An Improved NSGA-II Algorithm for Transit Network Design and Frequency Setting Problem

. The transit network design and frequency setting problem is related to the generation of transit routes with corresponding frequency schedule. Considering not only the influence of transfers but also the delay caused by congestion on passengers’ travel time, a multi-objective transit network design model is developed. The model aims to minimize the travel time of passengers and minimize the number of vehicles used in the network. To solve the model belongs to a NP-Hard problem and is intractable due to the high complexity and strict constraints. In order to obtain the better network schemes, a multi-population genetic algorithm is proposed based on NSGA-II framework. With the algorithm, network generation, modechoice, demand assignment, and frequency setting are all integrated to be solved. The effectiveness of the algorithm which includes the high global convergence and the applicability for the problem is verified by comparison with previous works and calculation of a real-size case. The model and algorithm can be used to provide candidates for the sustainable policy formulation of urban transit network scheme.


Introduction
e urban transit planning process includes the following phases: network design, frequency setting, timetable development, vehicle, and driver scheduling [1,2].e network design with corresponding frequency is related to the scienti c conguration of urban transit facilities and the improvement of network service level, which will a ect the development of the city and the evolution of urban space morphology. is paper focuses on the network design and frequency setting problem.
e cross-sectional passenger volume and the congestion situation of a line, which depends on the frequency setting, should be considered in the network design.Furthermore, both the demand of users and the pro t of managers should be considered at the same time, thus it is a complicated multi-objective transit network design problem (MTNDP) [3,4].e characteristics of the related studies are listed in the Table 1.As can be seen, generally, the problem is formulated by minimizing the travel cost [5][6][7][8], minimizing the construction and operation cost [7,9], and maximizing the passenger attraction or coverage [5,6,9,10] etc. e studies about the problem can be classi ed into the entire network design and the routes design based on existing network [7].At present, the multi-mode networks integration method [11] and the multi-phase integration method in a network mode [12,13] are engaged.e e ective estimation of passenger ow state [14], the congestion and the capacity limitation of transit network [15] will have a great in uence on the network design.
e optimal matching between the distribution of passenger ow and the network structure can lead to less travel time and higher transport e ciency.
e solving of MTNDP is usually treated as an intractable probem for its high degree of complexity, the multiple objectives against each other and strict constraits, etc. e constraints and variables will explode with the expansion of network size.With the rapid development of computer technology and operational research, mathematical computing so ware and heuristic algorithm have been applied in transit network planning [6,9,10,16].However, for large-scale network, its nonlinear problem cannot be easily solved by conventional optimization so ware.As the classical approximative method, heuristic methods are o en used, which includes constructive algorithm, local improvement algorithm, and the combinations of them [17][18][19].However, many traditional heuristic algorithms have poor global convergence, and the results sometimes depend largely on the initial solutions.If the lines are selected from a given line pool to form the network, the rationality of these lines will largely depend on the quality of the line pool [20].At present, the metaheuristic methods, such as genetic algorithm [8,21,22], simulated annealing algorithm [13], tabu search algorithm [23], swarm intelligence algorithm [24] and other metaheuristics [25,26] are usually applied in the transit network design problem.
However, the generating of the network and the setting of the frequency, which are interrelated with each other, are separated in some methods.And it is particularly important to use the algorithm with high global convergence ability to search for the better network scenarios when the exact solutions cannot be obtained.In this paper, an e ective algorithm based on improved NSGA-II is put forward, in which the network generation, the mode choice, the passenger demand assignment considering passenger trip rule, and the frequency setting are all integrated.e algorithm has high global convergence and good applicability to the problem, which are veri ed by numerical experiments and case application.e structure of this paper is as follows: e mathematical model of MTNDP is presented in Section 2. Section 3 describes the multi-objective algorithm for the MTNDP, followed by numerical experiments in Section 4 and case application in Section 5. Section 6 presents the conclusions.

