## Problem Description and Formulation

The problem requires maximizing the objective function: $3 \times \text{hours worked by Paul} + 1 \times \text{hours worked by Dale}$, subject to several constraints.

### Variables and Resources

- Variables: 
  - $x_0$: hours worked by Paul
  - $x_1$: hours worked by Dale

- Resources/Attributes:
  - $r_0$: likelihood to quit index
    - $x_0$'s coefficient: 14
    - $x_1$'s coefficient: 7
    - Upper bound: 171
  - $r_1$: work quality rating
    - $x_0$'s coefficient: 21
    - $x_1$'s coefficient: 13
    - Upper bound: 152

### Constraints

1. $14x_0 \leq 14$
2. $21x_0 \leq 21$
3. $7x_1 \leq 7$
4. $13x_1 \leq 13$
5. $14x_0 + 7x_1 \geq 77$
6. $21x_0 + 13x_1 \geq 39$
7. $9x_0 - 4x_1 \geq 0$
8. $14x_0 + 7x_1 \leq 130$
9. $21x_0 + 13x_1 \leq 116$

### Observations

- Constraints 1, 2, 3, and 4 are essentially $x_0 \leq 1$ and $x_1 \leq 1$ because they limit $x_0$ and $x_1$ to be less than or equal to 1.
- The problem does not explicitly require $x_0$ and $x_1$ to be integers, but it mentions the possibility of fractional hours worked by Paul and does not require an integer number of hours worked by Dale.

## Gurobi Code Formulation

```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="hours_worked_by_Paul", lb=0)  # Assuming non-negative hours
    x1 = model.addVar(name="hours_worked_by_Dale", lb=0)  # Assuming non-negative hours

    # Objective function: Maximize 3*x0 + x1
    model.setObjective(3*x0 + x1, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(14*x0 <= 14, name="quit_index_paul")
    model.addConstr(21*x0 <= 21, name="work_quality_paul")
    model.addConstr(7*x1 <= 7, name="quit_index_dale")
    model.addConstr(13*x1 <= 13, name="work_quality_dale")
    model.addConstr(14*x0 + 7*x1 >= 77, name="combined_quit_index_min")
    model.addConstr(21*x0 + 13*x1 >= 39, name="combined_work_quality_min")
    model.addConstr(9*x0 - 4*x1 >= 0, name="hours_worked_constraint")
    model.addConstr(14*x0 + 7*x1 <= 130, name="combined_quit_index_max")
    model.addConstr(21*x0 + 13*x1 <= 116, name="combined_work_quality_max")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Hours worked by Paul: ", x0.varValue)
        print("Hours worked by Dale: ", x1.varValue)
    else:
        print("No optimal solution found")

solve_optimization_problem()
```