Design Problem of Travel Sections for Shared Autonomous Vehicles Considering the Immaturity of Autonomous Vehicle Technology

Abstract

Most studies of shared autonomous vehicles (SAVs) have assumed established automated driving technology. In contrast, this study assumes that automated driving technology is not yet mature; it proposes two types of models to optimize the number of SAVs and the sections they travel. The first model (i.e., the ‘actual depot model’) does not consider the travel distance from the depot to the service start and end points. The second (i.e., the ‘virtual depot model’) considers the as-described travel distance. We showed in the actual depot model that under mild assumptions, the continuity condition of the SAV travel segments can be described by a simple constraint condition. The models were then applied to a hypothetical network; the performances of the models confirmed that the evaluation indicators for the virtual depot model tended to be better than the indicators for the actual depot model.

1 Introduction

The fourth Industrial Revolution, centered around the Internet of Things and artificial intelligence (AI), is expected to bring substantial innovation to the transportation sector. Ridesharing services such as Uber and Lyft are widely deployed overseas, and several municipalities in Japan are exploring the use of AI in on-demand shared-ride cabs. In the future, based on advancements in autonomous driving technology, shared autonomous vehicles (SAVs) are expected to provide mobility services at lower costs. The SAV is a new concept, constructed on the basis of booking type, sharing system, and level of integration with other transportation modes [1, 2]. Because there is no driver, SAVs offer several advantages over traditional car-sharing vehicles driven by humans [3]: (i) SAVs are able to self-drive to travelers anywhere, (ii) SAVs can continuously work and seamlessly accomplish travelers’ requests, and (iii) SAVs can self-drive to parking locations after travelers have been dropped off at destinations. However, because fully automated driving technology has not yet been established, accidents may occur due to machine error; thus, a new legal liability framework is needed [2]. As discussed below, studies of SAVs generally assume mature automated driving technology. However, until the technology matures, implementation plans should consider the safety of SAVs through methods such as limiting the travel lanes of SAVs to sections where magnetic markers and sensors are installed.

Matching between users, an important element of SAVs, has been extensively studied. Individual users and vehicle behavior are matched using a space–time extended network (STEN) as the system optimal condition, as described in previous studies (e.g., Agatz et al. [4], Najmi et al. [5], Aiko et al. [6], Shimamoto [7]. However, because these problems are formulated as mixed-integer programming problems, in which vehicle and user movements are represented by binary variables, their solutions tend to be computationally expensive. Therefore, to reduce computational costs, some models treat users and vehicles as aggregated quantities and represent them as real variables. For example, Shimamoto et al. [8] proposed an assignment model of three subjects, comprising the solo driver, ridesharing driver, and rider under circumstances where a ridesharing service is available.

Regarding studies of SAVs, Shimamoto [7] compared ridesharing systems with manually driven and autonomous vehicles via driver and rider matching using a STEN model; they confirmed that a ridesharing service involving autonomous vehicles would equalize opportunities for user activity but increase total travel time. Li et al. [3] proposed a time-dependent SAV system design problem by jointly optimizing the fleet size, parking infrastructure deployment, and daily system operation for long-term infrastructure planning. Seo et al. [9] proposed an optimization model for an SAV system as a multi-objective optimization problem that simultaneously describes traveler behavior and optimizes SAV equipment. Maruyama et al. [10] further modified the model of Seo et al. [9] to consider an integrated system of SAVs and bus rapid transit (BRT). In addition to describing traveler behavior and optimizing SAV equipment, their model can optimize BRT routes and schedules; however, the computational cost of this model tends to be very high due to the large number of binary variables used to describe the BRT routes [9]. Note that these earlier studies implicitly assumed that SAV systems used mature automated technology.

The optimal lane design problem has also been extensively studied in transportation planning on a variety of topics other than SAV lanes, such as transit priority provision (e.g., use of a BRT lane) (Mesbah et al. [11]), wireless charging lanes for electric vehicles (Mubark et al. [12], Tran et al. [13]), and dedicated connected autonomous vehicle lanes (Lin et al. [14], Tani et al. [15]). However, the above-mentioned studies did not explicitly consider the continuous condition of the lane. In the present study, because SAVs are assumed to only travel on specified sections as described later, they cannot return to the depot on their own if the SAV travel sections are not contiguous. Thus, if the continuous condition of the SAV travel sections is not considered, many infeasible solutions may be generated, resulting in an increase in computation time. Furthermore, studies regarding dedicated lanes for SAVs [14,15,16,17] tend to focus on the effect of increased capacity due to shorter distances between vehicles, and/or on changes in driving behavior (e.g., acceleration and lane changing); they do not focus on lane limitations due to the immaturity of automated driving technology.

