Which metric is based on the relationship between the revenue produced by a specific customer the expenses incurred in acquiring and servicing that customer and the expected life of the relationshi?

The metric you're referring to is Customer Lifetime Value (CLV or LTV). It quantifies the total revenue a business can expect from a customer throughout their relationship while considering the costs associated with acquiring and servicing that customer. By analyzing CLV, companies can make informed decisions about marketing expenses and customer retention strategies.