A Credit-Based Congestion Management Scheme in General Two-Mode Networks with Multiclass Users

Abstract

This paper examines the design of the credit-based congestion management schemes that achieve Pareto-improving outcome in general two-mode networks. It is assumed that transit is a slower but cheaper alternative to driving alone. The distributional welfare effects of congestion pricing on users with the different value of time (VOT) in Liu and Nie (Trans Res Board 2283:34–43, 2012) are used in developing Pareto-improving credit schemes. We show that, similar to the single-mode model, the sufficient and necessary condition for the existence of a discriminatory Pareto-improving credit scheme is the reduction in the total system cost. A sufficient condition for the existence of an anonymous Pareto-improving credit scheme is also derived. A cross-OD subsidization scheme is proposed when the sufficient condition is not satisfied for each origin-destination (O-D) pair. Numerical experiments on the expanded Sioux Falls networks with a log-normal VOT distribution demonstrate that the proposed Pareto-improving scheme can generate positive net revenue in the presence of good transit coverage.

Appendix

Proof

Proof of Proposition 3

Necessity

According to Definition 1, at the tolled equilibrium, for those who stay on highway, Pareto-improving scheme requires that:

$$\beta_{m}\mu_{rs}^{m}-\pi_{rs}^{m}=\beta_{m} c_{rs}^{k}+\nu_{rs}^{k}+o_{rs}^{C}-\pi_{rs}^{m}\leq \beta_{m}\bar{c}_{rs}+o_{rs}^{C}=\beta_{m}\bar{\mu}_{rs}^{m} \Rightarrow\nu_{rs}^{k} \le \beta_{m}\bar{c}_{rs} - \beta_{m} c_{rs}^{k}+ \pi_{rs}^{m},\forall f_{rs}^{km}>0$$

Multiplying $f_{rs}^{km} >0$ on both sides and summing all the inequalities over m and k yields

$$ R_{rs}= \sum\limits_{m}\sum\limits_{k} f_{rs}^{km} \nu_{rs}^{k} \le \bar{c}_{rs}{\sum}_{m} \beta_{m} \sum\limits_{k} f_{rs}^{km} - \sum\limits_{m}\beta_{m}\sum\limits_{k} f_{rs}^{km} c_{rs}^{k}+ \sum\limits_{m}\pi_{rs}^{m}\sum\limits_{k} f_{rs}^{km} $$

(27)

where $R_{rs}={\sum }_{m}{\sum }_{k} \nu _{rs}^{k}f_{rs}^{km}$, defined as the total credits collected between rs; the demand of class m $d_{rs}^{m}={\sum }_{k} f_{rs}^{km}$ here because those users choose highway ($q_{rs}^{m}=0$).

We now turn to those “priced-out” users, the transit users at the tolled equilibrium who use highway at NTE. $q_{rs}^{m}-\bar {q}_{rs}^{m}$ represents the number of class m users tolled off from highway, and the aggregated users can be denoted by $q_{rs} - \bar {q}_{rs}$, where $q_{rs}^{m}$ and $\bar {q}_{rs}$ specifies the number of transit users of class m at tolled and no-toll equilibrium. To ensure Pareto-improving result requires:

$$\beta_{m}\gamma_{rs} + o_{rs}^{T} - \pi_{rs}^{m} \le \beta_{m} \bar{c}_{rs} + o_{rs}^{C}, \forall m \; \text{such that} \; e\le m\le p \Rightarrow 0\le\beta_{m} \bar{c}_{rs}-\beta_{m}\gamma_{rs}+\pi_{rs}^{m}+{\Delta}_{rs} $$

where ${\Delta }_{rs} = o_{rs}^{C} - o_{rs}^{T}$. Multiplying $q_{rs}^{m}-\bar {q}_{rs}^{m}$ on both sides and adding all inequalities over m together yields

$$ 0\leq \sum\limits_{m}(q_{rs}^{m}-\bar{q}_{rs}^{m})\beta_{m} \bar{c}_{rs}-\sum\limits_{m}(q_{rs}^{m}-\bar{q}_{rs}^{m})\beta_{m}\gamma_{rs}+ \sum\limits_{m}\pi_{rs}^{m}(q_{rs}^{m} - \bar{q}_{rs}^{m}) +(q_{rs} - \bar{q}_{rs}){\Delta}_{rs} $$

(28)

Finally, for those who always use transit regardless of toll, the following is always satisfied,

$$\beta_{m}\gamma_{rs} - \pi_{rs}^{m} \le \beta_{m}\gamma_{rs}, \forall m \leq e $$

Multiplying $\bar {q}_{rs}^{m}$ on both sides and adding up all inequalities over all the classes sticking to transit:

$$ 0\leq \sum\limits_{m} \bar{q}_{rs}^{m} \beta_{m}\gamma_{rs}-\sum\limits_{m}\bar{ q}_{rs}^{m}\beta_{m}\gamma_{rs}+\pi_{rs}^{m} \bar{q}_{rs} $$

(29)

