2016 3 rd International Conference on Vehicle, Mechanical and Electrical Engineering (ICVMEE 2016) ISBN: 978-1-60595-370-0 Responsive Bus Bridging Service Planning Under Urban Rail Transit Line Emergency YUEDI YANG AND HUI ZHAO ABSTRACT As an important urban transportation mode, urban rail transit (URT) systems act as a key solution for high-density passengers. For the URT systems, even minor disruptions of can lead to severe harm of passenger safety, large traffic delay and widespread confusion because of the passenger density. In emergency circumstances, quick response and efficient substitution services like bus birding is of great importance for evacuation and accommodation of the affected passengers. In this paper, we propose a mathematical programming with indicator constraints model for designing a responsive bus bridging services under URT line emergency. The response time is integrated as an objective in the problem by considering the distance between the bus parking spots and the URT station as a starting point of a scheduled line. To solve the proposed model, a Genetic Algorithm (GA) based procedure is deployed because of the intractability. The results of some numerical experiments are also provided to demonstrate the proposed model. INTRODUCTION As an alternative transportation mode for passengers, urban rail transit (URT) systems are becoming increasingly important for their large capacity, good punctuality, high efficiency and low emission. However, as a mass transit mode, the URT systems may be evidently impacted by even minor disruptions. Once an accident happens, an effective response service for passengers is needed to mitigate the negative impacts of the incidents under emergency conditions. As another mass transit mode, bus service is more flexible. It can help the passengers to evacuate and meet their demand in an optimal way. This kind of service is often called bus bridging service. The key issues in bus bridging service may be bus route design, vehicle allocation and timetable scheduling. And the main goal for bus bridging service design should be finding the optimal way to meet the commuting demand as quick as possible. Motivated by the facts, in this paper, the bus route design problem considering the response time in bridging service is focused. For simplicity, here we only consider the case of emergency is occurred at in single line. Yuedi Yang, Hui Zhao, School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China
In literature, many research works have been done on the bus bridging service designing problem. For example, Kepaptsoglou and Karlaftis [1] propose a methodological framework for bus bridging network design problem, including route design and vehicle resources allocation. Latterly, Jin et al. [2] gives a more comprehensive way for bus bridging service design problem. Whereas the bus bridging routes in Wang et al. [3] are always set to complete the entire railway line, we formulate the bridging route design problem into a line planning one. Furthermore, by considering the distance of each relay point, the model proposed in this paper explicitly reflect the trade-offs between evacuation time and response time comprehensively. One may noticed that, among the line planning models, there is a class of location based models, which is very similar with the one proposed in this paper. For designing lines which are able to compete with private mode like cars or bicycles, Laporte et al. [4,5] proposed a passenger oriented approach called trip coverage model by maximizing the number of passengers using public transportation. For solving the trip coverage model, Marín and Jaramillo [6] deployed a Bender s decomposition technique. What should be pointed out is, though the objective of the location based model is very similar with one of ours, the constraints of the two types of models is quite different. In Laporte et al. [4,5], the location related constraints are alignment constraints and allocation constraints. But in this paper, the purpose of proposed model is to establish a bus bridging service plan. So, one of the key issue is to minimize the distance between the bus parking lots and the relay spots. Furthermore, the proposed model contains potential indicator constraints [7], like if facility is inactive, then no client can be assigned to it, as in the facility location [8], which is totally different with the previous ones. The rest of the paper could be organized into four parts. The main mathematical model is presented in the next section. For solving the proposed model, a metaheuristic based on Genetic Algorithm is given in Section 3. Numerical results for demonstrating the model is reported in Section 4. And conclusions and discussions are included in Section 5. MATHEMATICAL MODEL 1 2 3... n-1 n 1... m Relay Point Planned Bus Line URT Station Disrupted URT line Rally Line Bus Parking Lot Figure 1. Illustration of the topological structure for the bus bridging service for the URT systems.
