A Lagrangian relaxation algorithm for stochastic fixed interval scheduling problem with non-identical machines and job classes

This paper deals with operational fixed interval scheduling problems under uncertainty caused by random delays. This stochastic programming problem has a deterministic reformulation based on network flow under the assumption that the machines are identical and the multivariate distribution of random delays follows an Archimedean copula. In this paper, we focus on the problem with heterogeneous machines and jobs classes. The deterministic problem with no delays and more than one machine type has been shown to be NP-hard. We generalize the deterministic reformulation of the stochastic problem with random delays for non-identical (heterogeneous) machines. This formulation for more than one machine type loses an important property of total unimodularity that holds for identical machines reformulation using the network flow problem. Therefore, an algorithm based on the Lagrangian relaxation is derived and implemented together with an efficient upper bounding heuristic. Its efficiency is confirmed by solving a large number of simulated instances.