As discussed thus far, the introduction of SAVs has two effects: an increase in link capacity and a decrease in the number of vehicles due to vehicle sharing. Unlike previous studies, the present study assumes a situation in which automated driving technology is immature and SAVs travel only in sections where magnetic markers and other devices are installed. Accordingly, we propose a strategy that optimizes the number of SAVs and the SAV travel sections, then evaluate the relationship between the level of SAV service and the social benefits of reducing the vehicle number (Fig. 1). We propose two models to achieve our goals: one model does not consider the travel distance from the depot to the service start and end points, whereas the second model considers this as-described travel distance. In the second model, the depot and the nodes providing the service must be connected by SAV travel sections to allow the SAVs to complete their travel. We show that the continuous condition of the SAV travel sections can be described by simple constraint conditions under mild assumptions. Finally, we apply the proposed model to a hypothetical network, which enables evaluation of the model characteristics.

Fig. 1

Concept of this research

The contributions of this study to the related literature are as follows:

• We formulate the lane continuity condition as a simple constraint to reduce the computational cost for lane determination.

• We propose a model describing user choice between private cars and SAVs, then numerically clarify the relationship between private car ownership cost and the SAV penetration rate.

The remainder of this paper is organized as follows. Section 2 describes the assumptions, notations, and formulations of the proposed models. Section 3 then provides a numerical example. Finally, Section 4 draws conclusions and establishes directions for future research.

2 Model Formulation

2.1 Model Overview and Assumptions

2.1.1 Outline of the Model

This study assumes a society in which automated vehicles can only travel in designated sections due to the immaturity of automated driving technology. We constructed a model to determine the sections where SAVs travel and the number of SAVs that satisfy the system optimal conditions, where all travelers travel either in manually driven vehicles or SAVs. Hereinafter, we refer to a human-driven vehicle (HDV) and its driver as the ‘HDV driver’; the person traveling in an SAV as the ‘SAV passenger’; and both ‘HDV driver’ and ‘SAV passenger’ as ‘travelers’.

Because the SAVs must travel from the depot to the service start point and from the service end point to the depot, the total travel distance varies according to the locations of the depot and the service start and end points. Therefore, we constructed two models: the ‘virtual depot model’, which does not consider the travel distance from the depot to the service start and end points, and the ‘actual depot model’, which does consider the as-described travel distance. As indicated below, the actual depot model requires the location of the depot as a component of the input data. By comparing the solutions of the two models, it is possible to evaluate the effect of specifying depot location on model performance efficiency.

2.1.2 Assumptions of the Model

First, we describe assumptions common to the virtual and actual depot models. The following assumptions are made regarding the flow of travelers and SAVs.

• Travelers either drive an HDV or use SAVs; HDV drivers travel alone.

• HDV drivers and SAV passengers are not allowed to swap during the journey.

• All traveler demands (e.g., departure and arrival locations, and departure and arrival time constraints) are known. The departure and arrival time constraints are established in a manner that requires travelers to depart earlier than the departure time constraint and arrive before the arrival time constraint.

• The total number of HDVs and SAVs cannot exceed link capacity.

Note that the third assumption above, which corresponds to the absence of considering the early arrival penalty, is included to avoid an unrealistic solution of making a detour to satisfy the arrival time constraint.

Next, the following assumptions are made regarding SAV behavior.

• SAVs can travel only on designated sections.

• The driving behavior (e.g., acceleration, lane changing, and distance between vehicles) of SAVs is identical to the behavior of HDVs.

Note that, as mentioned above, several studies regarding autonomous vehicles (AVs) have considered changes in capacity with changes in AV penetration due to differences in driving behavior between HDVs and AVs. However, we included the second assumption above to focus on the effect of SAV introduction on reducing the number of vehicles on the road.

The following assumptions are made regarding the costs to HDV drivers, SAV passengers, and SAV vehicles.

• The decision-maker, an SAV operator who desires to reduce congestion, determines the number of SAVs to be deployed and the sections where SAVs can travel.

• The cost of designating the SAV travel sections is proportional to the link length.

• The decision-maker designates the SAV travel sections within a budget constraint.

• HDVs can travel all sections, including the designated SAV travel sections.

Note that the third assumption above considers that the cost of installing a device, such as magnetic markers, is proportional to the length of the SAV travel section/s. Additionally, the decision-maker may designate SAV travel sections to minimize the total system cost, including total travel time. Nevertheless, we included the fourth assumption above because it is difficult to appropriately set the unit cost of designating SAV travel sections.

