On-line scheduling with forbidden zones K Khammuang 1 , A Abdekhodaee 2 and A Wirth 1 * 1 The University of Melbourne, Victoria, Australia; and 2 Commonwealth Scientific and Industrial Research Organisation, Australia In various manufacturing and computing contexts there may be a certain period in each time interval, during which processing may continue but may not be initiated. We examine the problem of on-line scheduling in the presence of such forbidden zones, whose complements are starting time windows. We show that no on-line algorithm is better than 9 7 - competitive, when minimizing the number of intervals used (essentially the makespan), whereas list scheduling is shown to be 2-competitive. We also investigate adaptations of the first fit, next fit and harmonic bin packing algorithms and test all four empirically. Journal of the Operational Research Society advance online publication, 11 January 2006 doi:10.1057/palgrave.jors.2602124 Keywords: scheduling; on-line algorithms; competitive analysis Introduction On-line approaches have become increasingly important in the field of scheduling. We frequently encounter problems where only partial information is provided but fast and effective solutions are required. Such problems are particu- larly common in computing and manufacturing environ- ments. In computer science, there are many instances when tasks are to be scheduled on servers yet complete informa- tion such as their arrival times, lengths or due-dates are not available in advance. In manufacturing systems, there are also instances when accurate information about processes is not available or optimization methods are so complex as to make it undesirable for decision makers to implement them. Furthermore, in some situations it may be necessary to instantly inform customers placing orders of the completion time of their jobs without having any information about future incoming orders. In such cases, it may be preferable to have some simple rules to assist a decision maker, for example a shopfloor manager, to decide and prioritize a set of tasks virtually instantly. Dispatch rules are frequently mentioned in operations management literature, these provide fast, simple and effective solution methods for many problems. The problem outlined below is one example of a situation where on-line methods are highly useful. The motivation for considering scheduling with starting time windows, whose complements are forbidden zones, arose from work by Abdekhodaee on a supply chain problem. In that problem a certain activity (ship berthing) was not allowed to commence inside particular time intervals. More generally, in some scheduling problems processing may continue but may not be initiated during certain intervals. For example, this will occur if a particular service, such as an operator’s attendance is required at the initiation and the completion of certain tasks but not throughout the whole process. The objective is to carry out all tasks in minimum time, given that the operator’s attendance is limited to certain time periods. We begin by recalling some definitions and results from an earlier complementary paper (Abdekhodaee and Ernst, 2004a), which studied the off-line version of this problem. Denote the n jobs to be scheduled by J 1 y J n . Let p i be the processing time of J i . Furthermore, we shall assume from now on that p i p1 8i. We partition time into a set of abutting intervals I ¼ {I 1 , I 2 , y, I s } with I 1 ¼ [0,2], I 2 ¼ [2,4]. I s ¼ [2s2,2s]. Each I j includes a corresponding forbidden zone F j CI j 8 j , where F 1 ¼ (1,2], F 2 ¼ (3,4] y F s ¼ (2s1,2s]. We call the intervals I j \F j the allowed zones. Below, when we refer to intervals we shall mean the sets I j . Forbidden zones represent time intervals during which a job cannot be started, but can be processed (Figure 1). Furthermore, if a job is completed before the end of a forbidden zone, it will be released at the start of the next interval. Thus, the latest time a job may commence in I j is at t ¼ 2j1. The objective is to sequence the jobs so that the number of intervals used is minimized. (If the last forbidden zone contains a job, then this is equivalent to minimizing the maximum completion time or makespan.) Abdekhodaee and Ernst (2004a) showed, for example, that the decision version of the (off-line) problem when all intervals are of equal length (|I i | ¼ I, 8i) and all forbidden zones are equal (|F i | ¼ F, 8i), is NP-complete in the weak *Correspondence: A Wirth, Department of Mechanical and Manufactur- ing Engineering, The University of Melbourne, Victoria 3010, Australia. E-mail: wirtha@unimelb.edu.au Journal of the Operational Research Society (2006), 1–11 r 2006 Operational Research Society Ltd. All rights reserved. 0160-5682/06 $30.00 www.palgrave-journals.com/jors