The Theoretical Framework of the Optimization of Public Transport Travel

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1 The Theoretical Framework of the Optimization of Public Transport Travel Jolanta Koszelew # # Faculty of Computer Science, Bialystok Technical University, Wiejska A, - Bialystok, Poland jolka@ii.pb.bialystok.pl Abstract With the increase of cars on the streets, more and more people in big cities, decide to move from one place to another by public transport, such as tram, metro or bus. Public transport users need detail informations about network connections. Therefore, some urban portals offer to use systems of optimization of public transport travel. However, finding one or several optimal routes, according to user preferences, is a difficult problem. This paper presents an original method for determining optimal paths in public transportation network. Author describes a network model with timedependent and bi-modal links, defines the problem of finding optimal routes with users preferences and proposes algorithm, which can be a solution of this problem.. Introduction Plan journeys become more popular in public transport services. People, which use metro, bus or train, want to know, how to get, as quick as it possible, from specified origin to a destination. They also have various limits for their routes, for example: maximum travel time, maximum number of bus-changes, maximum walking distance (Walk to the nearest us stop is an additional kind of link in our network model), minimum probability of successful realization of the the route). Standard shortest path (or paths) algorithms [] find the shortest path(s) in networks with static and deterministic links, meanwhile algorithms for a scheduled transportation network are time-dependent. Edges in such networks can only be traversed at certain points in time and the weights of these edges change in a day and are associated with them. Moreover, standard algorithms considered graphs with one kind of links. Graph in our model has two kinds of edges: directed links which represent connections between bus-stops and undirected edges correspond to the travel between each pair of neighbors nodes (bus-stops) on foot. Additionally, with each node in graphs that represent transportation network, is concerned detail information about: timetables, coordinates of bus-stops, etc. This information is necessarily to determine weights of links during realization of the algorithm. Those three differences between graphs in standard shortest path(s) problem and public transportation networks cause that complexity of algorithms which solve routing problem in the such original network representation, increase. Special methods have to compiled for a construction of optimal paths in this model. New algorithms are needed to accommodate users preferences. In this paper, the author presents a new version of the certain labeling algorithm [], which is solution of K-shortest paths problem and cover users conditions. Before the presentation of the algorithm in section, the author define in the section the network model and in the section the problem of finding optimal routes with user preferences.. Network Model A public transportation network is represented as a directed graph G = V, E where V is a set of nodes and E is a set of edges. Each node in a graph G corresponds to a certain transport station. We assume, for simplification, that there is only one kind of the public transportation bus, so each node corresponds to a bus-stop. This assumption doesn t narrow applications of the presented methods. Each node (bus-stop) is labeled by natural number from to n, where n is a total number of busstops. The edge ( i, j) E exists when is at leas bus connection between the bus-stop number i as a source point and the bus stop number j as a destination. Each edge has a weight t ij, which is an integer number. A value of t ij is time-dependent and is determined during the realization of the algorithm, on the base of information, which is included in the i and j nodes. The following information is contained in the node number i (bus-stop number i): /07 $

2 - numbers of bus lines, which are stop at i. We denote those numbers by i, i,..., etc. - coordinates of the i: x, y (needed to calculate weights of walk-links). Coordinates determine the position of a bus-stop on the map of city. - timetables of each lines i, i,, etc.. We assume, for simplification, that all lines have only one directed route and timetables for work days and weekends are the same. Each element of timetable consists of two numbers: hours and minutes of the departure. Timetable for line i j is a sequence d ij of departure times for line number i j from bus-stop number i. Each element of the sequence is a pair of integer numbers: (h ij, m ij ) h ij hour of departure, m ij minutes of departure - list of nodes which are connected with node i; the list of neighbor bus-stops. Bus-stop j is a neighbor of bus-stop number i, if exists at least