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;

An EPTAS for Scheduling on Unrelated Machines of Few Different Types

In the classical problem of scheduling on unrelated parallel machines, a set of jobs has to be assigned to a set of machines. The jobs have a processing time depending on the machine and the goal is to minimize the makespan, that is the maximum machine load. It is well known that this problem is NP-hard and does not allow polynomial time approximation algorithms with approximation guarantees smaller than $1.5$ unless P$=$NP. We consider the case that there are only a constant number $K$ of machine types. Two machines have the same type if all jobs have the same processing time for them. This variant of the problem is strongly NP-hard already for $K=1$. We present an efficient polynomial time approximation scheme (EPTAS) for the problem, that is, for any $\varepsilon > 0$ an assignment with makespan of length at most $(1+\varepsilon)$ times the optimum can be found in polynomial time in the input length and the exponent is independent of $1/\varepsilon$. In particular we achieve a running time of $2^{\mathcal{O}(K\log(K) \frac{1}{\varepsilon}\log^4 \frac{1}{\varepsilon})}+\mathrm{poly}(|I|)$, where $|I|$ denotes the input length. Furthermore, we study three other problem variants and present an EPTAS for each of them: The Santa Claus problem, where the minimum machine load has to be maximized; the case of scheduling on unrelated parallel machines with a constant number of uniform types, where machines of the same type behave like uniformly related machines; and the multidimensional vector scheduling variant of the problem where both the dimension and the number of machine types are constant. For the Santa Claus problem we achieve the same running time. The results are achieved, using mixed integer linear programming and rounding techniques

NEH-based heuristics for the permutation flowshop scheduling problem to minimize total tardiness

Since Johnson׳s seminal paper in 1954, scheduling jobs in a permutation flowshop has been receiving the attention of hundreds of practitioners and researchers, being one of the most studied topics in the Operations Research literature. Among the different objectives that can be considered, minimising the total tardiness (i.e. the sum of the surplus of the completion time of each job over its due date) is regarded as a key objective for manufacturing companies, as it entails the fulfilment of the due dates committed to customers. Since this problem is known to be NP-hard, most research has focused on proposing approximate procedures to solve it in reasonable computation times. Particularly, several constructive heuristics have been proposed, with NEHedd being the most efficient one, serving also to provide an initial solution for more elaborate approximate procedures. In this paper, we first analyse in detail the decision problem depending on the generation of the due dates of the jobs, and discuss the similarities with different related decision problems. In addition, for the most characteristic tardiness scenario, the analysis shows that a huge number of ties appear during the construction of the solutions done by the NEHedd heuristic, and that wisely breaking the ties greatly influences the quality of the final solution. Since no tie-breaking mechanism has been designed for this heuristic up to now, we propose several mechanisms that are exhaustively tested. The results show that some of them outperform the original NEHedd by about 25% while keeping the same computational requirements.Ministerio de Ciencia e Innovación DPI2010-15573/DPIMinisterio de Ciencia e Innovación DPI2013-44461-P/DP