## Step 1: Define the optimization problem
The problem is to maximize the objective function: $7 \times \text{hours worked by Peggy} + 8 \times \text{hours worked by Bobby} + 1 \times \text{hours worked by Ringo} + 1 \times \text{hours worked by Laura}$.

## Step 2: List the variables
Let $P$, $B$, $R$, and $L$ represent the hours worked by Peggy, Bobby, Ringo, and Laura, respectively.

## 3: Define the constraints
1. $1P + 4B + 13R + 11L \geq 24$ (combined dollar cost per hour from $P$, $R$, and $L$)
2. $1P + 4B + 11L \geq 24$ (combined dollar cost per hour from $P$, $B$, and $L$)
3. $1P + 4B + 13R \geq 24$ (combined dollar cost per hour from $P$, $B$, and $R$)
4. $1P + 13R + 11L \geq 18$ (combined dollar cost per hour from $P$, $R$, and $L$)
5. $1P + 4B + 11L \geq 18$ (combined dollar cost per hour from $P$, $B$, and $L$)
6. $1P + 4B + 13R \geq 18$ (combined dollar cost per hour from $P$, $B$, and $R$)
7. $1P + 13R + 11L \geq 32$ (combined dollar cost per hour from $P$, $R$, and $L$)
8. $1P + 4B + 11L \geq 32$ (combined dollar cost per hour from $P$, $B$, and $L$)
9. $1P + 4B + 13R \geq 32$ (combined dollar cost per hour from $P$, $B$, and $R$)
10. $12P + 11R \geq 30$ (combined work quality rating from $P$ and $R$)
11. $10B + 10L \geq 23$ (combined work quality rating from $B$ and $L$)
12. $12P + 10B + 10L \geq 20$ (combined work quality rating from $P$, $B$, and $L$)
13. $1P + 4B \leq 110$ (combined dollar cost per hour from $P$ and $B$)
14. $13R + 11L \leq 54$ (combined dollar cost per hour from $R$ and $L$)
15. $1P + 4B + 13R + 11L \leq 54$ (combined dollar cost per hour from all)
16. $12P + 10L \leq 98$ (combined work quality rating from $P$ and $L$)
17. $12P + 11R \leq 50$ (combined work quality rating from $P$ and $R$)
18. $10B + 10L \leq 78$ (combined work quality rating from $B$ and $L$)
19. $11R + 10L \leq 110$ (combined work quality rating from $R$ and $L$)
20. $12P + 10B \leq 32$ (combined work quality rating from $P$ and $B$)
21. $12P + 10B + 10L \leq 109$ (combined work quality rating from $P$, $B$, and $L$)
22. $12P + 10B + 11R + 10L \leq 109$ (combined work quality rating from all)

## 4: Implement the optimization problem using Gurobi
```python
import gurobi as gp

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

# Define the variables
P = m.addVar(lb=0, name="hours_worked_by_Peggy")
B = m.addVar(lb=0, name="hours_worked_by_Bobby")
R = m.addVar(lb=0, name="hours_worked_by_Ringo")
L = m.addVar(lb=0, name="hours_worked_by_Laura")

# Define the objective function
m.setObjective(7 * P + 8 * B + R + L, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(1 * P + 4 * B + 13 * R + 11 * L >= 24)
m.addConstr(1 * P + 4 * B + 11 * L >= 24)
m.addConstr(1 * P + 4 * B + 13 * R >= 24)
m.addConstr(1 * P + 13 * R + 11 * L >= 18)
m.addConstr(1 * P + 4 * B + 11 * L >= 18)
m.addConstr(1 * P + 4 * B + 13 * R >= 18)
m.addConstr(1 * P + 13 * R + 11 * L >= 32)
m.addConstr(1 * P + 4 * B + 11 * L >= 32)
m.addConstr(1 * P + 4 * B + 13 * R >= 32)
m.addConstr(12 * P + 11 * R >= 30)
m.addConstr(10 * B + 10 * L >= 23)
m.addConstr(12 * P + 10 * B + 10 * L >= 20)
m.addConstr(1 * P + 4 * B <= 110)
m.addConstr(13 * R + 11 * L <= 54)
m.addConstr(1 * P + 4 * B + 13 * R + 11 * L <= 54)
m.addConstr(12 * P + 10 * L <= 98)
m.addConstr(12 * P + 11 * R <= 50)
m.addConstr(10 * B + 10 * L <= 78)
m.addConstr(11 * R + 10 * L <= 110)
m.addConstr(12 * P + 10 * B <= 32)
m.addConstr(12 * P + 10 * B + 10 * L <= 109)
m.addConstr(12 * P + 10 * B + 11 * R + 10 * L <= 109)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print("Hours worked by Peggy:", P.varValue)
    print("Hours worked by Bobby:", B.varValue)
    print("Hours worked by Ringo:", R.varValue)
    print("Hours worked by Laura:", L.varValue)
    print("Objective function value:", m.objVal)
else:
    print("No optimal solution found.")
```