Skip to main content
SpringerLink
Log in

Repulsive Assignment Problem

Journal of Optimization Theory and Applications volume 144, pages 255–273 (2010)Cite this article

Abstract

Standard assignment is the problem of obtaining a matching between two sets of respectively persons and positions so that each person is assigned exactly one position and each position receives exactly one person, while a linear decision maker utility function is maximized. We introduce a variant of the problem where the persons individual utilities are taken into account in a way that a feasible solution must satisfy not only the standard assignment constraints, but also an equilibrium constraint of the complementarity type, which we call repulsive. The equilibrium constraint can be, in turn, transformed into a typically large set of linear constraints. Our problem is NP-hard and it is a special case of the assignment problem with side constraints. We study an exact penalty function approach which motivates a heuristic algorithm. We provide computational experiments that show the usefulness of a heuristic mechanism inspired by the exact approach. The heuristics outperforms a state-of-the-art integer linear programming solver.

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

  1. Kuhn, A.W.: The Hungarian method for the assignment problem. Nav. Res. Logist. Q. 2, 83–97 (1955)

    Article  Google Scholar 

  2. Bertsekas, D.P.: A new algorithm for the assignment problem. Math. Program. 21, 152–171 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  3. Little, J., Murty, K., Sweeney, D., Karel, C.: An algorithm for the traveling salesman problem. Oper. Res. 11, 972–989 (1963)

    Article  MATH  Google Scholar 

  4. Koopmans, T.C., Beckmann, M.J.: Assignment problems and the location of economics activities. Econometrica 25, 53–76 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lawler, E.: The quadratic assignment problem. Manag. Sci. 9, 586–599 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burkard, R.E., Çela, E., Pardalos, P.M., Pitsoulis, L.S.: The quadratic assignment problem. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, pp. 241–337. Kluwer, Boston (1998)

    Google Scholar 

  7. Anstreicher, K.M.: Recent advances in the solution of quadratic assignment problems. Math. Program. 97, 27–42 (2003)

    MATH  MathSciNet  Google Scholar 

  8. Martello, S., Toth, P.: Knapsack Problems, Algorithms, and Computer Implementations. Wiley, New York (1974)

    Google Scholar 

  9. Fisher, M.L., Jaikumar, R., Van Wassenhove, L.N.: A multiplier adjustment method for the generalized assignment problem. Manag. Sci. 32, 1095–1103 (1986)

    Article  MATH  Google Scholar 

  10. Cattrysse, D.G., Van Wassenhove, L.N.: A survey of algorithms for the generalized assignment problem. Eur. J. Oper. Res. 60, 260–272 (1992)

    Article  MATH  Google Scholar 

  11. Savelsbergh, M.: A branch and price algorithm for the generalized assignment problem. Eur. J. Oper. Res. 60, 260–272 (1992)

    Article  Google Scholar 

  12. Nauss, R.M.: Solving the generalized assignment problem: An optimizing and heuristic approach. INFORMS J. Comput. 15, 249–266 (2003)

    Article  MathSciNet  Google Scholar 

  13. Yagiura, M., Ibaraki, T., Glover, F.: An ejection chain approach for the generalized assignment problem. INFORMS J. Comput. 16, 133–151 (2004)

    Article  MathSciNet  Google Scholar 

  14. Yagiura, M., Ibaraki, T., Glover, F.: A path relinking approach with ejection chains for the generalized assignment problem. Eur. J. Oper. Res. 169, 548–569 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gaudioso, M., Giallombardo, G., Miglionico, G.: On solving the Lagrangian dual of integer programs via an incremental approach. Comput. Optim. Appl. (2009, to appear)

  16. Pentico, D.W.: Assignment problems: A golden anniversary survey. Eur. J. Oper. Res. 176, 774–793 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mazzola, J.B., Neebe, A.W.: Resource-constrained assignment scheduling. Oper. Res. 34, 560–572 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  18. Cordeau, J.F., Gaudioso, M., Laporte, G., Moccia, L.: A memetic heuristic for the generalized quadratic assignment problem. INFORMS J. Comput. 18, 433–443 (2006)

    Article  MathSciNet  Google Scholar 

  19. Hahn, P., Kim, B.J., Guignard, M., Smith, J., Zhu, Y.R.: An algorithm for the generalized quadratic assignment problem. Comput. Optim. Appl. 40, 351–372 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Cordeau, J.F., Gaudioso, M., Laporte, G., Moccia, L.: The service allocation problem at the Gioia Tauro maritime terminal. Eur. J. Oper. Res. 176, 1167–1184 (2007)

    Article  MATH  Google Scholar 

  21. Moccia, L., Cordeau, J.F., Monaco, M.F., Sammarra, M.: A column generation heuristic for a dynamic generalized assignment problem. Comput. Oper. Res. 36, 2670–2681 (2009)

    Article  MATH  Google Scholar 

  22. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  23. Holland, J.: Adaptation in Natural and Artificial Systems. MIT Press, Cambridge (1992)

    Google Scholar 

  24. Moscato, P., Cotta, C.: A gentle introduction to memetic algorithms. In: Glover, F., Kochenberger, G.A. (eds.) Handbook of Metaheuristics, pp. 105–144. Kluwer, Boston (2003)

    Google Scholar 

  25. Drezner, Z.: A new genetic algorithm for the quadratic assignment problem. INFORMS J. Comput. 15, 320–330 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Elettronica Informatica e Sistemistica, Università della Calabria, Via P. Bucci 41C, 87036, Rende (CS), Italy

    M. Gaudioso & M. F. Monaco

  2. Istituto di Calcolo e Reti ad Alte Prestazioni, Consiglio Nazionale delle Ricerche, Via P. Bucci 41C, 87036, Rende (CS), Italy

    L. Moccia

Authors
  1. M. Gaudioso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Moccia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. F. Monaco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Gaudioso.

Additional information

Communicated by P.M. Pardalos.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gaudioso, M., Moccia, L. & Monaco, M.F. Repulsive Assignment Problem. J Optim Theory Appl 144, 255–273 (2010). https://doi.org/10.1007/s10957-009-9601-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-009-9601-9

Keywords

search

Navigation