## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and their corresponding natural language objects:
- $x_0$ = lemons
- $x_1$ = corn cobs
- $x_2$ = apples
- $x_3$ = ham sandwiches

## Step 2: Formulate the objective function
The objective function to maximize is: $2x_0 + 5x_1 + 7x_2 + 2x_3$

## Step 3: List all constraints
### Fiber constraints
- $10x_0 + 10x_1 + 25x_2 + 8x_3 \leq 475$ (total fiber)
- $10x_1 + 25x_2 \leq 240$ (fiber from corn cobs and apples)
- $25x_2 + 8x_3 \leq 447$ (fiber from apples and ham sandwiches)
- $10x_0 + 8x_3 \leq 130$ (fiber from lemons and ham sandwiches)
- $10x_0 + 25x_2 \leq 346$ (fiber from lemons and apples)
- $10x_0 + 10x_1 + 25x_2 + 8x_3 \leq 346$ (total fiber from all)

### Umami index constraints
- $24x_0 + 17x_2 + 5x_3 \geq 70$ 
- $24x_0 + 16x_1 + 17x_2 \geq 70$ 
- $24x_0 + 16x_1 + 5x_3 \geq 70$ 
- $24x_0 + 17x_2 + 5x_3 \geq 69$ 
- $24x_0 + 16x_1 + 17x_2 \geq 69$ 
- $24x_0 + 16x_1 + 5x_3 \geq 69$ 
- $24x_0 + 17x_2 + 5x_3 \geq 72$ 
- $24x_0 + 16x_1 + 17x_2 \geq 72$ 
- $24x_0 + 16x_1 + 5x_3 \geq 72$ 
- $24x_0 + 16x_1 \leq 270$ 
- $24x_0 + 17x_2 \leq 197$ 
- $16x_1 + 5x_3 \leq 74$ 
- $16x_1 + 17x_2 \leq 162$ 
- $16x_1 + 17x_2 + 5x_3 \leq 219$ 
- $24x_0 + 16x_1 + 17x_2 + 5x_3 \leq 219$

### Variable constraints
- $x_0$ is continuous
- $x_1$ is integer
- $x_2$ is integer
- $x_3$ is continuous

## Step 4: Provide symbolic representation
```json
{
    'sym_variables': [
        ['x0', 'lemons'], 
        ['x1', 'corn cobs'], 
        ['x2', 'apples'], 
        ['x3', 'ham sandwiches']
    ], 
    'objective_function': '2*x0 + 5*x1 + 7*x2 + 2*x3', 
    'constraints': [
        '10*x0 + 10*x1 + 25*x2 + 8*x3 <= 475',
        '10*x1 + 25*x2 <= 240',
        '25*x2 + 8*x3 <= 447',
        '10*x0 + 8*x3 <= 130',
        '10*x0 + 25*x2 <= 346',
        '10*x0 + 10*x1 + 25*x2 + 8*x3 <= 346',
        '24*x0 + 17*x2 + 5*x3 >= 70',
        '24*x0 + 16*x1 + 17*x2 >= 70',
        '24*x0 + 16*x1 + 5*x3 >= 70',
        '24*x0 + 17*x2 + 5*x3 >= 69',
        '24*x0 + 16*x1 + 17*x2 >= 69',
        '24*x0 + 16*x1 + 5*x3 >= 69',
        '24*x0 + 17*x2 + 5*x3 >= 72',
        '24*x0 + 16*x1 + 17*x2 >= 72',
        '24*x0 + 16*x1 + 5*x3 >= 72',
        '24*x0 + 16*x1 <= 270',
        '24*x0 + 17*x2 <= 197',
        '16*x1 + 5*x3 <= 74',
        '16*x1 + 17*x2 <= 162',
        '16*x1 + 17*x2 + 5*x3 <= 219',
        '24*x0 + 16*x1 + 17*x2 + 5*x3 <= 219'
    ]
}
```

## Step 5: Implement the optimization problem using Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="lemons", lb=0, ub=gp.GRB.INFINITY)
x1 = m.addVar(name="corn_cobs", lb=0, vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="apples", lb=0, vtype=gp.GRB.INTEGER)
x3 = m.addVar(name="ham_sandwiches", lb=0)

# Define the objective function
m.setObjective(2*x0 + 5*x1 + 7*x2 + 2*x3, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(10*x0 + 10*x1 + 25*x2 + 8*x3 <= 475, name="total_fiber")
m.addConstr(10*x1 + 25*x2 <= 240, name="fiber_corn_apples")
m.addConstr(25*x2 + 8*x3 <= 447, name="fiber_apples_sandwiches")
m.addConstr(10*x0 + 8*x3 <= 130, name="fiber_lemons_sandwiches")
m.addConstr(10*x0 + 25*x2 <= 346, name="fiber_lemons_apples")
m.addConstr(10*x0 + 10*x1 + 25*x2 + 8*x3 <= 346, name="total_fiber_limit")

m.addConstr(24*x0 + 17*x2 + 5*x3 >= 70, name="umami_lemons_apples_sandwiches")
m.addConstr(24*x0 + 16*x1 + 17*x2 >= 70, name="umami_lemons_corn_apples")
m.addConstr(24*x0 + 16*x1 + 5*x3 >= 70, name="umami_lemons_corn_sandwiches")
m.addConstr(24*x0 + 17*x2 + 5*x3 >= 69, name="umami_lemons_apples_sandwiches_69")
m.addConstr(24*x0 + 16*x1 + 17*x2 >= 69, name="umami_lemons_corn_apples_69")
m.addConstr(24*x0 + 16*x1 + 5*x3 >= 69, name="umami_lemons_corn_sandwiches_69")
m.addConstr(24*x0 + 17*x2 + 5*x3 >= 72, name="umami_lemons_apples_sandwiches_72")
m.addConstr(24*x0 + 16*x1 + 17*x2 >= 72, name="umami_lemons_corn_apples_72")
m.addConstr(24*x0 + 16*x1 + 5*x3 >= 72, name="umami_lemons_corn_sandwiches_72")

m.addConstr(24*x0 + 16*x1 <= 270, name="umami_lemons_corn")
m.addConstr(24*x0 + 17*x2 <= 197, name="umami_lemons_apples")
m.addConstr(16*x1 + 5*x3 <= 74, name="umami_corn_sandwiches")
m.addConstr(16*x1 + 17*x2 <= 162, name="umami_corn_apples")
m.addConstr(16*x1 + 17*x2 + 5*x3 <= 219, name="umami_corn_apples_sandwiches")
m.addConstr(24*x0 + 16*x1 + 17*x2 + 5*x3 <= 219, name="umami_total")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Lemons: ", x0.varValue)
    print("Corn Cobs: ", x1.varValue)
    print("Apples: ", x2.varValue)
    print("Ham Sandwiches: ", x3.varValue)
else:
    print("The model is infeasible")
```