## Step 1: Define the optimization problem
The problem requires minimizing an objective function subject to several constraints. The objective function to minimize is $5.22 \times \text{apple pies} \times \text{chicken thighs} + 7.67 \times \text{apple pies} \times \text{eggs} + 5.72 \times \text{chicken thighs} \times \text{hamburgers} + 3.72 \times \text{chicken thighs} \times \text{eggs} + 8.75 \times \text{apple pies} + 4.07 \times \text{eggs}$.

## 2: List all variables and their attributes
Variables: 
- apple pies
- chicken thighs
- hamburgers
- eggs

Attributes (resources):
- $r0$: healthiness rating
  - apple pies: 2.59
  - chicken thighs: 5.76
  - hamburgers: 4.74
  - eggs: 3.35
  - Upper bound: 118

- $r1$: grams of protein
  - apple pies: 9.41
  - chicken thighs: 6.55
  - hamburgers: 0.76
  - eggs: 4.79

## 3: Formulate constraints
1. Total healthiness rating from chicken thighs and hamburgers $\geq 17$: $5.76 \times \text{chicken thighs} + 4.74 \times \text{hamburgers} \geq 17$
2. Total healthiness rating from hamburgers squared plus eggs squared $\geq 26$: $(4.74 \times \text{hamburgers})^2 + (3.35 \times \text{eggs})^2 \geq 26$
3. Total healthiness rating from all $\geq 26$: $2.59 \times \text{apple pies} + 5.76 \times \text{chicken thighs} + 4.74 \times \text{hamburgers} + 3.35 \times \text{eggs} \geq 26$
4. Total grams of protein from apple pies and chicken thighs $\geq 41$: $9.41 \times \text{apple pies} + 6.55 \times \text{chicken thighs} \geq 41$
5. Total grams of protein from apple pies and hamburgers $\geq 53$: $9.41 \times \text{apple pies} + 0.76 \times \text{hamburgers} \geq 53$
6. Total grams of protein from apple pies and eggs $\geq 51$: $9.41 \times \text{apple pies} + 4.79 \times \text{eggs} \geq 51$
7. Total grams of protein from hamburgers and eggs $\geq 42$: $0.76 \times \text{hamburgers} + 4.79 \times \text{eggs} \geq 42$
8. Total grams of protein from chicken thighs and hamburgers $\geq 27$: $6.55 \times \text{chicken thighs} + 0.76 \times \text{hamburgers} \geq 27$
9. Total grams of protein from chicken thighs squared plus eggs squared $\geq 58$: $(6.55 \times \text{chicken thighs})^2 + (4.79 \times \text{eggs})^2 \geq 58$
10. Total grams of protein from apple pies squared plus chicken thighs squared plus eggs squared $\geq 32$: $(9.41 \times \text{apple pies})^2 + (6.55 \times \text{chicken thighs})^2 + (4.79 \times \text{eggs})^2 \geq 32$
11. Total grams of protein from all $\geq 32$: $9.41 \times \text{apple pies} + 6.55 \times \text{chicken thighs} + 0.76 \times \text{hamburgers} + 4.79 \times \text{eggs} \geq 32$
12. $-3 \times \text{apple pies} + 6 \times \text{eggs} \geq 0$
13. $-10 \times \text{apple pies} + 6 \times \text{chicken thighs} \geq 0$
14. Total healthiness rating from hamburgers and eggs $\leq 33$: $4.74 \times \text{hamburgers} + 3.35 \times \text{eggs} \leq 33$
15. Total healthiness rating from apple pies, chicken thighs, and hamburgers $\leq 61$: $2.59 \times \text{apple pies} + 5.76 \times \text{chicken thighs} + 4.74 \times \text{hamburgers} \leq 61$
16. Total healthiness rating from apple pies, chicken thighs, and eggs $\leq 48$: $2.59 \times \text{apple pies} + 5.76 \times \text{chicken thighs} + 3.35 \times \text{eggs} \leq 48$
17. Total healthiness rating from chicken thighs, hamburgers, and eggs $\leq 50$: $5.76 \times \text{chicken thighs} + 4.74 \times \text{hamburgers} + 3.35 \times \text{eggs} \leq 50$
18. Total grams of protein from apple pies and eggs $\leq 195$: $9.41 \times \text{apple pies} + 4.79 \times \text{eggs} \leq 195$

## 4: Implement in Gurobi
```python
import gurobi

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

    # Define variables
    apple_pies = model.addVar(name="apple_pies", lb=0)  # No lower bound specified, assuming 0
    chicken_thighs = model.addVar(name="chicken_thighs", lb=0, integrality=gurobi.GRB.INTEGER)
    hamburgers = model.addVar(name="hamburgers", lb=0)
    eggs = model.addVar(name="eggs", lb=0)

    # Objective function
    model.setObjective(5.22 * apple_pies * chicken_thighs + 7.67 * apple_pies * eggs + 5.72 * chicken_thighs * hamburgers + 
                       3.72 * chicken_thighs * eggs + 8.75 * apple_pies + 4.07 * eggs, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(5.76 * chicken_thighs + 4.74 * hamburgers >= 17)
    model.addConstr((4.74 * hamburgers)**2 + (3.35 * eggs)**2 >= 26)
    model.addConstr(2.59 * apple_pies + 5.76 * chicken_thighs + 4.74 * hamburgers + 3.35 * eggs >= 26)
    model.addConstr(9.41 * apple_pies + 6.55 * chicken_thighs >= 41)
    model.addConstr(9.41 * apple_pies + 0.76 * hamburgers >= 53)
    model.addConstr(9.41 * apple_pies + 4.79 * eggs >= 51)
    model.addConstr(0.76 * hamburgers + 4.79 * eggs >= 42)
    model.addConstr(6.55 * chicken_thighs + 0.76 * hamburgers >= 27)
    model.addConstr((6.55 * chicken_thighs)**2 + (4.79 * eggs)**2 >= 58)
    model.addConstr((9.41 * apple_pies)**2 + (6.55 * chicken_thighs)**2 + (4.79 * eggs)**2 >= 32)
    model.addConstr(9.41 * apple_pies + 6.55 * chicken_thighs + 0.76 * hamburgers + 4.79 * eggs >= 32)
    model.addConstr(-3 * apple_pies + 6 * eggs >= 0)
    model.addConstr(-10 * apple_pies + 6 * chicken_thighs >= 0)
    model.addConstr(4.74 * hamburgers + 3.35 * eggs <= 33)
    model.addConstr(2.59 * apple_pies + 5.76 * chicken_thighs + 4.74 * hamburgers <= 61)
    model.addConstr(2.59 * apple_pies + 5.76 * chicken_thighs + 3.35 * eggs <= 48)
    model.addConstr(5.76 * chicken_thighs + 4.74 * hamburgers + 3.35 * eggs <= 50)
    model.addConstr(9.41 * apple_pies + 4.79 * eggs <= 195)
    model.addConstr(2.59 * apple_pies <= 118)  # Healthiness rating upper bound

    # Update model
    model.update()

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Apple pies: ", apple_pies.varValue)
        print("Chicken thighs: ", chicken_thighs.varValue)
        print("Hamburgers: ", hamburgers.varValue)
        print("Eggs: ", eggs.varValue)
    else:
        print("No solution found")

optimize_problem()
```