Designing the Master Schedule for Demand-Adaptive Transit Systems

Transcription

1 Designing te Master Scedule for Demand-Adaptive Transit Systems Teodor Gabriel Crainic Fausto Errico Federico Malucelli Maddalena Nonato May 2008

2 Teodor Gabriel Crainic 1,2,*, Fausto Errico 1,3, Federico Malucelli 3, Maddalena Nonato 4 1 Interuniversity Researc Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) 2 Department of Management and Tecnology, Université du Québec à Montréal, C.P. 8888, succursale Centre-ville, Montréal, Canada H3C 3P8 3 Politecnico di Milano, DEI, Piazza Leonardo da Vinci, Milano, Italy, University of Ferrara, via Saragat 1, Ferrara 44100, Italy Abstract. Demand-Adaptive Systems (DAS) display features of bot traditional fixed-line bus services and purely on-demand systems suc as dial-a-ride, tat is tey offer demand-responsive services witin te framework of traditional sceduled bus transportation. A DAS bus line serves, on one and, a given a set of compulsory stops according to a predefined scedule specifying te time windows associated to eac, providing te traditional use of te transit line, witout requiring any reservation. On te oter and, passengers may also issue requests for transportation between two desired, optional, stops, wic induces detours in te veicle routes. Te design of a DAS line is a complex planning operation tat requires to determine not only its design in terms of selecting te compulsory stops, but also its master scedule in terms of te time windows associated to te compulsory stops. Designing a DAS tus combines elements of strategic and tactical planning. In tis paper we focus on determining a master-scedule for a single DAS line. We propose a matematical description and a solution framework based on te estimation of a number of statistical parameters of te demand and te DAS line service. Results of numerical experiments are also given and analyzed. Cooperation as problem-solving and algoritm-design strategy is widely used to build metods addressing complex discrete optimization problems. In most cooperative-searc algoritms, te explicit cooperation sceme yields a dynamic process not deliberately controlled by te algoritm design but inflecting te global beaviour of te cooperative solution strategy. Te paper presents an overview of explicit cooperation mecanisms and describes issues related to te associated dynamic processes and te emergent computation tey often generate. It also identifies a number of researc directions into cooperation mecanisms, strategies for dynamic learning, automatic guidance, and selfadjustment, and te associated emergent computation processes. Keywords. Public transit, demand-responsive systems, demand-adaptive systems, sceduling. Acknowledgements. Partial funding for tis researc was provided by Natural Sciences and Engineering Researc Council of Canada (NSERC) troug its Industrial Researc Cair and Discovery grant programs. Tis researc was also partially funded by Regione Lombardia, Italy, troug its INGENIO grants. Fausto Errico wises to tank Professor Stein W. Wallace for an enligtening conversation eld in Montreal in December Results and views expressed in tis publication are te sole responsibility of te autors and do not necessarily reflect tose of CIRRELT. Les résultats et opinions contenus dans cette publication ne reflètent pas nécessairement la position du CIRRELT et n'engagent pas sa responsabilité. * Corresponding autor: Teodor-Gabriel.Crainic@cirrelt.ca Dépôt légal Bibliotèque nationale du Québec, Bibliotèque nationale du Canada, 2008 Copyrigt Crainic, Errico, Malucelli, Nonato and CIRRELT, 2008

