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NEO 2016 pp 183-202 | Cite as

A New Local Search Heuristic for the Multidimensional Assignment Problem

  • Sergio Luis  Pérez Pérez
  • Carlos E. Valencia
  • Francisco Javier Zaragoza Martínez
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 731)

Abstract

The Multidimensional Assignment Problem (MAP) is a natural extension of the well-known assignment problem. The most studied case of the MAP is the 3-dimensional Assignment Problem (3AP), though in recent years some local search heuristics and a memetic algorithm were proposed for the general case. Until now, a memetic algorithm has been proven to be the best-known option to solve MAP instances and it uses some procedures called dimensionwise variation heuristics as part of the improvement of individuals. We propose a new local search heuristic, based on ideas from dimensionwise variation heuristics, which consider a bigger space of neighborhoods, providing higher quality solutions for the MAP. Our main contribution is a generalization of several local search heuristics known from the literature, the conceptualization of a new one, and the application of exact techniques to find local optimum solutions at its neighborhoods. The results of computational evaluation show how our heuristic outperforms the previous local search heuristics and its competitiveness against a state-of-the-art memetic algorithm.

Keywords

Multidimensional assignment problem Local search Exact technique Heuristic Neighborhood 

Notes

Acknowledgements

We would like to thank to M. Saltzman who provided us with the instances used for him and E. Balas many years ago and, for show us his interest in the work that we are doing about this problem. We also want to thank to D. Karapetyan and G. Gutin who provided us with the instances that they used at both of their papers of 2011 about the MAP and, for provided us with the source code that they used for the generation of such families of instances, which was also very useful to accomplish this work. A special thanks to M. Vargas for provided us with a state-of-the-art implementation that solves 2AP through an auction algorithm. Finally, we also thank to the Mexican institutions Conacyt and Comecyt who supported to the realization of this researching work.

