## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

- Let $x_1$ represent 'peanutbutter sandwiches', $x_2$ represent 'tomatoes', and $x_3$ represent 'ravioli'.

## Step 2: Translate the problem into a symbolic optimization problem
The objective function to maximize is $6x_1 + 7x_2 + 5x_3$.

## Step 3: List all the constraints
The constraints given are:
1. $18x_1 + 23x_2 + 18x_3 \geq 84$ (total combined umami index minimum)
2. $20x_1 + 12x_3 \leq 172$ (carbohydrates from peanutbutter sandwiches and ravioli)
3. $32x_2 + 12x_3 \leq 182$ (carbohydrates from tomatoes and ravioli)
4. $20x_1 + 32x_2 + 12x_3 \leq 182$ (total carbohydrates)
5. $18x_1 + 23x_2 \leq 229$ (umami index from peanutbutter sandwiches and tomatoes)
6. $23x_2 + 18x_3 \leq 164$ (umami index from tomatoes and ravioli)
7. $18x_1 + 23x_2 + 18x_3 \leq 164$ (total umami index maximum)

## 4: Define the symbolic representation in JSON format
```json
{
    'sym_variables': [('x1', 'peanutbutter sandwiches'), ('x2', 'tomatoes'), ('x3', 'ravioli')],
    'objective_function': '6*x1 + 7*x2 + 5*x3',
    'constraints': [
        '18*x1 + 23*x2 + 18*x3 >= 84',
        '20*x1 + 12*x3 <= 172',
        '32*x2 + 12*x3 <= 182',
        '20*x1 + 32*x2 + 12*x3 <= 182',
        '18*x1 + 23*x2 <= 229',
        '23*x2 + 18*x3 <= 164',
        '18*x1 + 23*x2 + 18*x3 <= 164'
    ]
}
```

## 5: Implement the optimization problem using Gurobi
Now, let's implement this problem using Gurobi in Python.

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='peanutbutter_sandwiches', lb=0)  # No lower bound specified, assuming 0
    x2 = model.addVar(name='tomatoes', lb=0)  # No lower bound specified, assuming 0
    x3 = model.addVar(name='ravioli', lb=0)  # No lower bound specified, assuming 0

    # Define the objective function
    model.setObjective(6 * x1 + 7 * x2 + 5 * x3, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(18 * x1 + 23 * x2 + 18 * x3 >= 84)  # Minimum umami index
    model.addConstr(20 * x1 + 12 * x3 <= 172)  # Carbohydrates from peanutbutter sandwiches and ravioli
    model.addConstr(32 * x2 + 12 * x3 <= 182)  # Carbohydrates from tomatoes and ravioli
    model.addConstr(20 * x1 + 32 * x2 + 12 * x3 <= 182)  # Total carbohydrates
    model.addConstr(18 * x1 + 23 * x2 <= 229)  # Umami index from peanutbutter sandwiches and tomatoes
    model.addConstr(23 * x2 + 18 * x3 <= 164)  # Umami index from tomatoes and ravioli
    model.addConstr(18 * x1 + 23 * x2 + 18 * x3 <= 164)  # Total umami index maximum

    # Optimize the model
    model.optimize()

    # Print the status
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print(f"Peanutbutter sandwiches: {x1.varValue}")
        print(f"Tomatoes: {x2.varValue}")
        print(f"Ravioli: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.Status.INFEASIBLE:
        print("No feasible solution exists.")
    else:
        print("Optimization failed.")

solve_optimization_problem()
```