Optimization of Shared Autonomous Vehicles Routing Problem: From the View of Parking

Abstract

Shared autonomous vehicles (SAVs) have great potential for achieving beneficial changes to the society. Although recent studies have explored the traffic safety, economic benefits and environmental impact, parking decisions of SAVs is rarely considered. SAVs need to park to avoid cruising during the interval between services. Appropriate parking decisions can contribute to reduce the vehicle kilometers traveled (VKT). This study synergistically considers routing and parking of SAVs for system optimization. Since the problem is NP-hard, we develop a variable neighborhood search (VNS) heuristic to solve it. The heuristic aims to minimize the VKT, the number of SAVs and the parking cost by systematic changes of neighborhood. A series of experiments based on the Anaheim network prove the high solving efficiency and quality of the heuristic. Results also indicate that the marginal cost of the system decreases with the increase in travel demand and the VKT increases with the increase in parking fees.

1. Introduction

Industry and academia have been working to develop autonomous driving technology [1], which will greatly influence socio-economic structure, urban organizational form, and so on [2]. For the transportation system, current research suggests that autonomous vehicles (AVs) will help to reduce traffic accidents [3,4], road congestion [5,6], parking spaces [7], and improve travel convenience [8]. AVs may also have disadvantages, such as the higher vehicle miles traveled (VMT) [9], empty VMT [10,11], and the higher energy consumption that comes with them. Overall, most of the research supports that the benefits of AVs outweigh the disadvantages [2].

Shared autonomous vehicles (SAVs) [12] go further. Compared with privately-owned autonomous vehicles (PAVs), SAVs have greater potential for sustainable urban development. SAVs can reduce vehicle usage by providing the mobility-on-demand (MoD) service [13]. During the operation, SAVs assignment and requests allocation is a problem worth considering. However, current research tends to ignore the parking problem. In fact, during the interval between services, SAVs must cruise or park themselves to wait for the next passenger. Parking requires payment, while cruising consumes energy. So, in this study, we consider a collaborative routing and parking decision to minimize the system cost.

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section 3 develops a linear programming formulation for the problem. Section 4 designs a heuristic solution framework. Section 5 presents computational experiments, and we conclude in Section 6.

2. Literature Review

2.1. Shared Autonomous Vehicles

According to the survey, a personal vehicle spends 95% of its time parking, while its actual usage time only accounts for 5% [14]. This means that resources are wasted, and SAVs can largely avoid it. Current research [15,16,17,18,19] suggests that SAVs can significantly reduce the number of privately-owned vehicles. Besides the research on the fleet size of SAVs, current research also focuses on the travel cost [16,20,21], traffic safety [3,22,23], environmental impacts [24,25,26], and so on.

However, the parking problem of SAVs has relatively lacked attention; the existing research mainly focuses on parking demand and land use. For example, Zhang et al. [27] developed an agent-based model to estimate the impact of SAVs on urban parking demand. The simulation results indicate that SAVs can eliminate 90% of the parking demand. Moreover, Zhang et al. [28] simulated the parking reduction trend from 2020 to 2040 with various market penetrations of SAVs. The results also support that parking demand will decline as SAV occupancy rises. In these studies, parking strategies of SAVs tend to be simple and SAVs usually just park nearby; but in fact, parking decisions and orders allocation are related. During the interval between services, whether the SAV should be parked and where it should be parked depends on the drop-off location and end time of the previous order, the pick-up location and start time of the next order, and the locations of the parking lots. Conversely, parking decisions can affect order allocation as well. Therefore, the SAV parking and routing problem requires collaborative optimization, and the optimal solution could reduce energy consumption and parking cost.

2.2. Vehicle Routing Problem with Intermediate Stops

The SAV parking and routing problem is similar to the vehicle routing problem with intermediate stops (VRPIS) [29]. VRPIS is a developmental branch of the vehicle routing problem (VRP). It refers to the problem of how to plan the route each vehicle travels when it can or must visit intermediate stops during the operation of its service mission, such as vehicles—especially electric vehicles—requiring supplemental energy [30], drivers who need rest to prevent fatigue during long-distance driving [31], and so on. For the SAV, it has the option to park to avoid cruising between the service missions.

