## Step 1: Define the optimization problem and its constraints
The problem requires maximizing the objective function: $2 \times \text{hours worked by Hank} + 7 \times \text{hours worked by Paul} + 7 \times \text{hours worked by Mary}$, subject to various constraints on computer competence rating, likelihood to quit index, paperwork competence rating, organization score, and work quality rating for Hank, Paul, and Mary.

## 2: List all the constraints
1. $16 \times \text{hours worked by Hank} \leq 169$
2. $11 \times \text{hours worked by Hank} \leq 316$
3. $7 \times \text{hours worked by Hank} \leq 223$
4. $15 \times \text{hours worked by Hank} \leq 263$
5. $2 \times \text{hours worked by Hank} \leq 131$
6. $16 \times \text{hours worked by Paul} \leq 169$
7. $17 \times \text{hours worked by Paul} \leq 316$
8. $21 \times \text{hours worked by Paul} \leq 223$
9. $5 \times \text{hours worked by Paul} \leq 263$
10. $4 \times \text{hours worked by Paul} \leq 131$
11. $3 \times \text{hours worked by Mary} \leq 169$
12. $2 \times \text{hours worked by Mary} \leq 316$
13. $13 \times \text{hours worked by Mary} \leq 223$
14. $7 \times \text{hours worked by Mary} \leq 263$
15. $7 \times \text{hours worked by Mary} \leq 131$
16. $16 \times \text{hours worked by Hank} + 3 \times \text{hours worked by Mary} \geq 41$
17. $21 \times \text{hours worked by Paul} + 13 \times \text{hours worked by Mary} \geq 33$
18. $7 \times \text{hours worked by Hank} + 13 \times \text{hours worked by Mary} \geq 29$
19. $7 \times \text{hours worked by Hank} + 21 \times \text{hours worked by Paul} + 13 \times \text{hours worked by Mary} \geq 73$
20. $15 \times \text{hours worked by Hank} + 7 \times \text{hours worked by Mary} \geq 84$
21. $15 \times \text{hours worked by Hank} + 5 \times \text{hours worked by Paul} \geq 83$
22. $5 \times \text{hours worked by Paul} + 7 \times \text{hours worked by Mary} \geq 64$
23. $2 \times \text{hours worked by Hank} + 7 \times \text{hours worked by Mary} \geq 15$
24. $4 \times \text{hours worked by Paul} + 7 \times \text{hours worked by Mary} \geq 22$
25. $2 \times \text{hours worked by Hank} + 4 \times \text{hours worked by Paul} + 7 \times \text{hours worked by Mary} \geq 29$
26. $16 \times \text{hours worked by Hank} + 16 \times \text{hours worked by Paul} \leq 72$
27. $16 \times \text{hours worked by Hank} + 3 \times \text{hours worked by Mary} \leq 152$
28. $16 \times \text{hours worked by Paul} + 3 \times \text{hours worked by Mary} \leq 110$
29. $16 \times \text{hours worked by Hank} + 16 \times \text{hours worked by Paul} + 3 \times \text{hours worked by Mary} \leq 110$
30. $17 \times \text{hours worked by Paul} + 2 \times \text{hours worked by Mary} \leq 106$
31. $11 \times \text{hours worked by Hank} + 17 \times \text{hours worked by Paul} + 2 \times \text{hours worked by Mary} \leq 106$
32. $7 \times \text{hours worked by Hank} + 21 \times \text{hours worked by Paul} \leq 211$
33. $21 \times \text{hours worked by Paul} + 13 \times \text{hours worked by Mary} \leq 159$
34. $7 \times \text{hours worked by Hank} + 21 \times \text{hours worked by Paul} + 13 \times \text{hours worked by Mary} \leq 159$
35. $5 \times \text{hours worked by Paul} + 7 \times \text{hours worked by Mary} \leq 210$
36. $15 \times \text{hours worked by Hank} + 5 \times \text{hours worked by Paul} \leq 202$
37. $15 \times \text{hours worked by Hank} + 5 \times \text{hours worked by Paul} + 7 \times \text{hours worked by Mary} \leq 202$
38. $4 \times \text{hours worked by Paul} + 7 \times \text{hours worked by Mary} \leq 107$
