To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining the variables, the objective function, and the constraints using mathematical notation.

Let's denote:
- \(x_0\) as the number of ham sandwiches,
- \(x_1\) as the number of corn cobs.

The objective function is to maximize \(5x_0 + 6x_1\).

The constraints are as follows:
1. Calcium constraint: \(10.81x_0 + 1.92x_1 \geq 28\) and \(10.81x_0 + 1.92x_1 \leq 38\).
2. Umami index constraint: \(11.85x_0 + 6.3x_1 \geq 22\) and \(11.85x_0 + 6.3x_1 \leq 61\).
3. Iron constraint: \(2.72x_0 + 8.06x_1 \geq 22\) and \(2.72x_0 + 8.06x_1 \leq 44\).
4. Mixed constraint: \(9x_0 - 2x_1 \geq 0\).
5. Integer constraints: \(x_0, x_1 \in \mathbb{Z}\).

Symbolic representation:
```json
{
    'sym_variables': [('x0', 'ham sandwiches'), ('x1', 'corn cobs')],
    'objective_function': '5*x0 + 6*x1',
    'constraints': [
        '10.81*x0 + 1.92*x1 >= 28',
        '10.81*x0 + 1.92*x1 <= 38',
        '11.85*x0 + 6.3*x1 >= 22',
        '11.85*x0 + 6.3*x1 <= 61',
        '2.72*x0 + 8.06*x1 >= 22',
        '2.72*x0 + 8.06*x1 <= 44',
        '9*x0 - 2*x1 >= 0'
    ]
}
```

Now, let's implement this problem using Gurobi in Python:
```python
from gurobipy import *

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

# Define the variables
x0 = m.addVar(vtype=GRB.INTEGER, name="ham_sandwiches")
x1 = m.addVar(vtype=GRB.INTEGER, name="corn_cobs")

# Set the objective function
m.setObjective(5*x0 + 6*x1, GRB.MAXIMIZE)

# Add constraints
m.addConstr(10.81*x0 + 1.92*x1 >= 28, "calcium_lower")
m.addConstr(10.81*x0 + 1.92*x1 <= 38, "calcium_upper")
m.addConstr(11.85*x0 + 6.3*x1 >= 22, "umami_lower")
m.addConstr(11.85*x0 + 6.3*x1 <= 61, "umami_upper")
m.addConstr(2.72*x0 + 8.06*x1 >= 22, "iron_lower")
m.addConstr(2.72*x0 + 8.06*x1 <= 44, "iron_upper")
m.addConstr(9*x0 - 2*x1 >= 0, "mixed_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Ham Sandwiches: {x0.x}")
    print(f"Corn Cobs: {x1.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found")
```