## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to minimize the cost of a weight loss program that includes a minimum of 250 units of protein and 45 units of carbs. There are two cuisine options available: Vietnamese and Korean.

### Decision Variables

Let $x_V$ be the number of plates of Vietnamese food and $x_K$ be the number of plates of Korean food.

### Objective Function

The objective is to minimize the total cost of the program, which is given by $12.5x_V + 16.5x_K$.

### Constraints

1. **Protein Requirement**: The program must include a minimum of 250 units of protein. Since one plate of Vietnamese food contains 15 units of protein and one plate of Korean food contains 10 units of protein, the protein requirement can be expressed as $15x_V + 10x_K \geq 250$.
2. **Carb Requirement**: The program must include a minimum of 45 units of carbs. Since one plate of Vietnamese food contains 20 units of carbs and one plate of Korean food contains 14 units of carbs, the carb requirement can be expressed as $20x_V + 14x_K \geq 45$.
3. **Non-Negativity**: The number of plates of each cuisine cannot be negative, so $x_V \geq 0$ and $x_K \geq 0$.

## Gurobi Code

```python
import gurobi

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

    # Define decision variables
    x_V = model.addVar(name="Vietnamese", lb=0, vtype=gurobi.GRB.CONTINUOUS)
    x_K = model.addVar(name="Korean", lb=0, vtype=gurobi.GRB.CONTINUOUS)

    # Define objective function
    model.setObjective(12.5 * x_V + 16.5 * x_K, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(15 * x_V + 10 * x_K >= 250, name="protein_requirement")
    model.addConstr(20 * x_V + 14 * x_K >= 45, name="carb_requirement")

    # Optimize the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Minimum cost: ${model.objVal:.2f}")
        print(f"Vietnamese food: {x_V.x:.2f} plates")
        print(f"Korean food: {x_K.x:.2f} plates")
    else:
        print("The model is infeasible.")

solve_min_cost_program()
```