A Practical and Sustainable Approach to Determining the Deployment Priorities of Automatic Vehicle Identification Sensors

Abstract

Monitoring vehicles’ paths is important for the management and governance of smart sustainable cities, where traffic sensors play a significant role. As a typical sensor, an automatic vehicle identification (AVI) sensor can observe the whereabouts and movements of vehicles. In this article, we introduced an indicator called the deployment score to present the deployment priorities of AVIs for a better reconstruction of vehicles’ paths. The deployment score was obtained based on a programming method for maximizing the accuracy of a recurring vehicle’s path and minimizing the number of AVI sensors. The calculation process is data-driven, where a random-work method was developed to simulate massive path data (tracks of vehicles) according to travel characteristics extracted from finite GPS data. Then, for each simulated path, a path-level bi-level programming model (P-BPM) was constructed to find the optimal layout of the AVI sensors. The solutions of the P-BPM proved to be approximate Pareto optima from a data-driven perspective. Furthermore, the PageRank method was presented to integrate the solutions; thus, the deployment score was obtained. The proposed method was validated in Chengdu City, whose results demonstrated the remarkable value of our approach.

1. Introduction

1.1. Background

With the rapid development of intelligent transportation technology, there are increasingly diverse types of sensors and detectors in traffic operations, such as in intelligent transportation systems (ITSs), traffic management and control, and traffic safety analysis and prevention [1,2,3]. Two broad categories of sensors are generally considered: counting sensors and scanning sensors [4,5,6,7]. Here, we focus on a typical scanning sensor: the AVI sensor. AVI data show unique attributes [8]: (1) a large sample size—every passing vehicle’s information and attribute information will be captured, and an AVI system can provide 24/7 service without extra requirements for vehicles; (2) accurate observation—the accurate coordinates and the passing time of the tracked vehicle can be obtained, and the accurate travel time of the tracked vehicle can also be calculated based on the timestamps of observations from neighboring AVI sensors; (3) safety and stability—every detected vehicle has a virtual electronic license plate that is scanned. Specifically, AVI data can be adopted for origin–destination (OD) demand estimation, vehicle path reconstruction, link flow inference, and travel time estimation, though counting sensors cannot.

However, full coverage of AVI sensors over a whole network is more expensive than it is for loop detectors, and their deployment is also a long step-by-step process. In consequence, seeking the optimal locations and making a construction-priority recommendation for AVI sensors in a network are crucial for a powerful AVI system.

The AVI location problem (AVI-LP) has drawn sufficient attention in the areas of OD demand estimation, link flow inference, and travel time estimation, while few studies on the vehicle-path-reconstruction-based AVI-LP have been performed. Of particular concern is that, once each vehicle’s path is accurately and completely reconstructed, OD matrix estimation, link flow estimation, and travel time estimation will be naturally achieved. Questions have been raised about the enhancement of subjects of the path reconstruction model in the AVI-LP, whereas it is more difficult to realize an identified and applicable model compared to other methods. By using a data-driven method to simulate vehicles’ moving paths—in conjunction with the bi-level programming model of each path—this research provides a new indicator in order to recommend a path-reconstruction-oriented AVI sensor location plan.

1.2. Literature Review

The section essentially involves two viewpoints and perspectives, namely, the AVI-LP and its solution methods. Gentili [9] and Castillo [4] performed a state-of-the-art review of the network sensor location problem and outlined future challenges from different perspectives, including data, variables, constraints, objective functions, and examples. In what follows, we review previous studies related to our work.

To the best of our knowledge, limited attention has been paid to path reconstruction in sensor location problems compared to other objectives, such as OD estimation [10,11], link flow inference [12,13], and travel time estimation [14]. One primary problem is that path reconstruction is a complicated process, and the representation of the estimation of its error in an optimal location model is intractable, especially in the objective function. Gentili et al. [15] solved this problem by assuming that vehicles’ routes are accessible by AVI sensors, and a bi-level optimization model was presented. A more realistic scenario without the assumption was later adopted to infer the path [16,17]. It is worth noting that the estimation of OD demand and link flow was available in terms of the results of path reconstruction, which typically proved the powerful practicability of path reconstruction, despite the existing difficulty. The effects of sensors’ order on a path and path-differentiated congestion pricing were also considered [18,19]. Furthermore, both counting sensors and AVI sensors were located strategically for path reconstruction. The study showed that AVI sensors played a more critical role than counting sensors in the path reconstruction process [20]. Additional research considered the uncertainty of the link–path matrix for the path-reconstruction-oriented sensor location model [21]. Mostafa et al. recently solved the path-reconstruction-oriented sensor location problem in an environment of connected vehicles by using four different mixed-integer programming formulations for RSUs and AVIs [22]. In short, as far as we know, the research on the path-reconstruction-oriented sensor location problem is far from being desired for theoretical research and practical applications.

The traffic sensor location problem is a typical planning problem in realistic applications. In detail, modeling a traffic sensor location problem calls for multiple objectives, evaluation criteria, and complex subjections. Bianco et al. [23] proved that the sensor location problem is NP-complete on general directed and symmetric graphs and is polynomially solvable on directed and symmetric path; to the best of our knowledge, mathematical programming approaches and algebraic methods are the most commonly used solution methods. The stochastic programming model was demonstrated to be workable when solving stochastic vehicle routing problems [24], reliable facility location problems [25], and other transportation issues; hence, it has become the most commonly used model in the relevant areas.

The solution methods for stochastic programming models are divided into three groups: exact methods, approximate methods, and heuristic methods [26,27]. A heuristic method is a generally used algorithm that can find the optimal solutions to common problems, whereas it cannot guarantee global optimality. An approximate algorithm is merely suitable for a specific problem, resulting in infrequent application. An exact method, as a novel algorithm for solving location problems, is efficient in finding global optima, and can include the decomposition-based algorithms proposed in previous years [28,29]. Recently, Paul Rubin [30] was the first to address the exact algorithm for locating counting sensors in flow observability problems. Although the study overcame several limitations of the existing approaches and performed flexibly, it depended on many heuristic steps and some assumptions, and it showed limitations when locating AVI sensors. In addition, these studies were mainly concerned with the processes of building models and finding a robust methodology, while the practicability and generalizability in engineering practice were sometimes neglected.

1.3. Research Gaps and Contributions

To the best of our knowledge, existing methods show three main limitations:

(1) There is a limited number of studies formulizing a data-driven detection error when modeling the vehicle-path-reconstruction-based AVI location problem. On the one hand, the vast majority of AVI-LP models focus on traffic flow estimation (involving path flow, link flow, and OD flow) or travel time estimation, while the path reconstruction is seldom considered. On the other hand, the detection error of traffic sensors is merely used in evaluation, rather than in the model-building process.

(2) The existing implementation strategies for the AVI-LP consider mathematical exactness and accuracy rather than practical feasibility on the management side. Researchers prefer promoting the accuracy of the models. However, this gives rise to some problems, such as worse universalizability, greater complexity, and higher computational burden, which are undesired by administrators.

(3) The existing methods mostly contribute to a unique optimal solution for a specific scenario, but this solution is not practical in the real world. In academic research, authors devote efforts to finding the unique optimal solution mathematically with a fixed penetration of the layout, which makes little sense in practical applications. However, the deployment of traffic sensors is always city-wide and on a large scale, meaning that much greater costs and more time are needed. Thus, a unique optimal layout plan cannot meet the requirements for such a long-term engineering projects in transportation planning and construction.

