Optimizing Dial-a-Ride Services in Maryland

Transcription

1 0 0 0 Optmzng Dal-a-Rde Servces n Maryland Nola Marovć Department of Cvl & Envronmental Engneerng Unversty of Maryland College Par, MD Emal: nola@umd.edu Rahul Nar IBM Research Dubln, Ireland Emal: rahul.nar@e.bm.com Paul Schonfeld (correspondng author) Department of Cvl & Envronmental Engneerng Unversty of Maryland College Par, MD Tel: (0)0-0 Emal: pschon@umd.edu Else Mller-Hoos Department of Cvl & Envronmental Engneerng Unversty of Maryland College Par, MD Tel: (0)0-0 Emal: elsemh@umd.edu Matthew Mohebb IT Curves Gathersburg, MD 0 Emal: mmohebb@tcurves.net Word Count: + ( Tables + Fgures) * 0 = 0 Submtted for presentaton at the 0 TRB rd Annual Meetng and for publcaton n the Transportaton Research Record 0 TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

2 Marovć et al. 0 ABSTRACT Ths paper reports on a system developed to address the Dal-A-Rde problem and an mplementaton for Maryland where real-world practcal constrants are consdered n provdng customzed vehcle routng and schedulng for about 0 trp requests daly. The system, called Moble Resource Management System (MRMS), allows for dspatch operators to qucly study dfferent operatonal scenaros and, n a strategc settng, explore tradeoffs between level-ofservce and varous system characterstcs, ncludng fleet composton, fleet sze and dspatch rules. Such nsghts play a ey role n mang long-term nvestment decsons or estmatng cost of servcng contracts that have servce level agreements. Test comparson of manual and MRMS-based routes ndcated an estmated annual operatonal expense reducton of $0. mllon, or about % of the total annual expense. In addton to the cost benefts, mproved qualty of servce and the reducton n total vehcle-lometers traveled leadng to envronmental benefts are demonstrated. TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

3 Marovć et al INTRODUCTION Better accessblty to publc transportaton has become an mportant objectve for many transt systems across the world. Ths objectve has been partally acheved by ntroducng low-floor buses complemented wth neelng devces and specalzed ramps, nstallng elevators n ral statons, and ntroducng hgher platforms. Nevertheless, many handcapped and elderly people stll fnd t dffcult to use publc transt despte these recent enhancements. Some handcapped or elderly people need addtonal help, whle others may fnd the closest stop to be too far away, or the wat too long (). Transt systems across the world offer these people a specal demandresponsve door-to-door transportaton servce, whch s often called dal-a-rde (DAR) servce. The major dfference between DAR and tax servce s that DAR allows rdesharng; thus, multple unrelated passengers, wth dfferent orgns and destnatons, may be served smultaneously wth the same vehcles. Annual DAR rdershp growth of more than % has been reported n many ctes n the US and ths trend s expected to contnue because of the ever agng populaton (-). Due to ncreasng rdershp, the DAR servces have been among the fastest growng budget fractons of many transt agences. These trends pressed DAR servce provders to mprove the effcency of ther operatons and thereby lmt some of the budget ncreases. In response to these trends, DAR agences n North Amerca and Europe have started to mplement communcaton and computer technology n order to better control costs and manage ther growng operatons (). These technologes also serve as enablers for optmzaton algorthms to generate better schedules for drvers and vehcles. Ths paper addresses the vehcle routng and schedulng problem n DAR operatons and maes the followng contrbutons. Frst, ey practcal consderatons n real-world dspatchng operatons, such as loced-blocs and the heterogeneous nature of demand and supply, are dentfed and modeled. A loced-bloc refers to a seres of passenger demands that do not change from one day to the next, and as such, must be served n the same order every day. Heterogenety of supply manfests n two ways. Trp requests that have assocated wth t partcularly egregous costs due to spatal or tme crtera can be outsourced to a tax. Addtonally, the operator may have a fleet of vehcles wth varyng capacty. Ths capacty should be allocated based on the heterogenety of demand for partcular routes. For example, such demand varaton s lely to occur when there are wheelchar-bound passengers who requre addtonal space onboard. Such practcal consderatons, although dscussed n the lterature prevously, have not been adequately consdered n the algorthmc development of DAR solutons. The second contrbuton of the wor s n deployng the proposed algorthm to a sute of test nstances, and for a real-world deployment n Maryland. The paper examnes the benefts of mproved effcency and results from deployment n the feld, ncludng the mprovement n qualty of servce. Most of all, the results reported n ths paper ndcate that consderable savngs TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

