A shortest path algorithm in multimodal networks: a case study with time varying costs

Transcription

1 A shortest path algorithm in multimoal networks: a case stuy with time varying costs Daniela Ambrosino*, Anna Sciomachen* * Department of Economics an Quantitative Methos (DIEM), University of Genoa Via Vivali 5, Genova, Italy Abstract In this paper we eal with the problem of fining optimal routes in multimoal networks. Since moal change noes play a relevant role in the choice of origin estination paths, an consequently in the computation of multimoal shortest paths, we evaluate the performance of such noes with the aim of increasing their attractivity. We propose an algorithm that focuses on the moal change noes an forces as much as possible routings through those noes that coul be profitably selecte as commuting points. Preliminary results of a computational experimentation aime at valiating the propose algorithm with ranomly generate multimoal networks are reporte together with a case stuy relate to the city of Genoa, Italy. Keywors: multimoal networks, shortest path problems, moal change noes, generalize cost function. 1 Introuction an problem escription Intermoality, that consists of a change of carrier uring the travel, one or more times, using ifferent connections, is nowaays almost necessary in many kins of transports, both concerning freight an passenger mobility. There is hence the nee of establishing the proper sequence of means an commuting points that coul allow avantages for the travellers. The costs of multimoal routes are given by the cost of the travels relate to each moality plus the cost implie by the change of moality, that is the so calle transition costs. When using multimoal routes some travelling costs can be reuce since economies of scales can be obtaine when using ifferent transportation moalities, especially those relate to the mass transit moality. Moreover, the usage of a mass transit moality allows a great saving in terms of cost an pollution; unfortunately, this is not always available an can rarely reach many points of a multimoal network. Thus we have higher monetary an time costs ue to the change of moality, associate with a greater iscomfort pai by the users. When a user reaches an intermoal noe, the corresponing transition time is usually given by the time spent for fining a free an legal parking, the walking time require for reaching the train / bus stop, an the waiting time at the bus stop or train station. Generally these terms are functions consisting of a eterministic part plus a stochastic eviation (i.e. rivers on t know exactly the time require for looking for an available car park an reaching it once they arrive at the moal change noe). The monetary cost component is given by the parking fee. Most of the papers presente in the literature about intermoality focus on urban mobility, an in particular on the shortest path problem in urban multimoal transportation networks; see among others [8], [7], [6] an [3]. In this paper we focus on urban passenger mobility networks; however, the algorithm here escribe can be applie also to any kin of multimoal networks. The main goal of the propose approach is to fin origin estination (o-) multimoal least cost paths forcing as much as possible routings through those noes that coul be profitably selecte as commuting points. Our objective function consiers both time an monetary cost components; the require optimal paths are thus referre to herein as Pareto-optimal paths. Note that since such solutions are not generally equivalent to the ecision makers, they are evaluate also on the basis of a sort of rating of the moal change noes. Of course, ifferent priorities or orers in the selection of moal change noes can be use, accoring to several criteria, such as the buffer queue or some structural characteristics of the noes like the connectivity, as efine in [2]; another selection criterion can be to consier only primary an seconary moal change noes, as they have been efine in [1], that is moal change noes that guarantee very easy connections to the mass transit transportation network. In the search for optimal multimoal routes we first compute the require shortest paths by looking for the 1

