This paper considers two single-machine scheduling problems with outsourcing allowed where each job can be either processed on an in-house single-machine or outsourced. They include the problem of minimizing maximum Idleness and outsourcing costs, and that of minimizing total tardiness and outsourcing costs. Outsourcing is commonly required as a way to improve productivity in various companies including electronics industries and motor industries. The objective is to minimize the weighted sum of the outsourcing cost and the scheduling measure represented by either one of maximum lateness and total tardiness, subject to outsourcing budget. It is proved that the problem is NP-hard. Some solution properties are characterized to derive heuristic algorithms, and also a branch-and-bound algorithm. Numerical experiments are conducted to evaluate performance of the derived algorithms. (c) 2007 Elsevier B.V. All rights reserved.