Mathematical Model
2.1.Notations.Let be the set of all nodes which represent the stations, be the set of edges which represent the sections in the transit network, , ∈ represent a section between adjacent stations ∈ and ∈ , ̸ = .e notations in the model are de ned as follows.
First, variables associated with the connection among nodes in the network are de ned as follows.
is an e ective path between starting and ending points.(vi) is the set of e ective paths, ∈ .(vii) ( ) is the set of neighbour nodes of node .
(viii) is the route cycle time in line , min.en, the variables associated with the changes of travel path are de ned as follows.
(i) = 1 if the passenger ow corresponding to the OD pair uses the section , of line along the path , 0 otherwise.(ii) ὔ = 1 if the passenger ow corresponding to the OD pair transfers from line to line ὔ at station along the path , 0 otherwise.
is the total number of stations in path .(iv) is the transfer times needed along the path .(v) ℎ is the headway of vehicles at the starting point of path .(vi) ℎ is the headway of vehicles taken by passengers travelling a er the ℎ transfer in path .(vii) g is the transit passenger volume travelling along the path between the starting point and the ending point .
Finally, the known quantities in the model are de ned as follows.

(i)
is the number of lines in the transit network is the shortest distance between the starting point and the ending point .
is the dwell time of vehicles at the ℎ nontransfer station in path .(xvii) is the walking time for the ℎ transfer in path .(xviii) ℎ min is the minimum limited headway of vehicles.

Model Formulation.
e generalised travel cost function on paths in transit network is formulated to calculate the passenger travel time, including the waiting time at starting point, in-vehicle travel time and transfer time.e in-vehicle travel time includes the travel time in sections and the dwell time at intermediate stations.e transfer time includes the walking time and the waiting time at transfer stations.Assume that all the passengers arrive at the stations randomly and are able to board the rst vehicle on the route they encounter, the average waiting time of passengers can be considered as half of the headway [27][28][29].Let 1 be the in-vehicle travel time, 2 be the total transfer time, then they can be expressed by Equations ( 1) and (2), respectively.
Passengers have di erent sensitivities to the transfer times in the same path.Under the same travel time, the more the transfer times are, the less likely the path would be selected by passengers.In most previous works, such as in [8,25,[30][31][32], the xed penalty coe cient was used to describe the in uence of transfers on the travel time.In this paper, the sensitivity coe cient of transfer times and the adjustment parameter are introduced to adjust the transfer penalty, so the total transfer time can be expressed by where, represents passenger's sensitivity to the ℎ transfer during the travel.is the adjustment parameter, generally greater than 1. e larger the , the greater the di erence for sensitivity levels between di erent transfer times in the path.
Excessive passengers waiting at station and excessive load factor of incoming vehicle will have a great impact on the passenger travel delay, which can cause an increase in travel time.In order to describe the impact of congestion on travel time while entering a station or transferring, the travel delay coecient of the station is de ned, as shown in where, v and are the cross-sectional passenger ow and the capacity of next section of the station in path , respectively.and are the adjusting coe cients.Let be the generalized travel cost of the path between the two points and , then the can be expressed by ∈ ( ) ≤ 2, ∈ , ∈ , , ∈ , ∈ ( ) e mode choice model is used to calculate the transit passenger demand g between OD pair .Assume that any segment of the basic road network can be used by other modes, such as private car, bicycle, and walk.Since the travel cost is usually the most important in uencing factor of mode choice, it is assumed that the mode choice probability depends exclusively on the generalised travel costs using di erent modes.Compared to the absolute utility, the passengers are more concerned about their relative utility.So the choice probability of the transit can be simply calculated by where is the choice probability of public transport for passengers traveling between the OD pair , min is the shortest time traveling between the OD pair using public transport, min is the shortest travel time using the ℎ mode, min is the average shortest travel time of all the modes used.
From the perspective of passengers, the minimum travel time of passengers is regarded as one of objectives to improve the network service level and travel e ciency, which is shown in From the perspective of managers, the minimum number of vehicles used in the network is regarded as another objective to reduce operating costs and improve the economic bene t, which is shown in Equations ( 8)- (10).
where, the headway ℎ is limited to integer for the convenience of operation and management, as shown in Equation ( 9), and the maximum cross-sectional passenger volume is calculated by Equation (10).
Multiple constraints are included related to network topology relationship, connectivity between stations, limited transfer times, network size, etc., as shown in ≤ , ∈ , ∈ , , ∈ , where, is the crowding distance of the ℎ objective of the ℎ individual.+1 and −1 are the ℎ objective function values of the ( + 1) ℎ and ( − 1) ℎ individuals, respectively.,max and ,min are the maximum and minimum values of the ℎ objective function of the ℎ individual.
(3) Crowded comparison For two individuals and , if the ranks and are same, the crowding distances and are compared.In the comparison, if In this paper, a multi-population genetic algorithm is proposed based on the framework of NSGA-II. Figure 1 presents the owchart of the algorithm proposed in this paper for the MTNDP.

