Scheduling problems on a new setting of flexible flowshops: ℓ-Machine proportionate flowshops

The classical flexible flowshop ((Formula presented.)) environment consists of (Formula presented.) -stages in series with (Formula presented.) -machines in parallel at each stage, where each job has to undergo a particular series of actions. In this paper we consider a similar, yet different, and still a challenging real-life setting, that, to the best of our knowledge, has not been studied to date. We assume that there are (Formula presented.) -sets in parallel in which each of the sets is an (Formula presented.) -machine proportionate flowshop. In this new setting, each job can be processed on each one of the sets, but once a set is chosen, the job must be processed on all of its machines, in first-in-first-out method. We study several fundamental scheduling measures such as makespan, maximum tardiness with common due-date, total tardiness with common due-date, and total load. Moreover, we consider optional job-rejection and focus on minimising the total completion time. All problems are shown to be NP-hard, and pseudo-polynomial dynamic programming (DP) solution algorithms and Simulated Annealing (SA) metaheuristics are provided. The efficiency of all our proposed algorithms is validated through an extensive numerical study.