3 1 Introduction Traditional transit services are particularly suited to andle situations were te demand for transportation is strong, i.e., wen tere is a consistently ig demand over te territory and for te time period considered. Te ig degree of resource saring by a large number of passengers makes it ten possible to provide efficiently and economically ig quality, i.e., frequent, services operating generally ig-capacity veicles over fixed routes and scedules. Routes and scedules may and do vary during te day, but, in almost all cases, tey are not dynamically adjusted to te fluctuations of demand. In contrast, wen te demand for transportation is weak, e.g., during out of rus-our periods or in low-population density zones, operating a good-quality traditional transit system is very costly. In particular te fixed structure of traditional transit services cannot economically and adequately respond to significant variations in demand. Demand-responsive systems are a family of mass transportation services wic, as teir name suggests, are responsive to te actual demand for transportation in a specific time period. Suc services evolve toward a personalization of mass services: itineraries, scedules, and stop locations are variable and determined according to te needs for transportation as tey cange in time. Demand-responsive systems were introduced under te name of Diala-Ride (DAR) as door-to-door services for users wit particular needs or reduced mobility, suc as andicapped and elderly people [12, 8]. Te flexibility of DAR systems to respond to varying individual requests for transportation provides te means to offer more personalized services, wile still maintaining a certain degree of resource saring. Tis as lead certain transportation or city autorities to extend DAR services to more general transportation settings. DAR systems display, owever, a number of drawbacks, some of wic follow from te extreme flexibility inerent in te system definition. Tus, for example, because te supply of transportation service canges according to needs expressed for particular time periods, neiter te transit operator nor te users may predict te veicle itineraries, stop locations, and associated scedules. As a consequence, users are obliged to book te service well in advance of te actual desired time of utilization and te actual pick up time is very muc left to te discretion of te operator. For similar reasons, it is extremely difficult to integrate DAR and oter traditional transit services. A new type of demand-responsive systems, denoted Demand-Adaptive System (DAS) as been introduced to address some of tese issues [10, 11]. DASs are transit services displaying features of bot traditional fixed-line bus service and purely on-demand systems suc as DAR. In oter words, a DAS attempts to offer demand-responsive services witin te framework of traditional sceduled bus transportation. A DAS bus line serves, on one and, a given a set of compulsory stops according to a predefined scedule specifying te time windows associated to eac, providing te traditional use of te transit line, witout in-advance reservations. On te oter and, similarly to DAR services, passengers may issue requests for transportation between two desired, optional stops (not necessarily on te same 1

4 line), wic induces detours in te veicle routes. Similarly to most transportation systems dedicated to serve several demands wit te same veicle, traditional transit systems involve a complex planning system made up of many interrelated decisions. Scematically, te design of te system in terms of line routes is determined during te so-called strategic planning pase, timetables and veicle scedules and routes are part of te tactical planning pase, and crew scedules are built during operational planning (Ceder et al. [2]). Comparatively, purely on-demand services suc as DAR, need little strategic design, mainly to define service areas and te composition of te fleet (e.g., number and type of veicles). Te most important planning process for DAR is at te operational level wen routes and scedules are determined little time before actual operations and are possibly dynamically modified one service as begun. DAS services combining caracteristics of traditional and on-demand systems require bot a system-design pase and an operational, time and user request-dependent adjustment of veicles routes and scedules. Te latter as been addressed in Crainic et al. [6] and Malucelli et al. [9]. Te former forms te topic of tis paper. It is, in a certain sense, a more complex planning operation tan for traditional transit because it requires not only to determine te design of te line as te selection of te compulsory and optional stops, but also te determination of te time windows associated to te compulsory stops. Designing a DAS tus combines elements of strategic and tactical planning. To empasize tis caracteristics, we identify te output of te process and te master-route network and te master-scedule, respectively. In tis paper we focus on determining a master-scedule for a single DAS line. We propose a matematical description and a solution framework based on te estimation of a number of statistical parameters of te demand and te DAS line service. A sampling approac is used for te estimations. Te remaining part of te paper is organized as follows. We give a brief description of DAS services in Section 2, wile commonalities and differences among sceduling DAS, DAR, and traditional transit services are discussed in Section 3. Section 4 is dedicated to te description of te DAS line master-sceduling problem and te solution framework we propose. We discuss te effectiveness of te metod end computational results in Section 5 and conclude in Section 6. 2 Demand Adaptive Systems Demand adaptive systems were first introduced by Malucelli, Nonato, and Pallottino [10] and ten treated in a more general context by Crainic et al. [6] (see also, [5, 4, 9]). A similar type of service is also described in [11]. 2