one bus connection between the bus-stop number i as a source point and the busstop number j as a destination. Besides bus connections, graph has additional walk links. We assume, that there exists walk link beetween bus-stops i and j, if a time which is needed to walk from bus-stop i to j is less then fifteen minutes. We can calculate this time by the equation: ( i, j) = ( x x ) + ( x x ) dist walktime i j () i ( i, j) = dist( i, j) scaleofmap walkspeed where walkspeed is the input algorithm parameter an average walking speed in km/h and scaleofmap is a scale of map. A graph representation of such the network is shown in Fig.. It s a very simple example of the transportation network. In real world a number of nodes is equal to 00 for the city with 0 ts. citizens. In the next section we define problem of optimal routes with user preferences. The input data of this problem is a graph defined above and travelers additional conditions.. Problem of finding optimal routes with users preferences Optimal route finding is a shortest path problem. Dijkstra s algorithm [] is of course the most efficient and effective solution of this problem in its classical version. However, Dijkstra s algorithm does not allow for timedependent links. Dreyfus [] developed many methods for graphs with time-dependent links, but his algorithms don t accommodate users preferences. New approach is needed to allow the such conditions. Public transport users preferences may be various. We consider the most practical of them: maximum travel time, maximum number of bus-changes, maximum walking distance, minimum probability of successful realization of the route. j lines coordinates x =0, y =0 :7.:00, 7:0 :7:0, 7: :7:0, 7: :7:0, 7: timetables x =0, y =0 :7.:0, 7: :7:08, 7:0 :7:0, 7:0 x =0, y =0 :7.0, 7: :7:07, 7:9 walk links bus links Figure. Representation of a simple transportation network The last preference needs some explanations. What is the minimum probability of successful route realization? Assume, that there are two optimal routes, which include one bus-change, with the same travel time. Which of them should user choice? He can compare both routes in terms of his chance of a bus change. If the time interval between arrival and departure of buses for the first route is equal to four minutes and is equal to two minutes for the second path, then probability of successful realization of the first one is greater then for the second. User can define probability distribution for successful realization of bus-change, conditioned by the time interval between buses arrival and departure. Minimal set of users preferences consists of the following elements: v start source point of a travel, number of the started bus - stop, v stop destination point of travel, number of the finished bus-stop, t start time of a travel beginning, k the number of optimal paths, which algorithm must generate Additional preferences may be very special. Actually, it is unpractical to take every individual conditions into consideration. In this paper, the following preferences are considered: max changes upper limit of bus-changes count, max walking upper limit of walking distance, max time upper limit of travel time, min prob lower limit of a probability of successful realization. x =0, y =0 :7:08, 7: :7:, 7: :7: 7: x =0, y =0 :7:7, 7:7 :7:, 7: :7:, 7: x =0, y =0 :7:, 7:7 :7:, 7:0 :7:, 7: :7:, 7: /07 $

3 Probability of travel successful realization is original for each passenger, because it s determined on the base of probability distribution on the time for bus-change (). We will sign this distribution by BC_distribution. Algorithm, which is presented in the next section, determines at last k shortest paths for a given public transportation network. Each route has the same source and destination bus-stop and time of the travel beginning. Values of other parameters (additional preferences) may be different for different paths. There are two approaches to defining k-shortest paths problems. In classical version of this task (without additional preferences), we find k paths with the minimal travel time [7]. If the set of input parameters includes additional parameters (additional preferences), we can treat them in two different way. The first possibility lies in that all additional preferences have the same importance [8]. In this case algorithm generates k shortest routes with minimal travel time and each path satisfies all additional preferences, i.e. the number of bus changes is smaller then max changes, total walking distance is smaller then max walking value and probability of successful travel realization is greater then min prob. In the second approach we assume, that preferences may have different importance. In this paper we will consider this second possibility. It is practical to take the lowest weight for max changes value. If the path doesn t include buschanges or this value is small, automatically values of other parameters are good for traveler, i.e. travel time and walking distance