A Review on Popular TA Models

In the literature of traffic planning, many TA models have been proposed. Each is based on different certain assumptions. A number of models can be applied to various kinds of traffic systems, some others are developed for a specific traffic system. In this subsection some popular TA models, that are widely used in traffic planning, are briefly summarized.

One of the simplest models is the all-or-nothing (AON) model, which is based on the assumption that drivers choose a shortest length path to travel without con- sidering other factors. This assumption is reasonable in sparse and uncongested transportation networks where the traveling time on a link is approximately propor- tional to the distance of the link. However, in traffic systems where traffic congestion occurs frequently, a shortest path in terms of length may take more traveling time than other paths with less traffic flow do. In such traffic systems, the AON model does not describe exactly the routing behavior of drivers. The AON model can be solved by assigning all the traffic demands of each origin-destination (O-D) pair to a shortest path connecting them.

In the incremental model, the traffic demands are assigned on separated steps. On each step, a part of the traffic demands is assigned to the network based on the AON model. However, drivers in each step choose the best path in term of traveling

time instead of length of the path. This means drivers choose the path with shortest traveling time according to the current traffic flows on the links in the current step. After each step, the traveling time on each link is updated according to the current traffic flow on the link. The more traffic demands are assigned on a path, the more traveling time it takes on that path, thus drivers would not choose a path with a large traffic flow. This model does not yield an equilibrium, since it depends on the order of O-D pairs to be assigned, and also on the amount of demand to assign in each step. Thus, it is difficult to evaluate this model, however, the incremental model is closer to the real routing behavior than the AON model since it takes the traffic flows on the links into account while the AON does not.

s t t1(x1) = 50(1 + 0.15 (x1/200)4) t2(x2) = 100(1 + 0.15 (x2/300)4) t3(x2) = 300 + x3 Dst = 1000 0 100 200 300 400 500 600 700 800 900 Dst 0 50 100 200 300 400 500 t1(x) t2(x) Equilibrium point x3 = 0, x1 = 1000 − x2 (vehicles) T ra v eling time (s)

Figure 2.5: An example of the user equilibrium model.

Traffic assignment models satisfying the first principle of Wardrop are called user equilibrium (UE) models since they approach to an equilibrium by drivers, i.e., all drivers travel on a shortest traveling time path and they are not willing to change their paths. This assumption is close to the real traffic routing behavior, thus it is used widely in modern TA models. There are a number of models, developed from the UE model, such as the stochastic user equilibrium model and the dynamic user equilibrium model.

Figure 2.5 shows an example of the UE model on a transportation network containing only cars. There are three possible paths from s to t with the traveling time t1(x1),

2.6. TRAFFIC ASSIGNMENT MODELING 23

t2(x2), and t3(x3) where x1, x2, and x3 are the number of cars on the first, the

second, and the third paths, respectively. The number of traffic demands between s and t is Dst = 1000 PCU, i.e. there are in average 1000 cars traveling from the origin

s to the destination t every hour. Drivers choose one of the three possible paths to travel, such that their traveling time is minimum. The optimal solution is x1 ≈ 460

PCU, x2 ≈ 540 PCU, and x3 = 0 PCU. The traveling times on the first path and

second path are equal (around 260 seconds) while the traveling time on the third path is 300 seconds. No drivers choose the third path to travel since its traveling time is larger than those of other paths even when its traffic flow is free. Because all the drivers (either on the first path or on the second path) have to spend the same amount of traveling time from s to t and that is less than those on the third path, they are not willing to change their path to another possible path. This means the system has an equilibrium according to the drivers.

Traffic assignment models satisfying the second principle of Wardrop are called system optimal (SO) models. All SO models reach to a system equilibrium when all the drivers cooperate with each other, so that the total traveling time of all the drivers in the system is minimum. This assumption does not reflect the real traffic behaviors in many cases, however, it can be applied to traffic systems where drivers are willing to follow instructions of a traffic controlling center. The model is also used to estimate a lower bound on the total traveling time of drivers on the system assigned by other models.