To remedy these gaps, we proposed a novel approach. In particular, the contributions of this study are as follows:

(1) A data-driven approach was proposed to solve the vehicle-path-reconstruction-based AVI sensor location problem. In order to build a practical, universal, and implementable model, this study found the optimal layout of AVI sensors at the path level first, and then integrated the results of all paths to analyze the region-wide layout. In particular, we simulated massive path data (tracks of vehicles) according to travel characteristics extracted from finite GPS data by using a random walk method; then, for each path, a Path-level Bi-level Programming Model (P-BPM) was constructed to find the optimal layout of the AVI sensors; lastly, we integrated these layouts by using the PageRank algorithm.

(2) The detection error of the AVI sensors was calculated and applied as one of the optimization objectives when modeling for the AVI-LP. By deriving it from the tracks of vehicles, the traffic flow distribution of an intersection can be characterized. Then, the detection error can be calculated by finding the location of the AVI sensors in each path. The detection error was applied to an objective equation to ensure that the missing observations of AVI sensors were as few as possible.

(3) A novel indicator called the deployment score was introduced. The PageRank algorithm finally output the priority ranking set of the intersections to locate AVI sensors. As a consequence, the installation sequence of the AVI sensors was determined, which is more implementable and practical in decision making.

We introduced and formulated the traditional region-level BPM (R-BPM) and proved it to be non-deterministic polynomial hard (NP-hard) (Section 2). Our data-driven framework is described, and the random work method, the decomposed BPM, and a PageRank method for the computation of the deployment score are detailed (Section 3). Furthermore, experiments were performed and the computational results were summarized (Section 4). Lastly, Section 5 presents the conclusions and limitations.

2. Problem Description and Preliminaries

2.1. Problem Description and Assumptions

A road network can be defined as a directed graph $\mathrm{G} =\left(\mathrm{V}, \mathrm{E}\right)$, where $\mathrm{V}$ is the set of network nodes, $\mathrm{V}=\left\{v_1,v_2,\dots ,v_n\right\}$, representing the intersections, and $\mathrm{E}$ is the set of edges, $\mathrm{E}=\left\{e_1,e_2,\dots ,e_m\right\},$ representing road links.

In the AVI-LP, an AVI sensor is established at a candidate node $v_i$, monitoring the next node that the vehicle will pass. For example, in the network shown in Figure 1, it is assumed that there are three paths and 100% traffic flow between the origin ${\mathit{v}}_{\mathit{o}}$ and destination ${\mathit{v}}_{\mathit{d}}$, with Path-1 (${\mathit{v}}_{\mathbf{8}}$→${\mathit{v}}_{\mathbf{9}}$→${\mathit{v}}_{\mathbf{10}}$→${\mathit{v}}_{\mathbf{16}}$→${\mathit{v}}_{\mathbf{22}}$→${\mathit{v}}_{\mathbf{28}}$→${\mathit{v}}_{\mathbf{29}}$) conveying 60% of the traffic flow, Path-2 (${\mathit{v}}_{\mathbf{8}}$→${\mathit{v}}_{\mathbf{9}}$→${\mathit{v}}_{\mathbf{15}}$→${\mathit{v}}_{\mathbf{21}}$→${\mathit{v}}_{\mathbf{22}}$→${\mathit{v}}_{\mathbf{28}}$→${\mathit{v}}_{\mathbf{29}}$) conveying 35% of the traffic flow, and Path-3 (${\mathit{v}}_{\mathbf{8}}$→${\mathit{v}}_{\mathbf{9}}$→${\mathit{v}}_{\mathbf{11}}$→${\mathit{v}}_{\mathbf{17}}$→${\mathit{v}}_{\mathbf{23}}$→${\mathit{v}}_{\mathbf{29}}$) conveying 5% of the traffic flow. Intuitively, there should be thirteen AVI sensors to identify all paths, but this is unnecessary. As a matter of fact, with only four AVI sensors established at ${\mathit{v}}_{\mathbf{8}}$, ${\mathit{v}}_{\mathbf{9}}$, ${\mathit{v}}_{\mathbf{10}}$, and ${\mathit{v}}_{\mathbf{29}}$, these paths can be inferred entirely. ${\mathit{v}}_{\mathbf{8}}$ and ${\mathit{v}}_{\mathbf{29}}$ indicate the origin and destination, respectively, while ${\mathit{v}}_{\mathbf{9}}$ and ${\mathit{v}}_{\mathbf{10}}$ distinguish the shunt flows. If there are three AVI sensors established at ${\mathit{v}}_{\mathbf{8}}$, ${\mathit{v}}_{\mathbf{9}},$ and ${\mathit{v}}_{\mathbf{29}}$, Path-2 with 35% of the traffic flow can be truly inferred, while the other 65% of the traffic flow is in dispute. With the AVI sensor at ${\mathit{v}}_{\mathbf{9}}$, the flow through ${\mathit{v}}_{\mathbf{9}}$ can be identified, where 35% of the traffic flow will go to ${\mathit{v}}_{\mathbf{15}}$, while 65% of the traffic flow will go to ${\mathit{v}}_{\mathbf{10}}$, so Path-1 and Path-3 cannot be identified clearly. From a data-driven point of view, if a vehicle is observed to pass along the path ${\mathit{v}}_{\mathbf{8}}$→${\mathit{v}}_{\mathbf{9}}$→${\mathit{v}}_{\mathbf{10}}$→${\mathit{v}}_{\mathbf{29}}$, we can speculate that this vehicle may pass through Path-1 with a probability of $60\%/\left(60\%+5\%\right)$ or through Path-3 with a probability of $5\%/\left(60\%+5\%\right)$. That is to say, if 5% error is reasonably acceptable, these three AVI sensors are enough. Hence, we obtain the optimal location/allocation of AVI sensors within the allowed error range (>5%). Furthermore, together with one more AVI sensor established at ${\mathit{v}}_{\mathbf{10}}$, these traffic flows can be generally obtained.

To formulate the problem, the following assumptions are made:

(a) The basic topology of the task network and the information of the location candidates are known beforehand;

(b) The detection error rate of the AVI sensors is free;

(c) Each AVI sensors is placed at a node, observing the next node that the vehicle will pass;

(d) In each vehicle’s path, the first node observed to be passed in the study region is regarded as the origin node, while the last one is the destination node.

2.2. Preliminary Representation and Traditional Region-Level BPM

Here, from the region-level point of view, a simple traditional BPM can be formulated as:

$$\min f_1={\displaystyle \sum }_{i\in C}{x}_i$$

(1)

$$\min f_2={\displaystyle \sum }_{r\in \mathrm{R}}\left(1-{\displaystyle \prod }_{j,k\in {\mathit{N}}_{\mathit{r}}}{\left(1-\frac{q_{j\to k}^r·l_{j\to k}^r}{q^r·l^r}\right)}^{a_{j\to k}^r·\left(1-x_j\right)}\right)$$

(2)

$$x_{\mathrm{i}}, a_{j\to k}^r\in \left\{0,1\right\} \forall i\in C,j\in {\mathit{N}}_{\mathit{r}},k\in {\mathit{N}}_{\mathit{r}}$$

(3)

The objective function (1) is the upper level, which is devoted to determining the optimal locations of the AVI sensors by minimizing the count and, thus, minimizing the infrastructure investment. Objective function (2) is the lower level, which is devoted to maximizing the detection rate. Figure 2 introduces the calculation of the identified error in objective function (2).

