Synergistic sensor location for link flow inference without path enumeration: A node-based approach
Highlights
► The link observability problem is reformulated using a node-based approach. ► Path enumeration is shown to be unnecessary in the link observability problem. ► The minimum number of sensors needed for observability is theoretically derived. ► The proposed method can also be applied to networks that already have sensors.
Introduction
With the advent of Intelligent Transportation Systems (ITS), sensors are becoming increasingly critical elements of modern transportation systems (e.g. see Ban et al., 2009, Ng and Waller, 2010a). This growing need for (real-time) traffic information has resulted in an interesting class of problems collectively known as sensor location problems. In these problems, sensors are optimally located for purposes such as origin–destination (O–D) matrix estimation (e.g. see Yang and Zhou, 1998, Bianco et al., 2001, Chootinan et al., 2005, Gan et al., 2005, Castillo et al., 2008a, Mínguez et al., 2010), path flow estimation (e.g. see Gentili and Mirchandani, 2005, Castillo et al., 2008b) and, more recently, link flow inference (e.g., Hu et al., 2009, Castillo et al., 2010).
In a recent paper, Hu et al. (2009) introduced the rather interesting problem of determining the smallest subset of links in a network for counting sensors installation, in such a way that it becomes possible to infer the flows on all remaining, non-equipped links (this problem is referred to by Castillo et al., 2010 as the link observability problem). As the authors pointed out, numerous applications can potentially benefit from the ability to determine all link flows in a network, including dynamic traffic assignment (e.g. see Peeta and Ziliaskopoulos, 2001), evacuation planning (e.g. see Ng and Waller, 2010b) and pavement management systems (e.g. see Ng et al., 2011a). In addition, link flow measurements can also be tremendously useful in travel time reliability studies (e.g. see Chen et al., 2002, Lo et al., 2006, Ng and Waller, 2010c, Ng et al., 2011b). The proposed problem was particularly elegant because of its assumption-free character. Particularly, the link observability problem does not require any assumptions on turning proportions, rout choice behavior and O–D matrices, information that is hard – if not impossible – to obtain in practice. Unfortunately, the problem maintained the restrictive requirement of path enumeration that is common in traffic observability problems (Castillo et al., 2008c). In large-scale, real-world networks, it is clearly impossible to enumerate all paths. In such cases, one needs to resort to simplifying assumptions such as “most likely paths” (as indicated by Hu et al. (2009)), destroying the assumption-free nature of the problem. Path enumeration is particularly vexing since at this time it is not clear at all how sensor location decisions for large-scale networks are affected by the failure to enumerate every single path in a network.
The main contributions of this paper are as follows. First, we demonstrate that the link observability problem can be reformulated using a node-based approach. That is, instead of path enumeration, we only require node enumeration, which is obviously a much simpler task for large-scale networks. Second, using this alternative modeling approach, we prove (and refine) the conjecture made in Hu et al. (2009) that “there may be an upper bound on the number of basis links that is governed by the network topology irrespective of the total number of links in the network”. More specifically, we explicitly state this upper bound (see Proposition 2) and show that it is dependent on the total number of links. Third, recognizing that road networks typically already have sensors installed on them, it is demonstrated how the set of inferable link flows can be determined using a node-based approach (Castillo et al., 2010 addressed this question in terms of the path-link incidence matrix), and how all link flows can be made observable though the addition of the smallest number of sensors.
The remainder of this paper is organized as follows. In Section 2, the main results of the paper are developed. In addition to the numerical examples in Section 2, Section 3 provides some more case studies to demonstrate the proposed methods. Section 4 concludes and summarizes the main results in this paper.
Section snippets
A node-based modeling approach
Let G = (V*, E) denote a transportation network, where V* denotes the set of nodes and E the set of links. Following transportation planning practice (and without loss of generality), V* is here further sub-divided into centroids and non-centroids. Centroids are the nodes in V* where traffic originates/is destined to (Sheffi, 1985), and non-centroids denote all other nodes in V*. (Without loss of generality, it is assumed that each network has at least one centroid.) In this paper we shall refer
Another example
In this section we examine one more network to illustrate the power and ease of use of the proposed node-based approach. Consider the fishbone network introduced by Hu et al. (2009) in Fig. 2. The node-link incidence matrix and its reduced row echelon form are shown in Table 5, Table 6, respectively. Note that the node-link incidence matrix – as predicted by Proposition 1 – has full rank. Thus, only (18 − 6)/18 * 100% ≈ 67% of the links needs to be equipped with sensors (the same value as empirically
Summary and conclusions
With the advent of ITS, sensors are becoming increasingly critical elements in the nation’s road transportation network. In a recent paper, Hu et al. (2009) introduced the interesting problem of determining the smallest subset of links in a network for the installation of counting sensors, in such a way that it becomes possible to infer the flows on all remaining links in a traffic network. The problem is particularly elegant because of its limited number of assumptions, a property not found in
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