VRPIS, like VRP, is an NP-hard problem [32] and is difficult to solve. Existing research mainly solves the problem by exact and heuristic algorithms. Although the exact algorithm can find the optimal solution, it is usually limited to small-scale problems, and when the problem size increases, the exact algorithm is so time-consuming that it cannot even obtain a feasible solution in an acceptable time.

Several heuristic algorithms have been developed to solve the VRP or VRPIS. Especially in recent years, many hybrid algorithms have emerged, such as the hybrid beetle swarm optimization algorithm [33], the ant colony system–improved grey wolf optimization algorithm [34], the hybrid metaheuristic algorithm of Discrete Particle Swarm Optimization (DPSO), Harris Hawks Optimization (HHO) [35], and so on. In this study, we focus on a metaheuristic called variable neighborhood search (VNS). VNS is one of the most widely used metaheuristics for solving combinatorial and global optimization problems. Its core idea is a systematic change of neighborhood both within a descent phase to find a local optimum and in a perturbation phase to get out of the corresponding valley [36].

Many studies have employed VNS to solve VRPs. Mladenović and Hanson [37] first proposed the VNS and then applied it to the traveling salesman problem with backhauls (TSPB). For the VRPIS, Del Pia and Filippi [38] used the variable neighborhood descent algorithm for a real waste collection problem with mobile depots. Divsalar et al. [39] used a variant of VNS called Skewed VNS (SVNS) to solve the orienteering problem (OP) with hotel selection. In summary, current studies have proved the effectiveness of developing VNS to solve the VRP and VRPIS.

3. Formulation

The SAV routing and parking problem of this study consists of a base, multiple parking lots, multiple pick-up and drop-off locations, and multiple SAVs of the same type that can be inserted into the network. The base, parking lots, passenger pick-up and drop-off locations, and passenger travel time windows are known. The distance and travel time of any two points in the network are known. SAVs all depart from the base, serve several passengers, and finally all return to the base. After each SAV has completed a drop-off service, it can cruise to wait for the next passenger or it can park to avoid cruising (to avoid affecting the efficiency of urban traffic, SAVs can only be parked in the base or parking lots). Each passenger can only receive a single service from one SAV during its time window. In addition, SAV services in this study include only car-sharing and not ride-sharing. Figure 1 shows the whole process of the SAV service.

Figure 1. The SAV service process.

The goal of this study is to determine the parking decisions and routes for the SAVs so that the travel demands can be met and the total cost is the lowest. Definitions of sets, indexes, parameters, and variables are represented in Table 1. For the sake of simplicity, we omitted the drop-off point and converted the travel time from the pick-up point to the drop-off point into the service time of the SAV. We set up the dummy nodes for the parking lots and the base indicating the multiple visits of the real nodes.

Table 1. Notations of the model.

