1 article information
The article titled "Integrated optimization of bus bridging routes and timetables for rail disruptions" is an article about bus connection and subway timetable optimization, which was published in the European Journal of Operational Research.
2 Summary
Rail disruptions can lead to rapid degradation of public transport systems. This article focuses on the design of bus bridging under the interruption of railway traffic, and considers the route strategy combining express and short-distance routes. This paper proposes a comprehensive optimization framework based on time-varying demand, which jointly optimizes bus crossing routes and bus schedules. This paper proposes a brute-force search method to identify all candidate bus bridge routes, builds a mixed integer linear programming model, and simultaneously solves the problem of route selection, bus deployment, and bus scheduling. In order to solve the computational challenges brought about by this large-scale problem, this paper proposes a LP rounding heuristic algorithm based on tabu search. The article takes the Cranbourne line in Melbourne as an example, and numerical studies show the applicability of the method in practice. Experimental results demonstrate that the proposed framework can generate high-quality bus bridging solutions in a reasonable amount of time, allowing rapid response to railway line interruptions. A multi-route transit bridging strategy outperforms standard bridging routes in terms of bus route capacity and passenger travel time.
The contributions of the article are summarized as follows:
1. This study is based on the experience of Melbourne, aiming at the bus bridge crossing strategy for different routes. In addition to standard bus feeder routes, this paper also considers short-distance and express bus bridge routes, and studies the comprehensive optimization of bus bridge routes and timetables under time-varying demand and congestion conditions. In this case, in order to ensure The coordination between the various bus crossing services creates irregular timetables.
2. The article uses the mixed integer linear programming model to simulate the joint design problem of bus routes and dispatching, and pre-generates candidate bus bridge routes through the proposed brute force search method. A heuristic algorithm combining tabu search and LP rounding is designed to solve the MILP model.
3. The article conducts experiments on the actual track and verifies the applicability of the model in practice. In the experiment, an initial solution can be obtained within 30 seconds, and an approximate optimal solution can be obtained within 10 minutes. The efficiency of the solving algorithm guarantees a fast response to unexpected interruptions.
3 Problem Definition
The article focuses on the interruption of a railway line, and the train station between two transfer stations is disconnected, that is, only one line is considered, as shown in the figure above. The feeder bus is designed to restore connectivity between subway stations. The article notes 
indicate bus stops, and these bus stops are set around the interrupted subway stations, where station 1 and station S represent the transfer station (turnover), in addition, from station 1 to station S represents the uplink, and vice versa, by station S To station 1 means downlink.
The article assumes that interrupt recovery times can be predicted or predefined. Let [0, T] represent the interruption time, where time 0 is the initial time of the connecting bus service, and T represents the time for the interruption and recovery of the subway point. The connecting bus leaves the terminal station of the line within the time interval [0,T]. In addition, the article assumes that passengers arriving during the traffic interruption period [0, T] choose to connect to the bus to complete the trip, while passengers arriving after time T will choose the subway. In the problem, passenger demand includes three parts of passenger flow, namely stranded passengers, passengers arriving by subway and passengers arriving by other means.
Due to the maximum capacity limit of the connecting bus, when the passenger arrives at the connecting station, he may need to wait for the next bus or other lines, and may even be unable to be served due to passenger flow overload. If the passenger cannot get the service, the passenger will Leave the bus bridge system immediately to save waiting time. The article assumes that the longest waiting time for serving passengers is set to W. Therefore, the research time is extended to [0,T + W], as shown in the figure below. It is worth mentioning that although the article assumes that W is less than the total travel time of the bus line, the article points out that the proposed method is still effective when W is larger.
The feeder bus in the article allows buses to operate in full/short distance and local/express mode. Specifically, in the full route, the bus runs between two transfer stations, while the short route connects a transfer station (turnover) and an intermediate station. For the express/partial service mode, the express service mode is only provided at a few major stations where passenger traffic is high, while the local service mode stops at all stations. Further, the article divides a line into several subsegments, and each subsegment refers to a set of small stations between two adjacent main stations. For a line, some subsections are likely to call at every station, while stations in other subsections are skipped. Therefore, this route provides a mix of express and local services. The article calls it a limited express service (Limited express).
To sum up, a route is determined based on the destination, that is, whether it is a full journey or a short journey, and whether the main station is selected as a stop. Further, for each sub-section between two selected main stations, it should be determined whether to run local train or express train. A typical bus route across the bridge is shown in the diagram below.
Articles are recorded as a collection of candidate connecting bus lines, numbered by r. Due to the existence of multiple parallel connecting bus lines, passengers have many options for travel. This article assumes that in the travel of passengers, the transfer between buses and buses is not considered, and a standard route is opened as the basic route to provide direct services for passengers. .
