð7þ where k s the fxed cost for lnk. We use y (t) - y (t-1), (t) snce k s the cost whch only counts the frst tme when y changes from 0 to 1. We assume (n Assumpton 7 above) that some economes occur f several statons (and ther lnks) are bult together. In our numercal examples, the constructon cost savngs are set at 3 % for 2 statons, 6 % for 3 statons, 9 % for 4 statons, 12 % for 5 statons, 15 % for 6 statons, 18 % for 7 statons, 21 % for 8 statons, and 24 % for more than 9 statons. The operatng cost s the transt fleet sze F T multpled by the hourly operatng cost per car u T ($/vehcle-h) and the number of cars n C needed per tran. u T ncludes the equvalent hourly captal cost of the ral cars. Because the optmal headway changes as we extend the lne, we have to update the headway after every decson made. To obtan the fleet sze, the transt round trp tme R (t) s derved frst as follows: R ðtþ ¼ 2 X30 d þ1 V þ t d þ1 ; ð8þ where d?1 represents the staton spacng between statons? 1 and. Snce our demand functon s not elastc wth respect to headway (whch means that demand s fxed durng each teraton), the optmal headway h can be found by checkng the frst-order dervatve of the total cost (C) functon wth respect to h equal to zero and solvng t for h. The second dervatve of the total cost functon wth respect to h s also checked to nsure that the total cost functon s convex. oc oh ¼ 0 ð9þ oc oh 2 ¼ 2Rn Cu T h 3 [ 0 ð10þ The resultng optmal headway s vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u 2n C u T h ðtþ þ1 ¼ u t P u 30 W P 30 d þ1 V q ðtþ j þ t d ; ð11þ where h (t) [ h mn (.e., h) and h (t) \ h max = tran capacty/peak pont one-way passenger flow. The fleet sze F (t) s then the transt round trp tme dvded by the headway h: F ðtþ ¼ RðtÞ h : ð12þ ðtþ Wth the fleet sze, we can then determne the operatng cost: C ðtþ O ¼ FðtÞ n C u T : ð13þ Mantenance cost, C M, s expressed as the passenger mles traveled (PMT) multpled by a unt mantenance cost, u L ($/pass. mle):! C M ¼ 2 X30 X m X 30 y j q j X y q j m¼1 ¼1 d mþ1 V þ t d j¼þ1 y mþ1 u L : j¼1 ð14þ Therefore, the suppler cost s equal to the operatng cost plus mantenance cost:

6 232 Urban Ral Transt (2015) 1(4): C S ¼ C O þ C M þ C C : ð15þ Equatons (16) to(21) present a model for maxmzng the system s net present worth: Maxmze NPW ¼ X30 Subject to t ðb CÞð1 þ sþ t ð16þ ¼ 1or0 ð17þ y ðt1þ 0; for all ; t 1 ð18þ þ1 0; for all t; 1 ð19þ f revenue ðtþ C ðtþ O þ CðtÞ M ; for all t ð20þ ð1 f Þrevenue ðt1þ þ Subsd C ðtþ C ; for all t: ð21þ Equaton (16) s the objectve functon that maxmzes the system s NPW. The annual net beneft s equal to total beneft (B) mnus total cost (C). Total beneft ncludes user beneft; total cost ncludes suppler cost and user cost. We have to nclude the nterest rate n the model to obtan the NPW. In Eq. (17) the decson varables are bnary. Equaton (18) s the realstc constrant ensurng that after lnk s bult, t always remans n operaton. Equaton (19) s the precedence constrant that prevents any lnk from beng bult f any one of ts predecessors s not yet completed. The transt lne has to be bult sequentally, snce there would be fewer benefts f we randomly choose any segment to buld along the route. In transt operaton, some fracton of the fare collecton may be used for coverng operaton expenses, and the remanng fracton (f any) may be used for fundng the constructon of new transt lne extensons. Equaton (20) s the revenue constrant for coverng operatonal expenses,.e., operatng and mantenance costs. Due to uncertantes about the future, transt operators may try to balance ther operaton-related expendtures n each year. Thus, a fracton f of the revenue collected from fares s used to cover the operatng and mantenance costs n each year. Equaton (21) s the budget constrant for fundng the captal nvestments. It shows that the constructon costs n any year must not exceed the avalable captal funds plus some fracton (1 - f) of the fare revenues accumulated from the prevous year. 2.3 Smulated Annealng Smulated Annealng (SA) s a heurstc method, whch s very useful n optmzng objectve functons wth numerous local optma. It was orgnally developed by Metropols et al. [15] who descrbe ts detals. Unlke most of the earler search methods, SA may accept (wth a decreasng probablty) moves