Scheduling jobshops with some identical or similar jobs

We consider the following job shopscheduling p roblem: N jobs move through I machines, along R routes, with given processing times, and one seeks a schedule to minimize the latest job completion time. This problem is NP-hard. We are interested in the case where the number of routes and the number of machines are fixed, while the number of jobs varies and is large. We distinguish two cases: If jobs on the same route are identical we provide an approximation algorithm which is within constant of the optimum, no matter how many jobs there are. If jobs on the same route have different processing times, the job-shop scheduling problem can be crudely approximated by a continuous deterministic fluid scheduling problem, with buffers of fluid representing the jobs waiting for each operation. The fluid makespan problem is easily solved ∗ Part of this work was done as part of the author’s M.Sc. thesis, done under the supervision of Michal Penn and Gideon Weiss, in the Faculty of Industrial Engineering and Management, Technion, Haifa, Israel. † Research supported by the fund for promotion of research at the Technion, and by the fund for joint