Ambulance Dispatch Optimization

This project spans Sep 2025 - Apr 2026 and is actively developing

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Introduction

Objective

The optimization objective is:

$$ \min_{x^{(t)}} \sum_{a \in A} \sum_{i \in R_t} \frac{1}{w_i}\left( d(\text{pos}_a^{(t)}, i) + d(i, h^*(i)) \right) x_{ai}^{(t)} $$

Subject to

Each ambulance serves at most one incident:

$$ \sum_{i \in R_t} x_{ai}^{(t)} \le 1 \quad \forall a \in A $$

Each incident is served at most once:

$$ \sum_{a \in A} x_{ai}^{(t)} \le 1 \quad \forall i \in R_t $$

Exactly (k_t) assignments are made:

$$ \sum_{a \in A} \sum_{i \in R_t} x_{ai}^{(t)} = k_t $$

Binary decision variables:

$$ x_{ai}^{(t)} \in {0,1} \quad \forall a \in A, i \in R_t $$

The first constraint ensures each ambulance serves at most one incident, the second ensures each incident is served at most once, and the third ensures exactly (k_t) assignments per round.

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Where

• $A$: set of available ambulances, indexed by $a$
• $R_t$: set of unserved incidents at round $t$
• $H$: set of hospitals (fixed locations), indexed by $h$
• $\text{pos}_a^{(t)}$: current position of ambulance $a$ at round $t$
• $d(p, q)$: Google Distance Matrix API distance between coordinates $p$ and $q$
• $h^*(i)$: nearest hospital to incident $i$
• $w_i \in {1,2,4,8,16}$: priority weight (larger means higher priority)
• $k_t = \min(|A|, |R_t|)$: number of assignments in round $t$
• $x_{ai}^{(t)}$:
  • 1 if ambulance $a$ is assigned to incident $i$
  • 0 otherwise

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Interpretation

The optimization seeks to minimize the total weighted travel cost of all ambulance–incident–hospital assignments during round (t).

Each term in the objective contains:

$$ d(\text{pos}_a^{(t)}, i) + d(i, h^*(i)) $$

which is the distance from the ambulance to the incident and then to the nearest hospital.

The weighting factor (1/w_i) increases the influence of high-priority incidents by reducing their effective cost relative to lower-priority ones.

Simulation