Cooperation in traffic network problems via evolutionary split variational inequalities

Shipra Singh; Aviv Gibali; Xiaolong Qin

doi:10.3934/jimo.2020170

Article Contents

2022, Volume 18, Issue 1: 593-611. Doi: 10.3934/jimo.2020170

This issue Previous Article On the convexity for the range set of two quadratic functions Next Article ADMM-type methods for generalized multi-facility Weber problem

Cooperation in traffic network problems via evolutionary split variational inequalities

1.
Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 3200003, Israel
2.
Department of Mathematics, ORT Braude College, Karmiel, 2161002, Israel
3.
The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, Haifa, 3498838, Israel
4.
Department of Mathematics, Zhejiang Normal University, Zhejiang, China

^* Corresponding author: Xiaolong Qin
^* Corresponding author: Xiaolong Qin

Received: July 31, 2020

Revised: August 31, 2020

Early access: November 2020

Published: January 2022

Abstract Full Text(HTML) Figure(3) / Table(2) Related Papers Cited by

Abstract

In this paper, we construct an evolutionary (time-dependent) split variational inequality problem and show how to reformulate equilibria of the dynamic traffic network models of two cities as such problem. We also establish existence result for the proposed model. Primary numerical results of equilibria illustrate the validity and applicability of our results.

Keywords:

Evolutionary split variational inequality problem,
Wardrop condition,
traffic equilibrium problems,
equilibrium solutions,
projected dynamical system.

Mathematics Subject Classification: Primary:90C33, 90C39, 49J40.

Citation:

Full Text(HTML)

Figure 1. Transportation network patterns of the City $ X\; \text{and}\; Y $

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Figure 2. Traffic network pattern of two cities $ X $ and $ Y $

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Figure 3. Traffic network pattern of City $ Y $

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Table 1. Numerical Results

$ t_{i} $	$ x_{1}^{*}(t_{i}) $	$ x_{2}^{*}(t_{i}) $	$ x_{3}^{*}(t_{i}) $	$ x_{4}^{*}(t_{i}) $
$ 0 $	1	1	1.5	1.5
$ \frac{1}{8} $	1.125	1.125	1.6875	1.6875
$ \frac{1}{4} $	1.25	1.25	1.875	1.875
$ \frac{3}{8} $	1.375	1.375	2.0625	2.0625
$ \frac{1}{2} $	1.5	1.5	2.25	2.25
$ \frac{5}{8} $	1.625	1.625	2.4375	2.4375
$ \frac{3}{4} $	1.75	1.75	2.625	2.625
$ \frac{7}{8} $	1.875	1.875	2.8125	2.8125
$ 1 $	2	2	3	3
$ \frac{9}{8} $	2.125	2.125	3.1875	3.1875
$ \frac{5}{4} $	2.25	2.25	3.375	3.375
$ \frac{11}{8} $	2.375	2.375	3.5625	3.5625
$ \frac{3}{2} $	2.5	2.5	3.75	3.75
$ \frac{13}{8} $	2.625	2.625	3.9375	3.9375
$ \frac{7}{4} $	2.75	2.75	4.125	4.125
$ \frac{15}{8} $	2.875	2.875	4.3125	4.3125
$ 2 $	3	3	4.5	4.5

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Table 2. Numerical Results

$ t_{i} $	$ y_{1}^{*}(t_{i}) $	$ y_{2}^{*}(t_{i}) $	$ y_{3}^{*}(t_{i}) $	$ y_{4}^{*}(t_{i}) $	$ y_{5}^{*}(t_{i}) $
$ 0 $	2.5	2.5	2	2	2
$ \frac{1}{8} $	2.8125	2.8125	2.25	2.25	2.25
$ \frac{1}{4} $	3.125	3.125	2.5	2.5	2.5
$ \frac{3}{8} $	3.4375	3.4375	2.75	2.75	2.75
$ \frac{1}{2} $	3.75	3.75	3	3	3
$ \frac{5}{8} $	4.0625	4.0625	3.25	3.25	3.25
$ \frac{3}{4} $	4.375	4.375	3.5	3.5	3.5
$ \frac{7}{8} $	4.6875	4.6875	3.75	3.75	3.75
$ 1 $	5	5	4	4	4
$ \frac{9}{8} $	5.3125	5.3125	4.25	4.25	4.25
$ \frac{5}{4} $	5.625	5.625	4.5	4.5	4.5
$ \frac{11}{8} $	5.9375	5.9375	4.75	4.75	4.75
$ \frac{3}{2} $	6.25	6.25	5	5	5
$ \frac{13}{8} $	6.5625	6.5625	5.25	5.25	5.25
$ \frac{7}{4} $	6.875	6.875	5.5	5.5	5.5
$ \frac{15}{8} $	7.1875	7.1875	5.75	5.75	5.75
$ 2 $	7.5	7.5	6	6	6

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