The Depth-First Optimal Strategy Path Generation Algorithm for Passengers in a Metro Network

: Passenger behavior analysis is a key issue in passenger assignment research, in which the path choice is a fundamental component. A highly complex transit network o ﬀ ers multiple paths for each origin–destination (OD) pair and thus resulting in more ﬂexible choices for each passenger. To reﬂect a passenger’s ﬂexible choice for the transit network, the optimal strategy was proposed by other researchers to determine passenger choice behavior. However, only strategy links have been searched in the optimal strategy algorithm and these links cannot complete the whole path. To determine the paths for each OD pair, this study proposes the depth-ﬁrst path generation algorithm, in which a strategy node concept is newly deﬁned. The proposed algorithm was applied to the Beijing metro network. The results show that, in comparison to the shortest path and the K-shortest path analysis, the proposed depth-ﬁrst optimal strategy path generation algorithm better represents the passenger behavior more reliably and ﬂexibly.

is the optimal strategy, which has the minimum expected travel time. E + a is the link from the optimal strategy, which starts with node a. W(E + a ) is the average waiting time at node a for E + a . P (a,b) (E + a ) is the service probability for the link (a, b) from the optimal strategy E * . D is the destination set and d is the destination station. v represents the passenger volume, where v a is the passenger volume at station a and v (a,b) is the passenger volume at the link (a, b).
In a is the entry passenger at node a. u a is the expected travel time from a to the destination. f a is the accumulated serve frequency at node a.
x (a,b) is the binary variable that determines if the link (a, b) belongs to the optimal strategy or not. w a is the passenger total waiting time at node a.

Model and Algorithm
It is a common phenomenon that the passenger wants to complete their trip with the shortest travel time. The passenger choice model can be formulated as follows.

Min
In a ≥ 0, a ∈ I.
The object is to minimize the total travel time for all of the passengers in the network. The links which are satisfied with the model are contained in the optimal strategy set E * . Equation (2) is the passenger volume that is assigned to the link (a, b). Equation (3) is the flow conservation at node a. Equation (5) is the non-negative constraint for the passenger flow. This is a non-linear mixed-integer model. Equation (4) is the only non-linear constraint. Following the characteristic of the binary variable, Equation (4) can be relaxed as v (a,b) ≤ f (a,b) w a , (a, b ∈ E + a , a ∈ I. To simplify the model description, the variable w a , w a = α · v a / f (a,b) x (a,b) , a ∈ I, a, b) ∈ E + a . is introduced to represent the passenger total waiting time. The optimization model can be re-written as follows.

Min
In a ≥ 0, a ∈ I.
The new model is a mixed-integer model. To solve this model and find the optimal link for the optimal strategy, Spiess and Florian [28] already proposed the optimal strategy searching algorithm to find the best solution for the above model. The search algorithm is illustrated in Figure 1.
The optimal strategy algorithm is similar to the backward labeling searching algorithm. The algorithm only determines the links that are used by the passengers to shorten their travel time. However, the path information is not included in the proposed algorithm. To overcome this problem, the depth-first path generation algorithm is proposed in the next section to generate a passenger path from the optimal strategy set. The optimal strategy algorithm is similar to the backward labeling searching algorithm. The algorithm only determines the links that are used by the passengers to shorten their travel time. However, the path information is not included in the proposed algorithm. To overcome this problem, the depth-first path generation algorithm is proposed in the next section to generate a passenger path from the optimal strategy set.

Optimal Strategy Tree
The algorithm from Section 2 can determine the optimal strategy for each OD pair. Figure 2a presents the transit service map, which includes four kinds of service lines. The red and blue lines are the stop-stop transit services for two lines. The green line and the yellow line are the skip-stop transit services on the map. After applying the optimal strategy algorithm, the optimal strategy links are the remaining links as shown in Figure 2b.

