Abstract
In many cities and regions, decision makers independently develop Train Operation Plan (TOP) for each line in the rail transit network, resulting in a lack of TOP Synchronization (TOPS). Considering the entire network as a whole, researchers have realized that synchronous optimization is of great significance. In this paper, we formulate two Mixed-Integer Linear Programming (MILP) models to optimize demand-driven TOP in the network. The former is an Asynchronous TOP Optimization (ATOPO) model, while the latter is a Synchronous TOP Optimization (STOPO) model. The bi-objective models simultaneously determine train frequency, train timetable, and rolling stock circulation under small-granularity passenger demand to minimize trains’ total cost and passengers’ total time. Then, we propose the Nondominated Sorting Coevolutionary Memetic Algorithm (NSCMA) to solve the combinatorial optimization problems. The hybrid heuristic algorithm incorporates Coevolutionary Memetic Algorithm (CMA) into Advanced and Adaptive Nondominated Sorting Genetic Algorithm II (AANSGA-II) to ameliorate the evolution process for elite individuals. On this basis, we study the case of Shenyang Metro to verify the models and the algorithm. The results demonstrate that the STOPO model is better than the ATOPO model in reducing trains’ total cost and passengers’ total time. In addition, NSCMA is better than AANSGA-II in obtaining elite individuals.
1. Introduction
As carbon emissions become a global issue, governments have paid more and more attention to energy consumption [1]. In recent decades, rail transit has developed rapidly around the world as an environmentally friendly mode of public transport. The rail transit systems in many cities and regions have entered the network era. However, lines in most networks are connected only by transfer stations, and trains on most lines are organized independently.
Since researchers know the independence of lines, ATOPO has become a hot issue. In these problems, lines are separated from the network, and TOPs are separately optimized for each line. Several studies have made significant progresses on demand-driven ATOPO problems [2–11]. These studies considered nontransfer passengers but omitted transfer passengers. Therefore, the asynchronously optimized TOPs are probably optimal for nontransfer passengers but probably not optimal for transfer passengers.
Since researchers understand the importance of transfer passengers, TOPS has become another hot issue. In these problems, transfer stations are separated from the network, and TOPs are synchronized for the network. Several studies have made many contributions to TOPS problems. Wong et al. [12] combined a heuristic algorithm with CPLEX to solve a TOPS model aiming at minimizing passengers’ transfer waiting time. Wu et al. [13] presented a TOPS model to minimize the maximal transfer waiting time while limiting passengers’ transfer waiting time equitably over any transfer station. Guo et al. [14] constructed a TOPS model to maximize the number of transfer synchronization events for the transitional period (from peak to off-peak hours or vice versa). A hybrid heuristic algorithm combined particle swarm optimization with simulated annealing to obtain near-optimal solutions efficiently. Liu et al. [15] built a TOPS model for minimizing passengers’ transfer waiting time. Tian and Niu [16] developed a TOPS model to minimize passengers’ transfer waiting time and maximize the number of connections. A novel sequential search algorithm solved the bi-objective model. Cao et al. [17] proposed a Genetic Algorithm (GA for short) with a Local Search (LS for short) strategy to solve a TOPS model by maximizing the number of connections. These studies considered transfer passengers but omitted nontransfer passengers. Therefore, the synchronized TOPs are probably optimal for transfer passengers but probably not optimal for nontransfer passengers.
Since researchers realize the importance of all passengers, STOPO has become an acknowledged challenge. STOPO overcomes the drawbacks of ATOPO and TOPS. In these problems, the network is seen as a whole, and TOPs are simultaneously optimized for the network. Several studies have made some attempts at STOPO problems. Niu et al. [18] presented a demand-driven STOPO model aimed at minimizing passengers’ waiting time and crowding disutility. Robenek et al. [19] constructed a demand-driven STOPO model to maximize companies’ profit and passengers’ satisfaction. Shang et al. [20] built a demand-driven STOPO model for minimizing passengers’ travel time. Wang et al. [21] developed a demand-driven STOPO model to minimize passengers’ waiting time and the number of passengers with failed transfers. Han et al. [22] formulated a demand-driven STOPO model to minimize trains’ operation cost and passengers’ total time. AANSGA-II obtained an approximate Pareto Optimal Solution Set (POSS for short) efficiently. These studies considered train formation, train frequency, and train timetable but omitted rolling stock circulation. Therefore, the synchronously optimized TOPs lack practicality.
