Analysis of a Flexible Transit Network in a Radial Street Pattern

Traditional transit systems are usually composed of xed routes and stops, which are suitable in densely populated areas. is paper presents a reformulation of the exible transit model developed by Nourbakhsh and Ouyang (2012) to adapt it to many low demand cities in the world, especially those characterized by radial street patterns. Unlike traditional ones, buses of the proposed transit network are allowed to traverse in a predetermined service area and their precise trajectories hinge on the exact locations of passengers. To identify the optimal topology structure of the exible transit system, continuous approximation approaches are developed to explore the optimal value of design parameters of the whole system, dening the optimal network layout through minimizing its objective function. To exhibit its advantages, numerical experiments are conducted to compare the exible transit system with its two variants. e results show that the exible transit system proposed in this paper outperforms the other two variants. e higher the access cost is, the more it would tilt towards the exible transit system with a signicant margin. Besides, the exible transit system in a radial pattern competes more eectively than that in a grid structure. is is encouraging because the proposed transit system can be applied in a number of real-world cases.


Introduction
As a mode of collective transportation, public transit has been ourishing over the past few decades due to the severely congested tra c and polluted air condition in metropolitan cities. Traditional transit, functioning as a high-capacity vehicle, is favorable in densely populated areas. However, in well-developed countries, due to low demand density, operating the transit system exactly according to the traditional mode is indeed unwise, surely leading to waste of resources. To help the transit system stay above water, speci c e orts of planning, designing, and transforming transit system are desirable. Indispensable in this e ort is to transform the way by which the transit systems are designed and operated [1,2]. e overwhelming majority of transit design research focus on choosing certain xed routes and frequencies such that the whole system cost is minimized. To attract more passengers to transit, an e cient and e ective way is to inject new vitality to the operation of the transit service. To this end, an improvement in transit network topology and operations management is the main element. is paper is concerned with the optimal design of a novel exible-route transit system that couples the hybrid structure and the exibility of the demand responsive service (DRS). Some research needs to be conducted before a exible transit system envisioned above can be con gured with more details.
In this paper, a novel exible transit system designed is adapted to cities where the travel demand density is relatively low. To the best of our knowledge, traditional transit system is usually composed of buses operating along xed routes and stopping at xed bus stops. However, such an operating mode tends to be more suitable for high travel demand density areas, such as many cities in Asian countries. However, in some cities of Europe, America and many other well-developed countries, these traditional transit systems do not make good use of resources and will no doubt perform better with some adjustment. Otherwise, the optimal spacing between transit lines and stops are likely to be rather large, which will de nitely render passengers in an inconvenient and uncomfortable situation.
In the world, di erent cities have di erent street patterns, some form a grid structure, such as Beijing, Chicago and so on, while others may take on a radial pattern, composed of radial axes and concentric rings, such as Moscow, Paris, Madrid, Amsterdam, Milan, Berlin, and Palma [3]. As we know, the flexible transit system defined by Nourbakhsh and Ouyang [4] can be adapted to cities with grid street patterns nicely, but not to a radial one directly. e differences between the topology patterns necessitate the development of a different analytical model from the previously developed ones to determine the optimal design in a radial structure, which is what this paper intends to focus on. To this end, a continuous approximation approach is employed to establish the optimal design model, which is expressed as a mixed integer problem and then solved by an optimizer. e highlights of the proposed model are as follows: (i) A flexible transit system is initiated with no fixed routes and stops with a radial street pattern, which seems to have garnered less attention in the current literature. (ii) e flexible transit system bears some characteristics of demand responsive transit (DRT). However, it also has its own unique characteristics. Each vehicle in the flexible transit system operates in its own predetermined "bus tube" rather than "wandering" in the whole service area aimlessly. (iii) e flexible transit system shows great potential and benefits under certain demand levels. It has enough flexibility to adapt to the changes of travel demand density. For example, with the travel demand density exceeding a certain threshold, the transit system can run in line with the circular routes concentric to the center and the radial routes passing through the center, picking up, and dropping off passengers at designated positions rather than the exact position of passengers. (iv) To bypass the complexity in establishing and solving the problem, continuous approximation models are developed to obtain the optimal transit network topology.
e remainder of this paper is structured as follows: in Section 2, we give a literature review on related topics involving transit network design and continuous approximation approaches. In Section 3, the structure of the flexible transit system is initiated and described, along with the formulation of the optimal design problem. In Section 4, numerical experiments and comparisons with other two comparable transit systems are conducted. Sensitivity analysis is then conducted in Section 5. Eventually, we summarize the results, point out the limitations and put forward future research in Section 6.

