Scheduling unrelated machines with job splitting, setup resources and sequence dependency

We examine the parallel machine scheduling problem where a set of jobs are to be processed by a set of unrelated parallel machines. We examine the most general among the variations for which an exact method has been proposed regarding makespan minimisation. This is because, apart from unrelated machines, we allow for (i) job splitting: each job's quantity can be split and processed by multiple machines simultaneously; (ii) sequence- and machine-dependent setup times: the setup time when job j succeeds k is different than the time when k succeeds j and varies also per machine m; and (iii) setup resource constraints: the number of setups that can be performed simultaneously on different machines is restricted. We present novel lower bound formulations and a heuristic that solves instances of up to 1000 jobs in a few minutes at an average gap of less than $ 20\% $ 20%. Then, we propose a logic-based Benders decomposition, which, coupled with our heuristic, solves instances of up to 200 jobs and 20 machines to near optimality in less than two hours. Our method is used for a broad range of instances from textile manufacturing, thus yielding valuable managerial insights on makespan's versatility under varying machines or resources.