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1 USE OF A HYBRID ALGORITHM FOR MODELLING COORDINATED FEEDER BUS ROUTE NETWORK AT SUBURBAN RAILWAY STATION Shrivastava Prabhat, M.ASCE1 and O’Mahony Margaret2 ABSTRACT In the metropolitan cities of developed and developing countries, longer journeys are mostly performed by two or more modes. In the event of availability of suburban trains and public buses, commuters prefer to travel a longer stretch of their journeys by train, so as to avoid traffic congestion on roads, and the remaining part by buses to reach local areas if their final destination is not in close proximity to railway stations. Normally suburban trains have fixed corridors and buses have flexibility to serve remote local areas. Thus design of feeder routes from railway stations to various destinations and the transfer time from trains to buses play a very important role and can be controlled by transport planners. A considerable amount of research has been done on independent design of a bus route network without considering the effect of train services. Researchers have made attempts using heuristics, simulation, expert systems, artificial intelligence and optimization techniques for design of routes and schedules. So far, limited effort has been made in modelling coordinated operations. In this research, a new hybrid algorithm which exploits the benefits of Genetic Algorithms and a well tested heuristic algorithm for the study area is discussed. More convincing results in terms of feeder routes and coordinated schedules at the selected railway station are obtained by the proposed hybrid algorithm as compared to earlier approaches adopted by the authors for the same study area. CE DATABASE SUBJECT HEADINGS: Routing and scheduling, Genetic Algorithms, Optimisation, Hybrid Algorithm, public transportation, Intermodal coordination, Heuristic Approach, Transportation Planning 1 Professor (Transportation Engg), Civil Engg Dept., Sardar Patel College of Engg., Andheri (W), Mumbai, India 2 ,Professor & Head Dept. of Civil, Struct & Env Engg, Director , Centre for transp Research, Trinity College, Dublin-2, Ireland 2 INTRODUCTION In metropolitan cities of developed and developing countries significant growth of private and intermediate transport has taken place. Due to this growth and limited capacity of carriageways, traffic congestion, environmental pollution, poor level of service and longer travel times are very common. These problems can be solved only by reducing dependency on private vehicles. The dependency can be reduced only by making public transport more efficient. The efficiency of the public transport system can be enhanced by better integration of its various components. Most metropolitan cities have suburban railway and public buses as major public transport facilities. Feeder bus routes and coordinated schedules critically determine the performance of an integrated public transport system specially when longer journeys are made by suburban trains and public buses feed the local areas. The purpose of this paper is to develop a methodology for development of feeder routes for public buses and also to develop coordinated schedules of public buses for the existing schedules of suburban trains at a suburban railway station. The heuristic approach had been very popular for bus route network design problems. In places where practical realistic solutions are more important than optimal solutions one can opt for the heuristic approach. Using intuition and experience the analyst may be able to guide the heuristic search process far more efficiently than a predetermined set of rules. Good heuristics have obvious advantages over the more standard algorithms of combinatorial optimization (Reklaitis et al, 1983). Desirable characteristics of the heuristic process include execution in reasonable computational time, solutions which are close to optimality on average and the simplicity of both design and computational requirements. In the case of route design, the designer has to strike a balance between satisfaction of demand at various destinations and shorter lengths of routes. Shrivastava and O‟Mahony (2007) have discussed in detail about the application of an heuristic approach in development of routes or network of routes. The transit network design problems become more complex due to the difficulty in combining user costs and operator costs in a single objective function. Non-linearity and non-convexities are involved in the objective function along with the discrete nature of route design and various other constraints relating to route coverage, route duplication, route length and directness of service. Thus 3 complexities involved in network design problems necessitate intelligent searches and the use of robust optimization techniques. Application of Genetic Algorithms for route scheduling and network design problems has been discussed in detail in Shrivastava and O‟Mahony (2007) and Shrivastava and O‟Mahony (2006). . Limited efforts have been made for the design of feeder route