## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Mary', 'hours worked by Bill', 'hours worked by John', and 'hours worked by Bobby', which can be represented symbolically as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $9.41x_0 + 6.59x_1 + 3.37x_2 + 1.98x_3$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $5x_1 + 2x_2 + 2x_3 \geq 4$
- $4x_0 + 5x_1 + 2x_2 \geq 4$
- $4x_0 + 5x_1 + 2x_2 \geq 4$ (same as the previous one, redundant)
- $4x_0 + x_1 + 3x_2 + 3x_3 \geq 8$
- $4x_1 + 4x_2 \geq 7$
- $4x_1 + 4x_2 + 3x_3 \geq 5$
- $2x_0 + 4x_2 + 3x_3 \geq 5$
- $4x_1 + 4x_2 + 3x_3 \geq 7$
- $2x_0 + 4x_2 + 3x_3 \geq 7$
- $4x_0 + 2x_3 \leq 15$
- $4x_0 + 5x_1 \leq 14$
- $4x_0 + 5x_1 + 2x_3 \leq 10$
- $4x_0 + 5x_1 + 2x_2 + 2x_3 \leq 10$
- $x_1 + 3x_2 \leq 29$
- $3x_2 + 3x_3 \leq 19$
- $4x_0 + x_1 \leq 17$
- $x_1 + 3x_2 + 3x_3 \leq 25$
- $4x_0 + 3x_2 + 3x_3 \leq 13$
- $4x_0 + x_1 + 3x_3 \leq 22$
- $4x_0 + 5x_1 + 2x_2 + 2x_3 \leq 22$
- $4x_1 + 4x_2 \leq 22$
- $2x_0 + 3x_3 \leq 7$
- $4x_2 + 3x_3 \leq 14$
- $2x_0 + 4x_2 \leq 23$
- $2x_0 + 4x_1 + 4x_2 + 3x_3 \leq 23$

## 4: Determine the variable types
- $x_0$ can be fractional
- $x_1$ must be an integer
- $x_2$ must be an integer
- $x_3$ must be an integer

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

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

# Define the variables
x0 = m.addVar(lb=0, name="hours_worked_by_Mary")  # Can be fractional
x1 = m.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="hours_worked_by_Bill")  # Must be an integer
x2 = m.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="hours_worked_by_John")  # Must be an integer
x3 = m.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="hours_worked_by_Bobby")  # Must be an integer

# Define the objective function
m.setObjective(9.41*x0 + 6.59*x1 + 3.37*x2 + 1.98*x3, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(5*x1 + 2*x2 + 2*x3 >= 4, name="productivity_rating_Bill_John_Bobby")
m.addConstr(4*x0 + 5*x1 + 2*x2 >= 4, name="productivity_rating_Mary_Bill_John")
m.addConstr(4*x0 + x1 + 3*x2 + 3*x3 >= 8, name="organization_score_Mary_Bill_John_Bobby")
m.addConstr(4*x1 + 4*x2 >= 7, name="dollar_cost_Bill_John")
m.addConstr(4*x1 + 4*x2 + 3*x3 >= 5, name="dollar_cost_Bill_John_Bobby_1")
m.addConstr(2*x0 + 4*x2 + 3*x3 >= 5, name="dollar_cost_Mary_John_Bobby")
m.addConstr(4*x1 + 4*x2 + 3*x3 >= 7, name="dollar_cost_Bill_John_Bobby_2")
m.addConstr(2*x0 + 4*x2 + 3*x3 >= 7, name="dollar_cost_Mary_John_Bobby_2")
m.addConstr(4*x0 + 2*x3 <= 15, name="productivity_rating_Mary_Bobby")
m.addConstr(4*x0 + 5*x1 <= 14, name="productivity_rating_Mary_Bill")
m.addConstr(4*x0 + 5*x1 + 2*x3 <= 10, name="productivity_rating_Mary_Bill_Bobby")
m.addConstr(4*x0 + 5*x1 + 2*x2 + 2*x3 <= 10, name="productivity_rating_all")
m.addConstr(x1 + 3*x2 <= 29, name="organization_score_Bill_John")
m.addConstr(3*x2 + 3*x3 <= 19, name="organization_score_John_Bobby")
m.addConstr(4*x0 + x1 <= 17, name="organization_score_Mary_Bill")
m.addConstr(x1 + 3*x2 + 3*x3 <= 25, name="organization_score_Bill_John_Bobby")
m.addConstr(4*x0 + 3*x2 + 3*x3 <= 13, name="organization_score_Mary_John_Bobby")
m.addConstr(4*x0 + x1 + 3*x3 <= 22, name="organization_score_Mary_Bill_Bobby")
m.addConstr(4*x0 + 5*x1 + 2*x2 + 2*x3 <= 22, name="organization_score_all")
m.addConstr(4*x1 + 4*x2 <= 22, name="dollar_cost_Bill_John")
m.addConstr(2*x0 + 3*x3 <= 7, name="dollar_cost_Mary_Bobby")
m.addConstr(4*x2 + 3*x3 <= 14, name="dollar_cost_John_Bobby")
m.addConstr(2*x0 + 4*x2 <= 23, name="dollar_cost_Mary_John")
m.addConstr(2*x0 + 4*x1 + 4*x2 + 3*x3 <= 23, name="dollar_cost_all")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Mary: ", x0.varValue)
    print("Hours worked by Bill: ", x1.varValue)
    print("Hours worked by John: ", x2.varValue)
    print("Hours worked by Bobby: ", x3.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Mary'), 
        ('x1', 'hours worked by Bill'), 
        ('x2', 'hours worked by John'), 
        ('x3', 'hours worked by Bobby')
    ], 
    'objective_function': '9.41*x0 + 6.59*x1 + 3.37*x2 + 1.98*x3', 
    'constraints': [
        '5*x1 + 2*x2 + 2*x3 >= 4', 
        '4*x0 + 5*x1 + 2*x2 >= 4', 
        '4*x0 + x1 + 3*x2 + 3*x3 >= 8', 
        '4*x1 + 4*x2 >= 7', 
        '4*x1 + 4*x2 + 3*x3 >= 5', 
        '2*x0 + 4*x2 + 3*x3 >= 5', 
        '4*x1 + 4*x2 + 3*x3 >= 7', 
        '2*x0 + 4*x2 + 3*x3 >= 7', 
        '4*x0 + 2*x3 <= 15', 
        '4*x0 + 5*x1 <= 14', 
        '4*x0 + 5*x1 + 2*x3 <= 10', 
        '4*x0 + 5*x1 + 2*x2 + 2*x3 <= 10', 
        'x1 + 3*x2 <= 29', 
        '3*x2 + 3*x3 <= 19', 
        '4*x0 + x1 <= 17', 
        'x1 + 3*x2 + 3*x3 <= 25', 
        '4*x0 + 3*x2 + 3*x3 <= 13', 
        '4*x0 + x1 + 3*x3 <= 22', 
        '4*x0 + 5*x1 + 2*x2 + 2*x3 <= 22', 
        '4*x1 + 4*x2 <= 22', 
        '2*x0 + 3*x3 <= 7', 
        '4*x2 + 3*x3 <= 14', 
        '2*x0 + 4*x2 <= 23', 
        '2*x0 + 4*x1 + 4*x2 + 3*x3 <= 23'
    ]
}
```