Problem Statement

A company needs to transport goods between three warehouses $($Warehouse $1,$ Warehouse $2,$ and Warehouse $3)$ using two types of trucks: small trucks and large trucks. The transportation logistics team is trying to determine how many small and large trucks to allocate to each warehouse, subject to resource constraints. However, due to variability in the number of available trucks, not all constraints are binding, leading to free variables.

Truck Capacities:

$-$ Small trucks can carry $1$ unit of goods from each warehouse.

$-$ Large trucks can carry $2$ units of goods from each warehouse.

Resource availability:

$-$ Warehouse $1$ requires $4$ units of goods transported.

$-$ Warehouse $2$ requires $5$ units of goods transported.

$-$ Warehouse $3$ requires $7$ units of goods transported.

$-$ There are $4$ large trucks.

Tasks:

$1.$ Solve the system of equations for the number of small and large trucks allocated to each warehouse.

$2.$ Identify the free variables in your solution and express the parametric form of the solutions.

$3.$ Interpret the meaning of the free variables in terms of the flexibility in truck allocation.