# Evacuation Management System for Major Disasters
## 2 Mathematical Model
### Evacuation Routing  
A route refers to the way taken in getting from a starting point to a destination. Therefore, three main factors are needed to compute the evacuation routes. _Assembly points_ are the starting points (origins) where pedestrians board the vehicles. To represent assembly points, a hexagonal tiling is used to discretise the evacuation area. Assembly points are the centre of each hexagon \(C_{i}=\left(C_{\phi},\ C_{\theta}\right)\) (Figure 1). The hexagonal cells ensure two fundamental properties: (1) every evacue in the evacuation area is located at a maximum distance \(r\) (circumradius) from any assembly point; (2) the evacuation capacity of the assembly point does not exceed the expected population load in each hexagonal cell. To determine the optimal assembly locations, the external service OpenRouteService (ORS) ([[https://openrouteservice.org/](https://openrouteservice.org/)]([https://openrouteservice.org/](https://openrouteservice.org/)), accessed on 25 July 2022) can be used as well, in the case of managed evacuation. This service contains a database of places and points of interest that are classified so that they can be filtered according to the intended use. _Destination shelters_ are defined by geographical coordinates. Shelters refer to places giving emergency/temporary protection to people that fulfil three criteria: shelter purpose, facilities and spatial location. Three principles are required to address the spatial location:  
Figure 1: Hexagonal tiling (black) over evacuation area (red). * \(\forall\;e_{j}\in E_{i}\;\exists\;S_{ij}=\{s_{1},\;s_{2},\;\ldots,\;s_{m}\}\) where \(S_{ij}\). represents all feasible shelters for which the distance to an \(e_{j}\) geographical boundary ranges in \([d,\;d+\mu]\), where \(d+\mu\) is the maximum distance. * \(S_{ij}\) is a systematic uniform random (UR) generated on the circumference with radius \(d\), ensuring properly distributed locations. \[\alpha_{0}\;\sim\text{UR}(0,\;2\pi)\] \[\alpha_{l}=\alpha_{0}+l\cdot T\] \[T=2\pi/m\] where \(m\) is the total number of shelters (Figure 2). * \(S_{ij}\subset S_{ij}\|\;\forall\;s_{m}\in S_{ij}\), \(\forall\;l_{n}\in(E_{l}\cup A_{l})\), \(d\leq Dist(s_{m},l_{n})\leq d+\mu\), where the function \(Dist(s_{m},\;l_{n})\) is the minimum distance between a shelter geographical location \(s_{m}\) and the damaged asset or alternative evacuation \(l_{n}\) geographical boundary (Figure 2). These principles ensure that the presence of shelters is uniformly distributed within a safe distance. The geographical locations of shelters are slightly modified using ORS to well-known points of interest (Pois) nearby the optimal location (e.g., schools, sport centres, etc.) to fulfil sheltering purposes and facility requirements (e.g., safe location, people capacity). It is true that this adjustment requires precise knowledge of the locations of the different facilities and their current state, as a facility may be damaged during the disaster or may not be equipped with the necessary elements for the reception and accommodation of evacuees. In critical scenarios where none of the locations proposed in this distribution coincides with a possible shelter because it does not exist, it is necessary to deploy a temporary shelter system. _Evacuation routes_ are achieved using a dedicated routing service from assembly points to destination shelters avoiding \(A_{i}\) damaged assets, yielding to an OD matrix of routes. The optimal routes are filtered by prioritizing those with better characteristics (number of lanes, maximum speed limit, route overlapping) and reducing them as much as possible while ensuring two conditions: (1) at least one route starts from every assembly point; (2) the distribution of evacuees is uniform for shelters defined as destinations (Figure 3).