To solve this problem, we can define it as a linear programming optimization problem. The goal is to maximize profit, which depends on the number of carry-on and large suitcases produced, subject to constraints related to production rates and sewing machine capacity.

Let's denote:
- \(x_c\) as the number of carry-on suitcases produced per day,
- \(x_l\) as the number of large suitcases produced per day.

The profit from each carry-on suitcase is $100, and from each large suitcase is $150. Therefore, the total profit \(P\) can be represented as:
\[ P = 100x_c + 150x_l \]

There are three main constraints:
1. The production rate of carry-on suitcases cannot exceed 15 per day: \( x_c \leq 15 \).
2. The production rate of large suitcases cannot exceed 20 per day: \( x_l \leq 20 \).
3. The total number of suitcases (both types) that can be processed by the sewing machine cannot exceed 25 per day: \( x_c + x_l \leq 25 \).

Additionally, \(x_c\) and \(x_l\) must be non-negative since they represent quantities of suitcases.

To find out how many of each type of suitcase should be made to maximize profit, we can use the Gurobi solver in Python. Here's the code that captures this problem:

```python
from gurobipy import *

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

# Define variables
x_c = m.addVar(lb=0, vtype=GRB.INTEGER, name="carry_on_suitcases")
x_l = m.addVar(lb=0, vtype=GRB.INTEGER, name="large_suitcases")

# Set the objective function: Maximize profit
m.setObjective(100*x_c + 150*x_l, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x_c <= 15, "carry_on_production_limit")
m.addConstr(x_l <= 20, "large_production_limit")
m.addConstr(x_c + x_l <= 25, "sewing_machine_capacity")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Carry-on suitcases to produce: {x_c.x}")
    print(f"Large suitcases to produce: {x_l.x}")
    print(f"Maximum profit achievable: ${m.objVal}")
else:
    print("No optimal solution found. The problem might be infeasible.")
```