are the smallest and probability of successful travel realization is the greatest (is equal to ). Example. Sample input parameters for the transportation network in Fig. Base preferences : v start =, v stop =, t start = 7:00, k =, Additional preferences in order of importance: I. max time = (minutes), II: min prob = 0.7, III: max changes =, IV: max walking = 0 (meters), Above order of importance does it mean that if do not exist k routes, which satisfy all additional conditions, algorithm generates routes, which do not satisfy max walking preference. If do not exist k routes, which satisfy I. II. and III. preference, algorithm determines routes, which do not satisfy max changes condition, etc. 0, for 0 t < 0,7 for t < BC _ distribution() t = () 0,8 for t <,0 for t 8 where t is a time (in minutes) for a bus-change, walkspeed = (km/h) scaleofmap = 000 Table. Optimal routes for k= and the transportation network in Fig.. T, T, T optimal routes, L, L, L bus lines for corresponding routes routes T - L T - L T L buschanges time travel walk (km) probability of successful realization If there are two or more routes, which satisfy the same set of additional conditions, algorithm sorts routes according to the priorities of preferences.. Algorithm for optimal routes in public transportation network problem In order to find optimal paths (for k>=), it is important for the algorithms to choose different routes throughout the network []. It can be realized by labeling nodes and edges or by removal of a node or a edge []. Because it is easier to implement the labeling algorithms than the path deletion algorithms for the transportation network, the algorithm described in this paper is based on the label setting technique []. Moreover, the presented algorithm consists of four main steps. In the first step it generates paths without bus- changes and walk distances. Next, standard breath-first-search method (BFS) [] is applied to construct the shortest paths in a direct graph G. The length of paths which are generated in this step is equal to the number of edges on the path and bus-changes are allowed. Then, the algorithm determines the last kind of paths. They are include possibility of bus-changes and walk distance. This third part of the algorithm is based on the method presented in []. For each generated path all values of the additional users preferences parameters are determined: the number of bus-changes, walk distance, time travel and probability of succesfull realization. Finally, the algorithm divides the set of all generated paths to five subsets. The first subset includes paths, which satisfy all users preferences. Paths in this subset are sorted according to the priorities of preferences. Second subset consists of paths, which satisfy three the most important conditions and is also sorted in order of priority of those conditions. Next subset includes paths, which satisfy two main preferences. The fourth subset consists of paths, which come true only one main condition and the last subset includes other routes. First k /07 $

4 ranked paths from each subset is the result of algorithm. OPTIMAL_ROUTES (G, v start, v stop, t start, k, max changes, max walking,, max time,, min prob ) Step : Establish numbers of bus-lines, which stop on v start and v stop location. Determine values of all parameters (travel time, number of bus changes, walk distance and probability of successful realization) for each direct connection and add to a subset R D (direct routes) Step : Use standard BFS procedure to construct all shortest paths in a direct graph G from origin node v start to the destination node v stop. The length of path is equal to the number of edges on the route. Because each generated path must be different, do not use the same set of intermediate nodes (exclude one node in one path). Determine values of parameters for each route and add to a subset R BFS (routes with the minimal number of busstops). Step : Run below label setting algorithm LSA to get the ranked k-shortest paths from the origin to the destination node. 7:0 7:00 7:0 Step.: Find the closest node to the origin by walking. For the network in the Fig. node is the closest node to node by walking, because the time walking from to is equal to minutes. Step.: Compare the arrival time for bus and walk to the closest nodes. Time travel by bus to node is smaller then time walking from node to node. Label node. Step.: Upgrade the network according to the labeled node. In the Fig., there is a bus link from node to node, so the network is upgraded by adding a bus link from the node to the node. There are also walking links from the node to the node, the node and the node, so the network is upgraded by adding links from node to node, node and node. Since there is already a walking link between node and node, the walking link -- becomes the second walking link from node to. It must be added separately as a second walking link with time 9 (+). 