## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

- Let $x_0$ represent the number of 'security onions',
- Let $x_1$ represent the number of 'automatic alerts',
- Let $x_2$ represent the number of 'system administrators'.

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

## Step 2: List all the constraints in symbolic notation
Now, let's list all the constraints:

1. $11x_0 + 7x_1 + 14x_2 \geq 21$ (Total computational load from all should be at least 21 TFLOPs, corrected from the problem statement for accuracy)
2. $11x_0 + 7x_1 \geq 29$ (security onions and automatic alerts computational load)
3. $11x_0 + 7x_1 + 14x_2 \geq 21$ (Total computational load from all)
4. $7x_1 + 14x_2 \geq 20$ (computational load from automatic alerts and system administrators)
5. $6x_0 + 16x_1 + 20x_2 \geq 32$ (Total combined data integrity impact)
6. $6x_0 + 16x_1 \geq 49$ (Total combined data integrity impact from security onions and automatic alerts)
7. $6x_0 + 20x_2 \geq 46$ (Total combined data integrity impact from security onions and system administrators)
8. $16x_1 + 20x_2 \geq 29$ (Total combined data integrity impact from automatic alerts and system administrators)
9. $20x_0 + x_1 + 3x_2 \geq 16$ (Total combined data accessibility impact from all)
10. $x_1 + 3x_2 \geq 17$ (Total combined data accessibility impact from automatic alerts and system administrators)
11. $20x_0 + 3x_2 \geq 16$ (Total combined data accessibility impact from security onions and system administrators)
12. $-9x_1 + 9x_2 \geq 0$ (Relationship between automatic alerts and system administrators)
13. $-x_0 + 2x_2 \geq 0$ (Relationship between security onions and system administrators)
14. $16x_1 + 20x_2 \leq 142$ (Upper limit on data integrity impact from automatic alerts and system administrators)
15. $6x_0 + 20x_2 \leq 53$ (Upper limit on data integrity impact from security onions and system administrators)
16. $6x_0 + 16x_1 + 20x_2 \leq 154$ (Upper limit on total data integrity impact)

## 3: Define the symbolic representation in JSON format
```json
{
    'sym_variables': [
        ('x0', 'security onions'),
        ('x1', 'automatic alerts'),
        ('x2', 'system administrators')
    ],
    'objective_function': '4*x0 + 3*x1 + x2',
    'constraints': [
        '11*x0 + 7*x1 + 14*x2 >= 21',
        '11*x0 + 7*x1 >= 29',
        '7*x1 + 14*x2 >= 20',
        '6*x0 + 16*x1 + 20*x2 >= 32',
        '6*x0 + 16*x1 >= 49',
        '6*x0 + 20*x2 >= 46',
        '16*x1 + 20*x2 >= 29',
        '20*x0 + x1 + 3*x2 >= 16',
        'x1 + 3*x2 >= 17',
        '20*x0 + 3*x2 >= 16',
        '-9*x1 + 9*x2 >= 0',
        '-x0 + 2*x2 >= 0',
        '16*x1 + 20*x2 <= 142',
        '6*x0 + 20*x2 <= 53',
        '6*x0 + 16*x1 + 20*x2 <= 154'
    ]
}
```

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

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

    # Define the variables
    x0 = model.addVar(name='security_onions', vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name='automatic_alerts', vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name='system_administrators', vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(4 * x0 + 3 * x1 + x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(11 * x0 + 7 * x1 + 14 * x2 >= 21)
    model.addConstr(11 * x0 + 7 * x1 >= 29)
    model.addConstr(7 * x1 + 14 * x2 >= 20)
    model.addConstr(6 * x0 + 16 * x1 + 20 * x2 >= 32)
    model.addConstr(6 * x0 + 16 * x1 >= 49)
    model.addConstr(6 * x0 + 20 * x2 >= 46)
    model.addConstr(16 * x1 + 20 * x2 >= 29)
    model.addConstr(20 * x0 + x1 + 3 * x2 >= 16)
    model.addConstr(x1 + 3 * x2 >= 17)
    model.addConstr(20 * x0 + 3 * x2 >= 16)
    model.addConstr(-9 * x1 + 9 * x2 >= 0)
    model.addConstr(-x0 + 2 * x2 >= 0)
    model.addConstr(16 * x1 + 20 * x2 <= 142)
    model.addConstr(6 * x0 + 20 * x2 <= 53)
    model.addConstr(6 * x0 + 16 * x1 + 20 * x2 <= 154)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Security Onions: ', x0.varValue)
        print('Automatic Alerts: ', x1.varValue)
        print('System Administrators: ', x2.varValue)
    else:
        print('No solution found')

solve_optimization_problem()
```