## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Mary', 'hours worked by Dale', and 'hours worked by Laura', which can be represented symbolically as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $8.72x_1 + 5.55x_2 + 1.19x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
- $9.26x_1$ (no direct constraint, just a given index)
- $4.77x_2$ (no direct constraint, just a given index)
- $11.23x_3$ (no direct constraint, just a given index)
- $4.77x_2 + 11.23x_3 \geq 34$
- $9.26x_1 + 11.23x_3 \geq 29$
- $9.26x_1 + 4.77x_2 + 11.23x_3 \geq 29$
- $-4x_2 + 8x_3 \geq 0$
- $9.26x_1 + 4.77x_2 \leq 43$

## 4: Consider the bounds of the variables
Given that $x_1$, $x_2$, and $x_3$ can have fractional numbers of hours and there are no explicit upper bounds given for $x_1$, $x_2$, or $x_3$ except that the likelihood to quit index for each has an upper bound of 108, we consider:
- $9.26x_1 \leq 108$
- $4.77x_2 \leq 108$
- $11.23x_3 \leq 108$

## 5: Express the problem in a format suitable for Gurobi
The problem can be expressed as:
Minimize $8.72x_1 + 5.55x_2 + 1.19x_3$
Subject to:
- $4.77x_2 + 11.23x_3 \geq 34$
- $9.26x_1 + 11.23x_3 \geq 29$
- $9.26x_1 + 4.77x_2 + 11.23x_3 \geq 29$
- $-4x_2 + 8x_3 \geq 0$
- $9.26x_1 + 4.77x_2 \leq 43$
- $x_1 \leq 108 / 9.26$
- $x_2 \leq 108 / 4.77$
- $x_3 \leq 108 / 11.23$
- $x_1, x_2, x_3 \geq 0$

## 6: Calculate specific bounds for $x_1$, $x_2$, and $x_3$
- $x_1 \leq 11.66$
- $x_2 \leq 22.64$
- $x_3 \leq 9.62$

## 7: Provide the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'hours worked by Mary'), ('x2', 'hours worked by Dale'), ('x3', 'hours worked by Laura')],
'objective_function': '8.72*x1 + 5.55*x2 + 1.19*x3',
'constraints': [
    '4.77*x2 + 11.23*x3 >= 34',
    '9.26*x1 + 11.23*x3 >= 29',
    '9.26*x1 + 4.77*x2 + 11.23*x3 >= 29',
    '-4*x2 + 8*x3 >= 0',
    '9.26*x1 + 4.77*x2 <= 43',
    'x1 <= 11.66',
    'x2 <= 22.64',
    'x3 <= 9.62',
    'x1, x2, x3 >= 0'
]
}
```

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

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(name="hours_worked_by_Mary", lb=0)
    x2 = model.addVar(name="hours_worked_by_Dale", lb=0)
    x3 = model.addVar(name="hours_worked_by_Laura", lb=0)

    # Objective function
    model.setObjective(8.72 * x1 + 5.55 * x2 + 1.19 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(4.77 * x2 + 11.23 * x3 >= 34)
    model.addConstr(9.26 * x1 + 11.23 * x3 >= 29)
    model.addConstr(9.26 * x1 + 4.77 * x2 + 11.23 * x3 >= 29)
    model.addConstr(-4 * x2 + 8 * x3 >= 0)
    model.addConstr(9.26 * x1 + 4.77 * x2 <= 43)
    model.addConstr(x1 <= 11.66)
    model.addConstr(x2 <= 22.64)
    model.addConstr(x3 <= 9.62)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Mary: {x1.varValue}")
        print(f"Hours worked by Dale: {x2.varValue}")
        print(f"Hours worked by Laura: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```