## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' corresponds to 'hours worked by John'
- 'x2' corresponds to 'hours worked by Bobby'
- 'x3' corresponds to 'hours worked by Peggy'
- 'x4' corresponds to 'hours worked by Bill'

## Step 2: Define the productivity ratings
The productivity ratings are given as:
- 'r0': {'description': 'productivity rating', 'upper_bound': 41, 'x0': 5, 'x1': 7, 'x2': 5, 'x3': 1}

However, it seems there is a typo in the problem description. The productivity ratings should likely be associated with each person as follows:
- John has a productivity rating of 5
- Bobby has a productivity rating of 7
- Peggy has a productivity rating of 5
- Bill has a productivity rating of 1

So, we can assign:
- Productivity rating of John = 5
- Productivity rating of Bobby = 7
- Productivity rating of Peggy = 5
- Productivity rating of Bill = 1

## 3: Formulate the objective function
The objective function to minimize is: 
\[ 5.01x_1 + 5.97x_2 + 8.09x_3 + 5.78x_4 \]

## 4: Define the constraints
The constraints are:
1. \( 7x_2 + 5x_3 \geq 5 \)
2. \( 5x_3 + 1x_4 \geq 7 \)
3. \( 5x_1 + 5x_3 \geq 10 \)
4. \( 5x_1 + 7x_2 + 5x_3 + 1x_4 \geq 10 \)
5. \( 1x_1 - 1x_2 \geq 0 \)
6. \( 9x_3 - 10x_4 \geq 0 \)
7. \( 5x_3 + 1x_4 \leq 41 \)
8. \( 7x_2 + 5x_3 \leq 29 \)
9. \( 5x_1 + 7x_2 + 5x_3 \leq 28 \)
10. \( 5x_1 + 7x_2 + 1x_4 \leq 16 \)
11. \( 7x_2 + 5x_3 + 1x_4 \leq 30 \)

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by John'),
        ('x2', 'hours worked by Bobby'),
        ('x3', 'hours worked by Peggy'),
        ('x4', 'hours worked by Bill')
    ],
    'objective_function': '5.01*x1 + 5.97*x2 + 8.09*x3 + 5.78*x4',
    'constraints': [
        '7*x2 + 5*x3 >= 5',
        '5*x3 + x4 >= 7',
        '5*x1 + 5*x3 >= 10',
        '5*x1 + 7*x2 + 5*x3 + x4 >= 10',
        'x1 - x2 >= 0',
        '9*x3 - 10*x4 >= 0',
        '5*x3 + x4 <= 41',
        '7*x2 + 5*x3 <= 29',
        '5*x1 + 7*x2 + 5*x3 <= 28',
        '5*x1 + 7*x2 + x4 <= 16',
        '7*x2 + 5*x3 + x4 <= 30'
    ]
}
```

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

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

    # Define the variables
    x1 = model.addVar(name="x1", lb=0)  # hours worked by John
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Bobby
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Peggy
    x4 = model.addVar(name="x4", lb=0)  # hours worked by Bill

    # Define the objective function
    model.setObjective(5.01 * x1 + 5.97 * x2 + 8.09 * x3 + 5.78 * x4, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(7 * x2 + 5 * x3 >= 5)
    model.addConstr(5 * x3 + x4 >= 7)
    model.addConstr(5 * x1 + 5 * x3 >= 10)
    model.addConstr(5 * x1 + 7 * x2 + 5 * x3 + x4 >= 10)
    model.addConstr(x1 - x2 >= 0)
    model.addConstr(9 * x3 - 10 * x4 >= 0)
    model.addConstr(5 * x3 + x4 <= 41)
    model.addConstr(7 * x2 + 5 * x3 <= 29)
    model.addConstr(5 * x1 + 7 * x2 + 5 * x3 <= 28)
    model.addConstr(5 * x1 + 7 * x2 + x4 <= 16)
    model.addConstr(7 * x2 + 5 * x3 + x4 <= 30)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
        print("x3: ", x3.varValue)
        print("x4: ", x4.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```