Notation	Definition
Set
$O$	The initial depot, i.e., the base, $O=\left\{0\right\}$
$T$	The final depot, i.e., the base, $T=\left\{1\right\}$
$P$	Set of travel demands which indicate the pick-up points, $P=\left\{2,3,4,\cdots ,E+1\right\}$
$B$	Set of the dummy bases for park between service intervals, $B=\left\{E+2,E+3,E+4,\cdots ,E+W+1\right\}$
$U$	Set of parking lots, $U=\left\{1,2,3,\cdots ,Q\right\}$
$S_q$	Set of dummy parking lots in the parking lot $q$, $q\in U$, for example, $S_1=\{E+W+2,E+W+3,E+W+4,\cdots ,E+2W+1\}$
$S$	Set of all dummy parking lots, $S=S_1\cup S_2\cup S_3\cup \cdots \cup S_Q$
$N$	Set of all nodes, $N=O\cup T\cup P\cup B\cup S=\left\{1,2,3,\cdots ,E+(Q+1)W+1\right\}$
$V$	Set of SAVs
Parameter
$c_e$	Energy cost per unit distance
$c_c$	Basic usage cost per SAV
$c_s$	Parking cost per unit time in a parking lot
$w_e$	Weight coefficient of $c_e$
$w_c$	Weight coefficient of $c_c$
$w_s$	Weight coefficient of $c_s$
$E$	Size of the travel demand
$Q$	Number of parking lots, $Q=\left|U\right|$
$W$	Number of the dummy bases for the base and dummy parking lots for each parking lot, $W\le E+1$
$H$	Maximum number of available SAVs, $H=\left|V\right|$
$v_t$	Travel speed
$t_{ij}$	Travel time from $i\in N$ to $j\in N$. If $i\in P$, $t_{ij}$ contains the time for delivering passengers to the drop-off point i.e., $c_i$ and the time from the drop-off point to $j$
$a_i$	The earliest acceptable boarding time for the travel demand $i\in P$
$T_w$	The time window for travel demands
$b_i$	The latest acceptable boarding time for the travel demand, $i\in P$, $b_i=a_i+T_w$
$c_i$	Service time for SAVs to complete the travel demand $i\in P$
$T_s$	Start time
$T_t$	End time
$M$	A sufficiently large number
Variable
${\tau }_i^k$	Decision variable specifying the arrival time of the SAV $k\in V$ at the vertex $i\in N$
${\rho }_i^k$	Decision variable specifying the departure time of the SAV $k\in V$ at the vertex $i\in N$
${\omega }_i^k$	Decision variable specifying the starting serve time of SAV $k\in V$ for the travel demand $i\in P$
$x_{ij}^k$	Binary decision variable indicating if SAV $k\in V$ travel from vertex $i\in N$ to vertex $j\in N$
${\theta }^k$	Binary decision variable indicating if SAV $k\in V$ participated in the service mission

The objective function is given as follows:

$$\begin{array}{cc}\hfill \min \mathrm{Z}=& \{w_e{c}_e{{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{i\in N}{{\displaystyle \sum }}_{j\in N}{t}_{ij}{x}_{ij}^k+w_e{c}_e{{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{i\in P}\left({\omega }_i^k-{\tau }_i^k\right)+w_c{c}_c{{\displaystyle \sum }}_{k\in V}{\theta }^k+\hfill \\ & w_s{c}_s{{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{i\in S}\left({\rho }_i^k-{\tau }_i^k\right)\}\hfill \end{array}$$

(1)

$c_e{{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{i\in N}{{\displaystyle \sum }}_{j\in N}{t}_{ij}{x}_{ij}^k$ is the energy cost for SAVs. $c_e{{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{i\in P}\left({\omega }_i^k-{\tau }_i^k\right)$ is the cruising cost for SAVs while waiting for passengers. $c_c{{\displaystyle \sum }}_{k\in V}{\theta }^k$ is the basic usage cost for SAVs. $c_s{{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{i\in S}\left({\rho }_i^k-{\tau }_i^k\right)$ is the parking cost for SAVs while parking in the parking lot. It should be noted that there is no cost for SAVs parked at the base.

3.1. Route Constraints

Route constraints are given as (2)–(12).

$${{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{i\in N}{x}_{ij}^k=1 \forall j\in P$$

(2)

$${\displaystyle {\sum }_{k\in V}{\displaystyle {\sum }_{j\in (P\cup B\cup S\cup T)}{x}_{ij}^k}=1 }\forall i\in P$$

(3)

$${\displaystyle {\sum }_{k\in V}{\displaystyle {\sum }_{j\in (N-P)}{x}_{ij}^k}=0 }\forall i\in O$$

(4)

$${\displaystyle {\sum }_{k\in V}{\displaystyle {\sum }_{i\in (N-P)}{x}_{ij}^k}=0 }\forall j\in T$$

(5)

$${{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{j\in N}{x}_{ij}^k=0 \forall i\in T$$

