Computational Management Science

, Volume 14, Issue 1, pp 161–177 | Cite as

Flow-based formulations for operational fixed interval scheduling problems with random delays

  • Martin Branda
  • Štěpán Hájek
Original Paper


We deal with operational fixed interval scheduling problem with random delays in job processing times. We formulate two stochastic programming problems. In the first problem with a probabilistic objective, all jobs are processed on available machines and the goal is to obtain a schedule with the highest attainable reliability. The second problem is to select a subset of jobs with the highest reward under a chance constraint ensuring feasibility of the schedule with a prescribed probability. We assume that the multivariate distribution of delays follows an Archimedean copula, whereas there are no restrictions on marginal distributions. We introduce new deterministic integer linear reformulations based on flow problems. We compare the formulations with the extended robust coloring problem, which was shown to be a deterministic equivalent to the stochastic programming problem with probabilistic objective by Branda et al. (Comput Ind Eng 93:45–54, 2016). In the numerical study, we report average computational times necessary to solve a large number of simulated instances. It turns out that the new flow-based formulation helps to solve the FIS problems considerably faster than the other one.


Fixed interval scheduling Random delays Stochastic programming Flow-based formulation Archimedean copula Integer programming 

Mathematics Subject Classification

90C15 90B36 90C10 



The authors gratefully acknowledge that this project was supported by the Czech Science Foundation under the Grant P402/12/G097 (DYME). We would like to express our gratitude to the anonymous referees for their valuable comments which helped us to improve the paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Probability and Mathematical Statistics, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic

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