category: literaturenote citekey: zengdynamicevacuationoptimizationmodel2021 title: Dynamic evacuation optimization model based on conflict-eliminating cell transmission and split delivery vehicle routing authors: "Zeng, Ming-hua; Wang, Min; Chen, Yao; Yang, Zhenyu" year: 2021 date: 2021-00-00 2021 doi: 10.1016/j.ssci.2021.105166 publication: Safety Science url: "https://doi.org/10.1016/j.ssci.2021.105166" zotero_key: MK3PGKZF zotero_storage: WPQ2ZHNT collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Zeng, Ming-hua"

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Safety Science

journal homepage: www.elsevier.com/locate/safety

Dynamic evacuation optimization model based on conflict-eliminating cell transmission and split delivery vehicle routing

Ming-hua Zeng a,* , Min Wang ^a , Yao Chen ^b , Zhenyu Yang ^b

^a School of Transportation and Logistics, East China Jiaotong University, Nanchang 330013, Jiangxi, China
^b Department of Industrial Systems Engineering and Management, National University of Singapore, Engineering Drive 2, Singapore 117576, Singapore

ARTICLE INFO

Keywords: Dynamic evacuation Optimization model Cell transmission model Split delivery vehicle routing problem

ABSTRACT

This paper studies dynamic evacuation optimization in the context of a new cell transmission with the split delivery vehicle routing problem (SDVRP) of multiple modes of buses and cars. A bi-directional multilane conflict-eliminating cell transmission model (BCECTM) is proposed to simulate the mixed traffic dynamic evacuation process on links and at intersections. The method of SDVRP is applied to manage dynamic evacuation demands and optimize the routing of multiple modes. The mechanisms of vehicle transmission, traveling trajectory, and residual capacity updating are depicted based on the models of BCECTM and SDVRP; thus, a mathematical programming model aiming to minimize the total evacuation time is established by integrating the BCECTM and SDVRP. A genetic algorithm with a chromosome position-coding scheme is designed to solve the evacuation optimization model. Case studies show that the proposed model can reasonably assign evacuation demands that gather at some sites around the event to, and effectively evacuate people by, buses and cars so that the capacities of buses are fully utilized, and can generate optimal vehicle traveling trajectories to improve the evacuation performance. Furthermore, the evacuation and congestion degree under different demand levels and different arrival distributions are analyzed with meaningful conclusions.

1. Introduction

Under the emergent circumstances of fire accidents and flood disasters, etc., research on traffic evacuation has attracted extensive attention. However, restricted by many practical factors such as road network infrastructure, intersection conflicts, dynamically arriving demands, and dynamic traffic conditions, dynamically organizing buses to circulate between the event area and shelters to pick-up and deliver evacuees as well as evacuating cars with high efficiency is a complex problem. Intensive investigation on it is of great importance.

Evacuation route planning methods include mainly optimization models (Kim and Shekhar, 2008; Yang et al., 2018), simulation methods (Yuan et al., 2009; Xie and Turnquist, 2011), etc. Among the class of microscopic simulation models, the cell transmission model (CTM) (Guo et al., 2011; Li and Ozbay, 2015) is one of the most typical ones. Daganzo (1994) proposed the cell transmission model, in which a road link is divided into isometric cells on which vehicles run, and vehicle flow is constrained by upstream and downstream traffic conditions and conservation equations. The well-known CTM has been applied in the simulation of traffic evacuation (Kimms and Maassen, 2012; Li and

Ozbay, 2015). Li and Ozbay (2015) employed a cell-based model to study evacuation planning with endogenous transportation network degradations. As far as evacuation optimization and planning are concerned, some studies focus on designing optimal routes (Wang and Gong, 2014; Zhang et al., 2018). The particular kind of lane-based evacuation model that was suggested early by Cova and Johnson (2003), and followed by such as Xie and Turnquist (2011) maybe can be regarded as a mesoscopic model with larger granularity than CTM model.

In the CTM, a bi-directional road is usually viewed as two independent cells in opposite directions. Carey et al. (2015) extended the singlelane cell transmission model to multiple lanes in the same direction with lane changing. Assuming that lane direction can be reversed in small intervals of space–time, Levin and Boyles (2016) presented a cell transmission model formulation for dynamic lane reversal. However, in the literature, the original CTM cannot avoid the conflicts at intersections and can only roughly evaluate the traffic volumes of leftturning, right-turning, and straight-going movement in terms of turning ratios at an intersection. Therefore, when traffic flows out a roadway segment towards the adjacent intersection, the three different

^* Corresponding author at: School of Transportation and Logistics, East China Jiaotong University, No. 808, Shuanggang Road, Nanchang 330013, Jiangxi, China. E-mail address: zmhcsu@163.com (M.-h. Zeng).

directions of turning left, turning right, and going straight are considered in the respects of their traffic volumes and their potential conflicts with traffic from other directions. In our CTM, conflict avoidance at intersections is introduced to model interactions among different directions, and the outflow traffic volumes of the three different directions are calculated accurately. Thus, the traditional CTM is extended to a bidirectional multilane conflict-eliminating CTM (BCECTM).

During an evacuation, public transportation modes such as buses are useful for rapidly transferring evacuees from the place of an event. For evacuees, those who are not accessible to an automobile or do not use it will wait at the gathering sites for buses to pick them up; the rest having access to automobiles can drive away from the event area. Many studies focus on car evacuation; however, mass transit evacuation has not received much attention (Abdelgawad and Abdulhai, 2012). Furthermore, pick-ups at locations such as bus stations, geographic population centroids, have a significant impact on evacuation service quality of public transport, which has not attracted much attention until, recently, some works (Abdelgawad and Abdulhai, 2012; An et al., 2013; Goerigk et al., 2015) focused on bus evacuation. Abdelgawad and Abdulhai (2012) modeled a large-scale evacuation using subway and bus transit as a multi parking lot, time limit, and loading and unloading vehicle route problem (VRP). In contrast, Goerigk et al. (2015) assumed a vehicle scheduling problem, in which the paths from collection points to shelters and back are fixed, rather than a path choice problem to robustly schedule a bus fleet to evacuate persons from an endangered region.

A large number of evacuees gathering at bus stops have to be delivered by multiple buses. Therefore, the method of split delivery vehicle routing problem (SDVRP) can be used to complete bus path planning for evacuation. The SDVRP was first proposed by Dror and Trudeau (1989), aiming for an optimization solution to improve vehicle loading efficiency and reduce path cost (Archetti et al., 2006). A small number of studies have employed the SDVRP to investigate evacuation route planning problems. Shen et al. (2009) used the SDVRP method to generate evacuation routes in advance at the planning stage. Wang et al. (2014) set up a multi-objective nonlinear integer model optimized by SDVRP for selecting earthquake disaster sites considering vehicle routing after the Sichuan earthquake.

According to the previous studies, an optimization model and CTM each has its disadvantages. First, the transmission mechanism of traffic flow is not well handled in optimization models; moreover, the organization of reverse bus flow and coping with the conflict situation at an intersection is difficult for CTM. Second, the delivery of evacuees using buses is not depicted by optimization models such as lane-based models. To overcome these shortcomings, a feasible method is to use CTM and VRP to improve an optimization model. However, to the best of the authors' knowledge, no work has involved a combination of CTM and VRP except the work of Aziz and Ukkusuri (2013) who solved a capacitated vehicle routing problem (CVRP) using the CTM to update travel time. Therefore, we establish a dynamic and bus-evacuationoriented CTM-based evacuation optimization model from a macro-–micro integration perspective by gaining worthwhile insights from previous studies into the microscopic dynamic behavior of individual vehicles and macroscopic routing schedule for traffics. A BCECTM is proposed and integrated with the SDVRP to formulate an optimization model for dynamic evacuation with two modes of bus and car. The BCECTM is applied to better describe the dynamic propagation of vehicles and the cyclic picking-up and unloading evacuees of buses during evacuation. Furthermore, the BCECTM and SDVRP are integrated to optimize the evacuation route of buses and cars. The SDVRP is introduced to split all evacuation demands, and depict the pick-up and route choice decision of buses. Taking full advantage of buses' capacity to meet the evacuation demands at gathering sites will guarantee to complete evacuation in a period as short as possible.

The remainder of this paper is organized as follows. Section 2 presents a bi-directional conflict-eliminating cell transmission model, following which an evacuation optimization model integrated with the

BCECTM and SDVRP is formulated in Section 3. Section 4 designs a genetic algorithm based on a position-coding scheme for solving the proposed model. Case studies are finally executed in Section 5.

2. Bi-directional conflict-eliminating cell transmission model

Buses are used to transport evacuees who have no access to an automobile or do not use it from the event area to shelter points. However, the CTM cannot describe the organization of reverse bus flow and the conflict situation at an intersection. To handle this case, a BCECTM that can eliminate conflicts at intersections is formulated. In the BCECTM, each roadway segment is divided into several subsegments, in which a sub-segment is composed of a forward cell and a backward cell. Each forward and backward cell is identical in length in the same road segment but inconsistent and determined by free-flow speed from the network aspect. Other cells, such as cells of destination and gathering site, are considered to have no length. The sum of the number of lanes of the forward cell and the backward cell is equal to the total number of lanes of this road sub-segment. Traffic flows on the forward cell and the backward cell are prohibited from exchanging. Symbols for modeling are defined in Table 1.

The objective function of this evacuation optimization model is defined as the total evacuation time for all evacuees to stay in the cells, excluding the destination cells. The BCECTM model for evacuation is formulated as the following mathematical programming:

Table 1 Symbols.

Symbol Definition *V*1 Set of buses.

V2	Set of automobiles.
C	Set of forward cells and backward cells.
S	Set of evacuation destination cells.
R	Set of gathering site cells, and R = R1 ∪ R2, where R1 is the set of bus stop cells, R2 is the set of car parking lot cells.
A	Set of connectors, where a connector is defined as the connection of two adjacent cells.
AI	Set of connectors at intersections, AI⊂A.
AS I ,AL I	AS I ⊂AIis the set of all straight-going connectors at intersections; AL I ⊂AI is the set of all left-turning connectors at intersections.
Γ(i)	Set of adjacent upstream cells of cell i.
Γ− 1(i)	Set of adjacent downstream cells of cell i.
T	The time period that the evacuation should be completed.
Δt	The length of a time step.
α	Conversion coefficient of a bus to the number of passenger car unit (pcu)
Ni(t)	assuming that a car is equivalent to a pcu. The capacity of cell i within time step t.

ni(t)	The number of lanes of cell i within time step t.
kjam	Jam density of each cell.
Li	Length of cell i, Li = vi⋅Δt/3600, while vi is free flow speed in cell i.
βl	The capacity of a vehicle of type l, where the subscripts l = 1, 2 mean buses and cars, respectively.
xil(t)	The number of vehicles of type l in cell i within time step t.
yijl(t)	The volume of vehicles of type l on connector (i, j) within time step t.
di(t)	The number of evacuees in gathering site cell i ∈ R within time step t.
db	The number of evacuees picked up by bus b in cell i ∈ R1 within time step t.
i (t) ai(t)	The number of evacuees arriving at cell i ∈ R within time step t.
wb(t)	The number of passengers on the bus b within time step t.
zij(t)	A dummy parameter;zij(t) = 1if connector (i, j) has flows within time step t, zero otherwise.
zv ij(t)	A dummy parameter;zv ij(t) = 1 if vehicle v ∈ V1 ∪ V2 passes through
	connector (i, j) within time step t, zero otherwise.

step t, zero otherwise.

$$min \ Z = \sum{t=1}^{T} \sum{b \in V1} w^b(t) + \sum{t=1}^{T} \sum_{i \in C} \beta2 \cdot x{i2}(t) + \sum{t=1}^{T} \sum{i \in R} d_i(t)$$ (1)

$$x{il}(t+1) = x{il}(t) + \sum{k \in \Gamma(l)} y{kil}(t) - \sum{j \in \Gamma^{-1}(l)} y{ijl}(t) \ \forall i \in C \cup R \cup S, \ \forall l, \ \forall t < T \quad \textbf{(2)}$$

$$\sum{k \in \Gamma(i)} [\alpha \cdot y{ki1}(t) + y_{ki2}(t)] \leq Ni(t) - [\alpha \cdot x{i1}(t) + x_{i2}(t)] \ \forall i \in C \cup R \cup S, \forall (k, i)$$

$$\in A, \ \forall t$$ (3)

$$\sum{j \in \Gamma^{-1}(i)} y{ijl}(t) \leqslant x_{il}(t) \ \forall i \in C \cup R \cup S, \ \forall (i, j) \in A, \ \forall l, \ \forall t$$

$$\tag{4}$$

$$\sum{j \in \Gamma^{-1}(i)} y{ij2}(t) = 0 \ \forall i \in S, \ \forall t$$ (5)

$$Ni(t) = k{jam} \cdot L_i \cdot n_i(t) \ \forall i \in C \cup R \cup S, \ \forall t$$ (6)

$$x{il}(t) = \begin{cases} x{il}(0), \ t = 0 \ \geqslant 0, \text{ othersise} \end{cases} \forall i \in C \cup R \cup S, \ \forall l, \ \forall t$$ (7)

$$y_{ijl}(t) = \begin{cases} 0, \ t = 0 \ \ge 0, \ \text{otherwise} \end{cases} \forall (i, j) \in A, \ \forall l$$ (8)

$$z{ij}(t) = \begin{cases} 1, & \sum{i} y{ijt}(t) \neq 0 \ 0, & \sum{i} y_{ijt}(t) = 0 \end{cases} \quad \forall (i, j) \in A$$ (9)

$$z{rm}(t) + z{ph}(t) \le 1, \ \forall (r, \ m), \ (p, \ h) \in A_I^S$$ (10)

$$z{pth}(t) + z{nq}(t) + z_{hm}(t) \leq 1, { \forall (n, q), (k, m), (p, h) \in A_I^S \text{ or } A_I^L } \cap A_I^L \neq \emptyset$$ (11)

Formula (1) represents the total evacuation time that is composed of three parts: the first two items represent the total travel time of evacuees on buses and on automobiles, respectively, and the third item represents the total waiting time of evacuees at gathering sites. Formula (2) is the flow conservation equation. Formulae (3)–(4) are flow constraints, where α is set as 2.0 for a bus taking an automobile as reference. Formula (5) illustrates that automobiles no longer enter the evacuation network after reaching evacuation destination cells. It is notable that buses have no such restriction of (5) because they are required to return to pick up evacuees at gathering sites. Formula (6) calculates the capacity of each cell. Eqs. (7)–(8) are constraints of the initial distribution of vehicle traffic and nonnegative constraints, respectively. The meaning of formula (9) is obvious from the definitions of zij(t) and yijl(t). The constraints (10)–(11) are constraints aiming to eliminate conflicts at intersections. The potential conflicts at an intersection in a oneintersection CTM network are shown in Fig. 1. The following formulae can be used to control the number of lanes in a road segment:

Fig. 1. The bi-directional cell transmission network at an intersection.

$$n_u(t) + nv(t) = n{uv} \ \forall u, v \in C \cup R, \ \forall t$$ (12)

$$n_i(t) = n_u(t) = nm(t) \in [0, n{uv}], \ n_j(t) = n_v(t) = nn(t) \in [0, n{uv}] \ \forall i, u, m, j, v, n \in C \cup R, \ \forall t$$

where nuv(t) is the total number of lanes of cell u and v during the evacuation period. Formula (12) is the total lane constraint and formula

(13)

(14)

(13) requires that, for each direction, the lane number of each cell should be identical. Formulae (12)-(13) can realize the dynamic operation of lance reversal through adjusting the number of lanes in different directions according to real-time traffic volume.

Different from the traditional CTM, the proposed BCECTM can avoid conflicts at intersections and is capable of generating the outflows of left-turning, straight, and right-turning movements of any approach at interactions. This will significantly improve the efficiency of traffic evacuation at intersections as key locations. In addition, the BCECTM can describe a real bi-directional road network with multiple lanes and can reflect dynamic lane reversal.

3. Evacuation optimization model based on BCECTM and SDVRP

A basic model for evacuation based on the BCECTM has been established in Section 2. Although the BCECTM can simulate the dynamics of traffic flow, it cannot describe the characteristics of such as trajectory from cell to cell and effective (or remaining) capacity state of an individual vehicle in traffic flow. To be complementary, the method of SDVRP focuses on an individual vehicle for modeling. Therefore, the SDVRP method is employed to complement the BCECTM to model the complex dynamic evacuation completed by buses as well as automobiles.

Additional to the BCECTM presented in the previous section, the following formulae (14)–(17) are established for an arbitrary vehicle including buses and automobiles traversing in the BCECTM network.

$$\begin{split} z_i^{\boldsymbol{v}}(t+1) &= zi^{\boldsymbol{v}}(t) + \sum{k \in \Gamma(i)} z{ki}^{\boldsymbol{v}}(t) - \sum{j \in \Gamma^{-1}(i)} z_j^{\boldsymbol{v}}(t), \ \forall \boldsymbol{v} \in V_1 \cup V_2, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup S, \ \forall i \in C \cup R \cup S, \ \forall i \in C \cup R, \ \forall i \in C \cup R, \ \forall i \in C \cup R$$

$$\sum z_i^{\nu}(t) = 1, \ \forall \nu \in V_1 \cup V_2, \ \forall i \in C \cup R \cup S, \ \forall t$$ (15)

$$\sum \sum z_{ij}^{\nu}(t) \leqslant 1, \ \forall \nu \in V_1 \cup V_2, \ \forall (i, j) \in A, \ \forall t$$

$$\tag{16}$$

$$\mathbf{z}{i}^{v}(t) \in {0, 1}, \ \mathbf{z}{ij}^{v}(t) \in {0, 1}, \ \forall v \in V{1} \cup V{2}, \ \forall (i, j) \in A, \ \forall t$$ (17)

Formula (14) is the flow conservation equation. Formula (15) indicates that each vehicle can only be assigned to one cell at any time. Formula (16) means that any vehicle can only pass through at most one connector at any time. These formulae simulate the moving trajectory of a single vehicle in the BCECTM network according to the law of time and space of traffic flow.

The number of persons to be evacuated by buses waiting at gathering sites varies with time. The following formulae are designed to describe the process:

$$d_i^b(t) = z_i^b(t) \cdot min{\beta_1 - w^b(t-1), d_i(t)}, \ \forall b \in V_1, \ \forall t, \ \forall i \in R_1$$ (18)

$$w^{b}(t) = w^{b}(t-1) + \sum{i \in R{1}} d{i}^{b}(t), \ \forall b \in V{1}, \ \forall t$$ (19)

$$0 \leq w^{b}(t) \leq \left[1 - \sum{i \in V} z{i}^{b}(t)\right] \cdot \beta{1}, \ \forall b \in V{1}, \ \forall t$$ (20)

Assume that no overload is allowed. A bus passing by a gathering site

should pick up as many evacuees as possible that have been waiting there, so we get Eq. (18), which suggests that the number of passengers picked up by the bus takes the smaller value of its remaining capacity and the number of evacuees waiting there at the current time. Eq. (19) updates the number of passengers on a bus. Formula (20) indicates that a bus can drive to any evacuation destination cell and then the bus is reset to full capacity, thus a bus can shuttle passengers (back and forth); this formula also implies that the passengers on a bus cannot exceed the capacity of the bus as long as the bus has not arrived at a destination.

Among many of the response distribution in an evacuation, such as uniform distribution, Rayleigh distribution, Poisson distribution, Weibull distribution, and so on (Pel et al., 2012), the Weibull distribution with the following form is the most widely used and is closest to reality.

$$D(t) = 1 - exp(-\beta t^{\gamma}) \ \forall t$$ (21)

where D(t) is the cumulative percentage of evacuees who have arrived until time step t. β and γ are distribution shaping parameters that their higher values lead to a faster response (Pel et al., 2012). Thus, the Weibull distribution is applied to calculate the number of evacuees ai(t) arriving at cell i within time step t in the overall evacuation process:

$$a_i(t) = [D(t) - D(t-1)] \cdot p_i = [exp(-\beta(t-1)^{\gamma}) - exp(-\beta t^{\gamma})] \cdot p_i \ \forall t$$ (22)

4. The solution algorithm

Archetti et al. (2006) studied the complexity of the SDVRP problem. It has been proved that SDVRP is an NP-hard problem. Therefore, the evacuation optimization model based on the BCECTM and SDVRP is an NP-hard problem. A genetic algorithm is applied to solve the proposed model. A genetic algorithm is an intelligent computing method based on the theory of biological evolution. A basic genetic algorithm can be found in Huang and Chen (2009). The following key elements should be considered for completing the procedure of a genetic algorithm: chromosome representation and the genetic operators of crossover, mutation, and selection.

First, a position-coding scheme is designed to construct chromosomes. The structure of a chromosome generated by such a positioncoding scheme is shown in Fig. 2. A gene is represented by the location of a vehicle at a time interval. For example, a gene g^v(t) = i indicates that vehicle v is in cell i at time interval t. The length of a chromosome is equal to (|*V*1| + |*V*2|)⋅T.

A chromosome is related to the proposed model by decoding the neighboring genes. The decoding process is put as follows:

$$\begin{cases} \text{If } g^{\mathbb{V}}(t) = i, \ g^{\mathbb{V}}(t+1) = j, \ \text{then } z{i}^{\mathbb{V}}(t) = 1, \ z{j}^{\mathbb{V}}(t+1) = 1, \ \text{and } z{ij}^{\mathbb{V}}(t) = 1, \ \forall (i,j) \in A \ \text{If } g^{\mathbb{V}}(t) = i, \ g^{\mathbb{V}}(t+1) = i, \ \text{then } z{i}^{\mathbb{V}}(t) = 1, \ z{i}^{\mathbb{V}}(t+1) = 1, \ \text{and } z{ij}^{\mathbb{V}}(t) = 0, \ \forall (i,j) \in A \end{cases}$$

where pi is the total number of evacuees gathered at cell i.

The number of evacuees waiting at the bus stop is updated by formula (23). This formula implies that d^b ⁱ (t) is the split demand from the total evacuation demand gathered at site cell i ∈ *R*1 within time step t.

$$\begin{cases} d_i(t+1) = d_i(t) + ai(t) - \sum{\forall b \in v_1} d_i^b(t), \ \forall i \in R_1, \ \forall t < T \ d_i(t) \geqslant 0, \ \forall i \in R_1, \ \forall t \end{cases}$$

$$(23)$$

Similarly, the number of evacuees in an automobile parking lot cell i ∈ *R*2 is given by

$$\begin{cases} d_i(t+1) = d_i(t) + ai(t) - \sum{j \in \Gamma^{-1}(l)} \beta2 \cdot y{ij2}(t), \ \forall i \in R_2, \ \forall t < T \ d_i(t) \geqslant 0, \ \forall i \in R_2, \ \forall t \end{cases}$$

$$(24)$$

Formulae (18)–(24) are connected with the BCECTM evacuation model by the following formula (25).

$$y{ij1}(t) = \sum{r=1}^{\infty} z{ij}^{\nu}(t), \ y{ij2}(t) = \sum{r=1}^{\infty} z{ij}^{\nu}(t), \ \forall (i,j) \in A, \ \forall t$$ (25)

It is valuable to note that the proposed BCECTM can accurately calculate volumes of left-turning, right-turning, and straight-going traffic according to formula (25) while a traditional CTM cannot produce accurate traffic volumes of the three directions.

To summarize, the formulae (14)–(25) model the SDVRP optimization process in a BCECTM network. All formulae (1)–(25) constitute the overall BCECTM evacuation optimization model based on SDVRP.

Decoding a chromosome through formula (26) will generate the values of independent variable zv ij(t). A fragment composed of T sequential genes indicates an evacuation path of the corresponding vehicle.

According to the characteristics of the position-coding scheme, a recursive method is established to update the positions of vehicles. If g^v(t) = i, then select the next position for vehicle v according to the following operations:

If all downstream cells in the set Γ⁻ ¹(i) have no residual capacity or have conflicts, then set g^v(t + 1) = i; otherwise, set g^v(t + 1) = j, where j is randomly selected from Γ⁻ ¹(i). In detail, if vehicle v is in cell i at time step t, that vehicle v can select to drive to an adjacent downstream cell j of cell i is dependent on the premise that cell j will have positive remaining capacity at the next time step t +1 and will not trigger any intersection conflict when vehicle v drives to cell j within time step t + 1. If all downstream cells Γ⁻ ¹(i) have reached their capacity or at least one intersection conflict will happen when vehicle v is in the intersection at time t + 1, the vehicle v can only stay in the current cell i. After the next positions of all vehicles are identified, the network information is updated immediately to ensure that the constraints are satisfied.

Second, the genetic operators of crossover, mutation, and selection are discussed. The two-point crossover is used as a crossover operator to randomly choose pairs of chromosomes with a crossover probability, until the population of next generation is completed. In the first chromosome of each pair of chromosomes, two vehicles are randomly chosen, and their last genes are used as crossover points. Between the two

$$\begin{array}{c|cccccccccc$$

Fig. 2. A chromosome generated by the position-coding scheme.

crossover points, a gene segment is also selected in another chromosome. Then exchange gene segments of the two chromosomes. This operation will exchange the driving path of the related vehicles. The mutation operation is executed by the following means. Produce a random number for each chromosome. For a chromosome, if the random number is less than mutation probability, then perform mutation for the corresponding chromosome. In this chromosome, randomly choose five percent of buses and automobiles respectively, and restore the capacity of each cell in the original paths of these vehicles, then regenerate these vehicles' routes by the model and update the capacity of each cell in the currently selected paths. The elitist retention method and roulette selection method are adopted to select chromosomes for the next generation. The elitist retention method preserves the first two best chromosomes as individuals in the next generation of the population. In the roulette-wheel selection, for the detailed operation of which one can refer to Sivanandam and Deepa (2008), a chromosome is selected among the remaining population with a probability proportional to its fitness value. The process is ended if the next generation has been cloned.

Third, the fitness function of a chromosome cn is defined as follows:

$$fitness(c_n) = \frac{1}{Z(c_n) + pc \cdot penalty(c_n)}$$ (27)

For the proposed model, we designed algorithms and wrote programs in Microsoft Visual C++ to solve the model. Other parameters are set as follows: the population size takes 50, the maximum evolutionary generation takes 600, the crossover probability takes 0.9, the mutation probability takes 0.1, and the penalty coefficient in the fitness function takes 1 000 000. Note that, similar to the termination criterion in Huang and Chen (2009) that is 50 generations with no improvement in the objective value, we set termination generation as 600 since the fitness value remains unchanged since about the 400th generations.

5.1. Calculation and analysis of BCECTM evacuation model optimized by SDVRP

The best individual was achieved when the programs ran to the 387th generation. The optimal total evacuation time for evacuating all evacuees is 359 388 time steps. The trend of total evacuation time is shown in Fig. 4.

Routes of partial buses obtained by decoding the chromosome with the best evacuation performance are listed in Table 3. In the first round trip of shuttling passengers, buses 1–20 start from cell 225, and buses 21–40 starts from cell 227. Buses choose to go to any destination cell to unload evacuees and start from the shelter just arrived to shuttle evacuees. Some buses need to shuttle two times.

Take the bus stop cell 3 as an example to observe how the persons

$$penalty(cn) = \sum{t=1}^{T} \sum{i \in CUSUR} max{0, \sum{k \in \Gamma(i)} [a \cdot y{ki1}(t) + y{ki2}(t)] + [a \cdot x{i1}(t) + x{i2}(t)] - N_i(t)}$$

$$+ \sum{t=1}^{T} \sum{i \in CUSUR} max{0, \sum{j \in \Gamma^{-1}(i)} y{ji}(t) - x_{ii}(t)}$$ (28)

where Z(cn) is the value of objective function calculated by the formula (1) for individual cn; pc is the penalty coefficient; penalty(cn) is the penalty function aiming to eliminate illegal individuals that may appear in the crossover process.

5. Case studies

The Sioux Falls network is taken as the test network. Assume that a large-scale conference is being held in the lower middle area of the network and most of the persons who attended the conference need to be evacuated. The Sioux Falls network is transformed into a BCECTM network shown in Fig. 3. Cells 1 ~ 12 are bus stop cells, cells 225 ~ 228 are destination cells, and cells 229 ~ 231 are automobile parking lot cells. The capacity of each cell is given in Table 2 according to the length and free-flow speed of its corresponding roadway sub-segment. Automobile parking lot cells 229 ~ 231 and destination/shelter cells 225 ~ 228 are assumed to be not restricted by the road conditions and be sufficiently large to accommodate all automobiles and persons to be evacuated.

We assume that 8 400 persons should be evacuated in a time period of T = 120 time steps. 6 000 persons will be evacuated by automobiles, and others (2 400 persons) will be transported by buses to shelters. There are 1 000 cars in each automobile parking lot, and each evacuation destination has 20 buses that can be used to shuttle evacuees. Each automobile and bus can carry two persons (β² = 2) and 40 persons (β¹ = 40), respectively. Suppose that 2 000 persons and 200 persons will arrive at each automobile parking lot and each bus stop, respectively, in 40 time steps. According to Section 3, the number of persons arriving at parking lots and bus stops is given by:

$$a_i(t) = \left[ exp(-0.055(t-1)^{1.22}) - exp(-0.055t^{1.22}) \right] \cdot p_i$$ (29)

consecutively get to the stop will be loaded by buses. Each bus that has picked up evacuees and the number of evacuees picked up by the bus at the stop is put in Table 4. At time step t = 16, 156 persons had been waiting at cell 3, buses 9, 10, 14, 17, and 19 carried 32, 40, 40, 40, and 4 persons, respectively, so that no evacuees should still wait. At t = 25, 30 persons had been waiting at cell 3, buses 30 and 33 carried 22 and 8 persons, respectively, but 14 persons who had gathered to cell 3 later left there. The remaining 14 persons waited there and were taken away by bus 2 at t = 50. It is observed that the traffic evacuation plan obtained by this model can split the evacuation demands at a bus stop and reasonably assign persons arriving in succession at the bus stop platform to buses to guarantee evacuees' orderly evacuation and bus capacity restriction.

Take bus 2 as an example to observe the process of shuttling evacuees of buses. The route of bus 2 is shown by the blue dash line and red dotted line in Fig. 3. Bus 2 arrived at bus stop cell 1 at time t = 12 and picked up 40 persons, then traversed the network and reached destination cell 226 at t = 34. After recovery of bus capacity, bus 2 departed from cell 225, returned into the network, and arrived at bus stop cell 1 at t = 46, where all remaining 22 persons in cell 1 got on the bus. Then bus 2 continued driving and carrying the rest 14 persons in cell 3 at t = 50, and took 4 persons in cell 8 at t = 53, thus the bus is fully loaded and finally sent 40 persons to the destination cell 226 at t = 71. Therefore, the model has achieved the goal of fully utilizing bus capacity; also, it will improve the evacuation efficiency.

The cumulative number of persons evacuated by buses varies as time goes on, as shown in Fig. 5. It is noted from the results that the final total number evacuated by buses arrived at cell 226 is significantly higher than cell 228. This is because bus stop cells are closer to cell 226 than those to cell 228, and the buses are preferred to choose a path with a shorter distance. However, constrained by cell capacity and intersection

Fig. 3. The bi-directional cell transmission network for Sioux Falls network.

conflict on the path forward to cell 226, some buses still decide to drive to cell 228 to meet these constraints. Similarly, as shown in Fig. 6, the number of arriving automobiles in cell 226 is always significantly higher than that of cell 228.

To further observe the efficiency of traffic evacuation at intersections for the BCECTM, we carried out calculations and analyses of the evacuation model based on the CTM and SDVRP. Ignoring constraints (10) and (11) will result in an evacuation model based on the CTM and SDVRP.

For the evacuation model based on the CTM and SDVRP, the cumulative number of conflicts is 1134 during the evacuation in the road network, and the cumulative vehicles affected by conflicts are comprised of 1270 buses and 23,240 automobiles, i.e., 25,780 pcus.

Take some intersections for example to observe the conflicts. For the first intersection through which traffic flows from cell 231, conflicts at time step t = 4 can be stated as follows. 55 cars from cell 230 flow along connector (145,76). 35 cars from cell 231 flow along connector (14,76)

and 30 cars from cell 231 flow along connector (14,146). Thus, there occur conflicts between cars on connecters (145,76) and (14,146). 55 cars on connector (145,76) and 30 cars on connector (14,146) are affected. These conflicts exist because the CTM lacks conflict elimination mechanism. For the third intersection through which traffic from cell 231 goes straight, conflicts at time step t = 10 can be observed as follows. 5 buses flow along connector (87,85), 3 along (178,176), and 3 along (178,85). These buses are all from cell 227. At the same time step, 90 cars at cell 86 (include 55 cars from cell 230 and 35 cars from cell 231) break up into two groups of 30 and 60 cars and drive along connectors (86,88) and (86,177), respectively. Thus, there are 3 pairs of conflicts between connectors, i.e., (87,85) and (178,176), (178,176) and (86,88), and (178,85) and (86,88). Note that all the conflicts are eliminated by BCECTM.

The intersection conflicts will cause a high risk of much delay and even chaotic traffic order, which have not been calculated and may be difficult to calculate because it is impossible to predict the consequences

of a conflict. Thus, the time to complete evacuation through the CTM will be greatly delayed. Therefore, although the total network evacuation time is less using the evacuation model based on the CTM and SDVRP than that based on the BCECTM and SDVRP, the optimal BCECTM-based evacuation plan which can guarantee the evacuation to be completed reliably in a reasonable time period is more attractive than the CTM-based one with higher risk and possibly shorter time. A reasonable explanation for longer evacuation time based on the BCECTM is that some vehicles make detours to avoid conflicts and ensure high efficiency and reliability at intersections.

5.2. Calculation and analysis of BCECTM evacuation model based on VRP

If the formula (18) is replaced by formula (30):

$$d{i}^{b}(t) = \begin{cases} z{i}^{b}(t) \cdot min{\beta{1}, d{i}(t)}, & \text{if } w^{b}(t-1) = 0 \ 0, & \text{else} \end{cases} \forall b \in v{1}, \ \forall t, \ \forall i \in R{1} \quad \textbf{(30)}$$

then the evacuation demand less than *β*1 will also occupy a whole bus so that the bus can no longer carry more persons on other cells. This evacuation strategy indicates that evacuation demands are not split and are converted to the required number of buses without remaining capacity updating for buses. Therefore, formulae (9), (14), (19), (20), and (30) belong to the VRP model, and formulae (1)–(17), (19)–(25), (30) constitute the BCECTM evacuation optimization model based on the VRP (denoted as VRP + BCECTM).

Fig. 4. The evolution of total evacuation time.

Table 2 Capacity of cells.

