# Ambulance Locations in a Tiered Emergency Medical System in a City
## 2 Materials and Methods
### The Modular Capacitated Location Problem  
In this section we propose a modification of the modular capacitated location problem (MCLP). This location problem arose in healthcare strategic planning [26]. Many healthcare facilities may be built from modules of a certain size. The number of patients a facility can serve, as well as the capital and operating costs, depend on the facility size. The size of a facility must be chosen only from a finite and discrete set of possible capacities. Each aggregated customer (a zone or community) can be partially served by more than one facility. A variant of the MCLP for ambulance locations was presented in [7]. Both formulations [7; 26] minimize the total costs, consisting of the capital and operating costs of the facilities, as well as the costs induced by serving customers. In this paper we present an adjusted model of the MCLP, where the objective is the average response time to patients. Instead of the costs of opening and operating the facilities, the number of ambulances is limited directly by a given amount. In contrast to the models presented above, this model allows for differentiating patients' priorities and the service levels provided by the different rescue units. The capacity of an ambulance is expressed as the amount of time when the ambulance is ready for rescue trips. Every trip consumes a portion of this capacity. To model the trip time more precisely, we take into account not only the travel time to the site of the incident, but also the transport to a hospital, the drop-off time at the hospital, and the return to the base station. That is why additional sets and parameters must be introduced. According to practices in Slovakia and in many other countries, we will distinguish two patient priorities and two levels of EMS. The highest priority is assigned to patients who are in a life-threatening condition. The most severe patient diagnoses include cardiac arrest, stroke, severe respiratory difficulties, chest pain, severe trauma, and unconsciousness. These diagnoses are denoted as the First Hour Quintet (FHQ) because the first hour after a medical incident decides whether the patient survives. An advanced life support (ALS) ambulance is always dispatched when a call-taker at dispatch center assesses the emergency as critical. Non-critical calls may be responded to by a basic life support (BLS) ambulance. Our model also takes into account a reduced availability of ambulances caused by technical breaks and secondary transports. The secondary transport is a planned activity where an ambulance does not respond to an emergency call but transports patients or medical materials between two healthcare facilities. In addition to the symbols introduced at the beginning of this section, the following notations will be used in the MCLP model:  
_Indices_  
\(h\) hospital  
\(k\) ambulance type (_k_ = 1--ALS, \(k\) = 2--BLS)  
\(l\) patient priority (_l_ = 1--FHQ, \(l\) = 2--others)  
_Parameters_  
\(\alpha_{k}\) the proportion of the interventions of the type \(k\) ambulances, which include the transportation of patients to a hospital  
\(\beta_{k}\) a correction factor that reduces the availability of the type \(k\) ambulances by the time they are occupied with secondary trips and technical breaks  
\(\gamma_{kl}\) coefficient; \(\gamma_{kl}\) = 1 if a type \(k\) ambulance can serve the priority \(l\) patients and \(\gamma_{kl}\) = 0 otherwise  
\(p_{k}\) the maximum number of the type \(k\) ambulances  
\(b_{jl}\) the yearly demand of the priority \(l\) patients in demand zone \(j\) (the number of patients)  
_td_ the average drop-off time at the hospital (min)  
_tsu_ the average on-scene time of a type \(k\) ambulance serving a priority \(l\) patient (min)  
\(\overline{t}_{ij}\) the travel time of an ambulance driving at a normal speed from location \(i\) to its base station \(j\) (min)  
Since this model distinguishes ambulance types and patient priorities, the definitions of the location and allocation variables become slightly different in comparison to the _p_MP.