4 Marovć et al. can be acheved n provdng DAR servces through mplementaton of custom-desgned vehcle routng and schedulng algorthms. These results show a potental for reducng the governmental expendtures provded proper nvestments n computer-aded decson mang n DAR operatons. The paper s organzed as follows. The next secton presents the overall system archtecture and a descrpton of the components outlned n ths paper followed by the model and soluton algorthm. The paper concludes by presentng numercal results on test nstances and a real-world deployment n Maryland, followed by a summary of ey performance measures and benefts THE MRMS SYSTEM A system to manage DAR servces, from trp request management to dspatch and operatons was mplemented. At the core of the system s the algorthm for effcent schedulng and dspatchng of vehcles satsfyng varous operatonal and level-of-servce (LOS) constrants. The MRMS was desgned to provde operators wth many parameters that could be set for dfferent runs allowng the operators to experment and fnd the optmal tradeoff between LOS and operatng and captal expenses. Current wde avalablty of wreless data, tablets and wreless devces, bult-n navgaton tools, and powerful bac offce computers enable the role of optmzaton algorthms to mprove operatons. The MRMS development and mplementaton ncluded three dstnct parts:. In-vehcle navgaton and communcaton equpment that wor over standard G or G wreless data servces. Ths equpment allows montorng and controllng the locaton and status of the vehcles at all tmes. It also allows dynamc modfcatons of drver tas lsts, whch nclude the sequence of pc-up and drop-off locatons and tme wndows.. Cloud-based montorng, control and rule-based decson-mang that provdes the ntellgence for routng and performance optmzaton based on provded LOS and system constrants.. A presentaton layer that enables operators to vew n real-tme the performance of the operaton. It also provdes the operators the ablty to set up performance parameters, and modfy the constrants and optmzaton rules. The ey element of the MRMS s the algorthm used to effcently solve the vehcle routng and schedulng problem n DAR operatons. Snce a vast majorty of trps are scheduled between one and seven days n advance, the algorthm was based on the statc DAR problem. For largescale nstances exact solutons to DAR problem are computatonally prohbtve. So, a specalzed soluton heurstc was developed, ncorporatng the practcal consderatons such as, outsourcng of outler requests, loced-blocs, and heterogenety n fleet and requests. TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

5 Marovć et al. 0 0 STATIC DIAL-A-RIDE PROBLEM When the DAR requests are made long n advance, whch s the case wth 0% of requests, the problem of assgnng multple trp requests to be served concurrently on multple vehcles s called a statc DAR Problem (DARP) (). The statc DARP s usually formulated as a generalzed vehcle routng problem wth pc-up and delvery (). Snce DARP nvolves transportaton of people, tghter tme wndows and an addtonal maxmum rde tme constrant are consdered (). The latter ensures that a passenger does not spend too much tme onboard whle other passengers are pced up and dropped off. The most commonly defned objectve s to desgn m least cost vehcle routes to accommodate n requests under a set of constrants (-, -). The mathematcal formulatons assume that m s a gven nteger. The problem s repeatedly solved by mputng dfferent values for m to fnd the mnmum number of vehcles that can accommodate all the requests (). For a more detaled lterature survey, readers are referred to Berbegla et al. (0). In the followng secton, we provde the mathematcal formulaton of the statc DARP that explans the practcal decsons and ssues arsng n real-world DAR operatons. We formulate t as a fleet sze and mx vehcle routng problem () wth pc-up and delvery and trp outsourcng. Ths formulaton ncludes features le fleet sze, trp outsourcng to taxs, and no dlng of loaded vehcles that are very mportant n practcal applcatons and are commonly consdered n heurstc soluton methods (, ). Despte the practcal mportance of the aforementoned features, the mathematcal formulatons found n the lterature typcally assume a fxed fleet (.e. treated m as a constant), and few consder trp outsourcng () or dlng for loaded vehcles (). The proposed formulaton s bult upon () and follows most of Cordeau s notaton. Formulaton Let n denote the number of users (or requests) to be served. The DARP may be defned on a complete drected graph ( N, A) P D 0,n P,...,n, and G, where N, D n,..., n. Subsets P and D contan pc-up and drop-off nodes, respectvely, whle nodes 0 and n represent the depot. Wth each user are thus assocated an orgn node 0 and a destnaton node n. Let K be the set of vehcles. Each vehcle duraton of ts route cannot exceed nonnegatve servce duraton T. Wth each node K has a capacty Q, and the total N are assocated a load q and a d such that q 0 qn 0, q q n (,..., n), and d 0 dn 0. A tme wndow [ e, l ] s also assocated wth node N, where e and l represent the earlest and latest tme, respectvely, at whch servce may begn at node. Wth TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