to neghborng solutons whch worsen the objectve functon, n order to escape from locally optmal holes. Usng SA, f a neghborhood soluton s better than the prevous one, t s always accepted. To avod gettng stuck n a local mnmum or maxmum, occasonally solutons worse than the current one are also accepted but wth a probablty smlar to that n the dynamcs of the annealng process. As the temperature decreases, the probablty of acceptng a bad soluton s decreased and n the fnal stages the Smulated Annealng algorthm becomes smlar to gradent-based search. The smulated annealng process proceeds as follows: Step 1 randomly generate a feasble ntal soluton x 0 and calculate f(x 0 ). Step 2 from the current soluton x 0, jump to ts neghbor x 0 and calculate f(x 0 ). Step 3 compare f(x 0 ) and f(x 0 ). If f(x 0 ) [ f(x 0 ), x 0 replaces x 0 to be the current soluton. Otherwse, randomly generate a number z between 0.01 and If z\ exp f ðxþf ðx0 Þ T [8], x 0 becomes the current soluton. Otherwse, do nothng. Step 4 for every 5 teratons, reduce the temperature T by 1 %,.e., multplyng by Step 5 check termnaton rule. Maxmum teratons reached or stoppng crtera reached. If yes, algorthm stops; otherwse, return to Step 2. More detaled SA desgn and parameter tunng can be found n Cheng [9]. 2.4 Numercal Results The procedure was coded wth MATLAB and run on an IBM Laptop wth a 1.60 GHz Pentum R processor and 1.00 Ggabytes of RAM. Snce runnng a 30-staton route over a 30-year analyss perod takes consderable tme, a very large number of teratons are needed to converge whle searchng wth Smulated Annealng. In the numercal examples presented here, t s assumed for smplcty that the externally funded budget for captal mprovements s equally dstrbuted over all perods. Two problem cases were tested: an unconstraned case and a revenue-budget-constraned case. 2.5 Unconstraned Case Fgure 3a shows the resultng dscounted net benefts/year and the optmzed phases. Surprsngly, ths optmzed soluton has only one phase whch conssts of addng 23 lnks

7 Urban Ral Transt (2015) 1(4): n year 2. Snce t s assumed that we have unlmted funds for extensons, ths answer mples that we should add lnks as soon as possble f the demand s suffcent. The demand at statons 28, 29, and 30 s ntally too low, so the route stops at staton 27. The annual dscounted net benefts respond to the addton of lnks. In year 2, the negatve value s due to the constructon costs. Fgure 4a compares four alternatves. The green lne s the optmzed soluton found for the unconstraned case. The black lne s the case wthout addton of lnks, whch has only 4 statons n servce for the 30-year horzon. The drop n year 2 s due to captal costs for extenson. If the transt lne s extended to lnk 27 n year 2, the NPW wll ncrease much faster than wthout an extenson. Alternatve 1 (red) extends to lnk 27 n year 17; alternatve 2 (blue) extends to lnk 30 n year 2. None of them have a hgher objectve value than the green lne. 2.6 Constraned Case Two knds of constrants are added: a revenue constrant and a budget constrant. Penalty methods are used here for dealng wth constrants. A 5 % borrowng allowance s added nto both revenue and budget constrants. Addng such an offset s reasonable to avod delayng the constructon just because of small shortfalls. For the revenue-budget-constraned case, the stoppng crteron s ncreased to 100 k teratons and the objectve value s Fgure 3b shows the annual dscounted net benefts n each year and the optmzed phases. There are sx phases for ths case: Phase I adds 3 lnks n year 3; Phase II adds 2 lnks n year 5; Phase III adds 1 lnk n year 6; Phase IV adds 1 lnk n year 9; Phase V adds 3 lnks n year 13; and the last phase adds 1 lnk n year 14. The annual dscounted net benefts drop sgnfcantly when lnks are added but bounce back wth a hgher value the followng year. Fgure 4b shows the NPW for dfferent cases. In Fg. 4b, as more constrants are appled, NPW decreases, as expected. However, the dfferences n NPW between revenue-constraned case and revenue-budgetconstraned case are small. There are probably two reasons: frst, the 5 % borrowng allowance brngs the answers n our two cases farly close; second, the revenue constrant domnates n the numercal example. Addng a budget constrant does not bnd the soluton. Compared wth the unconstraned and revenue-budget-constraned cases, the NPW n the case constraned by revenue and budget s nearly one-thrd of that n the unconstraned case. NPW s sgnfcantly affected by the constructon phases. 