To solve this problem, let's first define the symbolic representation of the optimization problem.

### Symbolic Representation

We have four variables:
- $x_0$: water purification units
- $x_1$: airborne infantry companies
- $x_2$: medical platoons
- $x_3$: artillery batteries

The objective function to minimize is: $3x_0 + 4x_1 + 4x_2 + 5x_3$

## Constraints

1. Offensive capability ratings:
   - $2x_0 + 3x_1 + 16x_2 + 16x_3 \geq 49$
   - $3x_1 + 16x_2 + 16x_3 \geq 51$
   - $2x_0 + 3x_1 + 16x_2 \geq 71$
   - $2x_0 + 16x_2 + 16x_3 \geq 71$
   - $3x_1 + 16x_2 + 16x_3 \geq 71$
   - $2x_0 + 3x_1 + 16x_2 \geq 58$
   - $2x_0 + 16x_2 + 16x_3 \geq 58$
   - $3x_1 + 16x_2 + 16x_3 \geq 58$
   - $2x_0 + 3x_1 + 16x_3 \geq 58$
   - $2x_0 + 3x_1 + 16x_2 \geq 46$
   - $2x_0 + 16x_2 + 16x_3 \geq 46$
   - $3x_1 + 16x_2 + 16x_3 \geq 46$
   - $2x_0 + 3x_1 + 16x_3 \geq 46$
   - $2x_0 + 3x_1 + 16x_2 \geq 41$
   - $2x_0 + 16x_2 + 16x_3 \geq 41$
   - $3x_1 + 16x_2 + 16x_3 \geq 41$
   - $2x_0 + 3x_1 + 16x_3 \geq 41$
   - $2x_0 + 3x_1 + 16x_2 + 16x_3 \geq 41$

2. Deployment weights:
   - $16x_2 + 13x_3 \geq 30$
   - $5x_0 + 17x_1 \geq 19$
   - $5x_0 + 13x_3 \geq 47$
   - $17x_1 + 13x_3 \geq 50$
   - $17x_1 + 16x_2 \geq 42$
   - $5x_0 + 17x_1 + 16x_2 \geq 46$
   - $17x_1 + 16x_2 + 13x_3 \geq 46$
   - $5x_0 + 17x_1 + 16x_2 \geq 47$
   - $17x_1 + 16x_2 + 13x_3 \geq 47$
   - $5x_0 + 17x_1 + 16x_2 + 13x_3 \geq 47$

3. Logistics footprint:
   - $9x_1 + 10x_3 \geq 20$
   - $7x_0 + 13x_2 \geq 36$
   - $7x_0 + 9x_1 + 13x_2 + 10x_3 \geq 36$

4. Mobility ratings:
   - $9x_0 + 8x_1 \geq 37$
   - $9x_0 + 6x_2 \geq 36$
   - $8x_1 + 6x_2 + 15x_3 \geq 49$
   - $9x_0 + 8x_1 + 6x_2 + 15x_3 \geq 49$

5. Other constraints:
   - $-x_0 + 8x_3 \geq 0$
   - $-6x_2 + 3x_3 \geq 0$
   - $2x_0 + 16x_2 + 16x_3 \leq 165$
   - $5x_0 + 17x_1 \leq 77$
   - $16x_2 + 13x_3 \leq 107$
   - $7x_0 + 13x_2 \leq 92$
   - $9x_1 + 10x_3 \leq 157$
   - $8x_1 + 6x_2 \leq 180$
   - $8x_1 + 6x_2 + 15x_3 \leq 190$
   - $9x_0 + 8x_1 + 6x_2 \leq 144$