In general, if an emergency occurs in an URT system, the whole URT line will be influenced. Figure 1 shows the topological structure of the system we studies. As is shown, there are URT stations numbered form 1 to and bus parking lots numbered from 1 to that are responsible for the emergency. One may notice that, for simplicity, there is only one direction for the URT system which is from station 1 to station. Actually, it is not difficult to generalize the model proposed below to the case with double directions. In this paper, the following notation for the input data is used: Indices, : URT station : Frequency of a line : Bus parking lot, : URT station, and, is adjacent 1, if the line from station to is with frequency = 0, otherwise 1, if the line from station to contains adjacent station, = 0, otherwise Parameters : Maximal frequency of the lines : Number of URT stations : Number of bus parking lots : The travel time for URT between station and station : Distance from URT station to the nearest bus parking lot : Distance from URT station to bus parking lot : The carriage capacity for the URT system : Passenger demand from to The above described input data are determined by three aspects: the railway infrastructure, the passenger demand, and the distance between the URT stations and bus parking lots. Using the input, the bus bridging problem could be formulated as a mathematical programming with indicator constraints as follows. min + 1 (1) s.t.,, (2) 1,,> (3) 0,1,,> (4) >0 =min 1, (5) =0 =0, (6) The objective of the problem in Equation (1) represents the sum of the operation costs of the bus bridging system and the length of the relay line, which is the distance between the URT station as a relay point to the nearest bus parking lot.constraints Equation (2) ensure that the bus bridging service could match the demand of all
passengers, which means that all passengers affected by the emergency condition could be transported. Constraints in Equation (3) state that each bridging service must obtain a unique frequency. Constraints in Equation (4) state the binary restriction for the decision variable x. Constraints in Equation (5) and (6) are indicator constraints to define the distance. As is presented, if >0, which means that station is a relay point, the distance from URT station to the nearest bus parking lot should be taken into consideration. Otherwise, if =0, which means that no bus should relay at point, the distance is set to be 0. SOLUTION ALGORITHM For solving the common mathematical programming problem with indicator constraints, it is usually to transform the indicator constraints into the explicit ones. But it is difficult to solve using the traditional methods like big technique. The heuristic algorithm adopted is based on the Genetic Algorithm (GA). As a widely used meta-heuristic procedure, one may refer to many references for details [9]. Specifically, the chromosomes for the GA procedure used in the paper are integercoded, and all decision variables in obtained feasible solutions are rounded to integer values also. To evaluate the fitness, it is better to use a maximization problem as the objective function. Fortunately, an upper bound of the objective function in Equation (1) could be expected, which is the case with maximum frequency for the bridging transit between all adjacent URT station pairs. So, the fitness could be computed by the difference between the upper bound and the value of the original objective function in Equation (1). NUMERICAL ANALYSIS In this example, there are 12 URT stations distributed along a rail. If an unexpected emergency occurs, buses from 4 bus parking spots are available for evacuation and rescue works. The numbers of evacuees getting on and off at each station are listed in Table 1. As discussed in the previous section, the traffic demand between each neighbor URT station pair could be calculated. And the results are also listed in the last line in Table 1. The distance form bus parking spot to station is given in Table 2, while the travel time between station to station is given in Table 3. The parameter is set as 0.1. TABLE 1. THE NUMBERS OF EVACUEES GETTING ON AND OFF, AND THE TRAFFIC DEMAND BETWEEN EACH NEIGHBOR URT STATION PAIR. Stations 1 2 3 4 5 6 7 8 9 10 11 12 1800 1200 1000 800 700 1300 800 1200 600 400 200 0 0 100 600 900 800 1400 1200 1300 1200 900 1000 600 1800 2900 3300 3200 3100 3000 2600 2500 1900 1400 600 -