2.7 Relablty The relablty of the obtaned soluton s an mportant concern. Snce the exact optmal soluton to ths problem s not known (note that no exstng methods guarantee fndng the global optmum for a large RCPSP), t s dffcult to prove the goodness of the soluton found by the proposed Smulated Annealng algorthm. Therefore, an experment s desgned to statstcally test the effectveness of the algorthm. In ths experment, the ftness value s evaluated for each randomly generated soluton to the problem. Frst, numerous soluton samples are generated and tested. The next step compares the random sample solutons wth the SA optmzed soluton. We frst create a random sample of 1,000,000 observatons. The best ftness value (.e., NPW) n ths sample s , whle the worst one s The sample mean s and the standard devaton s , as shown n Fg. 5. In the experment, the optmzed soluton obtaned ( ) s approxmately 8 % hgher (.e., better n NPW) than the hghest value n the random sample ( ). In other words, the soluton found by the SA algorthm domnates by a consderable margn all the solutons n the dstrbuton. In fact, the random sample does not cover the range of the ftness values for all possble solutons n the search space. The number of possble solutons for the unconstraned case s 27 29, whch ncludes nfeasble solutons. Ths number comes from the soluton vector whch has 30 elements. Besdes the base year (year 1), n each year the number of statons n servce can change from 4 to 30, so there are dfferent permutatons. It s dffcult to calculate the exact number of possble feasble solutons, snce the problem s dynamc. Ths suggests that an even larger sample mght be worth testng. However, the optmzed soluton value s consderably better than any of the 1 mllon random solutons sampled. The result shows that the best soluton found by the SA algorthm, although not necessarly globally optmal, s stll remarkably good when compared wth other possble solutons n the search space and s unlkely to be sgnfcantly mproved upon by the globally optmal soluton. We can conclude that the soluton qualty wll be lmted by the varous uncertantes regardng the nputs rather than the capabltes of the Smulated Annealng algorthm. Ths analyss ndcates a very promsng performance for the proposed optmzaton model. 2.8 Computaton Tme One of the man drawbacks of the Smulated Annealng approach s ts computaton tme. As the problem sze changes from ten statons and 10 years to thrty statons and 30 years, respectvely, the computaton tme ncreases sgnfcantly, as shown n Fg. 6. Varous computatons such as computaton of the net present worth functon and computaton of the probablty of acceptng bad solutons

8 234 Urban Ral Transt (2015) 1(4): Statons.# dscounted NB 30 2,000 #. statons n servce , ,000-2,000-3,000 dscounted NB (mllon $) year (a) Unconstraned Case -4,000 Statons.# dscountd NB #. statons n servce dscounted NB (mllon $) year (b) Constraned Case Fg. 3 Dscounted net benefts and optmzed phases -500 ncrease the computaton tme when the problem sze grows. Also, for better results the coolng schedule has to be carred out very slowly and ths sgnfcantly ncreases the computaton tme. 2.9 Senstvty Analyss The followng senstvty analyss s desgned to nvestgate the effects of one nput parameter (.e., the nterest rate) on the resultng optmzed values (.e., constructon phases and total net benefts). If the model s very senstve to changes n a partcular nput parameter, that parameter should be predcted as accurately as possble and decsons should be made more cautously Interest Rate The nterest rate plays an mportant role n project schedulng, especally n large nvestment projects. Theoretcally, projects tend to be postponed when the nterest rate s hgh. If the nterest rate ncreases, then nvestment decreases due to the hgher cost of borrowng. Although transt planners cannot control the nterest rate, senstvty analyss can show them how extenson decsons are affected by nterest rates. To evaluate the effects of dfferent nterest rates (s) on phasng decsons and NPW n ths secton, s, whose base value s 5 %, s vared between 0 and 30 %. Table 2 shows the dfferences n optmzed values and phases. Not only s

