## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by George' and 'hours worked by Bill', which we can denote as $x_0$ and $x_1$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $1.69x_0 + 5.4x_1$.

## Step 3: List the constraints
The constraints given are:
- $14x_0 \leq 145$
- $1x_0 \leq 78$
- $4x_0 \leq 60$
- $6x_0 \leq 133$
- $2x_1 \leq 145$
- $12x_1 \leq 78$
- $13x_1 \leq 60$
- $13x_1 \leq 133$
- $14x_0 + 2x_1 \geq 63$
- $1x_0 + 12x_1 \geq 31$
- $4x_0 + 13x_1 \geq 24$
- $6x_0 + 13x_1 \geq 56$
- $-6x_0 + 9x_1 \geq 0$
- $14x_0 + 2x_1 \leq 77$
- $1x_0 + 12x_1 \leq 65$
- $4x_0 + 13x_1 \leq 37$
- $6x_0 + 13x_1 \leq 115$
- $x_0 \geq 0$ and $x_1 \geq 0$ and are integers.

## 4: Correct and simplify constraints based on the problem description
Upon reviewing, notice that some constraints directly relate to individual ratings and are not actually constraints on $x_0$ and $x_1$ but rather given conditions. The actual constraints from the problem description that limit $x_0$ and $x_1$ are:
- $x_0, x_1 \geq 0$ and are integers.

## 5: Formulate the problem in Gurobi
We will use Gurobi to solve this linear programming problem.

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

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

# Define the variables
x0 = m.addVar(name="hours_worked_by_George", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="hours_worked_by_Bill", vtype=gp.GRB.INTEGER)

# Objective function
m.setObjective(1.69 * x0 + 5.4 * x1, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(14 * x0 + 2 * x1 >= 63, name="productivity_rating")
m.addConstr(x0 + 12 * x1 >= 31, name="likelihood_to_quit_index")
m.addConstr(4 * x0 + 13 * x1 >= 24, name="organization_score")
m.addConstr(6 * x0 + 13 * x1 >= 56, name="computer_competence_rating")
m.addConstr(-6 * x0 + 9 * x1 >= 0, name="combined_productivity")
m.addConstr(14 * x0 + 2 * x1 <= 77, name="max_productivity_rating")
m.addConstr(x0 + 12 * x1 <= 65, name="max_likelihood_to_quit_index")
m.addConstr(4 * x0 + 13 * x1 <= 37, name="max_organization_score")
m.addConstr(6 * x0 + 13 * x1 <= 115, name="max_computer_competence_rating")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Hours worked by George: ", x0.varValue)
    print("Hours worked by Bill: ", x1.varValue)
    print("Objective: ", m.objVal)
else:
    print("The model is infeasible")
```

## 7: Symbolic representation
```json
{
    'sym_variables': [('x0', 'hours worked by George'), ('x1', 'hours worked by Bill')],
    'objective_function': '1.69*x0 + 5.4*x1',
    'constraints': [
        '14*x0 + 2*x1 >= 63',
        'x0 + 12*x1 >= 31',
        '4*x0 + 13*x1 >= 24',
        '6*x0 + 13*x1 >= 56',
        '-6*x0 + 9*x1 >= 0',
        '14*x0 + 2*x1 <= 77',
        'x0 + 12*x1 <= 65',
        '4*x0 + 13*x1 <= 37',
        '6*x0 + 13*x1 <= 115',
        'x0 >= 0',
        'x1 >= 0'
    ]
}
```