Multi-Population Dynamic Coding.
In the process of chromosome coding, not only the connections among stations, but also the lines to which the stations belong should be considered.In order to e ectively store the topological information and improve the convenience of coding for transit networks, a multi-population dynamic coding method is proposed.
In this paper, the integer coding method is adopted, and the concepts of "main population" and "a liated population" are proposed.e information about connections among stations is stored in the chromosomes of "main population", and the information about lines to which the stations belong is stored in the corresponding chromosomes of "a liated population", as shown in Figure 2.
In Figure 2, alleles on chromosome in the 'main population' represent the stations while the alleles in its "a liated population" represent the line labels, and the stations that belong to the same line are connected sequentially.An individual which is a characteristic entity of a chromosome represents a network scheme.If a station appears multiple times in a chromosome of "main population", it represents a transfer station.e chromosomes in the same population have di erent lengths because of di erent number of transfer stations in networks.In the iteration process, the chromosomes in the "main population" and its "a liated population" whose lengths are dynamically changed have one-to-one correspondence.

Generation of Initial Populations.
For large network, it is di cult to generate a large number of initial solutions with high diversity under strict constraints.At present, the main initial solution generation procedures can be classi ed as follows.
Random connection method of adjacent nodes.In this method, the routes are generated by the connection between adjacent nodes, such as the probability-based IRSG procedure proposed by Jha et al. [26].e method usually has high calculation e ciency, but cannot guarantee that the travel time between OD pairs will not be too long along the routes.
e terms ( 11)-( 15) are the network topology constraints, indicating the relationship between nodes and edges.e term (16) indicates that the ow corresponding to the OD pair can use the section , of a line along the path only if the section , belongs to the line .e term (17) indicates that line has no branch at point .e terms (18) and (19) indicate that line is continuous and loop-free, where ∈ is an arbi- trary node set.Constraint (20) makes di erent lines have different nodes or connections.e term (21) indicates that the isolated nodes are not allowed in the network, where ⊂ is a candidate node set.e term (22) is the network connectivity constraint, which indicates that if there is passenger demand between two stations, the two stations must be connected in the network.Constraints ( 23) and ( 24) limit the transfer times for a path.e term (25) indicates that the passenger ow corresponding to the OD pair only has the in ow at the starting point and the out ow at the ending point , and the in ow is equal to the out ow at the intermediate stations of the path .e term (26) indicates that the passenger ow entering the line is equal to that exiting the line .e term (27) indicates that re ow is not allowed in the path.e term (28) limits the capacity of sections.Constraint (29) limits the number of stations in line .Constraint (30) limits the length of line .Constraint (31) limits the number of lines in the network.Constraint (32) limits the minimum value of the integral headway.