(6)

$${{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{j\in (N-P)}{x}_{ij}^k=0 \forall i\in B\cup S$$

(7)

$${{\displaystyle \sum }}_{k\in V}{{\displaystyle \sum }}_{i\in N}{x}_{ij}^k\le 1 \forall j\in B\cup S$$

(8)

$${{\displaystyle \sum }}_{j\in N}{x}_{ij}^k-{{\displaystyle \sum }}_{j\in N}{x}_{ji}^k=0 \forall i\in P\cup B\cup S, \forall k\in V$$

(9)

$${{\displaystyle \sum }}_{i\in O}{{\displaystyle \sum }}_{j\in N}{x}_{ij}^k={{\displaystyle \sum }}_{i\in N}{{\displaystyle \sum }}_{j\in T}{x}_{ij}^k \forall k\in V$$

(10)

$$x_{ij}^k\le {\theta }^k\forall i\in N, \forall j\in N, \forall k\in V$$

(11)

$${\displaystyle {\sum }_{i\in O}{{\displaystyle \sum }}_{j\in P}{x}_{ij}^k}={\theta }^k \forall k\in V$$

(12)

Constraint (2) ensures that each travel demand is met once.

Constraint (3) ensures that after the SAV completes a service, it can continue to serve the next travel demand, or park in the parking lot or the base temporally, or return to the base without further service.

Constraint (4) ensures that the SAV from the base can only travel to the pick-up point. Set $(N-P)$ includes all nodes except set $P$ (travel demands).

Constraint (5) ensures that the SAV can only return to the base after completing all services. It also avoids parking at the dummy bases or dummy parking lots before returning to the final depot.

Constraint (6) ensures that the SAV cannot travel to any node after it return to the final depot.

Constraint (7) ensures that the SAV from temporary parking can only travel to the pick-up node.

Constraint (8) ensures that each dummy base or parking lot can only be visited once.

Constraint (9) ensures flow conservation.

Constraint (10) ensures that the SAV departing from the initial depot must return to the final depot.

Constraints (11) and (12) are used to confirm whether the SAV is inserted into the network.

3.2. Time Constraints

Time constraints are given as (13)–(23).

$${\tau }_i^k\le {\rho }_i^k \forall i\in N, \forall k\in V$$

(13)

$${\tau }_i^k=T_s \forall i\in O, \forall k\in V$$

(14)

$${\rho }_i^k=T_t \forall i\in T, \forall k\in V$$

(15)

$${\omega }_i^k\ge {\tau }_i^k \forall i\in P, \forall k\in V$$

(16)

$${\omega }_i^k\ge a_i \forall i\in P, \forall k\in V$$

(17)

$${\omega }_i^k\le b_i \forall i\in P, \forall k\in V$$

(18)

$${\omega }_i^k+c_i={\rho }_i^k \forall i\in P, \forall k\in V$$

(19)

$${\rho }_i^k+x_{ij}^k{t}_{ij}-M\cdot \left(1-x_{ij}^k\right)\le {\tau }_j^k \forall i\in N, \forall j\in N, \forall k\in V$$

(20)

$${\rho }_i^k+x_{ij}^k{t}_{ij}+M\cdot \left(1-x_{ij}^k\right)\ge {\tau }_j^k \forall i\in N, \forall j\in N, \forall k\in V$$

(21)

$${\rho }_i^k+x_{ij}^k{t}_{ij}-M\cdot \left(1-x_{ij}^k\right)\le a_j \forall i\in O\cup B\cup S, \forall j\in P, \forall k\in V$$

(22)

$${\rho }_i^k+x_{ij}^k{t}_{ij}+M\cdot \left(1-x_{ij}^k\right)\ge a_j \forall i\in O\cup B\cup S, \forall j\in P, \forall k\in V$$

(23)

Constraint (13) ensures that the time when the SAV arrives at any point is earlier than the time it leaves.

Constraints (14) and (15) set the start and end time.

