## Problem Description and Formulation

The problem is an optimization problem with the goal to maximize the objective function:

\[ 2 \times \text{hot dogs} + 9 \times \text{slices of pizza} + 6 \times \text{chicken drumsticks} \]

subject to various constraints on nutritional values and healthiness ratings.

## Constraints

The constraints can be summarized as follows:

1. **Nutritional Constraints:**
   - Fat: \(4.97 \times \text{hot dogs} + 0.94 \times \text{slices of pizza} + 0.45 \times \text{chicken drumsticks} \leq 53\)
   - Healthiness Rating: \(0.69 \times \text{hot dogs} + 3.25 \times \text{slices of pizza} + 1.51 \times \text{chicken drumsticks} \geq 11\)
   - Carbohydrates: \(2.7 \times \text{hot dogs} + 2.23 \times \text{slices of pizza} + 4.0 \times \text{chicken drumsticks} \leq 36\)
   - Protein: \(1.64 \times \text{hot dogs} + 3.26 \times \text{slices of pizza} + 4.61 \times \text{chicken drumsticks} \leq 35\)
   - Iron: \(4.59 \times \text{hot dogs} + 0.72 \times \text{slices of pizza} + 3.06 \times \text{chicken drumsticks} \leq 43\)

2. **Specific Constraints:**
   - Fat from slices of pizza and chicken drumsticks: \(0.94 \times \text{slices of pizza} + 0.45 \times \text{chicken drumsticks} \leq 37\)
   - Fat from hot dogs and chicken drumsticks: \(4.97 \times \text{hot dogs} + 0.45 \times \text{chicken drumsticks} \leq 41\)
   - Fat from hot dogs and slices of pizza: \(4.97 \times \text{hot dogs} + 0.94 \times \text{slices of pizza} \leq 28\)
   - Total Fat: \(4.97 \times \text{hot dogs} + 0.94 \times \text{slices of pizza} + 0.45 \times \text{chicken drumsticks} \leq 28\)

   - Healthiness Rating from hot dogs and chicken drumsticks: \(0.69 \times \text{hot dogs} + 1.51 \times \text{chicken drumsticks} \leq 41\)
   - Healthiness Rating from slices of pizza and chicken drumsticks: \(3.25 \times \text{slices of pizza} + 1.51 \times \text{chicken drumsticks} \leq 29\)
   - Total Healthiness Rating: \(0.69 \times \text{hot dogs} + 3.25 \times \text{slices of pizza} + 1.51 \times \text{chicken drumsticks} \leq 29\)

   - Carbohydrates from slices of pizza and chicken drumsticks: \(2.23 \times \text{slices of pizza} + 4.0 \times \text{chicken drumsticks} \leq 18\)
   - Total Carbohydrates: \(2.7 \times \text{hot dogs} + 2.23 \times \text{slices of pizza} + 4.0 \times \text{chicken drumsticks} \leq 18\)

   - Protein from slices of pizza and chicken drumsticks: \(3.26 \times \text{slices of pizza} + 4.61 \times \text{chicken drumsticks} \leq 22\)
   - Protein from hot dogs and chicken drumsticks: \(1.64 \times \text{hot dogs} + 4.61 \times \text{chicken drumsticks} \leq 26\)
   - Total Protein: \(1.64 \times \text{hot dogs} + 3.26 \times \text{slices of pizza} + 4.61 \times \text{chicken drumsticks} \leq 26\)

   - Iron from hot dogs and slices of pizza: \(4.59 \times \text{hot dogs} + 0.72 \times \text{slices of pizza} \leq 18\)
   - Iron from slices of pizza and chicken drumsticks: \(0.72 \times \text{slices of pizza} + 3.06 \times \text{chicken drumsticks} \leq 25\)
   - Total Iron: \(4.59 \times \text{hot dogs} + 0.72 \times \text{slices of pizza} + 3.06 \times \text{chicken drumsticks} \leq 25\)

