## Problem Description and Formulation

The company needs to determine the optimal number of tea and coffee tins to sell in order to maximize profit, given the constraints on filling and labeling time.

Let's define the decision variables:
- \(T\): the number of tea tins to sell
- \(C\): the number of coffee tins to sell

The objective function to maximize profit is:
\[ \text{Maximize:} \quad 11T + 13C \]

Subject to the constraints:
1. Filling time constraint: \( 4T + 3C \leq 500 \)
2. Labeling time constraint: \( T + 2C \leq 600 \)
3. Non-negativity constraints: \( T \geq 0, C \geq 0 \)

## Gurobi Code

To solve this linear programming problem using Gurobi in Python, we can use the following code:

```python
import gurobi

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

    # Define the decision variables
    T = model.addVar(lb=0, name="T")  # Number of tea tins
    C = model.addVar(lb=0, name="C")  # Number of coffee tins

    # Objective function: Maximize profit
    model.setObjective(11*T + 13*C, gurobi.GRB.MAXIMIZE)

    # Filling time constraint
    model.addConstr(4*T + 3*C <= 500, name="Filling_Time")

    # Labeling time constraint
    model.addConstr(T + 2*C <= 600, name="Labeling_Time")

    # Optimize the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution found. Tea tins: {T.varValue}, Coffee tins: {C.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("The model is infeasible.")

# Run the function
solve_tea_coffee_problem()
```

This code defines the problem in Gurobi, solves it, and prints out the optimal number of tea and coffee tins to sell in order to maximize profit, along with the maximum achievable profit. If the problem is infeasible, it indicates that as well.