This paper focuses on an integrated demand-driven TOP Optimization (TOPO for short) problem in the rail transit network. Two bi-objective MILP models called the ATOPO model and the STOPO model simultaneously determine train frequency, train timetable, and rolling stock circulation under small-granularity passenger demand to minimize trains’ total cost and passengers’ total time. A hybrid heuristic algorithm called NSCMA efficiently solves the bi-objective problem by ameliorating the evolution process for elite individuals based on AANSGA-II. A case study of Shenyang Metro verifies that the STOPO model is better than the ATOPO model and that NSCMA is better than AANSGA-II.
This paper is organized as follows: Section 2 states the demand-driven TOPO problem. Section 3 formulates the ATOPO and STOPO models. Section 4 proposes NSCMA. Section 5 studies the case of Shenyang Metro. Section 6 presents the conclusions.
2. Problem Statement
We focus on a network formed by a set of bidirectional lines . Each line contains a set of stations and a set of transfer stations , as illustrated in Figure 1. Each physical station on line refers to and in both directions, and each physical transfer station on line refers to and in both directions.
We use station to reindex transfer station . Each transfer station refers to station . On this basis, the set of transfer stations corresponds to a set of stations . Binary parameter equals to one if transfer corridor is valid.
We schedule a set of trains with capacity on each line . Limited by the maximum transport capacity, at most trains run on line with headways of at least . Required by the minimum service level, at least trains run on line with headways of at most . Each active train on line runs from station to station . We preset dwell time (at transfer station ), travel time (from station to station ), the earliest departure time and the latest departure time of each train, essential cycle time (on line ) of each connection as well as transfer walking time (in transfer corridor ) of each transfer passenger.
Assumption 1. We assume that no line adopts overtaking, skip-stop, cross-line, multi-routing, and multi-marshalling strategies.
We construct a set of time slices (also known as a set of time points) with length to express the time period of line . Combining the time periods of all lines, the time period of the network is expressed as . Notably, the time periods of all stations on line are normalized. The normalization of time periods reduces the dimension of variables effectively [22].
We describe dynamic passenger demand as small-granularity cumulative number of arriving passengers (at station at time ), alighting ratio of loaded passengers (at station at time ) and transferring ratio of alighting passengers (to transfer corridor at time ). The data are processed from small-granularity origin-destination matrices for simplicity [22].
Assumption 2. We assume that passengers from the outside arrive evenly during each time slice.
We design three binary variables and three integer variables for the demand-driven TOPO problem, as represented in formulas (1)–(6). Binary variable equals to one if train departs at time . Binary variable equals to one if train connects train . Binary variable equals to one if transfer passengers in transfer corridor from train arrive at time . Integer variable indicates the number of loaded passengers on train in section . Integer variable denotes the number of boarding passengers at station at time . Integer variable represents the number of transfer passengers in transfer corridor at time .
3. Mathematical Models
3.1. Asynchronous Train Operation Plan Optimization Model
The ATOPO model separately optimizes the TOP for each line in the network. It is formulated as a MILP model. Although binary variables are more than integer variables in formulating the same problem, the MILP model is linear without processing [23].
3.1.1. Objective Function
The ATOPO model aims to minimize generalized cost of line for companies and passengers in (7). On the one hand, we select trains’ total cost to express companies’ cost of line . On the other hand, we select passengers’ total time under weight to represent passengers’ cost of line .
Trains’ total cost of line includes trains’ operation cost and trains’ depreciation cost , as represented in (8).