Literature Review
Relevant research on the transit network design problem (TNDP) is furnished in the following section. Transit network design problems have been intensively explored for quite a long time since 1960s. Various models have been developed. Among them, the majority deal with the problem by choosing a system of fixed routes and corresponding frequencies to serve a certain travel demand (see [5]). e problems are always nonlinear, nonconvex. us they are solved by heuristics (see e.g., [5][6][7][8][9][10][11][12][13]). e current models can be classified based on route structure, which includes (1) linear corridors that connect residential areas with business districts (CBD) [14][15][16]; (2) parallel corridors with perpendicular feeder lines [17][18][19][20][21][22]; (3) rectangular grid [23][24][25]; (4) radial network with CBD at the center [3,[26][27][28][29][30]; (5) hub-and-spoke networks where the hub is a large street [31]. As regards the optimal design models, these models can be split into two categories in general. In the first category, input parameters and decision variables take values that are discrete. Travel demands are expressed as origin-destination (OD) matrix and the locations available for transit stops and routes are constrained by detailed street topology. Models of this type can obtain exact design solutions. Yet the realism embedded in these discrete models adds great complexity. It may sometimes only be tackled using mathematical approaches to obtain optimal results in a small network, however, with the enlarged size of the network, the problem tends to be quite difficult and cannot be solved using traditional methods. us heuristics are developed to solve it. Nevertheless, the results that are obtained donot guarantee an optimal or near-optimal design, which indeed limits its application in real cases. See e.g., grid structure [32], radial structure [27], radial/circular structure, hub-and-spoke structure [31]. Another class of models is continuous approximation models, which use "continuously changing" variables as their inputs. It omits most of the real world complex details existing in discrete models in favor of simplicity and compact. Given their advantages, they are widely used in large size applications. In the CAs, only a small number of continuous functions are established; e.g., travel demand is expressed as a density function, while lines and stations are specified in terms of the spacing between them [33][34][35].
Unlike discrete models, the continuous approximation models unveil the relations between inputs and outputs, which can instruct and support real world applications greatly [14,20,24,28]. Tracing back to the beginning of transit network design study, [23] did the pioneering work to explore a grid transit system, followed by a pile of works [5,35]. ompson [36] made a simple comparison between four different structures: radial, ubiquitous, grid, and timed transfer. It was found that a transfer-based system offers a better service for peripheral trips than the traditional radial scheme, and, at the same time, it is cheaper than a ubiquitous scheme. Newell [31] argued that a central area, where the major part of activities is concentrated, is generally found in cities due to economies of scale. erefore, he maintained the idea that a radial system concentrated around a central corridor would be the best strategy. Jara-Díaz and Gschwender [37] studied the same problem in a simple spatial system, only composed of five nodes that form a cross. Two different strategies to serve the demand are compared: direct trips by means of lines that connect the different pairs O/D demanded, or by only two perpendicular corridors with a transfer point at their intersection. eir conclusion was that working with two corridors became more convenient with the demand disparity, and it is conditioned by the level of demand and unit user and agency costs. ompson and Mato [38] and Brown and ompson [39] justi ed the change of transit system approach to transfer-based by comparing the supply, ridership, e ciency, and e ectiveness of di erent American metropolitan areas. ey concluded that those that had implemented a transfer-based network had a better performance on most measures, and had not lost productivity. Mees [40] made an exhaustive comparison about urban form and transit service between two cities, Melbourne and Toronto. e main conclusion was that the main reason why ridership in Toronto was higher than Melbourne was not related to the characteristics of their urban form, which were similar, but rather to the transit service orientation and its consequences. Toronto presented a grid transit network structure following its street pattern. Other comparisons between Boston and Toronto [41] or Broward (Florida) and Tarrant (Texas), made by Brown and ompson [42], gave similar conclusions. ere was also a similar proposal for Barcelona that is currently being implemented [34].
Recently, Daganzo [33] examined a hybrid transit system that features both the ring and radial lines in the central area and only radial lines in the city's periphery band. Estrada et al. [34] further extended Daganzo's work to a rectangular area. Nourbakhsh and Ouyang [4] designed a novel exible transit system with no xed transit routes. Each bus is allowed to traverse in a designated area called "bus tube" to pick up and drop o passengers. ey found out that such a exible transit system is advantageous over other transit systems when the demand density is low to moderate. Sivakumaran et al. (2014) used CAs both to explore the in uence of access mode and TODs on the choice of transit technology. ey found that metro and rapid transit can become more competitive if accessed by fast-moving feeders. Yet, the in uence of more favorable land-use patterns tends to be negligible. Badia et al. [3] rebuilt the hybrid transit rst put forward by Daganzo [33] to adapt it to cities with radial street patterns. e core of his work is to introduce stops with single coverage to improve the accessibility, striking a balance between agency cost and spatial accessibility. Such a network structure nicely inherits the advantages of both a hub-and-spoke structure (e.g., low infrastructure investment) and a grid structure (e.g., low travel time). Chen et al. [30] compared the ring-radial and grid street networks using CA optimization models. Both the models take on a hybrid fashion: intersecting routes in the center and only radial lines in the periphery. e results showed that the ring-radial networks perform better than the grid patterns. Saidi et al. [35] explored the long-term planning for ring-radial urban rail transit. In their work, the optimal number of radial lines is rst examined and then the passenger route choice is analyzed according to the total travel cost rather than the shortest distance, which distinguishes their work from others' . It is found that the most important factors in uencing the ring line alignment are OD patterns and the current radial network con guration. Chen and Nie [1,2] analyzed two variants of an integrated e-hailing transit system: a zone-based transit one and a line-based one. ey found that the line-based one competes more e ciently than the zone-based one. Further, in 2018, they studied the optimal design of demand adaptive paired-line hybrid transit in a radial route structure. Two variants are considered di erentiated by the relative position of the adaptive transit lines and the xed transit line. e results veri ed that the radial type outperforms the grid one. Fan et al. [43] developed continuum models to investigate the bimodel transit system. In their study, trips are grouped into ve types. en a two-level optimization problem is formulated to solve the intersecting network. e results suggested that it is wise to design the local and express lines simultaneously.
From the above review, we can nd that there is a gap in the current literature on studying whether and when to establish a exible transit system in a radial pattern and, if it is feasible, how to obtain the optimal design. In light of this, we now use CAs to address the literature's above-cited limitations.