Improved NSGA-II Algorithm for the MTNDP
e multi-objective network design model developed in this paper belongs to the NP-hard problem mathematically.NSGA-II [33], which is the nondominated sorting genetic algorithm with elite strategy, is one of very e ective algorithms to solve the multi-objective problem.Compared with NSGA [34], the algorithm reduces the complexity and has the advantage of better robust performance.In NSGA-II, there are three important concepts that need to be explained as follows.For detailed description of the algorithm, please refer to the literature [33] by Deb et al.
(1) Fast nondominated sort If the objective functions of an individual is no worse than the others, the individual is the nondominated solution.In the iterative process, all the individuals are assigned to several fronts according to the noninferior relationships.e individuals in the next front is dominated by that in the previous front, and each individual is assigned a rank value .(2) Crowding distance In order to evaluate the distribution density of other individuals around a particular individual, the average distance between the two sides of the individual needs to be calculated, which is called the crowding distance of the individual.e distance for the two boundary individuals is assigned to in nity, and for the other individuals, the distances is calculated by the Equation ( 33).
Multipath-based method.In this method, the routes are generated based on e ective paths, such as the RGA and RFA procedures for initial solution skeleton proposed by Mahdavi Moghaddam et al. [38].With the method, the initial solutions can be generated from both the perspectives of passengers and managers, but the number of e ective paths between OD pairs to be searched will have a signi cant impact on the calculation time of the procedure.
In the existing procedures, the isolated nodes (i.e., nodes are in the connected basic road network but not served by the routes) cannot be e ectively circumvented, and the Shortest-path-based method.In this method, the routes are generated based on the shortest paths searching between OD pairs, and another procedure dealing with constraints is usually needed.For example, the construction and repair procedures used by Ahmed et al. [35] and Szeto and Jiang [36], the initial solution generation procedure based on greedy algorithm used by Nikolić and Teodorović [24] and Nayeem et al. [37], and the Floyd's algorithm and feasibility check used by Chew et al. [22].With the method, the direct demand between departure and destination stations can be met, but more vehicles on the network will be needed [38].
F 2: Multi-population chromosomes coding.Journal of Advanced Transportation the generated lines as intermediate nodes.In order to maintain the diversity of species, the repetitive chromosomes in the population are removed to ensure their uniqueness.e pseudo code for the initial population generation algorithm is shown in Algorithm 1.

Demand Assignment.
For comparison, the assumption about paths selection is the same as in the literature [40] by Arbex and Cunha, that is, assuming that passengers only select the paths with no more than two transfers.e passenger initially selects the direct path to avoid transfer.If the direct path is not available, only a single-transfer path will be selected.If there is no single-transfer path, then the two-transfer path is selected.Ultimately, if there is no two-transfer path, it is considered that the passengers' travel demand cannot be covered.characteristics of transit network structure are not well re ected in the initial solutions, such as the phenomenon of "rich club", that is, nodes with larger degree values ("rich nodes") tend to be connected to each other [39].For transit network, the endpoints of lines have the smallest average degree value while the transfer stations have the largest.erefore, in order to generate initial schemes with actual network characteristics, an initial population generation method is proposed in this paper.
(1) Generation of the endpoint set of lines e node with smaller degree is more likely to be selected as the endpoint which represents the departure or the terminal station of a line.In this paper, the line endpoints are randomly selected based on the roulette method, and the selection probability can be calculated by the Equation (34).
where, is the probability that node is selected as the endpoint of a line, satisfying ∑ ∈ = 1, is the degree value of candidate node .
(2) Generation of lines e remaining candidate nodes are randomly selected as intermediate stations of a line a er the endpoints are selected.Firstly, the -shortest path search method in nonweighted road network is used to get the top paths between two endpoints of a line.All the top paths should meet the constraint about nodes number which can be expressed by the distance between the two endpoints of each path, as shown in Equation (35).
where, is the distance between endpoints and in the nonweighted network.
en, a path is selected from the paths as the rst line according to a certain probability.e larger the average degree value of the intermediate nodes in the path, the greater the probability that the path is selected.e roulette method is also used and the probability can be expressed by Equation (36).
where, is the probability that the candidate path is selected, satisfying 0 < < 1 and ∑ ∈ = 1, is the average degree value of intermediate nodes in the path .
A er the rst line is generated, the paths between the two endpoints of next line are searched with the same method to constitute the set ὔ .e paths in ὔ that contains the isolated nodes in the road network are stored in the new set as the candidate paths of the next line.So, with the increase of generated lines, the number of candidate paths and isolated nodes decrease gradually.
Finally, if isolated nodes are not allowed and still exist in the generated network, they will be randomly inserted into

