Tariff Zones Design in Integrated Transport Systems: a case study for the Žilina Municipality

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1 Tariff Zones Design in Integrated Transport Systems: a case study for the Žilina Municipality MICHAL KOHÁNI Department of Mathematical Methods and Operations Research Univeity of Žilina Univerzitná 8215/1, Žilina SLOVAKIA kohani@frdsa.fri.uniza.sk Abstract: Sustainable transportation and transport integration are connected with a large numbe of challenges, especially in public transportation within the cities and suburban regions. One of the ways how motivate citizens to use public transportation is an effective integrated transport system (ITS), which integrates several transportation means (buses, trams, trains, etc.). Designing of tariff system in ITS is also an important issue. Tariff system should be undetandable and motivating for passenge, but, on the other hand, effective for transport operato and municipalities. There are several approaches how to design a tariff system, for example a division the region into smaller areas - tariff zones can be used. Changing of fares can also affect the demand for traveling. Higher fare often occu a drop in demand and number of carried passenge, it also affects the total income from passenge. We will introduce a mathematical model of the tariff zones design problem based on counting zones that contains paramete modelling the impact of fare changes on the demand for traveling. Designed model is verified on the real data set from the Žilina municipality, where we have an OD matrix calculated using passenge smart card transactions for a period of one week (approx transaction records). Model is solved using a univeal optimization tool Xpress and the impact of changes in the demand to zone partitioning is studied in the paper. Key-Words: tariff zones design, zone partitioning problem, integrated transport system, exact method, demand, optimization 1 Introduction Sustainable transportation and transport integration are connected with a large numbe of challenges, especially in public transportation within the cities and suburban regions. Many regions all over the world are facing problems with increasing of individual motorized transport, causing many negative phenomena such as traffic jams, parking problems in city cente and pollution increased amount of the exhaust gases. Governments and local authorities are trying to solve these problems by increasing the attractiveness of public transport. One of the way how motivate citizens to use public transportation is effective integrated transport system (ITS), which integrates several transportation means (buses, trams, trains, etc.). Designing of effective public transport systems includes solving of various optimization problems such as line network planning, coordination of connections in transport nodes, optimization of connection supply, minimization of time losses related to the changing of travel connection etc. Designing of tariff system in ITS is also an important issue, tariff system should be undetandable and motivating for passenge, but, on the other hand, effective for transport operato and municipalities. [10] [12] There are several possibilities how to design tariff in public transportation. Hamacher and Schöbel in [3] and Schöbel in [11] described a basic division of tariff systems. Fit and frequently used approach is a distance tariff system, where the price for a trip depends only on the length of the trip. This tariff system can be considered as fair. The price for the trip is based on the distance between arrival and destination station of the trip. This type of tariff is frequently used in regional bus and rail transport. Second type of tariff is a unit tariff. All trips in this system cost the same price and are independent on their length. The unit tariff is frequently used in city public transport, but it is not very suitable for regional public transportation, because a short trip between two neighbouring stations has the same ticket price as a long trip through the whole ITS region. Third type of tariff, which combines the unit tariff and the distance tariff, is a zone tariff system. In this system the whole area has to be divided into smaller ISBN:

