## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. The variables are 'hours worked by Bill', 'hours worked by Laura', and 'hours worked by Mary', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Identify the objective function
The objective function to be maximized is $3x_0 + 4x_1 + 9x_2$.

## 3: List the constraints
The constraints given are:
- $2x_0 \leq 321$ (Bill's work quality rating constraint, but it's an attribute and not directly used as a constraint in this form)
- $15x_0 \leq 319$ (Bill's likelihood to quit index constraint)
- $13x_1 \leq 321$ (Laura's work quality rating constraint)
- $16x_1 \leq 319$ (Laura's likelihood to quit index constraint)
- $1x_2 \leq 321$ (Mary's work quality rating constraint)
- $11x_2 \leq 319$ (Mary's likelihood to quit index constraint)
- $13x_1 + 1x_2 \geq 94$ (Combined work quality rating from Laura and Mary)
- $2x_0 + 13x_1 + 1x_2 \geq 78$ (Combined work quality rating from Bill, Laura, and Mary)
- $15x_0 + 11x_2 \geq 83$ (Combined likelihood to quit index from Bill and Mary)
- $16x_1 + 11x_2 \geq 96$ (Combined likelihood to quit index from Laura and Mary)
- $15x_0 + 16x_1 + 11x_2 \geq 56$ (Combined likelihood to quit index from Bill, Laura, and Mary)
- $2x_0 + 1x_2 \leq 241$ (Combined work quality rating from Bill and Mary)
- $13x_1 + 1x_2 \leq 264$ (Combined work quality rating from Laura and Mary)
- $2x_0 + 13x_1 + 1x_2 \leq 264$ (Combined work quality rating from Bill, Laura, and Mary)
- $16x_1 + 11x_2 \leq 218$ (Combined likelihood to quit index from Laura and Mary)
- $15x_0 + 16x_1 \leq 126$ (Combined likelihood to quit index from Bill and Laura)
- $15x_0 + 11x_2 \leq 143$ (Combined likelihood to quit index from Bill and Mary)
- $15x_0 + 16x_1 + 11x_2 \leq 109$ (Combined likelihood to quit index from Bill, Laura, and Mary)

## 4: Create a symbolic representation of the problem
The symbolic representation is:
```json
{
    'sym_variables': [('x0', 'hours worked by Bill'), ('x1', 'hours worked by Laura'), ('x2', 'hours worked by Mary')],
    'objective_function': '3*x0 + 4*x1 + 9*x2',
    'constraints': [
        '13*x1 + x2 >= 94',
        '2*x0 + 13*x1 + x2 >= 78',
        '15*x0 + 11*x2 >= 83',
        '16*x1 + 11*x2 >= 96',
        '15*x0 + 16*x1 + 11*x2 >= 56',
        '2*x0 + x2 <= 241',
        '13*x1 + x2 <= 264',
        '2*x0 + 13*x1 + x2 <= 264',
        '16*x1 + 11*x2 <= 218',
        '15*x0 + 16*x1 <= 126',
        '15*x0 + 11*x2 <= 143',
        '15*x0 + 16*x1 + 11*x2 <= 109'
    ]
}
```

## 5: Implement the problem in Gurobi
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="hours_worked_by_Bill", lb=0)  # No upper bound given, assuming continuous
    x1 = model.addVar(name="hours_worked_by_Laura", lb=0)  # No upper bound given, assuming continuous
    x2 = model.addVar(name="hours_worked_by_Mary", lb=0)  # No upper bound given, assuming continuous

    # Define the objective function
    model.setObjective(3*x0 + 4*x1 + 9*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(13*x1 + x2 >= 94)
    model.addConstr(2*x0 + 13*x1 + x2 >= 78)
    model.addConstr(15*x0 + 11*x2 >= 83)
    model.addConstr(16*x1 + 11*x2 >= 96)
    model.addConstr(15*x0 + 16*x1 + 11*x2 >= 56)
    model.addConstr(2*x0 + x2 <= 241)
    model.addConstr(13*x1 + x2 <= 264)
    model.addConstr(2*x0 + 13*x1 + x2 <= 264)
    model.addConstr(16*x1 + 11*x2 <= 218)
    model.addConstr(15*x0 + 16*x1 <= 126)
    model.addConstr(15*x0 + 11*x2 <= 143)
    model.addConstr(15*x0 + 16*x1 + 11*x2 <= 109)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Bill: ", x0.varValue)
        print("Hours worked by Laura: ", x1.varValue)
        print("Hours worked by Mary: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```