## Problem Description and Symbolic Representation

The problem requires finding the minimum cost for 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. One plate of Vietnamese food contains 15 units of protein and 20 units of carbs. One plate of Korean food contains 10 units of protein and 14 units of carbs. Vietnamese food costs $12.5 per plate, and Korean food costs $16.5 per plate.

### Symbolic Representation

Let's define the symbolic variables:
- $x_1$ : number of plates of Vietnamese food
- $x_2$ : number of plates of Korean food

The objective function is to minimize the total cost:
Minimize $12.5x_1 + 16.5x_2$

The constraints are:
- Protein requirement: $15x_1 + 10x_2 \geq 250$
- Carb requirement: $20x_1 + 14x_2 \geq 45$
- Non-negativity: $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'plates of Vietnamese food'), ('x2', 'plates of Korean food')],
    'objective_function': '12.5*x1 + 16.5*x2',
    'constraints': [
        '15*x1 + 10*x2 >= 250',
        '20*x1 + 14*x2 >= 45',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="vietnamese_food", lb=0, vtype=gurobi.GRB.CONTINUOUS)
    x2 = model.addVar(name="korean_food", lb=0, vtype=gurobi.GRB.CONTINUOUS)

    # Define the objective function
    model.setObjective(12.5 * x1 + 16.5 * x2, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(15 * x1 + 10 * x2 >= 250, name="protein_requirement")
    model.addConstr(20 * x1 + 14 * x2 >= 45, name="carb_requirement")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of plates of Vietnamese food: {x1.varValue}")
        print(f"Number of plates of Korean food: {x2.varValue}")
        print(f"Minimum cost: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```