Trains’ operation cost of line equals to unit operation cost multiplied by the number of trains, as shown in (9). The number of trains on line is accumulated by binary variable .
Trains’ depreciation cost of line equals to unit depreciation cost multiplied by the number of rolling stocks, as displayed in (10). The number of rolling stocks on line equals to the number of trains minus the number of connections. The number of connections on line is accumulated by binary variable .
Passengers’ total time of line includes passengers’ waiting time and passengers’ penalty time , as represented in (11).
Passengers’ waiting time of line consists of passengers’ basic waiting time and passengers’ additional waiting time, as shown in (12). Passengers’ basic waiting time of line equals to half multiplied by the number of arriving passengers from the outside. Passengers’ additional waiting time of line equals to multiplied by the number of waiting passengers.
Passengers’ penalty time of line equals to unit penalty time multiplied by the number of finally stranded passengers, as displayed in (13).
3.1.2. Constraints
The ATOPO model is subject to train constraints, connection constraints, and passenger constraints.
(1) Train Constraints. Train constraints stipulate the uniqueness, priority, departure time, and headway of each train, as well as the number of trains.
Constraints (14) and (15) specify the uniqueness of each train. Each time point can only correspond to at most one active train. Meanwhile, each train can only correspond to at most one time point.
Constraint (16) states the priority of each train. A train should be inactive if the previous train is inactive.
Constraints (17) and (18) limit that the departure time of each active train should be between and .
Constraints (19) and (20) limit that the headway of each active train should be between and .
Constraints (21) and (22) limit that the number of trains should be between and .
(2) Connection Constraints. Connection constraints stipulate the uniqueness and cycle time of each connection.
Constraints (23) and (24) clarify the uniqueness of each connection. Each active train can only connect at most one active train. Meanwhile, each active train can only be connected by at most one active train.
Constraint (25) claims that the cycle time of each connection should be at least .
(3) Passenger Constraints. Passenger constraints stipulate the number of loaded and boarding passengers.
Constraint (26) limits that the number of loaded passengers should not exceed if the train is active and equals to zero otherwise.
Constraint (27) limits that the number of boarding passengers should not exceed if an active train departs at the time point and equals to zero otherwise.
Constraints (28)–(31) declare the quantitative relationship between loaded passengers and boarding passengers. Constraints (28) and (29) are only for station , while Constraints (30) and (31) are only for the other stations. The number of loaded passengers should correspond to the number of boarding passengers and the number of alighting passengers. The number of alighting passengers should be equal to the alighting ratio multiplied by the number of loaded passengers.
Constraint (32) specifies the quantitative relationship between boarding passengers and arriving passengers. The cumulative number of boarding passengers should not exceed the cumulative number of arriving passengers.
In summary, the ATOPO model consists of the objective function and constraints as follows:
3.2. Synchronous Train Operation Plan Optimization Model
The STOPO model simultaneously optimizes the TOPs for all lines in the network. It is also formulated as a MILP model.
3.2.1. Objective Function
The STOPO model aims to minimize generalized cost of the network in (34).
Trains’ total cost of the network includes trains’ operation cost and trains’ depreciation cost , which are accumulated by trains’ operation cost and trains’ depreciation cost of each line, respectively, as represented in (35)-(37).
Passengers’ total time of the network includes passengers’ waiting time and passengers’ penalty time , which are accumulated by passengers’ waiting time and passengers’ penalty time of each line, respectively, as represented in (38)–(40). Since arriving passengers from the transfer corridors also wait for trains, passengers’ waiting time and passengers’ penalty time should also consider arriving passengers from the transfer corridors.
3.2.2. Constraints
The STOPO model not only inherits all the constraints in the ATOPO model but also supplements several new constraints.
Constraints (41) and (42) declare the quantitative relationship between boarding passengers and arriving passengers to replace constraint (32). Constraint (41) is only for the nontransfer stations, while constraint (42) is only for the transfer stations.