The purpose of the feeder bus service design problem is to first determine the set of all candidate bridging routes R, and select an optimal subset of these routes, which contains several routes that are related to bus resource deployment and bus departure time scheduling Relatedly, the optimization objective is to minimize the total delay for all affected passengers given a time-varying demand pattern.
4 models
Candidate Route Generation (CRG)
We design a brute-force search method to identify all candidate bus bridge routes. For each site s, the article introduces parameters such that each candidate route is associated with a unique dummy value 
. The brute force search method iteratively generates candidate routes according to the ascending order of dummy values. The candidate route generation (CRG) model generates a new candidate route in each iteration, and the parameters of the model are shown in the following table.
Candidate feeder bus routes are uniquely identified by the bus terminal, the main stop at which the bus stops, and the stop schedule (i.e., local/express) for the sub-segment between any two selected main stops. The CRG model formula for generating route r is shown below.
Among them, the objective function is to minimize the virtual value, which is achieved by constraining (1k) to ensure that the newly generated route has a higher virtual value than any existing route in the current set. Any line that satisfies constraints (1b)-(1j) is a candidate connecting bus line.
Constraints (1b)-(1e) represent the constraints added when selecting the bus terminal of route r, and constraints (1b) and (1c) ensure that the start station and the end station are selected from the set of candidate terminal destinations, respectively. Constraint (1d) ensures that only two feeder stations can be selected as bus terminals. Constraint (1e) guarantees that bus routes start and/or end at transfer stations.
Constraints (1f) and (1j) are related to the route's parking schedule. Constraint (1f) ensures that buses can only stop at stops between the first and last stops. Constraint (1g) ensures that vehicles must stop at the origin and destination stations. Constraints (1h)-(1j) determine whether a line skips subsections or not.
The CRG model proposed in the article is an integer linear programming. The number of binary decision variables is 3|S|, so the problem can be easily solved using a solver. The article states that with |S| = 13, the solver can complete the solution in 1 second. Based on the CRG model, this article proposes a brute force search algorithm to obtain all connecting bus lines until no feasible path can survive.
Route selection, route allocation and timetable formulation (RST)
After obtaining all feasible routes, the article proposes the RST model to be used to realize route selection, route allocation and timetable formulation. This model is a mixed integer linear programming model. The parameters and variables of the model are defined as follows.
In order to facilitate the formulation of the model, the article puts forward four assumptions, which are briefly summarized as follows:
(1) Time-varying demand: Passengers' demand for connecting bus service changes with time is definite. The passenger arrival rate of each OD pair may be different at different times
(2) Passenger assignment: Passengers get on the bus according to the principle of system optimization, that is, the total delay is the smallest.
(3) Transfer between connecting buses: The route marked in the article replaces the transfer behavior between buses.
(4) Bus service: The bus running time and bus capacity are given.
Based on the above assumptions, the RST model is as follows.
Among them, the objective function aims to minimize the total travel time of passengers. The first item is the total waiting time of passengers at the station, the second item is the total boarding time of passengers, and the third item represents the punishment time for dissatisfied passengers.
Constraint (2b) states that at most R lines are chosen, where R is a parameter. Constraint (2c) states that the standard route must be chosen as the basic route. Constraint (2d) expresses the restriction on the number of buses running. Constraint (2e) ensures that buses can only be deployed on selected routes.
Constraints (2f) and (2g) ensure that the fleet sizes assigned upstream and downstream of route r are sufficient to perform bus departures respectively.
Constraints (2h) and (2i) ensure that the total number of bus services on each route should be within a predefined range.
Constraints (2j)-(2o) are constraints imposed on the assignment of passengers to bus services for each OD pair. Constraints (2j) and (2k) ensure that the passenger demand of su with an OD request can be assigned to the connecting bus route r only when the connecting bus route stops at stations s and u. Constraints (2l) and (2m) are bus capacity constraints upstream and downstream of the route, respectively. Constraints (2n) represent flow balance constraints, as shown in the figure below.
Solving Algorithm
Due to the large number of integer variables in the RST model, this paper proposes a heuristic algorithm based on tabu search and LP rounding method to solve the model. In order to improve the efficiency of the algorithm, the search space of the tabu heuristic algorithm only contains the decision variables of the candidate route selection, not the bus deployment and schedule variables. Given a set of selected routes generated by a tabu search, solutions for bus deployment and schedule and passenger assignment can be determined by solving RST subproblems. However, the RST subproblem is still difficult to solve. This paper proposes an LP intake method to solve the subproblem by linearly relaxing some integer constraints. This paper proposes an LP rounding method for solving subproblems by relaxing integer constraints. This allows a quick assessment of the quality of the solutions in the search space, thus speeding up the process of the Tabu Search. After the tabu search process, the optimal solution of the current candidate route selection is obtained, and then the RST sub-problem is solved by CPLEX to obtain a feasible bus deployment and timetable.
