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Speedy evacuation plan to leave the Louvre When a large building is in danger, it is the most important to guide the inside crowd to evacuate safely as quickly as possible. By employing the obtained information of the Louvre, we want to design an evacuation plan to allow inside crowd egress through an optimal exit in order to empty the building as quickly as possible. Our aim is to identify the bottlenecks in the process of visitors’ movement. Once the museum face an emergency, our model should be employed to direct the people evacuate from the danger building. In accordance with the structure of the Louvre, we draw a directed graph to indicate the path of evacuation based on the graph theory. In order to have all occupants to leave the building as quickly and safely as possible, Floyd-Warshall algorithm is used to search for the shortest path from pavilion groups to exits. By running the algorithm, we get the evacuation matrix, which can be used by staﬀ to guide visitors safely reach the exits through each planned paths. Now visitors will queue up at the exits when evacuation, in view of queuing theory, the queuing parameters such as the queue length and the waiting time matter a lot. Apart from exits, 46th/53th/101th stairs with large ﬂow of people are also bottlenecks. For dredging these bottlenecks, particle swarm optimization is used for optimization of the model. After optimization, the ultimate average evacuation time is 3 seconds, the average queue length at exits is 3 meters, and the waiting time of queues is 3 seconds, which are 10% less than before. Moreover, it is necessary to make this evacuation model capable of dealing with various threats. In the face of ﬁre, we need to let ﬁre ﬁghters get to the ﬁre point as quickly as possible to protect tourists and collections, so it is necessary to open ﬁre ﬁghter channels at the four main entrances and make use of additional unknown entrances. In addition, provided that an explosion occurs on the stairs or if evacuation exits are destroyed artiﬁcially, we can use the evacuation model to redesign the evacuation routes quickly and eﬃciently. After the sensitivity analysis of the model, we can conﬁdently provide a few of evacuation guidance to leave Louvre or other crowded large structures, so that more people can survive after an emergency. Keywords: Graph Theory; Floyd-Warshall Algorithm; Queuing Theory; Particle Swar- m Optimization Algorithm

Contents 1 Introduction 2 Assumptions 3 Notations 4 Evacuation Model 4.1 Basic Information of Louvre . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Floyd-Warshall Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Queuing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Bottlenecks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 2 4 5 8 9 4.6 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Responses for Threats 5.1 Conﬂagration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Blockage of exits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Sensitivity Analysis 7 Strengthens and Weaknesses of Models 7.1 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Recommendations for Emergency Management 8.1 The Application Scope of Evacuation plan . . . . . . . . . . . . . . . . . . . 8.2 Reporting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Emergency measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Training and Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13 14 15 16 17 17 17 17 17 17 18

Team # 1917800 Page 1 of 19 1 Introduction Because of the recent terrorist attacks in France, our team is helping to design evacuation plans at the Louvre in Paris. There are four main entrances and other entrances that are not safe but helpful for the evacuation plans. After identifying potential bottlenecks and security threats in the museum, our team establish a evacuation model in order to allow the museum leaders to explore a range of options to evacuate visitors from the museum, while also allowing emergency personnel to enter the building as quickly as possible. The overview of our work: • According to the characteristics of the interior structure of the museum, we draw a directed graph to indicate the path of evacuation, and then use Freud algorithm to search for the shortest path. Because visitors will arrive at the exit in a short period of time, the exit is bound to line up, this time the use of queuing theory knowledge, calculate the four main exits of the queuing team length and queuing waiting time. In order to prevent the staircase and outlet blockage time is too long, we use particle swarm optimization algorithm to clear these two bottlenecks, to achieve the initial optimization of the model. • To make the models we build more adaptable, we analyzed three of threats and made e- vacuation plans in diﬀerent threat situations, and showed that evacuation plans changed dynamically when threats changed. • Because the evacuation plan is aﬀected by too many factors, we analyze the sensitivity of the model and enumerate the inﬂuence of diﬀerent factors on the model, thus reﬂecting the superiority of our model. • Based on the results of our work, we propose policy and procedural recommendations for emergency management of the Louvre. 2 Assumptions • Visitors evacuate from the nearest stairs. • Do not use the elevator when visitors evacuate. • The Louvre buildings meet the basic requirements of France ﬁre safety regulations

Team # 1917800 3 Notations Page 2 of 19 Notations vn vsi xij lij tij µ λj s Lj tj hij xk Pbest Gbest Vk w nmax ni c1, c2 M vm nj RSET dj tj Table 1: Notation Deﬁnition n pavilions exit nodes the total population of the nth Pavilion the distance from node i to node j he time for visitors to evacuate from ith pavilion to jth exit the number of passing visitors per time the number of arrival visitors per time at exit i the maximum number of simultaneous imports and exports the queue length at exit i the longest evacuation time at exit j the shortest path from node i to exit j the k-th particle(k = 1, 2, ..., m) position the best position where particles went through the best position where the entirety went through the velocity of particles the weighting coeﬃcient the whole iteration the current iteration earning factors any large positive integer the velocity of ﬁreﬁghters the entry rate of ﬁreﬁghters the safe evacuation time the distance from the exit j to the ﬁre point the required time from exit j to the ﬁre point 4 Evacuation Model 4.1 Basic Information of Louvre In Louvre, there are four main entrances: the Pyramid entrance, the Passage Richelieu entrance, the Portes Des Lions entrance and the Carrousel du Louvre entrance which could all be treated as the evacuation exits shown in Figure 1.

