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

## Step 2: Define the variables
Let $x_0$ be the hours worked by Dale, $x_1$ be the hours worked by Peggy, and $x_2$ be the hours worked by Bill.

## 3: Define the constraints
1. $12x_1 + 11x_2 \geq 24$ (paperwork competence rating from Peggy and Bill)
2. $20x_0 + 12x_1 \geq 36$ (paperwork competence rating from Dale and Peggy)
3. $20x_0 + 12x_1 + 11x_2 \geq 36$ (paperwork competence rating from all)
4. $13x_0 + 18x_2 \geq 15$ (work quality rating from Dale and Bill)
5. $15x_1 + 18x_2 \geq 33$ (work quality rating from Peggy and Bill)
6. $13x_0 + 15x_1 + 18x_2 \geq 23$ (work quality rating from all)
7. $10x_1 + 15x_2 \geq 32$ (dollar cost per hour from Peggy and Bill)
8. $20x_0 + 10x_1 \geq 23$ (dollar cost per hour from Dale and Peggy)
9. $20x_0 + 10x_1 + 15x_2 \geq 24$ (dollar cost per hour from all)
10. $-10x_1 + 10x_2 \geq 0$ (relationship between Peggy and Bill's hours)
11. $13x_0 + 15x_1 \leq 103$ (work quality rating limit from Dale and Peggy)
12. $13x_0 + 18x_2 \leq 66$ (work quality rating limit from Dale and Bill)
13. $13x_0 + 15x_1 + 18x_2 \leq 115$ (work quality rating limit from all)
14. $10x_1 + 15x_2 \leq 95$ (dollar cost per hour limit from Peggy and Bill)
15. $20x_0 + 15x_2 \leq 42$ (dollar cost per hour limit from Dale and Bill)

## 4: Implement the problem in Gurobi
We will use the Gurobi Python API to model and solve this linear programming problem.

```python
import gurobi

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

# Define the variables
x0 = m.addVar(name="hours_worked_by_Dale", lb=0)  # hours worked by Dale
x1 = m.addVar(name="hours_worked_by_Peggy", lb=0)  # hours worked by Peggy
x2 = m.addVar(name="hours_worked_by_Bill", lb=0)  # hours worked by Bill

# Define the objective function
m.setObjective(x0 + 3 * x1 + 6 * x2, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(12 * x1 + 11 * x2 >= 24)  # paperwork competence rating from Peggy and Bill
m.addConstr(20 * x0 + 12 * x1 >= 36)  # paperwork competence rating from Dale and Peggy
m.addConstr(20 * x0 + 12 * x1 + 11 * x2 >= 36)  # paperwork competence rating from all
m.addConstr(13 * x0 + 18 * x2 >= 15)  # work quality rating from Dale and Bill
m.addConstr(15 * x1 + 18 * x2 >= 33)  # work quality rating from Peggy and Bill
m.addConstr(13 * x0 + 15 * x1 + 18 * x2 >= 23)  # work quality rating from all
m.addConstr(10 * x1 + 15 * x2 >= 32)  # dollar cost per hour from Peggy and Bill
m.addConstr(20 * x0 + 10 * x1 >= 23)  # dollar cost per hour from Dale and Peggy
m.addConstr(20 * x0 + 10 * x1 + 15 * x2 >= 24)  # dollar cost per hour from all
m.addConstr(-10 * x1 + 10 * x2 >= 0)  # relationship between Peggy and Bill's hours
m.addConstr(13 * x0 + 15 * x1 <= 103)  # work quality rating limit from Dale and Peggy
m.addConstr(13 * x0 + 18 * x2 <= 66)  # work quality rating limit from Dale and Bill
m.addConstr(13 * x0 + 15 * x1 + 18 * x2 <= 115)  # work quality rating limit from all
m.addConstr(10 * x1 + 15 * x2 <= 95)  # dollar cost per hour limit from Peggy and Bill
m.addConstr(20 * x0 + 15 * x2 <= 42)  # dollar cost per hour limit from Dale and Bill

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Dale: {x0.varValue}")
    print(f"Hours worked by Peggy: {x1.varValue}")
    print(f"Hours worked by Bill: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```