Emergency evacuation problem for a multi-source and multi-destination transportation network: mathematical model and case study

Abstract

Disasters such as earthquake or tsunami can easily take the lives of thousands of people and millions worth of property in a fleeting moment. A successful emergency evacuation plan is critical in response to disasters. In this paper, we seek to investigate the multi-source, multi-destination evacuation problem. First, we construct a mixed integer linear programming model. Second, based on K shortest paths and user equilibrium, we propose a novel algorithm (hereafter KPUE), whose complexity is polynomial in the numbers of nodes and evacuees. Finally, we demonstrate the effectiveness of algorithm KPUE by a real evacuation network in Shanghai, China. The numerical examples show that the average computation time of the proposed algorithm is 95% less than that of IBM ILOG CPLEX solver and the optimality gap is no more than 5%.

Appendices

Proof

Proof of Proposition 1

We define $T_a^*$ as the optimal solution value and $N_a^{k,*}$ as the corresponding number of evacuees that are allocated to path $p_a^k$. According to the definition of $T_a$, we have $T_a=\max _{1 \le k \le \hat{K}(a)}\{t_a^k(D_0)-1+\left\lceil \frac{N_a^{k}}{f_a^k} \right\rceil \}$. Hence, we have

$$\begin{aligned} \begin{aligned} T_a^* \ge t_a^k(D_0)-1+\left\lceil \frac{N_a^{k,*}}{f_a^k} \right\rceil ; \ \\ T_a^* \cdot f_a^k \ge (t_a^k(D_0)-1)f_a^k+N_a^{k,*}. \end{aligned} \end{aligned}$$

$\square$

Summing over k, we can get $T_a^* \cdot \sum _{k=1}^{\hat{K}(a)}{f_a^k} \ge \sum _{k=1}^{\hat{K}(a)}(t_a^k(D_0)-1)f_a^k+{ ICS}_a$. We have

$$\begin{aligned} \begin{aligned} T_a^* \ge \frac{\sum _{k=1}^{\hat{K}(a)}{\left( t_a^k(D_0)-1\right) f_a^k}+{ ICS}_a}{\sum _{k=1}^{\hat{K}(a)}{f_a^k}}. \end{aligned} \end{aligned}$$

We denote

$$\begin{aligned} \begin{aligned} \overline{T}_a&=\frac{\sum _{k=1}^{\hat{K}(a)}{(t_a^k(D_0)-1)f_a^k}+{ ICS}_a}{\sum _{k=1}^{\hat{K}(a)}{f_a^k}}; \\ \overline{N}_a^k&=f_a^k \cdot \overline{T}_a-(t_a^k(D_0)-1)\cdot f_a^k \\&=f_a^k \cdot \left( \overline{T}_a-t_a^k(D_0)+1\right) . \end{aligned} \end{aligned}$$

In the following, we proceed the analysis according two cases: (a) $\overline{T}_a$ is integral; (b) $\overline{T}_a$ is fractional.

Case (a) If $\overline{T}_a$ is integral, $\overline{N}_a^k$ can be evacuated in $\overline{T}_a$, and a total of $\sum _{k=1}^{\hat{K}(a)}{\overline{N}_a^k}={ ICS}_a$ people can be evacuated.

Case (b) If $\overline{T}_a$ is fractional, denote $\hat{N}_a^k=\left\lceil \overline{N}_a^k \right\rceil$, and $\hat{N}_a^k$ people can be evacuated in $\left\lceil \overline{T}_a \right\rceil$.

Because $\left\lceil \frac{\overline{N}_a^k}{f_a^k}\right\rceil \ge \frac{\overline{N}_a^k}{f_a^k}$, we can get $\left\lceil \frac{\overline{N}_a^k}{f_a^k} \right\rceil \cdot f_a^k \ge \overline{N}_a^k$. Hence, we have $\left\lceil \frac{\overline{N}_a^k}{f_a^k} \right\rceil \cdot f_a^k \ge \left\lceil \overline{N}_a^k\right\rceil =\hat{N}_a^k$ and $\left\lceil \frac{\overline{N}_a^k}{f_a^k} \right\rceil \ge \frac{\hat{N}_a^k}{f_a^k}$. So, we also have

Because $\overline{N}_a^k \le \hat{N}_a^k$, we have

According to (A.1) and (A.2), we can get $\left\lceil \frac{\overline{N}_a^k}{f_a^k} \right\rceil = \left\lceil \frac{\hat{N}_a^k}{f_a^k} \right\rceil$. Hence, sending $\hat{N}_a^k$ people to super destination node $D_0$ need $\left\lceil \frac{\overline{N}_a^k}{f_a^k} \right\rceil +t_a^k(D_0)-1=\left\lceil \overline{T}_a \right\rceil$. All $\sum _{k=1}^{\hat{K}(a)}{N_a^k}> { ICS}_a$ people can be evacuated in $\left\lceil \overline{T}_a \right\rceil$.

To sum up case (a) and (b), $T_a^*=\left\lceil \frac{\sum _{k=1}^{\hat{K}(a)}{(t_a^k(D_0)-1)f_a^k}+{ ICS}_a}{\sum _{k=1}^{\hat{K}(a)}{f_a^k}}\right\rceil$ is the optimal solution.

Data and evacuation networks of the literature (Chalmet et al. 1982; Lu et al. 2005)

In Sect. 5, we present different scenarios for numerical study. The evacuation network of Scenario S01 and S02 are presented in Fig. 6. The number of evacuees at source nodes [N1, N2, N3] from S01 are [10, 5, 15], while the number of evacuees at source nodes [N1, N2, N3] from S02 are [220, 240, 450]. The evacuation network of Scenarios S11, S12, S13, S14, S15 and S16 are presented in Fig. 1. The number of evacuees at source nodes [N1, N2, N3] from S11 are [220, 240, 450]. The number of evacuees at source nodes [N1, N2, N3, N4] from scenarios S12, S13, S14 and S15 are [150, 350, 305, 105], [300, 700, 600, 200], [500, 1150, 1050, 300], [850, 1800, 1550, 400], respectively. The number of evacuees at source nodes [N1, N2, N3, N4, N5, N6, N7, N8] from scenario S16 are [100, 100, 200, 100, 110, 100, 100, 100]. The evacuation network of Scenario S21 are presented in Fig. 7. The number of evacuees at source nodes [N1, N2, N3] from S21 are [220, 240, 450].

Fig. 6

Evacuation network from Lu et al. (2005)

Fig. 7

Evacuation network from Chalmet et al. (1982)