Integrated Public Transportation System with Shared Autonomous Vehicles and Fixed-Route Transits: Dynamic Traffic Assignment-Based Model with Multi-Objective Optimization

The concept of time-expanded network [17], shown in Fig. 1, is employed to describe the dynamic traffic assignment of SAVs, BRTs, and travelers in a network. A network is composed of nodes and links. SAVs can travel the network under capacity constraints. BRTs can only travel fixed routes in the network. Travelers can travel the network by riding SAV or BRT.

Each link has its traffic capacity, and the excess of SAVs over the capacity cannot enter the link. Each node also has its storage capacity, and the excess of SAVs over the capacity cannot wait at the node. Both link traffic capacity and node storage capacity have maximum and minimum values, and they can be increased from the minimum to the maximum by infrastructure investment in the network planning phase.

BRTs travel the network using links with lanes for BRTs only (i.e., BRT lanes). BRT lanes are designated network planning phase, and they will not change during operation. BRT lanes limit the link traffic capacity because they occupy a part of the link.

SAVs move for travelers’ transportation or parking, and SAVs stop when they are parked at the node or when they are caught in traffic congestion. The model for SAV is basically identical to that of Seo and Asakura [17].

The traffic congestion is represented by the point queue model with limited queue length. SAVs always run on the links at the free flow speed, and SAVs cannot enter the link when the traffic volume of SAVs reaches the link traffic capacity. SAVs which cannot enter a link must wait at the node which is the starting point of the link. When the volume of waiting SAVs at the node also reaches the node storage capacity, SAVs cannot enter the node. That is, congestion is expanded. The dynamic route is determined by solving the optimization problem.

SAVs have pre-determined passengers’ capacity, and SAVs can transport one or more travelers with ridesharing under this constraint. How and which passengers are loaded by SAVs is determined by solving the optimization problem.

BRTs run along the BRT lanes. Thus, BRTs are free from traffic congestion and always run according to the schedule.

BRTs’ routes design and decision of BRTs’ schedules are determined by solving the optimization problems. BRTs’ routes design means both the decision of the number of BRT routes and routing of BRTs. BRTs’ schedules mean the decision of the frequency of BRTs and BRTs running intervals.

This model can describe a balance between the use of SAVs and BRTs. Because BRTs are mass transportation, passengers’ capacity of BRTs is larger than that of SAVs generally. Thus, passengers’ capacity of SAVs and BRTs are given separately. However, considering that BRTs have to wait at every stop as the conventional transportation, the average speed of BRTs is slower than that of SAVs. Thus, the average speed of SAVs and BRTs are also given separately. In this way, the balance of SAVs and BRTs which have both merit and demerit is expressed in this model.

Time-dependent origin-destination (OD) demand of all travelers is defined by origin and destination nodes, departure time, and allowable maximum travel time. All travelers need to finish their travel within their own allowable maximum travel time.

Travelers move to their destination by SAV or BRT, not on foot. Thus, when travelers cannot find both SAVs and BRTs or when SAVs and BRTs are filled with passengers, travelers must wait at the node without moving.

Travelers can mutually transfer between SAVs and BRTs at station nodes by spending given transfer time.

The Pareto frontiers of the three scenarios are shown in Fig. 6. Note that an actual Pareto frontier is 7-dimensional object (i.e., relation among \(T, D, \widehat {D}, N, \widehat {N}, C, G\)), which is impossible to illustrate. In order to show important characteristics of the Pareto frontiers, we show 2-dimensional relation (i.e., cross section of the 7-dimensional Pareto frontier) between D and another objective function. The objective function D, total distance traveled by SAVs, is chosen because the mode choice between SAVs and BRT is one of the important points in the model. Note that Pareto frontiers of the other two objectives also have similar convex shapes, and thus it is not very informative to show all of the possible combinations of two objectives out of seven objectives. Also note that the number of actual Pareto solution is infinity, thus the results shown in Fig. 6 is just an approximation.

The Pareto frontiers were obtained by the following procedure. The value of α_D, the weight parameter for D, was varied over a sufficiently wide range (roughly 0.0003–30), whereas the rest of weights were fixed to the values shown in Table 4 for all scenarios.

The meanings of the values of weights shown in Table 4 are not strictly realistic. The aim of the multi-objective optimization employed by our study is to derive the trade-off relations between different objectives, not to derive a single optimal solution with specific weights. The trade-off relation is represented as Pareto frontier, a set of Pareto efficient solutions. Pareto frontiers can be derived by changing the weight parameters from 0 to infinity. The Pareto frontiers we obtained are summarized in Fig. 6. The specific value for weights has little meaning in the context of investigating the trade-off relations, and thus we have not specified the weight values considering rigorous consistency to the reality.