5 A DAS targets low-density/volume demand areas and attempts to conjugate te advantages of traditional transit transportation services and te flexibility of on-demand personalized services. It is based on te observation tat even in suc areas tere are locations were a relatively important part of te overall demand may be consistently found: railway and underground stations, sopping centers, ospitals, etc. Tis leads to te possibility to economically design a backbone transit service covering tese most attractive stops, wile allowing veicles to detour as needed to pick up and drop off passengers at te oter stops. Te latter capability, combined to an on-request booking system, increases customer satisfaction and te dimension of te potential user group. In its most general form, a DAS is made up of several lines and is connected to te lines of te traditional transit system. Several veicles operate on eac DAS line providing service among a sequence of compulsory stops. Eac compulsory stop is served witin a predefined time window. Te collection of time windows corresponding to te compulsory stops, including te start and end of te line, makes up te master scedule of te DAS line. Tis makes up te traditional part of a DAS. Additional service and flexibility is provided by allowing customers to request service from and to optional stops, tat is, stops wic are served only if a request is issued and it is accepted. We identify users wic request service at an optional stop as active, wile users moving only between compulsory stops are identified as passive. Figure 1: A Basic DAS Line Serving te Compulsory Stops To serve optional stops, te veicle must generally deviate from te sortest pat joining two successive compulsory stops. Te set of optional stops tat it is possible to visit between two consecutive compulsory stops is part of te design of te DAS line and is denoted segment. An optional stop cannot belong to more tan one segment. Figure 1 depicts te basic DAS service of te compulsory stops, wile Figure 2 illustrates te same DAS line wen user requests for optional stops are present. Transfers between DAS lines and between tese and regular transit lines take place at compulsory stops. Time windows play an important role in tis context because tey establis time relations among different DAS and traditional lines wic sare te same com- 3

6 Figure 2: Te DAS Line Serving Optional Stops pulsory stops. Te time windows in te master scedule also influence te flexibility te service may provide for user requests at optional stops. Te wider tey are, te more flexibility tere is available. Yet, it is not possible to increase teir widt arbitrarily, because te service would slow down excessively, loosing attractiveness. Notice finally, tat te time windows and te segment specification provide an a priori guarantee relative to te longest time users migt ave to spend traveling on te line. In any case, te detours associated to optional stops must be consistent wit te time windows at compulsory stops. 3 Sceduling Issues in DAS, DAR and Traditional Transit Systems Sceduling is a fundamental planning activity for any transportation system, in particular for DASs, DAR and traditional transit systems. Te nature of te sceduling process canges significantly, owever, according to te type of service at and. In traditional transit systems, a scedule indicates te passing times at eac stop of eac line. Veicles are supposed to follow tese times as strictly as possible, since users of te system base teir trip plans on te publised scedules. Sceduling problems in traditional transit system belong to te so-called tactical planning level, te line definitions and te service frequencies being usually assumed known. Once te scedule as been establised, it remains uncanged for medium-term periods, suc as six monts or one year. For a more in-dept discussion of sceduling issues in traditional transit systems, te interested readers are referred to [2]. Te situation is different for DAR systems. Scedules are still indicating veicle itineraries, stops, and passing times, but tese particular to eac veicle tour according to te actual requests for transportation accepted for te corresponding time period. Tis corresponds 4

7 to an operational planning level activity tat decides on all scedule components (i.e., line itinerary, stops, and passing times), wic are valid only for te duration of te specific service. See [3] for a review of te topic. Te case of DASs is more complex. Because DAS aims to provide demand-responsive services witin te framework of traditional sceduled transit transportation, its sceduling combines te two planning processes briefly sketced above. Two scedules are tus built. A master scedule defines te partial line (veicle) itineraries, te sequence of compulsory stops, and time windows at tese stops. Tis scedule plays te same role for te transit autority and te passive users of te system as te scedule in traditional transit systems. At operation time, te actual scedule is built to incorporate te additional, optional stops corresponding to te accepted active-user requests, wile respecting te time windows constraints imposed by te master scedule. Te problem of finding a DAS scedule at operational level was addressed in [9, 6]. Building te master scedule is a tactical planning level activity, were actual service times at compulsory stops are modified according to te season. It is also an important component of te strategic planning process, as te definition of te segments making up te line requires te specification of time windows at compulsory stops. Te next section is dedicated to te tese issues for a single DAS line. 4 Te DAS Line Master Sceduling Problem Tis section is dedicated to te issue of determining te master scedule of a single DAS line, tat is, determining te time windows for te compulsory stops of te line. Tis so-called DAS Line Master Sceduling Problem (DLMSP) may be viewed as te last stage of te DAS line design problem wic is addressed in more dept in [7]. Te single-line DAS design problem assumes tat te territory to be covered by te DAS line as been determined, te travel times between any pair of potential stops in te territory (tese include transfer points to oter lines or transportation systems) ave been accurately estimated, and tat a measure of te transportation demand among te potential stops is available. For a given time orizon were demand is assumed stable (e.g., morning rus our), te DAS line design problem is made up of several interrelated decisions regarding te selection of compulsory stops among all te potential stops in te territory, teir sequencing, te partitioning of te optional stops into segments, and te determination of te master scedule, tat is te definition of te time windows veicles will ave to respect at compulsory stops. Te first tree components make up te so-called topological-design pase of te problem and a number of metodological approaces are proposed in [7] to address various problem settings, e.g., objectives to be satisfied (veicle cost, travel time experienced by users, a combination tereof, etc. - we use a combination), weter all potential stops sould be reacable by te designed line, and so on. Te last component of 5

