Scheduling parallel dedicated machines with the speeding‐up resource

Abstract

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

Scheduling jobs with time-resource tradeoff via nonlinear programming

“…As a special case, this comprises linear time-resource tradeoff functions. Notice that our result generalizes the (3 + ε)-approximation of [10]. Although we obtain the same performance bound, we stress that we derive the first polynomial time approximation algorithms for problems with succinctly encodable time-resource tradeoff functions.…”

“…More recently, such problems with the assumption that jobs are distributed over the machines beforehand have been discussed by Kellerer and Strusevich [9][10][11]. They use the term dedicated machine scheduling.…”

“…Although we obtain the same performance bound, we stress that we derive the first polynomial time approximation algorithms for problems with succinctly encodable time-resource tradeoff functions. Previous approaches such as [6,10] generally do not yield polynomial time algorithms.…”

“…They show weak NP-hardness for the case where the number of machines is fixed, and strong NP-hardness for an arbitrary number of machines. In [10], Kellerer and Strusevich derived a (3 + ε)-approximation algorithm again using a linear program with nk variables. For the problem when the number of machines is constant, they derive a polynomial time approximation scheme.…”

Task scheduling system for UAV operations in indoor environment

“…Several researchers have studied various scheduling problems with different constraints. Kellerer and Strusevisch [20] showed that the two-machine problem with one additional resource type is NP-hard. Hou and Guo [19] proposed an MIP model of parallel machine scheduling with resource constraints and sequence-dependent setup times, which was also reported to be NP-hard.…”