## Step 1: Define the optimization problem and its components
The problem requires minimizing the objective function $1.67 \times (\text{hours worked by Paul}) \times (\text{hours worked by Ringo}) + 6.79 \times (\text{hours worked by Ringo})$. The variables are 'hours worked by Paul' ($x_0$) and 'hours worked by Ringo' ($x_1$). The constraints involve various ratings and costs associated with Paul and Ringo.

## 2: List all given resources/attributes
Given resources/attributes:
- $r_0$: Paperwork competence rating
- $r_1$: Dollar cost per hour
- $r_2$: Organization score
- $r_3$: Computer competence rating

With their respective values for Paul ($x_0$) and Ringo ($x_1$):
- $r_0$: $x_0 = 19$, $x_1 = 2$, upper bound = 148
- $r_1$: $x_0 = 18$, $x_1 = 7$, upper bound = 100
- $r_2$: $x_0 = 14$, $x_1 = 14$, upper bound = 158
- $r_3$: $x_0 = 13$, $x_1 = 18$, upper bound = 112

## 3: Formulate the objective function
The objective function to minimize is $1.67x_0x_1 + 6.79x_1$.

## 4: Define the constraints
1. $19^2x_0^2 + 2^2x_1^2 \geq 44$
2. $19x_0 + 2x_1 \geq 44$
3. $18x_0 + 7x_1 \geq 31$
4. $14x_0 + 14x_1 \geq 56$
5. $13x_0 + 18x_1 \geq 46$
6. $6x_0 - 10x_1 \geq 0$
7. $19x_0 + 2x_1 \leq 94$
8. $18x_0 + 7x_1 \leq 82$
9. $14^2x_0^2 + 14^2x_1^2 \leq 119$
10. $13x_0 + 18x_1 \leq 93$
11. $x_1$ is an integer
12. $x_0$ is a non-integer (continuous)

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

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

    # Define variables
    x0 = model.addVar(lb=0, name="hours_worked_by_Paul", vtype=gurobi.GRB.CONTINUOUS)
    x1 = model.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="hours_worked_by_Ringo")

    # Objective function
    model.setObjective(1.67 * x0 * x1 + 6.79 * x1, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(19**2 * x0**2 + 2**2 * x1**2 >= 44, name="paperwork_competence_rating_squared")
    model.addConstr(19 * x0 + 2 * x1 >= 44, name="paperwork_competence_rating")
    model.addConstr(18 * x0 + 7 * x1 >= 31, name="dollar_cost_per_hour")
    model.addConstr(14 * x0 + 14 * x1 >= 56, name="organization_score")
    model.addConstr(13 * x0 + 18 * x1 >= 46, name="computer_competence_rating")
    model.addConstr(6 * x0 - 10 * x1 >= 0, name="six_times_paul_minus_ten_times_ringo")
    model.addConstr(19 * x0 + 2 * x1 <= 94, name="paperwork_competence_rating_upper_bound")
    model.addConstr(18 * x0 + 7 * x1 <= 82, name="dollar_cost_per_hour_upper_bound")
    model.addConstr(14**2 * x0**2 + 14**2 * x1**2 <= 119, name="organization_score_squared_upper_bound")
    model.addConstr(13 * x0 + 18 * x1 <= 93, name="computer_competence_rating_upper_bound")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Paul: {x0.varValue}")
        print(f"Hours worked by Ringo: {x1.varValue}")
    else:
        print("No optimal solution found.")

optimize_problem()
```