In addition, passenger constraints also stipulate the uniqueness and arriving time of each transfer passenger, and the number of transfer passengers.
Constraint (43) states the uniqueness of each transfer passenger. Each transfer passenger should alight from an active train, walk in a valid transfer corridor, and arrive at a time point.
Constraint (44) clarifies that the arrival time of each transfer passenger should correspond to the departure time of the train from which the transfer passenger alights.
Constraint (45) limits that the number of transfer passengers should not exceed if an active train departs at the time point and equals to zero otherwise.
Constraints (46) and (47) claim the quantitative relationship between transfer passengers and loaded passengers. The number of transfer passengers should be equal to the transferring ratio multiplied by the number of alighting passengers.
In summary, the STOPO model consists of the objective function and constraints as follows:
4. Nondominated Sorting Coevolutionary Memetic Algorithm
4.1. Algorithm Scheme
With the progress in artificial intelligence, machine learning algorithms have been widely used in transportation research [24–26]. However, heuristic algorithms are still the most common method to solve TOPO problems as they have been proven to be NP-hard problems [27].
The demand-driven TOPO models in this paper aim to minimize generalized cost for companies and passengers, in which weight is undetermined. Since decision makers expect different in different situations, a novel heuristic algorithm that can obtain an approximate POSS is appropriate. We select AANSGA-II to build the algorithm scheme.
AANSGA-II originates from NSGA-II. NSGA-II is a classical heuristic algorithm for solving multi-objective problems [28]. AANSGA-II improves NSGA-II in three aspects for TOPO problems. In local sorting, AANSGA-II proposes neighborhood distance instead of crowding distance to sequence the individuals in each frontier appropriately. In crossover and mutation, AANSGA-II introduces a scoring mechanism and alternative operators to produce the offspring population effectively. In population initialization, AANSGA-II adopts boundary individuals to generate the initial population reasonably. [22].
Despite these improvements, AANSGA-II also shows a drawback. Due to the nature of local sorting, AANSGA-II tends to pursue the first frontier rather than the elite individuals. We select CMA to improve the algorithm mechanism.
CMA originates from MA. MA combines GA with LS to improve computation efficiency and solution quality. Excellent individuals generated by LS participate in GA instead of original individuals [29, 30]. CMA improves MA by encoding LS settings (i.e., position, direction, step, strategy, and other parameters) as memes for coevolution [31, 32].
Inspired by MA based on NSGA-II, NSCMA incorporates CMA into AANSGA-II [33, 34]. It consists of population initialization, local search, nondominated sorting, local sorting, tournament selection, crossover and mutation, population combination, population replacement, POSS extraction, operator scoring, and termination judgement, as demonstrated in Table 1.
4.2. Algorithm Improvements
We only focus on the improvements in NSCMA since AANSGA-II was introduced comprehensively in our previous work [22]. The improvements are reflected in chromosome construction, local search, and local sorting.
4.2.1. Improvements in Chromosome Construction
AANSGA-II encodes real variables (i.e., headway of train ) instead of binary variables as genes to express the GA information. The chromosome in AANSGA-II is .
Differing from AANSGA-II, NSCMA encodes integer variable and binary variables , , and as memes to express the LS settings. Integer variable represents the position (i.e., line) of LS, which is between and . Binary variable expresses the direction of LS. means to broad the headway, while means to narrow the headway. Binary variable represents the step of LS. means to move , while means to move . Binary variable expresses the strategy of LS. means overall movement, while means local movement. The chromosome in NSCMA is , as illustrated in Figure 2.
As a module of the chromosome, the LS settings participate in crossover and mutation just like the GA information. In each crossover operation, the LS settings of the two individuals are partially exchanged. In each mutation operation, the LS settings of the individual are partially replaced.
4.2.2. Improvements in Local Search
AANSGA-II adopts all individuals in the hybrid population without processing. Due to the nature of crossover and mutation, the child population is generated without an optimization guarantee. The optimization mechanism in AANSGA-II is entirely nondeterministic.