6 Marovć et al. 0 each arc (, j) A and vehcle K are assocated a routng cost c j and a travel tme t j whch s assumed to be dentcal for all the vehcles. Let f denote the fxed cost for usng vehcle K, and let g and v denote the fxed and varable tax cost, respectvely. Fnally, let r denote the maxmum rde tme coeffcent such that user P. For each arc to node j. For each node begns servce at node, and r, represents the maxmum rde tme for t n (, j) A and each vehcle K, let x f vehcle travels from node N and each vehcle K, let vehcle s non-empty after vstng node. For each user, vehcle, and y f the customer s outsourced to a tax. j B be the tme at whch vehcle Q be the load of vehcle after vstng node. Let z f L be the rde tme of user on The statc DARP arsng n the real-world operatons can be formulated as the followng mxed nteger program: j Mn f x, ) 0, j cj xj ( g v t n y (eq) K N j N subject to K j N x j y P, (eq) j N x j x j N n, j 0 P, K, (eq) j N x0 j K, (eq) j N x j x j N j 0 P D, K, (eq) N x, (eq), n K 0 B Q Q B j j B d t M( x ) N, j N, K, (eq) j j M z N, K, (eq) M( z ) N, K, (eq) B d t M( x z ) N, j N, K, (eq0) j j TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

7 Marovć et al. Q L j Q q M( x ) N, j N, K, (eq) j j B ( B d ) N, j N, K, (eq) n B B0 T, (eq) n K e B l N, K, (eq) t L r t, n P,, (eq), n K Q Q N, K, (eq) x, z, y {0,}; B, L, s R ; Q Z N, j N, K. j (eq) 0 0 The objectve functon (eq) mnmzes fxed and varable vehcle routng and trp outsourcng costs. Constrants (eq) ensure that each request s served exactly once or s outsourced to a tax. Constrants (eq) ensure that the orgn and destnaton nodes are vsted by the same vehcle. Constrants (eq)-(eq) guarantee that the route of each used vehcle starts and ends at the depot. Constrants (eq) ensure consstency of the tme wndows, whle (eq)-(eq0) prevent a loaded vehcle from dlng. Consstency of the load varable s ensured by constrants (eq). Equaltes (eq) defne the rde tme of each user, whch s bounded by constrants (eq). The latter also act as precedence constrants, because the non-negatvty of the varables ensures that node wll be vsted before node n for every user. Inequaltes (eq) bound the duraton of each route. Fnally, constrants (eq) and (eq) mpose tme wndows and capacty lmts, respectvely. Ths formulaton bulds on the math program n Cordeau (), but dffers n that t ncludes outsourcng of trps (eq and eq), varable fleet sze (eq and eq), and precludes dlng of loaded vehcles (eq-eq). L Soluton Method Implemented n MRMS The optmzaton model presented above can be solved drectly only for small nstances. For medum and large nstances, several approaches that are scalable have been presented n the lterature as summarzed n Table. TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

