Exact Algorithms for Two-Machine Job-Shop Scheduling Problem with One Joint Job Considering Machine Repetition and Transportation Times

We consider the two-machine job-shop scheduling problem with the makespan criterion when one job is joint and the others are non-joint, where a joint (resp., non-joint) job is defined as a job whose operations are to be processed by different machines (resp., same machine). In this research, the machine repetition and transportation times between machines were considered. This problem is associated with real-world applications in the production planning and supply chains. We demonstrate that this problem is NP-hard when the joint job has more than two operations. We propose polynomial-time algorithms based on dynamic programming for cases with a fixed number of jobs. We also propose lemmas to reduce the number of states in dynamic programming without loss of optimality, so that the time complexity is improved. Other methodologies, including preprocessing and the two-pointers method, are also embedded. Our algorithm has better time complexity than a well-known algorithm that can be applied to our problem.