Optimal Strategy Tree
The algorithm from Section 2 can determine the optimal strategy for each OD pair. Figure 2a presents the transit service map, which includes four kinds of service lines. The red and blue lines are the stop-stop transit services for two lines. The green line and the yellow line are the skip-stop transit services on the map. After applying the optimal strategy algorithm, the optimal strategy links are the remaining links as shown in Figure 2b. The optimal strategy can be considered as the combination of the alternative links for each node. In Figure 3, the optimal strategy can be written as (A, B), H) and (H, I). If we take the link node as the decision tree node, then the link is the tree branch. The optimal strategy tree is shown in Figure 3.  The optimal strategy can be considered as the combination of the alternative links for each node. In Figure 3, the optimal strategy can be written as (A, B), (A, I), (B, D), (D, E), (E, F) (E, H), (F, H) and (H, I). If we take the link node as the decision tree node, then the link is the tree branch. The optimal strategy tree is shown in Figure 3. The optimal strategy can be considered as the combination of the alternative links for each node. In Figure 3, the optimal strategy can be written as (A, B), (A, I), (B, D), (D, E), (E, F) (E, H), (F, H) and (H, I). If we take the link node as the decision tree node, then the link is the tree branch. The optimal strategy tree is shown in Figure 3.  When passenger travel behavior is based on the optimal strategy, they will receive the first arrived services from the optimal strategy. The passenger behavior searches along the optimal tree. The complexity of the optimal tree is determined by the number of nodes and the alternative links that are connected with the node. In this way, the passenger's choice can be taken as the path generated from the optimal strategy tree.

Strategy Node
There are two kinds of nodes in the optimal strategy tree. Some of the nodes are connected with only one link, such as Node B and Node D in Figure 3. For some of the nodes such as Node A and Node E, there is more than one connecting link. This means that the passengers could make their choice at these nodes according to the train arriving time or the link service frequency. This also follows the assumptions that were described by Spiess and Florian in 1989 [26]. To better distinguish these multi-link nodes, we refer to these nodes as the strategy node.
The strategy node is similar to the transfer station in the transit network, in which there is more than one attractive line. Although it is different from the physical transfer node, the strategy node can be in a real transfer station or a normal station that has multiple transit services. The number of When passenger travel behavior is based on the optimal strategy, they will receive the first arrived services from the optimal strategy. The passenger behavior searches along the optimal tree. The complexity of the optimal tree is determined by the number of nodes and the alternative links that are connected with the node. In this way, the passenger's choice can be taken as the path generated from the optimal strategy tree.

Strategy Node
There are two kinds of nodes in the optimal strategy tree. Some of the nodes are connected with only one link, such as Node B and Node D in Figure 3. For some of the nodes such as Node A and Node E, there is more than one connecting link. This means that the passengers could make their choice at these nodes according to the train arriving time or the link service frequency. This also follows the assumptions that were described by Spiess and Florian in 1989 [26]. To better distinguish these multi-link nodes, we refer to these nodes as the strategy node.
The strategy node is similar to the transfer station in the transit network, in which there is more than one attractive line. Although it is different from the physical transfer node, the strategy node can be in a real transfer station or a normal station that has multiple transit services. The number of optimal strategy paths is highly dependent on the number of strategy nodes for the given OD pair. Based on the characteristic of the strategy node, a depth-first path generation algorithm is proposed.

