Machine Scheduling with Resource Dependent Processing Times

We consider several parallel machine scheduling settings with the objective to minimize the schedule makespan. The most general of these settings is unrelated parallel machine scheduling. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource. A given amount of that resource, e.g. workers, can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes classical machine scheduling problems, adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of integer programming formulations for relaxations of the respective problems, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham''s list scheduling, we thus prove the existence of constant factor approximation algorithms. Our performance guarantee is 6.83 for the most general case of unrelated parallel machine scheduling. We improve this bound for two special cases, namely to 5.83 whenever the jobs are assigned to machines beforehand, and to (5+e), e>0, whenever the processing times do not depend on the machine. Moreover, we discuss tightness of the relaxations, and derive inapproximability results.operations research and management science;

A PTAS for a resource scheduling problem with arbitrary number of parallel machines

In this paper we study a parallel machine scheduling problem with non-renewable resource constraints. That is, besides the jobs and machines, there is a common non-renewable resource consumed by the jobs, which has an initial stock and some additional supplies over time. Unlike in most previous results, the number of machines is part of the input. We describe a polynomial time approximation scheme for minimizing the makespan

Scheduling Parallel Jobs with Linear Speedup

We consider a scheduling problem where a set of jobs is distributed over parallel machines. The processing time of any job is dependent on the usage of a scarce renewable resource, e.g., personnel. An amount of k units of that resource can be allocated to the jobs at any time, and the more of that resource is allocated to a job, the smaller its processing time. The dependence of processing times on the amount of resources is linear for any job. The objective is to find a resource allocation and a schedule that minimizes the makespan. Utilizing an integer quadratic programming relaxation, we show how to obtain a (3+e)-approximation algorithm for that problem, for any e>0. This generalizes and improves previous results, respectively. Our approach relies on a fully polynomial time approximation scheme to solve the quadratic programming relaxation. This result is interesting in itself, because the underlying quadratic program is NP-hard to solve in general. We also briefly discuss variants of the problem and derive lower bounds.operations research and management science;

Approximation schemes for parallel machine scheduling with non-renewable resources

In this paper the approximability of parallel machine scheduling problems with resource consuming jobs is studied. In these problems, in addition to a parallel machine environment, there are non-renewable resources, like raw materials, energy, or money, consumed by the jobs. Each resource has an initial stock, and some additional supplies at a-priori known moments in time and in known quantities. The schedules must respect the resource constraints as well. The optimization objective is either the makespan, or the maximum lateness. Polynomial time approximation schemes are provided under various assumptions, and it is shown that the makespan minimization problem is APX-complete if the number of machines is part of the input even if there are only two resources. © 2016 Elsevier B.V