Constraints (16)–(18) ensure that the SAV starts service within the time window and is later than its arrival time.

Constraint (19) ensures that the service end time of the SAV is its departure time from the vertex.

Constraints (20) and (21) ensure the time feasibility of SAVs.

Constraints (22) and (23) ensures that SAVs departing from the base or parking lots need to reach the pick-up point in time.

4. Proposed Heuristic Solution Framework

For large-scale SAV parking and routing problems, exact algorithms are not applicable. Therefore, we focus on the metaheuristic VNS to obtain high-quality solutions efficiently. The procedure of the VNS in this study is shown in Algorithm 1. The basic idea of the VNS is to find the local optimum by local search (line 2, 6) and get out of the corresponding valley by shaking (line 5).

Algorithm 1. The procedure of the VNS.
Input	the set of neighborhood structures $N_k, \mathrm{for} k=1,2,\dots ,k_{\max }$, that will be used in the shaking phase
Output	The optimized solution $x^{*}$
1	Construct the initial solution $x_0$
2	$x^{*}\leftarrow$ local search $\left(x_0\right)$
3	$k\leftarrow 1$
4	repeat
5	$\hspace{1em}\hspace{1em}{x}^{\prime }=$ shake $\left(x^{*},N_k\right)$
6	$\hspace{1em}\hspace{1em}{x}^{″}=\mathrm{local} \mathrm{search}\left(x^{\prime }\right)$
7	if $f\left(x^{″}\right)<f\left(x^{*}\right)$
8	$\hspace{1em}\hspace{1em}\hspace{1em}{x}^{*}\leftarrow x^{″}$, $k\leftarrow 1$
9	else
10	$\hspace{1em}\hspace{1em}\hspace{1em}k\leftarrow k+1$
11	end if
12	until $k>k_{\max }$

4.1. Label Sequences of Routes

As shown in Figure 2, we use label sequences to represent the routes of SAVs. B, U, and P denote the base, parking lots, and travel demands, respectively. If all constraints are satisfied, then combining all label sequences is a feasible solution. Furthermore, according to the time constraints, we can obtain the arrival and departure times of SAVs at each node and thus we can calculate the system cost.

Figure 2. Label sequences of routes.

4.2. Construction Algorithm

Algorithm 1 starts with an initial solution. In this study, we construct the initial solution based on the greedy idea. Travel demands will be added to an empty route, and each travel demand added is the one who makes the system has the smallest increase in system cost until the route cannot accommodate more travel demands. We then create a new empty route to accommodate the remaining travel demands. This cycle continues until no travel demands are left.

4.3. Shaking

Shaking is the key to get out of the valley. A series of operators are designed and applied to generate the neighborhood structures for this study:

• Parking label removal operator. This kind of operator randomly removes several parking labels on several randomly selected sequences. The removed parking labels include parking lots and the base that is not in the first and last position. Figure 3 shows an example. In this study, there are five operators of this type, each operating on 1, 2, 3, 4, and 5 parking labels.
  

  Figure 3. Parking label removal operator.

2.

Parking label replacement operator. This kind of operator randomly replaces several parking labels on several randomly selected sequences. The replaced parking labels include parking lots and the base that is not in the first and last position. Figure 4 shows an example. In this study, there are five operators of this type, each operating on 1, 2, 3, 4, and 5 parking labels.

Figure 4. Parking label replacement operator.

3.

2-opt operator [40]. This kind of operator randomly reverses several adjacent travel demands on a randomly selected sequence. Figure 5 shows an example. In this study, there are four operators of this type, each operating on 2, 3, 4, and 5 adjacent travel demands.

Figure 5. 2-opt operator.

4.

Or-opt operator [41]. This kind of operator randomly selects a section of labels from a randomly selected sequence and inserts the section into a random position of the sequence. Figure 6 shows an example. In this study, there are four operators of this type, each operating on the section with 2, 3, 4, and 5 adjacent travel demands.

Figure 6. Or-opt operator.

5.

