1. Introduction
With the rapid development of global urbanization, the scale of cities is becoming larger and larger, and the internal structure of buildings is becoming more and more complex, which makes indoor emergency evacuation more difficult. Once an emergency such as a fire occurs indoors, it will cause huge property losses and casualties if there is no efficient evacuation strategy. Therefore, indoor emergency evacuation has been paid more and more attention. Existing methods are mainly divided into two categories: evacuation drill and computer simulation [
1,
2,
3,
4]. The former can provide more accurate and richer information about emergency evacuation, but it has obvious drawbacks such as high cost, time-consuming, and immorality. So, this kind of method is usually used to verify the effectiveness of evacuation strategies in the real world. Computer-aided simulation has become the dominant method for studying emergency evacuations [
5].
Nowadays, building information modeling (BIM) and geographic information systems (GIS) are widely used for indoor emergency evacuation. BIM-GIS system has two meanings: one is the integration of the underlying technologies, and the other is the integration at the application level. This paper discusses BIM-GIS at the application level to solve the emergency evacuation problem. Before path planning, a route network model can be generated from the BIM model. Then, all evacuation strategies are obtained by network analysis of GIS. When an emergency occurs indoors, all occupants need to be evacuated immediately and decision-makers are required to formulate the best evacuation strategy in a short time. Therefore, an efficient evacuation strategy must ensure not only the shortest TET but also the shortest formulating time of it. At present, evacuation strategies can be roughly divided into optimization-oriented and simulation-oriented [
6]. The research in this paper belongs to the former. Generally, it is easy to cause congestion around the emergency exits or in the corridors when there are many evacuees in a building. In existing studies, although some strategies can reduce the congestion around exits, congestion is inevitable in the evacuation process. Occupants at the congested locations can only run slowly towards exits. The staged evacuation strategy transforms the stagnation time of escape groups into their delay time to solve congestion, but it lacks the optimization of safety exit selection.
Based on staged evacuation, to improve the utilization rate of exits and strengthen the applicability of the algorithm in different scenarios, an indoor emergency evacuation algorithm based on time equalization is proposed. In general, indoor emergency evacuation is a multi-exit evacuation problem that needs to clarify the escape path and departure time of every occupant. To simplify the problem, the multi-exit problem is transformed into multiple single-exit problems by partitioning the whole evacuation zone into several evacuation zones, each of which has one safety exit. The proposed algorithm allocates all occupants to each zone and balances the evacuation time of each zone. Our contributions include: (1) The occupancy time of each exit is introduced as the basis of the evacuation partition, and the evacuation time of each zone is balanced to get the shortest TET; (2) The method of group merging is proposed to further optimize the result of evacuation zone partitioning; (3) The proposed algorithm is generally applicable to various evacuation scenarios.
The rest of this paper is organized as follows.
Section 2 reviews related work.
Section 3 describes the problem and presents our method of solving it.
Section 4 gives the proposed algorithm and its time complexity.
Section 5 illustrates the results of the algorithm, evaluates its performance and effectiveness by a series of tests, and gives a testing simulation.
Section 6 concludes the paper.
2. Related Work
Crowd evacuation models are mainly divided into macro-simulation models and micro-simulation models from the perspective of simulation. Similarly, evacuation strategies can also be divided into macro and micro. Macro strategies focus on controlling network flow while micro ones focus on simulating individual behaviors. Micromodels including the cellular automata, social force, lattice gas, and agent-based models are mainly used to simulate some common self-organization phenomena such as transcendence behavior, herding behavior, re-entry phenomenon, small group phenomenon, and so on [
7,
8,
9,
10]. Some guiding strategies can rationally evacuate the crowd and reduce congestion [
11,
12,
13]. Lei et al. studied the impact of patience on evacuation efficiency and proved that the strategy of queuing to evacuate is best when the crowd density is low, otherwise choosing the exit with lower density to escape is the best choice [
14]. Therefore, TET and crowd density should be balanced among all safety exits during the evacuation [
15]. To resolve the problem of underutilization of exits, many strategies have been proposed. Xu et al. proposed an optimized regional division [
16]. Yue et al. integrated many strategies based on distance and time [
17]. Kurdi et al. assigned the same number of occupants to each exit [
18]. Jin et al. classified occupants according to the degree of danger to avoid the congestion caused by the excessive crowd density [
19]. Rozo et al. designed an evacuation strategy that considered various behaviors and multiple paths of occupants [
4]. Other scholars discussed the evacuation strategies of high-rise buildings by using stairs and elevators [
20,
21]. All the evacuation strategies above are based on the micro models and their time complexity is large, which causes the slow simulation.
The macro models focus on the overall characteristics of the crowd. Network-based models usually abstract the evacuation problem into a network flow problem [
22]. The time-extended network can dynamically allocate evacuees and make evacuation strategy more reasonable but its time complexity is very high [
23,
24]. A method was proposed to keep the evacuation process going smoothly by controlling the speed and flow rate of occupants [
25]. Noh et al. planned evacuation routes for occupants with different speeds to minimize the blocking effect [
26]. In addition, there are many strategies for path planning to reduce the exposure time of occupants in a hazardous environment [
27,
28]. Liu et al. proposed a dynamic guidance method based on digital twin [
29]. Zheng et al. used the Vickrey model to reduce queuing time and to improve the utilization rate of all exits [
30]. Yang et al. adopted the principle of “balancing evacuation time” and prioritized evacuation of floors that required a long time to escape [
31]. Xiong et al. also proposed a distinguishing model that can avoid congestion and stagnation [
32]. Based on the Dijkstra algorithm, Cao et al. proposed an intelligent path optimization method for crowd emergency evacuation [
33]. Jin et al. suggested a method of finding congestion and gave a more targeted evacuation strategy [
34]. Taneja et al. proposed an optimization model of a two-layer network to ensure the change of optimal capacity and more efficient evacuation [
35].
When a large number of occupants escape from the building where a fire breaks out, path conflicts are easy to occur. It is necessary to adopt reasonable methods to solve these conflicts because they will cause a severe reduction in evacuation efficiency. Simultaneous evacuation increases the congestion on paths within a short period, which makes it difficult for escape groups to evacuate according to the prescribed plan. This behavior further aggravates the path conflicts with other escape groups. The partitioned and staged evacuation planning (PSEP) algorithm [
36] adopted an evacuation partitioning strategy to assign a large number of occupants to each emergency exit and a staged evacuation strategy to compute out the escape path and delay time of each group. The algorithm is better than the distance-based staged algorithm in evacuation efficiency and superior to the algorithm proposed by Li [
37] in computation efficiency. However, the PSEP algorithm can achieve better results only in the condition of high density and even spatial distribution of occupants and takes no consideration of low density of occupants, the uneven spatial distribution of occupants, and capacity of exits. Therefore, an improved partitioned and staged evacuation algorithm based on time equalization is proposed in this paper, which overcomes the shortcomings of the current staged evacuation algorithms and further reduces TET.
3. Methodology
To evacuate all occupants as quickly as possible is what all decision-makers need to consider when an emergency occurs indoors. In general, indoor population density is a key factor affecting evacuation strategy. Occupants can choose the nearest exit to escape without any congestion when the density is low. However, the same choice will cause congestion in the evacuation process when the density is high. So, the utilization ratio of each exit should be improved to reduce congestion.
Figure 1a is an illustration of the evacuation problem. Occupants are grouped according to their proximity. Emergency evacuation planning is carried out in a group, that is, a group of evacuees is seen as a whole. The related variables used in this paper are listed in
Table 1.
Evacuation strategies can be divided into simultaneous evacuation and staged evacuation. Simultaneous evacuation means that in case of emergency, all pedestrians will escape at the same time. But in case of congestion, they need to wait until the front groups pass through. Waiting in a crowded place is not a wise choice because increased congestion can slow the evacuation speed of occupants. The situation will reduce traffic efficiency and even lead to serious accidents such as stampedes. Staged evacuation is an evacuation strategy based on evacuation partitioning, the advantage of which is that occupants in different zones have no path conflicts with each other. Meanwhile, occupants in the same zone can avoid path conflicts by setting a delayed departure time for each group. The delay time of each group should be as short as possible to reach the minimum TET, as shown in
Figure 1b. Therefore, staged evacuation is chosen as the basic strategy for emergency planning in this paper.
We mainly study the impact of the density of occupants, the spatial distribution of occupants, and the capacity of exits on evacuation partitioning and TET respectively. To simplify the problem, it is assumed that the speed of all groups is equal. The order of evacuation is in accordance with the principle of “first in first out, second delay” [
36]. That is, the delay time of each group in the same zone is calculated according to the path length in ascending order. The formula of delay time is as follows:
where,
denotes group
in zone
. If the delay time of the group is less than 0, it is set to 0. If
, it means that group
and group
are evacuated end-to-end. The simultaneous evacuation strategy uses the following equation to calculate the evacuation time of group
.
It is assumed that escape groups evacuate immediately, namely . is the time from the group beginning to escape to the last member of the group leaving the exit, regardless of the congestion condition. The larger is, the longer the queueing time will be. Obviously, it is very prone to congestion during the evacuation in the situation.
During the staged evacuation, the delay time of groups can avoid waiting in the queue during the evacuation, namely
. Therefore, the evacuation time of groups using the staged strategy is equal to the delay time plus the travel time. In fact, the delay time of groups takes the place of their queuing time. The evacuation time of each zone is determined by the maximum evacuation time of all groups in the zone. The equation of
is as follows:
Similarly,
is determined by the maximum evacuation time of all zones. The equation is as follows:
Each zone has a safety exit. TET is one of the key factors to determine the quality of evacuation strategies. It can be seen from Equation (5) that the TET of the staged evacuation strategy is jointly determined by the evacuation time of all zones. Therefore, the partitioning method is the key to the staged evacuation strategy and is one of the important factors to determine evacuation efficiency. Equation (6) is the objective function, indicating that a good staged strategy should make the maximum evacuation time of all zones as small as possible. To shorten TET, a reasonable partitioning method taking into account the density and spatial distribution of occupants and the capacity of exits should be proposed to make the evacuation time of each zone approximately equal.
The partitioning methods of staged evacuation are generally divided into two categories: the distance-based partitioning method and the partitioning method of balancing the number of occupants at all exits. The former is suitable for low crowd density, while the latter is suitable for evacuation partitioning with high density. The PSEP algorithm is a typical staged evacuation algorithm based on population equalization. In this paper, a partitioning method of balancing the evacuation time at all exits is suggested, to make it better applicable to various indoor evacuation scenarios.
According to Equation (4) and the principle of “first in first out, second delay” [
36], the evacuation time of each zone is determined by the evacuation time of the group with the longest path length in each zone. Dijkstra algorithm is used to determine which group in the zone is the group currently extended. The evacuation time of the group is the current evacuation time of this zone. As the zone is expanded, the current evacuation time of the zone is updated accordingly. When the last group is extended, the evacuation time of the group is the final evacuation time of the zone. According to Equation (6), the evacuation time of the last group extended in the current zone is taken as a function of the occupancy time of each exit for expansion, which includes the delay time and travel time of the group. The travel distance of the group consists of the length and path length of the group (
Figure 2). Then the exit with the minimum occupancy time should be chosen to expand its evacuation zone each time. Finally, the process of evacuation partitioning based on time equalization is finished. The function of
is
In the process of evacuation partitioning, the Dijkstra algorithm may cause an imbalance of evacuation partitioning if there are branch paths. The following pictures of a simple partial route network illustrate this problem and the proposed solution for it, as shown in
Figure 3.
In
Figure 3, node D is the branch of node A. The serial numbers of nodes indicate the order of path length. For example, the path length of group B is longer than that of group A. The path length of group D is longer than that of groups B and C. According to the principle of “first in first out, second delay”, groups B and C are extended by zone E before group D. If the Dijkstra algorithm is used, group D must pass through node A. Group D is extended by default when the node A is being extended. If the evacuation time of group A is used as the evacuation time of zone E, it must be shorter, because the occupancy time represents the evacuation time of the zone. Therefore, the evacuation time of group D should be used as the current occupancy time of zone E, so that some groups, such as groups B and C, may not be extended by zone E. The group merging method is used to increase further the balance of expansion of zones. Node operation needs to be executed before the group merging method as the following steps:
Add non-exit nodes to array M, and create an empty array B;
Add a virtual node, which connects to all exits, and the length of new arcs is set to 0;
The virtual node is set as the search starting node;
Use Tarjan’s algorithm to search a cut vertex in the unvisited nodes, then mark the visited nodes;
If there is a cut vertex, go to the next step; otherwise, break;
Except for the cut vertex, add all nodes in the bi-connected component to the array B, return step (4).
If there are sink nodes in the bi-connected component, all groups can choose their own exits belonging to the bi-connected component to escape without going through the cut vertex. Step (2) excludes these bi-connected components.
The general steps for the group merging method are as follows:
Determine the currently extended node g and the corresponding exit E (details about this are described in the next section);
Select the group closest to the exit E in the array B;
If , go to the next step;
Calculate the delay time of the group , and update the occupancy time of the exit E with the evacuation time of the group ;
Remove the group from arrays M and B, return step (2).
The expansion results partitioned by the group merging strategy are shown in
Figure 3b. When node A is extended, the evacuation time of group A is actually the evacuation time of group D because group D is merged by node A. The continuity of groups will not increase the evacuation time of the current zone. So, the evacuation time of group D can be used as the occupancy time of the exit
e. The PSEP algorithm can also use the group merging method to solve the problem of the lack of local information on buildings. It only needs to add the population on branch nodes to their own trunk node.
4. Algorithm
Based on the problem and solution proposed in
Section 3, a staged evacuation algorithm based on the balance of occupancy time of exits is proposed in this paper. The algorithm considers the change of different densities and spatial distribution of occupants and the capacity of exits.
4.1. Algorithm Idea
The PSEP algorithm is a multi-exit emergency evacuation strategy proposed for indoor crowded conditions, but it uses the partitioning strategy of “balanced evacuation” based on the population of each exit without consideration of the density and spatial distribution of indoor population and capacity of exits. Therefore, we proposed the partitioning strategy of “balanced evacuation” based on the occupancy time of each exit to improve the evacuation efficiency and to shorten TET. The proposed method uses the improved Dijkstra algorithm to assign all groups to different exits. During the partitioning process, the evacuation time of each group is taken as the occupancy time of the safety exits to form evacuation zones. The basis of partitioning is to ensure that the occupancy time of each exit is approximately equal. For all groups in the same zone, the “delayed waiting” strategy is implemented, and the time window for each group to occupy the safety exit is calculated according to the path length to ensure that no path conflicts occur. In addition, it is proposed to transfer the information of branch nodes to the trunk nodes for group merging. Therefore, the basic idea of the proposed algorithm for constructing zones is: (1) Taking each exit as the starting point, the Dijkstra algorithm and the group merging method are used to expand the evacuation zone according to the path length, and the occupancy time of each exit is updated; (2) Choose the exit with the minimum occupancy time for the next expansion of zone until all nodes are assigned to different exits. If all nodes are extended, the process of evacuation partitioning is completed.
4.2. Algorithm Description
Different from the PSEP algorithm, the proposed algorithm carries out the partitioning and the calculation of the delay time at the same time, while the PSEP algorithm performs the partitioning first and then calculates the delay time. The detailed algorithm flow is described as follows:
Input: indoor route network, the location and capacity of exits, the number and spatial location of escape groups
Output: the escape path and delay time of each group, TET
Algorithm procedure:
Node operation, get arrays M and B;
Initialize the occupancy time of each exit as zero;
Select the exit E with the smallest occupancy time value, then use the Dijkstra algorithm to find the group g closest to E in M;
Calculate the delay time of g, remove g from arrays M and B, then update the occupancy time of the exit E;
If there is a node in the array B, go to the next step; otherwise, go to (7);
Group merging, seen as the previous section for specific steps;
If there is a node in the array M, go to (3); otherwise, go to the next step;
Calculate TET.
The flow chart of the algorithm above is shown in
Figure 4.
4.3. Performance Evaluation Index
Optimal Performance Statics (OPS) and Mean Non-flow Statics (MNS) are usually used to quantify the efficiency of multi-exit evacuation. Especially, OPS is mainly used to evaluate the utilization ratio of all exits [
13,
38], whose definition is as follows:
where,
is the evacuation time of the last group in zone
, which is equal to the evacuation time of zone
. The value of OPS is in the interval [0,1]. The closer OPS is to 0, the better the utilization ratio of each exit is. On the contrary, the closer it is to 1, the lower the utilization ratio of other exits is, which indicates that most groups escape to the same exit. In this paper, OPS is mainly used in scenarios with a large population. It should be noted that the OPS of the evacuation in the group may be slightly larger than that in individuals. In addition, TET is used to judge the quality of evacuation strategies. The average path length of all evacuees is used to evaluate the safety during evacuation.
4.4. Time Complexity
Before the group merging method is carried out, the improved Tarjan’s algorithm is required first to find out all branch nodes with the time complexity .
Each time the evacuation zone is expanded, the exit with the minimum occupancy time will be determined. The time complexity is . It is assumed that there are nodes in array B. Each node is extended by the Dijkstra algorithm, and the path length of all nodes in the network needs to be compared times. In the best case, all nodes in array B are extended in the first loop, whose time complexity is . In the worst case, no group is extended by group merging, whose time complexity is . As , the final time complexity is .
After the process of evacuation partitioning is completed, the evacuation time of each zone needs to be compared to get TET, and the time complexity is .
6. Conclusions
The density and spatial distribution of occupants and the capacity of exits have an impact on emergency evacuation indoors. Based on the analysis of staged evacuation algorithms, a novel multi-zone staged indoor emergency evacuation algorithm based on time equalization is proposed. It can perform excellently in various evacuation scenarios and significantly shorten TET. In the partitioning process, a group merging method is proposed to overcome the drawbacks of the PSEP algorithm that causes unreasonable expansion of evacuation zones. In addition, a “delayed waiting” strategy is adopted to ensure that there will be no path conflicts during the evacuation process. The strategy makes the indoor emergency evacuation more orderly, safe, and efficient. In addition, the proposed method can also be applied to outdoor emergency evacuation, which has strong operability and robustness.
As a staged evacuation method considering the balance of time, the proposed algorithm can get better planning results. However, it does not take into account the real-time state of the route network. In the case of an actual fire disaster, some nodes and arcs in the network may become impassable. Therefore, how to combine the staged evacuation strategy with dynamic path planning needs further research. In addition, the strategy of waiting in the original place of emergency needs to be further discussed, because it may be unsafe and does not apply to the occurrence of local disasters such as an indoor fire.