To solve this optimization problem, we first need to define the decision variables and the objective function. Let's denote the number of coats as \(C\) and the number of skirts as \(S\). The objective is to maximize profit, which is $12 for each coat and $16 for each skirt.

Given:
- Each coat requires 1.5 hours of designing time and 2.5 hours of printing time.
- Each skirt requires 3 hours of designing time and 3.5 hours of printing time.
- Designers are available 45 hours a week.
- The printing machine is available 70 hours per week.

The constraints can be formulated as follows:
1. Designing time constraint: \(1.5C + 3S \leq 45\)
2. Printing time constraint: \(2.5C + 3.5S \leq 70\)

The objective function to maximize profit is: \(12C + 16S\)

We will use Gurobi, a Python optimization library, to solve this linear programming problem.

```python
from gurobipy import *

# Create a new model
m = Model("X-Luxury_Cloth")

# Define the decision variables
C = m.addVar(vtype=GRB.CONTINUOUS, name="Coats")
S = m.addVar(vtype=GRB.CONTINUOUS, name="Skirts")

# Define the objective function
m.setObjective(12*C + 16*S, GRB.MAXIMIZE)

# Add constraints
m.addConstr(1.5*C + 3*S <= 45, "Designing_Time")
m.addConstr(2.5*C + 3.5*S <= 70, "Printing_Time")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Coats: {C.x}")
    print(f"Skirts: {S.x}")
    print(f"Maximum Profit: {m.objVal}")
else:
    print("No optimal solution found")
```