In addition to the assumptions described above, the following assumptions are made in the actual depot model:

• Only one SAV depot is assumed, and its location is given exogenously.

• Both directions between adjacent nodes are simultaneously designated as SAV travel sections (unless they are one-way).

Note that the second assumption above is included to express in a simple equation the constraint that SAVs can only travel the specified sections and reach the depot after providing service. The formulation of the constraint is described in Section 2.5.

2.2 Space–time Extended Network (STEN)

To represent the flows of travelers and SAVs, an ‘original network’ (i.e., a graphical representation of an actual road network), shown in Fig. 2(a), is converted into two types of networks: a ‘travelers’ network’ and an ‘SAV network,’ as shown in Fig. 2(b). Note that HDV drivers’ and ‘SAV passengers’ cannot be interchanged during a journey, as described in the previous section. Thus, the ‘travelers’ network’ is further categorized into two network types: an ‘HDV driver network’ and an ‘SAV passenger network’. The centroid in the ‘SAV network’ is represented as a dummy node (i.e., an SAV depot node). The virtual depot model creates access and egress links from SAV depot nodes to all nodes in the ‘original network’, whereas the actual depot model creates access and egress links from SAV depot nodes only to nodes with depots in the ‘original network’. The network in Fig. 2(b) is then extended to the STEN, [18] as shown in Fig. 2(c).

Fig. 2

Network transformation

A node in the STEN is defined as a node in the two-dimensional network for each time period and is denoted as $\left(i,t\right)$, where $i$ is the node number in the two-dimensional network and $t$ is the time period of the node. This study defines three node types: an ‘origin node’, a ‘destination node’ where demand is generated and concentrated, and an ‘intermediate node’ corresponding to all other nodes. Three types of links are also defined: an ‘in-vehicle link’, an ‘access/egress link,’ and a ‘waiting link’. An ‘in-vehicle link’ represents travel by vehicles between two nodes. When both tail and head nodes of an in-vehicle link are expressed as $\left(i,{t}_{1}\right)$ and $\left(j,{t}_{2}\right)$, respectively, ${t}_{2}={t}_{1}+{c}_{ij}$, where ${c}_{ij}$ is the travel time between $i$ and $j$. Note that the interaction between the travelers' network and the SAV network is described by the relationship between flows in the HDV driver network and SAV passenger network and corresponding flows in the SAV network. A ‘waiting link’ connects two adjacent nodes $\left(i,t\right)$ and $\left(i,t+\Delta t\right)$, representing travelers or SAVs waiting at node $i$ from $t$ to $t+\Delta t$. Finally, an ‘access/egress link’ represents access from or egress to an origin or depot node.

2.3 Notations

2.3.1 Notations Related to the STEN

D:

Set of drivers

P:

Set of SAV passengers

SAV:

Set of SAVs

T:

Set of travelers $(D\cup P)$ 

X:

Type of STEN, consisting of ‘travelers’ network’ (T)  or ‘SAV network’ (SAV)

$\Delta t$:

Duration of time step 

NT:

Set of time steps

$A_1^X$ :

Set of in-vehicle links of type X

$A_2^X$ :

Set of waiting links of type X

$A_3^X$ :

Set of access links of type X

$A_4^X$ :

Set of egress links of type X

A:

Set of all links

$I^X$ :

Set of nodes of type X

I:

Set of all nodes

R:

Set of origin nodes

S:

Set of destination nodes

${i}_{a}$:

Tail node of link $a\in A$

$j_a$ :

Header node of link $a\in A$

$OUT(i)$ :

Set of links that lead out of node $i\in I$  

IN(i):

Set of links that lead into node $i\in I$

${q}_{rs}$ :

Travel demand between r and s

${d}_{1},{d}_{2}$ :

Nodes representing SAV origins and destinations, respectively

2.3.2 Notations Related to the Correspondence Between the STEN and the ‘Original Network’

L:

Set of links in the original network

N:

Set of nodes in the original network

n(i):

Identity (ID) of node $i\in I$ in the original network

$a_{1,lt}^X$ :

In-vehicle link of type X in the STEN that leads out link l at time step t

$a_{2,nt}^X$ :

Waiting link of type X in the STEN that leads out of node n at time step t

2.3.3 Notations Related to Decision Variables

$x_a^s$ :

Passenger flow of link a with destination s

${y}_{a}$ :

SAV flow of $a\in A$

z:

Number of SAVs

${\phi }_{l}$ :