2 sub-regions - tariff zones. The price for a trip in the zone tariff system is calculated using the information about the starting and the ending zone of the trip and on the number of travelled zones. If the price is given arbitrarily for each pair of zones, we call the tariff system a zone tariff with arbitrary prices. Second possibility of the zone tariff system is a counting zone tariff system. The price of trip depends only on the number of travelled zones in the trip. Trips passing the same number of zones have the same price. The example of a counting zone tariff system is in Figure 1. experts in [3] and [11], the goal is to design the zones in that the new and the old price for most of the trips are as close as possible. This means that neither the public transportation company nor the custome will have major disadvantages when changing the current tariff system to a zone tariff. Changing of fares can change the demand for traveling. Higher fare often occur a drop in demand and number of carried passenge and it also affect the total income from passenge. However, if the price will be lower, it can motivate travelle to use public transport and may lead to the increasing of the number of transported passenge. Commonly used models do not take into account these facto. In this contribution we are dealing with the robust tariff zones design. In the next chapter we introduce a mathematical model of the tariff zones design problem based on counting zones which includes paramete modelling the impact of price changes on the demand for traveling and shortly describe solution method. Proposed model is verified in the chapter 3, where we make a case study for the Žilina City. We use real data about passenge trips and we are able to calculate OD matrix, what is briefly described. The case study is focused on the impact of changes in the demand for travelling on the tariff zones design and the division of stations into the tariff zones. All numerical experiments are performed by a general optimisation tool Xpress. 2 Problem Formulation Fig.1: Example of the counting zones tariff system in Bratislava region, Tariff zones design problem can be divided into two sub-problems zone partitioning problem and optimal pricing problem. Several approaches for designing zone tariff system can be found in the literature. Hamacher and Schöbel in [3], Schöbel in [11] and Babel and Kellerer in [1] proposed exact solution approaches for the counting zones tariff system. A note on fair fare tariff on the bus line was mentioned also by Palúch in [8]. Exact algorithm to solve the problem was mentioned also in [4] and [5]. In addition to previously mentioned problem and issues, there are many important facto with major influence on the solution of the tariff zones design. According to [9], important facto which has major influence on demand in public transport are price, speed of travel, information about transport, safety, comfort and availability. Prices for travelling are one of the major facto. According to advises of 2.1 Mathematical Model of the Problem Let all stations/stops in the network of public transport constitute the set I and the symbol E denotes the set of edges in the transport network. The station i and j from set I are connected by the edge (i,j) E, if there is direct connection by public transport line between these two stations. For each pair of stations i and j is c the current price of travelling between these two stations. The number of passenge between stations i and j is b (OD matrix). To describe passenge journeys, we introduce parameter a, where the used paths is observed. a is equal to 1 if the edge (r,s) will be used for travelling between i and j and 0 otherwise. The current or fair price between stations i and j is denoted by c. We assume that each station/stop can be assigned to one zone exactly. We introduce binary variables y i, which represent the fictional centre of the zone. Variable y i is equal to 1 if there is a centre of the zone in node i and 0 otherwise. For each pair of ISBN:

3 stations i and j we introduce variables z. Variable z is equal to 1 if the station j is assigned to the zone with centre in the node i and 0 otherwise. We expect to create at most p tariff zones. If we want to calculate new price of the trip between nodes i and j in the counting zones tariff system, we need to calculate the number of zones crossed on this trip. The calculation of the number of crossed zones can be replaced by the calculation of crossed zone borde as was used in [3] and [11]. As was mentioned earlier, we assume that the stop can be assigned only to one zone, so the border between zones is on the node. We introduce a binary variable w for each existing edge (r, s) E, which is equal to 1 if stations r and s are in different zones and is equal to 0 otherwise. For calculation of the number of crossed borde we need to determine the used path for travelling between stations i and j When we want to calculate new prices for travelling in the counting zones tariff system, there are more possibilities. Various approaches were mentioned in [1], [3] and [11]. As was mentioned in [4] and [5], we can use two possibilities, how to set a new price for travelling. In the fit case, the unit price f for travelling in one zone is established. This means that the new price is calculated according to the number of travelled zones and multiplied it by the unit price per one zone. In the second case, two different unit prices will be determined price f 1 per travelling in the fit zone and unit price f 2 for travelling in each additional zone. The final price will be calculated as a sum of the basic price for the fit zone and number of other travelled zones multiplied by the unit price for additional zones. Second approach is more natural and according to the numerical experiments performed in [4] it gave better results, so we use this approach in the formulation of our mathematical model. New price for travelling between stations i and j is calculated using (1): n = f 1 + ( r, s) f 2 E a w (1) Another issue, which should be solved when creating the mathematical model of the problem, is the objective function of the problem. In the literature we can find various approaches to objective function such as minimisation of the maximal deviation between the current or fair price and new price determined by the number of crossed minimisation of the average deviation between current and new price for all passenge as in [4] [11] and maximising the revenue of the transportation companies as in [1] [3]. According to the previous research [4] and advices of experts in [11], in our model we use the average deviation approach. Objective function of the model can be written in the form (2): c Minimize devavg = (2) b n b Nevertheless, designed objective function (2) is not a linear function. To be able to solve the problem using an IP solver, we have to reformulate the objective function. Reformulation of the objective function was described in [6]. We introduce new nonnegative real variables u, v. Variables u represent the calculated prices for travelling in case that new price is lower than current and variables v represent the calculated prices for travelling in opposite case. Reformulated objective function can be written in the form (3) and we need to add to model binding constrains (10) to describe relation between original and new variables. b + b Minimize dev = (3) avg u As was mentioned earlier, the increase in prices can cause reducing the number of transported passenge. On the contrary, a price reduction may affect the attractiveness of transport and hence the slight increase in the demand and the number of transported passenge. To describe an influence of the changes in the demand for travelling, we introduce coefficients d and e. [7] Coefficient d represents the percentage increase in number of passenge in the case of lover new prices, coefficient e represents the percentage decrease in number of passenge in the case of higher new prices. The mathematical model of the tariff zones partitioning problem with the impact of demand on the solution can be written in the following form: Minimize dev avg = subject to (1 + d). u b + b z = 1, for v b (1 e). v b (4) (5) ISBN:

