Skip to main content

Stable multi-skill workforce assignments

Annals of Operations Research volume 213, pages 95–114 (2014)Cite this article

Abstract

This paper analyzes stability in multi-skill workforce assignments of technicians and jobs. In our stability analysis, we extend the notion of blocking pairs as stated in the Marriage model of Gale-Shapley to the multi-skill workforce assignment. It is shown that finding stable assignments is NP-hard. A special case turns out to be solvable in polynomial time. For the general case, we give a characterization of the set of stable assignments by means of linear inequalities involving binary variables. We propose an integer programming (IP) model to construct optimal stable assignments with several objectives. In the computational results, we observe that it is easier to attain stability in instances with easy jobs and we consider a range of instances to show how fast the solution time increases. Open questions and further directions are discussed in the conclusion section.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  • Avramidis, N. A., Chan, W., Gendreau, M., L’Ecuyer, P., & Pisacane, O. (2010). Optimizing daily agent scheduling in a multi-skill call center. Eur J Oper Res, 200(3), 822–832.

    Article  Google Scholar 

  • Baïou, M., & Balinski, M. (2000a). The stable admissions polytope. Math Program, 87(3), 427–439.

    Article  Google Scholar 

  • Baïou, M., & Balinski, M. (2000b). Many-to-many matching: stable polyandrous polygamy (or polygamous polyandry). Discrete Appl Math, 101, 1–12.

    Article  Google Scholar 

  • Bellenguez, M. O., & Neron, E. (2004). Lower bounds for the multi-skill project scheduling problem with hierarchical levels of skills. In Proceedings of practice and theory of automated timetabling (PATAT2004), Pittsburgh, PA, USA (pp. 429–432).

  • Bellenguez, M. O., & Neron, E. (2007). A Branch-and-bound method for solving multi-skill project scheduling problem. RAIRO Rech Opér, 41, 155–170.

    Article  Google Scholar 

  • Cordeau, J. F., Laporte, G., Pasin, F., & Ropke, S. (2010). Scheduling technicians and tasks in a telecommunication company. J Sched, 13(4), 393–409.

    Article  Google Scholar 

  • Estellon, B., Gardi, F., & Nouioua, K. (2009). High-performance local search for task scheduling with human resource allocation. In Lect Notes Comput Sci (Vol. 5752, pp. 1–15).

    Google Scholar 

  • Fırat, M., & Hurkens, C. A. J. (2012). An improved MIP-based approach for a multi-skill workforce scheduling problem. J Sched, 15(3), 363–380.

    Article  Google Scholar 

  • Fleiner, T., Irving, R. W., & Manlove, D. F. (2007). Efficient algorithms for generalized stable marriage and roommates problems. Theor Comput Sci, 381, 162–176.

    Article  Google Scholar 

  • Gale, D., & Shapley, L. S. (1962). College admissions and the stability of marriage. Am Math Mon, 69(1), 9–15.

    Article  Google Scholar 

  • Gale, D., & Sotomayor, M. (1985). Some remarks on the stable matching problem. Discrete Appl Math, 11, 223–232.

    Article  Google Scholar 

  • Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: a guide to the theory of NP-completeness (p. 221). San Francisco: Freeman.

    Google Scholar 

  • Gelain, M., Pini, M. S., Rossi, F., Venable, K. B., & Walsh, T. (2010). Local search for stable marriage problem. In Proceedings of COMSOC 2010, Düsseldorf, Germany.

    Google Scholar 

  • Heimerl, C., & Kolisch, R. (2010). Scheduling and staffing multiple projects with a multi-skilled workforce. IEEE Spectr, 32(2), 343–368.

    Google Scholar 

  • Hurkens, C. A. J. (2009). Incorporating the strength of MIP modeling in schedule construction. RAIRO Rech Opér, 43, 409–420.

    Article  Google Scholar 

  • Iwama, K., Manlove, D. F., Miyazaki, S., & Morita, Y. (1999). Stable marriage with incomplete lists and ties. In Lecture notes in computer science: Vol. 1644. Proceedings of ICALP 99 (pp. 443–452). Berlin: Springer.

    Google Scholar 

  • Iwama, K., Miyazaki, S., & Okamoto, K. (2004). A (2-clogN/N)-approximation algorithm for the stable marriage problem. In Lecture notes in computer science: Vol. 3111. Proceedings of SWAT 2004 (pp. 349–361). Berlin: Springer.

    Google Scholar 

  • Iwama, K., Miyazaki, S., & Yamauchi, N. (2008). A (2-\(c 1 / \sqrt{N}\))-approximation algorithm for the stable marriage problem. Algorithmica, 51(3), 902–914.

    Article  Google Scholar 

  • Li, H., & Womer, K. (2009). Scheduling projects with multi-skilled personnel by a hybrid MILP/CP benders decomposition algorithm. J Sched, 12(3), 281–298.

    Article  Google Scholar 

  • Valls, V., Perez, A., & Quintanilla, S. (2009). Skill workforce scheduling in service centers. Eur J Oper Res, 193(3), 791–804.

    Article  Google Scholar 

  • Vande Vate, J. H. (1989). Linear programming brings marital bliss. OR Lett, 8, 147–153.

    Google Scholar 

Download references

Acknowledgements

This research is supported by France Telecom/TUE Research agreement No. 46145963. Thanks are due to Gerhard J. Woeginger for helpful discussions on complexity results and due to Judith C.M. Keijsper for her remarks on the notation of this paper. The reviewers deserve special thanks due to their comments on our proofs and on our IP models. Their comments helped a lot in improving the content of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Murat Fırat & C. A. J. Hurkens

  2. France Telecom/R&D/BIZZ/DIAM 905, rue Albert Einstein, 06921, Sophia-Antipolis Cedex, France

    Alexandre Laugier

Authors
  1. Murat Fırat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. A. J. Hurkens
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexandre Laugier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Murat Fırat.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fırat, M., Hurkens, C.A.J. & Laugier, A. Stable multi-skill workforce assignments. Ann Oper Res 213, 95–114 (2014). https://doi.org/10.1007/s10479-012-1224-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-012-1224-0

Keywords

  • Multi-skill workforce schedules
  • Stable assignments
  • Instability
  • Blocking pair
  • University admissions problem
  • Three-dimensional matching problem
  • Optimal stable assignments