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On the efficient modeling and solution of the multi-mode resource-constrained project scheduling problem with generalized precedence relations


For variants of the single-mode resource-constrained project scheduling problem, state-of-the-art exact algorithms combine a Branch and Bound algorithm with principles from Constraint Programming and Boolean Satisfiability Solving. In our paper, we propose new exact approaches extending the above principles to the multi-mode RCPSP (MRCPSP) with generalized precedence relations (GPRs). More precisely, we implemented two constraint handlers cumulativemm and gprecedencemm for the optimization framework SCIP. With the latter, one can model renewable resource constraints and GPRs in the context of multi-mode activities, respectively. Moreover, they integrate domain propagation and explanation generation techniques for the above problem characteristics. We formulate three SCIP-models for the MRCPSP with GPRs, two without and one with our constraint handler gprecedencemm. Our computational results on instances from the literature with 30, 50 and 100 activities show that the addition of this constraint handler significantly strengthens the SCIP-model. Moreover, we outperform the state-of-the-art exact approach on instances with 50 activities when imposing time limits of 27 s. In addition, we close (find the optimal solution and prove its optimality for) 289 open instances and improve the best known makespan for 271 instances from the literature.

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Fig. 1


  1. Note that gprecedencemm captures the explanation generation variant which uses the potentially stronger explanations (see Sect. 5.2). Preliminary experiments have shown that this version outperforms the weaker explanation generation variant in all categories.

  2. GPRs can be enforced by (15) in a valid way, i.e. it can be used in a standalone manner to model GPRs. Therefore, we speak of an “exact” formulation of GPRs through gprecedencemm.

  3. These are available Online (2014).

  4. For the 30-job instances, the respective upper bounds vary on average by at least 80 % from the best known makespan from the literature.

  5. For the 50-job instances, the respective upper bounds vary on average by at least 55 % from the best known makespan from the literature.

  6. For the 100-job instances, the upper bounds used in SCIPExStrBest20 deviate on average by at least 25 % from the best known makespan.


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We would like to thank the SCIP team and especially Stefan Heinz and Jens Schulz for their helpful information about the implementation of constraint handlers. Moreover, we are very grateful to the associate editor and to the anonymous reviewers for their highly valuable comments. These helped a lot to improve the presentation and the results of our paper.

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Schnell, A., Hartl, R.F. On the efficient modeling and solution of the multi-mode resource-constrained project scheduling problem with generalized precedence relations. OR Spectrum 38, 283–303 (2016).

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  • Project scheduling
  • Constraint programming
  • Multi-mode
  • Generalized precedence relations
  • SAT solving