Differing from AANSGA-II, NSCMA introduces LS to enhance the optimization guarantee. Several representative weights are preset to obtain all elite individuals in the first frontier since only elite individuals are valuable for decision makers. Each elite individual performs LS according to its LS settings. The processed individual is accepted if an improvement is achieved, and the original individual is retained otherwise. As the nature of LS, the processed population is optimized with an optimization guarantee. The optimization mechanism in NSCMA is partially deterministic.
4.2.3. Improvements in Local Sorting
AANSGA-II sequences all individuals in each frontier by local sorting. Local sorting is based on neighborhood distance. The individuals in AANSGA-II have descending original neighborhood distances.
Differing from AANSGA-II, NSCMA proposes a reference point to improve local sorting. The reference point is composed of the maximum acceptable objective values for decision makers. An individual is valid if it dominates the reference point and invalid otherwise. On this basis, all individuals in each frontier are classified into valid individuals and invalid individuals. Since valid individuals are more valuable than invalid individuals, the neighborhood distances of valid individuals are retained, while the neighborhood distances of invalid individuals are cleared. The individuals in NSCMA have descending processed neighborhood distances.
Since the benchmark individual with trains’ total cost and passengers’ total time may be too excellent to serve as the reference point, NSCMA proposes tolerance factor to expend the maximum acceptable objective values to and , as illustrated in Figure 3.
5. Case Study
5.1. Case Setup
Shenyang Metro in Northeast China operated a cruciform rail transit network from December 30, 2013, to April 7, 2018, as illustrated in Figure 4.
Dynamic passenger demand is strictly processed from the historical data on December 9, 2016. The distribution of passenger demand is bimodal, as illustrated in Figure 5.
The actual TOPs of the two lines are encoded together as the Benchmark Solution (BS for short). The time period of each line is normalized to [4 : 50, 22 : 20]. Table 2 demonstrates the parameters of the case.
The case focuses on the approximate POSS based on the STOPO model (SS for short) and the approximate POSS based on the ATOPO model (AS for short). Notably, AS of Line 1 (AS-1 for short) and AS of Line 2 (AS-2 for short) are optimized independently. Each solution in AS-1 and each solution in AS-2 form an integrated solution together. AS refers to the POSS in all integrated solutions.
The case applies NSCMA and AANSGA-II for comparisons. Both heuristic algorithms are encoded in MATLAB R2019a. All computations were performed on a personal computer. Table 3 demonstrates the parameters of the heuristic algorithms.
5.2. Case Results
Firstly, we solve the STOPO model with NSCMA and AANSGA-II. NSCMA obtains SS at generation 8491 within 123 min, while AANSGA-II obtains at generation 9773 within 127 min. We use trains’ total cost and passengers’ total time as indicators to compare the performances of SS, , and BS, as illustrated in Figure 6. Notably, only the solutions better than BS in both costs, also known as the valid solutions, are drawn in Figure 6.
According to Figure 6, SS outperforms in progressiveness, while SS does not outperform in diversity. SS dominates , which means that NSCMA is better than AANSGA-II in the evolution process for the elite individuals. However, presents more small cracks but fewer large cracks than SS, which means that NSCMA is not better than AANSGA-II in the evolution process for the first frontier.
Secondly, we solve the ATOPO model by NSCMA. NSCMA obtains AS-1 at generation 1142 within 7 min and AS-2 at generation 304 within 2 min. AS is selected from all combinations of AS-1 and AS-2 within 1 min. We use trains’ total cost and passengers’ total time as indicators to compare the performances of SS, AS, and BS, as illustrated in Figure 7.
In the light of Figure 7, SS contains thirty-four solutions, while AS contains fifteen solutions. SS dominates AS, obviously, which means that the STOPO model is better than the ATOPO model.
