To address the problem described, we first need to identify the variables and constraints involved. The variables are 'hours worked by Hank' (let's denote this as $x_1$) and 'hours worked by Paul' (denoted as $x_2$). 

The objective function is given as minimizing $8x_1 + x_2$, subject to several constraints involving the dollar cost per hour and productivity ratings of Hank and Paul, along with additional linear constraints.

Here's a breakdown of how we can represent this problem symbolically:

- **Variables**: 
  - $x_1$: hours worked by Hank
  - $x_2$: hours worked by Paul

- **Objective Function**:
  - Minimize: $8x_1 + x_2$

- **Constraints**:
  1. Total combined dollar cost per hour from hours worked by Hank and Paul should be at minimum 37: $1.88x_1 + 5.85x_2 \geq 37$
  2. The total combined productivity rating from hours worked by Hank and Paul should be at minimum 40: $0.66x_1 + 2.36x_2 \geq 40$
  3. $10x_1 - 10x_2 \geq 0$
  4. The total combined dollar cost per hour from hours worked by Hank and Paul has to be less than or equal to 96: $1.88x_1 + 5.85x_2 \leq 96$
  5. The total combined productivity rating from hours worked by Hank plus hours worked by Paul must be 165 at a maximum: $0.66x_1 + 2.36x_2 \leq 165$
  6. Non-fractional (integer) amount of hours worked by Hank: $x_1$ is an integer
  7. An integer number of hours worked by Paul: $x_2$ is an integer

In symbolic representation, this problem can be summarized as:

```json
{
  'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Paul')],
  'objective_function': 'Minimize: 8*x1 + 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 is an integer',
    'x2 is an integer'
  ]
}
```

Now, let's implement this problem in Gurobi:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Add variables to the model
x1 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Hank")
x2 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Paul")

# Set the objective function
m.setObjective(8*x1 + x2, GRB.MINIMIZE)

# Add constraints to the model
m.addConstr(1.88*x1 + 5.85*x2 >= 37, "dollar_cost_min")
m.addConstr(0.66*x1 + 2.36*x2 >= 40, "productivity_rating_min")
m.addConstr(10*x1 - 10*x2 >= 0, "balance_constraint")
m.addConstr(1.88*x1 + 5.85*x2 <= 96, "dollar_cost_max")
m.addConstr(0.66*x1 + 2.36*x2 <= 165, "productivity_rating_max")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Hank: {x1.x}")
    print(f"Hours worked by Paul: {x2.x}")
else:
    print("No optimal solution found")
```