Assumes the value of 1 if link $l\in L$  is designated as the SAV travel section, otherwise 0

2.3.4 Notations Related to Parameters

$\alpha_1$ :

Value of in-vehicle time

${\alpha }_{2}$ :

Value of waiting time

${t}_{a}, {t}_{l}$ :

Travel time of link $a\in A$ and $l\in L$

${\beta }^{TC}$ :

Travel cost per unit distance (e.g., fuel expenses)

${\theta }_{HDV}$ :

Ownership cost of an HDV

${\theta }_{SAV}$ :

Ownership cost of SAVs

$BGT$ :

Budget limit for specifying SAV travel sections

$\kappa$ :

Capacity of SAVs

$C{R}_{l}$ :

Capacity of an in-vehicle link on link l in the original network

$C{P}_{n}$ :

Capacity of a waiting link on node n in the original network

2.4 Costs in the STEN Links

2.4.1 In-vehicle Links

Based on the above assumptions, the cost of an in-vehicle varies among HDV drivers, SAV passengers, and SAVs, as given by (1).

$${c}_{a}=\left\{\begin{array}{cc}{\alpha }_{1}{t}_{a}+{\beta }^{TC}{t}_{a}& \left(\forall a\in {A}_{1}^{D}\right)\\ {\alpha }_{1}{t}_{a}& \left(\forall a\in {A}_{1}^{P}\right)\\ {\beta }^{TC}{t}_{a}& \left(\forall a\in {A}_{1}^{SAV}\right)\end{array}\right.$$

(1)

2.4.2 Waiting Links

Because the early arrival penalty is not considered, the cost of a waiting link is given as follows:

$${c}_{a}^{s}=\left\{\begin{array}{cc}{w}_{a}^{s}& \left(\forall a\in {A}_{2}^{D}\cup {A}_{2}^{P}\right)\\ 0& \left(\forall a\in {A}_{2}^{SAV}\right)\end{array}\right.$$

(2)

where

$${w}_{a}^{s}=\left\{\begin{array}{cc}{\alpha }_{2}\Delta t& \left(n\left(s\right)\ne n\left({j}_{a}\right)\right)\\ 0& \left(n\left(s\right)=n\left({j}_{a}\right)\right)\end{array}\right.$$

(3)

Note that the cost of a waiting link in the travelers’ network is destination-dependent, whereas the cost in the SAV network is destination-independent. Therefore, in Section 2.5, the cost in the travelers’ network is described as ${c}_{a}^{s}$ for each destination, whereas the cost in the SAV network is described as ${c}_{a}$.

2.4.3 Access/Egress Links

The cost of owning an HDV (vehicle price and maintenance cost per day) is included in the access links of the HDV driver, as described in Eq. (4). The costs for egress links are all 0, as shown in Eq. (5).

$${c}_{a}=\left\{\begin{array}{cc}{\theta }_{HDV}& (\forall a\in {A}_{3}^{D})\\ 0& (\forall a\in {A}_{3}^{P}\cup {A}_{3}^{SAV})\end{array}\right.$$

(4)

$${c}_{a}=0 (\forall a\in {A}_{4}^{D}\cup {A}_{4}^{P}\cup {A}_{4}^{SAV})$$

(5)

Note that the cost of owning an SAV is not included in the access links because the number of SAVs, a decision variable, is explicitly included in the objective function, as shown in the next section.

2.5 Formulation

2.5.1 Virtual Depot Model

The virtual depot model, which determines the number of SAVs and the SAV travel sections without considering the travel distance from the depot to the service start and end points, can be formulated as a mixed-integer linear programming problem that minimizes the total cost:

$$\underset{\mathbf{x},\mathbf{y},z, {\varvec{\upphi}}}{\mathrm{min}}Z=\sum_{s\in S}\sum_{a\in {A}^{T}}{c}_{a}^{s}{x}_{a}^{s}+\sum_{a\in {A}^{SAV}}{c}_{a}{y}_{a}+{\theta }_{SAV}z$$

(6)

Subject to:

$$\sum_{a\in OUT\left(r\right)}{x}_{a}^{s}-\sum_{a\in IN\left(r\right)}{x}_{a}^{s}={q}_{rs}, \forall r\in R,s\in S$$

(7)

$$\sum_{a\in OUT\left(s\right)}{x}_{a}^{s}-\sum_{a\in IN\left(s\right)}{x}_{a}^{s}=-\sum_{r\in R}{q}_{rs}, \forall s\in S$$