7: :00 7:0 7:0 7:0 8 Figure. Step. of LSA algorithm for network in Fig. 7:0 8 8 Figure. Simple transport network with bus links and departure time (in minutes) for two bus lines (routes: - - and ---) and walk links with travel time. LSA algorithm: 7:0 7:09 7:0 Step.: Find the closest node to the v start by bus. Node j is the closest to i node iff the time taken to travel between nodes is minimal. For the network in the Fig. the closest node to v start = is the node, because the time travel from to at t start =7:00 is equal to minutes. If suppose in Step. time walk from the node to the node would be equal to minutes, then the node would be labeled as the closest to node the and the network would be upgraded using the same rules. Step.: If the labeled node is the destination node, add route to the subset R LSA (subset of routes which are generated as a result of LSA algorithm). Step.: Repeat step. to step. until the k ranked shortest path has been added to the subset R LSA. Step.7: Determine values of all parameters for each the shortest route and add to subset R LSA Step : Determine set R of all generated routes. R = R R R { } D BFS LSA /07 $

5 Step : Divide the set R to five new subsets R, R, R, R, R. Subset R includes all routes from R, which satisfy all additional user preferences, the subset R is consists of paths, which satisfy three the most important conditions. Next subset R includes paths, which satisfy two main preferences, the subset R consists of paths, which come true only one main condition and the last subset R includes other routes. 7:0 7:00 7 7:0 7:0 Figure. Step of the LSA algorithm, assuming that the network in Fig. has the time walk from node the to node the equals to minutes. Step : Sort each subset R i (i=...) according to the set users preferences, which are, correspond to the given subset. Step 7: First k ranked paths from R in the first order, from R in the second order, R and R, R in the last order are the result of this algorithm. The designed algorithm generates a list of ranked shortest paths and those paths are compared with users preferences. The graph, which is an input data structure of the algorithm is generated on the base of timetables (bus links) and coordinates of bus stops (walk links). The complexity of the algorithm for all steps behind the Step is not high, but the execution time of labelsetting algorithm, which is realized in Step is still long. For the graph with about 00 nodes (bus-stops), the algorithm returns, five paths with all user preferences after about three minutes. The next stage of the research will focus on reducing the overlap rate by carefully embedding the constraints into the algorithm investigating different methods to ensure credible execution time. In the next paper will be introduced the concept of stochasticity to the above algorithm based on methods presented in [9]. Stochasticity greatly increases the complexity of the rout finding problem, so greater algorithmic efficiency becomes imperative. Different kinds of heuristics [8] will 7:0 be investigated and implemented one after another until satisfying performance is gained.. Conclusions This paper presents some of the approaches to solving of the shortest paths problem in a time-dependent scheduled transportation network with bi-modal links. The algorithm described in the previous section works on random generated, non-realistic network now. The next step of the research will be implementation of the algorithm in a full functional system, which will help travelers to find the optimal routes. In addition we may provide a graphic user interface []. Our system needs to work with a geographic information about location of bus-stops. The algorithm and system will finally be implemented for Bialystok (about 0 nodes) and Warsaw (about 00 nodes) public transport network. Various kinds of Warsaw public transport will be take into consideration: buses, trams and underground. Acknowledgement This paper was supported by the Rector of Bialystok Technical University (grant no.s/wi//0) References [] R. K. Ahuja, Dynamic shortest path minimizing travel times and costs, citeseer.nj.nec.com/ 8.htm, 00. [] I. Chabini, Discrete dynamic shortest path problem in transportation applications: Complexity and algorithms with optimal run time, Transportation Research Records,, 998, pp [] H.K. Chen, G. Feng G, Heuristics for the dynamic useroptimal route choice problem, European Journal of Operational Research,, 000, pp. -0. [] E. Dijkstra, A note on two problems in connection with graphs, Mumerische Mathematik,, 99, pp. 9-7 [] S. E. Dreyfus, An Appraisal of Some Shortest-path Algorithms, Operations Research, 7, 99, pp. 9-. [] P.J. Elkins, Service management systems for public transport the German approach, Proceedings of the IEE Colloquium on Vehicle Location and Fleet Management Systems, 99, pp [7] J. K. Hartley, A. Bargiela, Decision Support for Planning Multi-Modal Urban Travel, Proceedings of th European Simulation Symposium, Marseille, 00, pp /07 $

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