Minimizing the makespan in a two-machine cross-docking flow shop problem

This paper studies a two-machine cross-docking flow shop scheduling problem in which a job at the second machine can be processed only after the processing of some jobs at the first machine has been completed. The objective is to minimize the makespan. We first show that the problem is strongly NP-hard. Some polynomially solvable special cases are provided. We then develop a polynomial approximation algorithm with an error-bound analysis. A branch-and-bound algorithm is also constructed. Computational results show that the branch-and-bound algorithm can optimally solve problems with up to 60 jobs within a reasonable amount of time.