## Step 1: Define the optimization problem
The problem requires maximizing the objective function: $5 \times \text{hamburgers} + 1 \times \text{eggs} + 7 \times \text{protein bars} + 6 \times \text{pickles}$, subject to various constraints on cost, umami index, and healthiness rating.

## 2: List all variables and their attributes
- Variables: ['hamburgers', 'eggs', 'protein bars', 'pickles']
- Attributes:
  - 'r0': {'description': 'dollar cost', 'upper_bound': 159, 'x0': 11, 'x1': 7, 'x2': 7, 'x3': 1}
  - 'r1': {'description': 'umami index', 'upper_bound': 87, 'x0': 8, 'x1': 14, 'x2': 1, 'x3': 4}
  - 'r2': {'description': 'healthiness rating', 'upper_bound': 132, 'x0': 5, 'x1': 11, 'x2': 10, 'x3': 7}

## 3: Formulate the objective function
The objective function to maximize is: $5 \times \text{hamburgers} + 1 \times \text{eggs} + 7 \times \text{protein bars} + 6 \times \text{pickles}$.

## 4: Define the constraints
### Cost Constraints
- $11 \times \text{hamburgers} + 7 \times \text{eggs} + 7 \times \text{protein bars} + 1 \times \text{pickles} \leq 159$
- $7 \times \text{eggs} + 1 \times \text{pickles} \geq 14$
- $7 \times \text{eggs} + 7 \times \text{protein bars} \geq 34$
- $11 \times \text{hamburgers} + 7 \times \text{eggs} \geq 22$
- $11 \times \text{hamburgers} + 1 \times \text{pickles} \leq 68$
- $7 \times \text{eggs} + 1 \times \text{pickles} \leq 46$
- $11 \times \text{hamburgers} + 7 \times \text{protein bars} + 1 \times \text{pickles} \leq 61$
- $11 \times \text{hamburgers} + 7 \times \text{eggs} + 7 \times \text{protein bars} \leq 39$
- $11 \times \text{hamburgers} + 7 \times \text{eggs} + 7 \times \text{protein bars} + 1 \times \text{pickles} \leq 39$

### Umami Index Constraints
- $1 \times \text{protein bars} + 4 \times \text{pickles} \geq 20$
- $14 \times \text{eggs} + 1 \times \text{protein bars} \geq 14$
- $8 \times \text{hamburgers} + 4 \times \text{pickles} \geq 9$
- $14 \times \text{eggs} + 1 \times \text{protein bars} + 4 \times \text{pickles} \geq 21$
- $8 \times \text{hamburgers} + 1 \times \text{protein bars} + 4 \times \text{pickles} \geq 21$
- $8 \times \text{hamburgers} + 14 \times \text{eggs} + 1 \times \text{protein bars} \geq 21$
- $8 \times \text{hamburgers} + 14 \times \text{eggs} + 4 \times \text{pickles} \geq 21$
- $14 \times \text{eggs} + 1 \times \text{protein bars} + 4 \times \text{pickles} \geq 10$
- $8 \times \text{hamburgers} + 1 \times \text{protein bars} + 4 \times \text{pickles} \geq 10$
- $8 \times \text{hamburgers} + 14 \times \text{eggs} + 1 \times \text{protein bars} \geq 10$
- $8 \times \text{hamburgers} + 14 \times \text{eggs} + 4 \times \text{pickles} \geq 10$
- $14 \times \text{eggs} + 1 \times \text{protein bars} + 4 \times \text{pickles} \geq 15$
- $8 \times \text{hamburgers} + 1 \times \text{protein bars} + 4 \times \text{pickles} \geq 15$
- $8 \times \text{hamburgers} + 14 \times \text{eggs} + 1 \times \text{protein bars} \geq 15$
- $8 \times \text{hamburgers} + 14 \times \text{eggs} + 4 \times \text{pickles} \geq 15$
- $14 \times \text{eggs} + 1 \times \text{protein bars} + 4 \times \text{pickles} \leq 41$
- $8 \times \text{hamburgers} + 1 \times \text{protein bars} \leq 31$
- $1 \times \text{protein bars} + 4 \times \text{pickles} \leq 56$
- $8 \times \text{hamburgers} + 14 \times \text{eggs} \leq 40$
- $8 \times \text{hamburgers} + 14 \times \text{eggs} + 1 \times \text{protein bars} \leq 41$
- $8 \times \text{hamburgers} + 14 \times \text{eggs} + 1 \times \text{protein bars} + 4 \times \text{pickles} \leq 41$