8 Marovć et al. Table : Overvew of soluton methods for the DARP Group Specfc Soluton Methods Pros Cons. Dynamc programmng Optmalty s Exact. Mathematcal optmzaton usng: guaranteed Methods Branch & Cut, Branch & Prce or nteger L-shaped method Heurstc Methods Metaheurstc Methods. Cluster frst, chan second. Inserton heurstc. Tabu search. Smulated annealng. Genetc algorthms. Varable neghborhood search. Ant colony optmzaton Qucly bulds a feasble soluton Can be bult on top of heurstc methods Cannot solve real-world problem nstances (max about 0 requests) Optmalty s NOT guaranteed Optmalty s NOT guaranteed; typcally taes more tme than heurstc 0 0 Due to ts effcency relatve to other methods, an nserton-based heurstc was mplemented n the MRMS. In partcular, the parallel nserton heurstc () wth mprovement operators (, ) was mplemented and modfed to ncorporate several operatonal requrements specfed by potental users. These requests ncluded worng wth blocs of trps that may or may not be modfed, consderng dfferent passenger needs, and dfferent specfcatons of the objectve functon. The nserton algorthm outlned next dffers from other nserton heurstcs mentoned prevously as follows.. Several changes n trp nserton are made to account for groups of trps that are often referred to as blocs and loced-blocs n the DAR ndustry. The MRMS allows a user to specfy a sequence of trps that may or may not be changed wth subsequent nsertons. Blocs and loced-blocs are often formed for repettve trps that are requested from day-to-day or wee-to-wee. Ths s also the case for groups of people wth common orgns or destnatons. Loced-blocs refer to groups of trps that are not only requred to travel together, but also for contractual reasons, operators are not allowed to add addtonal trps to these groups.. Early mornng, late afternoon, or long trps are fltered and exempt from the nserton. The operator s notfed of these trps and may decde whether to outsource them to taxs. Outsourcng requests to taxs s common practce n DAR operatons. It occurs when dspatchers assess that some trps have partcularly egregous costs due to spatal or tme crtera demand and, thus, would be cheaper to outsource to a thrd party rather than force nto rdesharng tours.. Dfferent weghts may be assocated wth passengers dependng on the requred capacty (e.g. passengers n wheelchars need more space than others).. The vehcles n the depot are heterogeneous n ther capacty and number of wheelchar passengers they can carry, and each route has a pre-specfed start tme. TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

9 Marovć et al. Outlne of the algorthm mplemented n MRMS Intalze: Sort requests accordng to desred pc-up tme and ntroduce the frst vehcle step 0. Notfy operator of early, late, and long trps Preprocessng step. Consder the next unassgned request step. For each ntroduced vehcle: Generate all feasble nsertons of the request nto the schedule and compute the change n the objectve functon step. If there exsts a feasble nserton of the request, then the nserton wth the mnmum ncrease n the objectve functon s selected and the request s nserted whle updatng the schedule. If a feasble nserton does not exst, a new vehcle s ntroduced and the request s assgned to t. step. If there are unassgned requests, then go to step, otherwse go to step step. Apply the followng two operators through desred number of teratons: Trp renserton operator: remove trp from ts current route and try rensertng t nto all the vehcle routes. Mae the renserton wth the mnmum assocated cost (the fnal route may be the same as the current one); Trp exchange operator: remove trp from ts route r and remove trp j from ts route s ( r s) ; nsert the two stops of trp n the best postons of route s and nsert the two stops of trp j n the best postons of route r. If the exchange yelds a cost reducton, mae the swap. Constructs Feasble Soluton Performs Local Search The above modfcatons enabled an algorthm wth academc orgns to be mplemented n the feld, allowng ts applcaton n the daly operatons of several DAR companes. Others, ncludng Madsen et al. () and Toth and Vgo (), also extended Jaw s parallel nserton algorthm to account for practcal aspects arsng n the applcatons they consdered. 0 Testng the Soluton Method on Benchmar Problems To demonstrate the valdty of the mplemented soluton, the algorthm was run on several benchmar nstances where solutons from exact methods were nown. The benchmar problems ncludng nstances were taen from Cordeau (). He used a branch-and-cut algorthm to fnd m least cost vehcle routes to accommodate n requests, assumng that m s gven. The benchmar problems nclude up to trps. The largest nstance solved optmally has requests. Table compares Cordeau s solutons and lower bounds wth solutons obtaned by the mplemented heurstc method. Note that the mplemented heurstc ensures hgher LOS snce t prevents loaded vehcles from dlng. Here, we do not consder outsourcng of requests to tax n order to mae the heurstc comparable wth Cordeau s settng. The heurstc was mplemented n Matlab 00a on a PC wth an AMD Athlon 00 GHz processor and the computaton tmes are compared wth those reported n (). 0 TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

