Advertisement

A Local Search Heuristic for Solving the Maximum Dispersion Problem

  • Mahdi Moeini
  • David Goerzen
  • Oliver Wendt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10751)

Abstract

In this paper, we are interested in studying the Maximum Dispersion Problem (MaxDP). In this problem, a set of objects are given such that each object has a non-negative weight. The objective of the MaxDP consists in partitioning the given set of objects into a predefined number of classes. The partitioning is subject to some conditions. First, the overall dispersion of objects, assigned to each class, must be maximized. Second, there is a predefined target weight assigned to each class and the total weight of each class must belong to an interval surrounding its target weight. It has been proven that the MaxDP is NP-hard and, consequently, difficult to solve by classical exact methods. In this paper, we provide a Variable Neighborhood Search (VNS) algorithm for solving the MaxDP. In order to evaluate the efficiency of the introduced VNS, we carried out numerical experiments on randomly generated instances. Then, we compared the results of our VNS algorithm with those provided by the standard solver Gurobi. According to our results, our VNS algorithm provides high-quality solutions within a short computation time and dominates the solver Gurobi.

Keywords

Maximum Dispersion Problem Heuristic Variable Neighborhood Search Local Search 

Notes

Acknowledgements

The authors acknowledge the chair of Business Information Systems and Operations Research (BISOR) at the TU-Kaiserslautern (Germany) for the financial support, through the research program “CoVaCo”.

References

  1. 1.
    Baker, K.R., Powell, S.G.: Methods for assigning students to groups: a study of alternative objective functions. J. Oper. Res. Soc. 53(4), 397–404 (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    Balas, E., Mazzola, J.: Nonlinear 0–1 programming: I. Linearization techniques. Math. Program. 30(1), 1–21 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brimberg, J., Mladenović, N., Urošević, D.: Solving the maximally diverse grouping problem by skewed general variable neighborhood search. Inf. Sci. 295, 650–675 (2015)CrossRefGoogle Scholar
  4. 4.
    Erkut, E.: The discrete p-dispersion problem. Eur. J. Oper. Res. 46(1), 48–60 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fernández, E., Kalcsics, J., Nickel, S., Ríos-Mercado, R.Z.: A novel maximum dispersion territory design model arising in the implementation of the WEEE-directive. J. Oper. Res. Soc. 61(3), 503–514 (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fernández, E., Kalcsics, J., Nickel, S.: The maximum dispersion problem. Omega 41(4), 721–730 (2013)CrossRefGoogle Scholar
  7. 7.
    Glover, F., Wolsey, R.: Converting the 0–1 polynomial programming problem to a 0–1 linear program. Oper. Res. 22(1), 455–60 (1974)CrossRefGoogle Scholar
  8. 8.
    Glover, F., Ching-Chung, K., Dhir, K.: A discrete optimization model for preserving biological diversity. Appl. Math. Model. 19(11), 696–701 (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Goeke, D., Moeini, M., Poganiuch, D.: A variable neighborhood search heuristic for the maximum ratio clique problem. Comput. Oper. Res. 87, 283–291 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hansen, P., Mladenović, N., Brimberg, J., Moreno Prez, J.A.: Variable neighborhood search. In: Gendreau, M., Potvin, J.Y. (eds.) Handbook of Metaheuristics, pp. 61–86. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-1-4614-6940-7_12 CrossRefGoogle Scholar
  11. 11.
    Martí, R., Gallego, M., Duarte, A.: A branch and bound algorithm for the maximum diversity problem. Eur. J. Oper. Res. 200(1), 36–44 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Moeini, M., Wendt, O.: A heuristic for solving the maximum dispersion problem. In: Fink, A., Fügenschuh, A., Geiger, M.J. (eds.) Operations Research Proceedings 2016. ORP, pp. 405–410. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-55702-1_54 CrossRefGoogle Scholar
  14. 14.
    Palubeckis, G., Karčiauskas, E., Riškus, A.: Comperative performance of three metaheuristic approaches for the maximally diverse grouping problem. Inf. Technol. Control 40(4), 277–285 (2011)Google Scholar
  15. 15.
    Prokopyev, O., Kong, N., Martinez-Torres, D.: The equitable dispersion problem. Eur. J. Oper. Res. 197(1), 59–67 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.BISOR, University of KaiserslauternKaiserslauternGermany

Personalised recommendations