MATHEMATICS OF OPERATIONS RESEARCH Vol. 37, No. 2, May 2012, pp. 379–398 ISSN 0364-765X (print)  ISSN 1526-5471 (online) http://dx.doi.org/10.1287/moor.1110.0520 © 2012 INFORMS The Power of Preemption on Unrelated Machines and Applications to Scheduling Orders José R. Correa Departamento de Ingeniería Industrial, Universidad de Chile, República 701, Santiago, Chile, jcorrea@dii.uchile.cl, http://www.dii.uchile.cl/ ~ jcorrea Martin Skutella, José Verschae Fakultät II, Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany {martin.skutella@tu-berlin.de, http://www.coga.tu-berlin.de/people/skutella; verschae@math.tu-berlin.de, http://www.coga.tu-berlin.de/people/verschae} Scheduling jobs on unrelated parallel machines so as to minimize makespan is one of the basic problems in the area of machine scheduling. In the first part of the paper, we prove that the power of preemption, i.e., the worst-case ratio between the makespan of an optimal nonpreemptive and that of an optimal preemptive schedule, is at least 4. This matches the upper bound proposed in Lin and Vitter [Lin, J.-H., J. S. Vitter. 1992. -approximations with minimum packing constraint violation. Proc. 24th Annual ACM Sympos. Theory of Comput. (STOC), ACM, New York, 771–782] two decades ago. In the second part of the paper, we consider the more general setting in which orders, consisting of several jobs, have to be processed on unrelated parallel machines so as to minimize the sum of weighted completion times of the orders. We obtain the first constant factor approximation algorithms for the preemptive and nonpreemptive cases, improving and extending a recent result by Leung et al. [Leung, J., H. Li, M. Pinedo, J. Zhang. 2007. Minimizing total weighted completion time when scheduling orders in a flexible environment with uniform machines. Inform. Processing Lett. 103 119–129]. Finally, we study this problem in a parallel machine environment, obtaining a polynomial-time approximation scheme for several special cases. Key words : machine scheduling; approximation algorithms MSC2000 subject classification : Primary: 90B35, 68W25; secondary: 68Q25, 90C10 OR/MS subject classification : Primary: production/scheduling, approximations/heuristic History : Received October 22, 2010; revised August 19, 2011. Published online in Articles in Advance December 22, 2011. 1. Introduction. We consider the classical scheduling problem of minimizing the makespan on unrelated parallel machines. In this problem we are given a set of jobs J = 811 : : : 1 n9 and a set of machines M = 811 : : : 1 m9 to process the jobs. Each job j ∈ J has a processing requirement of p ij units of time on machine i ∈ M . Every job has to be scheduled without interruption on exactly one machine, and each machine can process at most one job at a time. The objective is to minimize the makespan C max 2= max j ∈J C j , where C j denotes the completion time of job j . In the standard three-field scheduling notation (see, e.g., Lawler et al. [21]) this problem is denoted by R  C max . In a seminal work, Lenstra et al. [23] present a 2-approximation algorithm for R  C max and show that the prob- lem is NP-hard to approximate within a factor better than 3/2. On the other hand, Lawler and Labetoulle [20] consider a linear relaxation of an integer programming formulation of R  C max , and show that it is equivalent to its preemptive version. In this setting, denoted RpmtnC max , jobs can be interrupted and resumed later on the same or a different machine. Naturally, the corresponding linear program can be used as a lower bound for designing approximation algorithms for R  C max . Indeed, Shmoys and Tardos (cited as personal communication in Lin and Vitter [27]) present a rounding procedure showing that the integrality gap is at most 4. Equivalently, this result shows that the power of preemption (Canetti and Irani [6], Shachnai and Tamir [32], Schulz and Skutella [31]), the worst-case ratio between the makespan of an optimal preemptive and an optimal nonpreemp- tive schedule, is at most 4. Interestingly, it has been unknown whether this bound is tight. In this paper we answer this question on the positive, devising a lower bound showing that the power of preemption is exactly 4. The proof of the lower bound relies on a recursive construction, where in each iteration the gap of the instance is increased. In the second part of the paper, we apply a variant of the rounding procedure of Shmoys and Tardos [33] to a problem of scheduling orders of jobs. In this setting, clients place orders, consisting of several products, to a manufacturer owning m unrelated parallel machines. Each product has a machine-dependent processing requirement. The manufacturer has to find an assignment of products to machines (and a schedule on each machine) so as to give the best possible service to his clients. More precisely, we are given a set of machines M = 811 : : : 1 m9, a set of jobs J = 811 : : : 1 n9, and a set of orders O ⊆ 2 J , such that  L∈O L = J . Each job j ∈ J takes p ij units of time to be processed on machine i ∈ M , and each order L has a weight w L depending on its importance. Also, job j has machine-dependent release dates r ij , i.e., it can only be processed on machine i 379