10 Marovć et al. 0 Table. Testng the mplemented heurstc on the benchmar problems Instance Input Heurstc Cordeau 00 Heurstc vs. Cordeau 00 n T Q L Veh Drvng CPU Drvng CPU Veh (hr) (sec) (hr) sec ΔVeh a * a * a * a * a * a * a * a * a * a a a b- 0. * b * b * b- 0. * b- 0. * b * b- 0.0 * b- 0. * b- 0. * b- 0. * b b Relatve gap n drvng hr 0 Note: n, T, Q, L stand for the number of requests, route duraton, capacty, and maxmum rde tme for each user, respectvely. Values mared wth * represent optmal total drvng tme for m vehcles. Other values for drvng tme represent lower bounds. The last column shows the percentage gap n drvng tme between the soluton obtaned through heurstc and the lower bound. The mplemented heurstc provdes compettve results wthn no more than seconds computaton tme. In % of the cases, the mplemented heurstc provded results wth the same or fewer vehcles. The drvng tme was wthn to % of the optmal soluton or bound reported n Cordeau 00. Moreover, computaton tmes needed to solve optmally the nstances n Table justfy the use of the heurstc. For example, nstance b- ncludes requests and s solved optmally n about hours wth the branch-and-cut algorthm. Thus, soluton of the problem nstances wth several hundred requests that face mdsze or large DAR companes would be computatonally ntractable. Even wth recent developments n computer technology, fndng exact solutons would be formdable and heurstcs or metaheurstcs must be appled. TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

11 Marovć et al. 0 0 REGENCY TAXI CASE STUDY Regency Tax (RT) s a tax company and a DAR servce provder located n Maryland, US. Currently, RT has 0 employees and 0 drvers who are fully traned n handlng transportaton needs of dsabled and senor ctzens. The company s fleet conssts of 0 vehcles, some of whch are equpped wth ramps or lfts that facltate access for dsabled passengers travelng n wheelchars. Moreover, the vehcles are equpped wth G or G wreless smart devces,.e. tablets, whch allow regular communcaton wth the dspatch center. In RT s operatons, about 0% of DAR requests are made n advance. A passenger schedules a trp by specfyng pc-up and delvery locatons, as well as the desred pc-up or drop-off tme. After the trp requests are gathered, the dspatchers use the afternoon to desgn routes and schedules for the followng day. A route manfest s electroncally sent to drvers who can start ther assgnments on the followng mornng wth the schedules avalable on ther tablets. The routes are gven n the form of a sequence of pc-up and drop-off locatons wth requested arrval tmes, and drvers are ased to serve the routes n exactly the same sequence that s provded to them. Another 0% of requests are made for the same day and those trps are qucly nserted nto prevously bult routes whle consderng the GPS postons of the vehcles. When dspatchers determne that a request s an outler,.e. does not ft well wth the rest of the demand, they dspatch these trps as a sngle tax trp. As RT s DAR operatons have grown over the past several years, ts management determned that manual dspatch not only produced neffcent routes but also became mpractcal. Therefore, they decded to computerze the dspatchng process. The management envsoned that computerzed routng wll consderably reduce dspatchng effort and provde an even hgher LOS to ther passengers than through manual routng. LOS s an essental part of any DAR contract. Consderng ts daly routne, the company sought a soluton method capable of fndng good feasble solutons to problems of formng routes/manfests wthn less than an hour. The management studed the specfcatons of several commercally avalable software solutons and decded that the MRMS would be the best ft for ther operaton. In the followng secton we evaluate the benefts of mplementng the MRMS n the descrbed operatons. 0 Benefts of mplementng the MRMS: Operatonal Level The mmedate benefts of mplementng the MRMS system were observed n reducng dspatchng, transportaton, and external costs. RT receves about 0 requests on a daly bass. The MRMS was able to route ths number of trps wthn mnutes and thereby save the dspatch staff about dspatcher-hr/day that were prevously needed to buld the manfests manually. Moreover, the transportaton cost was reduced by decreasng the number of vehcles and drvers needed to serve all requests. The MRMS-based routes were compared wth the manually bult route manfest for one day of RT s operatons (Table ). Test comparsons showed that the TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