25: return
time is set to an initial nonzero value.If there are multiple paths, the relative utility of the path is used to calculated the selection probability , as shown in Equation ( 37). ( e multipath incremental assignment method is used in this paper.e OD passenger ow is divided into multiple parts equally that is presented in multiple OD matrices and distributed to the network successively.Before an OD matrix is assigned, the selection of paths and lines for passengers are calculated according to the assignment result of the previous OD matrix.In the initial state, there is no passengers on each line and the waiting where, is the calibration parameter, is the average travel cost for the e ective paths between the OD pair .

Two Types of Crossover.
According to the characteristics of transit network, the crossover operators of chromosomes can be classi ed into two types: inter-chromosome crossover and intra-chromosome crossover.
(1) Inter-chromosome crossover Inter-chromosome crossover is the replacement and recombination of nodes and connections between di erent parent individuals.According to the number of changed lines in a chromosome, it can be divided into single-line crossover and multi-line crossover, as shown in Figures 3 and 4. Taking the single-line crossover between two parents as an example, the steps are as follows.
Step 1.If there are common endpoints between lines belonging to the two parents, the common endpoints are selected to constitute the set , and the two lines are selected to constitute the set 1 and 2 respectively, then turn to Step 2. Otherwise, turn to Step 5.
Step 2. If the lines 1 ∈ 1 and 2 ∈ 2 have common inter- mediate nodes, the intermediate nodes are selected to constitute the set , and turn to Step 3. Otherwise, turn to Step 4.
F 6: Exchange of lines without common endpoints.
Line N 1 Line N 1 F 8: Mutation.Step 3. Randomly select a node ∈ to replace the node 2 .e corresponding chromosome in the "a liated population" remains unchanged.

Dealing with Strict Constraints.
O spring population is generated by selection, crossover and mutation of parents.It is necessary to ensure the diversity of the o spring.However, the generated o spring may not satisfy the strict constraints a er genetic operation.erefore, an auxiliary algorithm is designed to deal with the strict constraints.e genetic operators are executed according to certain probabilities.In this paper, intra-chromosome crossover will be executed when the inter-chromosome crossover is nished with ineligible o spring, and the mutation operator will also be executed when the two types of crossover operators are nished but none of eligible o spring is generated.If the size of the o spring does not meet the requirement, the genetic operators will be executed cyclically, and the new eligible ospring will be merged until the size meets the requirement.
e pseudo code for the auxiliary algorithm is shown in Algorithm 2.
Step 3. Randomly select two lines 1 ∈ 1 and 2 ∈ 2 which have the endpoint ∈ and intermediate node ∈ . All the nodes from to of the two lines exchange with each other, and update the corresponding chromosomal alleles in the "a liated population".If the o spring individuals meet the other constraints, the crossover ends.Otherwise, turn to Step 4.
Step 4. e two lines which have common endpoints are exchanged as a whole with each other between the two parents, and update the corresponding chromosomal alleles in the "a liated population" (as shown in Figure 5).If the o spring individuals meet the other constraints, the crossover ends.Otherwise, turn to Step 5.
Step 5. Randomly select a line from each parent and exchange them as a whole, then update the corresponding chromosomal alleles in the "a liated population" (as shown in Figure 6).
By executing the inter-chromosome crossover, not only the network size which includes the total number of stations and the total mileage of the network, but also the line characteristics which include the starting and ending points, the alignment, the mileage, the number of stations of each line and the transfer relationship between lines can be adjusted to get di erent network schemes.
(2) Intra-chromosome crossover Intra-chromosome crossover is the replacement and recombination of nodes and connections between lines in the same parent individual, as shown in Figure 7.It can't change the network size, but the line characteristics.e steps of the intra-chromosome crossover in a parent are as follows: Step 1. Search for the transfer nodes in the parent to constitute the set .
Step 2. Randomly select a node ∈ , and search for the lines with node to constitute the set .
Step 3. Randomly select two lines 1 , 2 ∈ 2 .All the nodes from to an endpoint of the two lines exchange with each other, and update the corresponding chromosomal alleles in the "a liated population".

