To solve the given optimization problem, we first need to define the decision variables and the objective function. The decision variables are \(x_1\) (the number of bottles of regular wine made per day) and \(x_2\) (the number of bottles of premium wine made per day). The objective is to maximize profit, with profits of $20 per bottle for regular wine and $50 per bottle for premium wine.

The constraints are as follows:
1. Demand constraints: \(x_1 \leq 80\) (at most 80 bottles of regular wine) and \(x_2 \leq 50\) (at most 50 bottles of premium wine).
2. Supply constraint: \(x_1 + x_2 \leq 120\) (the company can make at most 120 bottles in total per day).
3. Non-negativity constraints: \(x_1, x_2 \geq 0\) (the number of bottles cannot be negative).

Given these conditions, the problem can be formulated as a linear programming problem where we aim to maximize profit under the given constraints.

Here's how this translates into Gurobi code in Python:

```python
from gurobipy import *

# Create a new model
m = Model("WineProduction")

# Define the decision variables
x1 = m.addVar(name="RegularWine", lb=0)  # Number of bottles of regular wine
x2 = m.addVar(name="PremiumWine", lb=0)  # Number of bottles of premium wine

# Define the objective function: Maximize profit
m.setObjective(20*x1 + 50*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 <= 80, name="DemandRegular")  # Demand constraint for regular wine
m.addConstr(x2 <= 50, name="DemandPremium")  # Demand constraint for premium wine
m.addConstr(x1 + x2 <= 120, name="SupplyConstraint")  # Total supply constraint

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: Produce {x1.x} bottles of regular wine and {x2.x} bottles of premium wine.")
else:
    print("No optimal solution found.")
```