networks for feeder buses at railway stations (Shrivastava and Dhingra, 2000). However, Wirasinghe (1980), Geok and Perl (1988) attempted routing and scheduling problems for coordinated operations using analytical models. They had considered a highway grid which is assumed to be rectangular and parallel to a single railway line which may not always be true in practice. They had made an attempt to describe complex transit systems by approximate analytical models. Zhao and Ubaka (2004) presented a mathematical methodology for transit route network optimization. The objective was to minimize transfers and optimize route directness while maximizing service coverage. The authors have communicated research papers on development of feeder routes for the same study area using different approaches. This paper builds on the work of earlier research papers and contains significant differences from previous work. In other words it inherits the best qualities of earlier papers. In Shrivastava and Dhingra (2001) feeder routes were generated with the help of a heuri stic algorithm which was suitable for the typical case study. The heuristic algorithm does not promise an optimal solution but it may give a near optimal or suboptimal solution. In Shrivastava and O‟Mahony (2006) a Genetic Algorithm has been used for simultaneous development of feeder routes and schedules but it was found that demand at all nodes is not satisfied. This is because in the typical study area under consideration some of the nodes are connected in the form of a chain without any additional connectivity with any other node / nodes. Thus very limited alternatives are available for such nodes and even with higher penalties demand remains unsatisfied. In Shrivastava and O‟Mahony (2007), well scattered nodes are selected as potential nodes and an heuristic algorithm is used as the repair algorithm. This research work differs from earlier work of the authors in following ways: 4 The demand at nodes (destinations) is given priority for development of routes over the location of nodes. Thus nodes with more than average demand are identified as potential destinations. In the case of any network, if higher demand nodes are located on the outskirts of the study area then the heuristic algorithm acts as repair algorithm otherwise the genetic algorithm develops a partially optimized network giving due consideration to higher demand nodes. After development of the partially optimized network the heuristic algorithm further modifies the network. In the modification process using the heuristic algorithm again the demand of nodes is given priority for development of the route system. The proposed approach in this paper is oriented towards demand satisfaction of nodes but their locations are also given due consideration for modification of routes which is the key idea for development of networks especially in the context of the public transport system. A judicious use of both genetic algorithms and heuristic approaches can be seen in this research. If the higher demand nodes are close to the DART station then the heuristic algorithm plays a major role and if higher demand nodes are well scattered and away from DART station then the genetic algorithms plays the main role. In this paper, first potential destinations having more than average demands are identified and with k – path algorithm and Genetic Algorithms an optimized feeder route structure is developed. This route structure is modified with heuristic algorithms so that all other nodes having less demand are connected. Thus in this process an optimized feeder route structure is developed by Genetic and K – path algorithms which is desirable because nodes with higher demand should be given priority over lower demand nodes. Lower demand nodes are connected to the developed route structure by a heuristic algorithm. Thus a judicious use of both the algorithms is shown in this paper. In this research the Shrivastava - O‟Mahony Hybrid Feeder Route Generation Algorithm (SOHFRGA) which is a combination of Genetic Algorithms and Heuristic Approach has been developed. It has been found that the feeder routes developed by SOHFRGA are more efficient than those developed 5 by authors using other approaches. Dun Laoghaire is a rapidly growing suburb in Dublin city of Ireland and DART station is selected as the study area for coordination between Dublin buses and DART services. Assessment of the number of commuters currently using DART services, the percentage of commuters using Dublin buses after arriving by DART at Dun Laoghaire and the number of commuters who would shift from other modes to bus is discussed in previous research papers i.e. Shrivastava and O‟Mahony (2007) and Shrivastava and O‟Mahony (2006). Table 1 indicates potential demand to various destinations which includes current demand by buses and the expected shift of commuters from other modes. Readers are advised to refer to the above papers for further details of the study area and data collection PROPOSED METHODOLOGY The overall methodology for development of