2 most well performing moal change noes from both the origin an the estination noes; successively, we evaluate the corresponing possible multimoal paths with the aim of fining more convenient the intermoal solutions than the mono moal ones. We base our computations on the Dijkstra s algorithm [4], but any of the classical shortest path algorithms propose in literature can be use as well (see e.g. [5] for a survey of such algorithms). Our goal is to compute multimoal o- paths in a reasonable amount of CPU time, without having the nee of verifying exhaustively every possible o- multimoal connections. We are also intereste in analysing how the propose approach is sensible to any kin of upgraing of the ata concerning the network; in particular, we consier some perturbation in the ata about travelling times. Moreover, this analysis allows to face possible travel time changes over a ay ue to a variation of the traffic congestion in transportation networks. 2 The network moel In orer to be able to represent all the features of multimoal transportation problems we have to eal with moels that enable us to inclue all the parameters that can impact on the corresponing ecisional process, especially those relate to the moal change points. For this purpose, focusing on urban transportation networks, we consier igraph G = (V, A, C, I, D, M, R) with the following specifications. V is the set of noes that efines, as usual, relevant locations within the network; great attention will be evote to N V, that represents the subset of possible moal change noes. Let V = m an N = n. Set A efines the oriente arcs, or connections, between pair of noes; C is the set of weights (travelling times) associate with each arc (i,j) A, while I is the set of weights (transition costs) associate with each noe belonging to N. D is the set of attributes associate with the moal change noes that will be evaluate in the selection phase of the algorithm for etermining the shortest multimoal paths. A better explanation of costs an attributes of the network moel will be given in Subsections 2.1 an 2.2. M is the set of the transportation moalities that are allowable in the network uner consieration; we may split them into two main classes, namely: a) restricte moalities, that is private ones, such as car or motorcar, which can be use only by the ecision maker; b) mass transit moalities, that is share ones, such as train or bus, which can allow a multiple transport with a single carrier. We have to consier that people generally on t like more than one changes of mean in the same trip; in fact, moal changes are perceive uncomfortable by users an usually are not free. For this reason, once the change of moality from the private to a mass transit one is selecte, the presente algorithm tries to o up to two moal changes, that is we assume that no more than two means for each o- path is allowable; more precisely, we say that noe j is connecte to noe i in a mass transit transportation network if an only if there exists a path from i to j in G in which at most one change of a mass transit mean may occur. Finally, R is the set of the possible ecisional criteria use to evaluate the generalize cost of the whole journey in the multimoal network. The specification of any element r R that we consier in this work is follows: r = 1 is relate to the traveling time, r = 2 concerns the monetary cost, r = 3 can refer to some iscomfort factors, such as weather conitions in the case of urban mobility, reliability in case of ata transfer an so on. 2.1 Definition of the (arc an noe) costs in G We consier ifferent kins of weights, associate, respectively, with the arcs an noes of the network. Set C is the set of the costs associate with the arcs of the network; every element c(i,j) t,r C is the cost for travelling arc (i,j) A with moality t accoring to a given ecisional criterion r. In orer to better represent the user s preferences, we use a weighte value for each criterion, that is we compare the cost of ifferent routes by computing for every consiere o- path travelle with moality t the generalize R cost function C(o,) t given in (1) as C( o, ) t = c( i, j) t, rϖ (1) r r = 1 2

3 R where ω r is the weight that users assign to the r-th evaluation criterion, such that ω r = 1. r = 1 In literature, a lot of works eal with travelling time functions; assuming that every problem shoul have its own cost function, we legitimately assume that the travelling cost is a linear function; for example, we may suppose that, in a roa transportation network, the time function epens on the length of the street, the with, the traffic conition, the grae line an the level of bow-street (see e.g. [9]). I a relevant set in our moel an for our algorithm too, since it efines the transitions costs that users have to pay at the moal change noes. Set I of the cost associate with the noes of the network is a M R matrix. Let τ(i) ab,r be a generic element of set I representing the cost that the users have to pay for moving from moality a, that is the restricte one, to moality b, that is one of the possible mass transit moality, at the moal change noe i N accoring to a given ecisional criterion r. Note that we imply that τ(i) ab,r = + i N, that is we assume that a change of moality is not allowe anywhere. 