## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'oranges' and 'cheeseburgers', which can be represented symbolically as $x_1$ and $x_2$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $8x_1 + 8x_2$.

## 3: Define the constraints in symbolic notation
The constraints given are:
- $10.8x_1 + 3.74x_2 \geq 20$ (minimum umami index)
- $7.17x_1 + 0.2x_2 \geq 46$ (minimum grams of protein)
- $2x_1 - 3x_2 \geq 0$ (relationship between oranges and cheeseburgers)
- $10.8x_1 + 3.74x_2 \leq 34$ (maximum umami index)
- $7.17x_1 + 0.2x_2 \leq 156$ (maximum grams of protein)
- $x_1$ is an integer (oranges must be a whole number)
- $x_2$ can be any real number (cheeseburgers can be non-integer)

## 4: Create a symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'oranges'), ('x2', 'cheeseburgers')],
    'objective_function': '8*x1 + 8*x2',
    'constraints': [
        '10.8*x1 + 3.74*x2 >= 20',
        '7.17*x1 + 0.2*x2 >= 46',
        '2*x1 - 3*x2 >= 0',
        '10.8*x1 + 3.74*x2 <= 34',
        '7.17*x1 + 0.2*x2 <= 156',
        'x1 >= 0 and x1 is an integer',
        'x2 >= 0'
    ]
}
```

## 5: Write the Gurobi code to solve the optimization problem
```python
import gurobi

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

    # Define the variables
    oranges = model.addVar(name="oranges", vtype=gurobi.GRB.INTEGER)
    cheeseburgers = model.addVar(name="cheeseburgers")

    # Set the objective function
    model.setObjective(8 * oranges + 8 * cheeseburgers, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(10.8 * oranges + 3.74 * cheeseburgers >= 20)  # minimum umami index
    model.addConstr(7.17 * oranges + 0.2 * cheeseburgers >= 46)  # minimum grams of protein
    model.addConstr(2 * oranges - 3 * cheeseburgers >= 0)  # relationship between oranges and cheeseburgers
    model.addConstr(10.8 * oranges + 3.74 * cheeseburgers <= 34)  # maximum umami index
    model.addConstr(7.17 * oranges + 0.2 * cheeseburgers <= 156)  # maximum grams of protein

    # Optimize the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Oranges: {oranges.varValue}")
        print(f"Cheeseburgers: {cheeseburgers.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has no optimal solution.")

solve_optimization_problem()
```