## Step 1: Define the symbolic representation of the variables
The variables are 'oranges', 'apple pies', and 'hamburgers', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- $x_0$ : oranges
- $x_1$ : apple pies
- $x_2$ : hamburgers

## 3: Define the objective function in symbolic notation
The objective function to minimize is $2x_0 + 2x_1 + 5x_2$.

## 4: List the constraints in symbolic notation
The constraints are:
- $13x_1 + 6x_2 \geq 46$ (umami index from apple pies and hamburgers)
- $4x_0 + 13x_1 \geq 34$ (umami index from oranges and apple pies)
- $4x_0 + 6x_2 \geq 37$ (umami index from oranges and hamburgers)
- $4x_0 + 13x_1 + 6x_2 \geq 37$ (umami index from all)
- $8x_1 + 15x_2 \geq 30$ (grams of fat from apple pies and hamburgers)
- $13x_0 + 8x_1 \geq 43$ (grams of fat from oranges and apple pies)
- $13x_0 + 8x_1 + 15x_2 \geq 28$ (grams of fat from all, note: this is less restrictive than others and can be ignored if others are satisfied)
- $13x_0 + 15x_2 \geq 44$ (grams of carbohydrates from oranges and hamburgers)
- $15x_0 + 18x_1 \geq 22$ (grams of carbohydrates from oranges and apple pies)
- $15x_0 + 18x_1 + 18x_2 \geq 22$ (grams of carbohydrates from all)
- $-8x_0 + 8x_1 \geq 0$ (relationship between oranges and apple pies)
- $4x_0 + 13x_1 + 6x_2 \leq 154$ (total umami index)

## 5: Consider the bounds and types of variables
- $x_0$, $x_1$, $x_2$ are continuous variables.

## 6: Formulate the problem in Gurobi
We will use Gurobi to solve this linear programming problem.

## 7: Write the Gurobi code
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="oranges", lb=0, ub=gurobi.GRB.INFINITY)
x1 = m.addVar(name="apple_pies", lb=0, ub=gurobi.GRB.INFINITY)
x2 = m.addVar(name="hamburgers", lb=0, ub=gurobi.GRB.INFINITY)

# Define the objective function
m.setObjective(2*x0 + 2*x1 + 5*x2, gurobi.GRB.MINIMIZE)

# Add constraints
m.addConstr(13*x1 + 6*x2 >= 46, name="umami_ap_hb")
m.addConstr(4*x0 + 13*x1 >= 34, name="umami_o_ap")
m.addConstr(4*x0 + 6*x2 >= 37, name="umami_o_hb")
m.addConstr(4*x0 + 13*x1 + 6*x2 >= 37, name="umami_all")
m.addConstr(8*x1 + 15*x2 >= 30, name="fat_ap_hb")
m.addConstr(13*x0 + 8*x1 >= 43, name="fat_o_ap")
m.addConstr(13*x0 + 8*x1 + 15*x2 >= 28, name="fat_all")
m.addConstr(15*x0 + 18*x1 >= 22, name="carb_o_ap")
m.addConstr(13*x0 + 15*x2 >= 44, name="carb_o_hb")
m.addConstr(15*x0 + 18*x1 + 18*x2 >= 22, name="carb_all")
m.addConstr(-8*x0 + 8*x1 >= 0, name="oranges_apples")
m.addConstr(4*x0 + 13*x1 + 6*x2 <= 154, name="total_umami")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Oranges: ", x0.varValue)
    print("Apple Pies: ", x1.varValue)
    print("Hamburgers: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 8: Provide the symbolic representation in JSON format
```json
{
    "sym_variables": [
        ["x0", "oranges"],
        ["x1", "apple pies"],
        ["x2", "hamburgers"]
    ],
    "objective_function": "2*x0 + 2*x1 + 5*x2",
    "constraints": [
        "13*x1 + 6*x2 >= 46",
        "4*x0 + 13*x1 >= 34",
        "4*x0 + 6*x2 >= 37",
        "4*x0 + 13*x1 + 6*x2 >= 37",
        "8*x1 + 15*x2 >= 30",
        "13*x0 + 8*x1 >= 43",
        "13*x0 + 8*x1 + 15*x2 >= 28",
        "15*x0 + 18*x1 >= 22",
        "13*x0 + 15*x2 >= 44",
        "15*x0 + 18*x1 + 18*x2 >= 22",
        "-8*x0 + 8*x1 >= 0",
        "4*x0 + 13*x1 + 6*x2 <= 154"
    ]
}
```