Maximizing Network Throughput under Stochastic User Equilibrium with Elastic Demand

Abstract

Most of the existing studies adopt the fixed-demand equilibrium formulation to model drivers’ route choice when studying network throughput maximization problem. Travelers’ reactions to the increased origin-destination (O-D) travel cost and network congestion level are less considered in the problem. Note that travelers can cancel the trip or use other modes to travel if the road network is congested. This study aims to address this gap by analyzing the maximum network throughput problem using the formulation of Logit-based SUE with elastic demand (SUEED). The Logit-based SUEED problem not only models the drivers’ route choice according to the SUE principle, but also estimates the equilibrium O-D demand by factoring the effect of expected perceived O-D travel time on O-D demand. A bi-level programming problem is proposed to characterize the maximum network throughput based on the Logit-based SUEED problem. The sensitivity analysis for the Logit-based SUEED problem is presented and incorporated into the solution algorithm for the proposed problem. A numerical example demonstrates the effectiveness of the proposed sensitivity-based solution algorithm. This study finds that under the SUEED condition, the maximum network throughput decreases monotonically when travelers’ knowledge level of traffic conditions increases (less travel time perception error). It implies that promoting advanced traveler information system ATIS may not serve to foster more number of trips by travelers and make more use of physical network capacity.

Appendix 1: Proof of Lemma 1

Recall ${S}_{rs}=E\left[\underset{k\in {R}_{rs}}{\min \left\{{C}_k^{rs}\right\}}|{\mathbf{c}}^{rs}\right]$, then to prove Lemma 1 is equivalent to prove

$$ \sum \limits_{k\in {R}_{rs}}{P}_k^{rs}{c}_k^{rs}\ge E\left[\underset{k\in {R}_{rs}}{\min \left\{{C}_k^{rs}\right\}}|{\mathbf{c}}^{rs}\left(\mathbf{v}\right)\right] $$

(37)

The following two steps will be used to show inequality (37).

Step 1:

$$ E\left[\underset{k\in {R}_{rs}}{\min \left\{{C}_k^{rs}\right\}}|{\mathbf{c}}^{rs}\right]\le \underset{k\in {R}_{rs}}{\min}\left\{{c}_k^{rs}\right\} $$

(38)

Proof: If there is only one route between O-D pair r-s. Let l denote this route, i.e., R _rs  = {l}. Then we have

$$ E\left[\underset{k\in {R}_{rs}}{\min \left\{{C}_k^{rs}\right\}}|{\mathbf{c}}^{rs}\right]=E\left[{C}_l^{rs}|{c}_l^{rs}\right]={c}_l^{rs} $$

(39)

Hence, $E\left[\underset{k\in {R}_{rs}}{\min \left\{{C}_k^{rs}\right\}}|{\mathbf{c}}^{rs}\right]=\underset{k\in {R}_{rs}}{\min}\left\{{c}_k^{rs}\right\}$ and the inequality (38) holds.

Recall $E\left[\underset{k\in {R}_{rs}}{\min \left\{{C}_k^{rs}\right\}}|{\mathbf{c}}^{rs}\left(\mathbf{v}\right)\right]$ monotonically decreases (or at least non-increase) with respect to the number of alternative routes (page 279, Sheffi 1985). Then if a new route (for example route m) is added between the O-D pair r-s, we have

$$ E\left[\min \left\{{C}_m^{rs},{C}_l^{rs}\right\}\right]\le E\left[{C}_l^{rs}\right]={c}_l^{rs} $$

(40)

Similarly,

$$ E\left[\min \left\{{C}_m^{rs},{C}_l^{rs}\right\}\right]\le E\left[{C}_m^{rs}\right]={c}_m^{rs} $$

(41)

Thus the inequality (38) also holds under the two route cases. Through repeatedly applying the inequality (40) and (41), it can show that the inequality (38) also holds for an arbitrary number of routes between O-D pair r-s.

Step 2:

$$ \sum \limits_{k\in {R}_{rs}}{P}_k^{rs}{c}_k^{rs}\ge \underset{k}{\min}\left\{{c}_k^{rs}\right\} $$

(42)

Proof: If there is only one route between O-D pair r-s, then $\sum \limits_{k\in {R}_{rs}}{P}_k^{rs}{c}_k^{rs}=\underset{k}{\min}\left\{{c}_k^{rs}\right\}$. The inequality (42) holds obviously. Suppose the number of routes in set R _rs is larger than 1. Without loss of generality, let m ₁ Denote the route with minimum travel cost, i.e., $\underset{k\in {R}_{rs}}{\min}\left\{{c}_k^{rs}\right\}={c}_{m_1}^{rs}$. Note

$$ \sum \limits_{k\in {R}_{rs}}{P}_k^{rs}=1 $$

(43)

Then

$$ \sum \limits_{k\in {R}_{rs}}{P}_k^{rs}{c}_k^{rs}\ge \sum \limits_{k\in {R}_{rs}}{P}_k^{rs}{c}_{m_1}^{rs}={c}_{m_1}^{rs}=\underset{k\in {R}_{rs}}{\min}\left\{{c}_k^{rs}\right\} $$

(44)

Thereby, the inequality (42) also holds for an arbitrary number of routes between OD pair r-s.

Combine conclusions in step 1 and step 2, we have

$$ E\left[\underset{k\in {R}_{rs}}{\min \left\{{C}_k^{rs}\right\}}|{\mathbf{c}}^{rs}\right]\le \underset{k\in {R}_{rs}}{\min}\left\{{c}_k^{rs}\right\}\le \sum \limits_{k\in {R}_{rs}}{P}_k^{rs}{c}_k^{rs} $$

This completes the proof of Lemma 1.