feeder routes and coordinated schedules is indicated in figure 1. As discussed in Shrivastava and O‟Mahony (2006) and Shrivastava and O‟Mahony (2007) the potential demand is assessed, travel time matrices are developed, other parameters and values have been decided (refer to steps 1, 2 and 3 of proposed methodology in the above papers). The remaining methodology used in the research presented here can be explained in the following. (1) Feeder routes are developed by Shrivastava – O‟Mahony Hybrid Feeder Route Generation Algorithm (SOHFRGA). Figure II indicates the various steps involved in SOHFRGA which are described below. Step I: Identification of Potential Destinations From the traffic surveys, demand at various destinations (nodes) is identified. The destinations having more than average demand were identified as potential destinations. This has been done in order to develop an initial feeder route network for potential destinations. In doing so the demands at potential destinations get satisfied by shorter routes and then other nodes of which demand is less than average are inserted / attached to developed „K – Paths‟ using the heuristic algorithm. This is explained in later steps. 6 The preference is given to small number of routes with higher percentage of demand satisfaction with lower travel time. Large number of routes will be generated if K- Paths are developed for each destination. Large number of routes will require more buses and also some routes will be very short and some may be very long, which will pose problems in the scheduling process. Buses will have to be scheduled on many routes where demand may not be substantial which will lead to uneconomical operations. In view of this K – Paths were developed for destinations with demand more than average with upper limits on lengths of routes. Step II: Development of K –Paths between Origin (DART station) and Potential Destinations There were five potential destinations having more than average demand from the DART station. With the help of K-path algorithms (Eppstein, 1994) five short paths between origin (DART station) and each potential destination were developed. Thus between each pair of origin (DART station) and potential destination five alternative paths were available for further analysis as explained in the next step. As there were 16 destinations to which demand from DART station exists, out of these 16 destinations five had more than average demand and thus selected as potential destinations, remaining 11 nodes (destinations) were used for development of short paths to potential destinations. It can be seen in Table - 2 that variation in lengths between first (smallest) and fifth (largest) path for each pair is considerable therefore, if more than five paths are developed for each destinations there will be additional paths with higher travel times. Routes with higher travel times are not desirable as there will not be any improvement in result. In view of this five „k‟ paths are developed for each destination. The value of „k‟ can be selected higher if more nodes are available as there will be possibility of having more number of short paths without considerable variation in travel time between shortest and longest path. Step III: Determination of optimized K-Paths and Schedules A computer program to calculate the penalised objective function (summation of objective function and penalties due to violation of constraints) was developed in the „C++‟ environment. The developed „K‟ 7 paths are used with the objective function program. The provision is made so that each alternative path out of five „K‟ paths for each potential destination is selected with random frequencies generated by Genetic Algorithms. The alternative paths and frequencies corresponding to minimum penalized objective functions are selected as optimized feeder routes and frequencies. Thus frequencies ‘fj’ and a set of routes (with different lengths ‘lj’) are the decision variables. The time of departure of buses is decided on the basis of frequencies and the scheduled arrival time of DARTs from either direction at the DART station. The typical analysis in the paper has been carried out for peak hour only. From the traffic surveys it was found that from each train during the peak hour almost the same numbers of passengers alight and relatively similar percentages of passengers would select buses to their destinations for further travel. Due to this uniformity in the alighting pattern, the frequencies of connecting feeder buses are also found to be same, for example, on route number „1‟ each bus is scheduled at 6 minute headway starting from 8.07 am to 8.55 am. This frequency will change as per demand and with the distribution of commuters at any other time interval. In off peak hours, the headway between trains will increase due to lower demand. This frequency will change as per demand in other time intervals. During off-peak periods there will be less DART commuters and thereby headway between buses will increase which will lead to a lower requirement of buses. The