(8)

$$\sum_{a\in OUT\left(i\right)}{x}_{a}^{s}-\sum_{a\in IN\left(i\right)}{x}_{a}^{s}=0, \forall i\in \{{I}^{T}-R-S\}$$

(9)

$$\sum_{a\in OUT\left({d}_{1}\right)}{y}_{a}-\sum_{a\in IN\left({d}_{1}\right)}{y}_{a}=z$$

(10)

$$\sum_{a\in OUT\left({d}_{2}\right)}{y}_{a}-\sum_{a\in IN\left({d}_{2}\right)}{y}_{a}=-z$$

(11)

$$\sum_{a\in OUT\left(i\right)}{y}_{a}-\sum_{a\in IN\left(i\right)}{y}_{a}=0, \forall \mathrm{i}\in \{{I}^{SAV}-{d}_{1}-{d}_{2}\}$$

(12)

$$\sum_{s\in S}{x}_{{a}_{1,lt}^{D}}^{s}+{y}_{{a}_{1,lt}^{SAV}}\le C{R}_{l},\forall l\in L, t\in NT$$

(13)

$$\sum_{s\in S}{x}_{{a}_{2,nt}^{D}}^{s}+{y}_{{a}_{2,nt}^{SAV}}\le C{P}_{n},\forall n\in N, t\in NT$$

(14)

$$\sum_{s\in S}{x}_{{a}_{1,lt}^{P}}^{s}\le \kappa {y}_{{a}_{1,lt}^{SAV}}, \forall l\in L, t\in NT$$

(15)

$$y_{a_{1,lt}^{SAV}}\leq CR_l\cdot\phi_t,\forall l\in L,t\in NT$$

(16)

$$\sum_{l\in L}{t}_{l}{\phi }_{l}\le BGT$$

(17)

$${x}_{a}^{s}\ge 0, \forall a\in {A}^{T}, s\in S$$

(18)

$${y}_{a}\ge 0, \forall a\in {A}^{SAV}$$

(19)

$$z\ge 0$$

(20)

$${\phi }_{l}=\left\{\mathrm{0,1}\right\}, \forall l\in L$$

(21)

The first, second, and third terms in Eq. (6) represent the sum of the total travel time and total travel cost of travelers, the total travel cost of the SAVs, and the cost of introducing SAVs, respectively.

Constraints (7)–(9) and (10) − (12) represent the flow conservation laws for travelers and SAVs on the STEN, respectively. Constraints (13) − (15) represent capacity constraints. Constraint (13) states that the number of HDVs and SAVs in the in-vehicle link must be less than or equal to the link capacity, which describes the interaction between HDV and SAVs. Constraint (14) states that the number of HDVs and SAVs at a node must be less than or equal to the node capacity. Constraint (15) states that the flow on an in-vehicle link of the SAV passenger network is less than or equal to the total SAV capacity, which describes the interaction between SAV passengers and SAVs. Constraint (16) states that SAVs can only travel in designated sections; SAVs cannot travel if a link is not designated as an SAV travel section, and otherwise, the maximum number of SAVs can be the link capacity. Note that HDVs are allowed to travel on all sections because no constraints are imposed on the HDV travel sections. Constraint (17) represents the budget constraint whereby the designated cost of the SAV travel sections must be less than or equal to the limited budget. Constraints (18)–(20) represent the non-negative conditions for traveler flow, SAV flow, and number of SAVs, respectively. Finally, constraint (21) states that the design variables for the SAV travel sections are binary variables.

2.5.2 Actual Depot Model

For SAVs to complete their travel, the depot and the nodes providing service must be connected by the SAV travel sections. However, because the virtual depot model does not impose such constraints, it may generate a large number of infeasible solutions, reducing computational efficiency. Here, we discuss how to formulate the continuous condition of the SAV travel sections.

In the original network, there are two links between adjacent nodes in different directions, with the exception of one-way streets. However, as assumed in Section 2.1.2, both directions between adjacent nodes are simultaneously designated as SAV travel sections; thus, there is no requirement for distinguishing the direction of links between the nodes. The following procedure is defined to create a set of the forward links. Note that the forward and backward links are defined as links in a direction away from or closer to the depot node, respectively.

• Step 1: Compute the minimum length

Compute the minimum length from the depot node to all other nodes, where ${\varpi }_{i}$ is defined as the minimum length from the depot node to node $i$.

• Step 2: Generate a set of forward links

Remove backward links from the original network and generate a set of forward links $L\mathrm{^{\prime}}$. Note that we define ‘forward links’ as links such that ${\varpi }_{i(l)}\le {\varpi }_{j(l)}, \forall l\in L$, and ‘backward links’ as links such that ${\varpi }_{i(l)}>{\varpi }_{j(l)}, \forall l\in L$.

