Job-shop scheduling in a body shop

We study a generalized job-shop problem called the body shop scheduling problem (BSSP). This problem arises from the industrial application of welding in a car body production line, where possible collisions between industrial robots have to be taken into account. BSSP corresponds to a job-shop problem where the operations of a job have to follow alternating routes on the machines, certain operations of different jobs are not allowed to be processed at the same time and after processing an operation of a certain job a machine might be unavailable for a given time for operations of other jobs. As main results we will show that for three jobs and four machines the special case where only one machine is used by more than one job is already $mathcal NP $ -hard. This also implies that the single machine scheduling problem that asks for a makespan minimal schedule of three chains of operations with delays between the operations of a chain is $mathcal NP $ -hard. On the positive side, we present a polynomial algorithm for the two job case and a pseudo-polynomial algorithm together with an FPTAS  for an arbitrary but constant number of jobs. Hence for a constant number of jobs we fully settle the complexity status of the problem.