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

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $6.54x_1 + 6.17x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $18x_1 \geq 18$ is not correct, $x_1$ has a likelihood to quit index of 18, this is not a constraint but a parameter, 
- $8x_1 \geq 8$ is not correct, $x_1$ has a work quality rating of 8, this is not a constraint but a parameter,
- $7x_2 \geq 7$ is not correct, $x_2$ has a likelihood to quit index of 7, this is not a constraint but a parameter,
- $16x_2 \geq 16$ is not correct, $x_2$ has a work quality rating of 16, this is not a constraint but a parameter,
- $18x_1 + 7x_2 \geq 57$,
- $8x_1 + 16x_2 \geq 67$,
- $4x_1 - x_2 \geq 0$,
- $18x_1 + 7x_2 \leq 123$,
- $8x_1 + 16x_2 \leq 70$,
- $x_1$ is an integer.

## 4: Convert the problem into a Gurobi model
We will use Gurobi to solve this optimization problem.

## 5: Write the Gurobi code
```python
import gurobi as gp

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_Hank", vtype=gp.GRB.INTEGER) # integer hours
x2 = m.addVar(name="hours_worked_by_Dale") # continuous hours

# Define the objective function
m.setObjective(6.54 * x1 + 6.17 * x2, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(18 * x1 + 7 * x2 >= 57, name="likelihood_to_quit_index_constraint")
m.addConstr(8 * x1 + 16 * x2 >= 67, name="work_quality_rating_constraint")
m.addConstr(4 * x1 - x2 >= 0, name="hours_worked_constraint")
m.addConstr(18 * x1 + 7 * x2 <= 123, name="likelihood_to_quit_index_upper_bound_constraint")
m.addConstr(8 * x1 + 16 * x2 <= 70, name="work_quality_rating_upper_bound_constraint")

# Solve the model
m.optimize()

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

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Dale')],
    'objective_function': '6.54*x1 + 6.17*x2',
    'constraints': [
        '18*x1 + 7*x2 >= 57',
        '8*x1 + 16*x2 >= 67',
        '4*x1 - x2 >= 0',
        '18*x1 + 7*x2 <= 123',
        '8*x1 + 16*x2 <= 70',
        'x1 is an integer'
    ]
}
```