It is clear that the BPM includes O(|N|²·|R|) binary variables, O(|N|·|R|) non-negative variables, and O(|N|²·|R|) constraints, in which |N| is the number of candidate nodes and |R| is the number of vehicle paths in the network. Hence, there is a great number of variables and constraints with greater values of |N| and |R|, making it hard for us to easily obtain optimal solutions. To develop the greatest potential of AVI sensors, huge-magnitude quantities of vehicle paths are necessary, i.e., |R| will be too enormous. Then, with an analysis of the computational complexity, the relationship between the AVI-LP and the mixed-integer linear programming problem (MILPP) is given in the following remark.

Remark 1.

The BPM is one of the variants of the MILPP and is NP-hard. To be specific, if the heterogeneous distribution of paths of vehicles becomes homogeneous, the BPM degenerates into the MILPP. Since it extends the classic MILPP, the BPM is one of the variants of the MILPP. Therefore, the BPM is also an NP-hard problem.

There are mainly two barriers to the BPM. On one hand, it is difficult to solve most of NP-hard problems to optimality, especially in the real world. On the other hand, classic location programming problems focus on seeking the only optimal solution, which is not practical for recommendations for the deployment of AVI sensors. In consequence, we try to seek out a novel framework for the AVI-LP to conquer the solution and meet the real demand. The proposed framework is detailed in the following sections.

3. Model Formulation and Methodology

Figure 3 shows an overview of our approach. Firstly, we preprocessed the data and the aforesaid R-BPM was given, and it was proven to be complicated and unpractical. Secondly, a random-walk method was proposed to simulate massive paths, and for these simulated paths, a path-level BPM (P-BPM) was proposed. Thirdly, each P-BPM could be solved with the Gurobi solver. Thus, each path had an optimal layout of AVI sensors. Finally, the results of the P-BPMs of all of the paths were integrated using the PageRank method, and then we obtained the optimal AVI sensor allocation for the whole network. The remainder of this section discusses each step.

3.1. Random-Walk Method

Random-walk-based sampling methods have been widely employed to characterize social networks. In consideration of the heterogeneous distribution of vehicle trajectories, we developed a data-driven constrained random-walk method (D-RWM) to generate a certain number of vehicle routings, avoiding influential bias compared with the practical status.

The data-driven initialization step aimed at obtaining the initial path set $\mathrm{\Phi }=\left\{\varphi ,V_o,V_d\right\}$, ${\varphi }^i\in \varphi , v_o^i\in V_o,{\tilde{v}}_d^i\in V_d$, where ${\varphi }^i$ is the $i$th path generated from the D-RWM, $v_o^i$ is the origin node, and ${\tilde{v}}_d^i$ is the virtual destination node of ${\varphi }^i$, $v_o^i,{\tilde{v}}_d^i\in C$. The path of the ith sample ${\varphi }^i$ is a series of chronologically ordered nodes, $v_o^i,\dots ,\to ,v_x^i,\dots ,\to ,v_d^i$. ${\tilde{v}}_d^i$ may not be the true destination node $v_d^i$ of ${\varphi }^i$, as the random-walk process was partially constrained when derived from ${\tilde{v}}_d^i$. The steps were as follows. Firstly, a data-driven initialization method was introduced to initialize $v_o^i$ and ${\tilde{v}}_d^i$. Then, some characteristics of candidate nodes were extracted from traffic network structure data and massive trajectory data. Finally, we declared some constraints and regulations and expatiated the simulation process and results; thus, ${\varphi }^i$ was decided.

3.1.1. Transferring Probability Calculation

One of the key steps in random walking is to confirm the probability of moving to the next node while ‘walking’. We assumed that a walking agent starts a walk from one node to its neighbor based on a probability distribution function, which is called the transferring probability. The transferring probability was calculated using two indexes: betweenness centrality (denoted as BC, global and static) and transfer connectivity (denoted as TC, local and dynamic).

BC is a typical measure in complex network theory, and it considers both the travel path and the link weight. It represents the global importance of a node in connecting others through through-movement, which is also called global choice [31,32]. For node i ($v_i$), BC is defined as the number of the shortest paths passing $v_i$ compared to the total number of the shortest paths between any two nodes in the network [33]. The definition is represented as:

$${\mathrm{BC}}_i={\displaystyle \sum }_{i\ne j\ne k}^n\frac{pat{h}_{jk}\left(i\right)}{pat{h}_{jk}}$$

(4)

where $pat{h}_{jk}$ denotes the total number of the shortest paths from $v_j$ and $v_k$, and $pat{h}_{jk}\left(i\right)$ is the number of those paths that pass $v_i$.

TC is derived from complex network theory [34], and it considers traffic characteristics. TC of $v_i$ is defined as the number of traces passing $v_i$ compared to the total number of traces between any two nodes around $v_i$ within a specific topological connection range; three phases of consecutive nodes of node $i$ were used. The definition is represented as:

$${\mathrm{TC}}_i={\displaystyle \sum }_{\begin{array}{c}i\ne j\ne k\\ j,kϵ\mathbb{T}\left(v_i,𝓆\right)\end{array}}\frac{tri{p}_{jk}^{v_i,𝓆}\left(i\right)}{tri{p}_{jk}^{v_i,𝓆}}$$

(5)

where $tri{p}_{jk}^{v_i,𝓆}$ is the total number of traces from $v_j$ and $v_k$, and $tri{p}_{jk}^{v_i,𝓆}\left(i\right)$ is the number of those traces that pass $v_i$; $\mathbb{T}\left(v_i,𝓆\right)$ is a set of encoding nodes that link directly and indirectly to $v_i$, and $𝓆$ is the proximity of nodes to $v_i$ in $\mathbb{T}\left(i,𝓆\right)$.

Then, based on a logistic function, BC and TC were considered to calculate the transferring probability as follows:

$${\mathrm{p}}_{v_{now},v_{next}}^i=\frac{\exp \left({\widehat{\mathrm{BC}}}_i\right)+\exp \left({\widehat{\mathrm{TC}}}_i\right)}{{{\displaystyle \sum }}_{nextϵ\overline{\mathbb{T}}\left(v_i,1\right)}\left(\exp \left({\widehat{\mathrm{BC}}}_{next}\right)+\exp \left({\widehat{\mathrm{TC}}}_{next}\right)\right)}$$

(6)

$${\widehat{\mathrm{BC}}}_i=\frac{{\mathrm{BC}}_i}{{{\displaystyle \sum }}_{nextϵ\overline{\mathbb{T}}\left(v_i,1\right)}{\mathrm{BC}}_{next }}, {\widehat{\mathrm{TC}}}_i=\frac{{\mathrm{TC}}_i}{{{\displaystyle \sum }}_{nextϵ\overline{\mathbb{T}}\left(v_i,1\right)}{\mathrm{TC}}_{next }}$$

(7)

where ${\mathrm{p}}_{v_{now},v_{next}}^i$ is the probability of transferring from $v_{now}$ to $v_{next}$, $v_{now}$ is the node at which the agent stays at the current time, $v_{next}$ is the node adjacent to $v_{now}$, i.e., the agent can walk from $v_{now}$ to $v_{next}$ (but this does not mean that the agent can walk from $v_{next}$ to $v_{now}$), ${\widehat{\mathrm{BC}}}_i$ and ${\widehat{\mathrm{TC}}}_i$ are the normalized ${\mathrm{BC}}_i$ and ${\mathrm{TC}}_i,$ respectively, and $\overline{\mathbb{T}}\left(v_i,1\right)$ is a set of encoding nodes, as well as the nodes adjacent to $v_i$.