Methodology
is section formulates the exible transit design problem. Section 3.1 proposes the exible transit system with a radial route structure in details. e signi cant metrics are calculated in Section 3.2. e optimal transit network design problem is formulated in Section 3.3.
For clarity and compactness, all the assumptions involved are listed below: (1) e city is considered as an ideal circle of radius , and its street layout shows a ring-and-radial pattern; (2) Origins and destinations of trips are uniformly and independently distributed in the whole service area; (3) For simplicity and comparison purposes, the spacing of ring routes is uniform; (4) e service is provided by identical vehicles in the following operating parameters; (5) Passengers send their demands to the exible route vehicle dispatch center before the vehicle departs; (6) Passengers choose the routes with the shortest distance; (7) Passengers will not transfer to other transportation modes due to the decrease of service level. at is, the demand density is given in advance and unchanged; (8) Each bus serves its own "bus tube" and cannot cross the lines, which means the service area of buses has been determined at the very start; (9) No congestion e ect is considered in this paper as the background of this paper is low demand density cities.

Proposed Flexible Transit System with a Radial Route
Structure. Consider a circular region of radius with a radial street pattern. e travel demand density is passenger trips per hour per unit area. We assume that the origins and destinations of trips are uniformly and independently distributed in the whole service area. e exible transit network is composed of non xed transit routes. As is illustrated in Figure 1, two types of routes are included, radial routes (boundary-centerboundary) and circular routes (concentric to the center).
Journal of Advanced Transportation 4 (i) Vehicle distance:

User
Cost. e user costs relate with the route choice of passengers. Here we adopt the assumptions to simplify the optimal design problem: (1) passengers send their demands to the exible route vehicle dispatch center before the vehicle departs; (2) passengers choose the routes with the shortest distance.
For user costs, we consider three parts of costs: waiting time ( ), in-vehicle travel time ( ) and the transfer penalty ( ) for each transfer between two di erent transit services.
Note that the transfer penalty here talked about is the extra discomfort imparted on passengers by the transfer itself rather than the time needed to transfer from one route to another.
Let denote the number of radial transit routes which travel from one side of the area through the center to the other side of the area. us, the angle between two adjacent radial routes is / . Let denote the number of circular transit routes. In a similar way, the route spacing of circular transit routes is / . e headway of transit routes is . We further assume that the service is provided by identical vehicles in the following operating parameters: the cruising speed considering tra c signal and pedestrian interference, v(km/h); the stopping time per stop due to deceleration and acceleration, 1 s/stop ; the time to pick up and drop o a passenger, 2 s/pax . us the time per passenger stop is = 1 + 2 [44]. e street in the service area exhibits a ring-radial fashion. For the ease of reference, we borrow the notation "tube" originating from Nourbakhsh and Ouyang [4]. In our design, the buses make lateral movements to pick up and drop o passengers while sweeping longitudinally back and forth in their own designated "tubes", as depicted by the dash line in Figure 2.

Important Metrics of Agency and User Costs.
As we all know, the system cost consists of two components: agency costs and user costs. ese will be estimated for the so-called average-case user during his or her one-way trip. e details of the formulation of costs are furnished in this subsection and the Appendix. For convenience, we list the notations involved in this paper in Table 1.
e transit network con guration is de ned by three decision variables. Two of them are spatial variables that determine its topology. e third variable is the headway of service in the whole service area.

Agency Cost.
To derive the agency cost, we rst need to identify the agency cost components. Due to the fact that exible transit routes have no xed infrastructure, there is no xed cost. us the agency cost includes two components that depend on: the expected total vehicle distance traveled per hour of operation, and the expected total eet size of operation, .

Numerical Experiments
In the previous section, we have formulated the optimal exible transit network design problem. As regards the approach to solve the proposed model, according to Chen and Nie [1,2], Matlab's built-in generic algorithm (GA) (with default parameters on population/crossover/mutation) obtains satisfactory solutions. e constraints of the parameters used in Matlab's built-in function "ga" are the same as in the above formulation.
In this section, we compare the radial exible transit network system with the grid one [4]. e formula involved are listed in Table 2.
Solved by Matlab, the optimal decision variables and cost components are listed in Table 3.
Observed from Table 3, as expected, when the demand density increases, the average user cost reduces considerably, and the average agency cost drops relatively slightly, albeit the transfer cost remains almost intact. A plausible explanation is that with the increasing demand density, the number of transit routes rises. us the spacing between routes and stops is narrowed. For one thing, smaller spacing requires shorter detour distance in vehicles, which causes shorter travel distance for passengers. For another thing, passengers need to transfer a bit more, leading to the transfer cost increasing slightly.
Further, two batteries of comparisons are conducted to analyze the performance of three transit systems. Figure 4 plots the minimized system cost per passenger of the exible transit network versus travel demand levels: (1) a small radial city of radius = 10/ ; (2) a medium-scale radial city of radius = 20/ . We choose / denotes the side of a square city [33] as the radius of the service area because this ensures the same area as that of the grid one.
For the xed route transit system, the total system cost includes costs related to infrastructure establishment, vehicle distance traveled, vehicle operation, passengers' access and egress time, waiting time, in-vehicle travel time and transfer penalty. For comparison, we neglect the infrastructure cost. One may quibble over the assumption if it actually gives undue priority to the xed route transit system, however, this assumption indeed favors the xed route transit system and sets a high bar for success to the exible transit system proposed in this paper.
Reading from Figure 4, it is easy to nd that the exible transit system outperforms the xed route one with a signicant margin in the low demand density. is is intuitive, when the passenger demand is low, it is unnecessary to traverse along the xed route and stop at each xed transit stop since no passenger may be there for the service. To exhibit the We now rst turn our attention to an important user metric: the expected number of transfers for an average user of the exible transit services. e probability of requiring no transfer is equivalent to the probability that the origin and destination of a trip are within the same "tube" in uence area (e.g., see the trip with and labeled 1 in Figure 3).
According to the number of transfers, three types of trips can be distinguished: (1) the destination is in the same radial or circular line in uence area as the origin; (2) the origin and destination are not in the same line in uence area and the angular distance between them is less than 2; and (3) the origin and destination are not in the same line in uence area and the angular distance between them is larger than 2. Passengers in transit system are assumed to always take the shortest path. For Type 1 trips, no transfer is needed. For Type 2 trips, passengers will use both the radial line and the circular line, one transfer occurs at the intersection between the circular line and the radial line. For Type 3 trips, passengers will transfer once at the center. Hence, both Type 2 and Type 3 trips require one transfer.

