Scheduling of two-machine flowshop with outsourcing lead-time

•Considering two two-machine flowshop scheduling problems with outsourcing lead-time.•Proving the NP-hardness of the first problem.•Developing Heuristic HEU and B&B algorithm for the first problem;•Developing Heuristics HEU, NEH, PEH and B&B algorithm for the second problem;•Verifying the performance of the proposed algorithms through the numerical experiments. This paper considers a two-machine flowshop scheduling problem with outsourcing lead-time. While the first operation of jobs can be either processed in-house or outsourced to subcontractors, the second operation of jobs must be processed in-house. Outsourcing a job incurs a job-dependent outsourcing cost as well as outsourcing lead-time, which implies that the outsourced job becomes available for the second operation after its outsourcing lead-time. The objective is to minimize the weighted sum of the outsourcing costs and the scheduling costs which is either makespan or total completion time. The problems are shown to be NP-hard, and both exact and heuristic algorithms are developed for the problems. Finally, computational experiments show that the proposed algorithms provide efficient and effective solutions.