To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables for each type of company/battery and then expressing the objective function and constraints in terms of these variables.

Let's define:
- $x_1$ as the number of armored companies,
- $x_2$ as the number of airborne infantry companies, and
- $x_3$ as the number of air defense batteries.

The objective function to maximize is: $9x_1 + 5x_2 + 7x_3$.

The constraints are:
1. Defensive capability rating constraint for armored companies: $11x_1$
2. Defensive capability rating constraint for airborne infantry companies: $9x_2$
3. Defensive capability rating constraint for air defense batteries: $15x_3$
4. Constraint involving armored companies and air defense batteries: $-3x_1 + 6x_3 \geq 0$
5. Total defensive capability rating from armored and air defense companies: $11x_1 + 15x_3 \leq 75$
6. Total defensive capability rating from airborne infantry and air defense companies: $9x_2 + 15x_3 \leq 120$
7. Total defensive capability rating from all companies/batteries: $11x_1 + 9x_2 + 15x_3 \leq 135$ (Note: The original problem statement mentions an upper bound of 135 for 'r0', which seems to refer to the total defensive capability, but there's also a mention of not exceeding 120 in another constraint. We'll consider both constraints as per the detailed description.)
8. All variables must be non-negative integers.

The symbolic representation is:
```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 <= 135',
    'x1 >= 0', 'x2 >= 0', 'x3 >= 0',
    'x1 % 1 == 0', 'x2 % 1 == 0', 'x3 % 1 == 0'
  ]
}
```

Now, let's implement this optimization problem using Gurobi in Python:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="armored_companies")
x2 = m.addVar(vtype=GRB.INTEGER, name="airborne_infantry_companies")
x3 = m.addVar(vtype=GRB.INTEGER, name="air_defense_batteries")

# Objective function
m.setObjective(9*x1 + 5*x2 + 7*x3, GRB.MAXIMIZE)

# Constraints
m.addConstr(-3*x1 + 6*x3 >= 0, "armored_and_air_defense_constraint")
m.addConstr(11*x1 + 15*x3 <= 75, "total_armored_and_air_defense_rating")
m.addConstr(9*x2 + 15*x3 <= 120, "total_airborne_and_air_defense_rating")
m.addConstr(11*x1 + 9*x2 + 15*x3 <= 135, "total_defensive_capability")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Armored Companies: {x1.x}")
    print(f"Airborne Infantry Companies: {x2.x}")
    print(f"Air Defense Batteries: {x3.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found.")
```