Invited Review
Ambulance location and relocation models

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Abstract

This article traces the evolution of ambulance location and relocation models proposed over the past 30 years. The models are classified in two main categories. Deterministic models are used at the planning stage and ignore stochastic considerations regarding the availability of ambulances. Probabilistic models reflect the fact that ambulances operate as servers in a queueing system and cannot always answer a call. In addition, dynamic models have been developed to repeatedly relocate ambulances throughout the day.

Introduction

This review article traces the evolution of ambulance location and relocation models proposed over the past 30 years. This period was marked by an unprecedented growth not only in computer technology, but also in modeling and algorithmic sophistication, in the performance of mathematical programming solvers, and in the widespread adoption of computer software at several levels of decision making. The literature on ambulance positioning systems truly reflects this evolution. The first models proposed were unsophisticated integer linear programming formulations, but over time more realistic features were gradually introduced, and solution techniques also evolved.

Most of the early models dealt with the static and deterministic location problem. These were meant to be used at the planning stage and they ignored stochastic considerations. Several probabilistic models were then developed to reflect the fact that ambulances operate as servers in a queueing system and are sometimes unavailable to answer a call. Dynamic models are more recent. They address the problem of repeatedly relocating ambulances in the same day to provide better coverage. In recent years, the development of powerful local search algorithms, particularly tabu search (Glover and Laguna, 1997), coupled with the growth of parallel computing (Crainic and Toulouse, 1998) have given rise to a new stream of research that deals effectively with the dynamic nature of the problem. With the newest models and algorithms, large scale problems can be solved rapidly and dynamically in real time, with a high level of accuracy.

There exists a rich literature on emergency vehicles sitting models. The survey by Marianov and ReVelle (1995) provides an overview of the most important models published until that date. Our review is less general since it focuses on ambulance services, but it unavoidably covers some of the same material, albeit with a different emphasis. The Marianov–ReVelle survey ends with an indirect reference to dynamic relocation models integrated within geographic information systems: “rarely have ambulances been positioned at free standing stations” …, “Lastly we have a warming competition … The technique of GIS” (p. 223). Our survey precisely addresses this issue by devoting a section to dynamic relocation models which have just started to emerge. We also report on actual implementations of ambulance location and relocation models. Finally, we provide a synthetic overview, in table form, of all the models we discuss.

The article is structured as follows. In Section 2, we briefly describe the functioning of emergency medical services. Two early models developed for the static case are described in Section 3. An important shortcoming of these models is that they may no longer guarantee adequate coverage as soon as ambulances dispatched to a call become unavailable. Two types of models have been developed to handle the need to provide extra coverage: deterministic models and probabilistic models. These are presented in 4 Deterministic static models with extra coverage, 5 Probabilistic static models with extra coverage, respectively. We have chosen to concentrate on the most important models, leaving aside several minor variants already listed in the articles of ReVelle (1989), Swersey (1994) and Marianov and ReVelle (1995). In Section 6, we provide an account of some of the emerging research in the area of dynamic ambulance repositioning. A summary and conclusions follow in Section 7.

Section snippets

How emergency medical services operate

The chain of events leading to the intervention of an ambulance to the scene of an incident includes the following four steps: (1) incident detection and reporting, (2) call screening, (3) vehicle dispatching and (4) actual intervention by paramedics. Decisions made by emergency services managers are concerned with the second and third steps. The main function of the screening process is to determine the severity of the incident and its degree of urgency (e.g., on a one-to-four scale), and to

Two early models for the static ambulance location problem

Ambulance location models are defined on graphs. The set of demand points is denoted by V and the set of potential ambulance location sites is denoted by W. The shortest travel time tij from vertex i to vertex j of the graph is known. As is common in location theory, assigning demands to a discrete set can be achieved through an aggregation process which unavoidably results in a loss of accuracy. Various techniques have been proposed to measure and control the error bound (Erkut and Bozkaya,

Deterministic static models with extra coverage

Neither LSCM nor MCLP recognizes the fact that on occasions vehicles of several types may be dispatched to the scene of an incident. Also, even if only one vehicle type is used, solving MCLP alone may not provide a sufficiently robust location plan. We present in this section a number of deterministic models developed to deal with the issue of multiple coverage. Probabilistic models will be presented in Section 5.

One of the first models developed to handle several vehicle types is the tandem

Probabilistic static models with extra coverage

One of the first probabilistic models for ambulance location is the maximum expected covering location problem formulation (MEXCLP) due to Daskin (1983). In this model, it is assumed that each ambulance has the same probability q, called the busy fraction, of being unavailable to answer a call, and all ambulances are independent. The busy fraction can be estimated by dividing the total estimated duration of calls for all demand points by the total number of available ambulances. Thus, if vertex

A dynamic model

When siting emergency vehicles, relocation decisions must periodically be made in order not to leave areas unprotected. This was recognized by Kolesar and Walker (1974) who designed a relocation system for fire companies. The ambulance relocation problem is more difficult to tackle since it has to be solved more frequently at very short notice. More powerful solution methodologies are called for in this case. With the development of faster heuristics and advanced computer technologies, it is

Summary and conclusions

There has been an important evolution in the development of ambulance location and relocation models over the past thirty years. The first models were very basic and did not take into account the fact that some coverage is lost when an ambulance is dispatched to a call. Nevertheless, these early models served as a sound basis for the development of all subsequent models. The question of ambulance non-availability was addressed in two main ways. Deterministic models yield solutions in which

Acknowledgements

This research was partially funded by the Canada Research Chair in Distribution Management, by the Fonds de recherche en santé du Québec (FRSQ) under grant 980861, and by the Canadian Natural Sciences and Engineering Research Council (NSERC) under grants OGP0039682 and OGP0184123. This support is gratefully acknowledged. Thanks are due to two referees for their valuable comments.

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