Depth-first Based Optimal Strategy Path Generation Algorithm
There are two kinds of searching algorithms: breadth-first searching and depth-first searching. Breadth-first searching is an algorithm that can start at any one node of the graph. The search does not go to the next depth level until all of the neighbor nodes at the present depth have been searched and the visited nodes are recorded in the stack.
The depth-first algorithm is different, which first explores the highest-depth nodes before being forced to backtrack and expand the shallower nodes [31]. In the depth-first searching algorithm, only one node needs to be saved in the stack for each depth level. This reduces a lot of saving work compared with the breadth-first searching algorithm, especially in a high depth level graph.
Usually, passengers will pass a lot of stations or stops when they finally reach their respective destinations. Because of the limited kinds of transit services in the network, the number of strategy nodes is significantly smaller in comparison to the total nodes in the network. Considering the strategy node characteristics, the depth-first based path generation algorithm was proposed, as shown in  Step 1: Initialize the optimal strategy set (OSS) and strategy node list (SNL) for each destination station. The OSS is the exact algorithm given in Figure 1. The temporal strategy node list (TNL) and the path-set are null. The path number is 0. The previous node (LinkO) is the origin from the OD pair.
Step 2: Select the first link that started from the previous node from the OSS. Update the LinkO as the end of the selected link. Save the previous node in the path-set.
Step 3: If the SNL contains the LinkO, add it to the TNL.
Step 4: If the LinkO is the trip destination from the OD pair, go to Step 2. Otherwise, go to Step 5.
Step 5: The path number adds one. Print out the path number and the elements from the pathset.
Step 6: If the TNL is null, the algorithm stops. Otherwise, go to Step 7. Optimal strategy set (OSS). This refers to the definition by Spiess and Florian. For each OD pair, the selected optimal strategy links are recorded in this set and are ordered by the OD pair.
Optimal strategy node-list (SNL). For each OD pair, all of the optimal strategy nodes from the optimal strategy set are saved in this set.
Temporary optimal strategy node-list (TNL). When a strategy node is visited during the searching process, it is recorded in this set. This set saves and records the temporary strategy node.
Path-set list (PSL). The visited nodes are saved in this set by a sequence, which generates a path. Path number set. This save the temporary path number. The searching algorithm can be divided into four parts. The first part is the initialization section, which resets all of the sets that have been discussed for a specific OD pair. The second process is to find the optimal strategy link for the given link start node. The path and path number along the searching process are recorded in third part. Finally, if the visited node contains the optimal strategy node, the process will find the next optimal strategy part. The depth-first path generation algorithm is illustrated in Figure 4. These four parts are utilized in seven steps.
Step 1: Initialize the optimal strategy set (OSS) and strategy node list (SNL) for each destination station. The OSS is the exact algorithm given in Figure 1. The temporal strategy node list (TNL) and the path-set are null. The path number is 0. The previous node (LinkO) is the origin from the OD pair.
Step 2: Select the first link that started from the previous node from the OSS. Update the LinkO as the end of the selected link. Save the previous node in the path-set.
Step 3: If the SNL contains the LinkO, add it to the TNL.
Step 4: If the LinkO is the trip destination from the OD pair, go to Step 2. Otherwise, go to Step 5.
Step 5: The path number adds one. Print out the path number and the elements from the path-set.
Step 6: If the TNL is null, the algorithm stops. Otherwise, go to Step 7.
Step 7: Select the last added node (LN) from the TNL. Remove the first optimal strategy link started with LN from OSS. If there is more than one link, start from LN from the updated OSS. Update LinkO as LN and go to Step 2. Otherwise, remove the LN in TNL and go to Step 6.
Consider Figure 3 as a toy model to demonstrate the algorithm in detail. From the definition of the strategy node, the SNL is {A, E}. During the first recursion, the path {A, B, D, E, F, H, I} can be determined. The nodes A, E are added to the TNL. Node E is the last added node (LN) in the TNL and the first optimal link for node E is (E, F). During the second recursion, link (E, F) is deleted from the OSS. The first optimal link for node E is (E, G). The second path {A, B, D, E, G, H, I} is generated. During the third recursion, after removing the link (E, G), there is no link left in OSS for node E. The node E is removed from the TNL. Link (A, B) for node A is deleted in the OSS. The path {A, I} was found. After removing the link (A, I) from the OSS, node A will be deleted from TNL. Up to now, there is no element in TNL, the searching process stops and all three paths from A to I are found.
The optimal strategy aims to minimize passenger travel time. The link will be added to the strategy if it has the potential to reduce the total travel time while considering the link travel time and the link waiting time. When generating the path with the optimal strategy links, it also contains the path that can save passengers' total travel time. From this, the optimal strategy path and shortest path have something in common. If there is only one path generated in the optimal strategy, the path is the same as the shortest path. When there is more than one path in the optimal strategy, this means that there is more than the shortest path for the passengers when using the optimal strategy. There is no need to use the threshold to filter the path like the K-shortest path. This makes the path more feasible and workable.

The Application for the Beijing Metro Network
The Beijing metro network is a very busy transit system, and it serves more than 10 million passengers every day. During the morning peak hour, the minimal headway is about 90 s. Because of the high service frequency and the single service for a specific transit line, passengers do not care much about the timetable, and they would take the first coming train. This phenomenon is a similar behavior assumption in the optimal strategy model.
Meanwhile, the Beijing metro transit network is a very complex system that includes 17 lines, 344 stations and up to 574 km of transit lines by the end of 2017. In such a complex system, it is hard to determine if there should be three or five paths for each OD pair. A different OD pair may have a different number of feasible paths. After considering these, we used the optimal strategy and the proposed path generation algorithm to analyze the paths that were used by the passengers in the morning peak hour in Beijing.

The Topology of the Service Network
To represent the whole travel process in the network, the topology network of the Beijing metro was established. Take a part of Line 1 (blue) and Line 2 (red) as an example. In the partial network, there are five stations, Stations A-E, with one transfer Station B. As demonstrated in Figure 5, some dummy nodes that represent the platform and ticket gate are mapped to better illustrate the passenger's precise boarding, alighting, waiting and walking behavior.

