## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Mary', 'hours worked by Bobby', and 'hours worked by Paul', 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 maximize is $6.18x_0 + 2.63x_1 + 1.35x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $13x_0 \leq 108$ (not directly given but $x_0$ has an upper bound related to 'likelihood to quit index' which seems to be a misunderstanding since the index itself is given as 13 for Mary)
- $14x_0 \leq 167$ (similarly, not directly given but $x_0$ has an upper bound related to 'work quality rating')
- $5x_1 \leq 108$
- $11x_2 \leq 108$
- $13x_0 + 5x_1 + 11x_2 \geq 18$
- $14x_0 + 11x_1 \geq 19$
- $14x_0 + 5x_2 \geq 41$
- $-2x_1 + 7x_2 \geq 0$
- $5x_1 + 11x_2 \leq 67$
- $13x_0 + 5x_1 + 11x_2 \leq 67$
- $14x_0 + 5x_2 \leq 96$
- $14x_0 + 11x_1 \leq 160$
- $14x_0 + 11x_1 + 5x_2 \leq 159$
- $x_0 \in \mathbb{Z}$ (integer, but the problem states nonfractional which implies integer for all)
- $x_1 \in \mathbb{Z}$
- $x_2 \in \mathbb{Z}$

However, upon closer inspection, it seems there was an initial misunderstanding in directly translating the given attributes into constraints. The attributes given ('likelihood to quit index' and 'work quality rating') for each person seem to be specific values rather than bounds for $x_0, x_1, x_2$. Thus, we focus on the constraints directly provided or implied by the problem statement.

## 4: Correctly interpret the constraints based on the problem statement
Given:
- Mary's likelihood to quit index is 13.
- Mary's work quality rating is 14.
- Bobby's likelihood to quit index is 5.
- Bobby's work quality rating is 11.
- Paul's likelihood to quit index is 11.
- Paul's work quality rating is 5.

And the constraints:
- $13x_0 + 5x_1 + 11x_2 \geq 18$
- $14x_0 + 11x_1 \geq 19$
- $14x_0 + 5x_2 \geq 41$
- $-2x_1 + 7x_2 \geq 0$
- $5x_1 + 11x_2 \leq 67$
- $13x_0 + 5x_1 + 11x_2 \leq 67$
- $14x_0 + 5x_2 \leq 96$
- $14x_0 + 11x_1 \leq 160$
- $14x_0 + 11x_1 + 5x_2 \leq 159$

## 5: Symbolic Representation
```json
{
'sym_variables': [('x0', 'hours worked by Mary'), ('x1', 'hours worked by Bobby'), ('x2', 'hours worked by Paul')],
'objective_function': '6.18x0 + 2.63x1 + 1.35x2',
'constraints': [
    '13x0 + 5x1 + 11x2 >= 18',
    '14x0 + 11x1 >= 19',
    '14x0 + 5x2 >= 41',
    '-2x1 + 7x2 >= 0',
    '5x1 + 11x2 <= 67',
    '13x0 + 5x1 + 11x2 <= 67',
    '14x0 + 5x2 <= 96',
    '14x0 + 11x1 <= 160',
    '14x0 + 11x1 + 5x2 <= 159'
]
}
```

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

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

    # Define variables
    x0 = model.addVar(name="x0", vtype=gurobi.GRB.INTEGER)  # hours worked by Mary
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Bobby
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # hours worked by Paul

    # Objective function
    model.setObjective(6.18*x0 + 2.63*x1 + 1.35*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(13*x0 + 5*x1 + 11*x2 >= 18)
    model.addConstr(14*x0 + 11*x1 >= 19)
    model.addConstr(14*x0 + 5*x2 >= 41)
    model.addConstr(-2*x1 + 7*x2 >= 0)
    model.addConstr(5*x1 + 11*x2 <= 67)
    model.addConstr(13*x0 + 5*x1 + 11*x2 <= 67)
    model.addConstr(14*x0 + 5*x2 <= 96)
    model.addConstr(14*x0 + 11*x1 <= 160)
    model.addConstr(14*x0 + 11*x1 + 5*x2 <= 159)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Mary: {x0.varValue}")
        print(f"Hours worked by Bobby: {x1.varValue}")
        print(f"Hours worked by Paul: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```