On the aspect of the influence of trip purpose on random walking, we utilized the deviation angle to modify the transferring probability as follows:

$${\hat{\mathrm{p}}}_{v_{now},v_{next}}^i=f\left(\Delta {\theta }_{walking}^i\right){\ast \mathrm{p}}_{v_{now},v_{next}}^i$$

(8)

$$\begin{array}{cc}f\left(\Delta {\theta }_{walking}^i\right)\hfill & =\left\{\begin{array}{c}\exp \left(\cos \left({\theta }_{walking}^i\right)-1\right),0<{\theta }_{walking}^i\le {\theta }_{threshold}\\ 0, {\theta }_{walking}^i>{\theta }_{ threshold}^i\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{c}\exp \left(\frac{\left(x_{v_{next}}-x_{v_{now}}\right)\left(x_{{\tilde{v}}_d^i}-x_{v_{now}}\right)+\left(y_{v_{next}}-y_{v_{now}}\right)\left(y_{{\tilde{v}}_d^i}-y_{v_{now}}\right)}{l_{v_{now},v_{next}}+l_{v_{now},{\tilde{v}}_d^i}}-1\right),0<{\theta }_{walking}^i\le {\theta }_{ threshold}^i\\ 0, {\theta }_{walking}^i>{\theta }_{ threshold}\hfill \end{array}\right.\hfill \end{array}$$

(9)

where ${\hat{\mathrm{p}}}_{v_{now},v_{next}}^i$ is the modified transferring probability, ${\theta }_{walking}^i$ is the angle between the direction of the destination and the current walking direction, i.e., the angle between the vector from $v_{now}$ to $v_{next}$ and the vector from $v_{now}$ to ${\tilde{v}}_d^i$, ${\theta }_{threshold}$ is an angle threshold to ensure that the agent walks toward the destination and avoids a circular path, $x_{v_{now}}, y_{v_{now}}$ and $x_{v_{next}}, y_{v_{next}}$ are the position coordinates of $v_{now}$ and $v_{next},$ respectively, and $l_{v_{now},v_{next}}$ and $l_{v_{now},{\tilde{v}}_d^i}$ are the straight-line distances of $v_{now}$ to $v_{next}$ and $v_{now}$ to ${\tilde{v}}_d^i$, respectively. Thus, according to Equations (6)–(9), the modified transferring probability function could be rewritten as follows:

$$\begin{array}{c}{\hat{\mathrm{p}}}_{v_{now},v_{next}}^i\hfill \\ =\left\{\begin{array}{c}\exp \left(\begin{array}{c}\frac{\left(x_{v_{next}}-x_{v_{now}}\right)\left(x_{{\tilde{v}}_d^i}-x_{v_{now}}\right)+\left(y_{v_{next}}-y_{v_{now}}\right)\left(y_{{\tilde{v}}_d^i}-y_{v_{now}}\right)}{l_{v_{now},v_{next}}+l_{v_{now},{\tilde{v}}_d^i}}\\ -1\end{array}\right)\\ \ast \frac{\exp \left({\widehat{\mathrm{BC}}}_i\right)+\exp \left({\widehat{\mathrm{TC}}}_i\right)}{{{\displaystyle \sum }}_{nextϵ\overline{\mathbb{T}}\left(v_i,1\right)}\left(\exp \left({\widehat{\mathrm{BC}}}_{next}\right)+\exp \left({\widehat{\mathrm{TC}}}_{next}\right)\right)}, 0<{\theta }_{walking}^i\le {\theta }_{threshold}\\ 0, {\theta }_{walking}^i>{\theta }_{threshold}\end{array}\right.\hfill \end{array}$$

(10)

3.1.2. Simulation Process

We introduced the procedure of implementing the random-walk simulation proposed by Xuebin [35]. To begin with, an OD pair ($v_o^i,{\tilde{v}}_d^i$) in ${\varphi }^i$ will be selected in sequence, while $v_o^i$ is the starting node of the walk and ${\tilde{v}}_d^i$ is used to set the walking rules. Then, two stopping criteria for the agent were set: arriving at ${\tilde{v}}_d^i$ or reaching a length threshold for the walking journey. The length threshold is calculated with the following formula:

$$l_{max}=E\left(l\right)+S\left(l\right)\ast \sqrt{2\ast \log \left(\frac{1}{1-R_1}\right)}\ast \cos \left(2\ast \pi \ast R_2\right)$$

(11)

where $E\left(l\right)$ is the expectation and $S\left(l\right)$ is the standard deviation of the trip distance of the residents, and $R_1$ and $R_2$ are randomly generated numbers ranging from 0 to 1.

Next, the second point is selected according to the transferring probability and other rules. At the same time, the nodes contained in the walking path are checked for duplication to avoid a repeated selection of road segments. When a valid node—namely, not a repeated node or one in a circular path—is added to the walking path, the total length of the current walking track is calculated. On the condition that the total length exceeds the threshold length or the walking agent arrives at ${\tilde{v}}_d^i$, the random walking trip will stop. The nodes that the walking agent passes, one after the other, will be assigned to ${\varphi }^i$. Algorithm 1 shows the random-walk process in detail.

Algorithm 1: The pseudo-code for the random walk
Input: The surveyed OD matrix, observed trajectory data of the vehicles Output: The path set $\mathsf{\Phi }=\left\{\varphi ,V_o,V_d\right\}$
1:	Initialization of $V_o$ and $V_d$ in $\mathsf{\Phi }$, the length threshold $l_{max}$
2:	Calculate the transferring probability ${\mathrm{p}}_{v_{now},v_{next}}^i$ between each pair of nodes;
3:	for$i=1 to n_r$do
4:	initialize ${\varphi }^i$ to null and add $v_o^i$ into ${\varphi }^i$; walking agent begins at $v_o^i$; ${\hat{\mathrm{p}}}_{v_{now},v_{next}}^i$;
5:	while the total length of ${\varphi }^i$ does not exceed $l_{max}$ and the walking agent is not at $v_d^i$ do
6:	select one of the agent’s adjacent nodes according to the ${\hat{\mathrm{p}}}_{v_{now},v_{next}}^i$
7:	if ${\varphi }^i$ is not a repeated or circular path do
8:	add the selected adjacent node into ${\varphi }^i$;
9:	else do
10:	select another one of the agent’s adjacent nodes according to the ${\mathrm{p}}_{v_{now},v_{next}}^i$, return to line 7;
11:	end if
12:	let walking agent move to the last added node;
13:	return  ${\varphi }^i$
14:	end for

3.2. Formulating the P-BPM of Each Simulated Path

With the random-walk method, a mass of data-driven paths were produced. Then, a path-level BPM of these paths was built, instead of one at the region level. It is apparent that the P-BPM is simpler due to its lower dimensions, and it is more robust because the drawback of the heterogeneous distribution of paths no longer exists. The elements of the path were definite in the P-BPM, meaning that the node, link, and their topological relation were already known. A detailed description is provided in the subsequent section.

3.2.1. Problem Restatement

To better understand the P-BPM, we first introduce two data-driven parameters, the drop-fraction at the node (DN) and the drop-fraction at the link (DL), which are both calculated based on massive trajectory data to produce the ground-truth vehicle paths. The definitions are provided below, with Figure 4 as an illustration.

