## Step 1: Define the symbolic representation of the variables
The variables are 'armored companies', 'airborne infantry companies', and 'air defense batteries'. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function is to maximize $9 \times \text{armored companies} + 5 \times \text{airborne infantry companies} + 7 \times \text{air defense batteries}$. In symbolic terms, this becomes $9x_1 + 5x_2 + 7x_3$.

## 3: Define the constraints in symbolic representation
The constraints given are:
1. $-3x_1 + 6x_3 \geq 0$
2. $11x_1 + 15x_3 \leq 75$
3. $9x_2 + 15x_3 \leq 120$
4. $11x_1 + 9x_2 + 15x_3 \leq 120$
And $x_1, x_2, x_3$ are integers.

## 4: Express the problem in a JSON format for clarity
```json
{
'sym_variables': [('x1', 'armored companies'), ('x2', 'airborne infantry companies'), ('x3', 'air defense batteries')],
'objective_function': '9*x1 + 5*x2 + 7*x3',
'constraints': [
'-3*x1 + 6*x3 >= 0',
'11*x1 + 15*x3 <= 75',
'9*x2 + 15*x3 <= 120',
'11*x1 + 9*x2 + 15*x3 <= 120',
'x1, x2, x3 are integers'
]
}
```

## 5: 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="armored_companies", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="airborne_infantry_companies", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="air_defense_batteries", vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(9 * x1 + 5 * x2 + 7 * x3, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(-3 * x1 + 6 * x3 >= 0)
    model.addConstr(11 * x1 + 15 * x3 <= 75)
    model.addConstr(9 * x2 + 15 * x3 <= 120)
    model.addConstr(11 * x1 + 9 * x2 + 15 * x3 <= 120)

    # Set bounds for defensive capability ratings
    model.addConstr(x1 <= 135 / 11, name='r0_x1')
    model.addConstr(x2 <= 135 / 9, name='r0_x2')
    model.addConstr(x3 <= 135 / 15, name='r0_x3')

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Armored companies: {x1.varValue}")
        print(f"Airborne infantry companies: {x2.varValue}")
        print(f"Air defense batteries: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```