Ambulance Location Model¶

Definition¶

The ambulance location model selects a minimum number of station towns while ensuring that every town is within 18 minutes of a selected station. Let $y_j=1$ if a station is placed at town $j$ , and let $N_i=\{j\colon t_{ij}\le18\}$ $18}$ . Then

\begin{gathered} \min\sum_{j=1}^6y_j,\\ \sum_{j\in N_i}y_j\ge1\ (i=1,\ldots,6),\\ y_j\in\{0,1\}. \end{gathered}

Interpretation¶

The model answers one question: at how few towns must ambulance stations be placed so that every town has a station within 18 minutes? Each coverage constraint requires at least one sufficiently close station. A station may cover several towns, and a town may be covered by several stations.

Six towns as nodes, with dashed links joining the stations at towns 4 and 5 to the towns each reaches within 18 minutes; together the two stations cover every town.

The dashed links join each chosen station to the towns it reaches within 18 minutes. The station at town 4 covers towns 1, 2 and 6 as well as itself, and the station at town 5 covers towns 2 and 3 and itself, so all six towns are covered and town 2 is covered twice.

Read $N_i$ as the set of towns from which town $i$ can be served in time, always including $i$ itself. The sum $\displaystyle \sum_{j\in N_i}y_j$ then counts the stations that serve town $i$ , and the constraint says that this count is not zero. The two constraints in the Examples, $y_1+y_4\ge1$ and $y_3+y_5\ge1$ $1$ , share no variable, so every feasible choice opens at least two stations; the figure shows that two suffice.

This is a binary integer program: each $y_j$ is a binary variable. The bus scheduling model has the same kind of coverage constraints with integer counts instead.

A common slip is to write the coverage constraints with $=1$ instead of $\ge1$ $1$ . That would forbid any town from being served by two stations, and the choice of towns 4 and 5 would become infeasible because of town 2.

Key Results¶

Include the town itself because its travel time to itself is zero.
The threshold includes exactly 18 minutes.
This is the set-covering structure illustrated by the exercise; no station-capacity limit is specified.

Examples¶

The dashed links indicate travel times at most 18 minutes; the displayed positions are schematic rather than a geographical scale.

Town 1 can be covered from towns 1 or 4, giving $y_1+y_4\ge1$ $1$ .
Town 3 can be covered from towns 3 or 5, giving $y_3+y_5\ge1$ $1$ .

Prerequisites¶

Binary Integer Program

Sources¶

OR Modelling · Week 2

Keywords: Emergency ambulance service, Set covering model Symbols: element of, sum

Last updated: 20 October 2026 03:02