binary digit coding to represent routes and schedules together has been adopted (Shrivastava et al 2002). Details of Objective Function, Penalties and use of GAs in SOHFRGA The objective function is adopted as the minimization of user and operator costs. The user cost is the summation of the in-vehicle time cost and the transfer time cost between DARTs and buses. The operator cost is associated with the running cost (vehicle operation cost) of buses. Constraints are related to load factor, fleet size and unsatisfied demand. Readers are requested to refer Shrivastava and O‟Mahony (2007) as mathematical representation of objective function, constraints, and penalties along with detailed discussion is given there. In this paper, the same objective function and constraints are solved by using a different approach which differs in the way it selects potential destinations. 8 The developed objective function is used with LibGA software (Lance Chambers, 1995) for Genetic Algorithms in the Linux environment. The objective function and constraints pose a multi-objective problem. Some of the constraints are in favor of users and some are in favor of the operator. For example, lower load factor and higher load factor constraints are in favor of users and operators respectively. If the load factor is above „1‟ some of the users may not be able to get a seat but the operator will earn a profit. The fleet size constraint ensures that the scheduling of buses should be done within the limited fleet size and this is in favor of the operator. The unsatisfied demand constraint ensures that all commuters should be able to get on a bus, thus this constraint is in favor of the operator. Assigning a very high penalty to one of the constraint results in a biased solution. Thus these penalties are decided so as to keep the load factors between the minimum and maximum values, the fleet size within a specified limit and the unsatisfied demand to zero. The adopted set of penalties for the feasible solutions is decided so as to get a judiciously balanced solution. The demand satisfaction and load factors on various routes are two dominating factors for both users and operators. It has been found during the interviews with commuters that they prefer to have connecting buses within five minutes of waiting after arriving at bus stops but most of them even accept ten minutes of waiting as a reasonable time. Thus the variation of penalty coefficients for a minimum load factor is related to the percentage satisfaction of demand within ten minutes of waiting. The coefficient for the minimum load factor is selected because it is observed that the load factor frequently goes below 0.4 (minimum value) due to low demand which is not compatible to the adopted existing bus capacity (74). This typical variation is observed when the penalty coefficient corresponding to the minimum load factor (less than 0.4) is varied keeping other coefficients the same. A weighted factor is calculated by awarding equal weights to the overall load factor and the percentage demand satisfaction within ten minutes of waiting. The penalty coefficient corresponding to the higher weighted factor is selected for further analysis. Step IV: 9 Check whether the entire demand is satisfied and routes are within the specified length. If the entire demand is satisfied and routes are within the specified lengths then developed feeder routes and the frequencies are optimum. The frequencies are used to calculate coordinated schedules for the existing schedules of the DART. Step V: If the entire demand is not satisfied and routes are within specified lengths then these routes are used for modification. Destinations leading to unsatisfied demand are inserted in routes by node selection and insertion strategies. If the lengths of routes are not within specified limits (very small due to higher demands close to a railway station) then these routes are discarded. If the potential destinations are very close to a DART station then they lead to very short routes mushrooming near to a station (Shrivastava and Dhingra, 2001 & Baaj and Mahamassani, 1995). Such short routes are not acceptable in actual practice. Thus in the proposed case study a length of 2.5 km equivalent to travel time of 10 minutes is adopted as the minimum length of feeder routes and the routes less than or equal to this value are discarded. The nodes present on discarded routes and not duplicated in other routes are also used for heuristic insertion/attachment process along with other nodes with unsatisfied demand (destinations not included in any developed feeder route). Readers are requested to refer to Shrivastava and Dhingra (2001) where the full description of node selection and insertion / attachment process along with various insertion strategies are discussed in detail. Use of Genetic Algorithms for the objective function and constraints The proposed objective function is used with LibGA software (Lance Chambers, 1995) of Genetic Algorithms in the Linux environment to determine optimal routes and frequencies in SOHFRGA and thereafter for determination of final frequencies leading to coordinated schedules on developed feeder route network. The details of Genetic Algorithms, its application and adopted values of various operators etc is discussed in Shrivastava and O‟Mahony (2007). 