## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Jean', 'hours worked by Dale', and 'hours worked by Paul', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the problem description into a symbolic representation
The objective function to minimize is $4.5x_0 + 9.58x_1 + 5.64x_2$.

## 3: List the constraints in symbolic notation
The constraints are:
- $6x_1 + 10x_2 \geq 32$
- $16x_0 + 6x_1 \geq 54$
- $16x_0 + 6x_1 + 10x_2 \geq 54$
- $23x_1 + 20x_2 \geq 41$
- $12x_0 + 23x_1 + 20x_2 \geq 41$
- $6x_1 - 7x_2 \geq 0$
- $6x_0 - 5x_2 \geq 0$
- $6x_1 + 10x_2 \leq 169$
- $16x_0 + 10x_2 \leq 148$
- $12x_0 + 23x_1 \leq 116$

## 4: Define the symbolic representation of the problem
```json
{
'sym_variables': [
    ('x0', 'hours worked by Jean'), 
    ('x1', 'hours worked by Dale'), 
    ('x2', 'hours worked by Paul')
],
'objective_function': '4.5*x0 + 9.58*x1 + 5.64*x2',
'constraints': [
    '6*x1 + 10*x2 >= 32',
    '16*x0 + 6*x1 >= 54',
    '16*x0 + 6*x1 + 10*x2 >= 54',
    '23*x1 + 20*x2 >= 41',
    '12*x0 + 23*x1 + 20*x2 >= 41',
    '6*x1 - 7*x2 >= 0',
    '6*x0 - 5*x2 >= 0',
    '6*x1 + 10*x2 <= 169',
    '16*x0 + 10*x2 <= 148',
    '12*x0 + 23*x1 <= 116'
]
}
```

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

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Jean
x1 = m.addVar(name="x1", lb=0)  # hours worked by Dale
x2 = m.addVar(name="x2", lb=0)  # hours worked by Paul

# Define the objective function
m.setObjective(4.5 * x0 + 9.58 * x1 + 5.64 * x2, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(6 * x1 + 10 * x2 >= 32)
m.addConstr(16 * x0 + 6 * x1 >= 54)
m.addConstr(16 * x0 + 6 * x1 + 10 * x2 >= 54)
m.addConstr(23 * x1 + 20 * x2 >= 41)
m.addConstr(12 * x0 + 23 * x1 + 20 * x2 >= 41)
m.addConstr(6 * x1 - 7 * x2 >= 0)
m.addConstr(6 * x0 - 5 * x2 >= 0)
m.addConstr(6 * x1 + 10 * x2 <= 169)
m.addConstr(16 * x0 + 10 * x2 <= 148)
m.addConstr(12 * x0 + 23 * x1 <= 116)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Jean: ", x0.varValue)
    print("Hours worked by Dale: ", x1.varValue)
    print("Hours worked by Paul: ", x2.varValue)
else:
    print("The model is infeasible")
```