2.2 Attributes D is a particular set that is ifferent from the others since it contains the measure of the qualitative characteristics of the moal change noes. We think that for the ecisional process concerning the transportation moalities it is relevant the evaluation of the moal change noes from a structural point of view, that is to verify their connection capability to the mass transit transportation network from the restricte one. Therefore, we consier the following attributes. Connectivity. The connectivity is the first attribute that we verify in the evaluation of every possible moal change noe. This inex δ ) gives a measure of how a given moal change noe i N is connecte to the ( 1 i other noes of the network; it is hence given by the ratio between the number x i,t of noes that can be reache from i without any moal or vehicle change over the total number of noes of V. Note that if more than one moality is available, the connectivity inex is relative to the moality with higher connectivity. Usually the connectivity inex is compute for evaluating, as in our case, the gooness of the connection between the restricte moality, that is chosen at the origin noe, an the mass transit one, that can be selecte for reaching the estination noe. Accessibility. This inex ( δ 2 ) concerns the istance from all noes in V to noe i, i N. That is, the i accessibility inex gives a measure of how easily a moal change noe i N can be reache by the other noes of the network. Expecte time. The expecte time (e.g. queue or transfer) spent at a moal change noe may be calculate by an a-hoc function specifically for each case. Note that a too long waiting time may often iscourage the user an make him/her chooses another noe, thus implying a change in the strategy. For this reason, we may think to o not consier as possible commuting points those moal change noes that have an excessive waiting time to enter to. Generally, the waiting time associate with each noe may be compute as the prouct between the number of elements actually in the queue an the mean processing time; in this case the expecte queuing time at a moal change noe i N ( δ 3 ) can be given by the prouct between the average waiting time an the i number of users in the queue at noe i. Remember that set D is a component of the network an it is efine a priori at the beginning of the process, therefore users on t know precisely the queue time associate with the moal change noes. Thus set D can be now generally efine as 1 2 D = { δ i, δi,..., δi }, i N; in our case we have = 3. As previously escribe, the selection of the moal change noes is our main issue in fining the best intermoal path. We suppose that users on t know the effective generalize cost function associate with the commuting noes an thus make their choice in low information conitions. In fact, when users reach a moal change noe, they know the network status of a noe an the available connections from such noe to the others; however, they may only suppose the average waiting time at it basing the choice on a previous knowlege instea of calculating the real cost. For this reason, attributes are a set of information that users know an use to reuce the number of moal change noes to be consiere in the computation of the shortest paths, as we will see in the next paragraph. 3

4 3 The propose algorithm The presente algorithm is a heuristic approach to the problem of fining optimal multimoal o- routes in urban transportation networks. The algorithm looks for the best moal change noes an computes the minimum cost path using such noes for the given pair of o- noes. In particular, by using a Dijkstra like algorithm, we first compute the require shortest path by looking for the most well performing moal change noes from both the origin an the estination noes; successively, we evaluate the generalize cost of possible alternative multimoal paths with the aim of forcing as much as possible routings through those noes that coul be selecte as commuting points while reucing the previously etermine mono-moal solution. At the beginning of the ecision process, the user efines the upper an lower bouns for attribute δ i, = 1,, 3; these bouns are the threshol value for selecting a moal change noe in the search for the shortest path. In particular, let α 1, α 2,, α an β 1, β 2,, β be the upper an lower bouns, respectively, of valuesδ i, = 1,, 3; if i N an β δ i α D then noe i will be ae to the set of possible commuting points. For each kin of networks we may have ifferent evaluation parameters, as to better escribe the user s ecision process. We assume that attributes δ i, = 1,, 3, are alreay known at the beginning of the algorithm. In more etails, the propose algorithm procees by first executing the Dijkstra algorithm starting from the given origin noe. The selection process of the moal change noes occurs within the labelling phase of the algorithm. At each iteration every selecte noe is analyse; if the noe is a moal change one an the values of its consiere attributes fit into the allowable ranges then the noe is consiere as possible commuting noes. The Dijkstra algorithm stops as soon as