10 RESULTS AND DISCUSSION In the initial stage of SOHFRGA, nodes having more than average demand were selected as potential destinations and feeder routes with frequencies for feeder buses (leading to coordinated schedules) were developed simultaneously using Genetic Algorithms. There were six destinations having more than average demand. These destinations were Dun Laoghaire College (2), Sallynogin (3), Monkstown (4), Deans Grange (5), Stillorgan (8) and Loughlinstown (12). Selection of these nodes as potential destinations developed six routes with 12.85% unsatisfied demand. Selection of five nodes i.e. Dun Laoghaire College (2), Sallynogin (3), Deans Grange (5), Stillorgan (8) and Lough Linstown (12) also gave 12.85% unsatisfied demand with five feeder routes. Therefore these five destinations were selected as potential destinations and five k-paths as indicated in table 2 were developed for each of these potential destinations originating from the DART station. These k – paths were used with LibGA software of Genetic Algorithms in Linux and feeder routes with frequencies leading to coordinated schedules were developed simultaneously. The following feeder routes were obtained (with codes as given in table 1). Nodes in feeder routes Length in terms of travel time in „minutes‟ 1–2 8 1–3 7 1–4–5 15 1–6–7–8 25 1 – 3 – 17 – 11 – 12 30 Since the developed feeder route network does not satisfy 100% demand, the next stage of further modification of feeder routes using node selection and insertion strategies is adopted. The frequencies associated with feeder routes in the earlier stage are discarded since fresh frequencies are required to be determined due to the modification of routes. The travel time on the first two feeder routes is even less than ten minutes (minimum specified length of 2.5 km) Hence the first two routes are discarded and node „2‟ which leads to unsatisfied demand is selected for the insertion process in the next stage for modification of routes along with other unsatisfied nodes. Thus the 3rd, 4th and 5th routes were selected for 11 modification and nodes having unsatisfied demand are inserted priority-wise using a heuristic approach. Finally, the following three feeder routes, as indicated in figure 3 were developed. Nodes in feeder routes Length in „km‟ 1 – 4 – 2 – 5 – 10 – 9 – 16 10.76 1 – 6 – 7 – 8 – 15 – 14 – 13 14.60 1 – 3 – 17 – 11 – 12 7.72 It can be seen that the developed feeder routes are well within the specified minimum (2.5 km) and maximum (15 km) lengths of routes. These lengths of routes are decided based on locations of nodes to which demand exists as identified in typical traffic survey. The upper limit of the length of routes can be revised and shorter lengths can be adopted if a similar exercise of coordination is repeated at other DART stations also. This is due to the fact that a particular node may have connectivity with more than one DART station which may lead to shorter and better routes from one station, as compared to another one. For example, nodes 13, 14 and 15 are very close to the Blackrock DART station as compared to Dun Laoghaire. Thus feeder routes for these destinations from Blackrock will be shorter. In the next stage for determination of coordinated schedules for feeder buses for the existing schedules of DARTs these feeder routes were used. Genetic Algorithms are implemented for determining optimal frequencies and coordinated schedules were derived from these frequencies. Table 3 shows details of coordinated schedules of feeder buses for the existing schedules of DARTs and load factors. It can be seen in the table that the average load factor on each route is more than 0.40 and the overall load factor (average of load factors on all the three routes) is 0.47. The load factors on the routes and overall load factor have much improved values against the existing scenario in which the load factor hardly increases above 0.3. In the case study the capacity of feeder buses is taken as 74 (Scott Wilson, 2000) which is on higher side as compared to the number of DART commuters transfer to buses. The load factor will be more (greater than „1‟) if coordinating buses are held for longer time. Commuters from later DART trains will be able to seek transfer to a particular bus due to a longer holding period. Longer holding increases 12 transfer time between DARTs and feeder buses which is not desirable to users and will also increase the value of the objective function. Thus the two contradictory conditions regarding higher load factors in favor of the operator and the lower transfer time for users are satisfied by striking a balance between the two. As the load factor decreases below „1‟ penalties are imposed because the constraint for the minimum load factor is violated. Thus the load factors