Introducing fairness in Norwegian air ambulance base location planning

Abstract

Background: A primary task of the Norwegian helicopter emergency medical services (HEMS) is to provide advanced medical care to the critical ill and injured outside of hospitals. Where HEMS bases are located, directly influences who in the population can be reached within a given response time threshold and who cannot. When studying the locations of bases, the focus is often on efficiency, that is, maximizing the total number of people that can be reached within a given set time. This approach is known to benefit people living in densely populated areas, such as cities, over people living in remote areas. The most efficient solution is thus typically not necessarily a fair one. This study aims to incorporate fairness in finding optimal air ambulance base locations.

Methods: We solve multiple advanced mathematical optimization models to determine optimal helicopter base locations, with different optimization criteria related to the level of aversion to inequality, including the utilitarian, Bernoulli-Nash and iso-elastic social welfare functions. This is the first study to use the latter social welfare function for HEMS.

Results: Focusing on efficiency, a utilitarian objective function focuses on covering the larger cities in Norway, leaving parts of Norway largely uncovered. Including fairness by rather using an iso-elastic social welfare function in the optimization avoids leaving whole areas uncovered and in particular increases service levels in the north of Norway.

Conclusions: Including fairness in determining optimal HEMS base locations has great impact on population coverage, in particular when the number of base locations is not enough to give full coverage of the country. As results differ depending on the mathematical objective, the work shows the importance of not only looking for optimal solutions, but also raising the essential question of 'optimal with respect to what'.

Keywords: Air ambulance; Facility location problem; Fairness; HEMS.

Fig. 1

Population density heat map of Norway. Each municipality is depicted as a dot located in the population-weighted centroid of the municipality. The color indicates the fraction of the population living in that municipality in regards to the total Norwegian population. The twelve existing air ambulance bases are superimposed

Fig. 2

The optimal 8 base locations for a time threshold of 45 min using different objective functions. Utility is defined as the probability that a HEMS departing from the nearest base reaches the patient on time; green means an inhabitant may expect HEMS to always be on time; yellow means inhabitants should expect on-time arrival in 50% of the cases, and red 0% of cases

Fig. 3

Iso-elastic social welfare that is achieved by the utilitarian and iso-elasticoptimum for a = 0.9, for increasing number of base locations

Fig. 4

Optimal base locations when relocating one or two bases compared to the existing base structure for a 45 min threshold, using different objective functions. Utility is defined as the probability that a HEMS departing from the nearest base reaches the patient on time; green means an inhabitant may expect HEMS to always be on time; yellow means inhabitants should expect on-time arrival in 50% of the cases, and red 0% of cases

Fig. 5

Optimal base locations when adding one or two bases to the existing base structure for a 30 min threshold, using different objective functions. Utility is defined as the probability that a HEMS departing from the nearest base reaches the patient on time; green means an inhabitant may expect HEMS to always be on time; yellow means inhabitants should expect on-time arrival in 50% of the cases, and red 0% of cases

Fig. 6

Small example for the ambulance location problem, consisting of four demand locations (municipalities) which are all potential base locations