## 4 Materials and Methods
### Obtaining the Collapse Index at the Territorial Scale  
The aim of this phase is to obtain a map of collapse indices in the study area. The period selected for the calculation of these indices is one year, due to the seasonality of certain incidents such as forest fires, gales, and others. The results obtained in this phase are a raster map of collapse indices and an average and maximum global value of the index for the entire territory for a specific year. The cell size of this map is 2 \(\times\) 2 km to ensure that there is a communication route in each pixel of the territory in the Madrid Region and that the arrival time can be calculated from any FS to the pixel where the incident or incidents have occurred. For a specific kind of incident, the same type or types of vehicles will always arrive from one or more FS. Once they reach the pixel of the incident, the displacement from the road to the incident is considered the same for all vehicles regardless of the FS they come from, so this last displacement time does not affect the collapse index, as it is the same for all FS. The expression of the collapse index in pixel \(i\) is:  
\[RCI_{i}=n_{i}*\ t_{i} \tag{1}\]  
where \(n_{i}\) is the maximum number of resources (vehicles) that have worked simultaneously on one or more incidents in one year in pixel \(i\) and \(t_{i}\) is the maximum duration of simultaneous interventions on one or more incidents in one year in pixel \(i\). The map of the collapse indices is based on the emergencies database of the Madrid Region Fire Department for the year 2014, in which there were 25,221 incidents. This database provides the following information for each intervention: incident code, incident type, date, notification time, time of activation of each vehicle, departure time of each vehicle, arrival time of each vehicle at the incident, time of return from the incident for each vehicle, actual arrival time of each vehicle at the station, time of availability of the vehicle for another incident, duration of the intervention, stations, resources or vehicles dispatched, address of the incident, georeferencing of the incident, number of pixels in the 2 \(\times\) 2 km raster map, cost of the service, and number and type(s) of vehicles activated. Incidents coinciding in time are identified to calculate the overlap time in a pixel. The intersection of the time intervals from the time of activation of each vehicle to the time of its availability for another incident is calculated for these incidents. The value assigned is the highest value of all the coincidences in one pixel over one year. For example, if in pixel \(i\) on the 20 March 2014 it has been identified that 4 resources have worked simultaneously in that pixel, and after analysing the whole data set for that year 4 is the maximum number of resources that have worked simultaneously in pixel \(i\), then \(n_{i}\) = 4. Additionally, if on 7 July it has been identified in the same pixel \(i\) that 2 vehicles coincided in time in that pixel, the first activated at 18:00, becoming available for another incident at 19:00, and the second activated at 18:10, becoming available again at 18:50, the duration of the simultaneous interventions is 40 min. If this is the maximum duration of the simultaneous interventions in 2014 in pixel \(i\), then \(t_{i}\) = 40, meaning the collapse index in pixel \(i\) would be \(RCI_{i}\) = 4 \(\times\) 40 = 160 min. Determination of the Collapse Index Threshold  
One of the restrictions in the optimisation process is that the collapse index in every pixel in the territory must be less than a specific threshold. In the Madrid Region, there is a very high probability that with the current resources the organisation will crash if 20 vehicles intervene simultaneously for 500 min in different incidents, which is equivalent to a collapse index of 10,000 (oral communication from the chief inspector of the firefighting service in the Madrid Region).