6. Variable bounds:
   - $x_0, x_1, x_2, x_3 \geq 0$ and are integers.

### Gurobi Code

```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="water_purification_units", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="airborne_infantry_companies", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="medical_platoons", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="artillery_batteries", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(3 * x0 + 4 * x1 + 4 * x2 + 5 * x3, gurobi.GRB.MINIMIZE)

    # Offensive capability ratings
    model.addConstr(2 * x0 + 3 * x1 + 16 * x2 + 16 * x3 >= 49)
    model.addConstr(3 * x1 + 16 * x2 + 16 * x3 >= 51)
    model.addConstr(2 * x0 + 3 * x1 + 16 * x2 >= 71)
    model.addConstr(2 * x0 + 16 * x2 + 16 * x3 >= 71)
    model.addConstr(3 * x1 + 16 * x2 + 16 * x3 >= 71)
    model.addConstr(2 * x0 + 3 * x1 + 16 * x2 >= 58)
    model.addConstr(2 * x0 + 16 * x2 + 16 * x3 >= 58)
    model.addConstr(3 * x1 + 16 * x2 + 16 * x3 >= 58)
    model.addConstr(2 * x0 + 3 * x1 + 16 * x3 >= 58)
    model.addConstr(2 * x0 + 3 * x1 + 16 * x2 >= 46)
    model.addConstr(2 * x0 + 16 * x2 + 16 * x3 >= 46)
    model.addConstr(3 * x1 + 16 * x2 + 16 * x3 >= 46)
    model.addConstr(2 * x0 + 3 * x1 + 16 * x3 >= 46)
    model.addConstr(2 * x0 + 3 * x1 + 16 * x2 >= 41)
    model.addConstr(2 * x0 + 16 * x2 + 16 * x3 >= 41)
    model.addConstr(3 * x1 + 16 * x2 + 16 * x3 >= 41)
    model.addConstr(2 * x0 + 3 * x1 + 16 * x3 >= 41)
    model.addConstr(2 * x0 + 3 * x1 + 16 * x2 + 16 * x3 >= 41)

    # Deployment weights
    model.addConstr(16 * x2 + 13 * x3 >= 30)
    model.addConstr(5 * x0 + 17 * x1 >= 19)
    model.addConstr(5 * x0 + 13 * x3 >= 47)
    model.addConstr(17 * x1 + 13 * x3 >= 50)
    model.addConstr(17 * x1 + 16 * x2 >= 42)
    model.addConstr(5 * x0 + 17 * x1 + 16 * x2 >= 46)
    model.addConstr(17 * x1 + 16 * x2 + 13 * x3 >= 46)
    model.addConstr(5 * x0 + 17 * x1 + 16 * x2 >= 47)
    model.addConstr(17 * x1 + 16 * x2 + 13 * x3 >= 47)
    model.addConstr(5 * x0 + 17 * x1 + 16 * x2 + 13 * x3 >= 47)

    # Logistics footprint
    model.addConstr(9 * x1 + 10 * x3 >= 20)
    model.addConstr(7 * x0 + 13 * x2 >= 36)
    model.addConstr(7 * x0 + 9 * x1 + 13 * x2 + 10 * x3 >= 36)

    # Mobility ratings
    model.addConstr(9 * x0 + 8 * x1 >= 37)
    model.addConstr(9 * x0 + 6 * x2 >= 36)
    model.addConstr(8 * x1 + 6 * x2 + 15 * x3 >= 49)
    model.addConstr(9 * x0 + 8 * x1 + 6 * x2 + 15 * x3 >= 49)

    # Other constraints
    model.addConstr(-x0 + 8 * x3 >= 0)
    model.addConstr(-6 * x2 + 3 * x3 >= 0)
    model.addConstr(2 * x0 + 16 * x2 + 16 * x3 <= 165)
    model.addConstr(5 * x0 + 17 * x1 <= 77)
    model.addConstr(16 * x2 + 13 * x3 <= 107)
    model.addConstr(7 * x0 + 13 * x2 <= 92)
    model.addConstr(9 * x1 + 10 * x3 <= 157)
    model.addConstr(8 * x1 + 6 * x2 <= 180)
    model.addConstr(8 * x1 + 6 * x2 + 15 * x3 <= 190)
    model.addConstr(9 * x0 + 8 * x1 + 6 * x2 <= 144)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Water purification units: ", x0.varValue)
        print("Airborne infantry companies: ", x1.varValue)
        print("Medical platoons: ", x2.varValue)
        print("Artillery batteries: ", x3.varValue)
    else:
        print("No optimal solution found")

solve_optimization_problem()
```