obtained in the study are less than „1‟ but transfer time remains within desirable limits. Busses with lower seating capacity will improve the load factor. Also the load factor will be higher if the model is implemented to places where higher numbers of train commuters seek transfer to buses. Thus the decision of bus dispatch time not only depends on arrival time of DARTs but also on the number of passengers transferring from DARTs to buses. As discussed above, the dispatch time of buses is decided upon by means of striking a balance between transfer time and the acceptable value of the load factor. If the load factor is lower, then buses are held for more time so as to acquire more commuters from later DART trains. Other factors influencing the schedule of buses are the available fleet size and the percentage demand satisfied. The coordinated buses are dispatched only after five minutes of scheduled arrival of DART at station because it takes about five minutes for passengers to reach bus stops after arriving on a DART train. It can be seen in table 3 that the first bus is at 8.07 which is five minutes after the arrival of the south bound DART scheduled at 8.02; on route number 1, the second bus is at 8.13 which is five minutes after the arrival of the first north bound DART at 8.08. Some buses are scheduled later than five minutes after the arrival of a DART train. This is done in order to compromise with the load factor by holding the bus, as discussed above. The other coordinated schedules can be explained on the basis of the above reasoning. Table 4 gives a comparison between existing and proposed route network. It can be seen that on average 89 % of the total demand is satisfied within 5 minutes of effective waiting time and 99% of demand is satisfied within 10 minutes of effective waiting time. Only 1 % of commuters are required to wait up to 15 minutes. Route wise details indicate that 100 % demand on route 1 is satisfied within 5 minutes of effective waiting time and on route 2 and 3, 96 % of demand is satisfied within 10 minutes of 13 effective waiting time. Only 4 % of commuters have to wait up to 15 minutes. Thus if a monetary value is assigned to a saving in transfer time then about a 70% saving can be expected in transfer time due to the proposed system. In the existing scenario there are 6 bus routes travelling through identified nodes whereas in the proposed model three bus routes cover identified demand nodes. The saving in in-vehicle time and operating cost can be expected to be 50% due to the proposed model against the existing scenario. Table 4 briefly summarises the above discussions. CONCLUSIONS In the proposed research, feeder routes are developed using a hybrid approach which makes use of the benefits of optimisation using genetic algorithms and a heuristic approach. For the development of feeder routes, priority is given to nodes having higher demands. In view of this, nodes having higher demand (more than average) are considered as potential destinations and optimisation is carried out by GAs. Nodes having lower demand are attached to (or inserted in) developed routes keeping the length (or deviation) within reasonable limits. Coordinated schedules are determined by genetic algorithms. The following conclusions can be drawn from the proposed modelling exercise. 1. It has been observed that for the given influence area of a DART train station a combination of genetic algorithms and the proposed heuristic approach develops an improved feeder route structure. In the influence area of the railway station destinations closer to the railway stations have higher demands and other destinations are well-scattered with limited connectivity between them. By using this approach, a lower number of feeder routes is developed as compared to earlier approaches for the same influence area and demand. The proposed model also provides improved load factors with a higher percentage of demand satisfaction and lower waiting time. 2. The lengths of feeder routes are well controlled in the model. They are checked after the first stage i.e. after simultaneous development of routes and schedules using GAs and also using the heuristic approach route lengths are controlled by the „maximum demand deviation shorter path time criterion‟ and the „path extension time criterion‟. 