Journal of Optimization Theory and Applications

, Volume 144, Issue 2, pp 255–273 | Cite as

Repulsive Assignment Problem

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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.

Keywords

Generalized assignment problems Quadratic assignment problems Memetic algorithms Equilibrium constraints 

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References

  1. 1.
    Kuhn, A.W.: The Hungarian method for the assignment problem. Nav. Res. Logist. Q. 2, 83–97 (1955) CrossRefGoogle Scholar
  2. 2.
    Bertsekas, D.P.: A new algorithm for the assignment problem. Math. Program. 21, 152–171 (1981) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Little, J., Murty, K., Sweeney, D., Karel, C.: An algorithm for the traveling salesman problem. Oper. Res. 11, 972–989 (1963) MATHCrossRefGoogle Scholar
  4. 4.
    Koopmans, T.C., Beckmann, M.J.: Assignment problems and the location of economics activities. Econometrica 25, 53–76 (1957) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lawler, E.: The quadratic assignment problem. Manag. Sci. 9, 586–599 (1963) MATHCrossRefMathSciNetGoogle Scholar
  6. 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. 7.
    Anstreicher, K.M.: Recent advances in the solution of quadratic assignment problems. Math. Program. 97, 27–42 (2003) MATHMathSciNetGoogle Scholar
  8. 8.
    Martello, S., Toth, P.: Knapsack Problems, Algorithms, and Computer Implementations. Wiley, New York (1974) Google Scholar
  9. 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) MATHCrossRefGoogle Scholar
  10. 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) MATHCrossRefGoogle Scholar
  11. 11.
    Savelsbergh, M.: A branch and price algorithm for the generalized assignment problem. Eur. J. Oper. Res. 60, 260–272 (1992) CrossRefGoogle Scholar
  12. 12.
    Nauss, R.M.: Solving the generalized assignment problem: An optimizing and heuristic approach. INFORMS J. Comput. 15, 249–266 (2003) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Yagiura, M., Ibaraki, T., Glover, F.: An ejection chain approach for the generalized assignment problem. INFORMS J. Comput. 16, 133–151 (2004) CrossRefMathSciNetGoogle Scholar
  14. 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) MATHCrossRefMathSciNetGoogle Scholar
  15. 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) Google Scholar
  16. 16.
    Pentico, D.W.: Assignment problems: A golden anniversary survey. Eur. J. Oper. Res. 176, 774–793 (2007) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Mazzola, J.B., Neebe, A.W.: Resource-constrained assignment scheduling. Oper. Res. 34, 560–572 (1986) MATHCrossRefMathSciNetGoogle Scholar
  18. 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) CrossRefMathSciNetGoogle Scholar
  19. 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) MATHCrossRefMathSciNetGoogle Scholar
  20. 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) MATHCrossRefGoogle Scholar
  21. 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) MATHCrossRefGoogle Scholar
  22. 22.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979) MATHGoogle Scholar
  23. 23.
    Holland, J.: Adaptation in Natural and Artificial Systems. MIT Press, Cambridge (1992) Google Scholar
  24. 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. 25.
    Drezner, Z.: A new genetic algorithm for the quadratic assignment problem. INFORMS J. Comput. 15, 320–330 (2003) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Dipartimento di Elettronica Informatica e SistemisticaUniversità della CalabriaRende (CS)Italy
  2. 2.Istituto di Calcolo e Reti ad Alte PrestazioniConsiglio Nazionale delle RicercheRende (CS)Italy

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