12 Marovć et al. number of routes and vehcle-m traveled decreased by.% and.%, respectvely. Moreover, the number of routes and vehcle-m traveled decreased despte the mposton of tghter LOS constrants than those observed n manually bult routes. Table ndcates that the MRMS-based routes yeld estmated total annual savngs of approxmately $0. mllon. Ths represents roughly % of total operatng cost. Fnally, negatve external costs, such as emssons and traffc congeston, were smlarly substantally reduced as a result of sgnfcant antcpated reductons n the number of vehcle-hours drven. Table : Operatonal-Level Savngs acheved through mplementaton of MRMS Operatonal Level Costs Manual Routng & Schedulng MRMS Routng & Schedulng Estmated Daly Savngs Estmated Annual Savngs Dspatchng cost dspatcher-hr/day computer-mn/day $00 $0,000 Fxed trans. cost drvers & vehcles drvers & vehcles $0 $,00 Varable trans. cost,00 vehcle-m,0 vehcle-m $,0 $,00 total: $,0 0 0 Note: The savngs are computed by UMD and IT Curves based on feld data and assumng: $/dspatcherhr, $0/drver-day, $0/vehcle-day (vehcle deprecaton), $0./vehcle-m (varable cost), and 0 worng days/year. LOS and operator s constrants for computer-based routng correspond to those observed n manual routes. In addton to evdent cost reductons shown n Table, the MRMS allowed computaton of varous performance measures and vsualzaton of schedules and routes. These features were especally mportant for ganng trust n the methodologes and the ultmate adopton of the tool and ts operatons research-based technques. In manual routng and schedulng t s dffcult to eep trac of performance measures, because they requre tedous computatons. In computer-based routng, on the other hand, a varety of statstcs can be readly generated. These statstcs provde ncreased nsght nto overall system performance, whch s of partcular nterest to system managers. Moreover, statstcs allow better control of vehcle utlzaton and, thus, dscourage drvers from usng the vehcles for ther own needs. Some of the performance measures that the management found useful are shown n Table. Addtonally, the tool s user-frendly nterface (Fgure ) produced recognzable savngs n tme and effort for the dspatchers and drvers, thus also adng the tool s acceptance. 0 TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

13 Marovć et al. Table : Performance measures provde management wth many nsghts and better control over drvers Trps Rte Pass PasKm VehHr Utlz PaVHr PKmVHr Vehcles Total Passenger-Km Total Vehcle-Km Total Drvng Hours Total Vehcle Hours Average Utlzaton Average Vehcle Out-Depot Hours Average # of Passengers on Vehcle. Average # of Passengers on Loaded Vehcle Mnmum Devaton from DPT (mn) Average Devaton from DPT (mn) Maxmum Devaton from DPT (mn) Note: DPT = desred pc-up tme, PaVHr = pass/veh-hr, and PKmVhr = pass-m/veh-hr. Fgure : Compact menu for enterng new requests by specfyng orgn and destnaton locatons, number of passengers travelng, and whether the passengers are travelng n wheel chars. TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

14 Marovć et al. 0 0 Benefts of mplementng the MRMS: Tactcal Level Besdes cost reductons acheved at the operatonal level, the MRMS provded ts user wth a valuable tool that can be used for decson mang at the tactcal level. The MRMS allows operators to qucly study dfferent operatonal scenaros and explore tradeoffs between LOS and varous system characterstcs. Such nsghts play a ey role n mang long-term nvestment decsons n the company s fleet, or when sgnng a contract whch promses a certan LOS. Some of the nsghts that operators usng MRMS are fndng useful are provded next. One of the factors nfluencng LOS s the tme wndow promsed to customers. Intutvely, wder tme wndows provde more flexblty to the operator, but lower LOS for the customers. The MRMS allowed ts users to quantfy the cost of provdng dfferent tme wndows and explore these tradeoffs. The requred fleet sze ranged between and vehcles as the tme wndows were relaxed from 0 to 0 mn, respectvely, whle eepng a fxed hour lmt on route duraton (sold lne n Fgure a). Ths nsght s especally mportant n sgnng contracts wth transt agences that typcally specfy mnmum LOS to passengers n terms of tme wndows and maxmum system response tme. The correspondng operatng daly cost may be reduced from roughly $,0 to $0,00 as tme wndows are relaxed from 0 to 0 mnutes (sold lne n Fgure b). Operators usng MRMS were also able to quantfy the change n fleet needed to meet ts demand when the route duraton lmt was relaxed from to 0 and hours (Fgure a). Ths nsght s useful n negotatng general agreement contracts that may mpose a lmt on drver worng hours. It also allows operators to better estmate the value of drver overtme hours by assessng the correspondng operatng cost (Fgure b). Route Lmt hr Route Lmt 0 hr Route Lmt hr Fleet Sze (vehcles) mn mn 0 mn mn 0 mn 0 mn Tme Wndows Fgure a: Fewer vehcles are needed as the tme wndows and route duraton lmts are relaxed () TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

