Optimizing Firefighter Deployment to Minimize Forest Fire Costs

Problem

After a forest fire, the number of firefighters to dispatch must be decided: more crew reduces forest loss but raises rescue costs, while fewer crew reduces costs but increases loss. The goal is to balance loss cost and rescue cost to determine the optimal crew size.

Analysis

Let the variables be crew size, fire start time, fire fighting start time, fire extinguishing time, and forest burned area.

Loss cost is a decreasing function of burned area, determined by the area burned.

Rescue cost is an increasing function of crew size and firefighting duration; an appropriate crew size minimizes the sum of loss and rescue costs.

The key is to make reasonable simplifying assumptions, plot the approximate relationship between time and burned area, and identify critical time points (fire start, firefighting start, extinguishing).

Model Assumptions

Fire spread speed is proportional to a coefficient representing the fire's propagation rate.

The extinguishing speed per firefighter is reduced to a value representing the average extinguishing speed of a crew member.

Loss cost per unit area is proportional to a coefficient representing the damage cost of burning one unit area.

Each firefighter incurs a unit-time extinguishing cost and a one-time fixed cost.

Model Construction

Define the total cost function as the sum of loss cost (dependent on burned area) and rescue cost (dependent on crew size and firefighting duration). The objective is to minimize this total cost with respect to the crew size.

Model Solution

Solve the optimization problem to find the crew size that yields the minimum total cost.

Result Interpretation

The solution gives the minimum number of firefighters required to prevent further fire spread.

The model parameters include loss cost per unit area, per‑firefighter unit‑time extinguishing cost, one‑time firefighter cost, fire start time, fire spread speed, and average extinguishing speed per firefighter.