9 Urban Ral Transt (2015) 1(4): Fg. 5 Optmzed SA soluton compared to 10 6 random solutons Fg. 4 Cumulatve net benefts over years on dfferent alternatves and cases the extenson postponed but also the number of phases decreases when the nterest rate ncreases. When s s below 10 %, the transt lne s extended to lnk 15. When s ncreases to 15 %, the lne s extended to lnk 8. When s exceeds 30 %, the transt route merely extends to lnk 5. For lnks wth enough demand, delayng the constructon causes no problem. The margnal benefts of addng lnks wth enough demand are always postve, except when addng lnks n the last year of the analyss perod. However, lnks wth ntally low demand and enough hgh growth rates after mplementaton are only benefcal over the analyss perod f they are bult early. In order to acheve hgher cumulatve net benefts, the lnks whch would be economcally benefcal at the end of the analyss perod must be added as soon as possble. If some Fg. 6 Computaton tme constrants prevent the extensons at early stages, the lne cannot be extended as far as n the unconstraned case. 3 Conclusons A model s developed for optmzng the constructon phases of any ral transt lne that s bult wthout gaps from one end toward the other. It can be used to determne not only the constructon phases but also the economc feasblty of addtonal lnks under varous fnancal constrants. The optmzed soluton also avods overextenson of the proposed lne. In addton, tax-fundng polcy also can be optmzed through senstvty analyss, as demonstrated. The study leads to the followng conclusons: The numercal analyses show that for the unconstraned case, mmedately addng all lnks wth postve net

10 236 Urban Ral Transt (2015) 1(4): Table 2 Effects of nterest rates on NPW and optmzed phases # Statons n servce n each year # phases Cumulatve net benefts ($) Interest rate (%) E? E? E? E? E? E? E? E? benefts acheves the hghest objectve value. Ths result s consstent wth the one found n Kolsch and Padman [9], whch s to schedule jobs wth postve cash flows as soon as possble and to delay jobs wth negatve cash flows as much as possble. Wth ts gven nputs, the optmzed soluton has only one phase and, n the absence of a completon constrant, does not reach the end of the route. Therefore, those lnks wth negatve values are postponed ndefntely. If we nsst (through completon constrants) that outer lnks wth unjustfably low demand must be completed, then those lnks wth nsuffcent demand would be added n the last tme perod (The present worth of ther costs would thus be mnmzed). For the case n whch demand grows faster after an extenson, the economc feasblty of addng one lnk s affected sgnfcantly by the constructon tme. Compared wth varous fnancal constrants, the transt lne can be extended to lnk 27 for the unconstraned case, but t can only be extended to lnk 15 for the case constraned by external budget and route-generated revenue. If some lnks wth low demand and hgh growth rate after extenson cannot be added at early stages, they do not become justfed wthn the remanng 30-year span of our case study. That s due to the hgh captal costs of addng lnks. In our senstvty analyss, no extenson was justfed at later stages. Consequently, when analyzng the economc feasblty of a project wth hgh captal cost, constructon phases should be taken nto account. The results obtaned are reasonable, even for the possbly counterntutve results where demand growth accelerates after lnks and statons are added. Such a model s valuable because t quantfes the effects of extenson alternatves and fnds extremely good solutons for ths large combnatoral problem. Whle most of the results seem reasonable or even obvous qualtatvely, such a model can help quantfy and optmze the route development decsons. 3.1 Future Research The followng extensons are suggested for further studes: (1) The model desgned n ths study s determnstc. Based on uncertantes about the future, ths model could be mproved to consder probablstc factors. For nstance, the demand growth rate mght change over tme. Demand wll not necessarly ncrease n the future. Interest rates and nflaton rates also vary over tme. A probablstc model can address ths problem more realstcally than a determnstc model. (2) For ncreased realsm, a future model mght relax some smplfyng assumptons, such as that specfyng