DN: The accumulated proportion of the traffic flow that does not pass through node ${\mathit{v}}_{\mathit{i}}$ in a path is denoted as ${\phi }_i$. In Figure 4,

$${\phi }_1=\frac{0}{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_0}=0, {\phi }_2=\frac{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_0-\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_3}{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_0}=\frac{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_1}{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_0}, {\phi }_d=\frac{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_4}{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_0}=\frac{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_1\mathit{+}\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_2}{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_0}.$$

DL: The proportion of the traffic flow that does not pass through link ${\mathit{l}}_{i\to j}$ from node ${\mathit{v}}_{\mathit{i}}$ to node ${\mathit{v}}_{\mathit{j}+\mathbf{1}}$ is denoted as ${\omega }_{i\to j}$. In Figure 4,

$${\omega }_{o\to 1}=\frac{0}{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_0}=0,{\omega }_{1\to 2}=\frac{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_1}{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_0},{\omega }_{2\to d}=\frac{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_2}{\mathit{f}\mathit{l}\mathit{o}{\mathit{w}}_0}$$

Visibly, ${\phi }_i=\sum {\omega }_{i-1\to i}$, since ${\phi }_i$ is global and ${\omega }_{i-1\to i}$ is local. In particular, once an AVI sensor is established at ${\mathit{v}}_{\mathit{i}}$, ${\phi }_i$ will be reset to 0.

The developed P-BPM is formulated according to the example exhibited in Figure 5.

In this example, the selected path contains six nodes ($v_1, v_2, v_3, v_4, v_5,$and $v_6$) and five links ($l_{12},l_{23},l_{34},l_{45},$ and $l_{56}$), while the DL of each link is 0%, 10%, 8%, 9%, and 6%, and the distance of each link is 600, 800, 1500, 700, and 1000 m. Two allocation scenarios for AVI sensors are given as samples. In Scenario 1, an AVI sensor was established at$v_3$, since the distance of $l_{34}$ was at its maximum, while in Scenario 2, an AVI sensor was established at$v_4$, since ${\phi }_5$ was at its maximum. It is noticeable that these two scenarios lead to different detection performance, while the cumulative $\mathrm{RF}$s of each link are 100%, 90%, 100%, 91%, and 85.5% in Scenario 1 and 100%, 90%, 82.8%, 100%, and 94% in Scenario 2. As for the detection error, as shown in Figure 5, we get:

$$detection error \left(\mathrm{Scenario} 1\right)=4.36\%, detection error \left(\mathrm{Scenario} 2\right)=5.68\%$$

The allocation in Scenario 1 is superior to that in Scenario 2; thus, $v_1, v_3,$ and $v_6$ are the suggested nodes for locating the AVI sensors in this example. Therefore, we developed a P-BPM with the aim of finding the optimal allocation scenario with a minimal $detection error$, i.e., a maximal detection rate.

3.2.2. Formulating the P-BPM

In each definite artificial path obtained in Section 3.1, a P-BPM was reformulated as follows.

$$\min f=\left(1-{\displaystyle \prod }_{i\in {\mathit{N}}_{\mathit{r}}}{\left(1-{\phi }_i·\frac{l_{i\to i+1}^r}{l}\right)}^{a_{i\to i+1}^r·\left(1-x_i\right)}\right)+K·{\displaystyle \sum }_{i\in C}{x}_i$$

(12)

This was subject to:

$${\displaystyle \sum }_{i\in {\mathit{N}}_{\mathit{r}},i\ne 1}{x}_{\mathrm{i}}\ge 2$$

(13)

$${\phi }_i{}^{a_{i\to i+1}^r·\left(1-x_i\right)}>{\epsilon }_q,\forall i\in {\mathit{N}}_{\mathit{r}}$$

(14)

$${\left(1-\frac{l_{j\to k}^r}{l^r}\right)}^{a_{j\to k}^r·\left(1-x_i\right)}>{\epsilon }_l,\forall i,j\in {\mathit{N}}_{\mathit{r}}$$

(15)

$$\left({\phi }_{i-1}+{\omega }_{i-1\to i}-{\phi }_i\right)·\left(1-x_i\right)=0$$

(16)

$${\phi }_i·x_i=0$$

(17)

$$x_{\mathrm{i}}, a_{j\to k}^r\in \left\{0,1\right\} \forall i\in C,j\in {\mathit{N}}_{\mathit{r}},k\in {\mathit{N}}_{\mathit{r}}$$

(18)

Let $\mathrm{K}$ be the penalty factor in the objective function (12), and it decides the priority between the maximum performance and minimum cost. The objective function (12) aims at minimizing the detection rate and the count of AVI sensors, while minimum detection rate refers to the maximum performance, and the minimum count of AVI sensors refers to the minimum cost. The default value of $\mathrm{K}$ is set to $\frac{1}{2n_c}$, where $n_c$ is the number of candidate nodes.

Let ${\mathit{\epsilon }}_{\mathit{q}}$ represent the maximum reliable DN, and let ${\mathit{\epsilon }}_{\mathit{l}}$ represent the maximum of reliable links in proportion to the total links of path r. ${\mathit{\epsilon }}_{\mathit{q}}$ and ${\mathit{\epsilon }}_{\mathit{l}}$ are both constant and are the critical criteria for whether or not to place an AVI sensor. Constraints (13) ensure that there are at least two AVI sensors in each path to be used to detect the origin and destination, which is an essential requirement for observing where a vehicle enters and leaves the study area. Constraints (14) ensure that if, the drop-fraction at node $i$ is greater than ${\mathit{\epsilon }}_{\mathit{q}}$, an AVI sensor is required, and constraints (15) ensure that, if the distance between node j and node k exceeds ${\mathit{\epsilon }}_{\mathit{l}}$ (the distance of path r), an AVI sensor is required at node j. Constraints (16) indicate the correlations between the DN values of the consecutive points in order to determine if an AVI sensor is required. Generally speaking, ${\phi }_i={{\displaystyle \sum }}_{j=\max \left\{i|x_i=1\right\}}^i{\omega }_{j-1\to j}$. Constraints (17) indicate that, once an AVI sensor is established at${\mathit{v}}_{\mathit{i}}$, ${\phi }_i$ will be reset to 0. Constraints (18) define the binary variables $x_{\mathrm{i}}$ and $a_{\left(j,k\right)}^r$.

In particular, constraints (16) can be linearized using the big-M method [21].

$${\phi }_{i-1}+{\omega }_{i-1\to i}-{\phi }_i\le x_i·M$$

(19)

where M is a series of constants that are large enough to create no limits on the original constraint and variable.

Finally, as a powerful and user-friendly commercial solver in mixed-integer linear programming, the Gurobi solver was used to solve the P-BPM, with Python 3.6 as a modeling language [36].

3.3. Calculating the Deployment Score

Once candidate node ${\mathit{v}}_{\mathit{i}}$ is allocated an AVI sensor, the score of $v_i$ increase by one; after all of the random-walk paths have obtained their optimal allocations, each candidate node receives its final score, and the cumulative score is the deployment score. The one with the highest deployment-score should be given a higher priority for allocation of an AVI sensor. Let the vector $S=\left(s\left(v_1\right),s\left(v_2\right),\dots ,s\left(v_n\right)\right)$, where $s\left(v_i\right)$ represents the number of deployment scores of $v_i$.

We assumed that the simulation paths based on random walks closely approximated the realistic paths. Then, the set of the candidate nodes’ deployment scores was the solution with approximate Pareto global optimality. Theorem 1 and Proof are provided in the following.

Theorem 1.

Among a certain number of single-path AVI-LPs, the candidate node $v_i$(or a set of nodes denoted as $V\left(i to j\right)$) with the maximum time that is regarded as the optimal solution (i.e., $v_i$ is one of the nodes in the optimal layout) is the solution of this location problem at the regional level with approximate Pareto global optimality.

Proof 1.

