## Step 1: Define the optimization problem
The problem is to maximize the objective function: $3.85B + 5.18J + 2.03M + 3.6P$, where $B$, $J$, $M$, and $P$ represent the hours worked by Bill, John, Mary, and Peggy, respectively.

## 2: List all the constraints
The constraints are:
1. $B \geq 0$ and integer
2. $J \geq 0$ and integer
3. $M \geq 0$
4. $P \geq 0$ and integer
5. $3B + 8M \geq 33$
6. $3B + 8M + 22P \geq 82$
7. $3B + 17J + 8M \geq 82$
8. $3B + 8M + 22P \geq 53$
9. $3B + 17J + 8M \geq 53$
10. $22B + 13P \leq 186$
11. $15M + 13P \leq 217$
12. $22B + 15M \leq 262$
13. $4J + 13P \leq 162$
14. $4J + 15M \leq 255$
15. $4J + 15M + 13P \leq 247$
16. $22B + 4J + 15M \leq 200$
17. $22B + 15M + 13P \leq 192$
18. $22B + 4J + 15M + 13P \leq 192$
19. $8M + 22P \leq 254$
20. $17J + 8M \leq 209$
21. $17J + 22P \leq 242$
22. $3B + 22P \leq 192$
23. $3B + 17J + 22P \leq 329$
24. $17J + 8M + 22P \leq 291$
25. $3B + 17J + 8M + 22P \leq 291$

## 3: Implement the optimization problem using Gurobi
We will use the Gurobi Python API to model and solve this optimization problem.

```python
import gurobi as gp

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

# Define the variables
B = m.addVar(name="B", vtype=gp.GRB.INTEGER)  # hours worked by Bill
J = m.addVar(name="J", vtype=gp.GRB.INTEGER)  # hours worked by John
M = m.addVar(name="M")  # hours worked by Mary
P = m.addVar(name="P", vtype=gp.GRB.INTEGER)  # hours worked by Peggy

# Define the objective function
m.setObjective(3.85 * B + 5.18 * J + 2.03 * M + 3.6 * P, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(B >= 0)
m.addConstr(J >= 0)
m.addConstr(M >= 0)
m.addConstr(P >= 0)

m.addConstr(3 * B + 8 * M >= 33)
m.addConstr(3 * B + 8 * M + 22 * P >= 82)
m.addConstr(3 * B + 17 * J + 8 * M >= 82)
m.addConstr(3 * B + 8 * M + 22 * P >= 53)
m.addConstr(3 * B + 17 * J + 8 * M >= 53)

m.addConstr(22 * B + 13 * P <= 186)
m.addConstr(15 * M + 13 * P <= 217)
m.addConstr(22 * B + 15 * M <= 262)
m.addConstr(4 * J + 13 * P <= 162)
m.addConstr(4 * J + 15 * M <= 255)
m.addConstr(4 * J + 15 * M + 13 * P <= 247)
m.addConstr(22 * B + 4 * J + 15 * M <= 200)
m.addConstr(22 * B + 15 * M + 13 * P <= 192)
m.addConstr(22 * B + 4 * J + 15 * M + 13 * P <= 192)

m.addConstr(8 * M + 22 * P <= 254)
m.addConstr(17 * J + 8 * M <= 209)
m.addConstr(17 * J + 22 * P <= 242)
m.addConstr(3 * B + 22 * P <= 192)
m.addConstr(3 * B + 17 * J + 22 * P <= 329)
m.addConstr(17 * J + 8 * M + 22 * P <= 291)
m.addConstr(3 * B + 17 * J + 8 * M + 22 * P <= 291)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Bill: {B.varValue}")
    print(f"Hours worked by John: {J.varValue}")
    print(f"Hours worked by Mary: {M.varValue}")
    print(f"Hours worked by Peggy: {P.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```