MIP formulations for an application of project scheduling in human resource management



In the literature, various discrete-time and continuous-time mixed-integer linear programming (MIP) formulations for project scheduling problems have been proposed. The performance of these formulations has been analyzed based on generic test instances. The objective of this study is to analyze the performance of discrete-time and continuous-time MIP formulations for a real-life application of project scheduling in human resource management. We consider the problem of scheduling assessment centers. In an assessment center, candidates for job positions perform different tasks while being observed and evaluated by assessors. Because these assessors are highly qualified and expensive personnel, the duration of the assessment center should be minimized. Complex rules for assigning assessors to candidates distinguish this problem from other scheduling problems discussed in the literature. We develop two discrete-time and three continuous-time MIP formulations, and we present problem-specific lower bounds. In a comparative study, we analyze the performance of the five MIP formulations on four real-life instances and a set of 240 instances derived from real-life data. The results indicate that good or optimal solutions are obtained for all instances within short computational time. In particular, one of the real-life instances is solved to optimality. Surprisingly, the continuous-time formulations outperform the discrete-time formulations in terms of solution quality.


OR in the service industries Human resource management Project scheduling Mixed-integer programming Comparative analysis 


  1. Alvarez-Valdes R, Tamarit J (1993) The project scheduling polyhedron: dimension, facets and lifting theorems. Eur J Oper Res 67(2):204–220CrossRefMATHGoogle Scholar
  2. Ambrosino D, Paolucci M, Sciomachen A (2015) Experimental evaluation of mixed integer programming models for the multi-port master bay plan problem. Flex Serv Manuf J 27(2–3):263–284CrossRefGoogle Scholar
  3. Artigues C, Michelon P, Reusser S (2003) Insertion techniques for static and dynamic resource-constrained project scheduling. Eur J Oper Res 149(2):249–267MathSciNetCrossRefMATHGoogle Scholar
  4. Artigues C, Koné O, Lopez P, Mongeau M (2015) Mixed-integer linear programming formulations. In: Schwindt C, Zimmermann J (eds) Handbook on project management and scheduling, vol 1. Springer, Cham, pp 17–41Google Scholar
  5. Bianco L, Caramia M (2013) A new formulation for the project scheduling problem under limited resources. Flex Serv Manuf J 25(1–2):6–24CrossRefGoogle Scholar
  6. Bixby RE (2012) A brief history of linear and mixed-integer programming computation. Doc Math Extra ISMP:107–121MathSciNetMATHGoogle Scholar
  7. Chen X, Grossmann I, Zheng L (2012) A comparative study of continuous-time models for scheduling of crude oil operations in inland refineries. Comput Chem Eng 44:141–167CrossRefGoogle Scholar
  8. Christofides N, Alvarez-Valdés R, Tamarit JM (1987) Project scheduling with resource constraints: a branch and bound approach. Eur J Oper Res 29(3):262–273MathSciNetCrossRefMATHGoogle Scholar
  9. Collins JM, Schmidt FL, Sanchez-Ku M, Thomas L, McDaniel M, Le H (2003) Can basic individual differences shed light on the construct meaning of assessment center evaluations? Int J Sel Assess 11(1):17–29CrossRefGoogle Scholar
  10. Grüter J, Trautmann N, Zimmermann A (2014) An MBLP model for scheduling assessment centers. In: Huisman D, Louwerse I, Wagelmans A (eds) Operations research proceedings 2013. Springer, Berlin, pp 161–167Google Scholar
  11. Kaplan L (1988) Resource-constrained project scheduling with preemption of jobs. PhD thesis, University of MichiganGoogle Scholar
  12. Klein R (2000) Scheduling of resource-constrained projects. Kluwer, AmsterdamCrossRefMATHGoogle Scholar
  13. Koch T, Achterberg T, Andersen E, Bastert O, Berthold T, Bixby RE, Danna E, Gamrath G, Gleixner AM, Heinz S et al (2011) MIPLIB 2010. Math Progr Comput 3(2):103–163MathSciNetCrossRefGoogle Scholar
  14. Kolisch R, Sprecher A (1997) PSPLIB-a project scheduling problem library: OR software-ORSEP operations research software exchange program. Eur J Oper Res 96(1):205–216CrossRefMATHGoogle Scholar
  15. Koné O, Artigues C, Lopez P, Mongeau M (2011) Event-based MILP models for resource-constrained project scheduling problems. Comput Oper Res 38(1):3–13MathSciNetCrossRefMATHGoogle Scholar
  16. Koné O, Artigues C, Lopez P, Mongeau M (2013) Comparison of mixed integer linear programming models for the resource-constrained project scheduling problem with consumption and production of resources. Flex Serv Manuf J 25(1–2):25–47CrossRefGoogle Scholar
  17. Kopanos GM, Kyriakidis TS, Georgiadis MC (2014) New continuous-time and discrete-time mathematical formulations for resource-constrained project scheduling problems. Comput Chem Eng 68:96–106CrossRefGoogle Scholar
  18. Mingozzi A, Maniezzo V, Ricciardelli S, Bianco L (1998) An exact algorithm for the resource-constrained project scheduling problem based on a new mathematical formulation. Manag Sci 44(5):714–729CrossRefMATHGoogle Scholar
  19. Naber A, Kolisch R (2014) MIP models for resource-constrained project scheduling with flexible resource profiles. Eur J Oper Res 239(2):335–348MathSciNetCrossRefMATHGoogle Scholar
  20. Pritsker AAB, Waiters LJ, Wolfe PM (1969) Multiproject scheduling with limited resources: a zero-one programming approach. Manag Sci 16(1):93–108CrossRefGoogle Scholar
  21. Rihm T, Trautmann N (2016) A decomposition approach for an assessment center planning problem. In: Ruiz R, Alvarez-Valdes R (eds) Proceedings of the 15th international conference on project management and scheduling, Valencia, pp 206–209Google Scholar
  22. Stefansson H, Sigmarsdottir S, Jensson P, Shah N (2011) Discrete and continuous time representations and mathematical models for large production scheduling problems: a case study from the pharmaceutical industry. Eur J Oper Res 215(2):383–392MathSciNetCrossRefMATHGoogle Scholar
  23. Vanhoucke M, Coelho J, Debels D, Maenhout B, Tavares LV (2008) An evaluation of the adequacy of project network generators with systematically sampled networks. Eur J Oper Res 187(2):511–524CrossRefMATHGoogle Scholar
  24. Vielma JP (2015) Mixed integer linear programming formulation techniques. SIAM Rev 57(1):3–57MathSciNetCrossRefMATHGoogle Scholar
  25. Zapata JC, Hodge BM, Reklaitis GV (2008) The multimode resource constrained multiproject scheduling problem: alternative formulations. AIChE J 54(8):2101–2119CrossRefGoogle Scholar
  26. Zimmermann A, Trautmann N (2014) Scheduling of assessment centers: an application of resource-constrained project scheduling. In: Fliedner T, Kolisch R, Naber A (eds) Proceedings of the 14th international conference on project management and scheduling, Munich, pp 263–266Google Scholar
  27. Zimmermann A, Trautmann N (2015) A list-scheduling approach for the planning of assessment centers. In: Hanzálek Z, Kendall G, McCollum B, Šůcha P (eds) Proceedings of the multidisciplinary international scheduling conference: theory and application, Prague, pp 541–554Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Tom Rihm
    • 1
  • Norbert Trautmann
    • 1
  • Adrian Zimmermann
    • 1
  1. 1.University of BernBernSwitzerland

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