Competitive algorithms for multistage online scheduling - Hopf, Michael; Thielen, Clemens; Wendt, Oliver (European journal of operational research, 2017)
•We study online flow shop scheduling where jobs must be processed in several stages.•We show that a restricted version of the offline version of the problem is APX-hard.•We provide online algorithms and lower bounds for different variants of the problem. We study an online flow shop scheduling problem where each job consists of several tasks that have to be completed in t different stages and the goal is to maximize the total weight of accepted jobs. The set of tasks of a job contains one task for each stage and each stage has a dedicated set of identical parallel machines corresponding to it that can only process tasks of this stage. In order to gain the weight (profit) associated with a job j, each of its tasks has to be executed between a task-specific release date and deadline subject to the constraint that all tasks of job j from stages 1,⋯,i−1 have to be completed before the task of the ith stage can be started. In the online version, jobs arrive over time and all information about the tasks of a job becomes available at the release date of its first task. This model can be used to describe production processes in supply chains when customer orders arrive online. Even the basic version of the offline problem with a single machine in each stage, unit weights, unit processing times, and fixed execution times for all tasks (i.e., deadline minus release date equals processing time) is APX-hard and we show that the approximation ratio of any polynomial-time approximation algorithm for this basic version of the problem must depend on the number t of stages. For the online version of the basic problem, we provide a (2t−1)-competitive deterministic online algorithm and a matching lower bound. Moreover, we provide several (sometimes tight) upper and lower bounds on the competitive ratios of online algorithms for several generalizations of the basic problem involving different weights, arbitrary release dates and deadlines, different processing times of tasks, and several identical machines per stage.