## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill', 'hours worked by Hank', and 'hours worked by Peggy'. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $1.88x_1 + 3.92x_2 + 4.99x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $3x_1 \leq 311$ is not directly given but we have $x_0 = 3$, $x_1 = 24$, $x_2 = 14$ as work quality ratings for Bill, Hank, and Peggy respectively.
- The total combined work quality rating from hours worked by Bill and hours worked by Hank has to be as much or less than 246: $3x_1 + 24x_2 \leq 246$.
- The total combined work quality rating from hours worked by Hank and hours worked by Peggy should be no more than 146: $24x_2 + 14x_3 \leq 146$.
- The total combined work quality rating from hours worked by Bill, hours worked by Hank and hours worked by Peggy has to be 146 at maximum: $3x_1 + 24x_2 + 14x_3 \leq 146$.
- $x_1$ must be an integer.
- $x_2$ must be an integer.
- $x_3$ can be any real number.

## 4: Correct the understanding of the problem constraints
Upon reevaluation, it seems there was an initial misunderstanding in directly translating the given constraints. The work quality ratings given as $x_0 = 3$, $x_1 = 24$, $x_2 = 14$ seem to actually represent the ratings per hour for Bill, Hank, and Peggy respectively, not the total ratings. Thus, the correct constraints based on the problem statement should directly utilize these ratings in relation to the hours worked.

## 5: Formulate the problem in a standard optimization format
Let's denote:
- $x_1$ as the hours worked by Bill,
- $x_2$ as the hours worked by Hank,
- $x_3$ as the hours worked by Peggy.

The objective function is: $1.88x_1 + 3.92x_2 + 4.99x_3$.

The constraints are:
1. $3x_1 \leq 311$,
2. $24x_2 \leq 311$,
3. $14x_3 \leq 311$,
4. $3x_1 + 24x_2 \leq 246$,
5. $24x_2 + 14x_3 \leq 146$,
6. $3x_1 + 24x_2 + 14x_3 \leq 146$,
7. $x_1 \in \mathbb{Z}$,
8. $x_2 \in \mathbb{Z}$,
9. $x_3 \in \mathbb{R}$.

## 6: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'hours worked by Bill'), ('x2', 'hours worked by Hank'), ('x3', 'hours worked by Peggy')],
    'objective_function': '1.88*x1 + 3.92*x2 + 4.99*x3',
    'constraints': [
        '3*x1 <= 311',
        '24*x2 <= 311',
        '14*x3 <= 311',
        '3*x1 + 24*x2 <= 246',
        '24*x2 + 14*x3 <= 146',
        '3*x1 + 24*x2 + 14*x3 <= 146',
        'x1 ∈ ℤ',
        'x2 ∈ ℤ'
    ]
}
```

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

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

    # Define variables
    x1 = model.addVar(name="hours_worked_by_Bill", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="hours_worked_by_Hank", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="hours_worked_by_Peggy")

    # Objective function
    model.setObjective(1.88*x1 + 3.92*x2 + 4.99*x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(3*x1 <= 311, name="Bill_work_quality")
    model.addConstr(24*x2 <= 311, name="Hank_work_quality")
    model.addConstr(14*x3 <= 311, name="Peggy_work_quality")
    model.addConstr(3*x1 + 24*x2 <= 246, name="Bill_Hank_work_quality")
    model.addConstr(24*x2 + 14*x3 <= 146, name="Hank_Peggy_work_quality")
    model.addConstr(3*x1 + 24*x2 + 14*x3 <= 146, name="Total_work_quality")

    # Solve the model
    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Bill: {x1.varValue}")
        print(f"Hours worked by Hank: {x2.varValue}")
        print(f"Hours worked by Peggy: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```