14 3. The model strikes a balance between user need and operator requirements. The objective function incorporates user costs in terms of time spent in buses and the transfer time between DARTs and buses; the operators cost is the vehicle operation cost which is directly proportional to the distance travelled by buses. Similarly constraints are also as per the requirements of users and operators. The load factor constraint is kept within minimum and maximum values so as to maintain a better level of service for users and economic operation to satisfy operators. The fleet size constraint is also a realistic constraint from the operator‟s point of view. The constraint for unsatisfied demand increases the probability of availability of seats to commuters though it is not very important when the load factor remains less than a minimum value as has been experienced for the study area. 4. The same modelling exercise can be carried out at other DART stations with large scale data collection for the whole day. A fully integrated system with DART as the main line haul carrier and buses as feeder services can be developed. Schedules and thus the requirement of buses can be found for peak and off-peak periods of the day. Even the route structure can be appropriately designed as per demand at various destinations for different periods of the day. 5. The application of all three algorithms i.e. K – Path, Genetic and Heuristic Algorithms makes this approach acceptable for all types of networks i.e. networks with well connected nodes and networks in which all the nodes have limited connectivity. Planners have limited options available in the formation of alternative routes in the case of networks with limited connectivity. Combinations of Genetic Algorithms and K –Path algorithms have been successfully implemented for well connected networks. The heuristic approach is developed keeping in mind networks in which nodes are not well connected. Thus this method can be implemented for a network which is well connected and also for a network in which nodes do not posses better connectivity with other nodes. Thus this methodology is not developed only for a given case study but it can be implemented to other networks also. If the higher demand destinations are located away from the origin (railway station), well scattered and well connected then optimised feeder routes and coordinated schedules will be developed in the first stage of model in which optimisation of feeder routes and coordinated schedules is done simultaneously 15 using Genetic Algorithms. Thus the proposed model can be used for any influence area if demands at various destinations and network connectivity details are known. REFERENCES 1. Baaj M.H. and Mahmassani H.S. (1995). Hybrid Route Generation Heuristic Algorithm for the Design of Transit Networks, Transpn. Res.C, 3, 1, 31 – 50. 2. Eppstein, David (1994). Finding the k shortest paths. Tech. Report 94-26, Department of Information and Computer Science, University of California, USA. http://www.ics.uci.edu/~eppstein/pubs/Epp-TR-94-26.pdf accessed on 25.09.2004 3. Geok K, and Jossef P. (1988). Optimization of feeder bus routes and bus stop spacing. Journal of Transportation Engineering, 114, 3, ASCE, Reston, VA, 341-354. 4. Chambers, L. (1995). “Practical Handbook of Genetic Algorithms Applications. Volume I”, CRC Press, 555. 5. Reklaitis, G.V. Ravindran, A. and Ragsdell K.M. (1983), “Engineering Optimization – methods and applications”, Wiley, New York, N.Y. 6. Scott Wilson (2000). “Final Report on Bus Network Strategy Appraisal Report for Greater Dublin Area”. www.dublinbus.ie/about_us/pdf/swilson.pdf Accessed May 13, 2004. 7. Shrivastava P. and Dhingra S.L (2000). An overview of bus routing and scheduling techniques. Highway Research Bulletin, 62, 65 – 90. 8. Shrivastava P. and Dhingra S.L. (2001), Development of feeder routes for suburban railway stations using heuristic approach Journal of Transportation Engineering, ASCE, USA, July/August 2001, 127, 4, 334-341. 9. Shrivastava P., Dhingra S.L. and Gundaliya P.J. (2002), Application of Genetic Algorithm for Scheduling and Schedule co-ordination problems. Journal of Advanced Transportation. 36, 1, Winter 2002, 23 – 41. 10. Shrivastava P. and O‟Mahony M. (2006), A Model for Development of Optimized Feeder Routes and Coordinated Schedules – A Genetic Algorithms Approach, Transport Policy, 13, 5, 413-425. 16 11. Shrivastava P. and O‟Mahony M. (2007), Design of Feeder Route Network Using Combined Genetic Algorithm and Specialized Repair Heuristic, Journal of Public Transport, 10, 2, 99-123 12. Wirasinghe, S.C. (1980), Nearly optimal parameters for a rail feeder bus system on a rectangular grid. Transp. Sci. 14A (1), 33-40. 13. Zhao F. and Ubaka I. (2004), Transit network optimization – minimizing transfers and optimizing route directness, Journal of Public Transportation, 7, 67 – 82 17 Table 1 Potential Demand to Various Destinations Node Destinations Potential demand to various No. destinations (code) 7 – 8 a.m. 8 – 9 a.m. 7-9 a.m. 1 Dun Laoghaire DART Station 0 0 0 2 Dun Laoghaire College 39 202 241 3 Sallynoggin 17 103 120 4 Monks town 10 63 73 5 Deans Grange 16 93 109 6 Temple Hill 2 6 8 7 Black Rock 8 46 54 8 Stillorgan 13 77 90 9 Leopards town 2 8 10 10 Foxrock 2 8 10 11 Maple Manor / Cabinteely 2 4 6 12 Lough Linstown 13 78 91 13 Mount Merrion 2 15 17 14 University College of Dublin 4 23 27 15 Dundrum 6 31 37 16 Sandyford 3 15 18 17 Rouches Town Avenue 2 4 6 18 Table 2 Developed ‘K’ paths between DART station and Potential Destinations Origin Potential Destination Nodes in „k‟ paths Travel time ( k=5 ) min 1 2 1, 2 8 1,3,2 10 1,4,2 15 1,6,4,2 20 1,4,5,2 20 1 3 1,3 7 1,2,3 11 1,4,2,3 18 1,4,5,2,3 23 1,2,5,3 23 1 5 1,2,5 13 1,3,2,5 15 1,4,5 15 1,2,4,5 16 1,3,5 17 1 8 1,6,8 21 1,7,8 24 