The continuous condition of the SAV travel sections is represented by the following simple constraints:

$$\varphi_m\leq\sum_{n\in IN\left(i_m\right)}\varphi_n,\forall m\in L',i_m\neq n_{depo}$$

(22)

$${\varphi }_{m}=\left\{\mathrm{0,1}\right\},\forall m\in {L}{\prime}$$

(23)

$${\phi }_{l}={\Delta }_{m}^{l}{\varphi }_{m}, \forall l\in L,m\in {L}{\prime}$$

(24)

where ${n}_{depo}$ is the depot node in the original network and ${\Delta }_{m}^{l}$ is a binary variable with a value of 1 if a pair of nodes on links $l$ and $m$ are identical.

Constraint (22) states that for a forward link to be designated as the travel section, at least one of the forward links directly upstream of that link must be designated as an SAV travel section, representing the continuous condition of SAV travel sections, starting at the depot node (Fig. 3). Constraint (23) represents the requirement for the design variables to be binary. Constraint (24) converts the presence or absence of travel sections on "forward links" into the presence or absence of travel sections on links in the original network.

Fig. 3

Continuous condition of the SAV travel section

Therefore, the actual depot model can be formulated as a problem in which the continuous condition of the SAV travel sections is added to the constraints of the virtual depot model, as follows:

$$\underset{\mathbf{x},\mathbf{y},z, {\varvec{\upphi}}}{\mathrm{min}}Z=\sum_{s\in S}\sum_{a\in {A}^{T}}{c}_{a}^{s}{x}_{a}^{s}+\sum_{a\in {A}^{SAV}}{c}_{a}{y}_{a}+{\theta }_{SAV}z$$

(25)

subject to Eqs. (7)–(20) and Eqs. (22)–(24).

Note that SAVs may not reach the depot from some SAV travel sections that satisfy the continuous condition if there is a one-way street in the original network. However, the impact of the one-way streets is expected to be limited because the original network generally has few one-way sections.

3 Case Study

The two proposed models were applied to a hypothetical network to characterize the models. We solved the mixed-integer linear programming problem using MATLAB 2022 (MathWorks, Natick, MA, USA) and Gurobi 9.5.0. (Gurobi Optimization LLC, Beaverton, OR, USA). Section 3.1 describes the study network and computational conditions. Section 3.2 compares the solutions of the virtual depot model and the actual depot model when the HDV ownership cost is not considered. Finally, Section 3.3 presents a sensitivity analysis of the HDV ownership cost.

3.1 Test Network and Computational Conditions

3.1.1 Test Network

We used the Sioux Falls network [19], as shown in Fig. 4. The travel time is given by $[{d}_{a}/2]\times \Delta t$, where ${d}_{a}$ is the link distance (km) and $[x]$ is the largest integer smaller than or equal to $x$. We assumed that $\Delta t=15$ and the link capacities were set to half of the original values. Additionally, the location of the depot in the actual depot model was specified, such that two cases were compared: a case where the depot is located in a suburban area (Node 1) and a case where the depot is located in a central area (Node 10).

Fig. 4

The Sioux Falls network

3.1.2 Parameter Settings

The parameters of the proposed models are the value of time (${\alpha }_{1}$ and ${\alpha }_{2}$), the capacity of SAVs ($\kappa$), the running cost of the vehicle (${\beta }^{TC}$), and the ownership cost of SAVs and HDVs (${\theta }_{SAV}$, ${\theta }_{HDV}$). The value of times is assumed to be ${\alpha }_{1}=33.33(\mathrm{JPY}/\mathrm{min})$ and ${\alpha }_{2}=66.6 (\mathrm{JPY}/\mathrm{min})$; the SAV capacity is assumed to be $\kappa =3$. The travel cost per distance (${\beta }^{TC}$) was calculated by converting the vehicle running cost into hourly units using the values shown in Table 1, resulting in ${\beta }^{TC}=2.92 (\mathrm{JPY}/\mathrm{min})$. The SAV ownership cost (purchase and maintenance costs) was set to ${\theta }_{SAV}=5000 (\mathrm{JPY}/\mathrm{day})$, based on the daily cost ($4406.4\mathrm{ JPY}/\mathrm{yen}$) estimated using values shown in Table 2(a). The HDV cost was set to two ways: ${\theta }_{HDV}=1500 (\mathrm{JPY}/\mathrm{day})$ and ${\theta }_{HDV}=2500 (\mathrm{JPY}/\mathrm{day})$, referring to the estimated maintenance cost ($1666.7\mathrm{ JPY}/\mathrm{yen}$) and the purchase and maintenance cost ($2488.5\mathrm{ JPY}/\mathrm{yen}$) of HDVs, which were estimated using values shown in Table 2(b). Note that ${\theta }_{HDV}=1500$ corresponds to the case where only the maintenance cost of an HDV is considered, while ${\theta }_{HDV}=2500$ corresponds to the case where both purchase and maintenance costs are considered.