8 te design process constitutes te object of tis paper. A formal model for DLMSP is presented in Section 4.1. In Section 4.2 we focus on te Single Segment Master Sceduling Problem, a core subproblem in addressing te DLMSP. Section 4.3 presents te complete solution approac we propose for te DLMSP. 4.1 Problem description and modeling Te problem of building te DAS line master scedule, tat is, to fix te time windows at all compulsory stops of te line, assumes two inputs. Te first consists in te topological design of te line: te ordered set of compulsory stops and te associated set of optional stops partitioned into segments. Te demand for transportation between te stops of te line makes up te second input. Te coice of time windows must be performed taking into account several conflicting goals. First, te master scedule sould provide sufficient time between compulsory stops suc tat, during actual operations, veicles may serve all requests for service at optional stops. Second, for economical reasons, te total maximum time of te line, travel and stops, to be as sort as possible. Finally, quality of service criteria also induce conflicting actions: wile users already on te bus prefer narrow time windows, to avoid long delays at compulsory stops, and sort travel times between consecutive compulsory stops, to avoid being on te bus for long, users at optional stops prefer longer travel times tat allow veicles to detour by teir stop. To illustrate te incompatibility of tese goals, consider tat simultaneously enforcing small time windows and ig probability of being able to serve all potential requests implies a rater long travel time for eac segment compared to te corresponding sortest pat. But, since te actual number of requests is usually small compared to te total potential number, suc a strategy would result in veicles arriving at compulsory stops well before te earliest departure time, significant dead times at compulsory stops for users, and long ride times for te line. To avoid tis, time windows ave to be smaller, resulting in a smaller probability of being able to serve te wole set of issued requests. Tis is indeed general DAS operational policy (Malucelli, Nonato, and Pallottino [10] and Crainic et al. [6]), requests tat cannot be accommodated being eiter lost, served by a later veicle or by taxi, according to te policy of te transit company. We tus assume a maximum widt for time windows at compulsory stops and aim to select teir actual widts to minimize te maximum veicle ride time, wile guaranteeing to serve te set of requests wit a given probability. Te maximum widt and te service probability are, of course, managerial decisions and tus application dependent. Demand for transportation is usually described as te number of potential trips tat migt be requested during te time period considered for eac pair of stops. Based on tis 6

9 information, it is straigtforward to compute te probability of at least a request being issued for a given pair of stops, as well as te probability of eac optional stop of being requested for service eiter to board or to aligt a veicle. We work wit tis last set of probabilities in te model we propose. Consequently, te goal of serving te wole set of requests wit a given probability becomes serving te wole set of requested stops wit a given probability. Tis makes it easier to address te problem. To formally write te model, consider a sequence of compulsory stops H = {f 0,f 1,...,f n }. Sets of optional stops F, wit = 1,...,n are associated wit eac pair of consecutive compulsory stops f 1,f. Te sets F are mutually disjoint. We define a directed grap G = (N,A ) for any pair f,f +1 suc tat N = F {f 1,f } and A = N N. We call G a segment and G = G. A traversing (travel) time c ij is associated to eac arc (i,j) A. A probability p i of being requested for service is associated to eac optional stop s i F. Te DLMSP consists in associating to eac compulsory stop f a time window [a,b ] suc tat te veicle must not leave te compulsory stop f before a, nor after b ; it is allowed, owever, to arrive to f before a. Our goal is to find te best sequence { a 1,b 1, a 2,b 2,..., a n,b n } wic, wit a given probability P ǫ, guarantees service for every optional stop wic can be requested for service. We define te best sequence as te one displaying te smallest value of b n. For te sake of simplicity of exposition, we consider te case were all time windows are of equal widt b a = δ, wic reduces te problem to tat of finding te best sequence {b 0,b 1,...,b n }. Te procedure we propose extends straigtforwardly, owever, to te cases wen 1) te time windows are fixed but different for te compulsory stops, and 2) time windows are bounded by a maximum widt value but are allowed to be smaller. Te latter case could also consider finding, for a given compulsory stop f, te maximum a wic guarantees wit a given probability no veicle dead time at f. We now focus on a core subproblem tat estimates te travel time, and tus te time window at te destination compulsory stop, of a single segment. Te full algoritm will ten bring togeter te sequence of segments making up te route. 4.2 Te Single Segment Subproblem How long does it take to travel a segment? Te answer obviously depends on wat optional stops ave to be serviced and tis is usually different eac time a veicle travels te line and te segment, because service at optional stops follows particular user requests tat ave been accepted. Consequently, suc operational information must be estimated for te tacticalplanning purpose of building a master scedule. We propose a statistical estimation of te travel time of a given segment based on an efficient sampling procedure. 7

