# Ambulance Locations in a Tiered Emergency Medical System in a City
[PERSON] and [PERSON] [6] assessed several ambulance location models in the view of the performance indicators frequently used in practice. Simulation experiments with real-world case studies showed that the MEXCLP and the expected response time model (ERTM) performed the best. However, the ERTM model is rather complicated, and it takes a long computing time. That is why the average response time model is recommended as a good alternative. The average response time model is an application-related name for the model of the well-known \(p\)-median problem (\(p\)MP) that seeks the location of \(p\) facilities, so that the average travel time between customers and their closest facility is as short as possible. Several other studies proved the practical applicability of the \(p\)MP. For example, [PERSON] et al. [14] proposed optimal locations for ambulances in the city of Niigata, Japan. [PERSON] and [PERSON] [16] achieved a significant improvement in the average response time by using the \(p\)MP for two sub-regions of the Perth Metropolitan area. The \(p\)MP was also used in a study by [PERSON] and [PERSON] [24] to locate the helicopter emergency medical service bases in the state of New South Wales, Australia. The weak point of the cited models is the way of modelling ambulance availability. The models either ignore the fact that the closest ambulance may not be idle when it is needed, or they assume that all ambulances are unavailable for the same amount of time. Both these assumptions lead to an oversimplified model. [PERSON] and [PERSON] [7] increased the accuracy of the location model by explicitly considering the ambulance trip time that included not only the response time, but also the on-scene time, the time of transportation to the hospital, the drop-off time, and the travel time from the hospital to the station. Their modular capacitated location model decides on the locations of the stations, the number of ambulances to be assigned to each of them, and the demand allocation to the stations. The objective function is the total system costs, including the station-opening costs, the costs of allocating ambulances to the existing stations, and the transportation costs. This model inspired us, but we had to adapt it to the Franco-German style of EMS delivery that is common in European countries. [PERSON] and [PERSON]'s model reflects the Anglo-American system, where all patients are rapidly transported to the hospital with fewer pre-hospital interventions. On the other hand, emergency teams in the Franco-German model are qualified to treat patients in their homes or at the scene. This results in many EMS users being treated at the site of an incident and being transported to hospitals less frequently. Only few of the mentioned publications compared several location models mutually. If they did, they provided only analytical results that suffered from the same simplifying assumptions that were used in the modelling phase. The only exception is the study by [PERSON] and [PERSON] [6] who use computer simulation to assess the models in realistic situations. In regard to the solution methods, most ambulance location problems are modelled as mathematical programming problems and are solved by a general-purpose optimization solver [2]. Only a few studies propose such complicated models that cannot be solved to optimality, and require using a metaheuristic method, such as the tabu search [11], variable neighborhood search [10], or ant colony optimization [25]. In this study we propose a modification of the modular capacitated location model [26] and compare it to the most successful models reported in [6; 14; 16; 20; 24]; namely, the \(p\)-median model, the maximum expected coverage model, and the expected response time model. Firstly, we will evaluate the solutions of the models using the criteria that correspond to their objective functions, namely, the average response time, the expected response time, and the expected coverage within two time thresholds.