## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill', 'hours worked by Mary', 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 minimize is $8.53x_1 + 1.19x_2 + 3.23x_3$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
1. $11x_1 \leq 144$ (Bill's likelihood to quit index constraint, but this seems to be an upper bound on $x_1$ rather than a direct likelihood constraint. Assuming it's $11 \leq r_0$ for Bill, but $r_0$ is given as 11, so $x_1$ has an upper bound based on $r_0$),
2. $3x_2 \leq 144$ (Mary's likelihood to quit index constraint),
3. $8x_3 \leq 144$ (Peggy's likelihood to quit index constraint),
4. $3x_2 + 8x_3 \geq 30$,
5. $11x_1 + 8x_3 \geq 30$,
6. $11x_1 + 3x_2 + 8x_3 \geq 30$,
7. $6x_2 - 2x_3 \geq 0$,
8. $11x_1 + 3x_2 + 8x_3 \leq 122$.

However, upon closer inspection, it seems that $r_0$ represents a 'likelihood to quit index' and has specific values for each person ($x_0 = 11$, $x_1 = 3$, $x_2 = 8$) which might directly relate to $x_1, x_2, x_3$ as coefficients in constraints rather than separate variables. The correct interpretation seems to be:
- The likelihood to quit index for Bill is 11,
- For Mary is 3,
- For Peggy is 8.

So, constraints related to likelihood to quit index should directly use these values.

## 4: Correctly formulate the problem with given information
Given:
- $x_1$ is 'hours worked by Bill',
- $x_2$ is 'hours worked by Mary',
- $x_3$ is 'hours worked by Peggy',

And the resources/attributes:
- $r_0$ with $x_0 = 11$, $x_1 = 3$, $x_2 = 8$ and an upper bound of 144.

The problem can be formulated as:
Minimize $8.53x_1 + 1.19x_2 + 3.23x_3$

Subject to:
- $x_1 \leq 144 / 11$,
- $x_2 \leq 144 / 3$,
- $x_3 \leq 144 / 8$,
- $3x_2 + 8x_3 \geq 30$,
- $11x_1 + 8x_3 \geq 30$,
- $11x_1 + 3x_2 + 8x_3 \geq 30$,
- $6x_2 - 2x_3 \geq 0$,
- $11x_1 + 3x_2 + 8x_3 \leq 122$,
- $x_3$ is an integer.

## 5: Symbolic Representation
```json
{
'sym_variables': [('x1', 'hours worked by Bill'), ('x2', 'hours worked by Mary'), ('x3', 'hours worked by Peggy')],
'objective_function': '8.53*x1 + 1.19*x2 + 3.23*x3',
'constraints': [
    'x1 <= 144 / 11',
    'x2 <= 144 / 3',
    'x3 <= 144 / 8',
    '3*x2 + 8*x3 >= 30',
    '11*x1 + 8*x3 >= 30',
    '11*x1 + 3*x2 + 8*x3 >= 30',
    '6*x2 - 2*x3 >= 0',
    '11*x1 + 3*x2 + 8*x3 <= 122'
]
}
```

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

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

    # Define variables
    x1 = model.addVar(name="hours_worked_by_Bill", lb=0)
    x2 = model.addVar(name="hours_worked_by_Mary", lb=0)
    x3 = model.addVar(name="hours_worked_by_Peggy", lb=0, integrality=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(8.53 * x1 + 1.19 * x2 + 3.23 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x1 <= 144 / 11)
    model.addConstr(x2 <= 144 / 3)
    model.addConstr(x3 <= 144 / 8)
    model.addConstr(3 * x2 + 8 * x3 >= 30)
    model.addConstr(11 * x1 + 8 * x3 >= 30)
    model.addConstr(11 * x1 + 3 * x2 + 8 * x3 >= 30)
    model.addConstr(6 * x2 - 2 * x3 >= 0)
    model.addConstr(11 * x1 + 3 * x2 + 8 * x3 <= 122)

    # Optimize
    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 Mary: {x2.varValue}")
        print(f"Hours worked by Peggy: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The problem is infeasible.")
    else:
        print("No solution found.")

solve_optimization_problem()
```