11 Urban Ral Transt (2015) 1(4): sequental lnk addton. Currently the model can be used for radal networks. For some other cases, the assumpton that only adds lnks sequentally should be relaxed. (3) Addtonal factors that would be affected by transt extensons mght be modeled, such as mult-modal access to statons. External benefts and costs can be added nto the model f they are correctly estmated, ncludng employment opportuntes, land values, travel tme savngs, and envronmental mpacts. (4) Some operatonal varables (e.g., transt fare and cruse speed) can also be optmzed by a modfed model at varous tmes nstead of keepng them fxed. In order to optmze these varables, prce and travel tme elastcty of the demand would have to be consdered. (5) Ths model optmzes the constructon phases for a sngle route. It mght be mproved to deal wth more complex networks that nclude branched routes. (6) Other metaheurstc algorthms, such as genetc algorthms and tabu search, mght be tred for ths problem n attemptng to reduce the computaton tme. Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense ( tvecommons.org/lcenses/by/4.0/), whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made. References 1. Chen S, Schonfeld P (1998) Jont optmzaton of a ral transt lne and ts feeder bus system. J Adv Transp 32(3): DJoseph P, Chen S (2013) Optmzng sustanable feeder bus operaton consderng realstc networks and heterogeneous demand. J Adv Transp 47(5): Fan W, Machemehl RB (2006) Optmal transt route network desgn problem wth varable transt demand: genetc algorthm approach. J Transp Eng 132(1): Farahan RZ, Mandoabch E, Szeto WY, Rashd H (2013) A revew of urban transportaton network desgn problems. Eur J Oper Res 229(2): Guan JF, Yang H, Wrasnghe S (2006) Smultaneous optmzaton of transt lne confguraton and passenger lne assgnment. Transp Res Part B 40(10): Guhare V, Hao JK (2008) Transt network desgn and schedulng: a global revew. Transp Res Part A 42(10): Km M, Schonfeld P, Km E (2013) Comparson of vertcal algnments for ral transt. J Transp Eng ASCE 139(2): Km M, Schonfeld P (2014) Integraton of conventonal and flexble bus servces wth tmed transfers. Transp Res Part B 68B 2: Kolsch R, Padman R (2001) An ntegrated survey of determnstc project schedulng. OMEGA Int J Manag Sc 29: La X, Schonfeld P (2012) Optmzaton of ral transt algnments consderng vehcle dynamcs. Transp Res Record 2275: L ZC, Lam WHK, Wong SC, Sumalee A (2012) Desgn of a ral transt lne for proft maxmzaton n a lnear transportaton corrdor. Transp Res Part E 48(1): Lo HK, Szeto WY (2009) Tme-dependent transport network desgn under cost-recovery. Transp Res Part B 43(1): Markovc N, Mlnkovc S, Tkhonov KS, Schonfeld P (2015) Analyzng passenger tran arrval delays wth support vector regresson. Transp Res Part C 58: Matszw TC, Murray AT, Km C (2006) Strategc route extenson n transt networks. Eur J Oper Res 171: Metropols NA, Rosenbluth A, Rosenbluth M, Teller E (1953) Equaton of state calculatons by fast computng machnes. J Chem Phys 21: Shayanfar E, Abaneh AS, Schonfeld P, Zhang L (2015) Prortzng nterrelated road projects usng meta-heurstcs. J Infrastruct Syst ASCE 17. Spasovc LN, Schonfeld P (1993) Method for optmzng transt servce coverage. Transp Res Rec 1402: Sun Y, Schonfeld P (2015) Stochastc capacty expanson models for arport facltes. Transp Res Part B 80: Szeto WY, Jaber X, O Mahony M (2010) Tme-dependent dscrete network desgn frameworks consderng land use. Computer-Aded Cvl Infrastruct Eng 25(6): Tsa FM, Chen S, We CH (2013) Jont optmzaton of temporal headway and dfferental fare for transt systems consderng heterogeneous demand elastcty. J Transp Eng ASCE 139(1): Valadares Tavares L (1987) Optmal resource profles for program schedulng. Eur J Oper Res 29: Vuchc V (2005) Urban transt: operatons, plannng and economcs. John Wley & Sons Inc, Hoboken, NJ 23. Wang S, Schonfeld P (2007) Demand elastcty and beneft measurement n a waterway smulaton model. TRB 86th annual meetng compendum of papers 24. Zhou Y, Km HS, Schonfeld P, Km E (2008) Subsdes and welfare maxmzaton tradeoffs n bus transt systems. Ann Reg Sc 42 3: