Emergency evacuation is an important issue in public security. To make a considerate plan, various situations are presented including blocking the accident area and letting the emergency access path available. In order to offer dynamic evacuation routes due to different circumstances, a multistory building evacuation model is proposed. Firstly, to analyse the patency of the building, an evacuation formula is applied after binary processing. The function of evacuation time and some other parameters is given by means of regression analysis. Secondly, the cellular automata (CA) algorithm was applied to illustrate the effect of the bottleneck. The response of evacuation time could be approximately optimized through calculating time step of the CA simulation. Finally, the value of maximum evacuation population density could be determined according to the analysis of CA simulation results, which was related to the switch state of the emergency channel. The emergency evacuation model was simulated by using the Louvre museum as an example. The results of the simulation presented some feasible evacuation routes in all kinds of situations. Furthermore, the functional relationship would also be given among evacuation time with the diversity of tourists, pedestrian density, and width of exits. It can give a different perspective that the multistory building evacuation model shows excellent adaptability to different circumstances.
In recent years, there have been many terrorist attacks around the world, which poses a great threat to the safety of residents and tourists [
This paper would discuss the evacuation of multistory buildings based on the Louvre terrorist attack [
In this paper, the summary and unnoticed things of related work were studied and discussed in Section
The evacuation system in the building has been a hot issue over recent decades [
Early works proposed a classic problem which is a highly transient, stochastic, nonlinear, integer programming problem [
A preliminary model was proposed by Liu et al. [
Ma et al. [
Xiong et al. [
An accurate evacuation plan needs a schematic diagram of the floor plan, which thoroughly described the architectural details of Louvre. It is reasonable to put the structure map of Louvre into binarization as a pretreatment after analysing the map [
The binarization of topographic distribution on the first floor of the Louvre.
When an emergency situation happened, exits would be filled with evacuees immediately. The number of people in each exit of the exhibition area will increase sharply due to the panic sensation of the visitors. Visitors will peak the human density at the exit which exceeds the maximum operational capacity. Such disturbance will finally result in congestion. This kind of phenomenon, which limited people’s activity space, is defined as the bottleneck in this paper. As the maximum allowable pedestrian movement is not enough for evacuation flow of the exit, a semiellipseshaped crowded area will appear at the exit. Hence, this paper investigated the extent of bottlenecks by examining the characteristics of the semiellipse area.
As shown in Figure
An illustrative diagram of the bottleneck.
The bottleneck is a dynamic process which starts from generation to enlargement and then disappearance. The approximate semielliptical shadow area near the exit in Figure
In terms of the patency rate, after obtaining the binary matrix of Louvre’s topographic distribution, all the possible bottlenecks in the certain floor were analysed on the basis of the binary graph. Consequently, the
As for the index of the similarity degree, individual with larger correlation coefficient “
Euclidean distance was used as the evaluation index in terms of the planar structure of the Louvre in this paper. Exponential attenuation function can establish the discriminant function based on the Euclidean distance. The exponential function (base
Then, the evaluation system was applied to the binary matrix of the Louvre terrain distribution which obtained the patency coefficient of each point in the matrix. The distribution is shown in Figure
Patency distribution map of each area on the first floor of the Louvre.
The channel may be congested, as shown in Figure
If turning the image into 3D, congestion would be more likely to occur when the contribution rate was lower. The potential point of the bottleneck signed as “
3D diagram of regional patency on the first floor of the Louvre.
The Louvre is a multistory building. The model building and algorithm design are quite different compared to singlestory buildings. There must be some common parts due to the limited path capacity and common areas among floors. Higher source points need to share certain sections with lower source points. The actual flow of these shared sections is generally affected by the convergence of staff on each floor. Hence, the pedestrian movement “
Before construction, the rules for the multistory building evacuation algorithm were defined as follows [
Rule 1: if
Rule 2: let
Rule 3: let
In order to optimize the whole evacuation model of Louvre, the minimization of evacuation time was taken as the objective function. Considering the existence of the common section, the floor requiring longer evacuation time should be evacuated first. Meanwhile, the shorter evacuation route should be saturated to optimize utilization of time.
The following is a partial description of the algorithm for the evacuation model of multistory buildings [
Step 0: initialize the input floor set
Step 1: use Dijkstra algorithm to find the shortest path
Step 2: calculate the maximum capacity
on each arc. If
Step 3: determine the path set
Algorithm flow chart of model two.
In actual situations, the number of evacuees in the Louvre is possible to be greater than the sum of the maximum capacity per unit time of all paths [
To solve the problem, modifying the evacuation time aggregation
Regarded as a dynamic system in a cellular space, CA is composed of discrete, finite state cells. CA evolves in discrete time dimension according to certain local rules. Each cell has one or more states as a specific parameter. The state of the cell was selected in the finite state set such as the spatial characteristic “occupied and vacant.” In the cellular automata model, the evacuation pedestrian can only move the length of one cell within each time step
Simulation flow chart based on CA.
Some significant parameters will be explained in the following sections.
Due to the difference of age and physical fitness of individual based on the cellular automata model, the competitiveness value
Taking into consideration that personal physiological aspects have gender
The simulation model of pedestrian evacuation is built in a 2D discrete cellular grid system with the size of
In a moving area where evacuees are located in the center, the value of the corresponding directional parameter matrix element is shown as follows:
Pedestrians are not allowed to cross fences or obstacles; at the same time, they can get out of the cellular space only through the safe exit of the room. The value of the center cellular position is 0. If the cell is not occupied by a person or an obstacle, its occupation value is 1; otherwise, is −1. If the cell is −1, pedestrians in the center cell cannot reach this cell:
Attraction
The plan of a static floor field for the doors.
In Figure
Simulation process of bottleneck formation and disappearance.
Based on the multistory building evacuation model established in the previous sections, the model area partition and connection channel were shown in the schematic diagram (Figure
Multilayer structure planarization process diagram of the Louvre.
Each floor of the Louvre is divided into areas depending on where the bottleneck may occur [
To get the optimal evacuation routes, path capacity and other parameters need to be calculated. Using the algorithm introduced in previous sections, the actual evacuation routes of each floor could be determined. After finding out various solutions of all feasible paths, an optimal evacuation plan can be determined. The simulated results of the evacuation scheme are shown in Table
The Louvre evacuation plan table.
Zone  Initial number  Path 
Path length 

Evacuation zone 
600 

61.11 

94.44  


Evacuation zone 
700 

50.00 

50.00  


Evacuation zone 
1000 

23.61 

51.79  

59.29  

76.79  


Evacuation zone 
800 

25.00 

29.17  

54.17 
The evacuation scheme obtained by simulation was expressed by the 3D multistory structure model of the Louvre. As shown in Figure
Simulated evacuation scheme.
Obviously, the capacity of passages was saturated. For example, in the evacuation scheme of the
The path scheme would change considering some special circumstances in the museum. There were some different situations discussed: The accident area needs to be sealed off, and evacuees cannot pass through a certain area as the source of fire when the fire occurs; emergency personnel is required to enter the stadium as soon as possible. The simulation design is shown as follows.
When an emergency would not occur, the initial number of evacuees in
Path planning table in a normal emergency.
Zone  Initial number  Path 
Path length 

Evacuation zone 
600 

61.11 

61.11  


Evacuation zone 
700 

50.00 

50.40 
When the accident happened in zone
The Louvre evacuation plan table.
Zone  Initial number  Path 
Path length 

Evacuation zone 
600 

75.00 

89.29  

108.33  


Evacuation zone 
700 

50.00 

73.57 
The evacuation route simulations compare the situation one to the general case by the 3D multistory structure model as shown in Figure
Evacuation route contrast diagram one: evacuation route map of (a) areas
When the emergency occurs in
Path planning table in situation two.
Zone  Initial number  Path 
Path length 

Evacuation zone 
0 

50.00 


Evacuation zone 
600 

61.11 

94.44  

127.78  


Evacuation zone 
700 

50.00 

50.40 
The evacuation routes simulations results are shown in Figure
Evacuation route contrast diagram two: evacuation route map of (a) areas
As mentioned above, the Dijkstra algorithm in the multistory building evacuation algorithm could calculate well the length of the shortest path. However, because of the bottleneck caused by congestion, the evacuation time cannot be calculated from the path length directly. Therefore, simulation based on cellular automata is necessary. Then, the functional relationship among evacuation time and other factors was fitted according to the multigroup data further by simulation data analysis.
The program based on Matlab was used to simulate [
Simulation images of cellular automata on the first floor of the Louvre.
Through the simulation of cellular automata, the evacuation situation could be observed in a certain area in the multistory building structure model. The simulation of cellular automata could measure and analyse the time of bottleneck occurrence, the area of bottleneck, and time step by giving the planned paths.
In view of the intricate internal structure of the Louvre, the basic topographic structures and simulation results of different populations were discussed in this paper.
The size of the cellular space is
Simulation diagram of rectangular cellular automata.
Table
Simulation results table of Ltype channels.
Times of experiments  1  2  3  4  5  6  7  8  9  10 

Evacuation time (unit step 0.5 s)  363  350  341  351  376  334  345  339  364  372 
Maximum bottleneck area (grid)  340  370  364  397  354  384  364  312  336  387 
The evacuation model of the Lshaped channel spatial structure with the size of
Evacuation model diagram of the Lshaped channel spatial structure.
Path planning table in a normal emergency.
Number of experiments (times)  1  2  3  4  5  6  7  8  9  10 

Evacuation time (unit step 0.5 s)  382  378  391  381  376  394  401  412  384  392 
Maximum bottleneck area (grid)  160  210  154  124  154  178  184  162  196  157 
Through ten times of simulation results, the standard deviation of evacuation time was 10.6719, which shows the range of experimental data was small and the simulation was stable. Compared to the rectangular spatial structure, the bottleneck was not easy to occur. The average value of the maximum bottleneck area obtained by simulation was 167.9, as the evacuation results of the Lshaped channel were more stable.
Figure
Evacuation model diagram of the compound structure.
The Grubbs criterion [
According to the Grubbs critical value table, when the test data were 11 groups, the average value of evacuation time data is 353.5, and the standard deviation is 13.8076. When the abnormal probability of its data is 95%,
The competitiveness of ordinary tourists and disabled tourists in the cellular automata model was different. In the simulation model of this paper, competitiveness discrimination was simply divided into matrices
Functional relationship between evacuation time and the ratio of disabled tourists.
It is shown that the evacuation time would increase nonlinearly with the increase in the proportion of tourists of disabilities.
For the simulation data, regression analysis was carried out further based on the polynomial regression function
Based on the model, there may be a functional relationship among the evacuation time
Line
Line
Line
Evacuation model diagram of a rectangular spatial structure: (a)
Functional relationship between evacuation time and pedestrian density at different exit widths.
The theoretical analysis showed that the evacuation time tends to infinity when the channel length is 0, and the evacuation time was larger than zero when the channel length tends to infinity. Hence, the relationship among the functions should be an inverse proportional function. Assume that the function relation was
The function relation with egress time
This paper proposed a novel evacuation model instruction. Main contributions are as follows. First, the evaluation formula was proposed to analyse the patency of the buildings. The routes on the images of each floor in the buildings can be directly shown by threedimensional architectural model after obtaining the binary image of the active region in a multistory building. The multistory building evacuation algorithm was established to provide the schemes of optimal evacuation routes. The result of evacuation routes in different emergency situations was discussed and compared in this paper. Simulation in CA illustrated the efficiency of the model that the stability and reliability of the simulation results were verified by Grubbs criterion. Then, different factors were considered, such as diversity of tourists, pedestrian density, and width of exit. Furthermore, a functional relationship was clearly given among the evacuation time with those factors, and the functional relationship has a good correlation. It was applied to optimize the initial evacuation routes. Based on these, the algorithm can adapt to different circumstances, and an accurate time of leaving the buildings can be calculated accurately.
No data were used to support this study.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
YZ planned the study. YZ and ZF performed the research. YZ wrote the first manuscript. YZ, TW, GZ, and ZF read and approved the final manuscript.
The authors wish to express their gratitude for the financial support from Fundamental Research Funds for the Central Universities (no. XDJK2019C035).