Link Parameter Calibration
The cost and frequency are two characteristics of every link in the optimal strategy model. The link cost is walking or travel time to pass the link. The link frequency is the link service frequency. In Figure 5, there are five kinds of links. The link calibration work is described below.

Link Cost
(1) In-vehicle link For a single transit line, the timetable is homogeneous. The running time for the same section is fixed. The in-vehicle link cost (in seconds) from the timetable is fixed and is presented in Table 1.

ID Section Name
In-Vehicle Time/s 1 Muxidi-Nanlishilu 120 2 Nanlishilu-Fuxingmen 80 3 Fuxingmen-Xidan 120 4 Xidan-Tiananmen West 90 5 Tiananmen West-Tiananmen East 90 6 Tiananmen East-Wangfujing 90 7 Wangfujing-Dongdan 80 8 Dongdan-Jianguomen 110 9 Jianguomen-Yonganli 120 10 Yonganli-Guomao 70 (2) Access/Exit link The access/exit links are the walking links that start from the ticket gate to the platform dummy Take Line 1, Station A as an example. For one station, there are three kinds of nodes. Node 1A is the ticket gate where the passengers enter the station. Node 1Aa is a platform dummy node to split the direction for the upstream and downstream. 1A-U and 1A-D are the upstream platform and downstream platforms, respectively. The links connected with the different nodes are access links, waiting links, alighting links and in-vehicle links.
For a transfer station, one thing should be clarified in advance. In Beijing, the transfer station is a combination of several individual stations that are connected with transfer tunnels. The platforms and entrances/exits are not shared. Take Figure 5 as an example. If a passenger enters Station 1B and wants to take Line 2, he or she should go to the transfer tunnel and go to Station 2B. From this, there is a transfer link that is connected to the dummy Nodes 1Bb and 2Bb.

Link Parameter Calibration
The cost and frequency are two characteristics of every link in the optimal strategy model. The link cost is walking or travel time to pass the link. The link frequency is the link service frequency. In Figure 5, there are five kinds of links. The link calibration work is described below.

Link Cost (1) In-vehicle link
For a single transit line, the timetable is homogeneous. The running time for the same section is fixed. The in-vehicle link cost (in seconds) from the timetable is fixed and is presented in Table 1. Fuxingmen-Xidan 120 4 Xidan-Tiananmen West 90 5 Tiananmen West-Tiananmen East 90 6 Tiananmen East-Wangfujing 90 7 Wangfujing-Dongdan 80 8 Dongdan-Jianguomen 110 9 Jianguomen-Yonganli 120 10 Yonganli-Guomao 70 (2) Access/Exit link The access/exit links are the walking links that start from the ticket gate to the platform dummy node. We investigated the access/ exit link for every station in the network during rush hour and the off-peak hour. For each station, we followed more than 15 passengers at each period, workday and weekend, which includes males and females, young and old and with or without luggage. We used the median number as the link walking time. Table 2 lists the access and exit walking time for the first 10 stations on Line 1 during the workdays. The access and exit walking times highly depend on the station structure. The stations on Line 1 are mostly island platforms. Their station structure and station scale are similar. In this way, some stations, as presented in Table 2, share nearly the same access and exit walking times.

(3) Transfer link
In total, there are 52 transfer stations and 112 transfer links in the Beijing metro network. Similar to the access/exit links, we obtained the transfer walking time. Take Xierqi Station as an example. Xierqi is a transfer station for Line 13 and the Changping Line. The transfer time for Xierqi at each transfer direction during the workdays are described in Table 3. The waiting link frequency is determined by the train frequency. We read the timetable provided by the operation agency and the service frequency was calculated by the train that served for each station every 30 min. Table 4 shows the waiting link frequency for Fuxingming-upstream from 5:00 to 7:30 am during the workday. For walking links such as the access/exit link and transfer link, the passengers do not have to wait for the service. The frequency of these links is infinite, which means that the service waiting time is 0.

The Number of Optimal Strategy Paths
In this research, we selected the OD pairs during the morning peak time, which is between 7:30 and 7:45, to analyze how the passengers choose their path in the network. In total, 39,317 OD pairs represent the network. For each OD pair, the optimal strategy and the optimal paths were applied from the algorithm that is represented above. Table 5 shows the number of optimal strategy paths for the OD pairs. As shown in Table 5, it could be determined that nearly 80% of the OD pairs contain only one path. This means that, for nearly four-fifths of the OD pairs, the passengers use the shortest path.
There are two reasons for this result. First, most of the lines serve only one transit route in Beijing. The passengers do not have a lot of alternative transit services in the network. Second, for some of the lines, they even have long-short loop service. This is due to the same stopping time plan and the same travel time; thus, the passenger will not switch their loop during the trip. These services relate to most of the passengers who choose only one path during their trip.
Although nearly 80% of the OD pairs contain only one path, approximately 20.91% of the OD pairs have more than one path. The passengers can alternatively choose different paths during their trip.

