Article Contents
Article Contents

# Cooperation in traffic network problems via evolutionary split variational inequalities

• * Corresponding author: Xiaolong Qin
• 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.

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

 Citation:

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

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

Figure 3.  Traffic network pattern of City $Y$

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

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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