Adaptive Mutation. Each iteration can result in Pareto solutions and may contain only partial solutions of the original
problem. e nondominated solutions may be dominated by new solutions generated a er successive iterations.In this paper, if new nondominated solutions cannot be generated by multiple iterations, the mutation operator will be triggered to jump out of the local optimal solutions, as shown in Figure 8.
e steps for mutation of a parent are as follows.
Step 1. Randomly select a line from the parent and three adjacent nodes 1 , 2 , and 3 in order from the selected line.
Step 2. Search for the nodes that can be connected to the nodes 1 and 3 to constitute the set .
1: Selection based on tournament selection process 2: while g_ < _ 3: do Randomly select two parent chromosomes 4: 1 ← _ ∈ (0, 1) Execute inter-chromosome crossover 7: if O spring chromosomes meet the constraints then 8: Execute intra-chromosome crossover 13: if O spring chromosomes meet the constraints then 14: and mutation is set to 0.9, 0.9, and 0.1, respectively.e maximum number of iterations is set to 1000.Table 4 presents the transfer penalty coe cients under di erent sensitivities.

Results and Discussion
. Figure 10 presents the comparison among buses, user cost, and direct trips of the networks for the scenarios with 4 lines under di erent transfer sensitivities .e results show that the larger the , the smaller the passengers' willingness to transfer in general, and the higher the user cost under the same direct rate.e solutions with better objective values can be obtained when = 1 under the given transfer sensitivities, but the direct rates of the solutions are obviously lower.Taking into account the objective values and the direct rates at the same time, the smaller number of buses or the lower user cost can be obtained when = 3 under a certain direct rate.Taking the four solutions including the solution 1 with = 1, the solution 2 with = 2, the solution 3 with 3 and the solution 4 with = 3 (as shown in Figure 10) as examples, the network scheme corresponding to the solution 1 requires the least number of buses and the lowest user cost, but the direct rate which is 90.95% has the least advantage.ere is little di erence for the direct rate among the solutions 2 , 3 , and 4 , but the solution 2 requires 3 more buses than the solution 4 , and 0.1 × 10 5 min more cost than the solution 3 .
erefore, the scenarios with di erent line numbers are calculated when = 3. Figures 11-14 present the compari- son between the optimal Pareto solutions obtained by the algorithm and the results in previous literatures.As can be e road network of Switzerland given by Mandl in [32] is used as the basic road network, as shown in Figure 9.In the road network, each node represents a city and each edge represents a road segment between two cities. e number on each edge represents the travel time between two cities. e OD matrix of the network are shown in Table 2.