Theoretically, it is possible to determine the values of weights and derive a single solution of the model. However, this approach has a limitation. That is, we cannot determine the weight values precisely, because it should be determined by the society’s political decision making (some societies prioritize traveler’s benefit, others prioritize environment, etc.). Our multi-objective approach avoids this issue by obtaining various possible solutions. By deriving such various possible solutions, future policy planners would be able to choose the most suitable solution that matches the society’s goal.

The values of weight parameters shown in Table 4 are determined so that most of the objective functions are not negligible in the optimization problem. For example, T (total travel time of ten-thousands passengers) is always substantially larger than \(\hat {N}\) (number of BRTs). Thus, if we choose similar values for their weights, we cannot obtain reasonable solutions as some of the elements become completely negligible. This is not useful to demonstrate the effects of other objectives. The weight values in Table 4 were determined in order to maintain balance among objective function values so that we can avoid such inappropriate solutions. Please note that Pareto frontiers of the other two objectives also have similar convex shapes, and thus it is not very informative to show all of the possible combinations of two objectives. Please also note that the weight on G is exception and is set as small as possible. This is a technical objective function that eliminates mathematical errors called “subtour” and does not have physical meaning.

Approximated Pareto optimal solution can be obtained with high accuracy by using the weighted-sum method. The reasons are as follows. First, the continuous relaxation of the problem is a linear programming (which is convex), and therefore we can expect that a lower envelope (i.e., Pareto frontier) of the solutions is also convex. Thus, an accurate solution obtained by the weighted-sum method can be considered as an accurate Pareto optimal solution. We confirmed that the obtained solution was actually accurate (i.e., small duality gap). Second, the true Pareto frontier is expected to be convex. The obtained solutions in Fig. 6 are almost convex. This is supporting evidence that the solution is reasonable.

According to Fig. 6, various trade-off relations in the SAV-BRT system are clarified. Pareto frontiers are obtained as monotonic curves with negative slopes. This means that the two variables are in a trade-off relation.

In the T–D relation, the total travel time of travelers T and the total travel distance by SAVs D are in trade-off relation for each scenario. It means that the more actively SAVs are used, the faster transportation systems can be realized. By comparing the scenarios, it is clear that the travel time generally increased by introduction of BRTs, while the distance traveled by SAVs decreased. This is because the more BRTs are used, the fewer travelers use SAV, but BRTs are generally slower than SAVs; this usage pattern is evident in \(\hat {D}\)–D relation. Thus, introduction of BRT enables efficient transportation systems in terms of traffic congestion, but it slightly increases average travel time of travelers.

The N–D relation shows that the introduction of BRTs decreases the number of required SAVs N. In SAV only case (n = 0), the allowable range to the number of SAVs is narrow: roughly 4600–5400 SAVs are required regardless of operation pattern. When BRT routes are introduced, the allowable ranges are significantly increased: roughly 4050–5300 SAVs for 1 route scenario, and 3800–5100 SAVs for 2 route scenario. There are 2 implications from this result: 1) the number of SAVs can be significantly decreased by introducing 1 or 2 BRT routes; 2) the operational flexibility of the SAV system also is increased by introducing BRTs.

The \(\hat {N}\)–D relation confirms that the distance traveled by SAVs is significantly decreased by increasing the number of BRTs \(\hat {N}\). This is a reasonable result.

The C–D relation shows that the range of the infrastructure cost C is almost the same for all scenarios. However, the distance traveled by SAVs can be significantly decreased by BRTs without increasing the infrastructure cost. It means that the introduction of BRTs is a cost effective approach to reduce traffic congestion.

According to the above results, the introduction of BRT enables flexible operation of SAVs and efficient development of infrastructure, but tends to increase the travel time of travelers.

Figure 7 (a) and (b) show BRT routes in a Pareto optimal state in two scenarios with the same weight values shown in Table 4; hereafter, we use these parameter values, and the weight value for D is set to demonstrate situations where BRT is intensively utilized by travelers. In Fig. 7 (a) and (b), the red links and the blue links represent the BRT route, and the red nodes represent the destination nodes of travelers; especially the two red nodes in the center of the map corresponds to the area near the Omiya station, which attract many travelers.

According to the result, the BRT routes are determined to pass the central area regardless of the scenario. In addition, in the two routes scenario, both of the BRT routes go through the central area from the west. It means that the model automatically identifies where should be transfer point of the public transit. We considered this result reflects the actual road and travel patterns in the area.

