## Step 1: Define the optimization problem and identify the variables, objective function, and constraints.
The problem involves maximizing an objective function with variables representing the quantity of light infantry companies, signal platoons, logistics companies, and medical platoons. The objective function to maximize is: $2(light\ infantry\ companies)^2 + 6(light\ infantry\ companies)(signal\ platoons) + 3(light\ infantry\ companies)(logistics\ companies) + 4(signal\ platoons)^2 + 8(signal\ platoons)(medical\ platoons) + 6(logistics\ companies)^2 + 8(logistics\ companies)(medical\ platoons) + 4(signal\ platoons) + 6(logistics\ companies) + 9(medical\ platoons)$.

## 2: Translate the given resources/attributes into a usable format for the optimization problem.
Given resources/attributes:
- $r0$: logistical capacity with an upper bound of 136 and specific capacities for each unit type.
- $r1$: fuel demand with an upper bound of 176 and specific demands for each unit type.

## 3: Express the constraints based on the problem description.
Constraints:
1. Logistical capacity constraints:
- $13(light\ infantry\ companies) + 26(signal\ platoons) + 3(logistics\ companies) + 22(medical\ platoons) \leq 136$
2. Fuel demand constraints:
- $11(light\ infantry\ companies) + 15(signal\ platoons) + 10(logistics\ companies) + 20(medical\ platoons) \leq 176$
3. Minimum combined logistical capacity: $(light\ infantry\ companies)^2 + (logistics\ companies)^2 + (medical\ platoons)^2 \geq 30$
4. Minimum fuel demand constraints:
- $11(light\ infantry\ companies) + 20(medical\ platoons) \geq 38$
- $10(logistics\ companies) + 20(medical\ platoons) \geq 31$
- $15(signal\ platoons) + 10(logistics\ companies) \geq 33$
5. Maximum combined logistical capacity constraints:
- $13(light\ infantry\ companies) + 3(logistics\ companies) \leq 113$
- $3(logistics\ companies) + 22(medical\ platoons) \leq 110$
- $13(light\ infantry\ companies) + 22(medical\ platoons) \leq 105$
- $13(light\ infantry\ companies) + 26(signal\ platoons) + 3(logistics\ companies) + 22(medical\ platoons) \leq 105$
6. Maximum fuel demand constraints:
- $(11(light\ infantry\ companies))^2 + (15(signal\ platoons))^2 \leq 161$
- $(15(signal\ platoons))^2 + (20(medical\ platoons))^2 \leq 133$
- $(15(signal\ platoons))^2 + (10(logistics\ companies))^2 \leq 92$
- $10(logistics\ companies) + 20(medical\ platoons) \leq 174$
- $11(light\ infantry\ companies) + 20(medical\ platoons) \leq 61$
- $15(signal\ platoons) + 10(logistics\ companies) + 20(medical\ platoons) \leq 168$
- $(11(light\ infantry\ companies))^2 + (15(signal\ platoons))^2 + (10(logistics\ companies))^2 \leq 82$
- $11(light\ infantry\ companies) + 15(signal\ platoons) + 10(logistics\ companies) + 20(medical\ platoons) \leq 82$

## 4: Implement the optimization problem using Gurobi.
To solve this problem, we will use the Gurobi Python library. We need to define the variables, the objective function, and all the constraints.

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
light_infantry_companies = m.addVar(name="light_infantry_companies", vtype=gp.GRB.INTEGER)
signal_platoons = m.addVar(name="signal_platoons", vtype=gp.GRB.INTEGER)
logistics_companies = m.addVar(name="logistics_companies", vtype=gp.GRB.INTEGER)
medical_platoons = m.addVar(name="medical_platoons", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(2 * light_infantry_companies**2 + 6 * light_infantry_companies * signal_platoons + 
               3 * light_infantry_companies * logistics_companies + 4 * signal_platoons**2 + 
               8 * signal_platoons * medical_platoons + 6 * logistics_companies**2 + 
               8 * logistics_companies * medical_platoons + 4 * signal_platoons + 
               6 * logistics_companies + 9 * medical_platoons, gp.GRB.MAXIMIZE)

# Logistical capacity constraint
m.addConstr(13 * light_infantry_companies + 26 * signal_platoons + 3 * logistics_companies + 
             22 * medical_platoons <= 136, name="logistical_capacity")

# Fuel demand constraint
m.addConstr(11 * light_infantry_companies + 15 * signal_platoons + 10 * logistics_companies + 
             20 * medical_platoons <= 176, name="fuel_demand")

# Minimum combined logistical capacity
m.addConstr(light_infantry_companies**2 + logistics_companies**2 + medical_platoons**2 >= 30, 
             name="min_logistical_capacity")

# Minimum fuel demand constraints
m.addConstr(11 * light_infantry_companies + 20 * medical_platoons >= 38, name="min_fuel_demand_1")
m.addConstr(10 * logistics_companies + 20 * medical_platoons >= 31, name="min_fuel_demand_2")
m.addConstr(15 * signal_platoons + 10 * logistics_companies >= 33, name="min_fuel_demand_3")

# Maximum combined logistical capacity constraints
m.addConstr(13 * light_infantry_companies + 3 * logistics_companies <= 113, name="max_logistical_capacity_1")
m.addConstr(3 * logistics_companies + 22 * medical_platoons <= 110, name="max_logistical_capacity_2")
m.addConstr(13 * light_infantry_companies + 22 * medical_platoons <= 105, name="max_logistical_capacity_3")
m.addConstr(13 * light_infantry_companies + 26 * signal_platoons + 3 * logistics_companies + 
             22 * medical_platoons <= 105, name="max_logistical_capacity_4")

# Maximum fuel demand constraints
m.addConstr(11 * light_infantry_companies**2 + 15 * signal_platoons**2 <= 161, name="max_fuel_demand_1")
m.addConstr(15 * signal_platoons**2 + 20 * medical_platoons**2 <= 133, name="max_fuel_demand_2")
m.addConstr(15 * signal_platoons**2 + 10 * logistics_companies**2 <= 92, name="max_fuel_demand_3")
m.addConstr(10 * logistics_companies + 20 * medical_platoons <= 174, name="max_fuel_demand_4")
m.addConstr(11 * light_infantry_companies + 20 * medical_platoons <= 61, name="max_fuel_demand_5")
m.addConstr(15 * signal_platoons + 10 * logistics_companies + 20 * medical_platoons <= 168, 
             name="max_fuel_demand_6")
m.addConstr(11 * light_infantry_companies**2 + 15 * signal_platoons**2 + 10 * logistics_companies**2 <= 82, 
             name="max_fuel_demand_7")
m.addConstr(11 * light_infantry_companies + 15 * signal_platoons + 10 * logistics_companies + 
             20 * medical_platoons <= 82, name="max_fuel_demand_8")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Light Infantry Companies:", light_infantry_companies.varValue)
    print("Signal Platoons:", signal_platoons.varValue)
    print("Logistics Companies:", logistics_companies.varValue)
    print("Medical Platoons:", medical_platoons.varValue)
    print("Objective:", m.objVal)
else:
    print("The model is infeasible.")
```