4 n z c = f1 + f 2a ( r, s) E w, for i, (6) z yi, for, i (7) zik w jk, for i I, ( j, k) E (8) y i i I p (9) n = u v, for i, (10) z w y i n u v {,1}, for i 0, (11) {,1} for i I 0, (12) {,1}, for ( i, j E 0 ) (13) 0, for i, (14) 0, for i, (15) 0, for i, (16) Conditions (5) ensure that each stop/station is assigned exactly to one zone. Conditions (7) ensure that each stop/station is assigned only to the existing zone (centre of the zone). Conditions (8) are binding between variables for allocation of the station to the zone and the variables for determining the zone border on the edge (j,k). Condition (9) ensures that we create the most p tariff zones. Conditions (10) are binding constrains to describe relation between original and new variables due to the linearization of the objective function. Subsequently, three different algorithms were used to calculate the zone partitioning. Fit algorithm is based on the clustering theory, second algorithm is a greedy algorithm and the last algorithm is based on the spanning tree approach. To be able to test robustness of our model, we use modified two-stage method mentioned in [4] and [5]. Fitly, we calculate optimal setting of paramete p, f 1 and f 2 with paramete values d and e set to value 1 (no change in the demand when the price changes). Subsequently we calculate zone partitioning problem with different values of paramete d and e and we compare assignment of stops/stations in all cases. 3 Case Study To test proposed mathematical model and the influence of the changing of the demand on the zone partitioning, we use the test network of public transport in Žilina Municipality with approx. 85 thousand inhabitants that consists of 120 stops. Schematic map of the network is shown in the Fig.2. The diameter of the circles represent approximate number of passenge using the stop for starting or ending their journeys. 2.2 Solution Method Proposed linearized model can be solved using any general IP solver. To determine the optimal values of paramete in the model, a two-phase procedure was used, as was proposed in [4]. In the fit phase the optimal number of zones was determined. In the second phase, the model with different settings of paramete f 1 and f 2 and with given number of zones p was solved. The optimal parameter setting is determined as the parameter setting of the solution with the smallest value of the objective function. Another approach to solve the tariff zones design problem was proposed in [3] and [11]. In the fit stage the optimal price of travelling was calculated. Fig.2: Zilina Municipality To calculate OD matrix we use data set provided by Žilina public transport operator DPMŽ which consist from records of smart cards in the time period between October 6 and October 12, We calculate OD matrix b and track passenge journey to be able to calculate values in the matrix a. 3.1 OD matrix calculation In cases where passenge in transportation use the smart cards, we can obtain more accurate data ISBN:

5 about the passenge journeys even in cases where these data are incomplete. In our case we are dealing with the municipal public transportation, where we usually know just information about the boarding stop and the destination stop must be estimated. Since all passenge with smart cards are obliged to validate their card when boarding the vehicle, we can get following information from smart card transactions: serial number of passenger's smart card, name and ID number of boarding stop, date and time of boarding, bus line and connection (trip) number and route variant. To obtain OD matrix based on these data, we use a trip-chaining algorithm similar to the algorithms mentioned in [2], [13] and [15]. The basic idea of the algorithm assumes that the boarding stop of passenger's immediately following journey is the destination stop of current journey. To be able to follow the idea of chaining passenge trips, we need to sort all transaction data by unique serial card number, day and time. Trip-chaining algorithm involves also assumptions that some transaction records can represent not a return trip, but can be a record of an interchange between lines. To be able to handle this, algorithm includes conditions for evaluating transfe (time limits between consecutive smart card records, distance between possible transfer stops in the case of walking between different stops). Estimation of destination stop can also handle the possibility of different boarding/destination stops on the return journey. We evaluate various setting for transfe in tripchaining algorithm to get OD matrix with the highest percentage of the successfully processed records from the data set. We use values 30 and 60 minutes as the value of parameter maximum transfer time and values 300, 600 and 900 mete as the value of maximum distance between stops. We calculated OD matrix with approximately 80 % of successfully calculated destination stops. In our case study we solve the problem for parameter p = {3, 4, 5, 6, 7}. Regarding to the current prices, we use for parameter f 1 values 0.55 and 0.60 and for parameter f 2 values 0.05 and 0.1. Values of parameter d = {0, 1, 2, 3} what represent increase in demand up to 3 % and e = {0, 2, 4, 6} what represent decrease in demand up to 6 %. All numerical experiments were performed using optimization tool Xpress [14]. In all cases the best results are obtained for values f 1 = 0.55 and f 2 = 0.1. Resulting zone partitioning of the Žilina Municipality area for all values of parameter p are showed on the figures Fig.3 Fig.7. Area colored by darker colo represents set of stops which belong to the zone in all cases. Stops shaded by lighter shades of the zone color belong to the designated zone only in some cases. Fig.3: Zone partitioning for p=3 3.2 Numerical experiments With calculated OD matrix we are able to test proposed mathematical model of the zone partitioning problem. As the value of the parameter c we use current prices of travelling which depend partly on the distance. For all journeys up to 5 stops without transfer passenger pays 0.55, for all journeys for more than 5 stops without transfer 0.65 and for a journey with transfer Fig.4: Zone partitioning for p=4 ISBN:

6 Fig.5: Zone partitioning for p=5 changes on the demand for travelling. We made the case study for the Žilina Municipality based on real data about passenge trips. From the results it is obvious that newly created zones are mostly in suburban parts of the network with smaller demand for travelling and higher distances to the centre, as it is common in most transportation system. By comparing of obtained results in various settings of parameter p we can see that the changes in the demand do not have big influence on the zone partitioning. In the wot case only 6 out of 120 stops are assigned to the different zones. This allows us to conclude that the proposed solution are enough robust to be used in the case of changing the tariff system. The major drawback of this process was the time complexity of the problem. Computation time varies from few second up to 2 hou for one particular parameter setting. Acknowledgment This work was supported by the research grants VEGA 1/0339/13 Advanced microscopic modelling and complex data sources for designing spatially large public service systems and APVV Designing Fair Service Systems on Transportation Networks. We would like to thank the transport operator DPMŽ for providing necessary data for this research. Fig.6: Zone partitioning for p=6 Fig.7: Zone partitioning for p=7 4 Conclusion In this contribution we studied the robust tariff zones design based on counting zones which includes paramete modelling the impact of price References: [1] Babel, L., Kellerer, H., Design of tariff zones in public transportation systems: Theoretical and practical results. Technical report, Faculty of Economics, Univeity of Graz, Austria, 2001 [2] Bagchi, M., White, P.R., The Potential of Public Transport Smart Card Data, Transport Policy 12 (5), 2005, pp [3] Hamacher, H. W., Schöbel, A., Design of Zone Tariff Systems in Public Transportation. Operations Research 52, 2004, pp [4] Koháni, M., Designing of zone tariff in integrated transport systems, Communications: scientific lette of the Univeity of Žilina 15, 2013, pp [5] Koháni, M., Exact approach to the tariff zones design problem in public transport, Mathematical methods in economics: proceedings of the 30th International Conference, Silesian Univeity in Opava, Karviná, 2012, pp [6] Koháni, M., Zone partitioning problem with given prices and number of zones in counting zones tariff system, SOR 13: proceedings of the 12th International Symposium on ISBN:

7 Operational Research: Dolenjske Toplice, Slovenia., Vol. 13, 2013, pp [7] Koháni, M., Impact of changes in fares and demand to the tariff zones design, Mathematical methods in economics: proceedings of the 31 th International Conference, College of Polytechnics Jihlava, 2013, pp [8] Palúch, S., On a fair fare rating on a bus line, Communications: scientific lette of the Univeity of Žilina 15, 2013, pp [9] Paulley, N.; Balcombe, R.; Mackett, R.; Titheridge, H.; Preston, J.M.; Wardman, M.R.; Shires, J.D.; White, P., The demand for public transport: The effects of fares, quality of service, income and car ownehip. Transport Policy, 13 (4), 2006, pp [10] Ribeiro, P., Santos, P.: Public Transport towards Sustainability in Midsized Municipalities, Proceedings of the 6th International Conference on Urban Planning and Transportation (UPT '13), NAUN, Cambridge, UK, pp [11] Schöbel, A.: Optimization in Public Transportation: Stop Location, Delay Management and Tariff Zone Design in a Public Transportation Network, Springer, [12] Szubartowski, M.: Availability of City Buses in the Municipal Transport System, Proceedings of the 2nd International Conference on Sustainable Cities, Urban Sustainability and Transportation (SCUST '13), NAUN, Baltimore, MD, USA, pp [13] Trépanier, M., Tranchant, N., Chapleau, R., Individual Trip Destination Estimation in a Transit Smart Card Automated Fare Collection System, Journal of Intelligent Transportation Systems 11 (1), 2007, pp [14] XPRESS-Mosel User guide. Fair Isaac Corporation, Birmingham, [15] Wang, W., Attanucci, J. P., Wilson N. H. M., Bus Passenger Origin-destination Estimation and Related Analyses using Automated Data Collection System, Journal of Public Transportation 14 (4), 2011, pp ISBN: