## Step 1: Define the optimization problem and the goal
The goal is to minimize the objective function: $5 \times \text{hours worked by Ringo} + 4 \times \text{hours worked by Paul} + 4 \times \text{hours worked by Hank} + 5 \times \text{hours worked by Peggy}$, subject to various constraints.

## 2: List all the constraints
1. Ringo's organization score is 17.
2. Ringo has a work quality rating of 7.
3. Paul has an organization score of 23.
4. Paul has a work quality rating of 23.
5. Hank's organization score is 6.
6. Hank has a work quality rating of 18.
7. Peggy's organization score is 17.
8. Peggy's work quality rating is 19.
9. $17 \times \text{hours worked by Ringo} + 23 \times \text{hours worked by Paul} \geq 60$.
10. $17 \times \text{hours worked by Ringo} + 6 \times \text{hours worked by Hank} \geq 35$.
11. $23 \times \text{hours worked by Paul} + 6 \times \text{hours worked by Hank} + 17 \times \text{hours worked by Peggy} \geq 77$.
12. $17 \times \text{hours worked by Ringo} + 23 \times \text{hours worked by Paul} + 6 \times \text{hours worked by Hank} + 17 \times \text{hours worked by Peggy} \geq 77$.
13. $7 \times \text{hours worked by Ringo} + 18 \times \text{hours worked by Hank} \geq 19$.
14. $23 \times \text{hours worked by Paul} + 19 \times \text{hours worked by Peggy} \geq 24$.
15. $7 \times \text{hours worked by Ringo} + 19 \times \text{hours worked by Peggy} \geq 36$.
16. $7 \times \text{hours worked by Ringo} + 23 \times \text{hours worked by Paul} + 19 \times \text{hours worked by Peggy} \geq 35$.
17. $7 \times \text{hours worked by Ringo} + 23 \times \text{hours worked by Paul} + 18 \times \text{hours worked by Hank} \geq 35$.
18. $7 \times \text{hours worked by Ringo} + 23 \times \text{hours worked by Paul} + 18 \times \text{hours worked by Hank} + 19 \times \text{hours worked by Peggy} \geq 37$.
19. $4 \times \text{hours worked by Paul} - 8 \times \text{hours worked by Hank} \geq 0$.
20. $-8 \times \text{hours worked by Paul} + 10 \times \text{hours worked by Peggy} \geq 0$.
21. $-7 \times \text{hours worked by Ringo} + 9 \times \text{hours worked by Hank} + 10 \times \text{hours worked by Peggy} \geq 0$.
22. $17 \times \text{hours worked by Ringo} + 23 \times \text{hours worked by Paul} + 17 \times \text{hours worked by Peggy} \leq 231$.
23. $17 \times \text{hours worked by Ringo} + 6 \times \text{hours worked by Hank} + 17 \times \text{hours worked by Peggy} \leq 104$.
24. $23 \times \text{hours worked by Paul} + 6 \times \text{hours worked by Hank} + 17 \times \text{hours worked by Peggy} \leq 96$.
25. $17 \times \text{hours worked by Ringo} + 23 \times \text{hours worked by Paul} + 6 \times \text{hours worked by Hank} \leq 276$.
26. $7 \times \text{hours worked by Ringo} + 23 \times \text{hours worked by Paul} \leq 58$.
27. $23 \times \text{hours worked by Paul} + 18 \times \text{hours worked by Hank} \leq 118$.
28. All variables must be integers.

## 3: Formulate the problem in Gurobi
We will use Gurobi's Python API to formulate and solve this problem.

```python
import gurobi

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

# Define the variables
ringo = model.addVar(name="Ringo", vtype=gurobi.GRB.INTEGER)
paul = model.addVar(name="Paul", vtype=gurobi.GRB.INTEGER)
hank = model.addVar(name="Hank", vtype=gurobi.GRB.INTEGER)
peggy = model.addVar(name="Peggy", vtype=gurobi.GRB.INTEGER)

# Objective function
model.setObjective(5 * ringo + 4 * paul + 4 * hank + 5 * peggy, gurobi.GRB.MINIMIZE)

# Constraints
model.addConstr(ringo >= 0, name="Ringo_non_negative")
model.addConstr(paul >= 0, name="Paul_non_negative")
model.addConstr(hank >= 0, name="Hank_non_negative")
model.addConstr(peggy >= 0, name="Peggy_non_negative")

# Organization and work quality ratings (not directly constraints but given)
# Ringo's organization score is 17, work quality rating is 7
# Paul's organization score is 23, work quality rating is 23
# Hank's organization score is 6, work quality rating is 18
# Peggy's organization score is 17, work quality rating is 19

# Constraints from problem description
model.addConstr(17 * ringo + 23 * paul >= 60)
model.addConstr(17 * ringo + 6 * hank >= 35)
model.addConstr(23 * paul + 6 * hank + 17 * peggy >= 77)
model.addConstr(17 * ringo + 23 * paul + 6 * hank + 17 * peggy >= 77)
model.addConstr(7 * ringo + 18 * hank >= 19)
model.addConstr(23 * paul + 19 * peggy >= 24)
model.addConstr(7 * ringo + 19 * peggy >= 36)
model.addConstr(7 * ringo + 23 * paul + 19 * peggy >= 35)
model.addConstr(7 * ringo + 23 * paul + 18 * hank >= 35)
model.addConstr(7 * ringo + 23 * paul + 18 * hank + 19 * peggy >= 37)
model.addConstr(4 * paul - 8 * hank >= 0)
model.addConstr(-8 * paul + 10 * peggy >= 0)
model.addConstr(-7 * ringo + 9 * hank + 10 * peggy >= 0)
model.addConstr(17 * ringo + 23 * paul + 17 * peggy <= 231)
model.addConstr(17 * ringo + 6 * hank + 17 * peggy <= 104)
model.addConstr(23 * paul + 6 * hank + 17 * peggy <= 96)
model.addConstr(17 * ringo + 23 * paul + 6 * hank <= 276)
model.addConstr(7 * ringo + 23 * paul <= 58)
model.addConstr(23 * paul + 18 * hank <= 118)

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objval)
    print("Ringo: ", ringo.varValue)
    print("Paul: ", paul.varValue)
    print("Hank: ", hank.varValue)
    print("Peggy: ", peggy.varValue)
else:
    print("The model is infeasible")
```