1. Obtain the initial solution
The initial solution has a significant impact on the performance of the heuristic. In order to get a good initial solution, the paper builds a simplified RST model to get the solution of candidate routing. The simplified RST model formula is shown below.
In the simplified RST subproblem, the objective is reduced to minimize the ride time of passengers and the penalty travel time for not serving passengers. The simplified RST model takes total demand as input and solves for the selected routes and the total bus frequency on each route. The simplified model can be solved efficiently, and the result of the selected set of paths is used as the initial solution.
2. LP rounding
Given the set of selected routes in the initial solution, the RST subproblem determines the deployment and schedule of buses and the corresponding travel times for passengers. The RST subproblem is an RST model with a given set of candidate connecting bus routes, specifically expressed as follows.
Although the above model can significantly reduce the size of the RSP subproblem, it is still time-consuming to solve repeatedly for different combinations of bus routes. Therefore, this article proposes an LP intake method to further improve the solution efficiency. Specifically, the article first proposes a linear relaxation model of the subproblems, as follows.
After solving the relaxed RST subproblem, a fractional solution to the bus scheduling problem is obtained. Then we round it up to get a solution. The rounding method is shown in the figure below. Further, the article proves that the rounded solution is feasible for the RST subproblem, and proves that the target value of the relaxed RST subproblem provides the lower bound of the RST subproblem, and the corresponding target value of the rounded solution provides the RST subproblem. Upper Bound.
3. Taboo search
In this paper, the tabu search algorithm is used to search for the optimal set of connecting bus lines to achieve the goal of minimizing passenger travel time. The setting of tabu search can be summarized as neighborhood structure, tabu and expectation criteria, and its algorithm is shown below.
5 Numerical experiments
The article uses Melbourne's real subway dataset as an experimental dataset, and considers a railway interruption on the Caulfield-Dandenong section of the Cranbourne line, as shown in the figure above. The table below gives the bus running times between two adjacent stations. The stop time of the bus is set to 1 minute, which includes additional acceleration and deceleration time.
The total passenger demand for each OD pair during this period is shown in the figure below.
Based on the above dataset, in order to study the feeder strategy, the paper first compares the optimized solution with the standard feeder solution (i.e., only one route with all stops). The figure below shows the corresponding results for the optimized solution.
For the feeder strategy, the article compares the optimized solution with the actual solution, as shown in the figure below.
Further, in order to verify the effectiveness of different bus connection strategies (ie short-turn/express), the article compares the situation of three connection routes under different bus connection strategies, as shown in the figure below. The results indicate a synergy between short-turn and express services, both of which should be considered as candidate types.
In order to evaluate the effectiveness of the irregular schedule, the article compares the difference between the feeder solution with uneven lanes and the feeder solution with uniform lanes, and the results are shown below. The results show that the performance of bus bridging services is greatly improved by introducing irregular timetable optimization.
In addition, the article also conducted several parameter sensitivity analysis, including bus fleet size and different penalty sizes, the results are shown in the figure below.
Further, the article analyzes the computational efficiency from two perspectives, the fleet size and the length of the route set. The article first compares the computational efficiency of the tabu search algorithm with simulated annealing (SA) and genetic algorithm (GA), and the results are shown in the table below.
Further, the article compares the computational efficiency of the LP rounding algorithm with that directly solved by CPLEX, and the results are shown in the table below.
6 Summary
The article mainly aims at the problem of bus bridge service design, and optimizes the connecting bus routes and subway timetables together. In order to improve service efficiency, this paper studies a multi-route feeder bus strategy. A brute-force search method is proposed to identify all candidate connecting bus routes. On this basis, a mixed integer linear programming (MILP) model for candidate route selection, bus scheduling and bus departure time planning is established. Aiming at the large-scale MILP model, an algorithm is proposed to solve it.
Numerical research is carried out on the actual track to verify the effectiveness of the proposed model and algorithm. Compared with standard feeder bus lines, the article's strategy of adding short-distance and express routes significantly increases transportation capacity and reduces travel delays for passengers. The operation effect of the optimized bus line is better than that of the actual line. An irregular timetable further enhances the quality of service. Computational experiments show that tabu search is more suitable for the bus connection problem and has better performance than SA and GA. The LP rounding method improves the performance of the tabu search.
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