Mathematical Problems in Engineering

Volume 2016, Article ID 9012724, 10 pages

http://dx.doi.org/10.1155/2016/9012724

## Sensor Location Problem for Network Traffic Flow Derivation Based on Turning Ratios at Intersection

^{1}Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai 200092, China^{2}China Airport Construction Group Corporation, Beijing 100101, China

Received 28 August 2015; Accepted 28 January 2016

Academic Editor: Luca D’Acierno

Copyright © 2016 Minhua Shao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The sensor location problem (SLP) discussed in this paper is to find the minimum number and optimum locations of the flow counting points in the road network so that the traffic flows over the whole network can be inferred uniquely. Flow conservation system at intersections is formulated firstly using the turning ratios as the prior information. Then the coefficient matrix of the flow conservation system is proved to be nonsingular. Based on that, the minimal number of counting points is determined to be the total number of exclusive incoming roads and dummy roads, which are added to the network to represent the trips generated on real roads. So the task of SLP model based on turning ratios is just to determine the optimal sensor locations. The following analysis in this paper shows that placing sensors on all the exclusive incoming roads and dummy roads can always generate a unique network flow vector for any network topology. After that, a detection set composed of only real roads is proven to exist from the view of feasibility in reality. Finally, considering the roads importance and cost of the sensors, a weighted SLP model is formulated to find the optimal detection set. The greedy algorithm is proven to be able to provide the optimal solution for the proposed weighted SLP model.

#### 1. Introduction

Link flow data in the road network is the valuable information in many applications of traffic planning and management, such as the road network planning and the congestion analysis. Placing traffic sensors on every link is usually not realistic, because there are thousands of links in the urban road network normally and the investment on sensors installation and maintenance will be huge. So it is important to identify the subset of links so that the traffic flow on every link in the network can be inferred if sensors are installed on this subset of links to collect flow data. The sensor location problem (SLP) was proposed to address this problem.

The SLP has attracted increasing interest in the past few years for its importance in the area of transportation system analysis. Two main classes of SLP were summarized by Gentili and Mirchandani [1]: sensor location flow-observability problems and sensor location flow-estimation problems. The objective of the first-class SLP is to find the optimal sensor locations so that all link flows (or partial link flows) can be determined uniquely. The second-class SLP is trying to improve the accuracy of the estimation by optimizing the sensors locations. Early research in the literature mainly focused on the second-class SLP as a subproblem of origin-destination (OD) estimation. The “OD Covering Rule” was given by Yang et al. [2, 3] to guarantee the upper-bounded OD estimation error. Several other rules were also proposed to improve the estimation accuracy, like the “maximal flow fraction rule,” the “maximal flow-intercepting rule,” and so on (for details, see [1–5]). The first-class SLP was not formally defined until 2001. Bianco et al. [6] defined SLP as determining the minimum number and locations of counting points to infer all traffic flows in the network. We will focus on the first-class SLP in this paper.

The first-class SLP was proposed in the context of traffic planning and also was developed in this domain in most of the following research. So, essentially, it is a static optimization problem addressing the method of network sensor deployment under the steady-state traffic condition. Generally, in the sensor location flow-observability problems, a system of linear equations will be formulated at first, which represents the relationship of different link flows or the relationship between link flows and route flows (or OD trip) under the law of flow conservation. This system of equations is defined as the flow conservation system in this paper. Then, by deploying the sensors, equations representing the detected link flows will be added. These equations are defined as the detecting equations. The number of the detecting equations is equal to the number of sensors. Basically, the main idea of the first-class SLP is to find the optimal detecting equations so that all the variables are solvable. According to this idea, the SLP was normally designed as an integer linear programming model in the existing studies. The difference among these studies lies in the assumptions on the prior information, which lead to different formulations of the flow conservation system. According to the prior information, three kinds of flow conservation systems are formulated: OD-link based system, route-link based system, and link-link based system [1]. The OD-link based system uses the link choice proportions as the prior information, which is defined as the proportion of trips between one OD pair that uses a given link in the network. Similarly, the route-link based system uses the link-route coefficients as the prior information. When a link is used by a route, the corresponding coefficient will be 1; otherwise, the coefficient will be zero [7]. These two kinds of systems use the OD trips or route flows as the bridge between detected link flows and undetected link flows. So the OD matrix and route flows will be obtained at the same time. However, in the scenarios when the link flow is the only information we care about, the effort used in calculating the OD trips and route flows is not necessary. In particular for the route-based system, enumerating all routes is not easy for a large-scale network. At the same time, if we are only interested in the link flows of one small local network, the whole network still needs to be considered in the SLP since the OD trip and route flow are defined over the network. The third kind of system uses the relationship of the link flows directly. Without using the concept of OD trip or route flow, this kind of system can be used to formulate the SLP for local network. Bianco et al. [6] proposed the first system of this kind in 2001, using the split ratios of outgoing roads at intersections as the prior information. The split ratio of an outgoing link at an intersection is defined as the fraction of the flow on the outgoing link over the total incoming flow of the intersection. With the assumption of symmetrical network and node-based detection (the sensors are located on every in- and out-link of the intersection), a necessary condition was given to guarantee that the number of equations in the system is not less than the number of variables. They also presented a couple of greedy heuristics to find the lower and upper bounds on the number of sensors for random networks. After that, Bianco et al. [8] proved that the SLP proposed in 2001 is NP-hard in the computation complexity. Linear algorithms were invented for several special graphs, including the paths, circles, and combs. Based on the work of Bianco, Morrison et al. [9, 10] gave a stricter necessary condition for general graphs and proved the condition to be also sufficient for the special network with trees as its unmonitored components. Essentially, the network structure-driven logic is used in the link-link based sensor location flow-observability problems. So the traffic flows over the network are calculated based on the network topology and any behavior assumption of the road users is not required [6].

Until now, most discussions on the link-link based sensor location problems used the split ratios as the prior information. The network was assumed to be symmetrical and the detection was assumed to be node based. However, in reality, unidirectional roads normally exist, which leads to an asymmetrical network. At the same time, road-based detection is more reasonable. In this paper, the network is allowed to be asymmetrical and the detection is assumed to be road based. Turning ratios at intersections are used as the prior information. The turning ratio is defined as the ratios of flow turning from an incoming link to an outgoing link over the total flow of the incoming link. Turning ratio is used as a popular parameter in many applications. Many methods were designed to obtain the turning ratios (for details, see [11–13]), which make the estimation of turning ratios from historical data possible. Using turning ratios as the prior information, an integer linear programming model is designed in this paper to address the sensor location flow-observability problem under steady-state traffic conditions.

The present paper is organized as follows. In Section 2, the definition and assumptions of the road network used in this paper are specified. Then, the flow conservation system is formulated in Section 3 based on the turning ratios. After that, the coefficient matrix of the flow conservation system is proven to be full rank in row space. Based on that, in Section 5, the SLP model is given and the greedy algorithm is set up to solve the model. Finally, some conclusions are summarized.

#### 2. Definitions and Assumptions

Suppose the road network is composed of directed roads. The roads are numbered by . Represent the roads set as . Six assumptions are used in this paper.

*Assumption 1. *The road network which we are interested in is taken from a bigger one so that on the boundary there are directed roads whose upstream or downstream intersections are missing.

*Definitions for Three Categories of Roads.* Under Assumption 1, there must be three categories of roads in a network: incoming/outgoing road, which connects two adjacent intersections; exclusive incoming road, whose downstream intersection is in the network but the upstream intersection is not; exclusive outgoing road, whose upstream intersection is in the network but the downstream intersection is not. Figure 1 gives an example of a simple network including only two intersections.