## Step 1: Define the variables and their attributes
We have six variables representing the hours worked by Hank, Paul, Mary, Ringo, Jean, and John. Each variable has attributes based on the given resources/competence ratings.

## Step 2: Formulate the objective function
The objective function to minimize is: $1.73H + 1.02P + 2.71M + 3.81R + 9.75J + 3.15Y$, where $H, P, M, R, J, Y$ represent the hours worked by Hank, Paul, Mary, Ringo, Jean, and John, respectively.

## 3: List all constraints
1. $5H \leq 192$
2. $19H \leq 202$
3. $17P \leq 192$
4. $18P \leq 202$
5. $1M \leq 192$
6. $12M \leq 202$
7. $8R \leq 192$
8. $8R \leq 202$
9. $7J \leq 192$
10. $17J \leq 202$
11. $5Y \leq 192$
12. $14Y \leq 202$
13. $17P + 8R \geq 32$
14. $5H + 17P \geq 22$
15. $8R + 7J \geq 18$
16. $5H + 8R \geq 19$
17. $17P + 1M \geq 13$
18. $7J + 5Y \geq 29$
19. $1M + 5Y \geq 31$
20. $17P + 5Y \geq 10$
21. $5H + 5Y \geq 13$
22. $5H + 7J \geq 12$
23. $8R + 5Y \geq 29$
24. $5H + 17P + 1M + 8R + 7J + 5Y \geq 29$
25. $12M + 14Y \geq 16$
26. $12M + 8R \geq 27$
27. $19H + 18P + 12M + 8R + 17J + 14Y \geq 27$
28. $2P - 3J \geq 0$
29. $8H - 2J \geq 0$
30. $6P - 3Y \geq 0$
31. $17P + 1M \leq 122$
32. $5H + 1M \leq 149$
33. $5H + 7J \leq 110$
34. $17P + 8R \leq 36$
35. $8R + 5Y \leq 84$
36. $1M + 5Y \leq 37$
37. $17P + 5Y \leq 72$
38. $8R + 7J \leq 38$
39. $18P + 12M + 17J \leq 69$
40. $18P + 17J + 14Y \leq 81$
41. $19H + 17J + 14Y \leq 83$
42. $19H + 18P + 17J \leq 200$
43. $19H + 18P + 8R \leq 45$
44. $19H + 12M + 17J \leq 59$

## 4: Define variable types
- $H, P, J$ can be non-integer
- $M, R, Y$ must be integer

## 5: Implement in Gurobi
```python
import gurobi

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

    # Define variables
    H = model.addVar(lb=0, name="H")  # Hank
    P = model.addVar(lb=0, name="P")  # Paul
    M = model.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="M")  # Mary
    R = model.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="R")  # Ringo
    J = model.addVar(lb=0, name="J")  # Jean
    Y = model.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="Y")  # John

    # Objective function
    model.setObjective(1.73 * H + 1.02 * P + 2.71 * M + 3.81 * R + 9.75 * J + 3.15 * Y, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(5 * H <= 192)
    model.addConstr(19 * H <= 202)
    model.addConstr(17 * P <= 192)
    model.addConstr(18 * P <= 202)
    model.addConstr(1 * M <= 192)
    model.addConstr(12 * M <= 202)
    model.addConstr(8 * R <= 192)
    model.addConstr(8 * R <= 202)
    model.addConstr(7 * J <= 192)
    model.addConstr(17 * J <= 202)
    model.addConstr(5 * Y <= 192)
    model.addConstr(14 * Y <= 202)

    model.addConstr(17 * P + 8 * R >= 32)
    model.addConstr(5 * H + 17 * P >= 22)
    model.addConstr(8 * R + 7 * J >= 18)
    model.addConstr(5 * H + 8 * R >= 19)
    model.addConstr(17 * P + 1 * M >= 13)
    model.addConstr(7 * J + 5 * Y >= 29)
    model.addConstr(1 * M + 5 * Y >= 31)
    model.addConstr(17 * P + 5 * Y >= 10)
    model.addConstr(5 * H + 5 * Y >= 13)
    model.addConstr(5 * H + 7 * J >= 12)
    model.addConstr(8 * R + 5 * Y >= 29)
    model.addConstr(5 * H + 17 * P + 1 * M + 8 * R + 7 * J + 5 * Y >= 29)

    model.addConstr(12 * M + 14 * Y >= 16)
    model.addConstr(12 * M + 8 * R >= 27)
    model.addConstr(19 * H + 18 * P + 12 * M + 8 * R + 17 * J + 14 * Y >= 27)

    model.addConstr(2 * P - 3 * J >= 0)
    model.addConstr(8 * H - 2 * J >= 0)
    model.addConstr(6 * P - 3 * Y >= 0)

    model.addConstr(17 * P + 1 * M <= 122)
    model.addConstr(5 * H + 1 * M <= 149)
    model.addConstr(5 * H + 7 * J <= 110)
    model.addConstr(17 * P + 8 * R <= 36)
    model.addConstr(8 * R + 5 * Y <= 84)
    model.addConstr(1 * M + 5 * Y <= 37)
    model.addConstr(17 * P + 5 * Y <= 72)
    model.addConstr(8 * R + 7 * J <= 38)

    model.addConstr(18 * P + 12 * M + 17 * J <= 69)
    model.addConstr(18 * P + 17 * J + 14 * Y <= 81)
    model.addConstr(19 * H + 17 * J + 14 * Y <= 83)
    model.addConstr(19 * H + 18 * P + 17 * J <= 200)
    model.addConstr(19 * H + 18 * P + 8 * R <= 45)
    model.addConstr(19 * H + 12 * M + 17 * J <= 59)

    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hank: ", H.varValue)
        print("Paul: ", P.varValue)
        print("Mary: ", M.varValue)
        print("Ringo: ", R.varValue)
        print("Jean: ", J.varValue)
        print("John: ", Y.varValue)
    else:
        print("No solution found")

optimize_problem()
```