Continuous-time formulations for multi-mode project scheduling

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This paper reviews compact continuous-time formulations for the multi-mode resource-constrained project scheduling problem. Specifically, we first point out a serious flaw in an existing start-end-event-based formulation owing to inconsistent mode choices. We propose two options to formulate the missing constraints and consider an equivalent reformulation with sparser constraint matrix. Second, we formulate an aggregate variant of an existing model that relies on on-off-events, and we clarify the role of mode consistency issues in such models. Third, we suggest two variants of an existing network flow formulation. We enhance our models by adapting several techniques that have been used previously, e.g., in cases with only a single mode. A large set of benchmark instances from the literature provides the basis for an up-to-date and fair computational study with an out-of-the-box solver package. We compare our models against two models from the literature. Our experiments assert confidently that network flow formulations prevail in the test bed, and they provide a hint on why event-based models become less competitive in multi-mode settings.