1,6,7,8 25 1,7,6,8 26 1,2,5,8 26 1 12 1,3,17,12 23 1,2,3,17,12 27 1,3,11,12 28 1,2,17,12 30 1,3,17,11,12 30 19 Table 3 Details of Bus Schedules with Load Factors Train Timings Bus Timings Load Factors Over all load factor North South Bound Bound Route Route Route Route Route Route DARTS DARTS 1 2 3 1 2 3 8.08 8.02 8.07 8.07 8.07 0.46 0.23 0.22 8.15 8.09 8.13 8.19 8.19 0.16 0.31 0.30 8.23 8.20 8.19 8.31 8.31 0.46 0.62 0.60 8.29 8.25 8.25 8.43 8.43 0.62 0.70 0.67 8.33 8.31 8.31 8.55 8.55 0.62 0.39 0.38 0.47 8.38 8.36 8.37 - 0.62 - 8.43 8.45 8.43 - 0.78 - Load factor 8.49 8.53 8.49 - Buses to 0.16 - for Buses 8.58 - 8.55 - be 0.62 - to be Buses to be scheduled Load factor for scheduled Trains after 9 a.m. scheduled after after 9 Buses to be after 9 a.m. 9 a.m. a.m. scheduled after 9 a.m. Average load factors on individual routes 0.50 0.45 0.43 20 Table 4: Comparison between existing and proposed Route Network Property for comparison Existing Route network in the Proposed route network for the study area study area Type of route network Not a feeder route network Feeder route network Average load factor on routes Less than 0.3 Greater than 0.4 on all routes, over all load factor is 0.47 Waiting time / percentage Average waiting time is more 89% demand is satisfied with in demand satisfaction than 20 minutes 5 minutes, 99 % with in 10 minutes and 100 % is satisfied with in 15 minutes of effective waiting. Transfer Cost About 70% saving in transfer cost can be achieved per head due to proposed network and coordinated scheduling In vehicle cost and Operating There will be about 50 % saving in in-vehicle cost and operating cost Cost Total Cost The total saving can be expected to be more than 50% 21 Identification of data requirement Details of existing bus and DART network - Coded bus and DART network - Link lengths & link travel time - Characteristics of Dublin buses TRAFFIC SURVEYS Surveys for assessing Assessment of potential existing distribution of demand to different DART Commuters on destinations from DART Different Modes station using willingness to shift surveys Existing Road Network Potential O - D Matrix SHRIVASTAVA – O‟MAHONY HYBRID FEEDER ROUTE GENERATION ALGORITHM (SOHFRGA) Feeder routes for DART station SCHEDULE OPTIMISATION MODEL Minimisation of distance travelled by Dublin buses (Operator Cost) and Transfer time between DART and coordinating feeder public buses (User cost) with the constraints related to load factor, transfer time and Existing unsatisfied demand. DART timings Coordinated Schedules of Public Buses Figure 1: Proposed Methodology for development of feeder routes and coordinated Schedule 22 Figure 2: Proposed Shrivastava – O’Mahony Hybrid Feeder Route Generation Algorithm (SOHFRGA) Demand Matrix obtained Existing Road Network by Traffic surveys Link connectivity matrix Selection of Potential destinations based on average demand Development of K –paths between DART station and potential destinations Optimization of penalized objective function using Genetic Algorithms Objective function: Minimization (Transfer time between DARTs and Buses + in Vehicle time + Vehicle operating cost) Constraints: Related to Minimum and Maximum load factors, fleet size and unsatisfied demand Existing DART Optimized Feeder routes and coordinated schedules timings Yes No Is entire demand Satisfied? Yes A Are the lengths of feeder routes with in acceptable limits? Print optimized feeder routes and coordinated schedules 23 A Discard the routes having less than minimum specified Length / Travel time (2.5 km / 10 min for case study). Available routes for modification Sort all the available nodes due to following in decreasing order of demand: Could not be used by developed feeder routes and give rise to unsatisfied demand Available due to discarding of smaller feeder routes. Select first node among arranged in decreasing order of demand Stop, Yes Is last node has been Print Routes inserted? No Node selection and Find out the route in which selected node has to be insertion strategies inserted /attached as per node selection and insertion strategies and Insert / attach the selected node in the identified route. Select next Is route length with in route specified limit? Insert the node in selected route and delete from node list Take next node Figure 2 : Proposed Shrivastava – O’Mahony Hybrid Feeder Route Generation Algorithm (SOHFRGA) (Continued) 24 Figure 3 is available on request from the authors List of Tables Table 1 Potential Demand to Various Destinations Table 2 Developed „K‟ paths between DART station and Potential Destinations Table 3 Details of Bus Schedules with Load Factors Table 4 Comparison between existing and proposed Route Network List of Figures Figure 1 Proposed Methodology for development of feeder routes and coordinated schedule Figure 2 Proposed Shrivastava – O‟Mahony Hybrid Feeder Route Generation Algorithm (SOHFRGA) Figure 2 Proposed Shrivastava – O‟Mahony Hybrid Feeder Route Generation Algorithm (SOHFRGA) (Continued) Figure 3 Developed feeder route network for Dun Laoghaire DART station