2.1. Macromodel of the Evacuation Path Problem
Macromodel research can study the model and algorithm of evacuation path planning. A good evacuation path planning model and algorithm can give a better evacuation planning scheme to effectively shorten the evacuation time. Hamacher [
9] presented a detailed review of the macromodels of evacuation problems and introduced several models based on network flow. For example, minimum turnstile cost dynamic network flow models are used to estimate the average evacuation time of each evacuation individual; maximum dynamic flows and universally maximum dynamic flow models are used to calculate the maximum number of people evacuated within a given time limit; the fastest flow method (QFM) is used to estimate the minimum time to evacuate a certain number of people [
10]. Most of these models assume that travel time is a fixed constant. If the nonlinear relationship between crowd density and evacuation time is considered, the complexity of the model will be greatly increased [
11]. In this paper, the method to deal with the nonlinear relation replaces the nonlinear relation with an approximately linear function.
From the point of view of solving methods and algorithms of the macro evacuation model, it can be divided into a polynomial algorithm based on linear programming to solve the optimal evacuation scheme and various heuristic algorithms based on nonlinear programming [
12,
13,
14,
15]. The first method mainly uses the network flow method, especially the dynamic network flow method, to give the polynomial algorithm for various evacuation problems. A dynamic network [
16] is defined on a directed graph
, in which vertex set
V contains source point sources, endpoint sinks, and a medium point set. Every arc
has a non-negative capacity uxy and a non-negative transmission time
. The two classical dynamic network flow problems are the maximum dynamic flow problem (MDFP) and the quick flow problem (QFP). MDFP refers to the dynamic flow of sinks to the endpoint under a given time limit T. QFP refers to moving a predetermined amount of traffic from a source point to an endpoint [
17]. Burkard [
18] presented the first strong polynomial algorithm for the QFP problem in the case of a single source point and single endpoint. Hoppe and Tardos [
19] gave a polynomial algorithm for the evacuation problem with a fixed number of source points and endpoints. For a multi-source point and multi-endpoint QFP evacuation problem, they first used the binary search method to give an upper bound of optimal time T. Then, a polynomial algorithm of QFP through multi-source multi-end sinks was given by the ellipsoid method [
20].
In addition to the dynamic network flow method, many studies use classical network flow methods, such as the algorithm of the least short circuit, least cost flow, and maximum flow to solve the evacuation path optimization problem. Dunn and Newton [
21] used the maximum flow method to solve the problem of evacuation path allocation, aiming to transfer the evacuation crowd to the optimal path as far as possible within the capacity of the road network from the dangerous area to the safe area. Yamada [
22] applied the least cost flow problem to allocate evacuation traffic paths and proposed the shortest evacuation plan (SEP), aiming at minimizing the total journeys of all the people to be evacuated to the designated shelter. Therefore, according to this optimization goal, each vehicle in the traffic network will choose the exit of the area nearest to it for evacuation. However, in a complex traffic network, such allocation rules can easily cause traffic congestion. In the lane-based evacuation network flow model proposed by Cova [
23], the extended minimum cost flow model was established to prevent intersection conflicts and limit intersections while minimizing the total travel distance. Finally, the model outputs the possible road map of each intersection.
In summary, most researchers describe the evacuation problem as a network flow graph with nodes and edges with capacity constraints, in which some nodes are called source points, and each source point has a certain number of evacuees; and some nodes are called endpoints, and each endpoint can accommodate a certain number of evacuees. The edges in the figure represent evacuation paths. The evacuation problem is to evacuate all the people to the destination by allocating a reasonable number of people to each evacuation path to minimize the total evacuation time. The advantage of using the network flow theory to solve the problem is that the evacuation problem is transformed into a classic network flow problem and can be solved using the mature polynomial network flow algorithm, which is convenient and effective. However, the network flow problem can generally only solve the linear programming problem. The abovementioned articles can only solve linear problems; that is, the possible nonlinear relationship between crowd density and evacuation time cannot be considered in the model. Therefore, the model cannot reflect some evacuation traffic phenomena, such as slowing down due to traffic congestion.
2.2. Nonlinear Evacuation Planning
The nonlinear programming of the evacuation problem mainly considers the nonlinear relationship between crowd density and evacuation speed. At present, there is no polynomial algorithm for solving the nonlinear programming of evacuation problems, only a variety of heuristic algorithms. Pursals [
24] considered building evacuation when crowd movement speed was affected by crowd density. First of all, referring to Nelson and McLennan’s analysis of crowd movement speed in an emergency, they gave the convex function relation between crowd evacuation time and crowd number at each exit [
25]. This function is related to the length and width of the exit, the area occupied by the crowd before entering the exit, and the walking speed of the crowd under normal conditions. Then, the inverse function is the function of the time required to evacuate a specific number of people. In this paper, the inverse function of the evacuation function of all exits is summed, and the functional relationship between the total number of evacuations and evacuation time is obtained. Therefore, given the total number of people evacuated, K, the time required to evacuate them can be found by the function, M. Meanwhile, Carey [
26] used a piecewise linear function to approximate the nonlinear relationship between evacuation time and the number of people and established a linear model to solve it.
As mentioned before, there are two types of methods for solving the evacuation problem in the literature. One is to model the problem as nonlinear programming and design a heuristic algorithm to solve it. The other is to model the problem as a network flow problem. The first type of method generally obtains an approximate solution, and the approximation ratio is not guaranteed. The advantage of the second method is that the evacuation problem is transformed into a classic network flow problem, and can be solved by using the mature polynomial network flow algorithm, which is convenient and effective. In this paper, we adopted the second method, and the nonlinear evacuation time with respect to the crowd density is considered in the model.