To solve this problem, we first need to define the decision variables and the objective function. Let's denote the number of tea tins as \(T\) and the number of coffee tins as \(C\). The profit per tea tin is $11, and the profit per coffee tin is $13. Thus, the total profit can be represented as \(11T + 13C\), which we want to maximize.

The constraints are based on the time available for filling and labeling. For filling, each tea tin takes 4 minutes, and each coffee tin takes 3 minutes. The company has 500 minutes available for this task, leading to the constraint \(4T + 3C \leq 500\). For labeling, each tea tin takes 1 minute, and each coffee tin takes 2 minutes, with 600 minutes available, giving us the constraint \(T + 2C \leq 600\).

Additionally, we must ensure that the number of tins for both products is non-negative since selling a negative number of items does not make sense. Thus, we have \(T \geq 0\) and \(C \geq 0\).

With these definitions, we can formulate the linear programming problem as follows:

Maximize: \(11T + 13C\)

Subject to:
1. \(4T + 3C \leq 500\)
2. \(T + 2C \leq 600\)
3. \(T \geq 0\)
4. \(C \geq 0\)

This problem can be solved using Gurobi, a powerful linear and mixed-integer programming solver. Below is the Python code that formulates this problem and solves it using Gurobi:

```python
from gurobipy import *

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

# Define the decision variables
T = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Tea_Tins")
C = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Coffee_Tins")

# Set the objective function
m.setObjective(11*T + 13*C, GRB.MAXIMIZE)

# Add constraints
m.addConstr(4*T + 3*C <= 500, "Filling_Time")
m.addConstr(T + 2*C <= 600, "Labeling_Time")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Tea Tins: {T.x}")
    print(f"Coffee Tins: {C.x}")
    print(f"Total Profit: ${11*T.x + 13*C.x:.2f}")
else:
    print("No optimal solution found")
```