$\mathsf{\chi }$ denotes the number of single-path AVI-LPs (denoted as single problems). Assume that each candidate node receives the same score after solving $\mathsf{\chi }-1$ single problems. In the $\mathsf{\chi }\mathrm{th}$ single problem, ${\mathrm{v}}_{\mathrm{i}}$ (or $\mathrm{V}\left(\mathrm{i} \mathrm{to} \mathrm{j}\right)$) is the optimal solution at the path level. If ${\mathrm{v}}_{\mathrm{i}}$ (or $\mathrm{V}\left(\mathrm{i} \mathrm{to} \mathrm{j}\right)$) is not chosen as the optimal solution at the regional level, the objective function’s value will be worse because the objective function’s value for the $\mathsf{\chi }\mathrm{th}$ single problem is worse, while the values of the objective function for the other $\mathsf{\chi }-1$ single problems remain the same. Hence, ${\mathrm{v}}_{\mathrm{i}}$ or $\mathrm{V}\left(\mathrm{i} \mathrm{to} \mathrm{j}\right)$) with the maximum deployment-score is the Pareto optimal solution for these $\mathsf{\chi }$ paths at the regional level. We call ${\mathrm{v}}_{\mathrm{i}}$ (or $\mathrm{V}\left(\mathrm{i} \mathrm{to} \mathrm{j}\right)$) the approximate Pareto optimal solution due to the assumption that the simulation paths closely approximated realistic paths. □

To avoid ignoring the effects of the topological characteristics of the road network in the AVI-LP, the well-known PageRank algorithm was employed for further enhancement. Though it is widely known in search engine optimization, PageRank can also be applied to the measurement of a road network as follows:

$${\gamma }^j=\frac{1-q}{n}\ast 1_n+qA{\gamma }^{j-1}$$

(20)

where ${\gamma }^j=\left(P{r}^j\left(v_1\right),P{r}^j\left(v_2\right),\dots ,P{r}^j\left(v_n\right)\right)$ is the PageRank vector in the $j$th iteration, $P{r}^j\left(v_i\right)$ is the deployment score of $v_i$ in the $j$th iteration, n is the total number of candidate nodes, $q$ is the decay factor (this normally takes the value of 0.85), $1_n$ is an n-dimensional vector whose entries are 1 and $A$ stands for the adjacency matrix indicating the adjacent relations between candidate nodes. Letting ${\gamma }^0=S$, we can get the final $\gamma$, i.e., the modified deployment score of each candidate node. Thus, the priority of candidate nodes for which an AVI sensor is suggested can be obtained.

4. Experiments

4.1. Dataset

(1) Road Network: In our experiments, a part of the road network of Chengdu city in China, with a total length of 257 km, was used, as shown in Figure 6. This city-wide network graph covers a 3.5 × 5.2 km spatial range containing 587 nodes and 1397 road links.

(2) GPS Dataset: The GPS dataset was collected in Chengdu city from October 1st to October 31st, 2016, with support from Didi Chuxing. In the third quarter of 2016, Didi Intelligent Travel generated a total of 1 billion GPS trajectory records, which contained 35,450 cars and 202,505 orders in a region of 65 km² in the northeast section of the Second Ring Area in Chengdu. Each point of the GPS trajectories of the trips was matched to a real road to ensure that the data could correspond to actual road information, as shown in Figure 7.

(3) AVI Dataset: Some virtual AVI sensors could be assumed to be located at the intersections of the road network by reference to the GPS trajectory data. If an intersection was assumed to be a location for an AVI sensor, the recorded GPS data within this intersection were valid as observed AVI data; otherwise, they were abandoned.

4.2. Evaluation Methodology

4.2.1. Implementation Procedure

A path-inferring algorithm used to infer the whole path is presented in Algorithm 2. Some path sets were defined as follows (as shown in Figure 8), where $\psi =1,2,..,n_{\mathrm{evaluation}}$, $n_{\mathrm{evaluation}}$ is the total number of evaluation samples, ${\varphi }_{\mathrm{shortest}}^{\psi },{\varphi }_{\mathrm{popular}}^{\psi }$, and ${\varphi }_{\mathrm{inferred} }^{\psi }$ denote the $\psi$th path, and $\forall v_i, v_j\in C$.

The ground-truth path set: ${\mathsf{\Phi }}_{true }=\left\{{\varphi }_{true }^{\psi }\right\}$ contains the true paths of vehicles from the GPS data, and it is used as a reference for evaluation in the next section;

The shortest-path path set: ${\mathsf{\Phi }}_{\mathrm{shortest}}=\left\{{\varphi }_{\mathrm{shortest}}^{v_i\to v_j}\right\}$ contains the shortest paths between two arbitrary candidate nodes derived from the road map, and it serves as supplementary path data;

The most-popular path set: ${\mathsf{\Phi }}_{\mathrm{popular}}=\left\{{\varphi }_{\mathrm{popular}}^{v_i\to v_j}\right\}$ contains the most common paths between two arbitrary candidate nodes derived from ${\varphi }_{true }^{\psi }$, and it serves as supplementary path data;

The detected path set: ${\mathsf{\Phi }}_{\mathrm{detected} }=\left\{{\varphi }_{\mathrm{detected} }^{\psi }\right\}$ contains the paths detected by AVI sensors, and it is obtained from ${\mathsf{\Phi }}_{true }$, which may be incomplete and may need to be complemented using ${\mathsf{\Phi }}_{\mathrm{shortest}}$ or ${\mathsf{\Phi }}_{\mathrm{popular}}$;

The inferred path set: ${\mathsf{\Phi }}_{\mathrm{inferred} }=\left\{{\varphi }_{\mathrm{inferred} }^{\psi }\right\}$ contains the paths inferred from ${\mathsf{\Phi }}_{\mathrm{detected} }$, ${\mathsf{\Phi }}_{\mathrm{shortest}}$, and ${\mathsf{\Phi }}_{\mathrm{popular}}$, where the missing part of path ${\varphi }_{\mathrm{detected} }^{\psi }$ is interpolated with the path in ${\mathsf{\Phi }}_{\mathrm{shortest}}$ or ${\mathsf{\Phi }}_{\mathrm{popular}}$. The details are shown in Algorithm 2.

