## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank' and 'hours worked by Paul', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $8x_1 + 1x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
- The total combined dollar cost per hour from hours worked by Hank plus hours worked by Paul should be at minimum 37: $1.88x_1 + 5.85x_2 \geq 37$
- The total combined dollar cost per hour from hours worked by Hank plus hours worked by Paul must be 37 at minimum (same as the previous one): $1.88x_1 + 5.85x_2 \geq 37$
- The total combined productivity rating from hours worked by Hank, and hours worked by Paul should be 40 at minimum: $0.66x_1 + 2.36x_2 \geq 40$
- The total combined productivity rating from hours worked by Hank, and hours worked by Paul should be 40 at minimum (same as the previous one): $0.66x_1 + 2.36x_2 \geq 40$
- $10x_1 - 10x_2 \geq 0$
- The total combined dollar cost per hour from hours worked by Hank and hours worked by Paul has to be less than or equal to 96: $1.88x_1 + 5.85x_2 \leq 96$
- The total combined productivity rating from hours worked by Hank plus hours worked by Paul must be 165 at a maximum: $0.66x_1 + 2.36x_2 \leq 165$
- $x_1$ must be an integer
- $x_2$ must be an integer

## 4: Remove redundant constraints
Removing the redundant constraints, we have:
- $1.88x_1 + 5.85x_2 \geq 37$
- $0.66x_1 + 2.36x_2 \geq 40$
- $10x_1 - 10x_2 \geq 0$
- $1.88x_1 + 5.85x_2 \leq 96$
- $0.66x_1 + 2.36x_2 \leq 165$
- $x_1$ must be an integer
- $x_2$ must be an integer

## 5: Write the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Paul')],
'objective_function': '8*x1 + 1*x2',
'constraints': [
    '1.88*x1 + 5.85*x2 >= 37',
    '0.66*x1 + 2.36*x2 >= 40',
    '10*x1 - 10*x2 >= 0',
    '1.88*x1 + 5.85*x2 <= 96',
    '0.66*x1 + 2.36*x2 <= 165',
    'x1 % 1 == 0',
    'x2 % 1 == 0'
]
}
```

## 6: Convert the problem into Gurobi code
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='hours_worked_by_Hank', vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name='hours_worked_by_Paul', vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(8 * x1 + 1 * x2, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(1.88 * x1 + 5.85 * x2 >= 37)
    model.addConstr(0.66 * x1 + 2.36 * x2 >= 40)
    model.addConstr(10 * x1 - 10 * x2 >= 0)
    model.addConstr(1.88 * x1 + 5.85 * x2 <= 96)
    model.addConstr(0.66 * x1 + 2.36 * x2 <= 165)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Optimal solution found.')
        print(f'Hours worked by Hank: {x1.varValue}')
        print(f'Hours worked by Paul: {x2.varValue}')
        print(f'Objective function value: {model.objVal}')
    else:
        print('No optimal solution found.')

solve_optimization_problem()
```