## Problem Description and Formulation

The problem is an optimization problem with two variables: `bananas` and `granola bars`. The objective is to maximize the function:

\[ 1 \times \text{bananas} + 8 \times \text{granola bars} \]

subject to several constraints:

1. The cost of bananas is $14 each, and granola bars are $8 each.
2. The healthiness rating of bananas is 11 each, and granola bars are 9 each.
3. The total spending on bananas and granola bars must be at least $38.
4. The total healthiness rating must be greater than or equal to 145.
5. The constraint \(-4 \times \text{bananas} + 9 \times \text{granola bars} \geq 0\) must be satisfied.
6. The total spending on bananas and granola bars must not exceed $166.
7. The total healthiness rating must not exceed 335.
8. Bananas can be a non-whole number, but granola bars must be a whole number.

## Gurobi Code Formulation

Given the problem description, we can formulate the Gurobi code as follows:

```python
import gurobi

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

    # Define the variables
    bananas = model.addVar(name="bananas", lb=0)  # No lower bound specified, assuming 0
    granola_bars = model.addVar(name="granola_bars", lb=0, integrality=gurobi.GRB.INTEGER)

    # Objective function: Maximize 1 * bananas + 8 * granola_bars
    model.setObjective(bananas + 8 * granola_bars, gurobi.GRB.MAXIMIZE)

    # Constraints
    # Cost of bananas and granola bars: 14 * bananas + 8 * granola_bars >= 38
    model.addConstr(14 * bananas + 8 * granola_bars >= 38, name="min_spending")

    # Healthiness rating: 11 * bananas + 9 * granola_bars >= 145
    model.addConstr(11 * bananas + 9 * granola_bars >= 145, name="min_healthiness")

    # Constraint: -4 * bananas + 9 * granola_bars >= 0
    model.addConstr(-4 * bananas + 9 * granola_bars >= 0, name="special_constraint")

    # Maximum spending: 14 * bananas + 8 * granola_bars <= 166
    model.addConstr(14 * bananas + 8 * granola_bars <= 166, name="max_spending")

    # Maximum healthiness rating: 11 * bananas + 9 * granola_bars <= 335
    model.addConstr(11 * bananas + 9 * granola_bars <= 335, name="max_healthiness")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution found.")
        print(f"Bananas: {bananas.varValue}")
        print(f"Granola Bars: {granola_bars.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```