2-optstar operator [42]. This operator randomly selects a position from two randomly selected sequences separately and swap all the labels after these two positions. Figure 7 shows an example.

Figure 7. 2-optstar operator.

6.

Relocate operator [43]. This operator randomly selects a travel demand label from a randomly selected sequence and insert it into a random position of another randomly selected sequence. Figure 8 shows an example.

Figure 8. Relocate operator.

7.

Exchange operator. This operator swaps two random travel demand labels from two randomly selected sequences. Figure 9 shows an example.

Figure 9. Exchange operator.

8.

Cross operator. This operator swaps two random sections of labels from two randomly selected sequences. The section does not include the first and last base labels. Figure 10 shows an example.

Figure 10. Cross operator.

9.

Icross operator. This operator swaps two random sections of labels from two randomly selected sequences in reverse order. The section does not include the first and last base labels. Figure 11 shows an example.

Figure 11. Icross operator.

In this study, there are a total of 23 operators (5 + 5 + 4 + 4 + 1 + 1 + 1 + 1 + 1 = 23). Continuous parking labels may appear after the operator execution. They should be removed until there are no continuous parking labels.

4.4. Local Search

In order to find solutions more efficiently, we use the time-efficient first descent heuristic [36] for local search. It is summarized in Algorithm 2.

Algorithm 2. First improvement (first descent) heuristic.
Input	the set of neighborhood structures $N_l$, the solution $x$
Output	The optimized solution $x$
1	repeat
2	$\hspace{1em}\hspace{1em}{x}^{\prime }\leftarrow x$; $i\leftarrow 0$
3	repeat
4	$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i\leftarrow i+1$
5	$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x\leftarrow \arg \min \left\{f\left(x\right),f\left(x^i\right)\right\}$, $x^i\in N_l\left(x\right)$
6	until $f\left(x\right)<f\left(x\prime \right)$ or $i>=\left|N_l\left(x\right)\right|$
7	until $f(x)\ge f(x\prime )$

The operators used for local search include 2-opt operator, 2-optstar operator, cross operator, and the following:

• Single-sequence parking label removal operator. This operator removes one parking label (including the parking lot and the base that is not in the first and last position) on one sequence. Figure 12 shows an example.
  

  Figure 12. Single-sequence parking label removal operator.

2.

Single-sequence parking label replacement operator. This operator replaces one parking label on one sequence and the operator execution should avoid continuous parking labels. The replaced parking labels include parking lots and the base that is not in the first and last position. Figure 13 shows an example.

Figure 13. Single-sequence parking label replacement operator.

3.

Single-sequence parking label insertion operator. This operator inserts one parking label into one sequence. Figure 14 shows an example.

Figure 14. Single-sequence parking label insertion operator.

Continuous parking labels may appear after the operator execution. They should be removed until there are no continuous parking labels.

5. Numerical Results

This section presents the numerical results used to test the heuristic. The road network is based on Anaheim (available at https://github.com/bstabler/TransportationNetworks accessed on 17 May 2022). All instances (available at https://github.com/memojune/SAV-routing-and-parking-instances accessed on 3 July 2023.) were run on an Intel(R) Core (TM) i5-9400F (2.90 GHz) CPU with 16.0 GB RAM and Windows 10 operating system. Values of parameters are shown in Table 2. However, the coefficients can be chosen properly by the SAV operators depending on the impact of all kinds of cost.

Table 2. Values of parameters.

Parameter	Value	Unit	Parameter	Value	Unit
$c_e$	0.4	¥/min	$w_e$	1
$c_c$	60	¥	$w_c$	0.3
$c_s$	0.1	¥/min	$w_s$	1
$T_w$	10	min	$v_t$	30	km/h
$T_s$	0	min	$T_t$	1440	min

5.1. Small-Size Instances

There are three groups of small-size instances with 10, 20, and 30 travel demands. Each group has five instances and there are four parking lots in each instance. Locations of pick-up nodes, drop-off nodes, and parking lots are all randomly generated. The base is fixed at point 6 of the Anaheim road network. The earliest acceptable boarding time is randomly generated between [420, 600].

