In many heavily loaded manufacturing systems, managers routinely make use of outsourcing options in order to maintain reasonable Quality of Service for customers. Thus, there is a strong need to provide tools for managers to economically coordinate sourcing and scheduling decisions. Our main aim is to provide such tools for an important set of flow-shop scheduling problems where rejection (outsourcing) is allowed and processing times are machine-independent. Our scheduling problems are essentially bicriteria problems, which combine a scheduling objective and the total outsourcing cost. We study several problems which differ according to the scheduling criterion considered. Moreover, each problem is divided into four different variations depending on the way the two criteria are dealt with. For example, in one variation the two criteria are aggregated into a single objective function; in two other variations the aim consists of minimizing one criterion subject to ensuring that the value of the other criterion will not exceed a predefined threshold. From a theoretical point of view, a computational complexity classification is provided for all variations of the problems under consideration. Moreover, optimization algorithms have been constructed to solve all problem variations, and approximation schemes have been developed for solving hard variations. Those schemes enable managers to solve large instances of hard variations while controlling the maximal gap between the obtained solution and the (unknown) optimal solution.