31 research outputs found
Lower bounds for Smith's rule in stochastic machine scheduling
We consider the problem to minimize the weighted sum of completion times in nonpreemptive parallel machine scheduling. In a landmark paper from 1986, Kawaguchi and Kyan [5] showed that scheduling the jobs according to the WSPT rule -also known as Smith's rule- has a performance guarantee of . They also gave an instance to show that this bound is tight. We consider the stochastic variant of this problem in which the processing times are exponentially distributed random variables. We show,somehow counterintuitively, that the performance guarantee of the WSEPT rule, the stochastic analogue of WSPT, is not better than 1.229. This constitutes the first lower bound for WSEPT in this setting, and in particular, it shows that even with exponentially distributed processing times, stochastic scheduling has somewhat nastier worst-case examples than deterministic scheduling. In that respect, our analysis sheds new light on the fundamental differences between deterministic and stochastic scheduling
Stochastic scheduling on unrelated machines
Two important characteristics encountered in many real-world scheduling problems are heterogeneous machines/processors and a certain degree of uncertainty about the actual sizes of jobs. The first characteristic entails machine dependent processing times of jobs and is captured by the classical unrelated machine scheduling model.The second characteristic is adequately addressed by stochastic processing times of jobs as they are studied in classical stochastic scheduling models. While there is an extensive but separate literature for the two scheduling models, we study for the first time a combined model that takes both characteristics into account simultaneously. Here, the processing time of job on machine is governed by random variable , and its actual realization becomes known only upon job completion. With being the given weight of job , we study the classical objective to minimize the expected total weighted completion time , where is the completion time of job . By means of a novel time-indexed linear programming relaxation, we compute in polynomial time a scheduling policy with performance guarantee . Here, is arbitrarily small, and is an upper bound on the squared coefficient of variation of the processing times. We show that the dependence of the performance guarantee on is tight, as we obtain a lower bound for the type of policies that we use. When jobs also have individual release dates , our bound is . Via , currently best known bounds for deterministic scheduling are contained as a special case
Scheduling under Linear Constraints
We introduce a parallel machine scheduling problem in which the processing
times of jobs are not given in advance but are determined by a system of linear
constraints. The objective is to minimize the makespan, i.e., the maximum job
completion time among all feasible choices. This novel problem is motivated by
various real-world application scenarios. We discuss the computational
complexity and algorithms for various settings of this problem. In particular,
we show that if there is only one machine with an arbitrary number of linear
constraints, or there is an arbitrary number of machines with no more than two
linear constraints, or both the number of machines and the number of linear
constraints are fixed constants, then the problem is polynomial-time solvable
via solving a series of linear programming problems. If both the number of
machines and the number of constraints are inputs of the problem instance, then
the problem is NP-Hard. We further propose several approximation algorithms for
the latter case.Comment: 21 page
The robust single machine scheduling problem with uncertain release and processing times
In this work, we study the single machine scheduling problem with uncertain
release times and processing times of jobs. We adopt a robust scheduling
approach, in which the measure of robustness to be minimized for a given
sequence of jobs is the worst-case objective function value from the set of all
possible realizations of release and processing times. The objective function
value is the total flow time of all jobs. We discuss some important properties
of robust schedules for zero and non-zero release times, and illustrate the
added complexity in robust scheduling given non-zero release times. We propose
heuristics based on variable neighborhood search and iterated local search to
solve the problem and generate robust schedules. The algorithms are tested and
their solution performance is compared with optimal solutions or lower bounds
through numerical experiments based on synthetic data
Approximation Results for Preemptive Stochastic Online Scheduling
We present first constant performance guarantees for preemptive stochastic scheduling to minimize the sum of weighted completion times. For scheduling jobs with release dates on identical parallel machines we derive policies with a guaranteed performance ratio of 2 which matches the currently best known result for the corresponding deterministic online problem. Our policies apply to the recently introduced stochastic online scheduling model inwhich jobs arrive online over time. In contrast to the previously considered nonpreemptivesetting, our preemptive policies extensively utilize information on processing time distributions other than the first (and second) moments. In order to derive our results we introduce a new nontrivial lower bound on the expected value of an unknown optimal policy that we derive from an optimal policy for the basic problem on a single machine without release dates. This problem is known to be solved optimally by a Gittins index priority rule. This priority index also inspires the design of our policies.computer science applications;