10 Consider te generic segment G = (N,A ), and recall tat N = F {f 1,f } f 1 and f are te initial and terminal compulsory stops of te segment, respectively, F is a set of optional stops, and A = N N. Let L 1 represent te departure time from compulsory stop f 1. As indicated previously, eac optional stop s i F as a positive known probability p i of being active, tat is, of being requested for service during a particular veicle run. We assume tese probabilities to be mutually independent. Te set of optional stops tat are simultaneously active during a veicle run, S N, is denoted te active set (wit every s i S being active, wile every s i N \ S not being active). Te probability of any set S to be active, p S, is positive and may be easily derived from te probabilities p i of its active optional stops. Te time required by a veicle to travel from te initial to te terminal compulsory stop of segment serving a given set S of active optional stops in S is denoted H and is called te service time of set S. Assuming an efficient operation of te line, te service time H may be approximated at planning level as te duration of te sortest pat starting in f 1, ending in f, and passing by all te stops of te active set S. We tus used a Minimum Hamiltonian Pat solver to compute H for our experimentations. (More sopisticate procedures may be implemented to take advantage of particular application attributes, but tis does not cange te general beavior of te proposed metodology.) Te service time associated to segment is of course a random variable at planning level and we denote it H (ω). Te goal of te Single Segment Meta Scedule Problem (SSMST) is to determine te lowest value b wic guarantees, wit probability 1 ǫ, tat te veicle as sufficient time to serve an active set. Tat is, fix b = H 1 ǫ tat P{H (w) H 1 ǫ } 1 ǫ. + L 1, were H 1 ǫ is suc Te computation of H 1 ǫ requires te knowledge of te Cumulative Distribution Function (CDF) and tus of te Probability Mass Function (PMF) of te random variable H (ω). Since te latter requires 2 N Minimum Hamiltonian Pat computations, tis approac is not computationally affordable in most cases. Consequently, we estimate te PMF and sampling appears as te metod of coice. It is difficult to estimate ow large a sample tat represents adequately te population of active sets sould be, but we suppose it could become quite large. Ten, for computing efficiency reasons, we propose instead te very simple following algoritm: Take a number {r 1,r 2,...,r l } of random samples of relatively small cardinality; For eac sample r k, compute its PMF k and CDF k, as well as te value of b k ; Compute te mean value and standard deviation of b k ; If te standard deviation is close to te mean value, tat is if te solution is precise, stop; Oterwise, increment te cardinality of te samples and iterate te previous steps. 8