Chapter 3

Shortest Path Problem

This chapter presents the classical shortest path (SP) problem, that has a wide range of applications in both research and real-world problems. Various algorithms for the SP problem have been proposed. Among them, Dijkstra’s and A-Star algorithms are most investigated due not only to their efficiency but also to their ability to be implemented from two directions. Algorithms for the SP problem mentioned in this chapter are also reused frequently in a number of algorithms for the k shortest paths problem presented in Chapter 4.

The chapter is divided into 6 sections. Section 3.1 gives a general introduction on the SP problem including a review of popular algorithms for it. Section 3.2 and Section 3.3 present algorithms following the one-directional and the bidirectional approaches, respectively. A comparison between the implementations of various algorithms on real city maps is also presented in Section 3.4. Section 3.5 introduces routing methods on large maps and the last section, i.e. Section 3.6, are conclusions.

3.1 Introduction

Given a graph G(V, E), the SP problem is to find a path in G with the shortest length (weight) from a given source node s to a given destination node t, i.e.

Minimize W (p)

Subject to p ∈ Pst,

where Pst is the set of all possible paths from s to t and W (p) is the length of the

path p. As mentioned in Subsection 2.5.4, the SP problem can be formulated as an IP and solved by an IP solver, however, in this thesis we investigate some dedicated algorithms for it.

If the graph is connected, then there is always a finite shortest path between any pair of distinct nodes. This is stated in Theorem 3.1.1.

Theorem 3.1.1. Given a connected graph G(V, E), there is a finite shortest path from a source node s to a destination node t if and only if there are no negative cycles in the graph G.

Proof. Firstly, we prove that if there is a finite shortest path from s to t, say p∗, then there are no negative cycles in G. Indeed, for a proof by contradiction, we assume that there is a negative cycle c = (v1, v2, . . . , vl, v1), i.e. W (c) < 0. Because

G is connected, there is a path from s to v1, say p1, and a path from v1 to t, say p2.

The path p = p1⊕ p2 is also a possible path from s to t, thus W (p) ≥ W (p∗). Let

k = W (p) − W (p

∗₎

|W (c)| 

,

where dxe gives the next bigger integer of x. The path defined as ¯

p = p1⊕ (c⊕)h⊕ p2

is a path from s to t including h > k times the cycle c. We have W (¯p) = W (p1) + W (p2) + kW (c) < W (p∗).

We can create a sequence of paths from s to t with decreasing lengths, and all of those are less than the length of p∗. This conflicts with the assumption that p∗ is the path with smallest weight. Hence, there is no negative cycle in G.

Secondly, if there are no negative cycles in G, we prove that the kth shortest path is finite, where k ≥ 1. Indeed, if there are no cycles in G, then the number of nodes of any path from s to t does not exceed |V |. This means that all the possible paths are finite. If G contains (positive) cycles, let c = (v1, v2, . . . , vl, v1) is a cycle in G,

then we have a list of k finite paths D = {qi = p1⊕ (c⊕)i ⊕ p2 | i = 1, 2, . . . , k}

where p1 is a path from s to v1 and p2 is a path from v1 to t. We have

W (q1) < W (q2) < . . . < W (qk),

thus the length of the kth _{shortest path, say p}∗

k, should not exceed W (qk). If p∗k is

infinite, then there must be a (positive) cycle, say ¯c appear on pk unlimited number of times. This leads to W (p∗_k) = +∞. This conflicts with the statement that the length of p∗_k does not exceed W (qk).

Principle 3.1.2. (Optimality Principle) If v is a node on a shortest path p from s to t, then the subpath of p from v to t is a shortest path from v to t. In other words, a subpath of a shortest path is also a shortest subpath.

Proof. We have p = RootP ⊕ SubP1 where RootP and SubP1 are the subpaths of p