We consider the two-machine open shop and two-machine flow shop scheduling problems in which each machine has to be maintained exactly once during the planning period, and the duration of each of these intervals depends on its start time. The objective is to minimize the maximum completion time of all activities to be scheduled. We resolve complexity and approximability issues of these problems. The open shop problem is shown to be polynomially solvable for quite general functions defining the length of the maintenance intervals. By contrast, the flow shop problem is proved binary NP-hard and pseudopolynomially solvable by dynamic programming. We also present a fully polynomial approximation scheme and a fast 3/2-approximation algorithm.

This paper considers two-machine flow shop scheduling problems with machine availability constraints. When the processing of a job is interrupted by an unavailability period of a machine, we consider both the resumable scenario in which the processing can be resumed when the machine next becomes available, and the semi-resumable scenarios in which some proportion of the processing is repeated but the job is otherwise resumable. For the resumable scenario, problems with non-availability intervals on one of the machines are shown to admit fully polynomial-time approximation schemes that are based on an extended dynamic programming algorithm. For the problem with several non-availability intervals on the first machine, we present a fast 3/2-approximation algorithm. For the problem with one non-availability interval under the semi-resumable scenario, polynomial-time approximation schemes are developed.

We study a two-machine flow shop scheduling problem with no-wait in process, in which one of the machines is subject to mandatory maintenance. The length of the maintenance period is defined by a non-decreasing function that depends on the starting time of that maintenance. The objective is to minimize the completion time of all activities. We present a polynomial-time approximation scheme for this problem.

Abstract:We study a two-machine flow shop scheduling problem with no-wait in process, in which one of the machines is not available during a specified time interval. We consider three scenarios of handing the operation affected by the nonavailability interval. Its processing may (i) start from scratch after the interval, or (ii) be resumed from the point of interruption, or (iii) be partially restarted after the interval. The objective is to minimize the makespan. We present an approximation algorithm that for all these scenarios delivers a worst-case ratio of 3/2. For the second scenario, we offer a 4/3-approximation algorithm.

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