a given number k n of reachable moal change noes are positively evaluate or we are not able to fin any other caniate noe. Finally, we execute the Dijkstra algorithm having the estination noe as our starting noe, as before, until k moal change noes are selecte. Let Ω an enote the sets of the ientifie moal change noes starting from the origin an estination noes, respectively. The Dijkstra algorithm is execute in both irection for every restricte moality. Note that, ue to the multi objective nature of the problem, every cost comparison has to be mae relatively to the generalize cost function; that is, we look for the path having the minimum value of equation (1) consiering as interchanging noes only those belonging to Ω an with respect to the chosen criterion r, r = 1,, 3. Note that in the selection of moal change noes ifferent priorities or orers can be consiere. The algorithm takes into account only moal change noes that guarantee very easy connections from the private network to the mass transit one, an vice versa. For setting up our attributes before the execution of the 1 2 algorithm, we fix the minimum acceptable value of δ i an δ i, to 40%; this means that those noes with both connectivity an attractivity values lower than 40% are not consiere as commuting points an hence as possible origin noes. Note that the connectivity value is evaluate for the selection of the origin interchange noes belonging to Ω, while the attractivity is relevant for the selection of the estination interchange noes belonging to. 4 Computational experimentation An extensive computational effort evote at valiating the propose algorithm is in progress. A massive number of trials with ranomly generate instances have been use as a test be for verifying both the gooness of the obtaine solutions an the corresponing CPU time. We are also applying our algorithm to a urban passenger transportation network relate to the central area of the city of Genova, Italy. The algorithm has been implemente in C++ an we are testing it on a 3200 Mhz ual core with 512 Mb RAM platform. In Table 1 we report some preliminary computational results relate to ranom instances of ifferent sizes. Each entry of Table 1 is the average value of the objective function (1) compute on the basis of 100 ifferent ranomly generate graphs of the same size. Values in column Optimality gap has been obtaine as the ifference of the generalize cost (1) between the optimal solution foun with the exhaustive algorithm an the solution foun by our algorithm. Note that on the average the optimality gap is only about 6,75% for the non optimal solutions; more precisely, in at most 34% of the instances the non optimal paths are averagely between 5% an 8,5% more expensive than the optimal 4

5 solutions. In fact, the propose algorithm was able to fin the Pareto optimal solution in a very high percentage of cases; in particular, from the observe results, we can see that the algorithm fins the optimal solutions about two times every three trials. Combine with the number of successes, this means that the algorithm is able to fin on the average a solution that is less than 5% worst than the Pareto-optimal solution. As far as the CPU time is concerne, we can note that, while the CPU time of the exact algorithm grows very quickly, our heuristic algorithm reaches the solution in a very small CPU time. Instance type Table 1. Computational results with ranom instances # noes # arcs CPU time our algorithm CPU time exaustive algorithm # optimal solutions (%) <0,01 1,49 66% 6.52% <0,01 7,17 64% 7.91% ,016" 35,89" 64% 8.40% ,04" 1'07" 68% 5.51% ,055 2'16 62% 5.32% Optimality gap (%) Referring to the city of Genova an assuming that the mass transportation eman level is given, we look for the best intermoal solution for the most travelle o- paths. Starting from the efinition of G an the available o- matrices referring to a rush hour time perio, i.e. from 6.30 to 9.30 a.m., we have erive the corresponing network moel that represents the central area of the city an is relate to a linear extension of about 5 kilometres. The size of the network is the following: m = 56, n = 46, A = 1141, whose 997 refer to the mass transit moalities, that consist in turn of a multimoal network since it is split into bus, subway, train, elevator an cableway connections. We focus on 3 noes that are the most relevant origin noes in terms of traffic flow, since they represent the access to the centre for rivers coming from the west, east an north sie of the city, respectively, an on 2 noes in the centre of the town where it is concentrate the commercial heart of the city, that represent the most frequent estinations. In Tables 2-3 we present the values of the most relevant o- shortest paths; columns Ω an represent the possible moal change noes from the origin an the estination noes, respectively, in the consiere path; the remaining