## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Bill', 'hours worked by John', 'hours worked by Mary', 'hours worked by Peggy']. Let's denote them 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 $3.85x_0 + 5.18x_1 + 2.03x_2 + 3.6x_3$.

## Step 3: List the constraints in symbolic notation
1. $3x_0 \geq 33$ (total combined dollar cost per hour from hours worked by Bill and Mary)
2. $3x_0 + 22x_3 \geq 82$ (total combined dollar cost per hour from hours worked by Bill, Mary, and Peggy)
3. $3x_0 + 17x_1 + 8x_2 \geq 82$ (total combined dollar cost per hour from hours worked by Bill, John, and Mary)
4. $3x_0 + 22x_3 \geq 53$ (total combined dollar cost per hour from hours worked by Bill, Mary, and Peggy)
5. $3x_0 + 17x_1 + 8x_2 \geq 53$ (total combined dollar cost per hour from hours worked by Bill, John, and Mary)
6. $22x_0 + 13x_3 \leq 186$ (total combined paperwork competence rating from hours worked by Bill and Peggy)
7. $15x_2 + 13x_3 \leq 217$ (total combined paperwork competence rating from hours worked by Mary and Peggy)
8. $22x_0 + 15x_2 \leq 262$ (total combined paperwork competence rating from hours worked by Bill and Mary)
9. $4x_1 + 13x_3 \leq 162$ (total combined paperwork competence rating from hours worked by John and Peggy)
10. $4x_1 + 15x_2 \leq 255$ (total combined paperwork competence rating from hours worked by John and Mary)
11. $4x_1 + 15x_2 + 13x_3 \leq 247$ (total combined paperwork competence rating from hours worked by John, Mary, and Peggy)
12. $22x_0 + 4x_1 + 15x_2 \leq 200$ (total combined paperwork competence rating from hours worked by Bill, John, and Mary)
13. $22x_0 + 15x_2 + 13x_3 \leq 192$ (total combined paperwork competence rating from hours worked by Bill, Mary, and Peggy)
14. $22x_0 + 4x_1 + 15x_2 + 13x_3 \leq 192$ (total combined paperwork competence rating from hours worked by all)
15. $8x_2 + 22x_3 \leq 254$ (total combined dollar cost per hour from hours worked by Mary and Peggy)
16. $17x_1 + 8x_2 \leq 209$ (total combined dollar cost per hour from hours worked by John and Mary)
17. $17x_1 + 22x_3 \leq 242$ (total combined dollar cost per hour from hours worked by John and Peggy)
18. $3x_0 + 22x_3 \leq 192$ (total combined dollar cost per hour from hours worked by Bill and Peggy)
19. $3x_0 + 17x_1 + 22x_3 \leq 329$ (total combined dollar cost per hour from hours worked by Bill, John, and Peggy)
20. $17x_1 + 8x_2 + 22x_3 \leq 291$ (total combined dollar cost per hour from hours worked by John, Mary, and Peggy)
21. $3x_0 + 17x_1 + 8x_2 + 22x_3 \leq 291$ (total combined dollar cost per hour from hours worked by all)

## Step 4: Define the variable types
- $x_0$ (hours worked by Bill) and $x_1$ (hours worked by John) and $x_3$ (hours worked by Peggy) are integers.
- $x_2$ (hours worked by Mary) can be a fraction.

## 5: Write down the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Bill'), 
        ('x1', 'hours worked by John'), 
        ('x2', 'hours worked by Mary'), 
        ('x3', 'hours worked by Peggy')
    ], 
    'objective_function': '3.85x0 + 5.18x1 + 2.03x2 + 3.6x3', 
    'constraints': [
        '3x0 + 8x2 >= 33',
        '3x0 + 22x3 >= 82',
        '3x0 + 17x1 + 8x2 >= 82',
        '3x0 + 22x3 >= 53',
        '3x0 + 17x1 + 8x2 >= 53',
        '22x0 + 13x3 <= 186',
        '15x2 + 13x3 <= 217',
        '22x0 + 15x2 <= 262',
        '4x1 + 13x3 <= 162',
        '4x1 + 15x2 <= 255',
        '4x1 + 15x2 + 13x3 <= 247',
        '22x0 + 4x1 + 15x2 <= 200',
        '22x0 + 15x2 + 13x3 <= 192',
        '22x0 + 4x1 + 15x2 + 13x3 <= 192',
        '8x2 + 22x3 <= 254',
        '17x1 + 8x2 <= 209',
        '17x1 + 22x3 <= 242',
        '3x0 + 22x3 <= 192',
        '3x0 + 17x1 + 22x3 <= 329',
        '17x1 + 8x2 + 22x3 <= 291',
        '3x0 + 17x1 + 8x2 + 22x3 <= 291'
    ]
}
```

## 6: Implement the problem using Gurobi
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="x0", vtype=gurobi.GRB.INTEGER)  # hours worked by Bill
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by John
    x2 = model.addVar(name="x2")  # hours worked by Mary
    x3 = model.addVar(name="x3", vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy

    # Objective function
    model.setObjective(3.85 * x0 + 5.18 * x1 + 2.03 * x2 + 3.6 * x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(3 * x0 + 8 * x2 >= 33)
    model.addConstr(3 * x0 + 22 * x3 >= 82)
    model.addConstr(3 * x0 + 17 * x1 + 8 * x2 >= 82)
    model.addConstr(3 * x0 + 22 * x3 >= 53)
    model.addConstr(3 * x0 + 17 * x1 + 8 * x2 >= 53)
    model.addConstr(22 * x0 + 13 * x3 <= 186)
    model.addConstr(15 * x2 + 13 * x3 <= 217)
    model.addConstr(22 * x0 + 15 * x2 <= 262)
    model.addConstr(4 * x1 + 13 * x3 <= 162)
    model.addConstr(4 * x1 + 15 * x2 <= 255)
    model.addConstr(4 * x1 + 15 * x2 + 13 * x3 <= 247)
    model.addConstr(22 * x0 + 4 * x1 + 15 * x2 <= 200)
    model.addConstr(22 * x0 + 15 * x2 + 13 * x3 <= 192)
    model.addConstr(22 * x0 + 4 * x1 + 15 * x2 + 13 * x3 <= 192)
    model.addConstr(8 * x2 + 22 * x3 <= 254)
    model.addConstr(17 * x1 + 8 * x2 <= 209)
    model.addConstr(17 * x1 + 22 * x3 <= 242)
    model.addConstr(3 * x0 + 22 * x3 <= 192)
    model.addConstr(3 * x0 + 17 * x1 + 22 * x3 <= 329)
    model.addConstr(17 * x1 + 8 * x2 + 22 * x3 <= 291)
    model.addConstr(3 * x0 + 17 * x1 + 8 * x2 + 22 * x3 <= 291)

    # Solve the model
    model.optimize()

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

solve_optimization_problem()
```