Numerical Experiments
For comparison, the number of seats in each bus, the maximum limited load factor and the minimum number of stations min allowed in a line are the same as those used in [40] by Arbex and Cunha.In this calculation, because the passenger demand is static, the mode choic e model calculated by Equation ( 6) is not included.Since the headway of vehicles is not constrained to an integer in the literatures for comparison, the Equation ( 9) is calculated without rounding down, and the constraint (32) is not considered in this calculation.e parameter values in the model are shown in Table 3. e probabilities of inter-chromosome crossover, intra-chromosome crossover seen, the solutions obtained by the algorithm can dominate the results calculated in the previous literatures, which indicates the better solutions can be obtained by the algorithm.e objectives obtained in [40] by Arbex and Cunha is the closest to the solution obtained by the algorithm, but with the increase of transfer sensitivity, the advantage of the   : Average user cost in minutes per transit user min , comprising travel time and transfer penalties.
As can be observed from Table 5, the selected nondominated solutions obtained by the algorithm requires fewer buses, lower user cost, and lower transfer rate.erefore, the algorithm has high convergence in solving the MTNDP.        is included.It should be mentioned that as the travel distance increases, the probability of walking decreases the fastest until it drops to zero.e parameter values used in the case are shown in Table 6.
Figure 16 presents the Pareto frontier for the case obtained by the algorithm, and the percentages of direct and transfer passengers in each solution are shown in Figure 17.We can see that with the increase of buses on the network, the proportion of transfers decreases in general, and most of passengers can reach their destinations within one transfer.e two single objectives can achieve optimal in the boundary solutions respectively, as shown in Figure 16, and the bus networks for the two boundary solutions are shown in Figures 18 and 19, respectively.e boundary solution with fewer buses has more stations and smaller average cross-sectional passenger volume, and the maximum load rate of the lines for both solutions are less than 1, as shown in Tables 7 and 8.All the optimal Pareto solutions can be used as candidates for the city's bus network schemes.As there is no "one best solution" in the Pareto solutions and the determination of the network scheme is also in uenced by the policies such as urban politics, economy and environment, the choice of the best compromise scheme is not within the scope of the paper.

Conclusions
Considering both the travel e ciency of passengers and the bene t of managers, the multi-objective network design model

Case Application with Real-Size Network
e model and the algorithm are used for a real-size bus network calculation in Baotou city, Inner Mongolia.e predicted population density in long term and the trunk roads in the city centre area are shown in Figure 15.In the centre area, 10 bus lines will be planned on the trunk roads and the locations of 50 stops are determined in the trunk roads.In this case, the mode choice among bus, private car, bicycle and walk     is developed with strict constraints.For passenger travel behaviour, the transfer times and passenger congestion can a ect the selection of travel paths in the model.As the MTNDP is highly complex and di cult in solving, an e ective algorithm is proposed based on improved NSGA-II.e high global convergence of the algorithm is tested by comparing with previous works while its applicability for the MTNDP is veri ed by the calculation of a real-size case.

BeginF 1 :
Generate initial populations (main and its a liated populations)Chromosomes decodingDemands assignment and calculate objective function valuesIs o spring generated?Chromosomes of the parents are copied to the o spring Are parents generated?Flowchart of the proposed algorithm for the MTNDP.

F 5 :
Exchange of lines with the common endpoints.

F 10 :
Pareto solutions for the scenarios with 4 lines under di erent values of .

F 11 :
Pareto frontier for the scenario with 4 lines when = 3.

0
: Percentage of demand satis ed without any transfer.1 : Percentage of demand satis ed with one transfer.2 : Percentage of demand satis ed with two transfers.: Percentage of demand unsatis ed.: Average in-vehicle travel time in minutes per transit user, min .

F 12 :
Pareto frontier for the scenario with 6 lines when = 3.

F 13 :
Pareto frontier for the scenario with 7 lines when = 3.

F 14 :
Pareto frontier for the scenario with 8 lines when = 3.

T 5 :
Comparison between selected nondominated solutions and the results in previous literatures.

10 F 18 :
e bus network for boundary solution A.

10 F 19 :T 7 :
e bus network for boundary solution B. Detailed results of the lines for the boundary solution A. Line label Node sequence Fleet Headway (min) Maximum load factor Minimum load factor is the maximum cross-sectional passenger volume of line at peak hour.(xi) ℎ is the average headway of vehicles in line .
T1: e characteristics of the related studies on transit network design problem.max can dominate the solutions in the previous literatures are selected, as shown in Table5.e following indicators are usually used in many literatures to test the quality of results: Pareto frontier for the case obtained by the algorithm.
Detailed results of the lines for the boundary solution B.