columns report the total cost an that relate to the corresponing moality, incluing the walking time. In the case of the path from 56 to 18, synthesise in Table 2, the solution foun by our algorithm is Paretooptimal. Note that even if noe 56 is a moal change noe, it has been iscare by the algorithm ue to its ba attributes; in this case, the moal change noe is 27. Table 2. Cost of the multimoal path Ω Total Private Mass transit Interchange Walking ,31 1,53 9,55 10, ,84 1,53 11,06 10, ,97 1,53 8,54 10,23 2, ,17 1,63 7,85 4, ,68 1,63 9,36 4, ,83 1,63 6,84 4,69 2, ,03 2,19 5,94 6, ,54 2,19 7,45 6, ,09 2,19 4,93 6,9 2,67 A ifferent situation arises in the case of the path from 42 to 18, reporte in Table 4; this case is interesting to be analyze, since noe 42 is a moal change noe an it is not iscare by the algorithm. Having a look at the solution, that is the Pareto-optimal one, we can see that noe 42 is not the interchanging noe selecte by the solution; inee the change of moality occurs at noe 27. Table 3. Cost of the multimoal path Ω Total Private Mass transit Interchange Walking ,96 0 9, , , ,37 0 8,7 0 2, ,66 0,88 9,55 10, ,17 0,88 11,06 10, ,32 0,88 8,54 10,23 2, ,08 1,54 7,85 4, ,59 1,54 9,36 4,69 2 5

6 5 Sensitivity analysis ,74 1,54 6,84 4,69 2,67 As a final valiation of the propose algorithm we test the stability of the foune optimal solutions in the presence of noise in the ata, an more precisely of the weight of the arc set A. In fact, consiering the technical ifficulties usually encountere in getting precise ata, in particular information about ata flows, it is important that the algorithm is stable enough to allow to fin goo solutions also when a high ata accuracy level cannot be guarantee. For this purpose, we implemente a ranom noise generator that perturbe the arc costs. More precisely, we implemente the function anoise X that as to the arc weights a perturbation up to X% of the corresponing given input value; that is every arc cost has been moifie up to the efine percentage of noise (more or less) before running again the shortes path algorithm. Table 4 reports the synthesis of our computational results relative to ifferent levels of noise on ifferent kins of ranomly generate graphs. Table 4. Percentage of optimal solution in presence of perturbation on the arc cost Noise # optimal solutions (%) Optimality gap (%) 5% 100% 0% 10% 90% 3.52% 15% 90% 3.52% 20% 80% 5.23% 25% 75% 6.21% Each row of Table 4 reports the average value of 10 ifferent networks, each one of the type given in Table 1 for every noise level, an the corresponing comparison with the solution obtaine with the exact algorithm without any noise. We can see that results are very goo up to 15% of noise, while starting from a 20% noise level, the solution s reliability egenerates quickly. Preliminary results show that the algorithm has a goo tolerance to the noise. The main reason is that the noe selection is one also on the basis of the qualitative information about the structural characteristics of the same noes. 6 Conclusions an outlines for future works In this paper we have presente an algorithm for fining optimal paths in multimoal networks having ecisional weights on both arcs an noes. Preliminary tests shown that the algorithm has goo performances both in terms of CPU time an optimality gap, an it is able to fin moal change noes when such noes are well connecte to the other moalities. The authors inten to exten the computational experiments an the sensitivity analysis. References [1] Ambrosino D., Sciomachen A. Selection of moal choice noes in urban intermoal networks in Urban an transport XII, Brebbia C.A., Dolezel V. Es., WIT Press, (2006) pp [2] Benacchio M., Musso E., Sciomachen A. Intermoal transport in urban passenger mobility - in Urban transport an the environment for the 21st century, IV, Southampton-Boston: Wessex Institute of Technology Press, Computational Mechanics Publications, [3] Bielli M., Boulmakoul A., Mouncil H. Object moelling an path computation for mutimoal travel systems. European Journal of Operational Research, 175/3, (2005) pp [4] Dijkstra, E. W. A note on two problems in connexion with graphs. Numerische Mathematik, 1, (1959) pp [5] Gallo G., Pallottino S. Shortest path algorithms, Annals of Operations Research, 13, (1998) pp [6] Lozano A., Storchi G.. Shortest viable path in multimoal networks. Transportation Research A, 35, (2001) pp [7] Moesti, P., Sciomachen, A. A utility measure for fining multiobjective shortest paths in urban multimoal transportation networks, European Journal of Operational Research, 111/3, (1998) pp [8] Mote, J., Murthy, I., Olson, D.L.. A parametric approach to solving bicriterion shortest path in urban multimoal transportation networks, European Journal of Operational Research, 53, (1991) pp [9] Sheffy Y. Urban transportation networks. Prentice Hall, Englewoos Cliffs, NY,