TABLE 2. DISTANCE FORM BUS PARKING SPOTS AND URT STATIONS. Parking Station spot 1 2 3 4 5 6 7 8 9 10 11 12 1 30 27 25 23 20 23 25 30 36 42 50 60 2 28 25 20 24 27 33 38 42 47 50 57 63 3 83 77 65 60 51 46 32 26 21 24 28 32 4 42 36 34 31 29 26 24 21 18 15 21 26 TABLE 3. MINIMUM JOURNEY TIME BETWEEN STATIONS (MIN). Sta Station tion 1 2 3 4 5 6 7 8 9 10 11 12 1 3 6 9 12 15 18 21 24 27 30 33 2 3 6 9 12 15 18 21 24 27 30 3 3 6 9 12 15 18 21 24 27 4 3 6 9 12 15 18 21 24 5 3 6 9 12 15 18 21 6 3 6 9 12 15 18 7 3 6 9 12 15 8 3 6 9 12 9 3 6 9 10 3 6 11 3 12 The GA procedure are performed on an Lenovo Thinkpad X1 Carbon laptop with 2.2 GHz CPU and 8 GB memory, and the algorithms are implemented in Matlab 2011a on a Windows 7 OS platform. The number of population in the GA is set to 100, and the generation is set to be 1000. The probability of the crossover and mutation are set to be 0.8 and 0.2 accordingly. The result of the problem is shown in Table 4. TABLE 4. RESULTS FOR THE BRIDGING LINE AND FREQUENCY. 2 3 4 5 6 7 8 9 10 11 12 o F rom 1 6 5 3 1 2 2 2 5 6 5 1 1 2 3 3 1 6 5 1 7 3 5 1 4 2 7 3 1 3 5 7 9 1 3 2 5 6 2 7 2 8 9 10 8 10 3 2 11 CONCLUSIONS Reliable URT system is of great importance for dense passengers. Even minor emergencies could have an unexpected impact on the passenger safety and cause severe delays because of the density. To give an immediate response of the unexpected emergency, bus bridging services may be provided for the passengers.
Based on the consideration, this paper presented an optimization-based approach for the bus bridging service design in response to disruptions of URT systems. Specifically, a mathematical programming with indicator constraints model is established and a GA meta-heuristic procedure is also presented for solving the problem. The focus in this study is to evolve the optimal bus bridging services that are developed by the mathematical programming with indicator constraints modeling methodology. The model could integrate the response time to unexpected emergencies as an objective. For the bus bridging service design problem considered in this paper, the effect of emergency is global, which means the whole URT system is completed unusable when the emergency occurs. But in many cases, the emergency could be local, which means some segments in the URT system still works in emergency. So, passengers could be directed for evacuation with more flexibility. To design a bus bridging service in this case could also be studied in future work. ACKNOWLEDGEMENTS This research was partly funded by the National Natural Science Foundation of China (No. 71371028) and Fundamental Research Funds for the Central Universities (2015JBM049). REFERENCES 1. Kepaptsoglou, K., and M. G. Karlaftis. The bus bridging problem in metro operations: conceptual framework, models and algorithms. Public Transport, Vol. 1, No. 4, 2009, pp. 275-297. 2. Jin, J. G., K. M. Teo, and A. R. Odoni. Optimizing bus bridging services in response to disruptions of urban transit rail networks. Transportation Science, 2015. Published online in Articles in Advance. 3. Wang, Y., X. Yan, Y. Zhou, J. Wang, and S. Chen. Study of the bus dynamic coscheduling optimization method under urban rail transit line emergency. Computational intelligence and neuroscience, Vol. 2014, 174369. 4. Laporte, G., J. A. Mesa, F. A. Ortega, and I. Sevillano. Maximizing trip coverage in the location of a single rapid transit alignment. Annals of Operations Research, Vol. 136, No. 1, 2005, pp. 49-63. 5. Laporte, G., Á. Marín, J. A. Mesa, and F. A. Ortega. An integrated methodology for the rapid transit network design problem. In Algorithmic methods for railway optimization, Springer, 2007. pp. 187-199. 6. Marín, Á. G., and P. Jaramillo. Urban rapid transit network design: accelerated Benders decomposition. Annals of Operations Research, Vol. 169, No. 1, 2009, pp. 35-53. 7. Bonami, P., A. Lodi, A. Tramontani, and S. Wiese. On mathematical programming with indicator constraints. Mathematical programming, Vol. 151, No. 1, 2015, pp. 191-223. 8. Cornuejols, G., M. L. Fisher, and G. L. Nemhauser. Exceptional paper-location of bank accounts to optimize float: An analytic study of exact and approximate algorithms. Management science, Vol. 23, No. 8, 1977, pp. 789-810. 9. Kim, E., M. K. Jha, and B. Son. Improving the computational efficiency of highway alignment optimization models through a stepwise genetic algorithms approach. Transportation Research Part B: Methodological, Vol. 39, No. 4, 2005, pp. 339-360.