Here, it's not about making free delivery profitable, but rather about encouraging buyers to choose faster (and therefore paid) delivery.
While it may sound appealing, offering only free shipping is not the most cost-effective solution available to you. Receiving an order for free is a privilege in the eyes of buyers, so you should take advantage of it.
Here’s another very common technique: the shipping costs are free for the buyer, but the merchant has already included them in the selling price of the product.
Your delivery strategy can also allow you to increase the average cart of your buyers.
As an e-merchant, you know that shipping costs are a budget to take into account when calculating your margins. And it’s sometimes possible to considerably reduce these costs by offering slower delivery methods.