## Step 1: Define the optimization problem and the objective function
The objective function to minimize is $3 \times \text{hours worked by Hank} + 1 \times (\text{hours worked by Peggy} + 1) + 1 \times \text{hours worked by Dale} + 6 \times \text{hours worked by Laura}$.

## 2: List all the variables
Let $H$, $P$, $D$, and $L$ represent the hours worked by Hank, Peggy, Dale, and Laura, respectively.

## 3: Formulate the objective function in terms of the variables
The objective function is $3H + P + 1 + D + 6L$.

## 4: Define the constraints based on the given information
- The dollar cost per hour for Hank is $9.78H$, for Peggy is $10.71P$, for Dale is $1.77D$, and for Laura is $0.25L$.
- The productivity rating for Hank is $9.7H$, for Peggy is $7.36P$, for Dale is $6.5D$, and for Laura is $5.48L$.
- The computer competence rating for Hank is $1.73H$, for Peggy is $5.19P$, for Dale is $7.26D$, and for Laura is $7.28L$.

## 5: Translate given constraints into mathematical expressions
1. $9.78H + 10.71P \geq 6$
2. $10.71P + 0.25L \geq 6$
3. $9.78H + 0.25L \geq 18$
4. $9.78H + 1.77D \geq 11$
5. $9.78H + 10.71P + 1.77D + 0.25L \geq 11$
6. $9.7H + 5.48L \geq 8$
7. $7.36P + 6.5D \geq 16$
8. $9.7H + 6.5D \geq 19$
9. $6.5D + 5.48L \geq 12$
10. $7.36P + 5.48L \geq 15$
11. $7.36P + 6.5D + 5.48L \geq 9$
12. $9.7H + 7.36P + 6.5D \geq 9$
13. $7.36P + 6.5D + 5.48L \geq 17$
14. $9.7H + 7.36P + 6.5D \geq 17$
15. $9.7H + 7.36P + 6.5D + 5.48L \geq 17$
16. $1.73H + 7.28L \geq 15$
17. $1.73H + 5.19P \geq 8$
18. $5.19P + 7.26D \geq 14$
19. $1.73H + 7.26D \geq 15$
20. $1.73H + 5.19P + 7.26D + 7.28L \geq 15$
21. $H - 8D \geq 0$
22. $10.71P + 0.25L \leq 28$
23. $6.5D + 5.48L \leq 72$
24. $9.7H + 5.48L \leq 58$
25. $7.36P + 5.48L \leq 25$
26. $9.7H + 7.36P \leq 39$
27. $1.73H + 7.26D \leq 30$

## 6: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define variables
    H = model.addVar(lb=0, name="H")  # hours worked by Hank
    P = model.addVar(lb=0, name="P")  # hours worked by Peggy
    D = model.addVar(lb=0, name="D")  # hours worked by Dale
    L = model.addVar(lb=0, name="L")  # hours worked by Laura

    # Objective function
    model.setObjective(3*H + P + D + 6*L + 1, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(9.78*H + 10.71*P >= 6)
    model.addConstr(10.71*P + 0.25*L >= 6)
    model.addConstr(9.78*H + 0.25*L >= 18)
    model.addConstr(9.78*H + 1.77*D >= 11)
    model.addConstr(9.78*H + 10.71*P + 1.77*D + 0.25*L >= 11)
    model.addConstr(9.7*H + 5.48*L >= 8)
    model.addConstr(7.36*P + 6.5*D >= 16)
    model.addConstr(9.7*H + 6.5*D >= 19)
    model.addConstr(6.5*D + 5.48*L >= 12)
    model.addConstr(7.36*P + 5.48*L >= 15)
    model.addConstr(7.36*P + 6.5*D + 5.48*L >= 9)
    model.addConstr(9.7*H + 7.36*P + 6.5*D >= 9)
    model.addConstr(7.36*P + 6.5*D + 5.48*L >= 17)
    model.addConstr(9.7*H + 7.36*P + 6.5*D >= 17)
    model.addConstr(9.7*H + 7.36*P + 6.5*D + 5.48*L >= 17)
    model.addConstr(1.73*H + 7.28*L >= 15)
    model.addConstr(1.73*H + 5.19*P >= 8)
    model.addConstr(5.19*P + 7.26*D >= 14)
    model.addConstr(1.73*H + 7.26*D >= 15)
    model.addConstr(1.73*H + 5.19*P + 7.26*D + 7.28*L >= 15)
    model.addConstr(H - 8*D >= 0)
    model.addConstr(10.71*P + 0.25*L <= 28)
    model.addConstr(6.5*D + 5.48*L <= 72)
    model.addConstr(9.7*H + 5.48*L <= 58)
    model.addConstr(7.36*P + 5.48*L <= 25)
    model.addConstr(9.7*H + 7.36*P <= 39)
    model.addConstr(1.73*H + 7.26*D <= 30)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("H: ", H.varValue)
        print("P: ", P.varValue)
        print("D: ", D.varValue)
        print("L: ", L.varValue)
    else:
        print("The model is infeasible")

optimize_problem()
```