Table 1 Parameters for setting the travel cost per distance

Table 2 Parameters for SAVs and HDVs

3.1.3 Demand Settings

Because only the static OD demand is published as benchmark data in the Sioux Falls network, we consider the conversion of static demand into dynamic demand. As shown in Fig. 5, we assume that there is a specific variation between the baseline time period and the departure time limit. The arrival time limit is then set to ‘departure time limit + minimum travel time between ODs + time window’. In the following section, variation in the departure time limit is represented by a Poisson distribution with a mean value $\lambda =1$, and the time window is uniformly set to $4\times \Delta t$ (1 h (h)) for all OD pairs.

Fig. 5

Setting departure and arrival time constraints

3.1.4 Valuation Indicators

This study utilized three valuation indicators: the number of SAVs, the SAV passenger ratio, and the degree of improvement in the objective function value. The number of SAVs and the SAV passenger ratio are indicators of the extent of SAV penetration, whereas the degree of improvement in the objective function value is an indicator used to evaluate the efficiency of SAV introduction. Note that the SAV passenger ratio and degree of improvement in the objective function value are defined as follows:

$${r}_{SAV}=\frac{\sum_{s\in S}\sum_{a\in {A}_{1}^{D}}{{t}_{a}x}_{a}^{s}}{\sum_{s\in S}\sum_{a\in {A}_{1}^{D}\cup {A}_{1}^{P}}{{t}_{a}x}_{a}^{s}}$$

(26)

$$IR=1-\frac{{Z}_{OPT}}{{Z}_{HDV}}$$

(27)

where ${Z}_{OPT}$ and ${Z}_{HDV}$ represent objective function values with and without consideration of SAVs, respectively.

3.2 A Case without Considering the Ownership Cost of Human-driven Vehicles (HDVs)

This section compares the solutions between the two models when the cost of HDV ownership, ${\theta }_{HDV}$, is 0. Figure 6(a)–(c) show the relationships among the number of SAVs, the SAV passenger ratio, and the degree of improvement in the objective function value and the budget constraint specifying the HDV travel sections, respectively. In the actual depot model, the number of SAVs increased with the budget constraint but stabilized after the budget limit reached 70. The SAV passenger ratio and the degree of improvement in the objective function value increased up to the budget limit of 90. Increasing the budget limit above 70 did not increase the number of SAVs; however, increasing the budget limit to 90 increased the SAV passenger ratio and the degree of improvement in the objective function, because this increase led to an expanded area in which SAVs can travel.

Fig. 6

Relationships between valuation indicators and the budget constraint specifying the HDV travel section. VD: virtual depot model; AD: actual depot model. a Number of SAVs, b SAV passenger ratio, c Degree of improvement in the objective function value

In the virtual depot model, when the depot was located at Node 1, all indicator values were 0, indicating that the solution without SAV introduction and SAV travel sections was optimal. In contrast, when the depot was located at Node 10, the solution in which SAVs are introduced was optimal, but the number of SAVs stabilized after the budget limit reached 20. All indicator values in the virtual depot model were significantly lower than indicator values in the actual depot model. This difference is possibly because the virtual depot model does not consider the cost of travel from the depot to the service start points; it may also be related to the wider range of feasible solutions for the SAV travel section.

Figures 7(a) and (b) and 8(a) and (b) show the SAV travel sections for the virtual depot model and the actual depot model (assuming the depot is located at Node 10) when the budget limit was 10 and 50, respectively. When the budget limit was 10, the general locations of the SAV travel sections were similar for both models. However, in the actual depot model, the SAVs can return to the depot only by traveling on the SAV travel sections; in the virtual depot model, the SAVs could not return to the depot only by traveling on the SAV travel sections. When the budget limit was 50, the SAV travel sections were more extensive in the virtual depot model than in the actual depot model. However, as in the case when the budget limit was 10, the SAVs could not return to the depot only by traveling on SAV travel sections in the virtual depot model.

