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

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $4x_0 + 9x_1 + 9x_2$.

## 3: List the constraints in symbolic notation
The constraints given are:
- $3x_0 \leq 260$
- $5x_0 \leq 271$
- $18x_0 \leq 103$
- $26x_0 \leq 288$
- $19x_0 \leq 248$
- $12x_1 \leq 260$
- $15x_1 \leq 271$
- $6x_1 \leq 103$
- $6x_1 \leq 288$
- $24x_1 \leq 248$
- $2x_2 \leq 260$
- $24x_2 \leq 271$
- $18x_2 \leq 103$
- $15x_2 \leq 288$
- $9x_2 \leq 248$
- $12x_1 + 2x_2 \geq 39$
- $3x_0 + 2x_2 \geq 73$
- $3x_0 + 12x_1 + 2x_2 \geq 64$
- $5x_0 + 15x_1 \geq 33$
- $5x_0 + 15x_1 + 24x_2 \geq 33$
- $18x_0 + 18x_2 \geq 16$
- $6x_1 + 18x_2 \geq 32$
- $18x_0 + 6x_1 + 18x_2 \geq 32$
- $6x_1 + 15x_2 \geq 74$
- $26x_0 + 6x_1 + 15x_2 \geq 74$
- $24x_1 + 9x_2 \geq 68$
- $19x_0 + 24x_1 \geq 27$
- $19x_0 + 24x_1 + 9x_2 \geq 27$
- $6x_1 - 4x_2 \geq 0$
- $3x_0 - x_2 \geq 0$
- $-2x_0 + 3x_1 \geq 0$
- $3x_0 + 12x_1 \leq 143$
- $19x_0 + 24x_1 \leq 137$
- $19x_0 + 9x_2 \leq 114$

## 4: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [('x0', 'hours worked by Bill'), ('x1', 'hours worked by Peggy'), ('x2', 'hours worked by Bobby')],
    'objective_function': '4*x0 + 9*x1 + 9*x2',
    'constraints': [
        '3*x0 <= 260',
        '5*x0 <= 271',
        '18*x0 <= 103',
        '26*x0 <= 288',
        '19*x0 <= 248',
        '12*x1 <= 260',
        '15*x1 <= 271',
        '6*x1 <= 103',
        '6*x1 <= 288',
        '24*x1 <= 248',
        '2*x2 <= 260',
        '24*x2 <= 271',
        '18*x2 <= 103',
        '15*x2 <= 288',
        '9*x2 <= 248',
        '12*x1 + 2*x2 >= 39',
        '3*x0 + 2*x2 >= 73',
        '3*x0 + 12*x1 + 2*x2 >= 64',
        '5*x0 + 15*x1 >= 33',
        '5*x0 + 15*x1 + 24*x2 >= 33',
        '18*x0 + 18*x2 >= 16',
        '6*x1 + 18*x2 >= 32',
        '18*x0 + 6*x1 + 18*x2 >= 32',
        '6*x1 + 15*x2 >= 74',
        '26*x0 + 6*x1 + 15*x2 >= 74',
        '24*x1 + 9*x2 >= 68',
        '19*x0 + 24*x1 >= 27',
        '19*x0 + 24*x1 + 9*x2 >= 27',
        '6*x1 - 4*x2 >= 0',
        '3*x0 - x2 >= 0',
        '-2*x0 + 3*x1 >= 0',
        '3*x0 + 12*x1 <= 143',
        '19*x0 + 24*x1 <= 137',
        '19*x0 + 9*x2 <= 114'
    ]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="hours_worked_by_Bill", lb=0)
x1 = m.addVar(name="hours_worked_by_Peggy", lb=0)
x2 = m.addVar(name="hours_worked_by_Bobby", lb=0)

# Define the objective function
m.setObjective(4*x0 + 9*x1 + 9*x2, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(3*x0 <= 260)
m.addConstr(5*x0 <= 271)
m.addConstr(18*x0 <= 103)
m.addConstr(26*x0 <= 288)
m.addConstr(19*x0 <= 248)

m.addConstr(12*x1 <= 260)
m.addConstr(15*x1 <= 271)
m.addConstr(6*x1 <= 103)
m.addConstr(6*x1 <= 288)
m.addConstr(24*x1 <= 248)

m.addConstr(2*x2 <= 260)
m.addConstr(24*x2 <= 271)
m.addConstr(18*x2 <= 103)
m.addConstr(15*x2 <= 288)
m.addConstr(9*x2 <= 248)

m.addConstr(12*x1 + 2*x2 >= 39)
m.addConstr(3*x0 + 2*x2 >= 73)
m.addConstr(3*x0 + 12*x1 + 2*x2 >= 64)
m.addConstr(5*x0 + 15*x1 >= 33)
m.addConstr(5*x0 + 15*x1 + 24*x2 >= 33)
m.addConstr(18*x0 + 18*x2 >= 16)
m.addConstr(6*x1 + 18*x2 >= 32)
m.addConstr(18*x0 + 6*x1 + 18*x2 >= 32)
m.addConstr(6*x1 + 15*x2 >= 74)
m.addConstr(26*x0 + 6*x1 + 15*x2 >= 74)
m.addConstr(24*x1 + 9*x2 >= 68)
m.addConstr(19*x0 + 24*x1 >= 27)
m.addConstr(19*x0 + 24*x1 + 9*x2 >= 27)
m.addConstr(6*x1 - 4*x2 >= 0)
m.addConstr(3*x0 - x2 >= 0)
m.addConstr(-2*x0 + 3*x1 >= 0)
m.addConstr(3*x0 + 12*x1 <= 143)
m.addConstr(19*x0 + 24*x1 <= 137)
m.addConstr(19*x0 + 9*x2 <= 114)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Bill: {x0.varValue}")
    print(f"Hours worked by Peggy: {x1.varValue}")
    print(f"Hours worked by Bobby: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```