Summing up inequalities (27 – 29):

$$\begin{array}{@{}rcl@{}} \Rightarrow R_{rs} &\le& \left[ \bar{c}_{rs}{\sum}_{m} \beta_{m}\left( {\sum}_{k} f_{rs}^{km}+q_{rs}^{m}-\bar{q}_{rs}^{m}\right)+\gamma_{rs}{\sum}_{m} \beta_{m}\bar{q}_{rs}^{m}\right]- \left[ {\sum}_{m} \beta_{m}{\sum}_{k} f_{rs}^{km}c_{rs}^{k}\right. \end{array} $$

(30)

$$\begin{array}{@{}rcl@{}} &&\left.+\gamma_{rs}{\sum}_{m} \beta_{m}q_{rs}^{m}\right] +(q_{rs} - \bar{q}_{rs}){\Delta}_{rs} + {\Pi}_{rs} \end{array} $$

(31)

where ${\Pi }_{rs} \equiv {\sum }_{m}\pi _{rs}^{m} (q_{rs}^{m}+ {\sum }_{k} f_{rs}^{km}) = {\sum }_{m}\pi _{rs}^{m}d_{rs}^{m}$ is defined as the total credits issued between rs. Let $\bar {f}_{rs}^{km}$ specify the path flow of class m. Recalling the demand constraint (1) for both no-toll and tolled equilibrium: $d_{rs}^{m}={\sum }_{k} f_{rs}^{km}+q_{rs}^{m}={\sum }_{k}\bar {f}_{rs}^{km}+\bar {q}_{rs}^{m},\Rightarrow f_{rs}^{km}+q_{rs}^{m}-\bar {q}_{rs}^{m}={\sum }_{k}\bar {f}_{rs}^{km}$ , and the term $(q_{rs} - \bar {q}_{rs}){\Delta }_{rs}$ is equivalent to the difference of total operating cost between no-toll and tolled equilibrium: $\bar {w}_{rs}o_{rs}^{C}+\bar {q}_{rs}o_{rs}^{T}-w_{rs}o_{rs}^{C}-q_{rs}o_{rs}^{T}$, the inequality (31) can be simplified as:

$$\begin{array}{@{}rcl@{}} R_{rs} &\le& \left[\bar{c}_{rs}\sum\limits_{m}\beta_{m}\sum\limits_{k} \bar{f}_{rs}^{km}+\gamma_{rs}\sum\limits_{m}\beta_{m}\bar{q}_{rs}^{m}+\bar{w}_{rs}o_{rs}^{C}+\bar{q}_{rs}o_{rs}^{T}\right]\\ &&- \left[ \sum\limits_{m} \beta_{m}\sum\limits_{k} f_{rs}^{km}c_{rs}^{k}+\gamma_{rs}\sum\limits_{m} \beta_{m}q_{rs}^{m}+w_{rs}o_{rs}^{C}+q_{rs}o_{rs}^{T}\right] + {\Pi}_{rs}\\ \Rightarrow R_{rs} &\le& \bar{C}_{rs} - C_{rs} + {\Pi}_{rs} \end{array} $$

(32)

where C _{r s} and $\bar {c}_{rs}$ are defined in (8) as the system cost at tolled and no-toll equilibrium respectively. In the most favorable case where all revenues should be issued as credits i.e., R _{r s} =π _{r s} , Pareto-improving requires toll scheme reduces system cost:

$$\bar{C}_{rs} \ge C_{rs} $$

Sufficiency

The sufficiency can be proven by designing a Pareto-improving credit scheme, in which each class m user receives the equal lump-sum credits π _{r s} = R _{r s} /d _{r s} and the extra class-specific credits.

We now construct the following discriminatory credit scheme. Namely, in addition to π _{r s} , class m receives an extra subsidy $\phi _{rs}^{m}$ (positive or negative) and each user within the class receives $\phi _{rs}^{m}/d_{rs}^{m}$:

$$ \phi_{rs}^{m} = \beta_{m}\mu_{rs}^{m} d_{rs}^{m}-\pi_{rs}d_{rs}^{m}- \beta_{m}\bar{\mu}_{rs}^{m} d_{rs}^{m} + \alpha_{rs}^{m}(\bar{C}_{rs}-C_{rs}); \sum\limits_{m}\alpha_{rs}^{m} = 1, \alpha_{rs}^{m} \ge 0, \forall m $$

(33)

The reader can verify that ${\sum }_{m}\phi _{rs}^{m} = 0$, i.e., the class-specific scheme proposed is revenue-neutral (zero net revenue). Now for any class m in O-D pair rs, we have

$$\begin{array}{@{}rcl@{}} \beta_{m}\mu_{rs}^{m} -\pi_{rs} -\frac{ \phi_{rs}^{m}} {d_{rs}^{m}} &=& \beta_{m}\bar{\mu}_{rs}^{m} - \frac{\alpha_{rs}^{m}(\bar{C}_{rs} - C_{rs})}{d_{rs}^{m}} \le \beta_{m}\bar{\mu}_{rs}^{m} \end{array} $$

This completes the proof. □