We use the commercial solver Gurobi 9.5.1 to compare with the VNS. The time limit of Gurobi is set to 3600 s; the VNS was run five times for each instance to test the stability. The results are shown in Table 3.

Table 3. Results and comparison of Gurobi and VNS.

Demand	Gurobi		VNS
	Optimal Value	Runtime (s)	Optimal Value	Ave. Value	Ave. Runtime (s)
10	277.4	401.65	277.4	280.0	4.64
10	275.3	4.96	275.3	275.3	2.10
10	285.2	2.52	285.2	285.2	4.48
10	243.0	36.87	243.0	243.0	2.50
10	314.6	23.23	314.9	316.9	10.16
20	487.7	3600.00	462.3	462.3	7.28
20	-	3600.00	500.2	505.3	3.66
20	544.4	3600.00	474.6	477.7	4.77
20	688.4	3600.00	575.6	581.0	4.88
20	-	3600.00	497.5	498.9	4.12
30	-	3600.00	707.4	707.6	8.58
30	-	3600.00	643.5	650.4	19.90
30	-	3600.00	647.5	658.8	6.18
30	-	3600.00	698.4	703.1	25.57
30	-	3600.00	666.4	678.3	16.12

‘-’: the feasible solution was not found within the time limit. Bold: the better solution between Gurobi and VNS. Ave.: average.

As can be seen from Table 3, when travel demand is 10, Gurobi can solve all instances exactly. The VNS can find the exact solution for 4 of 5 instances in 5 runs, and the optimal value of the remaining instance is close to the exact solution. Overall, in this group, the VNS can find the feasible solution that is equal or close to the exact solution in a shorter runtime.

When travel demand increases to 20, all instances cannot be solved by Gurobi exactly under the time limit and 2 of 5 instances cannot even obtain the feasible solutions. In this group, the VNS can find better solutions in a shorter runtime.

When travel demand increases to 30, Gurobi cannot obtain the feasible solution for all instances under the time limit. However, the VNS can find solutions quickly.

5.2. Large-Size Instances

5.2.1. Effect of the Number of Operators in Shaking Procedure

In shaking procedure, the operators can help the algorithm get out of the valley. In this section, we tested the effect of the number of operators in shaking procedure. The experiments were conducted on an instance with 500 travel demands, whose earliest acceptable boarding time were randomly generated between [420, 1140]. The instance has 12 parking lots. Locations of pick-up nodes, drop-off nodes, and parking lots are all randomly generated. The base is fixed at point 6 of the Anaheim road network. We set the number of operators to 1, 12, and 23, respectively, i.e., in the shaking procedure, randomly selecting the corresponding number of operators for shaking.

Figure 15 shows the variation of system cost with the number of successful shakings under the different number of operators in the shaking procedure. It is evident that the algorithm reduces the system cost effectively compared to the initial solution. In addition, it can be seen that with the increase in the number of operators in the shaking procedure, the number of successful shakings increases, and the algorithm can achieve better system cost. This is because the more shaking operators bring more possibility to get out of the valley and restart the local search. Accordingly, the runtime also increases, with results of 188 s, 779 s, 2717 s, and 4182 s.

Figure 15. Variation of system cost with number of successful shakings under different number of operators in the shaking procedure.

5.2.2. Effect of the Travel Demand

There are large-size instances with 100, 200, 300, 400, and 500 travel demand. Each instance has 12 parking lots. Locations of pick-up nodes, drop-off nodes, and parking lots are all randomly generated. The base is fixed at point 6 of the Anaheim road network. The earliest acceptable boarding time is randomly generated between [420, 1140]. The VNS was run five times for each instance to test the stability and results are shown in Table 4. The VNS column records the relevant data for the lowest system cost in five runs. The $\overline{\mathrm{VNS}}$ column is the average data of five runs. From column CV, it can be seen that all CVs are less than 1.1%, indicating nice stability of the algorithm.

