I think Sidi gave a good answer to address your first question. I would add that the line of business should match the goal of the company: for example, if a company’s objective is to increase revenues and a division is profitable but with low contribution to revenues and high opportunity cost compared to other revenue-enhancing options, it’s not really helping the company to achieve its goal.
To address your second question, I believe there may be two reasons why you may want to shut down a profitable business:
- Cannibalization: It simply cannibalizes other lines of business
- Opportunity cost: It requires resources that invested in other segments could bring a better results and that you have no other option to obtain otherwise