References

  1. 1.
    Aiex, R., Resende, M., Pardalos, P., Toraldo, G.: GRASP with path relinking for three-index assignment. J. Comput. 17, 224–247 (2005). doi: 10.1287/ijoc.1030.0059 MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andrijich, S.M., Caccetta, L.: Solving the multisensor data association problem. Nonlinear Anal.: Theory Methods Appl. 47, 5525–5536 (2001). doi: 10.1016/S0362-546X(01)00656-3
  3. 3.
    Balas, E., Saltzman, M.: An algorithm for the three-index assignment problem. Oper. Res. 39, 150–161 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bekker, H., Braad, E., Goldengorin, B.: Using bipartite and multidimensional matching to select the roots of a system of polynomial equations. Comput. Sci. Appl. - ICCSA 2005, 397–406 (2005). doi: 10.1007/11424925_43
  5. 5.
    Bozdogan, A., Efe, M.: Ant colony optimization heuristic for the multidimensional assignment problem in target tracking. In: 2008 IEEE Radar Conference, pp. 1–6 (2008). doi: 10.1109/RADAR.2008.4720822
  6. 6.
    Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Crama, Y., Spieksma, F.: Approximation algorithms for three-dimensional assignment problems with triangle inequalities. Eur. J. Oper. Res. 60, 273–279 (1922). doi: 10.1016/0377-2217(92)90078-N CrossRefzbMATHGoogle Scholar
  8. 8.
    Dasgupta, S., Papadimitriou, C.H., Vazirani, U.V.: Algorithms. Mc Graw Hill Higher Education, New York (2008)Google Scholar
  9. 9.
    Edmonds, J., Karp, R.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 248–264 (1972). doi: 10.1145/321694.321699 CrossRefzbMATHGoogle Scholar
  10. 10.
    Ferland, J., Roy, S.: Timetabling problem for university as assignment of activities to resources. J. Comput. Oper. Res. 12, 207–218 (1985). doi: 10.1016/0305-0548(85)90045-0 CrossRefzbMATHGoogle Scholar
  11. 11.
    Frieze, A.: Complexity of a 3-dimensional assignment problem. Eur. J. Oper. Res. 13, 161–164 (1983). doi: 10.1016/0377-2217(83)90078-4 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  13. 13.
    Grundel, D., Pardalos, P.: Test problem generator for the multidimensional assignment problem. Comput. Optim. Appl. 30, 133–146 (2005). doi: 10.1007/s10589-005-4558-6 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gurobi Optimization, Inc.: Gurobi optimizer reference manual (2015). http://www.gurobi.com
  15. 15.
    Gutin, G., Goldengorin, B., Huang, J.: Worst case analysis of max-regret, greedy and other heuristics for multidimensional assignment and traveling salesman problems. Approximation and Online Algorithms. Lecture Notes in Computer Science, vol. 4368, pp. 214–225 (2007). doi: 10.1007/11970125_17
  16. 16.
    Huang, G., Lim, A.: A hybrid genetic algorithm for three-index assignment problem. Eur. J. Oper. Res. 172, 249–257 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jiang, H., Xuan, J., Zhang, X.: An approximate muscle guided global optimization algorithm for the three-index assignment problem. Evol. Comput. CEC 2008, 2404–2410 (2008)Google Scholar
  18. 18.
    Karapetyan, D.: Source codes and extra tables. http://www.cs.nott.ac.uk/~pszdk/?page=publications
  19. 19.
    Karapetyan, D., Gutin, G.: Local search heuristics for the multidimensional assignment problem. J. Heuristics 17, 201–249 (2011). doi: 10.1007/s10732-010-9133-3 CrossRefzbMATHGoogle Scholar
  20. 20.
    Karapetyan, D., Gutin, G.: A new approach to population sizing for memetic algorithms: a case study for the multidimensional assignment problem. Evol. Comput. 19, 345–371 (2011)CrossRefGoogle Scholar
  21. 21.
    Karapetyan, D., Gutin, G., Goldengorin, B.: Empirical evaluation of construction heuristics for the multidimensional assignment problem. London Algorithmics 2008: Theory and Practice, pp. 107–122. College Publications, London (2009)Google Scholar
  22. 22.
    Karp, R.: Complexity of Computer Computations: Reducibility Among Combinatorial Problems, pp. 85–103. Springer, Berlin (1972). doi: 10.1007/978-1-4684-2001-2_9
  23. 23.
    Koopmans, T., Beckmann, M.: Assignment problems and the location of economic activities. Cowles Found. Res. Econ. 25, 56–76 (1955)zbMATHGoogle Scholar
  24. 24.
    Kuhn, H.: The Hungarian method for the assignment problem. Naval Res. Logist. Q. 2, 83–97 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Magos, D.: Problem instances for multi-index assignment problems. http://users.teiath.gr/dmagos/MIAinstances
  26. 26.
    Magos, D., Mourtos, I.: Clique facets of the axial and planar assignment polytopes. Discret. Optim. 6, 394–413 (2009). doi: 10.1016/j.disopt.2009.05.001 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nguyen, D., Le Thi, H., Pham Dinh, T.: Solving the multidimensional assignment problem by a cross-entropy method. J. Comb. Optim. 27, 808–823 (2014). doi: 10.1007/s10878-012-9554-z
  28. 28.
    Pierskalla, W.: Letter to the editor - the multidimensional assignment problem. Oper. Res. 16, 422–431 (1968). doi: 10.1287/opre.16.2.422
  29. 29.
    Robertson, A.: A set of greedy randomized adaptive local search procedure (GRASP) implementations for the multidimensional assignment problem. Comput. Optim. Appl. 19, 145–164 (2001). doi: 10.1023/A:1011285402433 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Schell, E.: Distribution of a product by several properties. In: Directorate of Management Analysis, Second Symposium in Linear Programming, vol. 2, pp. 615–642 (1955)Google Scholar
  31. 31.
    Spieksma, F., Woeginger, G.: Geometric three-dimensional assignment problems. Eur. J. Oper. Res. 91, 611–618 (1996). doi: 10.1016/0377-2217(95)00003-8

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Sergio Luis  Pérez Pérez
    • 1
  • Carlos E. Valencia
    • 2
  • Francisco Javier Zaragoza Martínez
    • 3
  1. 1.Posgrado en OptimizaciónUniversidad Autónoma Metropolitana AzcapotzalcoMexico CityMexico
  2. 2.Departamento de MatemáticasCentro de Investigación y de Estudios Avanzados del IPNMexico CityMexico
  3. 3.Departamento de SistemasUniversidad Autónoma Metropolitana AzcapotzalcoMexico CityMexico

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