Groupe d’études et de recherche en analyse des décisions

# Optimal Ambulance Location with Tiered Response Time Standards: a Case Study for the Region of Waterloo

## Elizabeth M. Jewkes

The province of Ontario has announced a new land ambulance response time standard that distinguishes response time targets by patient acuity. For sudden onset cardiac arrest (SCA) patients, service providers are to report the percentage of calls where a defibrillator is on scene within 6 minutes. These first responders can be paramedics, fire fighters, police officers, or other qualified healthcare providers. EMS service providers are also to report on the percentage of urgent calls where an ambulance is on scene within 8 minutes. While response time thresholds for lower acuity patients have not been set by the province, service providers are to establish goals, and are to report each year on their performance.

The model discussed in this presentation is based on investigating the consequences of the new provincial standards on the Region of Waterloo EMS service providers. The new standards come into effect in the fall of 2012. The region has a fleet of 18 ambulances, three hospitals, 9 ambulance stations, in addition to 15 fire stations. It serves a growing population of approximately 500,000. For this study, we have used two years of EMS data, from July 2005 to June 2007 – a total of 57,000 calls. The 1400 km2 region is represented by a network of 378 nodes with high demand urban areas of the city being 1 km2 in size (corresponding to a mapping unit called a UTM), and lower demand rural areas of up to 25 km2. A model for the probability distribution of response times for inter-UTM travel (a separate study) has been developed so that we can determine call coverage for all nodes.

One of our goals has been to generate a compliance table for the region. Given $$K$$ ambulances on shift, a compliance table indicates where to locate $$0, 1,2,…,K$$ available ambulances. Note that the number available ambulances changes each time there is a call for service, or an ambulance returns from a call, so there is considerable need to reposition ambulances over the course of a shift to restore coverage. We have not studied the complex repositioning problem, although this is the subject of research by others.

We have formulated a mixed-integer nonlinear programming problem for locating ambulances throughout the city’s existing stations in order to maximize several different measures of coverage. The formulation is easily generalizable to place ambulances at any location in the city. The decision variables in our model are integer, and indicate the number of ambulances to locate at each station, given K, the number of ambulances on-shift. The constraints limit the number of ambulances to $$K$$, and define coverage for each node, for each of three categories of calls. We have solved several optimization problems with different objectives to gain a greater understanding of how the objective function affects the solution. The two primary objectives have been to maximize average coverage over all call categories, and to maximize average coverage of the SCA calls. These have produced different results, which we will comment on in the presentation.

Our theoretical contributions are in modelling the probability distribution of response time from station to UTM, and in the location model that captures several service level categories. It also represents a contribution to applied OR modelling in the healthcare sector.