1917800（提交）.pdf2019年美国大学生数学建模大赛D题逃离卢浮宫根据卢浮宫的结构，绘制

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详细说明：2019年美国大学生数学建模大赛D题逃离卢浮宫根据卢浮宫的结构，绘制了一个有向图，（指示基于图论的疏散路径），【Floyd-Warshall算法】用于搜索从亭组到出口的最短路径。通过运行算法，得到疏散矩阵，（工作人员使用疏散矩阵指导访客通过每条计划路径安全到达出口）；鉴于排队理论，当疏散时，访问者将在出口处排队，排队参数（例如队列长度和等待时间）。【粒子群优化用于优化模型】疏浚这些瓶颈，优化后，最终平均疏散时间为3秒，出口平均排队长度为3米，队列等待时间为3秒，比以前减少10％。在对模型进行敏感性分析后，以便博物馆能够在法国这个动荡的时期生存下来。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 e 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 aIgorithm to clear these two bottlenecks, to achieve the initia 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 e Do not use the elevator when visitors evacuate e The louvre buildings meet the basic requirements of france fire safety regulations Team#1917800 Page 2 of 19 3 Notations Table 1: Notation Notations Definition n pavilions exit nodes the total population of the nth Pavilion the distance from node i to node he time for visitors to evacuate from ith pavilion to j'th 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 the shortest path from node i to exit 3 the k-th particle(k= 1, 2, ..,m) position best 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 C1,C2 earning factors M any large positive integer the velocity of firefighters the entry rate of firefighters RSET the safe evacuation time the distance from the exit to the fire point t the required time from exit i 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 Interactive Floor plans ⑩ ACCESS VLA THE COUR CARREE → Enter the museun ACCESS VLA THE PASSAG ACCESS HE CARROUSEL CU⊥OLRE e四 YRAMID ENTRANCE POBTE DES LIONS ENTRANCE euse v Figure 1: Four Main Entrances of Louvre In order to calculate the flow of people in the evacuation channels, we need to know the nunber of people in each pavilion. Because the popularity of the pavilion varies, the How 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 oors Pi avallon Treasures Ground foor 345 Venus de milo 1st foor 703 Victoire de samothrace 1st foor 711 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 Pavilions Lower ground foor 102133169174183186 Ground hoor 219236403417 1st floor 503635700706708716717 2nd foor 811845848940 Team#1917800 Page 4 of 19 Of the 16 pavilions displayed above, each pavilion accounts for 2.5%c of the total number There are 31 other less popular pavilions, each of which accounts for 1% of the total number ]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 Cround Floor 1.0 0.5 losed Figure 2: Heat Map of ground Floor s正o 0.75 0.2 closed Figure 3: Heat Map of the lst 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 or 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 di irected graphs 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) U1, U2,.,145, Us1, vs2, 13, Us4. Among them, 1 N 45 represent 45 pavilion groups LUs1, 192, 033, 0's4 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 Vs2 v1 V3 VS1 v4 v 8 V6 VS4 k 8 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 dist ance from each node to the exit node and get the evacuation paths Floyd-Warshall algorithm is an algorithm to solve the shortest path be etween 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,)is the shortest path from node a to node i. For each node ki, we check Team#1917800 Page 6 of 19 whether Dis(i, k )+ Dis(hi, j)< Dis(i, j) 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 i directly, then we set Dis(i,i=Dis(i, k)+ Dis(h, i so that after we traverse all the nodes k, Dis(i,) 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, U4, 03. From the distance matrix, we can see that the shortest distance between U4>U3 is D (4, 3 )=10. According to its routing matrix, we can see that: R(4, 3)=1, meaning v4-703, passing through V1 first, then looking at R(1, 3)=2, indicating that we need to go through V2 again, so we look at R(2, 3)=3, at chis time we found that we finally reached v3, so let's sort out, the shortest path of 04>03 is: U4>01702>U3. 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 0413341.315210147 2510 90416398 D inf in f in f f in f 161574139.inf0280 inf inf inf inf inf 0 and the routing matrix R 47535 4 4 4744 in f inf in f…474849 161574139 474821 in f in,f in f 474849 where, matrix D and R are matrices of 4949, 49 equals 45 pavilion groups plus 4 exits indicates that this path is not feasible After obtaining the matrix D and R, we calculate the evacuation matrix of all evacuation Team#1917800 Page 7 of 19 paths show OW 11621474800 21621474800 4131742-60520 131742-605200 36-35-34-3342-6052 404245636400 151742-605200 92101103104000 69710110310400 97101103104000 Each row of the matrix, from left to right, represents the serial number of the path taken from a pavilion group to an exit. After the matrix is obtained, evacuation organizers can direct the evacuation plans through this matrix. The symbol of numbers in the matrix represents the direction of movement: from small pavilions to large pavilions, the direction is positive and vice versa. Then we draw a part of the path map shown in Figure 5 to illustrate it stair 2ND FLOOR pavilion group the most popular pavilion channel IND FLOOR Q40 GROUND FLOOR LOWER 2 GROUND FLOOR NAPOLEON HALL igure 5: a part of the path map The Exit. 1 represents the Carrousel du Louvre entrance, the exit 2 represents the Portes Des Lions entrance, the Exit 3 represents the Passage Richelieu entrance and the exit 4 represents the Pyramid entrance. Look at the two red-marked paths in the map. They all Team#1917800 Page of 19 start from the pavilion groups and go through several paths to different exits. In particular the visitors from the 39th pavilion group could evacuate through the 15th path, the 23th path, the 53th path, the 64th path to the exit 3 4. 4 Queuing theory After arriving at the exits, visitors need to queue up. At this time, our team uses the knowledge of queuing theory to calculate the length of queues and the waiting time Queuing theory, or stochastic service system theory, is a statistical study of the arrival and service time of service objects, and obtains the statistical rules of these quantitative indicators(waiting time, queuing length, busy period, etc ) Then according to these rules, the structure of service system or the reorganization of service objects are improved, so that the service system can not only meet the needs of service objects, but also make the cost of the organization the most economical or some indicators the best Queuing system consists of three parts: input process, queuing rules and exit organiza- tion The input process examines the law of customer arrival in the service system. It can be described by the number of customers arriving in a certain time or the interval between two customers arriving in succession. According to the characteristics of the evacuation How, we regard the input process of evacuation in Louvre as an average distribution Queuing rule in louvre is waiting system, which means when visitors arrives, all exits are occupied, then visitors queues, that is called the waiting system he exit organization in Louvre are several channels. Multiple channels can be arranged in parallel or in series. Considering that we need emergency personnel to quickly enter the building, we will set up several rescue channels which is emergency personnel only, the proportion is n. Obviously, the rest channels are the evacuation channels Set, Ti is the tot aI population of the nth Pavilion, lij is the distance frorn node i to node j vd, Va, Us are the evacuation velocity of the old and weak, groups and individuals. wd, w ws is the ratio of the old and weak, groups and individuals [8. Thus aunt +0 is the time for visitors to evacuate from ith pavilion to ith exit And 1s0d50 + d is the average time for visitors to pass through the exit. Where, u is the number of passing viSitors per time

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