Skip to main content
SpringerLink
Log in

The capacitated dispersion problem: an optimization model and a memetic algorithm

  • Regular Research Paper
  • Published:
Memetic Computing Aims and scope Submit manuscript

Abstract

The challenge of maximizing the diversity of a collection of points arises in a variety of settings, and the growing interest of dealing with diversity resulted in an effort to study these problems in the last few years. Generally speaking, maximizing diversity consists in selecting a subset of points from a given set in such a way that a measure of their diversity is maximized. Different objective functions have been proposed to capture the notion of diversity, being the sum and the minimum of the distances between the selected points the most widely used. However, in all these models, the number of points to be selected is established beforehand, which in some settings can be unrealistic. In this paper, we target a variant recently introduced in which the model includes capacity values, which reflects the real situation in many location problems. We propose a mathematical model and a heuristic based on the Scatter Search methodology to maximize the diversity while satisfying the capacity constraint. Scatter search is a memetic algorithm hybridizing evolutionary global search with a problem-specific local search. Our empirical analysis with previously reported instances shows that the mathematical model implemented in Gurobi solves to optimality many more instances than the previous published model, and the heuristic outperforms a very recent development based on GRASP. We present a statistical analysis that permits us to draw significant conclusions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Cotta C, Mathieson L, Moscato P (2018) Memetic Algorithms, In: Handbook of Heuristics (Martí, Pardalos, and Resende eds.), pp 607–638, Springer, Heidelberg

  2. Duarte A, Martí R (2007) Tabu search and GRASP for the MDP. Eur J Oper Res 178:71–84

    Article  Google Scholar 

  3. Duarte A, Sánchez-Oro J, Resende M, Glover F, Martí R (2015) GRASP with exterior path relinking for differential dispersion minimization. Inf Sci 296:46–60

    Article  Google Scholar 

  4. Feo T, Resende MGC (1995) Greedy randomized adaptive search procedures. J Global Optim 6:109–133

    Article  MathSciNet  Google Scholar 

  5. Gallego M, Duarte A, Laguna M, Martí R (2009) Hybrid heuristics for the maximum diversity problem. Comput Optim Appl 44(3):411–426

    Article  MathSciNet  Google Scholar 

  6. Ghosh JB (1996) Computational aspects of the maximum diversity problem. Operations Research Letters 19:175–181

    Article  MathSciNet  Google Scholar 

  7. Glover F (1977) Heuristics for integer programming using surrogate constraints. Decis Sci 8:156–166

    Article  Google Scholar 

  8. Glover F, Laguna M (1997) Tabu Search. Kluwer, Norwell, MA

    Book  Google Scholar 

  9. Glover F (1998) A template for scatter search and path relinking. In: Hao J-K, Lutton E, Ronald E, Schoenauer M, Snyers D (eds) Artificial evolution. Lecture Notes in Computer Science, vol 1363. Springer, Berlin, pp 13–54

    Google Scholar 

  10. Glover F, Kuo CC, Dhir KS (1995) A discrete optimization model for preserving biological diversity. Appl Math Model 19:696–701

    Article  Google Scholar 

  11. Glover F, Kuo CC, Dhir KS (1998) Heuristic algorithms for the maximum diversity problem. J Inf Optim Sci 19(1):109–132

    MATH  Google Scholar 

  12. Kuo CC, Glover F, Dhir KS (1993) Analyzing and modeling the maximum diversity problem by zero-one programming. Decis Sci 24:1171–1185

    Article  Google Scholar 

  13. Laguna M, Martí R (2003) Scatter search: methodology and implementations in C. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  14. Martí R, Duarte A (2010) The MDPLIB at Optsicom, http://grafo.etsii.urjc.es/optsicom/

  15. Martí R, Gallego M, Duarte A, Pardo E (2013) Heuristics and Metaheuristics for the maximum diversity problem. J Heuristics 19(4):591–615

    Article  Google Scholar 

  16. Martí R, Gallego M, Duarte A (2010) A branch and bound algorithm for the maximum diversity problem. Eur J Oper Res 200(1):36–44