Fig. 7

Illustration of SAV travel section (when the budget limit is 10). a VD model, b AD model

Fig. 8

Illustration of SAV travel section (when the budget limit is 50). a VD model, b AD model

3.3 Sensitivity Analysis of the Cost of HDV Ownership

This section analyzes the impact of the cost of HDV ownership. As stated previously, two cases of the cost of HDV ownership are assumed: a case in which only maintenance costs are considered (${\theta }_{HDV}=1500$) and a case in which both purchase and maintenance costs are considered (${\theta }_{HDV}=2500$). Note that the computational time for the actual depot model was ~ 2 min (min), whereas the computational time for the virtual depot model with ${\theta }_{HDV}=1500$ and a budget limit of 50 was > 72 h, using a computer with an Intel Xeon W-2102 central processing unit, with 2.90 GHz and 32.0 GB RAM. Therefore, only the actual depot model is considered in this section. The significantly higher computational cost of the virtual depot model, compared with the actual depot model, may be attributed to the simple constraint limitation of the feasible solution region of SAV travel sections in the actual depot model (Eq. (22)).

Figure 9(a)–(c) show the relationships among the number of SAVs, the SAV passenger ratio, and the degree of improvement in the objective function value with the budget constraint specifying the HDV travel sections, respectively, when the depot is located at Node 1. As stated previously, the solution that did not introduce SAVs and SAV travel sections was optimal when the cost of HDV ownership was 0. When ${\theta }_{HDV}$ was 1,500 and 2,500, there was no significant difference in the number of SAVs and the SAV passenger ratio for the same budget limit; however, both indicators were always larger when ${\theta }_{HDV}=2500$. In contrast, there was a significant difference in the degree of improvement in the objective function value when the cost of ownership was 1500 and 2000. This is presumably because a greater cost of HDV ownership led to a greater effect of reduction in the number of HDVs associated with SAV introduction.

Fig. 9

The relationship between the valuation indicators and the budget constraint specifying the HDV travel section when the depot is assumed to be Node 1. a Number of SAVs, b SAV passenger ratio, c Degree of improvement in the objective function value

Figure 10(a)–(c) show the relationships among the number of SAVs, the SAV passenger ratio, and the degree of improvement in the objective function value with the budget constraint specifying the HDV travel sections, respectively, when the depot is located at Node 10. The number of SAVs was always larger when the cost of ownership was 2500 than when the cost of ownership was 1500, and the difference between the two was larger than when the depot was located at Node 1. The degree of improvement in the objective function value also tended to increase with increasing cost of ownership.

Fig. 10

The relationship between the valuation indicators and the budget constraint specifying the HDV travel section when the depot is assumed to be Node 10. a Number of SAVs, b SAV passenger ratio, c Degree of improvement in the objective function value

Finally, all indicators were larger when the depot was located at Node 10 than when it was located at Node 1, indicating that the effect of introducing SAV travel sections is greater when the depot is located in a central area.

4 Conclusion

This study proposed a model for determining the optimal numbers of SAVs and SAV travel sections, assuming a situation in which travelers either drive an HDV or use SAVs. Two types of models were formulated as mixed-integer programming problems: a virtual depot model that does not consider the travel distance from the depot to the service start and end points, and an actual depot model that considers travel distance from the depot to the service start and end points. In particular, the actual depot model introduced the continuous condition for SAV travel sections with a simple constraint under mild assumptions about the SAV travel sections.

The proposed models were applied to a hypothetical network, and the following results were confirmed:

• Evaluation indicators for the virtual depot model tended to be higher than evaluation indicators for the actual depot model.

• In the actual depot model, the effect of SAV introduction was greater when the cost of HDV ownership was considered.

• In the actual depot model, the effect of SAV introduction depended on the depot location.

• The computation time of the virtual depot model was much greater than computation time of the actual depot model.

With regard to the second point above, when a manual driving vehicle is owned, its purchase and maintenance costs are not considered. Thus, the introduction of SAV travel sections should be considered before a drastic change occurs, such as the replacement of all internal combustion engine vehicles with electric vehicles.

There were several limitations in this study. First, because driving behavior is assumed to be identical between SAVs and HDVs, this study did not consider the effect of increased link capacity due to the increased SAV penetration. Therefore, this study may have underestimated the effect of SAV introduction. Second, because several assumptions are made in the virtual depot model, the model can be extended to a more general framework. Finally, although the proposed model can be solved by a commercial solver, the computational cost is high due to the inclusion of binary variables. Future work includes the application of efficient solution algorithms such as Bender’s decomposition algorithm, which decomposes binary and real variables (Mesbah et al. [8]).