## Problem Description and Formulation

The problem is a classic example of a linear programming problem. Sid needs to meet his daily requirements of various food groups while minimizing the cost. Let's define the decision variables:

- \(x_1\): Number of hamburgers Sid buys
- \(x_2\): Number of plates of pasta Sid buys

The objective is to minimize the total cost, which is \(3x_1 + 4x_2\).

The constraints are based on the daily requirements:

- Meat: \(x_1 \geq 2\)
- Dairy: \(0.5x_1 + x_2 \geq 1\)
- Vegetables: \(x_1 + x_2 \geq 4\)
- Grains: \(x_1 + 2x_2 \geq 3\)

Also, \(x_1, x_2 \geq 0\) and are integers, as Sid cannot buy fractions of a meal.

However, for the purpose of solving this with Gurobi using linear programming (which assumes continuous variables), we will initially relax the integer requirement and then discuss how to handle integrality if needed.

## Gurobi Code

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(lb=0, name="hamburgers")
    x2 = model.addVar(lb=0, name="plates_of_pasta")

    # Objective: Minimize cost
    model.setObjective(3*x1 + 4*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x1 >= 2, name="meat_requirement")
    model.addConstr(0.5*x1 + x2 >= 1, name="dairy_requirement")
    model.addConstr(x1 + x2 >= 4, name="vegetables_requirement")
    model.addConstr(x1 + 2*x2 >= 3, name="grains_requirement")

    # Optimize
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal cost: {model.objVal}")
        print(f"Hamburgers: {x1.varValue}")
        print(f"Plates of pasta: {x2.varValue}")
    else:
        print("No optimal solution found")

solve_optimization_problem()
```

## Handling Integrality

If we need \(x_1\) and \(x_2\) to be integers (as you might not be able to buy a fraction of a meal), we can modify the code to use Gurobi's integer programming capabilities:

```python
import gurobi

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

    # Define the variables as integers
    x1 = model.addVar(lb=0, type=gurobi.GRB.INTEGER, name="hamburgers")
    x2 = model.addVar(lb=0, type=gurobi.GRB.INTEGER, name="plates_of_pasta")

    # Objective: Minimize cost
    model.setObjective(3*x1 + 4*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x1 >= 2, name="meat_requirement")
    model.addConstr(0.5*x1 + x2 >= 1, name="dairy_requirement")
    model.addConstr(x1 + x2 >= 4, name="vegetables_requirement")
    model.addConstr(x1 + 2*x2 >= 3, name="grains_requirement")

    # Optimize
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal cost: {model.objVal}")
        print(f"Hamburgers: {x1.varValue}")
        print(f"Plates of pasta: {x2.varValue}")
    else:
        print("No optimal integer solution found")

solve_integer_optimization_problem()
```