39. $2 \times \text{hours worked by Hank} + 4 \times \text{hours worked by Paul} + 7 \times \text{hours worked by Mary} \leq 107$

## 3: Formulate the problem in Gurobi
We will use Gurobi to solve this linear programming problem. The code is as follows:

```python
import gurobi as gp

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

# Define the variables
hours_worked_by_Hank = m.addVar(name="hours_worked_by_Hank", lb=0)
hours_worked_by_Paul = m.addVar(name="hours_worked_by_Paul", lb=0)
hours_worked_by_Mary = m.addVar(name="hours_worked_by_Mary", lb=0)

# Define the objective function
m.setObjective(2 * hours_worked_by_Hank + 7 * hours_worked_by_Paul + 7 * hours_worked_by_Mary, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(16 * hours_worked_by_Hank <= 169)
m.addConstr(11 * hours_worked_by_Hank <= 316)
m.addConstr(7 * hours_worked_by_Hank <= 223)
m.addConstr(15 * hours_worked_by_Hank <= 263)
m.addConstr(2 * hours_worked_by_Hank <= 131)

m.addConstr(16 * hours_worked_by_Paul <= 169)
m.addConstr(17 * hours_worked_by_Paul <= 316)
m.addConstr(21 * hours_worked_by_Paul <= 223)
m.addConstr(5 * hours_worked_by_Paul <= 263)
m.addConstr(4 * hours_worked_by_Paul <= 131)

m.addConstr(3 * hours_worked_by_Mary <= 169)
m.addConstr(2 * hours_worked_by_Mary <= 316)
m.addConstr(13 * hours_worked_by_Mary <= 223)
m.addConstr(7 * hours_worked_by_Mary <= 263)
m.addConstr(7 * hours_worked_by_Mary <= 131)

m.addConstr(16 * hours_worked_by_Hank + 3 * hours_worked_by_Mary >= 41)
m.addConstr(21 * hours_worked_by_Paul + 13 * hours_worked_by_Mary >= 33)
m.addConstr(7 * hours_worked_by_Hank + 13 * hours_worked_by_Mary >= 29)
m.addConstr(7 * hours_worked_by_Hank + 21 * hours_worked_by_Paul + 13 * hours_worked_by_Mary >= 73)
m.addConstr(15 * hours_worked_by_Hank + 7 * hours_worked_by_Mary >= 84)
m.addConstr(15 * hours_worked_by_Hank + 5 * hours_worked_by_Paul >= 83)
m.addConstr(5 * hours_worked_by_Paul + 7 * hours_worked_by_Mary >= 64)
m.addConstr(2 * hours_worked_by_Hank + 7 * hours_worked_by_Mary >= 15)
m.addConstr(4 * hours_worked_by_Paul + 7 * hours_worked_by_Mary >= 22)
m.addConstr(2 * hours_worked_by_Hank + 4 * hours_worked_by_Paul + 7 * hours_worked_by_Mary >= 29)

m.addConstr(16 * hours_worked_by_Hank + 16 * hours_worked_by_Paul <= 72)
m.addConstr(16 * hours_worked_by_Hank + 3 * hours_worked_by_Mary <= 152)
m.addConstr(16 * hours_worked_by_Paul + 3 * hours_worked_by_Mary <= 110)
m.addConstr(16 * hours_worked_by_Hank + 16 * hours_worked_by_Paul + 3 * hours_worked_by_Mary <= 110)
m.addConstr(17 * hours_worked_by_Paul + 2 * hours_worked_by_Mary <= 106)
m.addConstr(11 * hours_worked_by_Hank + 17 * hours_worked_by_Paul + 2 * hours_worked_by_Mary <= 106)
m.addConstr(7 * hours_worked_by_Hank + 21 * hours_worked_by_Paul <= 211)
m.addConstr(21 * hours_worked_by_Paul + 13 * hours_worked_by_Mary <= 159)
m.addConstr(7 * hours_worked_by_Hank + 21 * hours_worked_by_Paul + 13 * hours_worked_by_Mary <= 159)
m.addConstr(5 * hours_worked_by_Paul + 7 * hours_worked_by_Mary <= 210)
m.addConstr(15 * hours_worked_by_Hank + 5 * hours_worked_by_Paul <= 202)
m.addConstr(15 * hours_worked_by_Hank + 5 * hours_worked_by_Paul + 7 * hours_worked_by_Mary <= 202)
m.addConstr(4 * hours_worked_by_Paul + 7 * hours_worked_by_Mary <= 107)
m.addConstr(2 * hours_worked_by_Hank + 4 * hours_worked_by_Paul + 7 * hours_worked_by_Mary <= 107)

# Solve the model
m.optimize()

# Print the results
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Hank: {hours_worked_by_Hank.varValue}")
    print(f"Hours worked by Paul: {hours_worked_by_Paul.varValue}")
    print(f"Hours worked by Mary: {hours_worked_by_Mary.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```