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In a recent paper [Mor, B., Mosheiov, G., 2012. Scheduling a maintenance activity and due-window assignment based on common flow allowance. International Journal of Production Economics 135, 222–230], a scheduling problem with job-dependent due-window based on common flow allowance is studied where the scheduler has the option to perform a maintenance activity to improve the processing times of the following jobs. However, as shown in this note, there exists an error in the paper. We will show the error by a counter-example, and correct it in this note.

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... Proof. For a given schedule = [ [1] , [2] , . . . Proof. ...

... Proof. See in Chen et al. [1]. ...

... ]. Obviously, for a given ℎ and the schedule = [ [1] , [2] , . . . , [ℎ] ], the due window assignment problem has a separable objective function. ...

This paper considers the SLK/DIF due window assignment methods for single machine scheduling problems with general job-dependent positional effect, where the actual processing time of a job is subject to a job-dependent and positional effect. Production schedulers may decide to outsource/reject some jobs by paying the corresponding costs. Based on whether to accept the tardiness jobs, we consider two different objectives under slack and different due window assignment, respectively. The first objective function is to minimize the weighted sum of earliness, tardiness, due window starting time, due window size and outsourcing costs, while the second objective function is to minimize the cost function that includes earliness, due window starting time, due window size and outsourcing costs. We study the structural properties of two due window assignment methods and develop polynomial-time solution algorithms for the considered problems. A numerical example proves the advantage of outsourcing decision and the distinction of two different objectives.

... In this section we present some properties for a schedule.The proofs of the following lemmas are similar to those in [3], [4], [5] and [6]. We use a conventional notation [r] to indicate the index of a job which is allocated at the rth position. ...

... The proofs of the following lemmas are similar to those in [14] and [15]. We use a conventional notation [r] to indicate the index of a job which is allocated at the rth position. ...

In this paper, we consider the slack due-window method and investigate single machine scheduling with a deteriorating rate-modifying activity, linear resource allocation and aging effect. The objective is to minimize the total cost caused by the due-window location, the due-window size, the earliness and tardiness with respect to a slack due-window, and resource consumption. We provide a polynomial-time algorithm to solve the corresponding problem.

... Mosheiov and Oron (2010) provide an O(n log n) solution for this problem. Mor and Mosheiov (2012) and Chen, Ji, and Ge (2013) extend this problem to the case including a maintenance activity. They consider several versions of this problem, including the duration of the maintenance activity is (i) a constant time, (ii) an increasing function of its starting time, and (iii) position-dependent. ...

This paper studies linear non-increasing processing times and the common/slack due window assignment problems on a single machine, where the actual processing time of a job is a linear non-increasing function of its starting time. The aim is to minimize the sum of the earliness cost, tardiness cost, due window location and due window size. Some optimality results are discussed for the common/slack due window assignment problems and two O(n log n) time algorithms are presented to solve the two problems. Finally, two examples are provided to illustrate the correctness of the corresponding algorithms.

We study due-date assignment problems with common flow allowance on a proportionate flowshop. We focus on both minsum and minmax objectives. Both cases are extended to a setting of a due-window. The proposed solution procedures are shown to be significantly different from those of the single machine problems. All the problems studied here are solved in polynomial time: the minsum problems in O(n2) and the minmax problems in O(nlog n), where n is the number of jobs.

We consider single machine scheduling problems with deteriorating jobs and SLK/DIF due window assignment, where the deteriorating rates of jobs are assumed to be job-dependent. We consider two different objectives under SLK and DIF due window assignment, respectively. The first objective is to minimise total costs of earliness, tardiness, due window location and due window size, while the second objective is to minimise a cost function that includes number of early jobs, number of tardy jobs and the costs for due window location and due window size. We study the optimality properties for all problems and develop algorithms for solving these problems in polynomial time. © 2016 Operational Research Society Ltd. All rights reserved. 0160-5682/16.

We consider group scheduling problem on a single machine with multiple due windows assignment. Jobs are divided into groups in advance according to their processing similarities, and all jobs of the same group are required to be processed contiguously on the machine in order to achieve production efficiency and save time/money resource. A sequence-independent setup time precedes the processing of each group. The goal is to determine the optimal sequence for both groups and jobs, together with an optimal combination of the due windows assignment strategy so as to minimize the total of earliness, tardiness and due windows related costs. We give an \(O(n\log n)\) time algorithm for the problem.

We consider a single-machine due window assignment and scheduling problem with batch deliveries, where all jobs have a common due window, and the start time and size of the due window are decision variables. Finished jobs are delivered in batches with unlimited batch capacity. The objective is to determine the due window, a job sequence, and the delivery times, so as to minimize the total cost which comprises earliness of delivery, job holding, start time of due window, size of due window, number of delivery batches, and tardiness penalty. We consider three different variants of the problem corresponding to different measurements of tardiness penalty. We present polynomial-time solution procedures for these variants with significantly lower computational complexities than those of known algorithms in the literature.

This paper deals with multiple due windows assignment scheduling problems and controllable processing times on a single machine. We assume that the actual processing time of a job can be controlled by the introduction of additional resource and any due window is not allowed to contain another due window as a proper subset. The objective is to determine the optimal due window positions and sizes, the set of jobs assigned to each due window, the optimal job compressions, and the optimal schedule to minimize a total cost function, which consists of the earliness, the tardiness, the processing time compressions, and the due windows related costs. We show that for the case when the number of jobs assigned to each due window is given in advance, the problem is polynomially solvable in O(3n)O(n3) time, where n is the total number of jobs; while if the number of jobs assigned to each due window is unknown, the problem can be optimally solved in O(nm+2)O(nm+2) time, where m is the number of due windows. Furthermore, we extend the problem by incorporating with the aging effect and prove that it remains polynomially solvable.

We study a due-window assignment problem on a single machine. The job-dependent due-windows are obtained by the common flow allowance criterion. The scheduler has the option to perform a maintenance activity which is rate modifying, i.e., improves the processing times of the following jobs. We consider a number of versions of this setting: (i) The maintenance requires a constant time, (ii) The maintenance duration is an increasing function of its starting time (linear deterioration), and (iii) The maintenance duration is position-dependent (general deterioration). We study the standard setting of regular job processing times, and investigate also the extension to position-dependent processing times. The set of potential optimal positions for the maintenance activity is fully characterized. Consequently, the problems based on all the combinations of these settings are shown to be solved in polynomial time.