    Article  Google Scholar 

  17. Martí R, Laguna M, Campos V (2005) Scatter search versus genetic algorithms: an experimental evaluation with permutation problems. In: Rego C, Alidaee B (eds) Metaheuristic optimization via adaptive memory and evolution: tabu search and scatter search. Kluwer Academic Publishers, Norwell, MA, pp 263–282

    Chapter  Google Scholar 

  18. Martí R, Laguna M, Glover F (2006) Principles of scatter search. Eur J Oper Res 169:359–372

    Article  MathSciNet  Google Scholar 

  19. Martínez-Gavara A, Campos V, Laguna M, Martí R (2017) Heuristic solution approaches for the maximum minsum dispersion problem. J Glob Optim 67(3):671–686

    Article  MathSciNet  Google Scholar 

  20. Neri F, Cotta C (2012) Memetic algorithms and memetic computing optimization: a literature review. Swarm and evolutionary computation, vol 2. Elsevier, Amsterdam, pp 1–14

    Google Scholar 

  21. Neri F, Cotta C (2012) A primer on memetic algorithms, handbook of memetic algorithms, chapter 4. In: Neri F, Cotta C, Moscato P (eds) Studies in computational intelligence, vol 379. Springer, Berlin, pp 43–54

    Google Scholar 

  22. Neri F (2012) Diversity management in memetic algorithms. In: Neri F, Cotta C, Moscato P (eds) Handbook of memetic algorithms, studies in computational intelligence, vol 379. Springer, Berlin, pp 153–165

    Chapter  Google Scholar 

  23. Palubeckis G (2007) Iterated tabu search for the maximum diversity problem. Appl Math Comput 189:371–383

    MathSciNet  MATH  Google Scholar 

  24. Parreño F, Álvarez-Valdés R, Martí R (2021) Measuring diversity. A review and an empirical analysis. Eur. J. Oper. Res. 289:515–532

    Article  MathSciNet  Google Scholar 

  25. Peiró J, Jiménez I, Laguardia J, Martí R (2021) Heuristics for the capacitated dispersion problem. International transactions in operational research 28:119–141

    Article  MathSciNet  Google Scholar 

  26. Resende MG, Martí C, Gallego M, Duarte A (2010) GRASP and path relinking for the max–min diversity problem. Comput Oper Res 37(3):498–508

    Article  MathSciNet  Google Scholar 

  27. Rosenkrantz DJ, Tayi GK, Ravi SS (2000) Facility dispersion problems under capacity and cost constraints. J Comb Optim 4:7–33

    Article  MathSciNet  Google Scholar 

  28. Sayyady F, Fathi Y (2016) An integer programming approach for solving the p-dispersion problem. Eur J Oper Res 253:216–225

    Article  MathSciNet  Google Scholar 

  29. Tirronen V, Neri F (2009) Differential evolution with fitness diversity self-adaptation. In: Chiong R (ed) Nature-inspired algorithms for optimisation, studies in computational intelligence, vol 193. Springer, Berlin, pp 199–234

    Chapter  Google Scholar 

  30. Wang Y, Hao J-K, Glover F, Lü Z (2014) A tabu search based memetic algorithm for the maximum diversity problem. Eng Appl Artif Intell 27:103–114

    Article  Google Scholar 

Download references

Acknowledgements

This research has been partially supported by the Spanish Ministry with grant ref. PGC2018-0953322-B-C21/MCIU/AEI/FEDER-UE and PGC2018-095322-B-C22, “Comunidad de Madrid” and “Fondos Estructurales” of European Union with grant refs. S2018/TCS-4566, Y2018/EMT-5062.

Author information

Authors and Affiliations

  1. Departamento de Estadística e Investigación Operativa, Universidad de Valencia, Valencia, Spain

    Rafael Martí & Anna Martínez-Gavara

  2. Department Computer Sciences, Universidad Rey Juan Carlos, Madrid, Spain

    Jesús Sánchez-Oro

Authors
  1. Rafael Martí
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Martínez-Gavara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jesús Sánchez-Oro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafael Martí.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martí, R., Martínez-Gavara, A. & Sánchez-Oro, J. The capacitated dispersion problem: an optimization model and a memetic algorithm. Memetic Comp. 13, 131–146 (2021). https://doi.org/10.1007/s12293-020-00318-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12293-020-00318-1

Keywords

Search

Navigation