Figure 7 (c) and (d) show BRT routes with the same weight values as Fig. 7 (a) and (b) except for α_D. In these two plots, α_D was set to 0.03, meaning that usage of SAVs is strongly encouraged. By comparing these results, we can confirm that the route length, the number of BRT vehicles, and passing nodes are significantly different. The length and number of BRT vehicles (roughly 20 vehicles per route in α_D = 30 case, whereas roughly 50 vehicles per route in α_D = 0.03 case) are smaller in α_D = 0.03 cases due to increased usage of SAVs. This is an intuitive result. On the other hand, the difference of passing nodes is not intuitive. In fact, this difference came from the passenger transportation capabilities of both modes and road capacity. In α_D = 30 case, BRT routes are placed at the central area of Omiya city where travel demand and road capacity are large. And by operating BRT and SAVs very frequently, it transports the large demand efficiently. In α_D = 0.03 case, however, BRT is not frequently operated. Such low frequency operation would not be sufficient to transport the large demand in the central area and decreases traffic capacity for SAVs (because we assume that BRT route blocks one of the lanes). Thus, BRT route is placed to low demand areas.

In summary, selection of BRT routes is affected by the mechanism that decreases traffic capacity of a road. Thus, if operational frequency of BRT is low, BRT routes are not placed at high demand areas, because otherwise BRT blocks SAV traffic too significantly. Contrarily, if operational frequency of BRT is high, BRT routes are placed at high demand areas so that roads can be utilized more efficiently.

In Fig. 8, the comparison of SAVs’ flow between the DSO assignment with and without BRTs is shown. In this comparison, the DSO assignment with BRT means the result of n = 2. The flow of SAVs means the sum of the SAVs through all time on each link. In Fig. 8, the red nodes mean the destination nodes. In Fig. 8a and b, the thickness of links means the traffic volume of SAVs. In Fig. 8c, the links on which the sum of SAVs in the DSO assignment with BRTs is more than that in the DSO assignment without BRTs are red, the links on which the value in the DSO assignment with BRTs is less than that in the DSO assignment without BRTs are blue, and the links on which the values are the same are black. Then, the thickness of red and blue links means the absolute value, and the thickness of the black links is fixed. Thus, the thicker the blue link is, the more the number of SAVs in the DSO assignment without BRTs is.

The number of SAVs in the area where the destination nodes are gathered, or the center of the network, is large in Fig. 8a and b. In Fig. 8, the flow of SAVs is large in the central areas, and the inflow to the central area from the west is especially large. We considered these results reflect the actual road and travel patterns in the area, at least to some extent.

Figure 8c indicates that the number of SAVs in the DSO assignment with BRTs is decreased in most links. Thus, less congestion is expected by introducing BRTs. Comparing Figs. 8c and 7b, the amount of decrease is especially larger in the links determined as the BRT routes. In addition, there are some links that are not determined as BRT routes in which the number of SAVs is increased. That’s because movements to and from the station nodes are increased. This result shows that the use of BRTs increases the number of SAVs in some links.

In Fig. 9, the comparison of travelers’ flow between the DSO assignment with BRTs and the DSO assignment without BRTs is shown. In this comparison, the DSO assignment with BRT means that of n = 2. The flow of travelers means the sum of the travelers through all time on each link. The color of links and the thickness of links in Fig. 9c have the same meaning as Fig. 8c. Thus, in Fig. 9c, the thicker the blue link is, the more the number of travelers in the DSO assignment without BRTs is.

Similar to Fig. 8a and b, the number of travelers is large in the center of the network in which the destination nodes are gathered in Fig. 9a and b. We considered the actual road and travel patterns in the area are reflected to the results because the flow of travelers is large in the central areas, and the inflow to the central areas from the west is especially large. Figure 9c indicates that the number of travelers in most links is decreased a little by introducing BRTs, whereas that of travelers in a part of links, which are determined as BRT routes or connected with the station nodes, is considerably increased by introducing BRTs. This result shows that travelers’ routing is gathered by introducing BRTs.

According to SAV and traveler flows in Figs. 8 and 9, the introduction of BRT causes traffic to be concentrated at BRT stations. It makes transfer at stations and rideshare matching easier. In addition, only limited links will be used intensively, making infrastructure investment efficient.

In order to investigate the dynamical features of the proposed model, time-dependent average speed and cumulative curves of travelers are derived. Figure 10 shows the average speed of travelers at each time step. The horizontal axis represents time, and the vertical axis represents the average speed of travelers.

According to the above results, the introduction of BRT stabilizes travelers’ speed, possibly due to the mitigation of traffic congestion.