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

## Step 2: Define the symbolic representation of the resources/attributes
The resources/attributes are given as:
- $r_0$: likelihood to quit index with $x_0 = 14$ for Paul and $x_1 = 7$ for Dale, and an upper bound of 171.
- $r_1$: work quality rating with $x_0 = 21$ for Paul and $x_1 = 13$ for Dale, and an upper bound of 152.

## 3: Formulate the objective function
The objective function to maximize is $3x_1 + x_2$.

## 4: Formulate the constraints
The constraints are:
1. $14x_1 \geq 14$ (Paul's likelihood to quit index is 14, but this seems to be an equality constraint, so $14x_1 = 14$ or simply $x_1 \geq 1$ and $x_1 \leq 1$ for it to be exactly 14, however, given $x_1$ represents hours and not the index directly, we interpret it as $14 \leq 14x_1$ which simplifies to $1 \leq x_1$),
2. $21x_1 \geq 21$ (Paul's work quality rating is 21, similarly, $x_1 \geq 1$),
3. $7x_2 \geq 7$ (Dale's likelihood to quit index is 7, $x_2 \geq 1$),
4. $13x_2 \geq 13$ (Dale's work quality rating is 13, $x_2 \geq 1$),
5. $14x_1 + 7x_2 \geq 77$,
6. $21x_1 + 13x_2 \geq 39$,
7. $9x_1 - 4x_2 \geq 0$,
8. $14x_1 + 7x_2 \leq 130$,
9. $21x_1 + 13x_2 \leq 116$.

## 5: Correct interpretation of constraints
Given that $x_1$ and $x_2$ represent hours worked, the constraints from the problem description should be directly translated into mathematical expressions without misinterpretation:
- The direct constraints on $x_1$ and $x_2$ from the problem statement seem to be misinterpreted. The correct interpretation should directly use the given numbers as they relate to $x_1$ and $x_2$:
  - $14 \leq 14x_1$ implies $1 \leq x_1$,
  - $21 \leq 21x_1$ implies $1 \leq x_1$,
  - $7 \leq 7x_2$ implies $1 \leq x_2$,
  - $13 \leq 13x_2$ implies $1 \leq x_2$.

## 6: Symbolic Representation
The symbolic representation is:
```json
{
'sym_variables': [('x1', 'hours worked by Paul'), ('x2', 'hours worked by Dale')],
'objective_function': '3*x1 + x2',
'constraints': [
    'x1 >= 1',
    'x1 >= 1',
    'x2 >= 1',
    'x2 >= 1',
    '14*x1 + 7*x2 >= 77',
    '21*x1 + 13*x2 >= 39',
    '9*x1 - 4*x2 >= 0',
    '14*x1 + 7*x2 <= 130',
    '21*x1 + 13*x2 <= 116'
]
}
```

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

def solve_optimization_problem():
    try:
        # Create a new model
        model = gurobi.Model()

        # Define variables
        x1 = model.addVar(lb=0, name="hours_worked_by_Paul")
        x2 = model.addVar(lb=0, name="hours_worked_by_Dale")

        # Objective function
        model.setObjective(3*x1 + x2, gurobi.GRB.MAXIMIZE)

        # Constraints
        model.addConstr(x1 >= 1, name="Paul_hours_min")
        model.addConstr(x2 >= 1, name="Dale_hours_min")
        model.addConstr(14*x1 + 7*x2 >= 77, name="quit_index_min")
        model.addConstr(21*x1 + 13*x2 >= 39, name="work_quality_min")
        model.addConstr(9*x1 - 4*x2 >= 0, name="work_balance")
        model.addConstr(14*x1 + 7*x2 <= 130, name="quit_index_max")
        model.addConstr(21*x1 + 13*x2 <= 116, name="work_quality_max")

        # Optimize
        model.optimize()

        # Print solution
        if model.status == gurobi.GRB.OPTIMAL:
            print("Objective: ", model.objVal)
            print("Hours worked by Paul: ", x1.varValue)
            print("Hours worked by Dale: ", x2.varValue)
        else:
            print("No optimal solution found")

    except gurobi.GUROBI_ERROR as e:
        print("Gurobi Error: ", e)

solve_optimization_problem()
```