A Link-Node Discrete-Time Dynamic Second Best Toll Pricing Model with a Relaxation Solution Algorithm

Abstract

Dynamic congestion pricing has become an important research topic because of its practical implications. In this paper, we formulate dynamic second-best toll pricing (DSBTP) on general networks as a bilevel problem: the upper level is to minimize the total weighted system travel time and the lower level is to capture motorists’ route choice behavior. Different from most of existing DSBTP models, our formulation is in discrete-time, which has very distinct properties comparing with its continuous-time counterpart. Solution existence condition of the proposed model is established independent of the actual formulation of the underlying dynamic user equilibrium (DUE). To solve the bilevel DSBTP model, we adopt a relaxation scheme. For this purpose, we convert the bilevel formulation into a single level nonlinear programming problem by applying a link-node based nonlinear complementarity formulation for DUE. The single level problem is solved iteratively by first relaxing the strick complementarity by a relaxation parameter, which is then progressively reduced. Numerical results are also provided in this paper to illustrate the proposed model and algorithm. In particular, we show that by varying travel time weights on different links, DSBTP can help traffic management agencies better achieve certain system objectives. Examples are given on how changes of the weights impact the optimal tolls and associated objective function values.

Appendices

Appendices

1.1 Appendix 1: Definition and properties of a point-to-set map

This appendix provides definitions and properties of a point-to-set map. Interested readers may refer to Facchinei and Pang (2003) for details and proofs.

Definition 1

A map Φ is a point-to-set map from ℝⁿ to ℝ^m if for any x ∈ ℝⁿ, Φ(x) is a subset of ℝ^m (possibly empty). The domain of Φ, denoted by domΦ, the range of Φ, denoted by ranΦ, and the graph of Φ, denoted by gphΦ, are defined as follows:

$$ \mbox{dom} \Phi \equiv \{x \in \mathbb{R}^n: \Phi(x) \neq \emptyset\} $$

(38)

$$\mbox{ran} \Phi \equiv \bigcup_{x \in dom \Phi} \Phi(x) $$

(39)

$$\mbox{ghp} \Phi \equiv \left\{(x,y) \in \mathbb{R}^n \times \mathbb{R}^m: y \in \Phi(x)\right\} $$

(40)

Definition 2

A point-to-set map Φ: ℝⁿ → ℝⁿ is said to be

• (a) closed at at point $\bar{x}$ if

  $$ \begin{array}{cccc} \begin{array}{c} {x^k} \longrightarrow \bar{x}\\ y^k \in \Phi(x^k) \mbox{ } \forall k\\ {y^k} \longrightarrow \bar{y} \end{array} & \Biggr\} & \Longrightarrow & \bar{y} \in \Phi(\bar{x}) \end{array};$$

• (b) closed on a set S if Φ is closed at every point of S.

• (c) upper semicontinuous at a point $\bar{x}$ if for every open set υ containing $\Phi(\bar{x})$, there exists an open neighborhood ${\cal N}$ of $\bar{x}$ such that, for each $x \in {\cal N}$, υ contains Φ(x).

• (d) lower semicontinuous at a point $\bar{x}$ if for every open set υ meeting $\Phi(\bar{x})$, there exists an open neighborhood ${\cal N}$ of $\bar{x}$ such that, for each $x \in {\cal N}$, υ meets Φ(x).

Theorem 3

The following statements are true for a point-to-set map Φ.

• (a) Suppose $\Phi(\bar{x})$ is a closed set. If Φ is upper semicontinuous at $\bar{x}$ , then Φ is closed at $\bar{x}$ ;

• (b) Φ is closed if and only if its graph is a closed set.

1.2 Appendix 2: Proof of Theorem 1

Under condition (b) of Theorem 1, the point-to-set mapping of SOL(y) is upper-semicontinuous (see Definition 2(c) in Appendix 1). To see this, assume $\upsilon = SOL(\bar{y}) + IB(0,\varepsilon)$ is an open set containing $SOL(\bar{y})$, where IB(0,ε) is an open ball with radius ε. We then define another open set ${\cal N} \equiv \{y|\|y-\bar{y}\|_2 < \delta\}$ containing $\bar{y}$. According to condition (b) in the theorem, for any ε > 0, there exists δ > 0 such that $\max_{\forall U \in SOL(y)} \mbox{ } \min_{\forall \bar{U} \in SOL(\bar{y})} \|U-\bar{U}\| < \varepsilon$, which is equivalent to $\bigcup_{y \in {\cal N}}SOL(y) \subseteq \upsilon$.

Then together with condition (a), it implies that the point-to-set mapping SOL(y) is closed on set K _y (see Theorem 3(a) and Definition 2(b) in Appendix 1). Therefore, the graph Φ(y, U) defined in (5) is closed (see Theorem 3(b) in the Appendix). Also under (a), SOL(y) is bounded for any y ∈ K _y . Since K _y is a bounded set, the graph Φ(y, U) is bounded as well. Thus, we proved that graph Φ(y, U) is compact. Together with (c) and the fact that K _y is compact, BiDSBTP must has at least one solution since it is an NLP with a continuous objective function defined on a compact constraint set (i.e. from Weierstrass’ Theorem). □