11 Te undeniable advantage of tis algoritm is its simplicity. On te oter side, one cannot guarantee an unbiased solution, nor tat te dimension of te samples will stay witin computationally reasonable dimensions. A number of parameters (e.g., te number of samples) must also be calibrated. Yet, as illustrated by te results of Section 5, te metod is very effective and adverse effects are not noticeable. 4.3 Solution Approac to DLMSP We now present te complete solution metod for te DAS line master scedule problem, were we need to sew segments togeter. In te previous subsection, we decoupled segments by assuming tat te veicle departure time from its initial compulsory stop f 1 was known for eac segment G. Actually, tis is true for te first segment only, te departure time from te first compulsory stop f 0 being ere arbitrarily denoted t = 0, wic also translates into P{L 0 (ω) = 0} = 1. For all subsequent segments, te departure time depends upon te arrival time of te previous segment, wic depends on its departure and service times, wic depends upon... Since service times are random variables, segment departure times for all segments but te first are also random variables. We terefore introduce te random variable L (ω) representing te veicle departure time from compulsory stop f, were a L (ω) b. We introduce also te random variable T (ω): T (ω) = L 1 (ω) + H (ω). (1) not constrained to belong to [a,b ]. Te veicle departure times at two consecutive compulsory stops and te service time for te segment to wic tey belong to are ten related as follows: T (ω) if ω a T (ω) b ; L (ω) = a if ω T (ω) < a ; (2) b if ω T (ω) > b. We assume in te following, witout loss of generality but wit a simplified notation, tat as long as a veicle arrives at a compulsory stop f during te interval [a,b ], te arrival and departure times coincide. Recall tat te value b for segment G as to be suc tat it is possible to serve all possible active sets wit a given probability. We must terefore compute te PMF (and consequently of te CDF) of T (ω), tat is, select b = T 1 ǫ, were T 1 ǫ is suc tat P {T (ω) T 1 ǫ } > 1 ǫ. Notice tat, by ypotesis, H (ω) and L 1 (ω) are independent. Consequently, te PMF of teir summation, T (ω), can be computed troug te simple convolution of te PMFs of H (ω) and L 1 (ω). Te problem of finding b ten reduces to te problem of estimating 9

12 te PMF of H (ω) and L 1 (ω). We sowed in te previous subsection ow to compute te CDF of H (ω). Te CDF of L 1 (ω) may be easily obtained from tat of T 1 (ω). We can now present te sceme of te algoritm we propose for te DLMSP. Te algoritm accepts as input te sequence of segments G = 1,2,...,n G and te service probability P ǫ = (1 ǫ) n, and proceeds as follows: 1. For every segment G, {1, 2,...,n} (a) Compute PMF and CDF of L (ω) (b) Compute PMF and CDF of H (ω) (c) Compute PMF and CDF of T (ω) as te convolution of te PMFs of L 1 and H (d) Compute T 1 ǫ and set b = T 1 ǫ. Te algoritm returns te best sequence {b 1,b 2,...,b n } of latest departure times for te segments, suc tat any randomly requested optional stop is served wit probability P ǫ = (1 ǫ) n. 5 Results Tis section is dedicated to a discussion of computational results relative to te estimation of te PMF of random variables H (w) wen considering decoupled segments, as well as of te corresponding values H 1 ǫ. Tis is in fact te core point of te solution metodology proposed. Experimental results support, in particular, te claim tat precise and unbiased values of H 1 ǫ are obtained even for relatively small sample dimensions. We tested our algoritm over square-saped segments wit initial and terminal compulsory stops located at te extremities of one of te diagonals. Optional stops are uniformly distributed on te square. Distances between optional stops are Euclidean and traveling times are proportional to distances wit proportionality constant 1. We generated instances wit a number of stops, including te two compulsory ones, varying from 20 to 50 by steps of 10. To eac optional stop is associated a probability included in te open interval (0, 1). For eac problem dimension we randomly generated tree instances, different bot in te possible locations of optional stops and te probabilities associated to tem. Table 1 displays te main features of te test problem instances: name, time lengt of te longest pat wit only one optional stop, te time lengt of te Hamiltonian pat, te sum of te probabilities associated to optional stops, and te number of optional stop wit probability grater tan 50%. Hamiltonian pats are computed wit a modified version of te Asymmetric TSP code available in [1]. 10