Formulation of the Optimal Design Problem.
e purpose of the exible transit network design problem is to nd the optimal combination of the design variables, namely, , , and , such that the cost of the whole transit system is minimized.
As is customary, we transform the cost components into equivalent travel time. According to Daganzo [33], we suppose that $ is the cost per vehicle distance, $ is the cost per vehicle hour and is the time value of passengers, thus = $ / 60% while the user cost drops slightly. e reason may be that when the travel demand density increases, the agency may bene t from economies of demand density, however, the user costs canot decrease endlessly beyond a critical level since the overcome distance exists. at is, no matter which level the travel demand reaches, a minimum value of travel distance must be overcome as the origin and the destination have a spatial distance which cannot be diminished.
Here comes more detailed analysis of di erent cost components. Figure 7 demonstrates that passengers su er more in-vehicle travel time relative to the total travel time, which is exactly more obvious in the grid exible transit system. is is not strange because the real transit routes between the origins and the destinations of trips in a grid transit system are more roundabout than those in radial ones.
is may be attributed to the manner that the radial network furnishes more direct routes to passengers compared with grid networks.
Analyzing the gures above in depth, it is obvious that apart from remarkable advantages, such as more convenient impact of the street structure on the exible transit design problem, we compare the exible transit system proposed in this paper with its counterpart in a grid network [4]. In their paper, the buses traverse in certain bus "tubes" in a grid street structure. To eliminate the external factors, we set the same area of the service areas in both the radial and grid patterns. e results demonstrate that the exible transit system in a radial structure also wins over the grid one slightly. Figure 5 plots the optimal decision variables of the transit networks versus demand levels. e number of both the radial routes and circular routes increases with the demand levels as we anticipated. It is striking to see that the increase of headway of the whole transit system tends to be more dramatic, compared with the other two decision variables. Figure 6 plots the minimized agency costs and user costs against demand levels. A notable observation can be found that the agency cost is more sensitive to the travel demand density than the user cost by a large margin. When the value of rises up to 100 from 1, the agency cost drops sharply about

Journal of Advanced Transportation 8
A few reasons can explain why the exible transit system dominates in low density areas. Firstly, it eliminates the access and egress time of passengers at xed stops and the cost savings from the former are more than the o set by the latter. Secondly, due to the low travel demand density, the operating vehicles can choose from minibuses instead of traditional high-volume ones, which, to a great extent, cut down the agency cost as well.
Note that in all the gures, user costs drops less than the agency costs.
is happens because agency costs can be reduced as much as possible by choosing large spacing and (or) headways, but user costs cannot be reduced below the time threshold to overcome the direct distance-between the origin and destination of a trip. and resource-saving, the exible transit system has its own defect. Unlike the xed route transit system, the exible transit operates in strict accordance with the realization of passengers (the realization of passengers means that the exact location of passengers), which may impose an extra detour distance on passengers especially those who get on the buses early and this can cause reduction of service quality.
Seeing from Figure 7(b), when the demand is low, the travel times of the exible transit system is slightly longer than that of the xed transit system. However, when the demand increases to a certain level, it incurs much more in-vehicle time than the xed route transit system, which is the product of extra detour brought by door-to-door service. access and egress time, to some extent, compensates for the longer in-vehicle travel time, which brings down the total system cost of the exible transit system. Figure 8(d) plots the minimized system cost versus the demand level when the value of time of passengers reaches up to four times as the former one. Compared with Figure 1, the bene ts of exible transit system are dwarfed. e more costly the in-vehicle time, the more unfavorable the exible transit system. Another observation is that the total cost of the xed route transit does not change perceptibly as the exible one, with discrepancies less than ten percentage.

