Here's the solution using Gurobi in Python:

We define the variables:

* `x`: Number of t-shirts produced.
* `y`: Number of hoodies produced.

We want to maximize the objective function:

Profit = 10 * x + 15 * y

Subject to the following constraints:

* Designing time: 1x + 2y <= 40
* Printing time: 2x + 3y <= 60
* Non-negativity: x >= 0, y >= 0


```python
import gurobipy as gp
from gurobipy import GRB

# Create a new model
m = gp.Model("clothing_production")

# Create variables
x = m.addVar(vtype=GRB.CONTINUOUS, name="t_shirts")  # Number of t-shirts
y = m.addVar(vtype=GRB.CONTINUOUS, name="hoodies")  # Number of hoodies

# Set objective function
m.setObjective(10 * x + 15 * y, GRB.MAXIMIZE)

# Add constraints
m.addConstr(1 * x + 2 * y <= 40, "design_constraint")
m.addConstr(2 * x + 3 * y <= 60, "printing_constraint")
m.addConstr(x >= 0, "t_shirts_non_negative")
m.addConstr(y >= 0, "hoodies_non_negative")

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print(f"Optimal Solution Found:")
    print(f"Number of T-shirts: {x.x}")
    print(f"Number of Hoodies: {y.x}")
    print(f"Maximum Profit: ${m.objVal}")
elif m.status == GRB.INFEASIBLE:
    print("The model is infeasible.")
else:
    print(f"Optimization terminated with status {m.status}")

```