OD with Multiple Strategy Paths
Life Science Park to Yonganli is the only OD pair that contains four optimal strategy paths. These paths are illustrated in Figure 6.
The passengers do not have a lot of alternative transit services in the network. Second, for some of the lines, they even have long-short loop service. This is due to the same stopping time plan and the same travel time; thus, the passenger will not switch their loop during the trip. These services relate to most of the passengers who choose only one path during their trip.
Although nearly 80% of the OD pairs contain only one path, approximately 20.91% of the OD pairs have more than one path. The passengers can alternatively choose different paths during their trip.

OD with Multiple Strategy Paths
Life Science Park to Yonganli is the only OD pair that contains four optimal strategy paths. These paths are illustrated in Figure 6. Life Science Park is on Line 13 and the destination Yonganli is on Line 1. The paths are represented in different colors. The red and blue paths share the same route until they reach Xizhimen Station, which is a transfer station for Lines 2 and 13. Line 2 is a ring line. Jianguomen is the transfer station for Lines 1 and 2. The two transfer stations are located diagonally in Line 2. The passengers can go clockwise or anticlockwise to reach their destination, which makes the red and blue paths. In the morning peak hour, the train frequency for the Line 13 clockwise direction is pretty high. Some passengers will choose the clockwise direction to reach Line 2 at Dongzhimen. After that, they finally Life Science Park is on Line 13 and the destination Yonganli is on Line 1. The paths are represented in different colors. The red and blue paths share the same route until they reach Xizhimen Station, which is a transfer station for Lines 2 and 13. Line 2 is a ring line. Jianguomen is the transfer station for Lines 1 and 2. The two transfer stations are located diagonally in Line 2. The passengers can go clockwise or anticlockwise to reach their destination, which makes the red and blue paths. In the morning peak hour, the train frequency for the Line 13 clockwise direction is pretty high. Some passengers will choose the clockwise direction to reach Line 2 at Dongzhimen. After that, they finally get to the destination, which is the yellow path. In addition, the transfer walking distance in Guloudajie is shorter than the other transfer stations. Some passengers go in the clockwise direction and they transfer to Line 8 at Zhuxinzhuan Station, which is on the green path.
For some of the other OD pairs, they also have alternative paths during their trip. Although it is different from the previous OD pair, the paths are not in the clockwise or anticlockwise direction for the ring line. Take Renmin University to Dongdan as an example, as shown in Figure 7. get to the destination, which is the yellow path. In addition, the transfer walking distance in Guloudajie is shorter than the other transfer stations. Some passengers go in the clockwise direction and they transfer to Line 8 at Zhuxinzhuan Station, which is on the green path. For some of the other OD pairs, they also have alternative paths during their trip. Although it is different from the previous OD pair, the paths are not in the clockwise or anticlockwise direction for the ring line. Take Renmin University to Dongdan as an example, as shown in Figure 7. Renmin University is on Line 4 and Dongdan is on Line 1. There are three paths in this OD pair. When the passengers reach Xizhimen by using Line 4, their total travel time may decrease if they transfer to Line 2. By doing this, some passengers might choose the blue path, and some will stay on Line 4 and transfer to Xidan Station, which is on the yellow path and it only transfers once. The train headway on Lines 5 and 10 is relatively small in the morning peak hour. Passengers may go upstream and transfer at Haidian Huangzhuang Station, which is the green path.
The optimal strategy path results show that many OD pairs have only one path, which is the shortest path. For some OD pairs, because of the train frequency and the transfer distance, the path that contains more transfers may have the potential to reduce the total travel time because the Renmin University is on Line 4 and Dongdan is on Line 1. There are three paths in this OD pair. When the passengers reach Xizhimen by using Line 4, their total travel time may decrease if they transfer to Line 2. By doing this, some passengers might choose the blue path, and some will stay on Line 4 and transfer to Xidan Station, which is on the yellow path and it only transfers once. The train headway on Lines 5 and 10 is relatively small in the morning peak hour. Passengers may go upstream and transfer at Haidian Huangzhuang Station, which is the green path.
The optimal strategy path results show that many OD pairs have only one path, which is the shortest path. For some OD pairs, because of the train frequency and the transfer distance, the path that contains more transfers may have the potential to reduce the total travel time because the reduced transfer path may have a longer waiting time. The optimal strategy path can better illustrate the passenger flexibility in the network compared with the shortest path and the K-shortest path.