Team # 1917800 Page 3 of 19 Figure 1: Four Main Entrances of Louvre In order to calculate the ﬂow of people in the evacuation channels, we need to know the number of people in each pavilion. Because the popularity of the pavilion varies, the ﬂow of people will vary, which will aﬀect the evacuation plan [10]. The following tables and explanations show the popularity of diﬀerent pavilions and the proportion of people in them. Table 2: The Proportion of Visitors in the Most Popular Pavilions Floors Ground ﬂoor 1st ﬂoor 1st ﬂoor Pavilions 345 703 711 Treasures Venus de Milo Victoire de Samothrace Mona Lisa The number of people in those pavilions where the above-mentioned treasures are located accounts for 10% of the total number. Table 3: The Proportion of Visitors in Popular Pavilions Floors Lower ground ﬂoor Ground ﬂoor 1st ﬂoor 2nd ﬂoor Pavilions 102 133 169 174 183 186 219 236 403 417 503 635 700 706 708 716 717 811 845 848 940

Team # 1917800 Page 4 of 19 Of the 16 pavilions displayed above, each pavilion accounts for 2.5% of the total number. There are 31 other less popular pavilions, each of which accounts for 1% of the total number [2] [9] [7]. Here is the heat map shown in Figure 2 and Figure 3. According to the cutline on the right, the more red the area, the more dense the visitors are. Figure 2: Heat Map of Ground Floor Figure 3: Heat Map of the 1st Floor From the heat map, we can see that the number and location of some pavilions are similar and can be packaged together. So we packed all the pavilions into 45 pavilion groups for analysis. 4.2 Graph Theory Imagine the whole evacuation process. When people hear the evacuation instructions, they need to evacuate quickly to the safe exits. In order to ensure the evacuation safety,

Team # 1917800 Page 5 of 19 we need to ﬁnd the shortest evacuation path for each pavilion. If we want to calculate the shortest evacuation path, we think of the knowledge of applying graph theory. It takes graph as its object of study. In graph theory, a graph consists of several given points and lines connecting two points. This graph is usually used to describe a speciﬁc relationship between certain things. Points are used to represent things, and lines connecting two points are used to represent the corresponding relationship between two things. Graphs have undirected graphs and directed graphs [5]. Each pavilion group with concentrated visitors is regarded as a node. The route be- tween nodes or stairs is a path. Our team construct a weighted directed graph G = (V, E), V = {v1, v2, ..., v45, vs1, vs2, vs3, vs4}. Among them, 1 ∼ 45 represent 45 pavilion groups. {vs1, vs2, vs3, vs4} denotes four exit nodes. Thus, we draw a directed graph of the 104 evac- uation paths in the Louvre. Because of the large number of pavilions, we cut part of the directed graph on for illustration shown in Figure 4. Figure 4: A Part of the Directed Graph Circles are all the nodes. The red circles in the picture represent the four main exits, the black circles represent the pavilions, and the arrow lines represent the directions and paths of evacuation movement. 4.3 Floyd-Warshall Algorithm We use Floyd-Warshall algorithm to ﬁnd the shortest distance from each node to the exit node and get the evacuation paths. Floyd-Warshall algorithm is an algorithm to solve the shortest path between any two points. For example, there are two possible shortest paths from any node i to any node j. One is directly from i to j, and the other is from i through several nodes k to j. So let’s assume that Dis (i, j) is the shortest path from node u to node v. For each node k, we check

Team # 1917800 whether Page 6 of 19 Dis (i, k) + Dis (k, j) < Dis (i, j) (1) is valid. If it is valid, we prove that the path from i to k to j is shorter than the path from i to j directly, then we set Dis (i, j) = Dis (i, k) + Dis (k, j) , (2) so that after we traverse all the nodes k, Dis (i, j) records the shortest path from i to j [4]. After calculation, we ﬁnally get the distance matrix D and routing matrix R. The distance matrix D means the shortest distances between 2 nodes. And the routing matrix R means the routes between 2 nodes. For example, v4 → v3. From the distance matrix, we can see that the shortest distance between v4 → v3 is D (4,3) = 10. According to its routing matrix, we can see that:R(4, 3) = 1, meaning v4 → v3, passing through V1 ﬁrst, then looking at R(1, 3) = 2, indicating that we need to go through V 2 again, so we look at R(2, 3) = 3, at this time we found that we ﬁnally reached V 3, so let’s sort out, the shortest path of v4 → v3 is: v4 → v1 → v2 → v3. In short, it is a ﬁxed column, which jumps in rows according to the routing matrix until it jumps to the corresponding point. Here is the distance matrix D:  D = and the routing matrix R:  413 341 ... 315 210 147 0 416 398 90 0 251 229 ... ... inf inf inf ... 161 574 139 ... inf inf inf ... 0 0 inf inf inf inf inf 280 0  1 4 5 2 5 4 ... 47 ... 47 ... 5 4 35 4 inf inf inf ... 47 48 49 161 574 139 ... 47 48 21 inf inf inf ... 47 48 49  R = where, matrix D and R are matrices of 49*49, 49 equals 45 pavilion groups plus 4 exits, “inf” indicates that this path is not feasible. After obtaining the matrix D and R, we calculate the evacuation matrix of all evacuation

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