Table 4. Results of large-size instances solved by VNS.

Demand	VNS					$\overline{\mathbf{VNS}}$		CV (%)
	Optimal Value	No. SAV	VKT	PTP (min)	PT (min)	Ave. Optimal Value	Ave. Runtime (s)
100	1858	9	1969	1203	1778	1859	233	1.02%
200	3375	16	3642	1735	3310	3399	534	0.79%
300	4874	23	5296	2233	3503	4898	1541	0.44%
400	6193	29	6813	2205	3989	6226	2017	0.66%
500	7598	34	8415	2544	4631	7656	3823	0.69%

No. SAV: Number of SAVs; VKT: Vehicle kilometers traveled; PTP: Parking time in the parking lots; PT: Total parking time including PTP and parking time in the base during service intervals; Ave.: average; CV: Coefficient of variation.

According to the VNS column of Table 4, system indicators per 100 travel demand can be converted. Figure 16 depicts the relevant results. It is evident that all indicators per 100 travel demand decrease with the increase in the size of travel demand. Figure 16 indicates that the marginal cost of the system is decreasing, which also indicates the potential of SAVs in urban sustainable development.

Figure 16. System indicators per 100 travel demand.

The travel frequency of private vehicles is 1.52 times a day [27]. If all travel demand is satisfied with the privately owned vehicles, the number of private vehicles should be size of travel demand/1.52. Figure 17 indicates that the required number of SAVs is much less than the number of private vehicles. This also demonstrates the significant role of SAV in saving resources and promoting sustainable urban development.

Figure 17. Number of vehicles required for travel demand.

5.2.3. Effect of the Parking Fee

With the travel demand size of 500, we check the effects of the parking fee. Figure 18 shows the effects of varying parking fee in the parking lots from 0 ¥/min to 0.2 ¥/min with a step of 0.04 ¥/min.

Figure 18. Effects of parking fee on system indicators, from 0 ¥/min to 0.2 ¥/min.

As can be seen from the Figure, with the increase in the parking fee, parking time in the parking lots decreases and the VKT increases. This is because the higher parking fee makes SAVs shift towards further bases for parking or just cruising in the road network. Parking cost (parking revenue for parking lots) first increases and then decreases as the parking fee increases. This also reminds city managers to strike a balance between parking revenue, energy consumption, and environmental protection.

6. Conclusions and Future Work

The SAV has great potential in reducing the use and number of privately owned autonomous vehicles. However, compared to human-driven vehicles, the route planning and parking decisions of SAV are more dependent on the centralized scheduling of the system. Therefore, in order to effectively improve the operational efficiency of SAV and reduce system costs, it is important to develop optimization solutions for the SAV routing and parking problem.

In this paper, we consider the SAV routing problem in the context of the parking decision. To solve the problem, we proposed a solution based upon the VNS. In the shaking and local search procedures, a series of neighborhood operators were designed. These operators enhance the problem-solving efficiency and quality. The proposed solution approach was tested on small- and large-size instances. The results showed that the marginal cost of the system decreases with the increase in travel demand size. It indicates the potential of SAV in urban sustainable development. We also examined the effect of parking fees and the results showed that with the increase in parking fees, the VKT increases continuously, while the parking cost first increases and then decreases. It reminds the social manager that parking fees should be reasonably priced.

The originality of the proposed approach is the consideration of route planning and parking decision. Following the work presented in this study, there are some areas that may be explored in future:

• The adaptive variable neighborhood search (AVNS) selects the operator in the shaking step by incorporating an adaptive mechanism. It helps to improve the efficiency and quality of the solution. This is the direction of improvement for our algorithm;
• Apply other heuristics to this problem and compare their advantages and disadvantages of problem-solving;
• This study only considers car-sharing. We hope to include ride-sharing in our research, which will reduce costs to a greater extent;
• Travel demand in this study is static, i.e., all demand is known in advance. In fact, previously unknown demand will appear after SAVs operations begin. This is also worth further exploration.