Algorithm 2: The pseudo-code for inferring the whole detected path
Input: The candidate graph $\mathrm{G}\prime$, ${\mathsf{\Phi }}_{true }$, ${\widehat{\mathrm{BC}}}_i$ and ${\widehat{\mathrm{TC}}}_i$ of each candidate node Output: ${\mathsf{\Phi }}_{\mathrm{inferred} }$
1:	Initialize ${\mathsf{\Phi }}_{\mathrm{shortest}}$ and ${\mathsf{\Phi }}_{\mathrm{popular}}$ from the road map and ${\mathsf{\Phi }}_{true}$, respectively
2:
3:	for$\psi =1 \mathrm{to} n_{\mathrm{evaluation} }$ do
4:	initialize ${\varphi }_{\mathrm{detected} }^{\psi }$ from ${\varphi }_{\mathrm{true} }^{\psi }$, ${\varphi }_{\mathrm{inferred} }^{\psi }$=${\varphi }_{\mathrm{detected} }^{\psi }$
5:	for each segment from $v_{{\mathrm{x}}_1 }^{\psi }$ to $v_{{\mathrm{x}}_2 }^{\psi }$ in ${\varphi }_{\mathrm{detected} }^{\psi }$ do
6:	if $v_{{\mathrm{x}}_1 }^{\psi }$ is not adjacent to $v_{{\mathrm{x}}_2 }^{\psi }$do
7:	remove the path ${\varphi }_{\mathrm{shortest}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}$ $\leftarrow v_{{\mathrm{x}}_1 }^{\psi }$ to $v_{{\mathrm{x}}_2 }^{\psi }$ from ${\mathsf{\Phi }}_{\mathrm{shortest}}$
8:	remove the path ${\varphi }_{\mathrm{popular}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}$ $\leftarrow v_{{\mathrm{x}}_1 }^{\psi }$ to $v_{{\mathrm{x}}_2 }^{\psi }$ from ${\mathsf{\Phi }}_{\mathrm{popular}}$
9:	calculate the total value $valu{e}_{\mathrm{shortest}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}$ of the nodes in ${\varphi }_{\mathrm{shortest}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}$ except $v_{{\mathrm{x}}_1 }^{\psi }$ and $v_{{\mathrm{x}}_1 }^{\psi }$
10:	calculate the total value $valu{e}_{\mathrm{popular}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}$ of the nodes in ${\varphi }_{\mathrm{popular}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}$ except $v_{{\mathrm{x}}_1 }^{\psi }$ and $v_{{\mathrm{x}}_1 }^{\psi }$
11:	if $valu{e}_{\mathrm{shortest}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}\ge valu{e}_{\mathrm{popular}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}$ do
12:	insert ${\varphi }_{\mathrm{shortest}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}$ to ${\varphi }_{\mathrm{detected} }^{\psi }$ between $v_{{\mathrm{x}}_1 }^{\psi }$ and $v_{{\mathrm{x}}_2 }^{\psi }$ as ${\varphi }_{\mathrm{inferred} }^{\psi }$
13:	else do
14:	insert ${\varphi }_{\mathrm{popular}}^{v_{{\mathrm{x}}_1 }^{\psi }\to v_{{\mathrm{x}}_2 }^{\psi }}$ to ${\varphi }_{\mathrm{detected} }^{\psi }$ between $v_{{\mathrm{x}}_1 }^{\psi }$ and $v_{{\mathrm{x}}_2 }^{\psi }$ as ${\varphi }_{\mathrm{inferred} }^{\psi }$
15:	end if
16:	end if
17:	end for
18:	return ${\varphi }_{\mathrm{inferred} }^{\psi }$

4.2.2. Performance Metrics

Metric 1,F1 score of paths: We evaluated the results based on the F1 measure, which indicates both accuracy and recall. Accuracy is defined as the percentage of the length of the matched segments of the total paths $S_{matched}$ to the length of the inferred segments of the total paths $S_{\mathrm{inferred}}$, where $S_{matched}$ will be the total length of the common edges between the inferred paths and the ground-truth paths $S_{\mathrm{truth}}$. The recall is measured according to the ratio between the length of the matched segments and the length of the ground-truth paths. Hence, the F1 score of the paths is computed as follows:

$$S_{matched}=\Vert S_{\mathrm{inferred} }\cap S_{\mathrm{truth}}\Vert$$

(21)

$$accuracy=\frac{S_{matched}}{\Vert S_{\mathrm{inferred} }\Vert }$$

(22)

$$recall=\frac{S_{matched}}{\Vert S_{\mathrm{truth}}\Vert }$$

(23)

$$F1 score=2\ast \frac{accuracy\ast recall}{accuracy+recall}$$

(24)

where $\Vert S\Vert$ is the total length of edges, $\mathrm{s}\in S$.

Metric 2,F1 score of nodes: Meanwhile, the F1 score of the nodes is also computed in the same was as Metric 1.

Metric 3,observability: Observability evaluates the scope of service of the allocation of AVI sensors, and it is computed as follows:

$$observability=\frac{S{c}_{matched}}{S{c}_{truth}}$$

(25)

where $S{c}_{matched}$ represents the quantities of the total matched nodes of all detected paths, and $S{c}_{truth}$ represents the quantities of the total nodes of all ground-truth paths.

4.3. Experimental Results

With Equations (4) and (5), the BC and TC of the candidate nodes were calculated, as shown in Figure 9, where the radius of the circle represents the value of the BC or TC.

The P-BPMs were solved by the commercial Gurobi solver. The time spent was about 20.34 s per 10,000 P-BPMs, running on a personal workstation with 3.50 GHz CPU and 16 GB of RAM. Then, the deployment scores of all candidates were globally optimized using PageRank. Finally, we estimated the performance when allocating AVI sensors with different coverage rates.

Figure 10d presents the distribution of the deployment scores of the candidates with a heat map. According to the deployment scores, we can make decisions on allocating diversified proportions of AVI sensors. Figure 10a–c provide examples of the results when deploying 10%, 30%, and 50% of the AVI sensors among the candidates, respectively. Some conclusions can be drawn.

The assessment metrics for the P-BPM are shown in

Table 1. Assessment metrics of the P-BPM with specific proportions of AVI sensors.

Deployment Proportion of AVI Sensors	Link			Node			Observability
	$\mathit{A}\mathit{c}\mathit{c}\mathit{u}\mathit{r}\mathit{a}\mathit{c}\mathit{y}$	$\mathit{R}\mathit{e}\mathit{c}\mathit{a}\mathit{l}\mathit{l}$	$\mathit{F}1 \mathit{s}\mathit{c}\mathit{o}\mathit{r}\mathit{e}$	$\mathit{A}\mathit{c}\mathit{c}\mathit{u}\mathit{r}\mathit{a}\mathit{c}\mathit{y}$	$\mathit{R}\mathit{e}\mathit{c}\mathit{a}\mathit{l}\mathit{l}$	$\mathit{F}1 \mathit{s}\mathit{c}\mathit{o}\mathit{r}\mathit{e}$
10%	0.9521	0.4747	0.6335	0.9781	0.3753	0.5425	0.2638
20%	0.9421	0.6092	0.7399	0.9697	0.5807	0.7264	0.4558
30%	0.9485	0.668	0.7839	0.9712	0.6836	0.8024	0.5644
40%	0.9572	0.7652	0.8505	0.9771	0.7979	0.8785	0.6878
50%	0.9543	0.8164	0.88	0.9763	0.8693	0.9197	0.7599
60%	0.9614	0.8483	0.9013	0.9788	0.9086	0.9424	0.8358
70%	0.9632	0.8647	0.9113	0.9791	0.9347	0.9564	0.8788
80%	0.967	0.9238	0.9449	0.9813	0.9695	0.9754	0.929
90%	0.9679	0.9569	0.9624	0.9814	0.9893	0.9853	0.9685

Note: The cells in the grid have a red background if the value is more than 0.80.

, including the three types of metrics mentioned above, from 10% to 90% at 10% intervals. For example, 10% means that 10% of the nodes with the top 10% of deployment scores are selected for deployment of AVI sensors. The fine-grained results are visualized in Figure 11, where the difference between each metric and the proportion of AVI sensors (i.e., the red line where y = x) is represented by a histogram. The following conclusions can be drawn.

$\mathrm{F}1 scores$: As seen in Table 1, the $F1 score$ of the link and node are both high, even with low proportions of AVI sensors; in other words, the performance of the deployment-score is very good. For example, 10% of the AVI sensors lead to $F1 \mathrm{scores}$ of 0.6335 and 0.5425 for the link and node, respectively, meaning that even when AVI sensors are placed at only 10% of the intersections, 63.35% of the total links and 54.25% of the total intersections of the passing vehicles’ paths can be inferred. Figure 11 indicates that the $F1 score$ of the link and node will exceed 0.8 if only approximately 25% or more of the AVI sensors are deployed. The proportion ranging from 10% to 45% of the AVI sensors shows significant superiority compared to the basic line at y = x, where the $F1 score$ is about 0.4 greater than the corresponding AVI sensor proportion, i.e., $\mathit{x}\%$ of the AVIs can monitor more than $\left(x+40\right)\%$ of the vehicles’ traces. This is somewhat counterintuitive, but is of great significance for the development of smart and sustainable cities.