### Healthiness Rating Constraints
- $11 \times \text{eggs} + 7 \times \text{pickles} \geq 28$
- $5 \times \text{hamburgers} + 10 \times \text{protein bars} \geq 17$
- $5 \times \text{hamburgers} + 7 \times \text{pickles} \geq 28$
- $5 \times \text{hamburgers} + 11 \times \text{eggs} \leq 55$
- $5 \times \text{hamburgers} + 10 \times \text{protein bars} + 7 \times \text{pickles} \leq 99$
- $5 \times \text{hamburgers} + 11 \times \text{eggs} + 7 \times \text{pickles} \leq 101$
- $5 \times \text{hamburgers} + 11 \times \text{eggs} + 10 \times \text{protein bars} + 7 \times \text{pickles} \leq 101$

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

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

    # Define variables
    hamburgers = model.addVar(name="hamburgers", lb=0)
    eggs = model.addVar(name="eggs", lb=0)
    protein_bars = model.addVar(name="protein_bars", lb=0)
    pickles = model.addVar(name="pickles", lb=0)

    # Objective function
    model.setObjective(5 * hamburgers + eggs + 7 * protein_bars + 6 * pickles, gurobi.GRB.MAXIMIZE)

    # Cost constraints
    model.addConstr(11 * hamburgers + 7 * eggs + 7 * protein_bars + pickles <= 159)
    model.addConstr(7 * eggs + pickles >= 14)
    model.addConstr(7 * eggs + 7 * protein_bars >= 34)
    model.addConstr(11 * hamburgers + 7 * eggs >= 22)
    model.addConstr(11 * hamburgers + pickles <= 68)
    model.addConstr(7 * eggs + pickles <= 46)
    model.addConstr(11 * hamburgers + 7 * protein_bars + pickles <= 61)
    model.addConstr(11 * hamburgers + 7 * eggs + 7 * protein_bars <= 39)
    model.addConstr(11 * hamburgers + 7 * eggs + 7 * protein_bars + pickles <= 39)

    # Umami index constraints
    model.addConstr(protein_bars + 4 * pickles >= 20)
    model.addConstr(14 * eggs + protein_bars >= 14)
    model.addConstr(8 * hamburgers + 4 * pickles >= 9)
    model.addConstr(14 * eggs + protein_bars + 4 * pickles >= 21)
    model.addConstr(8 * hamburgers + protein_bars + 4 * pickles >= 21)
    model.addConstr(8 * hamburgers + 14 * eggs + protein_bars >= 21)
    model.addConstr(8 * hamburgers + 14 * eggs + 4 * pickles >= 21)
    model.addConstr(14 * eggs + protein_bars + 4 * pickles >= 10)
    model.addConstr(8 * hamburgers + protein_bars + 4 * pickles >= 10)
    model.addConstr(8 * hamburgers + 14 * eggs + protein_bars >= 10)
    model.addConstr(8 * hamburgers + 14 * eggs + 4 * pickles >= 10)
    model.addConstr(14 * eggs + protein_bars + 4 * pickles >= 15)
    model.addConstr(8 * hamburgers + protein_bars + 4 * pickles >= 15)
    model.addConstr(8 * hamburgers + 14 * eggs + protein_bars >= 15)
    model.addConstr(8 * hamburgers + 14 * eggs + 4 * pickles >= 15)
    model.addConstr(14 * eggs + protein_bars + 4 * pickles <= 41)
    model.addConstr(8 * hamburgers + protein_bars <= 31)
    model.addConstr(protein_bars + 4 * pickles <= 56)
    model.addConstr(8 * hamburgers + 14 * eggs <= 40)
    model.addConstr(8 * hamburgers + 14 * eggs + protein_bars <= 41)
    model.addConstr(8 * hamburgers + 14 * eggs + protein_bars + 4 * pickles <= 41)

    # Healthiness rating constraints
    model.addConstr(11 * eggs + 7 * pickles >= 28)
    model.addConstr(5 * hamburgers + 10 * protein_bars >= 17)
    model.addConstr(5 * hamburgers + 7 * pickles >= 28)
    model.addConstr(5 * hamburgers + 11 * eggs <= 55)
    model.addConstr(5 * hamburgers + 10 * protein_bars + 7 * pickles <= 99)
    model.addConstr(5 * hamburgers + 11 * eggs + 7 * pickles <= 101)
    model.addConstr(5 * hamburgers + 11 * eggs + 10 * protein_bars + 7 * pickles <= 101)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hamburgers: ", hamburgers.varValue)
        print("Eggs: ", eggs.varValue)
        print("Protein Bars: ", protein_bars.varValue)
        print("Pickles: ", pickles.varValue)
    else:
        print("The model is infeasible")

optimize_problem()
```