A profit center is a division of a business where there is capability of measuring the Revenue and Expense solely related to that division of the business. The capability is a function of both how business is conducted the sophistication of the accounting system to capture the individual revenues and expenses.
Consider an oil change shop with a car wash attached. The employees of each division work only for one or the other. The accounting system can separate payroll costs, supplies and inventories for each. Certain costs, such as ground rent can be reasonable allocated by square feet. In this simple example it would be possible to measure separately the profits of each.
However, complications can arise. Some oil change shops will overcharge for their services, but then provide a free car wash. This would require an accounting mechanism to split out the car wash revenue from that derived from changing oil.
Human nature can make this more complicated than it sounds. Suppose market price in the area was $15.00 for an oil change and $10.00 for a car wash. If the combined price was $25.00, the revenue split would seem to be straight-forward. However, the oil change manager may argue that but for his work, the customer would not have gone to the car wash, and his profit centre should be allocated $16.00 of revenue.