## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'light infantry companies' and 'water purification units'. Let's denote 'light infantry companies' as $x_1$ and 'water purification units' as $x_2$. The objective function to minimize is $2x_1 + 2x_2$. The constraints are:
- $0.37x_1 + 1.05x_2 \geq 22$ (minimum logistics footprint)
- $0.37x_1 + 1.05x_2 \leq 65$ (upper bound of logistics footprint)
- $1.57x_1 + 1.74x_2 \geq 22$ (minimum offensive capability rating)
- $1.57x_1 + 1.74x_2 \leq 38$ (maximum offensive capability rating)
- $8x_1 - 8x_2 \geq 0$ (relationship between light infantry companies and water purification units)
- $x_1, x_2$ are integers (whole number requirement)

## Step 2: Convert the problem into a symbolic representation
The symbolic representation of the variables is:
- $x_1$ for 'light infantry companies'
- $x_2$ for 'water purification units'

The objective function in symbolic notation is: $2x_1 + 2x_2$

The constraints in symbolic notation are:
- $0.37x_1 + 1.05x_2 \geq 22$
- $0.37x_1 + 1.05x_2 \leq 65$
- $1.57x_1 + 1.74x_2 \geq 22$
- $1.57x_1 + 1.74x_2 \leq 38$
- $8x_1 - 8x_2 \geq 0$

## 3: Express the problem in the required JSON format
```json
{
    'sym_variables': [('x1', 'light infantry companies'), ('x2', 'water purification units')],
    'objective_function': '2*x1 + 2*x2',
    'constraints': [
        '0.37*x1 + 1.05*x2 >= 22',
        '0.37*x1 + 1.05*x2 <= 65',
        '1.57*x1 + 1.74*x2 >= 22',
        '1.57*x1 + 1.74*x2 <= 38',
        '8*x1 - 8*x2 >= 0'
    ]
}
```

## 4: Implement the problem using Gurobi
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="light_infantry_companies", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="water_purification_units", vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(2*x1 + 2*x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(0.37*x1 + 1.05*x2 >= 22, name="min_logistics_footprint")
    model.addConstr(0.37*x1 + 1.05*x2 <= 65, name="max_logistics_footprint")
    model.addConstr(1.57*x1 + 1.74*x2 >= 22, name="min_offensive_capability")
    model.addConstr(1.57*x1 + 1.74*x2 <= 38, name="max_offensive_capability")
    model.addConstr(8*x1 - 8*x2 >= 0, name="relationship_between_variables")

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Light infantry companies: {x1.varValue}")
        print(f"Water purification units: {x2.varValue}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status.")

solve_optimization_problem()
```