Conclusion
As we all know, a transit system can be more e cient (less system cost) and more e ective (more passengers) by eliminating the access and egress time. E orts in this aspect usually include three approaches: (1) providing high-speed feeders as connectors; (2) transforming the land-use patterns, such as Transit Oriented Development (TOD); (3) improving the vehicle operating mode. e former two approaches have been intensively talked about in the past. e last one is what this paper intends to focus on. To this end, this paper proposes a novel exible transit system with buses traversing in their own designated "tubes". rough analysis of all relevant agency costs and user costs, we utilize the continuous approximation approach with several variables to establish the optimal design problem as a mixed integer program, which derives a concise formulation and can be easily solved by an optimizer, such as Genetic Algorithm (GA) embedded in Matlab.
Owing to the inherent characteristics of the approach applied, the outcome obtained casts important insights into the internal relation of the input parameters and the outcome. e main ndings are listed as follows:

Sensitivity Analysis
In this section, we examine how the input parameters of the exible transit system in uence the optimal design. A wide range of scenarios are analyzed.
To test the impact of passenger stopping time, we choose a more conservative value of = 24 s. Intuitively, the longer the passenger stopping time, the more travel time passengers will have to endure. e e ect should be more obvious for the exible transit system as each passenger need a stop for both the get-on and get-o . To this end, we conduct a sensitivity analysis to see the real cases and the results are displayed in Figure 8. Figure 8(a) plots the minimized system cost versus the demand levels with the stop time per passenger increases from 12 s to 24 s. As clearly shown in Figure 6(b), all the three total cost increases compared with values in Figure 4(a), especially the exible transit system in a grid network. Figure 8(b) plots the minimized system cost versus the demand level when walking speed takes a low value. Such circumstances might occur when the weather is extremely terrible or on a dark night, which blocks passengers from accessing the transit system. us, the xed route transit system is forced to be put in an unfavorable situation because all passengers are assumed to walk to the xed transit stations without any other access approaches. e system cost of the exible transit also increases slightly, which may result from the increased transfer time. Figure 8(c) plots the minimized system cost versus the demand level when the walking speed rises by 50%. As can be read from Figure 8(c), all three curves follow the same trend, however, the gap between the exible transit system and the xed route system narrows. is is expected since the exible transit system is designed to eliminate the access and egress time. In the condition of high walking speed, the exible transit system will undoubtedly lose its superiority. Yet, the reduced due to the vehicle's lateral movements for picking up and dropping o passengers. When the travel demand density is relatively low, the extra lateral distance is negligible and the whole system is optimal. However, when the travel demand density is moderate to high, the extra lateral distance will impose high in-vehicle travel time on passengers, which is more than o set by the elimination of other cost savings and thus renders the exible transit system at a great disadvantage. (iii) When the demand is high, the xed route transit is more e cient. us the exible transit system can be converted into the traditional ones via operating along xed routes and stopping at designated transit stops, which can adapt to the changes of travel demand density easily and conveniently.
(i) e comparisons between the exible transit system proposed in the paper and its variant [4,44] demonstrate that in low travel demand density areas, the exible transit network designed in this paper is advantageous over the xed one with a signi cant margin. is is inspiring because it means the design can be applied to a list of real-world instances.
(ii) Compared with xed route transit network systems, the proposed exible transit system is advantageous in certain aspects. Above all, it enables passengers to gain access to the transit system easily without walking a long time to the designated transit stops. What is more, it reduces agency cost signi cantly as well because it has no infrastructure cost. However, passengers may have to travel a bit longer in vehicles at a cost thought here is to take the average cost of the whole transit system into consideration rather than speci c individual trips, which will exactly makes the optimal design problem too complicated. Now, we set out rst to explore the expected number of transfers. Result 1. e expected number of transfers is given by Derivation. To calculate the expected number of transfers, rst we need to classify all trips into three categories: (1) the origin and destination of a trip are in the same in uence area of a radial or circular line as depicted in Figure 9 (the solid dot (A.1) = 0 × P( = 0) + 1 × P( = 1) (iv) Also worth noting is that several factors, such as the city size, demand density level, value of time, in-vehicle travel time, are found to play an important role in the optimal alignment of the exible transit system. However, the most important factors are in-vehicle travel time and demand density levels.

