# Ambulance Locations in a Tiered Emergency Medical System in a City
The former class of problems uses the concept of coverage as an analogy to a typical performance measure of an EMS system, such as the proportion of patients responded to within a certain time standard [17,18]. If a patient is reached by an ambulance within the time limit, they are said to be covered by the service. The goal is either to cover the whole region with a minimum number of facilities, or to maximize the number of customers covered by a fixed number of facilities. In the latter group of problems, the so-called median problems, the quality of service is identified with the travel time that patients take to get to the closest facility, or the response time that ambulances need to reach the patients. Both types of problems admit several variations to allow for additional criteria and constraints from real-world applications. Optimization problems related to the emergency medical services are surveyed in [19]. [PERSON] et al. draws readers' attention to the shortcomings of various models for location of emergency centers. They point out that coverage or time-based performance measures are only a proxy for health outcomes. The drawback of the coverage concept is that it distinguishes only two states of a patient, covered or not covered. The response time does not play a role if it is below the time limit. Therefore, the same level of service is said to be provided if an ambulance comes in 1 min or in 15 min (in the event of a 15-min standard). In addition, patients who are not covered do not contribute to the objective function value at all, so their response times may be arbitrarily long. Another shortcoming of the former coverage models, such as the location set covering problem or the maximal covering location problem, is that they are based on a simplifying assumption that ambulances are always available when they are needed. The result is that the solution overestimates the real coverage. The maximum expected covering location problem (MEXCLP) offers a more realistic way of modelling the ambulance availability. It aims to locate a given number of ambulance stations in order to maximize the expected coverage, which depends on the busy fraction, defined as the probability that an ambulance is unavailable to immediately respond to an emergency call. The MEXCLP has been widely reported in the context of the EMS design. It is recommended by [PERSON] et al. [20] to be applied instead of the maximum availability/reliability models that also deal with the uncertain availability of ambulances but lead to a design that is too expensive. A modified MEXCLP model was used to optimize the distribution of ambulances in Edmonton, Canada [21]. [PERSON] et al. considered the variation in pre-trip preparation and travel times, and included correction factors that accounted for the dependence in the busy fractions between ambulances. Stochastic response times were also used in [22]. The uncertainty is expressed as a probability that the demand point, \(j\), will be reached by an ambulance from the base, \(i\), within the time threshold. [PERSON] et al. demonstrated their approach in the region of Amsterdam, Netherlands. The study [9] enhanced the MEXCLP by considering multiple customer types, as well as two different types of emergency vehicles. The objective was to maximize the total number of expected Priority 1 calls covered within a specified amount of time. Dispatching rules were included in the model. The experiments were conducted using real-world data collected from Hanover County, Virginia. [PERSON] and [PERSON] [23] proposed a hybrid model that combined the maximum expected coverage goal of the MEXCLP with the reliability of the service at every demand node. Although the authors reported the superiority of their model compared to the original MEXCLP model, the benefit of the model in practice has not been proved sufficiently, since the results were based on a rather small case study (55 demand nodes) and a simple simulation model.