## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic representation of the problem, including the variables, objective function, and constraints.

## Step 2: Identify the variables and their corresponding symbolic notation
The variables are 'fruit salads', 'pickles', and 'hot dogs', which we will denote as $x_0$, $x_1$, and $x_2$ respectively.

## Step 3: Define the objective function in symbolic notation
The objective function to minimize is $3.82x_0 + 7.34x_1 + 1.02x_2$.

## Step 4: List the constraints in symbolic notation
The constraints are:
- $26x_0 + 29x_1 + 6x_2 \leq 367$ (grams of fat)
- $x_0 + 26x_1 + 13x_2 \leq 512$ (grams of carbohydrates)
- $19x_0 + 28x_1 + 5x_2 \leq 447$ (umami index)
- $25x_0 + x_1 + 4x_2 \leq 389$ (dollar cost)
- $26x_0 + 6x_2 \geq 40$ (grams of fat from fruit salads and hot dogs)
- $29x_1 + 6x_2 \geq 107$ (grams of fat from pickles and hot dogs)
- $26x_0 + 29x_1 + 6x_2 \geq 107$ (grams of fat from all)
- $x_0 + 13x_2 \geq 168$ (grams of carbohydrates from fruit salads and hot dogs)
- $x_0 + 26x_1 \geq 97$ (grams of carbohydrates from fruit salads and pickles)
- $x_0 + 26x_1 + 13x_2 \geq 97$ (grams of carbohydrates from all)
- $19x_0 + 5x_2 \geq 119$ (umami index from fruit salads and hot dogs)
- $19x_0 + 28x_1 + 5x_2 \geq 106$ (umami index from all)
- $25x_0 + 4x_2 \geq 109$ (dollar cost from fruit salads and hot dogs)
- $25x_0 + x_1 \geq 51$ (dollar cost from fruit salads and pickles)
- $x_1 + 4x_2 \geq 106$ (dollar cost from pickles and hot dogs)
- $25x_0 + x_1 + 4x_2 \geq 122$ (dollar cost from all)
- $-10x_0 + 7x_1 \geq 0$
- $-4x_0 + 5x_2 \geq 0$
- $4x_1 - 8x_2 \geq 0$
- $x_0 + 26x_1 \leq 440$ (grams of carbohydrates from fruit salads and pickles)
- $28x_1 + 5x_2 \leq 375$ (umami index from pickles and hot dogs)
- $19x_0 + 28x_1 \leq 286$ (umami index from fruit salads and pickles)
- $25x_0 + 4x_2 \leq 277$ (dollar cost from fruit salads and hot dogs)

## Step 5: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [('x0', 'fruit salads'), ('x1', 'pickles'), ('x2', 'hot dogs')],
    'objective_function': '3.82x0 + 7.34x1 + 1.02x2',
    'constraints': [
        '26x0 + 29x1 + 6x2 <= 367',
        'x0 + 26x1 + 13x2 <= 512',
        '19x0 + 28x1 + 5x2 <= 447',
        '25x0 + x1 + 4x2 <= 389',
        '26x0 + 6x2 >= 40',
        '29x1 + 6x2 >= 107',
        '26x0 + 29x1 + 6x2 >= 107',
        'x0 + 13x2 >= 168',
        'x0 + 26x1 >= 97',
        'x0 + 26x1 + 13x2 >= 97',
        '19x0 + 5x2 >= 119',
        '19x0 + 28x1 + 5x2 >= 106',
        '25x0 + 4x2 >= 109',
        '25x0 + x1 >= 51',
        'x1 + 4x2 >= 106',
        '25x0 + x1 + 4x2 >= 122',
        '-10x0 + 7x1 >= 0',
        '-4x0 + 5x2 >= 0',
        '4x1 - 8x2 >= 0',
        'x0 + 26x1 <= 440',
        '28x1 + 5x2 <= 375',
        '19x0 + 28x1 <= 286',
        '25x0 + 4x2 <= 277'
    ]
}
```

## Step 6: Write the Gurobi code to solve the optimization problem
```python
import gurobi

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

# Define the variables
x0 = model.addVar(name="fruit_salads", lb=0)
x1 = model.addVar(name="pickles", lb=0)
x2 = model.addVar(name="hot_dogs", lb=0)

# Define the objective function
model.setObjective(3.82 * x0 + 7.34 * x1 + 1.02 * x2, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(26 * x0 + 29 * x1 + 6 * x2 <= 367)
model.addConstr(x0 + 26 * x1 + 13 * x2 <= 512)
model.addConstr(19 * x0 + 28 * x1 + 5 * x2 <= 447)
model.addConstr(25 * x0 + x1 + 4 * x2 <= 389)
model.addConstr(26 * x0 + 6 * x2 >= 40)
model.addConstr(29 * x1 + 6 * x2 >= 107)
model.addConstr(26 * x0 + 29 * x1 + 6 * x2 >= 107)
model.addConstr(x0 + 13 * x2 >= 168)
model.addConstr(x0 + 26 * x1 >= 97)
model.addConstr(x0 + 26 * x1 + 13 * x2 >= 97)
model.addConstr(19 * x0 + 5 * x2 >= 119)
model.addConstr(19 * x0 + 28 * x1 + 5 * x2 >= 106)
model.addConstr(25 * x0 + 4 * x2 >= 109)
model.addConstr(25 * x0 + x1 >= 51)
model.addConstr(x1 + 4 * x2 >= 106)
model.addConstr(25 * x0 + x1 + 4 * x2 >= 122)
model.addConstr(-10 * x0 + 7 * x1 >= 0)
model.addConstr(-4 * x0 + 5 * x2 >= 0)
model.addConstr(4 * x1 - 8 * x2 >= 0)
model.addConstr(x0 + 26 * x1 <= 440)
model.addConstr(28 * x1 + 5 * x2 <= 375)
model.addConstr(19 * x0 + 28 * x1 <= 286)
model.addConstr(25 * x0 + 4 * x2 <= 277)

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", model.objval)
    print("Fruit Salads: ", x0.varValue)
    print("Pickles: ", x1.varValue)
    print("Hot Dogs: ", x2.varValue)
else:
    print("The model is infeasible")
```