13 Name Longest 1-Pat Hamiltonian Pat Sum Prob > 50% A B C A B C A B C A B D Table 1: Features of Test Problem Instances Tables 2, 3, 4, and 5 report te computational results over te four sets of problem instances. Te columns display, respectively, te name of te instance and its dimension, te number of samples created, teir dimensions, te value 1 ǫ, te average of H 1 ǫ over te number of samples, te standard deviation, and te computing time in CPU seconds. Te experimental results indicate tat increasing te dimension of te sample yields more precise solutions. Te number of samples is relatively less important. Te results also indicate tat, in general, te standard deviations are very good, being average estimated values - standard deviation ratios smaller tan 2% even in te worst case. Regarding possible biases, we compared te values obtained by using our algoritm to tose obtained by computing H 1 ǫ using a single sample of cardinality As supported by te figures in Table 6, te values we found are very good. To conclude, notice tat te values of H 1 ǫ are remarkably smaller tan tose of te Hamiltonian pat caracterizing segments. Tey sould actually be even better in practice. Tis is because in our experimentation we considered probabilities associated to optional stops uniformly distributed in te interval (0, 1). Yet, in te real word, stops wit a value close to 1 would most likely be cosen as compulsory stops. In oter words, in an actual implementation of a DAS service, optional stops are not requested frequently and consequently we expect better H 1 ǫ - Hamiltonian pat-lengt ratios. 6 Conclusions In tis paper, we examined from a sceduling point of view a new type of semi-flexible transit service, te Demand-Adaptive Service. Comparing it to traditional transit and dial-a-ride services, we introduced te new sceduling requirements of DAS, wic we identified as te 11

14 Name&Dim. nsamples dimsamples 1-epsilon H 1 ǫ Dev-Strd Time (sec.) A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B C C C C C C C C C C C C C C C Table 2: Results for Segments wit 20 Nodes 12

15 Name&Dim. nsamples dimsamples 1-epsilon H 1 ǫ Dev-Strd Time (sec.) A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B B C C C C C C C C C C C C C C C Table 3: Results for Segments wit 30 Nodes 13

16 Name&Dim. nsamples dimsamples 1-epsilon H 1 ǫ Dev-Strd Time (sec.) A A A A A A A A A A A A A A A B B B B B B B B B B B B B B C C C C C C C C C C C C C C C Table 4: Results for Segments wit 40 Nodes 14

17 Name&Dim. nsamples dimsamples 1-epsilon H 1 ǫ Dev-Strd time (sec.) A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B C C C C C C C C C C C C C C C C Table 5: Results for Segments wit 50 Nodes Name&Dim. nsamples dimsamples 1-epsilon H 1 ǫ Dev-Strd Time sec. A A A A Table 6: Values of H 1 ǫ Computed over large Samples 15

18 construction of a master scedule. We formalized te master sceduling problem for a single DAS line and proposed a solution framework based on decoupling te origin-destination demand and using a particular sampling tecnique. Computational results sow tat te metod we propose is efficient and produces ig-quality results. Acknowledgments Partial funding for tis researc was provided by Natural Sciences and Engineering Researc Council of Canada (NSERC) troug its Industrial Researc Cair and Discovery grant programs. Tis researc was also partially funded by Regione Lombardia, Italy, troug its INGENIO grants. Fausto Errico wises to tank Professor Stein W. Wallace for an enligtening conversation eld in Montreal in December Wile working on tis project, Dr. Teodor Gabriel Crainic was Adjunct Professor at te Department of Computer Science and Operations Researc of te Université de Montréal, Canada, and at Molde University College, Norway. References [1] Carpaneto, G., Dell Amico, M., and Tot, P. Exact Solution of Large-scale, Asymmetric Traveling Salesman Problems. ACM Trans. Mat. Softw., 21(4): , [2] Ceder, A. and Wilson, H.M. Public Transport Operations Planning. In Design and Operation of Civil and Environmental Engineering systems, pages Jon Wiley & Sons, Inc., New York, [3] Cordeau, J.-F. and Laporte, G. Te Dial-a-Ride Problem (DARP): Variants, Modeling Issues and Algoritms. 4OR, 1(2):89 101, [4] Crainic, T.G., Malucelli, F, and Nonato, M. A Demand Responsive Feeder Bus System. In CD-ROM of 7t World Congress on Intelligent Transport Systems. 7WC-ITS, Torino, Italia, [5] Crainic, T.G., Malucelli, F, and Nonato, M. Flexible Many-to-few + Few-to-many = An Almost Personalized Transit System. In Preprints TRISTAN IV - Triennial Symposium on Transportation Analysis, volume 2, pages Faculdade de Ciências da Universidade de Lisboa and Universidade dos Açores, São Miguel, Açores, Portugal,

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