## Step 1: Define the optimization problem and its components
The problem is a linear optimization problem where we need to minimize the objective function: $1 \times \text{hours worked by Dale} + 8 \times \text{hours worked by Laura} + 7 \times \text{hours worked by Bobby}$. The problem has several constraints related to paperwork competence ratings and work quality ratings.

## 2: Identify the variables and their attributes
Let $x_0$, $x_1$, and $x_2$ represent the hours worked by Dale, Laura, and Bobby, respectively. The attributes for each variable are given as:
- $r_0$: paperwork competence rating with $x_0 = 3$, $x_1 = 7$, $x_2 = 6$, and an upper bound of 77.
- $r_1$: work quality rating with $x_0 = 9$, $x_1 = 3$, $x_2 = 2$, and an upper bound of 112.

## 3: Formulate the objective function
The objective function to minimize is: $x_0 + 8x_1 + 7x_2$.

## 4: List all constraints
1. $3x_0 \geq 0$ (Implicit, as $x_0 \geq 0$)
2. $9x_0 \geq 0$ (Implicit, as $x_0 \geq 0$)
3. $7x_1 \geq 0$ (Implicit, as $x_1 \geq 0$)
4. $3x_1 \geq 0$ (Implicit, as $x_1 \geq 0$)
5. $6x_2 \geq 0$ (Implicit, as $x_2 \geq 0$)
6. $2x_2 \geq 0$ (Implicit, as $x_2 \geq 0$)
7. $3x_0 + 6x_2 \geq 8$
8. $7x_1 + 6x_2 \geq 20$
9. $3x_0 + 7x_1 + 6x_2 \geq 20$
10. $9x_0 + 3x_1 \geq 31$
11. $9x_0 + 3x_1 + 2x_2 \geq 31$
12. $6x_0 - 4x_1 \geq 0$
13. $-9x_1 + 4x_2 \geq 0$
14. $3x_0 + 6x_2 \leq 27$
15. $3x_0 + 7x_1 \leq 68$
16. $7x_1 + 6x_2 \leq 45$
17. $9x_0 + 3x_1 + 2x_2 \leq 77$
18. $x_0 \in \mathbb{Z}$, $x_1 \in \mathbb{Z}$, $x_2 \in \mathbb{Z}$ (Integer constraints)

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

def optimize_problem():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="hours_worked_by_Dale", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="hours_worked_by_Laura", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="hours_worked_by_Bobby", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(x0 + 8 * x1 + 7 * x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(3 * x0 + 6 * x2 >= 8)
    model.addConstr(7 * x1 + 6 * x2 >= 20)
    model.addConstr(3 * x0 + 7 * x1 + 6 * x2 >= 20)
    model.addConstr(9 * x0 + 3 * x1 >= 31)
    model.addConstr(9 * x0 + 3 * x1 + 2 * x2 >= 31)
    model.addConstr(6 * x0 - 4 * x1 >= 0)
    model.addConstr(-9 * x1 + 4 * x2 >= 0)
    model.addConstr(3 * x0 + 6 * x2 <= 27)
    model.addConstr(3 * x0 + 7 * x1 <= 68)
    model.addConstr(7 * x1 + 6 * x2 <= 45)
    model.addConstr(9 * x0 + 3 * x1 + 2 * x2 <= 77)

    # Non-negativity constraints (implicit in Gurobi for continuous variables, but we require integers)
    model.addConstr(x0 >= 0)
    model.addConstr(x1 >= 0)
    model.addConstr(x2 >= 0)

    # Optimize
    model.optimize()

    # Solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Dale: {x0.varValue}")
        print(f"Hours worked by Laura: {x1.varValue}")
        print(f"Hours worked by Bobby: {x2.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The problem is infeasible.")
    else:
        print("The model status is: ", model.status)

optimize_problem()
```