## Gurobi Code

```python
import gurobi

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

    # Define variables
    hot_dogs = model.addVar(name="hot_dogs", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    slices_of_pizza = model.addVar(name="slices_of_pizza", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    chicken_drumsticks = model.addVar(name="chicken_drumsticks", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

    # Objective function
    model.setObjective(2 * hot_dogs + 9 * slices_of_pizza + 6 * chicken_drumsticks, gurobi.GRB.MAXIMIZE)

    # Constraints
    # Fat
    model.addConstr(4.97 * hot_dogs + 0.94 * slices_of_pizza + 0.45 * chicken_drumsticks <= 53, name="fat_constraint")
    model.addConstr(0.94 * slices_of_pizza + 0.45 * chicken_drumsticks <= 37, name="fat_slices_chicken_constraint")
    model.addConstr(4.97 * hot_dogs + 0.45 * chicken_drumsticks <= 41, name="fat_hotdogs_chicken_constraint")
    model.addConstr(4.97 * hot_dogs + 0.94 * slices_of_pizza <= 28, name="fat_hotdogs_slices_constraint")
    model.addConstr(4.97 * hot_dogs + 0.94 * slices_of_pizza + 0.45 * chicken_drumsticks <= 28, name="total_fat_constraint")

    # Healthiness Rating
    model.addConstr(0.69 * hot_dogs + 3.25 * slices_of_pizza + 1.51 * chicken_drumsticks >= 11, name="healthiness_rating_constraint")
    model.addConstr(0.69 * hot_dogs + 1.51 * chicken_drumsticks <= 41, name="healthiness_hotdogs_chicken_constraint")
    model.addConstr(3.25 * slices_of_pizza + 1.51 * chicken_drumsticks <= 29, name="healthiness_slices_chicken_constraint")
    model.addConstr(0.69 * hot_dogs + 3.25 * slices_of_pizza + 1.51 * chicken_drumsticks <= 29, name="total_healthiness_constraint")

    # Carbohydrates
    model.addConstr(2.7 * hot_dogs + 2.23 * slices_of_pizza + 4.0 * chicken_drumsticks <= 36, name="carbohydrates_constraint")
    model.addConstr(2.23 * slices_of_pizza + 4.0 * chicken_drumsticks <= 18, name="carbohydrates_slices_chicken_constraint")
    model.addConstr(2.7 * hot_dogs + 2.23 * slices_of_pizza + 4.0 * chicken_drumsticks <= 18, name="total_carbohydrates_constraint")

    # Protein
    model.addConstr(1.64 * hot_dogs + 3.26 * slices_of_pizza + 4.61 * chicken_drumsticks <= 35, name="protein_constraint")
    model.addConstr(3.26 * slices_of_pizza + 4.61 * chicken_drumsticks <= 22, name="protein_slices_chicken_constraint")
    model.addConstr(1.64 * hot_dogs + 4.61 * chicken_drumsticks <= 26, name="protein_hotdogs_chicken_constraint")
    model.addConstr(1.64 * hot_dogs + 3.26 * slices_of_pizza + 4.61 * chicken_drumsticks <= 26, name="total_protein_constraint")

    # Iron
    model.addConstr(4.59 * hot_dogs + 0.72 * slices_of_pizza + 3.06 * chicken_drumsticks <= 43, name="iron_constraint")
    model.addConstr(4.59 * hot_dogs + 0.72 * slices_of_pizza <= 18, name="iron_hotdogs_slices_constraint")
    model.addConstr(0.72 * slices_of_pizza + 3.06 * chicken_drumsticks <= 25, name="iron_slices_chicken_constraint")
    model.addConstr(4.59 * hot_dogs + 0.72 * slices_of_pizza + 3.06 * chicken_drumsticks <= 25, name="total_iron_constraint")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Hot Dogs: ", hot_dogs.x)
        print("Slices of Pizza: ", slices_of_pizza.x)
        print("Chicken Drumsticks: ", chicken_drumsticks.x)
    else:
        print("The model is infeasible.")

solve_optimization_problem()
```