Discussion
The empirical study showed that the optimal strategy could be used in the complex transit network to demonstrate passenger behavior. Passengers, trying to follow the shortest path during their trip, could be verified from the data in Table 5. The proportion of the OD pairs that only contain the shortest path exceeded our expectations. When we mapped these OD pairs on our network, we found that some OD pairs are on the same line, which means passenger enter and exit at the same transit line. In this case, it is obvious that there is only a single shortest path, because every other path that contains transfers will only increase the travel time.
On the other hand, the operation strategy in Beijing metro is simple and straightforward, where the lines are mostly of the stop-stop operation. The complex skip-stop and through service operation plan are seldom applied in Beijing metro, which also leads to the results that most of the OD pairs only contain one path.
For the OD pairs that contain more than one path, we found that these OD pairs generally contain more than one transfer station. Because of the different transfer time at different transfer stations and the various train frequency on different transit lines, it is possible for passengers to switch to other paths and thereby reduce their total travel time, which explained the multiple transit paths for these OD pairs. Furthermore, the number of reasonable paths is different for every OD pair. It is hard to determine a fixed K value in K-shortest path algorithm for all OD pairs in our transit network. On other hand, these results could better demonstrate the advantage of optimal strategy compared with the shortest path and K-shortest path. The optimal strategy could illustrate the passenger behavior in a more flexible way.

Conclusions
This study focused on the passenger's path choice behavior in a complex transit network. The main conclusions of this paper are fivefold: (1) We reformed the optimal strategy as the optimal strategy tree to analyze the passenger's behavior.
Based on the optimal strategy results in Section 2, the passenger behavior could be taken as searching for the path along the optimal strategy tree. (2) The strategy node is defined. There is more than one attractive line/link starts within the strategy node, which determines the number of optimal strategy path. (3) The depth-first based optimal strategy path generation algorithm is proposed to illustrate passenger behavior. The algorithm complexity is highly dependent on the number of strategy nodes. The algorithm combines four parts: (a) initialization; (b) find optimal strategy link for the given node; (c) record path and path number; and (d) find the next optimal strategy node. The optimal strategy path could contain more than one shortest path. (4) The topology of the service network helps illustrate passenger behavior. The service network proposed some dummy nodes that represent the platforms and ticket gates are mapped to better illustrate the passenger's precise boarding, alighting, waiting and walking behavior. The cost and frequency for each link is calibrated in seconds.
(5) The path choice behavior analysis with the optimal strategy. Almost 80% of the OD pairs only have shortest path and less than 0.4% OD pairs have more than two paths. This calibration helps planners understand passenger behavior better and could be used in determining the K-shortest path. Based on the detailed analysis presented in Section 4.3.2, we found that, because of the different train frequencies and transfer times for some long-distance OD pairs, passenger may switch their path during the trip to reduce the total travel time.
This research, using the real AFC record data and timetables, provided a new method to infer the passenger path choice behavior. We determined that most of the OD pairs in the Beijing metro network only have one optimal path, which means that the passengers only follow the shortest path during their trip. This is because of the simple and fixed operation plan, which does not have a skip-stop operation and through service. Comparing with the given shortest path and K-shortest path behavior analysis, this method is more flexible and the number of reasonable paths differs from the OD pairs. In this way, the use of the fixed K value in the K-shortest path for all of the OD pairs in the network is not always reasonable.
Some aspects of this study could be improved by future research. Passengers may take more transfers to reduce their travel time. However, the transfer penalty is not considered in the optimal strategy model, which should be further discussed and researched. Moreover, we should apply the optimal strategy and the path generation algorithm in other cities, such as Shanghai and Guangzhou, to illustrate the validity of the model and the proposed algorithm solution. Passenger path choice and other behavioral characteristics can and should be better understood. Finally, if possible, we should explore flexible operation plans to make the passenger path choice more complete and analyze passenger behavior in a more comprehensive way.  Acknowledgments: The authors would like to thank Alireza Khani from the University of Minnesota and Zijia Wang from Beijing Jiaotong University for helping us edit and polish the language of this paper. We would like to thank Editage (www.editage.cn) for English language editing.

Appendix A
The abbreviations in this research are listed as below.