15 Marovć et al. Route Lmt hr Route Lmt 0 hr Route Lmt hr Operatng Cost ($/day),000,000,000,000,000,000,000,000 0,000,000,000 0 mn mn 0 mn mn 0 mn 0 mn Tme Wndows 0 Fgure b: Operatng cost decreases as the tme wndows and route duraton lmts are relaxed. Note: The assumed costs are: $0/drver-day + $0/vehcle-day (vehcle deprecaton); $0./vehcle-m A factor that consderably nfluences operator costs s the dstrbuton of demand over tme. It s theoretcally expected that steader demand yelds better utlzaton of transportaton assets (). The MRMS allows ts users to quantfy effects of potental changes n demand dstrbuton. Fgure shows the orgnal and two slghtly modfed demand dstrbutons (Orgnal, Modfed, and Modfed ). The two modfed demand dstrbutons were obtaned by redstrbutng some of the requests orgnally scheduled durng the pea hours. Results n Fgure ndcate that fewer vehcles are needed to satsfy steader demand, whle eepng the same number of trps and ther orgn-destnaton locatons, 0 mnute tme wndows, and hour route lmt. Ths nsght s mportant n antcpatng potental savngs that could be acheved by spreadng some of the pea demand. To reach ths goal, the DAR operators could provde ncentves that encourage ther customers to travel durng off-pea hours, such as dscounts n co-pay pad by the customer and more drect servce due to fewer on-board passengers. TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

16 Marovć et al. 0 0 Orgnal Modfed Modfed Number of Trps Demand Dstrbuton: Orgnal Modfed Modfed Fleet Sze: vehcles vehcles vehcles Operatng Cost $, $, $, Fgure : More unform demand results n smaller fleet and operator s cost needed to serve all the requests () Through senstvty analyss, t was determned that ncreasng vehcle capacty from a base of passengers does not mprove the solutons. Ths occurs because other operatonal constrants, ncludng a tme wndow of between 0 and 0 mnutes, route duratons of between and hours and maxmum rde tmes per passenger prevent the algorthm from tang full advantage of vehcle capacty. Ths fndng s valuable n determnng long-term fleet composton. Tme 0 0 CONCLUSIONS A computerzed system, the MRMS, for managng DAR operatons was developed. The developed management system was mplemented n a number of mdsze transportaton companes yeldng consderable cost reductons. Test comparsons of manual and MRMS-based routes ndcated that a typcal md-sze operator performng approxmately 0 trps per day could reduce ts annual dspatchng and transportaton costs by an estmated 0. mllon dollars, nearly % of the total operatonal expense. The MRMS also provded the operators wth a powerful tool for qucly explorng system performance under dfferent operatonal scenaros. Insghts gleaned from studyng a varety of operatonal scenaros were found mportant for tactcal plannng where an operator needs to determne the cost of provdng a certan level-ofservce, long-term fleet composton, and utlty of drver overtme hours. These fndngs encourage further mplementaton of optmzaton methods wth realstc constrants n DAR TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