Thirdly, we list the Elite Solutions in SS (SS-Es for short) and the Elite Solutions in AS (AS-Es for short) under specific weights. Inspired by our previous work, the representative weights are [22]. These six numbers form a proportional sequence with a common ratio of 1.44. We use trains’ total cost , passengers’ total time , generalized cost , and improvement rates over BS , , and as indicators to compare the performances of SS-Es, AS-Es, and BS under different , as demonstrated in Table 4. Notably, all elitist solutions are marked in Figure 7.
According to Table 4, both elite solution sets consist of five distinct solutions, respectively. Compared with BS, SS-Es improve by 4.35% and by 5.51% on average, while AS-Es improve by 1.57% and by 1.39% on average. Compared with AS-Es, SS-Es improve by 2.83% and by 4.18% on average.
Fourthly, we use the load factor of each train as an indicator to analyze the utilization of trains in SS-Es and BS, as illustrated in Figure 8.
In the light of Figure 8, most SS-Es utilize trains more effectively than BS. Overall, SS-E-1 uses twenty-three fewer trains than BS, SS-E-2 uses eighteen fewer trains than BS, SS-E-3 uses four fewer trains than BS, SS-E-4 uses the same number of trains as BS, and SS-E-5 uses four more trains than BS. Specifically, no SS-E uses more trains on Line 1 than BS, while three SS-Es use more trains on Line 2 than BS. The difference demonstrates that the actual TOP of Line 1 has more space for optimization than that of Line 2. Besides, fewer trains generally mean higher average load factors.
Fifthly, we use the task order of each rolling stock as an indicator to analyze the utilization of rolling stocks in SS-Es and BS, as illustrated in Figure 9.
According to Figure 9, all SS-Es utilize rolling stocks more effectively than BS. Overall, SS-E-1 and SS-E-2 use four fewer rolling stocks than BS, SS-E-3 uses three fewer rolling stocks than BS, SS-E-4 uses two fewer rolling stocks than BS, and SS-E-5 uses one fewer rolling stocks than BS. Specifically, all SS-Es use fewer rolling stocks on Line 1 than BS, while no SS-E uses fewer rolling stocks on Line 2 than BS. The difference proves again that the actual TOP of Line 2 matches passenger demand better than that of Line 1. Besides, fewer rolling stocks generally mean more tasks per rolling stock.
In summary, the superiority of the STOPO model over the ATOPO model explains that TOPS is an important element in TOPO for the network. Besides, the superiority of NSCMA over AANSGA-II suggests that LS is a powerful complement to evolutionary algorithms.
6. Conclusion
This paper researched the demand-driven TOPO problem in the rail transit network. The bi-objective MILP models, the ATOPO model, and the STOPO model minimize trains’ total cost and passengers’ total time by simultaneously determining train frequency, train timetable, and rolling stock circulation. The hybrid heuristic algorithm, NSCMA, ameliorates the evolution process for elite individuals by incorporating CMA into AANSGA-II.
According to the case of Shenyang Metro, the STOPO model is better than the ATOPO model. The elite synchronous TOPs reduce trains’ total cost by 2.83% and passengers’ total time by 4.18% on average compared to the elite asynchronous TOPs. Besides, NSCMA is better than AANSGA-II. NSCMA outperforms AANSGA-II in obtaining the elite individuals, while NSCMA does not outperform AANSGA-II in obtaining the first frontier.
In the future, we will focus on the difficulties as follows. Firstly, the weight of passengers’ total time is discussed with an alternative set. It is meaningful to integrate all objectives into a unified dimension. Secondly, NSCMA does not show comprehensive improvement over AANSGA-II. It is optional to apply other heuristic algorithms.
Data Availability
The passenger demand data used to support the findings of this study have not been made available because of the secrecy agreement.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper was supported by the National Natural Science Foundation of China (71971019), the Fundamental Research Funds for the Central Universities (2020JBZD007), the Frontiers Science Center for Smart High-Speed Railway System, and the Beijing Natural Science Foundation (L201013).