A. Derivations
Please note that for an individual trip, it is likely to include some, not all of the travel time components. e design . As to the lateral distance per passenger, it is di cult to derive a compact formulation using analytical methods. Following Chen and Nie [44], we denote as the ratio between the total travel distance and the longitudinal distance of a trip per vehicle. As we know, the longitudinal distance of a round trip per vehicle in the th "tube" is 4 ( − (1/2)) , thus the total longitudinal distance per hour is = ∑ =1 4 ( − (1/2)) (1/ ) = 2 / . en, the ratio can be expressed as = / ὔ = 2 / + 2 3 /3 Approximately, the in-vehicle travel distance per passenger on average can be calculated via scaling up the longitudinal distance by the ratio .
at leads to = 2∑ =1 2 ⋅ 4 ( − (1/2)) In a similar way, for radial routes, the longitudinal distance to the transfer point in the th circular "tube" is (1/4) . e longitudinal distance for a round trip per vehicle of a radial "tube" is 4R, thus the total longitudinal distance is = 4 / . e ratio is expressed as = 0 × P( = 0) + 1 × P( = 1) (A.5) (A.7) represents the origin of a trip and the dash area represents the same "tube" in uence area of the origin); (2) the destination is not in the same "tube" in uence area of the origin but the angular between them is less than 2 (in radians); (3) the destination is not in the same "tube" in uence area of the origin and the angular between them is more than 2 (in radians). As regard to the rst category, no transfer is needed. For the second category, passengers need to transfer at the transfer point located in the intersecting point of one radial "tube" and one circular "tube".
Hence, the expected number of transfers is Let denote the angle between the origin and destination of a trip. Consideration shows that if > 2, passengers need to take two radial transit routes and a transfer is needed at the center. On the contrary, if < 2, passengers need to take one radial transit route, one circular route and transfer at the corresponding transfer point. Proofs can be found in Holroyd [23].
where P is the probability that the origin of a trip is in the th circular "tube" and P is the probability that the destination of the trip is in the same "tube" in uence area as the origin.
According to the analysis above, a trip has one transfer at most, thus the probability of having one transfer is = 0 × P( = 0) + 1 × P( = 1) e origin e same "tube" in uence area of the origin F 9: e same in uence "tube" area of an origin. . us, the vehicle size for the radial routes can be derived as In the same way, the vehicle size for the radial routes can be derived as

Data Availability
No data were used to support this study.

Additional Points
Future Work. In this paper, we assume the uniform distribution. One can assume other density patterns, for example, concentric demand or multi-pole demand patterns. is opens way for new research. It would also be intriguing to analyze if the objective function can be the total social bene t, taking the external cost into consideration although these further thoughts will undoubtedly add great complexity into the optimal design problem. Another line of work worthy to be done is to explore the hybrid structure of the exible transit system proposed in this paper. For example, one can consider combining the hybrid structure of the one proposed by Daganzo [33] with it, providing double coverage in the central service area and only radial service in the periphery band. With the rapid development of technology and sharing economy, adding e-hailing service into the transit system calls for further investigation. Limitations. Apart from remarkable advantages, on the downside, a exible transit system like the one proposed in this paper may impose signi cant in-vehicle travel time penalties on passengers. Happily, these penalties can be relieved by: (1) providing a nice and comfortable environment in the vehicles; (2) utilizing the in-vehicle time fully. As regards the second one, thanks to the rapid development of Internet and communication technology, passengers no longer need to waste their in-vehicle travel time. Rather, they can deal with their own business if they wish, which, to a large extent, reduces the disutility of the in-vehicle time. All of the above notwithstanding, we acknowledge that this kind of transit design proposed in this paper may be a bit divorced from reality. Actually, cities have two circular lines at most around the world. However, the work we do in this paper is not primarily to allocate its use in real cases. Instead, we choose this simple and compact form to obtain useful insights into the relation between the input parameters and the outcome of the optimal design topology.
erefore, the total travel distance per passenger in vehicles is In summary, the in-vehicle travel time per passenger is = + = /v ὔ + /v ὔ .

Result 4. Vehicle distance
Derivation. e total vehicle distance is given by the product of the number of transit routes, the expected travel distance per round trip and the service frequencies (the inverse of the headway).
For the circular transit routes: the expected travel distance per round trip consists of three components: (1) the overcome longitudinal distance. For the th = 1, 2, . . . , from the center, the longitudinal distance per round trip is 4 ( − (1/2)) ; (2) the lateral movement to pick up and drop o passengers. e expected lateral distance per passenger is approximately /3 and the passenger trips generated during a round trip is