17 Marovć et al. 0 operatons, because they can consderably reduce the costs of provdng these servces. Snce DAR operatons are heavly subsdzed, computerzed systems le MRMS could also help reduce some of these governmental expendtures. Future extensons of the MRMS wll nclude the development and mplementaton of a dynamc dal-a-rde algorthm to effcently serve requests that are receved wth slght advance notce, whle the DAR vehcles are already worng. Addtonal features wll be ncluded to contnuously reoptmze routes n real-tme and to manage vehcle devatons from the schedule. The dynamc algorthm wll be able to qucly rensert passengers ntally assgned to vehcles that ether fall behnd schedule or brea down. These mprovements wll further ncrease the effcency of the DAR operatons, as well as the compettveness of the proposed management system. Acnowledgments Ths wor was funded by the Maryland Industral Partnershp (MIPS) and the IT Curves company. Ths support s gratefully acnowledged, but mples no endorsement of the fndngs. 0 0 References. Borndoerfer, R., Groetschel, M., Klostermeer F., C. Kuettner.. Telebus Berln: Vehcle schedulng n a dal-a-rde system. Techncal Report SC -, Konrad-Zuse- Zentrum fuer Informatonstechn Berln.. Metro Moblty: Independence and opportunty for,000 metro resdents. Metro Councl Newsletter, 00. Revew of MetroAccess Rdershp, Costs and Polcy. WMATA Fnance Commttee Report. June, 00. Trends n Paratranst Technology: A Whte Paper by Trapeze Software Group. Trapeze Group 00. Cordeau, J.F., G. Laporte. 00. The dal-a-rde problem: Models and algorthms. Annals of Operatons Research,.. Cordeau, J.F. 00. A branch and cut algorthm for the dal-a-rde problem. Operatons Research, -. Luo, Y., Schonfeld P. 00. A rejected-renserton heurstc for the statc Dal-A-Rde Problem. Transportaton Research, B(),. Cordeau, J.K, Laporte, G. 00a. The dal-a-rde problem (DARP): Varants, modelng ssues and algorthms. OR-Quart. J. Belgan, French Italan Oper. Res. Soc., -0.. Cordeau, J.F., G. Laporte. 00b. A tabu search heurstc for the statc mult-vehcle dala-rde problem. Transportaton Research. B, -. TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author

18 Marovć et al Berbegla, G., Cordeau, J.F., Grbovsaa, I., G. Laporte. 00. Statc pcup and delvery problems: A classfcaton scheme and survey. Top,.. Golden, B., Assad, A., Levy, L., Gheysens, F.. The fleet sze and mx vehcle routng problem. Computers & Operatons Research, Volume, Issue, -. Jaw, J. J., Odon, A. R., Psarafts, H. N., N.H.M. Wlson.. A heurstc algorthm for the mult-vehcle advance-request dal-a-rde problem wth tme wndows. Transportaton Research, 0B(), -. Helporn, G., Cordeau, J.F., Laporte, G. 0. An nteger L-shaped algorthm for the dala-rde problem wth stochastc customer delays. Dscrete Appled Mathematcs, (), -. Parragh, S.N. 0. Introducng heterogeneous users and vehcles nto models and algorthms for the dal-a-rde problem. Transportaton Research Part C, July 00.. Toth, P., Vgo, D.. Fast local search algorthms for the handcapped persons transportaton problem. In: Osman, I.H., Kelly, J.P. (Eds.), Meta-Heurstcs: Theory and Applcatons. Kluwer, Boston, 0.. Luo, Y. Heurstcs and Performance Metamodels for the Dynamc Dal-a-Rde Problem. 00. PhD dssertaton, Unversty of Maryland, College Par.. Madsen, O.B.G., Ravn, H.F., Rygaard, J.M.. A heurstc algorthm for a dal-a-rde problem wth tme wndows, multple capactes, and multple objectves. Annals of Operatons Research, 0(), 0.. Toth, P., Vgo, D.. Heurstc algorthms for the handcapped persons transportaton problem. Transportaton Scence, (), 0.. Marovć, N., R. Nar, P. Schonfeld, E. Mller-Hoos, and M. Mohebb. Optmzng Dala-Rde Servces for Washngton, DC Regon. TSC Report 0-, Unversty of Maryland, College Par, July Ftzsmmons, J.A., Ftzsmmons, M.J. 00. Servce Management: Operatons, Strategy, and Informaton Technology, rd Edton, Irwn/McGraw-Hll TRB 0 Annual Meetng Orgnal paper submttal - not revsed by author