Scheduling parallel dedicated machines with the speeding‐up resource

We consider a problem of scheduling jobs on m parallel machines. The machines are dedicated, i.e., for each job the processing machine is known in advance. We mainly concentrate on the model in which at any time there is one unit of an additional resource. Any job may be assigned the resource and this reduces its processing time. A job that is given the resource uses it at each time of its processing. No two jobs are allowed to use the resource simultaneously. The objective is to minimize the makespan. We prove that the two-machine problem is NP-hard in the ordinary sense, describe a pseudopolynomial dynamic programming algorithm and convert it into an FPTAS. For the problem with an arbitrary number of machines we present an algorithm with a worst-case ratio close to 3/2, and close to 3, if a job can be given several units of the resource. For the problem with a fixed number of machines we give a PTAS. Virtually all algorithms rely on a certain variant of the linear knapsack problem (maximization, minimization, multiple-choice, bicriteria). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008

[1]  Hans Kellerer,et al.  Scheduling parallel dedicated machines under a single non-shared resource , 2003, European Journal of Operational Research.

[2]  Jan Karel Lenstra,et al.  Scheduling subject to resource constraints: classification and complexity , 1983, Discret. Appl. Math..

[3]  Hans Kellerer,et al.  Approximating Multi-objective Knapsack Problems , 2001, WADS.

[4]  Dvir Shabtay,et al.  Single and two-resource allocation algorithms for minimizing the maximal lateness in a single machine , 2004, Comput. Oper. Res..

[5]  Adam Janiak,et al.  Single machine scheduling subject to deadlines and resource dependent processing times , 1996 .

[6]  Eugeniusz Nowicki,et al.  A survey of results for sequencing problems with controllable processing times , 1990, Discret. Appl. Math..

[7]  Jacek Blazewicz,et al.  Scheduling in Computer and Manufacturing Systems , 1990 .

[8]  Hans Kellerer,et al.  Scheduling problems for parallel dedicated machines under multiple resource constraints , 2003, Discret. Appl. Math..

[9]  Alexander Grigoriev,et al.  Scheduling Parallel Jobs with Linear Speedup , 2005, WAOA.

[10]  Gerd Finke,et al.  Scheduling with Discrete Resource Constraints , 2004, Handbook of Scheduling.

[11]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[12]  Hans Kellerer,et al.  Approximating Multiobjective Knapsack Problems , 2002, Manag. Sci..

[13]  Mihalis Yannakakis,et al.  On the approximability of trade-offs and optimal access of Web sources , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.