## Step 1: Define the symbolic variables and their corresponding natural language objects.
The symbolic variables and their corresponding natural language objects are:
- $x_0$ : hours worked by Peggy
- $x_1$ : hours worked by Bobby
- $x_2$ : hours worked by Bill
- $x_3$ : hours worked by Jean

## Step 2: Express the objective function using symbolic variables.
The objective function to maximize is: $2x_0 + 4x_1 + 8x_2 + 2x_3$

## Step 3: List the constraints using symbolic variables.
The constraints are:
- $8x_0 \leq 377$ (Peggy's work quality rating constraint, but it's an equality for her rating, so $8x_0 = 377$ is not directly applicable; instead, we consider $x_0 \geq 0$ and the rating as a given)
- $19x_0 \leq 107$ (Peggy's likelihood to quit index constraint)
- $5x_1 \leq 377$ (Bobby's work quality rating constraint)
- $9x_1 \leq 107$ (Bobby's likelihood to quit index constraint)
- $x_2 \leq 377$ (Bill's work quality rating constraint)
- $11x_2 \leq 107$ (Bill's likelihood to quit index constraint)
- $4x_3 \leq 377$ (Jean's work quality rating constraint)
- $10x_3 \leq 107$ (Jean's likelihood to quit index constraint)
- $19x_0 + 9x_1 + 10x_3 \geq 15$
- $19x_0 + 11x_2 + 9x_1 \geq 15$
- $19x_0 + 11x_2 + 10x_3 \geq 15$
- $19x_0 + 9x_1 + 10x_3 \geq 21$
- $19x_0 + 9x_1 + 11x_2 \geq 21$
- $19x_0 + 11x_2 + 10x_3 \geq 21$
- $19x_0 + 9x_1 + 10x_3 \geq 18$
- $19x_0 + 9x_1 + 11x_2 \geq 18$
- $19x_0 + 11x_2 + 10x_3 \geq 18$
- $8x_0 + 5x_1 \leq 349$
- $5x_1 + x_2 \leq 292$
- $8x_0 + x_2 \leq 320$
- $5x_1 + 4x_3 \leq 277$
- $x_2 + 4x_3 \leq 274$
- $8x_0 + 4x_3 \leq 147$
- $8x_0 + 5x_1 + x_2 + 4x_3 \leq 147$
- $19x_0 + 10x_3 \leq 51$
- $9x_1 + 10x_3 \leq 63$
- $9x_1 + 11x_2 \leq 60$
- $19x_0 + 11x_2 \leq 32$
- $19x_0 + 9x_1 + 11x_2 \leq 63$
- $19x_0 + 9x_1 + 11x_2 + 10x_3 \leq 63$

## 4: Convert the problem into Gurobi code.

```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Peggy
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Bobby
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Bill
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Jean

    # Objective function: maximize 2*x0 + 4*x1 + 8*x2 + 2*x3
    model.setObjective(2*x0 + 4*x1 + 8*x2 + 2*x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(8*x0 <= 377)
    model.addConstr(19*x0 <= 107)
    model.addConstr(5*x1 <= 377)
    model.addConstr(9*x1 <= 107)
    model.addConstr(x2 <= 377)
    model.addConstr(11*x2 <= 107)
    model.addConstr(4*x3 <= 377)
    model.addConstr(10*x3 <= 107)

    model.addConstr(19*x0 + 9*x1 + 10*x3 >= 15)
    model.addConstr(19*x0 + 11*x2 + 9*x1 >= 15)
    model.addConstr(19*x0 + 11*x2 + 10*x3 >= 15)
    model.addConstr(19*x0 + 9*x1 + 10*x3 >= 21)
    model.addConstr(19*x0 + 9*x1 + 11*x2 >= 21)
    model.addConstr(19*x0 + 11*x2 + 10*x3 >= 21)
    model.addConstr(19*x0 + 9*x1 + 10*x3 >= 18)
    model.addConstr(19*x0 + 9*x1 + 11*x2 >= 18)
    model.addConstr(19*x0 + 11*x2 + 10*x3 >= 18)

    model.addConstr(8*x0 + 5*x1 <= 349)
    model.addConstr(5*x1 + x2 <= 292)
    model.addConstr(8*x0 + x2 <= 320)
    model.addConstr(5*x1 + 4*x3 <= 277)
    model.addConstr(x2 + 4*x3 <= 274)
    model.addConstr(8*x0 + 4*x3 <= 147)
    model.addConstr(8*x0 + 5*x1 + x2 + 4*x3 <= 147)

    model.addConstr(19*x0 + 10*x3 <= 51)
    model.addConstr(9*x1 + 10*x3 <= 63)
    model.addConstr(9*x1 + 11*x2 <= 60)
    model.addConstr(19*x0 + 11*x2 <= 32)
    model.addConstr(19*x0 + 9*x1 + 11*x2 <= 63)
    model.addConstr(19*x0 + 9*x1 + 11*x2 + 10*x3 <= 63)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("x0: ", x0.varValue)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
        print("x3: ", x3.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```

## 5: Provide the symbolic representation of the problem.

```json
{
    'sym_variables': [
        ('x0', 'hours worked by Peggy'),
        ('x1', 'hours worked by Bobby'),
        ('x2', 'hours worked by Bill'),
        ('x3', 'hours worked by Jean')
    ],
    'objective_function': '2*x0 + 4*x1 + 8*x2 + 2*x3',
    'constraints': [
        '8*x0 <= 377',
        '19*x0 <= 107',
        '5*x1 <= 377',
        '9*x1 <= 107',
        'x2 <= 377',
        '11*x2 <= 107',
        '4*x3 <= 377',
        '10*x3 <= 107',
        '19*x0 + 9*x1 + 10*x3 >= 15',
        '19*x0 + 11*x2 + 9*x1 >= 15',
        '19*x0 + 11*x2 + 10*x3 >= 15',
        '19*x0 + 9*x1 + 10*x3 >= 21',
        '19*x0 + 9*x1 + 11*x2 >= 21',
        '19*x0 + 11*x2 + 10*x3 >= 21',
        '19*x0 + 9*x1 + 10*x3 >= 18',
        '19*x0 + 9*x1 + 11*x2 >= 18',
        '19*x0 + 11*x2 + 10*x3 >= 18',
        '8*x0 + 5*x1 <= 349',
        '5*x1 + x2 <= 292',
        '8*x0 + x2 <= 320',
        '5*x1 + 4*x3 <= 277',
        'x2 + 4*x3 <= 274',
        '8*x0 + 4*x3 <= 147',
        '8*x0 + 5*x1 + x2 + 4*x3 <= 147',
        '19*x0 + 10*x3 <= 51',
        '9*x1 + 10*x3 <= 63',
        '9*x1 + 11*x2 <= 60',
        '19*x0 + 11*x2 <= 32',
        '19*x0 + 9*x1 + 11*x2 <= 63',
        '19*x0 + 9*x1 + 11*x2 + 10*x3 <= 63'
    ]
}
```