## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bobby', 'hours worked by Paul', and 'hours worked by Hank', which can be represented as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $8x_0^2 + 6x_0x_1 + 6x_1^2 + 3x_1x_2 + x_0 + 9x_1 + x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $x_0 = 5$ (Bobby's organization score is 5, but this is an attribute and not directly a constraint on $x_0$)
- $x_0 = 18$ (Bobby has a productivity rating of 18, but again, this is an attribute)
- $x_0 = 11$ (Bobby has a likelihood to quit index of 11)
- $x_0 = 13$ (Bobby has a dollar cost per hour of 13)
- $x_1 = 16$ (Paul has an organization score of 16)
- $x_1 = 15$ (Paul has a productivity rating of 15)
- $x_1 = 13$ (Paul's likelihood to quit index is 13)
- $x_1 = 6$ (Paul has a dollar cost per hour of 6)
- $x_2 = 15$ (Hank has an organization score of 15)
- $x_2 = 12$ (Hank's productivity rating is 12)
- $x_2 = 5$ (Hank has a likelihood to quit index of 5)
- $x_2 = 16$ (Hank's dollar cost per hour is 16)
- $x_0^2 + x_1^2 \geq 55$
- $x_0 + x_1 + x_2 \geq 55$
- $x_0^2 + x_2^2 \geq 50$
- $x_0 + x_1 \geq 22$
- $x_0 + x_1 + x_2 \geq 22$
- $x_1 + x_2 \geq 25$
- $x_0 + x_1 \geq 27$
- $x_0^2 + x_1^2 + x_2^2 \geq 27$
- $x_0 + x_1 + x_2 \geq 27$
- $13x_0 + 16x_2 \geq 48$
- $6x_1 + 16x_2 \geq 57$
- $13^2x_0^2 + 6^2x_1^2 + 16^2x_2^2 \geq 88$
- $13x_0 + 6x_1 + 16x_2 \geq 88$
- $x_1 - 4x_2 \geq 0$
- $x_0 + x_2 \leq 107 / 18$ (approximately, but we'll keep it as is for accuracy)
- $x_0 + x_1 + x_2 \leq 102 / 18$ (approximately, but we'll keep it as is for accuracy, however, given productivity ratings are 18, 15, 12, let's assume this translates directly)
- $x_1^2 + x_2^2 \leq 47$
- $x_0 + x_2 \leq 59$
- $x_0 + x_1 \leq 34$
- $x_0 + x_1 + x_2 \leq 59$
- $x_0$ is an integer

## 4: Correct and Clarify Constraints Based on Given Attributes
Given attributes for each person are:
- Bobby: organization score $r_0 = 5$, productivity rating $r_1 = 18$, likelihood to quit index $r_2 = 11$, dollar cost per hour $r_3 = 13$
- Paul: $r_0 = 16$, $r_1 = 15$, $r_2 = 13$, $r_3 = 6$
- Hank: $r_0 = 15$, $r_1 = 12$, $r_2 = 5$, $r_3 = 16$

## 5: Formulate Constraints with Resource/Attributes
The actual constraints from attributes are not directly limiting $x_0, x_1, x_2$ but rather are given. The constraints we have are:
- $5x_0 \leq 283$
- $18x_0 \leq 195$
- $11x_0 \leq 92$
- $13x_0 \leq 265$
- $16x_1 \leq 283$
- $15x_1 \leq 195$
- $13x_1 \leq 92$
- $6x_1 \leq 265$
- $15x_2 \leq 283$
- $12x_2 \leq 195$
- $5x_2 \leq 92$
- $16x_2 \leq 265$

And the previously defined constraints.

## 6: Symbolic Representation
```json
{
    'sym_variables': [('x0', 'hours worked by Bobby'), ('x1', 'hours worked by Paul'), ('x2', 'hours worked by Hank')],
    'objective_function': '8*x0^2 + 6*x0*x1 + 6*x1^2 + 3*x1*x2 + x0 + 9*x1 + x2',
    'constraints': [
        'x0^2 + x1^2 >= 55',
        'x0 + x1 + x2 >= 55',
        'x0^2 + x2^2 >= 50',
        'x0 + x1 >= 22',
        'x0 + x1 + x2 >= 22',
        'x1 + x2 >= 25',
        'x0 + x1 >= 27',
        'x0^2 + x1^2 + x2^2 >= 27',
        'x0 + x1 + x2 >= 27',
        '13*x0 + 16*x2 >= 48',
        '6*x1 + 16*x2 >= 57',
        '169*x0^2 + 36*x1^2 + 256*x2^2 >= 88',
        '13*x0 + 6*x1 + 16*x2 >= 88',
        'x1 - 4*x2 >= 0',
        '18*x0 <= 195',
        '15*x1 <= 195',
        '12*x2 <= 195',
        'x1^2 + x2^2 <= 47',
        'x0 + x2 <= 59',
        'x0 + x1 <= 34',
        'x0 + x1 + x2 <= 59',
        'x0 % 1 == 0' # x0 is an integer
    ]
}
```

## 7: Gurobi Code
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="x0", lb=0, type=gurobi.GRB.INTEGER) # hours worked by Bobby
    x1 = model.addVar(name="x1", lb=0) # hours worked by Paul
    x2 = model.addVar(name="x2", lb=0) # hours worked by Hank

    # Objective function
    model.setObjective(8*x0**2 + 6*x0*x1 + 6*x1**2 + 3*x1*x2 + x0 + 9*x1 + x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x0**2 + x1**2 >= 55)
    model.addConstr(x0 + x1 + x2 >= 55)
    model.addConstr(x0**2 + x2**2 >= 50)
    model.addConstr(x0 + x1 >= 22)
    model.addConstr(x0 + x1 + x2 >= 22)
    model.addConstr(x1 + x2 >= 25)
    model.addConstr(x0 + x1 >= 27)
    model.addConstr(x0**2 + x1**2 + x2**2 >= 27)
    model.addConstr(x0 + x1 + x2 >= 27)
    model.addConstr(13*x0 + 16*x2 >= 48)
    model.addConstr(6*x1 + 16*x2 >= 57)
    model.addConstr(169*x0**2 + 36*x1**2 + 256*x2**2 >= 88)
    model.addConstr(13*x0 + 6*x1 + 16*x2 >= 88)
    model.addConstr(x1 - 4*x2 >= 0)
    model.addConstr(18*x0 <= 195)
    model.addConstr(15*x1 <= 195)
    model.addConstr(12*x2 <= 195)
    model.addConstr(x1**2 + x2**2 <= 47)
    model.addConstr(x0 + x2 <= 59)
    model.addConstr(x0 + x1 <= 34)
    model.addConstr(x0 + x1 + x2 <= 59)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Bobby: ", x0.varValue)
        print("Hours worked by Paul: ", x1.varValue)
        print("Hours worked by Hank: ", x2.varValue)
    else:
        print("No solution found")

optimization_problem()
```