## Problem Description and Formulation

The problem requires minimizing the objective function: $4 \times \text{hours worked by Laura} + 3 \times \text{hours worked by Bobby} + 7 \times \text{hours worked by Dale} + 5 \times \text{hours worked by George}$.

The variables are:
- $x_0$: hours worked by Laura
- $x_1$: hours worked by Bobby
- $x_2$: hours worked by Dale
- $x_3$: hours worked by George

The constraints are:
1. $\text{dollar cost per hour for Laura} = 6$
2. $\text{dollar cost per hour for Bobby} = 2$
3. $\text{dollar cost per hour for Dale} = 5$
4. $\text{dollar cost per hour for George} = 3$
5. $2x_1 + 3x_3 \geq 9$
6. $5x_2 + 3x_3 \geq 13$
7. $6x_0 + 2x_1 + 5x_2 + 3x_3 \geq 13$
8. $10x_2 - 2x_3 \geq 0$
9. $4x_0 - x_2 \geq 0$
10. $6x_0 + 2x_1 + 5x_2 \leq 44$
11. $x_0, x_1, x_2, x_3$ are integers.

## Gurobi Code Formulation

```python
import gurobipy as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x0 = m.addVar(name="hours_worked_by_Laura", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="hours_worked_by_Bobby", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="hours_worked_by_Dale", vtype=gp.GRB.INTEGER)
x3 = m.addVar(name="hours_worked_by_George", vtype=gp.GRB.INTEGER)

# Objective function
m.setObjective(4*x0 + 3*x1 + 7*x2 + 5*x3, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(2*x1 + 3*x3 >= 9, name="constraint_1")
m.addConstr(5*x2 + 3*x3 >= 13, name="constraint_2")
m.addConstr(6*x0 + 2*x1 + 5*x2 + 3*x3 >= 13, name="constraint_3")
m.addConstr(10*x2 - 2*x3 >= 0, name="constraint_4")
m.addConstr(4*x0 - x2 >= 0, name="constraint_5")
m.addConstr(6*x0 + 2*x1 + 5*x2 <= 44, name="constraint_6")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Hours worked by Laura: {x0.varValue}")
    print(f"Hours worked by Bobby: {x1.varValue}")
    print(f"Hours worked by Dale: {x2.varValue}")
    print(f"Hours worked by George: {x3.varValue}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("The model is infeasible.")
```