Ni(t)(pcu)	Cells
60	1, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28; 3, 4, 39, 40, 41,
	42; 5, 6, 43, 44, 45, 46; 7, 8, 53, 54; 9, 10, 55, 56; 29, 30, 31, 32, 33, 34,
	35, 36, 37, 38; 57, 58, 59, 60, 61, 62, 63, 64; 87, 88, 89, 90; 91, 92, 93,
	94; 115, 116, 117, 118, 119, 120; 165, 166, 167, 168, 169, 170; 207,
	208, 209, 210, 211, 212.
75	65, 66, 67, 68, 69, 72; 213, 214, 215, 216, 217, 218.
90	11, 12, 143, 144, 145, 146; 13, 14, 73, 74; 48, 49, 50, 51, 52; 75, 76, 77,
	78; 79, 80, 81, 82, 83, 84, 85, 86; 95, 96, 97, 98, 99, 100, 101, 102; 103,
	104, 105, 106; 107, 108, 109, 110; 111, 112, 113, 114; 121, 122, 123,
	124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137,
	138; 139, 140, 141, 142; 147, 148, 149, 150, 151, 152; 153, 154, 155,
	156, 157, 158, 159, 160, 161, 162, 163, 164; 171, 172, 173, 174, 175,
	176; 177, 178, 179, 180, 181, 182; 183, 184, 185, 186, 187, 188; 189,
	190, 191, 192, 193, 194; 195, 196, 197, 198, 199, 200; 201, 202, 203,
	204, 205, 206; 219, 220, 221, 222, 223, 224.

Under the evacuation strategy of VRP, the total evacuation time is 390 916 time steps. Therefore, the evacuation efficiency of the BCECTM evacuation optimization model based on SDVRP (denoted as SDVRP +

Table 3 Routes of partial buses.

| Bus No. | Driving route |---------|----------------------------------------------------------------- | # | | 1 | 225-37-35-33-31-29-165-167-169-63-61-59-57-9-55-7-53-42-4-40-16- | | 18-2-20-22-24-26-28-30-32-34-36-38-226-225-37-35-33-31-29-165- | | 167-169-171-173-175-177-179-181-183-185-187-137-135-133-131- | | 129-127-125-123-121-52-50-48-74-14-146-12-144-56-10-58-60-62-64- | | 170-168-166-30-32-34-36-38-226; | 2 | 225-37-35-33-31-29-27-25-23-21-19-1-17-15-39-3-41-54-8-56-10-58- | | 60-62-64-170-168-166-30-32-34-36-38-226-225-37-35-33-31-29-27-25- | | 23-21-19-1-17-15-39-3-41-54-8-56-10-58-60-62-64-66-68-70-72-213- | | 215-217-219-221-223-120-228; | …
19 | …
225-37-35-33-31-29-27-25-23-21-19-1-17-15-39-3-41-43-5-45-47-49- | | | 51-96-98-100-102-104-106-108-110-182-180-178-88-90-195-197-199- | | 116-118-120-228-227-119-117-115-200-198-196-89-87-85-83-81-79- | | 77-75-13-73-46-6-44-42-4-40-16-18-2-20-22-24-26-28-30-32-34-36-38- | | 226; | 20 | 225-37-35-33-31-29-165-167-169-63-61-59-57-9-55-143-11-145-76-78- | | 158-156-154-58-60-62-64-171-173-175-177-173-181-112-114-116-118- | | 120-228-227-119-224-222-220-218-216-214-71-69-67-65-63-61-59-57- | | 9-55-7-42-4-40-16-18-2-20-22-24-26-28-30-32-34-36-38-226; | 21 | 227-119-224-222-22093-91-89-87-177-179-181-109-107-105-103-164- | | 162-160-77-75-13-73-46-6-44-42-4-40-16-18-2-20-22-24-26-28-30-32- | | 34-36-38-226-225-37-207-209-211-71-69-67-65-170-168-166-27-25-23- | | 21-19-1-17-15-39-3-41-43-5-45-47-49-51-122-124-126-128-130-132- | | 134-136-138-140-142-206-204-202-116-118-120-228; | 22 | 227-119-117-115-200-198-196-89-87-85-83-81-79-158-156-154-9-55-7- | | 53-43-5-45-47-49-51-96-98-100-102-104-106-108-110-112-114-116- | | 118-120-228-227-119-224-222-220-93-91-195-197-199-113-111-109- | | 107-105-103-101-99-97-95-52-50-48-46-6-44-54-8-56-10-58-60-62-64- | | 170-168-166-30-32-34-36-38-226; | …
39 | …
227-119-224-222-220-218-216-214-71-69-67-65-171-173-175-177-179- | | | 181-109-107-105-103-101-99-97-95-152-150-148-146-12-144-7-53-43- | | 5-45-74-14-146-12-144-56-10-153-155-157-77-75-13-73-46-6-44-42-4- | | 40-16-18-2-20-22-24-26-28-30-32-34-36-38-226; | 40 | 227-119-117-115-53-78-76-74-27-25-23-21-19-1-17-15-39-3-41-5-14- | | 67-46-48-72-38-40-76-32-34-88-94-225-37-35-33-31-29-27-25-23-21- | | 19-1-17-15-39-3-41-8-10-58-60-62-64-170-168-166-27-25-23-21-19-1- | | 17-15-39-3-41-54-8-143-11-145-13-73-46-6-44-54-8-56-10-69-71-50- | | | | | | | ------|--|--|--|--| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

52-79-66-86-116-118-120-228.

Fig. 5. Number of persons evacuated by buses.

Table 4 Dynamic traffic in cell 3.

Bus No.#	Arrival time step	Number of persons picked up	Bus No.#	Arrival time step	Number of persons picked up
9	16	32	19	16	4
10	16	40	30	25	22
14	16	40	33	25	8
17	16	40	2	50	14

BCECTM) is 8.1% higher than VRP + BCECTM.

Partial results of VRP + BCECTM are given by Figs. 7 and 8. The number of persons evacuated to the destinations by buses amounted to half of the total evacuation demand at t = 61. However, 432 persons still did not reach the destinations until t = 120. In contrast, according to the results from SDVRP + BCECTM, half of the total evacuees reached the destinations at t = 49, and all evacuees were transported to the destinations at t = 103; for automobiles, as shown in Fig. 8, the evacuation process based on VRP + BCECTM is also slower than SDVRP + BCECTM, the reason for which is that buses occupy more cells when adopting the

Fig. 6. Number of persons evacuated by cars.

Fig. 8. Number of persons evacuated by cars.

Fig. 9. Number of persons evacuated by buses.

evacuation strategy of VRP than selecting SDVRP.

Therefore, the new evacuation method based on the BCECTM and SDVRP is better than the evacuation method based on the BCECTM and VRP.