$\mathrm{Accuracy}$ and $\mathrm{recall}$: It is apparent from Table 1 and Figure 11b,c that the $accuracy$ of link and node is extremely high—on average, more than 0.95—regardless of how many AVI sensors are used, while the $recall$ is relatively low. Thus, the following conclusions can be drawn: (a) The suggested layouts hold a high identification accuracy regardless of how many AVI sensors there are (assuming that the proportion of AVI sensors is at least 10%); (b) some segments of the paths may be missed during identification if there are too few AVI sensors because fewer AVI sensors result in lower $recall$ but higher $accuracy$; (c) the marginal cost gets smaller and smaller as the number of AVI sensors increases because a larger AVI sensor proportion leads to a smaller value of $recall$ from which the proportion of AVI sensors is subtracted, i.e., the growth rate of $recall$ rapidly decays as the AVI sensor proportion increases.

$\mathrm{Observability}:$ On the one hand, similarly to the $recall$ and $F1 score$, fewer AVI sensors results in high $observability$, e.g., 10% vs. 0.2638 and 20% vs. 0.4558. As a result, the suggested layout is positive to increase the service coverage of AVI sensors. On the other hand, the general difference between the $observability$ and the proportion of AVI sensors is inconspicuous, ranging from about 15% to 20% greater. That is to say, unlike other metrics, the growth of $observability$ is steadier, meaning that, to rapidly improve the service coverage of AVI sensors, the deployment score is helpful, but not enough.

More specifically, these results confirm that the deployment score is positive to reinforce the identification power of AVI sensors, especially when inferring vehicle paths. If the AVI sensors are placed in the suggested layout, more than 95% of the inferred paths are consistent with the true paths. Counterintuitively, only a minority of AVI sensors can contribute to the identification of a great number of the paths, which is of great significance in practice.

4.4. Comparative Experiments

We compared the path identification performance of the deployment score with other indicators in terms of various purposes. The following indicators were used.

Scenario 1: Betweenness-centrality-based indicator: This method focuses on the priority of the shortest paths, with reference to the complex network theory. The candidate nodes are scored according to their betweenness centrality, and then the layout is determined according to the scores.

Scenario 2: Transfer-connectivity-based indicator: This method is derived from complex network theory. It focuses on the priority of local truth paths and the road network. In the same as in Scenario 1, the candidate nodes are scored according to their transfer connectivity, which was defined in the preceding sections.

Scenario 3: Traffic-volume-based indicator: This method focuses on the priority of traffic volume, the most significant indicator in urban traffic. The candidate nodes are scored according to the traffic volume.

Figure 12 compares the performance of these indicators with the deployment score (computed with the P-BPM), and each item is explained in the following.

Figure 12a shows the $F1 scores$ of these indicators. Obviously, the deployment score outperforms the other indicators in terms of the $F1 score$. It could achieve a high $F1 score$ with a low proportion of AVI sensors. What is striking in the figure is the dominance of the P-BPM at the beginning, indicating that the deployment score performs well in cases in which AVI sensors are sparse. We note that the estimated results can achieve $F1 scores$ for both the link and node of over 70% when the AVI sensor proportion is higher than 20%, while in other scenarios, an AVI sensor proportion of at least 30% is required.

Figure 12b describes the $accuracy$ of these methods. The figure shows that the deployment score is steady and high, while others have sharp fluctuations, especially if the AVI sensor proportion is low. The $accuracy$ of the deployment score maintains a high level, meaning that the layout suggested by the deployment score is robust and effective and can ensure the identification accuracy regardless of what the AVI sensor proportion is.

Figure 12c presents the $recall$ of these methods. As revealed in figure, the deployment score’s recall for both the link and node stands out higher than the others. Thus, its path-reconstruction ability is more powerful in the same condition, and the reconstruction ratio of both the link and node can also be guaranteed. In contrast, the other $recall$ scores are patchy with different proportions of AVI sensors, so the deployment score is more reliable.

Figure 12d shows the $observability$. Generally speaking, compared with other metrics, the $observability$ of all of these methods is low, and the growth rate is also slow. Even in this case, the deployment score still performs better than the others. The $observability$ of the traffic-volume-based method is the worst. That is to say, the traffic-volume-based method just ensures the service coverage at the node level, rather than at the regional level, leading to failure when inferring paths. Thus, the deployment score could generate a more reasonable and homogeneous spatial distribution for AVI sensors.

5. Conclusions and Discussions

AVI sensors show unique advantages for vehicle detection in traffic management. This study developed a data-driven method for solving the AVI-LP for better path reconstruction, the final solution of which is a novel indicator called the deployment score, which reflects the priority of deploying AVI sensors. In consideration of practical and universal feasibility, this research constructed a path-level bi-level programming model (P-BPM), rather than a region-level model, which is complicated and may be trapped in an NP-hard problem when applied. A random-walk method was introduced to simulate the tracks of vehicles according to travel characteristics extracted from finite GPS data. Accordingly, the P-BPM for each path was modeled to find the optimal layout of AVI sensors. Each P-BPM was solved by the Gurobi solver. The total scores of the candidates were obtained (i.e., the deployment scores) once all P-BPMs were solved, and the accumulated scores proved to be the approximate Pareto optimal solutions at the regional level from a data-driven perspective. The PageRank method was also introduced to modify the deployment scores with respect to the topological structural characteristics of the road network. Our proposed deployment score was compared with three baseline indicators based on the shortest paths, the true paths, and the traffic volume, respectively. The experimental results show that the deployment score outperforms the other indicators in terms of performance metrics. To be specific, the layout of AVI sensors suggested by the deployment score has excellent and steady routing detection accuracy, great and reliable routing recall, and satisfactory service coverage.

Enabled by the development of ITS, big data are more and more popular and important in smart and sustainable cities due to their indispensable contributions to the management and operation of environmentally sustainable development. The reasonable and effective deployment of traffic sensors is capable of providing comprehensive traffic monitoring data and, simultaneously, allowing lower costs. As some of the most important traffic sensors, AVI sensors are expected to be placed in the road network, i.e., at segments or intersections, to capture the detailed sequential information of passing vehicles. This study discussed the case of AVI sensors placed at intersections to better infer the full traces of the vehicles. This study can support the decision makers in deploying AVI sensors more practicably and effectively. Sustainability is the trend of future transportation development. Deploying AVI sensors more reasonably makes traffic operation and management more efficient, and thus provides motivation for the development of more sustainable transportation.

This research is a preliminary step toward solving the AVI-LP. Random-walk can effectively reduce the complexity and eliminate the deficiency of input data, while some traffic preferences can also be taken into consideration in the random-walk procedure according to various decision objectives. A limitation of this research is that the candidate nodes were manually selected and some assumptions were made, the intention of which was to make it convenient to model and develop solution methods. As we know, in some cases, we cannot grasp the situations of all of intersections. Thus, decision makers prefer to input the restricted conditions of every intersection into the model to decide on the candidate nodes. Hence, as a further research direction, we will consider taking each node as a candidate node, and their restricted conditions will be used to model the P-BPM.