5.3. Comparison of evacuation results under different levels of evacuation demand

Assume that the total number of cars to be evacuated is changed, the number of persons evacuated by cars will accordingly change to twice times the number of cars, while other settings remain unchanged, as stated at the beginning of Section 5. Then we analyze the effect of two levels of evacuation demand, i.e., there are 500 and 2000 cars waiting to be evacuated in each automobile parking lot.

If each automobile parking lot has 500 cars to be evacuated, the results are shown in Figs. 9 and 10. The decrease of car evacuation demand from 1 000 (in Sections 5.1 and 5.2) to 500 vehicles leads to relatively faster car evacuation. Another consequence is the reduction of interferences to buses; thus, more buses could reach the nearest cell 226 Fig. 7. Number of persons evacuated by buses. earlier, the gap of the number of persons transported by buses to cell 226

Fig. 10. Number of cars evacuated.

Fig. 11. Number of persons evacuated by buses.

and cell 228 is widened, and the evacuation completion time is shortened.

If the number of cars to be evacuated in each automobile parking lot is 2 000 that is two times as many as those in Sections 5.1 and 5.2, then the results are shown in Figs. 11 and 12. The speed of car evacuation slows down due to the substantial increase in the total number of cars. At the same time, buses face more interference, so more buses have to traverse a longer distance to cell 228, there is no significant difference between the final number of evacuees arriving at cell 226 and cell 228, and the final time to complete the evacuation is delayed.

We define the congestion degree of a cell as the ratio of the number of vehicles to the capacity of the cell at a specific time t. The frequency of occurrence of the λ-value intervals (different load levels) is shown in Table 5. When car evacuation demand is doubled, the occurrence of larger λ-value increases significantly.

5.4. Comparison of evacuation results from different arrival distributions

In this section, the results obtained from another arrival distribution — a uniform distribution are compared with those from the Weibull

Fig. 12. Number of cars evacuated.

Table 5 Frequency of occurrence of λ-value intervals.

	Value interval
Total number of cars	1	[0.9, 1)	[0.8, 0.9)	[0.7, 0.8)	[0.6, 0.7)	[0.5, 0.6)	[0,0.5)
1500 3000 6000	0 76 180	0 8 39	0 12 90	0 23 72	10 60 179	60 47 276	27,650 27,494 26,884

Fig. 13. Number of persons evacuated by buses.

distribution.

If the distribution of arriving at bus stops and car parking lots is changed to uniform distribution and other settings remain unchanged as stated at the beginning of Section 5, the total network evacuation time is 347 420 time steps, and the results are shown in Figs. 13 and 14. For uniform distribution, the residence time of evacuees staying at bus stops is less, but the number of persons evacuated to the destinations by buses is more dispersed between the destinations, and the evacuation time required is longer. For uniform distribution, before the completion of

Fig. 14. Number of cars evacuated.

Table 6 Frequency of occurrence of λ-value intervals for uniform distribution.

Value	1	[0.9,	[0.8,	[0.7,	[0.6,	[0.5,	Under
interval		1)	0.9)	0.8)	0.7)	0.6)	0.5
Frequency	0	0	14	44	44	70	27,548

automobiles evacuation, they arrived at destination evenly during the whole evacuation; while for Weibull distribution, the number of arriving cars increased rapidly at early stages and slowed down in the later stages. Although the total evacuation time when using uniform distribution is less than that using Weibull distribution, cell congestion shown in Table 6 is slightly increased compared with 1500 total car demands and Weibull distribution.

6. Conclusions

Dynamic evacuation problems using multiple modes including bus is a significant practical problem for large-scale activities. To this end, this paper formulizes an optimization model in the framework of micromacroscopic integration, i.e., a bi-directional conflict-eliminating CTM integrated with an SDVRP model.

For a two-way, multilane, and lane direction reversible road network with potential intersection traffic conflicts, the BCECTM is presented based on the typical CTM. The BCECTM has several advantages. It is capable of avoiding conflicts at intersections and can yield accurate outflows of straight-going, left-turning, and right-turning movements from any approach to an intersection. It is convenient to organize forward and reverse bus traffics. However, it is difficult to track an individual vehicle. Considering the advantages of SDVRP, i.e., the effective split of evacuation demands and tracking every vehicle (especially shuttle buses), the BCECTM and SDVRP are integrated to formulate a new evacuation optimization model. A genetic algorithm based on a position-coding scheme is designed to solve the proposed optimization model. Experiments show that the proposed model can well describe the trajectory and propagation mechanism of vehicles, and can improve bus capacity utilization. The evacuation model based on the BCECTM and SDVRP is compared with that based on the BCECTM and VRP, demonstrating that the former is superior to the latter. Comparing the CTM and BCECTM evacuation models optimized by SDVRP shows that the BCECTM is superior to the CTM because it eliminates conflictions at intersections thus improve intersection evacuation efficiency and whole evacuation reliability. Analyses of evacuation results under different levels of and different arrival distribution of evacuees demand further validate the proposed model. In this work, the evacuation demand is assumed fixed; however, in reality, it may be uncertain, thus a direction for future study is robust evacuation planning. Although the proposed model can depict lane reversal, about which, however, no further calculation is executed, a prospective study can investigate the impact of lane reversal and road degradation on robust planning. Last but not the least, the two factors of time of day (Fu and Wilmot, 2004; Fu et al., 2007; Yazici and Ozbay, 2010) and risk distribution of the evacuation area (Chiu and Zheng, 2007; Chiu and Mirchandani, 2008; Madireddy et al., 2015; Hsu and Peeta, 2015) are important for the evacuation efficiency and priority evacuation problem, respectively. According to the method about how these factors are modeled in previous evacuation models, the proposed model can be modified and expected to generate meaningful results.

Acknowledgements

This work was supported by the National Natural Science Foundation of China [grant number 51468020]; the Natural Science Foundation of Jiangxi Province of China [grant number 20181BAB206044]; and the General Project of Key R&D Program of Jiangxi Province [grant number 20192BBG70076 & 20203BBG73072].