Skip to main content
SpringerLink
Log in

Hybrid meta-heuristics with VNS and exact methods: application to large unconditional and conditional vertex \(p\)-centre problems

  • Published:
Journal of Heuristics Aims and scope Submit manuscript

Abstract

Large-scale unconditional and conditional vertex \(p\)-centre problems are solved using two meta-heuristics. One is based on a three-stage approach whereas the other relies on a guided multi-start principle. Both methods incorporate Variable Neighbourhood Search, exact method, and aggregation techniques. The methods are assessed on the TSP dataset which consist of up to 71,009 demand points with \(p\) varying from 5 to 100. To the best of our knowledge, these are the largest instances solved for unconditional and conditional vertex \(p\)-centre problems. The two proposed meta-heuristics yield competitive results for both classes of problems.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • Al-Khedhairi, A., Salhi, S.: Enhancements to two exact algorithms for solving the vertex \(p\)-center problem. J. Math. Model. Algorithm 4, 129–147 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Avella, P., Boccia, M., Salerno, S., Vasilyev, I.: An aggregation heuristic for large scale p-median problem. Comput. Oper. Res. 39(7), 1625–1632 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Barua, S., Sander, J.: Mining statistically sound co-location patterns at multiple distances. In: Proceedings of the 26th International Conference on Scientific and Statistical Database Management, Article No. 7 (2014)

  • Berman, O., Drezner, Z.: A new formulation for the conditional \(p\)-median and \(p\)-center problems. Oper. Res. Lett. 36, 481–483 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Berman, O., Simchi-Levi, D.: The conditional location problem on networks. Transp. Sci. 24, 77–78 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Bozkaya, B., Tansel, B.: A spanning tree approach to the absolute \(p-\)center problem. Locat. Sci. 6, 83–107 (1998)

    Article  Google Scholar 

  • Calik, H., Tansel, B.C.: Double bound method for solving the \(p\)-center location problem. Comput. Oper. Res. 40, 2991–2999 (2013)

    Article  MathSciNet  Google Scholar 

  • Caruso, C., Colorni, A., Aloi, L.: Dominant, an algorithm for the \(p\)-center problem. Eur. J. Oper. Res. 149, 53–64 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Casillas, P.: Aggregation problems in location-allocation modeling. In: Gosh, A., Rushton, G. (eds.) Spatial analysis and location-allocation models, pp. 327–344. Van Nostrand Reinhold, New York (1987)

    Google Scholar 

  • Chen, D., Chen, R.: New relaxation-based algorithms for the optimal solution of the continuous and discrete \(p\)-center problems. Comput. Oper. Res. 36, 1646–1655 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, D., Chen, R.: A relaxation-based algorithm for solving the conditional p-center problem. Oper. Res. Lett. 38, 215–217 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, R.: Conditional minisum and minimax location-allocation problems in euclidean space. Transp. Sci. 22, 158–160 (1990)

    MathSciNet  Google Scholar 

  • Cheng, T.C.E., Kang, L., Ng, C.T.: An improved algorithm for the p-center problem on interval graphs with unit lengths. Comput. Oper. Res. 34, 2215–2222 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Current, J.R., Schilling, D.A.: Elimination of source A and B errors in \(p\)-median location problems. Geogr. Anal. 19, 95–110 (1987)

    Article  Google Scholar 

  • Daskin, M.S.: Network and discrete location: models, algorithms, and applications. Wiley, New York (1995)

    Book  MATH  Google Scholar 

  • Daskin, M.S.: A new approach to solving the vertex \(p\)-center problem to optimality: algorithm and computational results. Commun. Oper. Res. Soc. Jpn. 45, 428–436 (2000)

    Google Scholar 

  • Davidovic, T., Ramljak, D., Selmic, M., Teodorovic, D.: Bee colony optimization for the \(p\)-center problem. Comput. Oper. Res. 38, 1367–1376 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Dongarra, J.J.: Performance of various computers using standard linear equation software. http://www.netlib.org/benchmark/performance.pdf. Accessed 15 April 2013

  • Drezner, Z.: The \(p\)-Centre problem-heuristic and optimal algorithms. J. Oper. Res. Soc. 35, 741–748 (1984)

    MATH  Google Scholar 

  • Drezner, Z.: Conditional \(p\)-center problems. Transp. Sci. 23, 51–3 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Drezner, Z.: On the conditional p-median problem. Comput. Oper. Res. 22, 525–530 (1995)

    Article  MATH  Google Scholar 

  • Elloumi, S., Labbe, M., Pochet, Y.: A new formulation and resolution method for the \(p\)-center problem. INFORMS J. Comput. 16, 84–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Fortney, J., Rost, K., Warren, J.: Comparing alternative methods of measuring geographic access to health services. Health Serv. Outcomes Res. Methodol. 1, 173–184 (2000)

    Article  Google Scholar 

  • Francis, R.L., Lowe, T.J., Tamir, A.: Demand Point Aggregation Analysis for a Class of Constrained Location Models: A Penalty Function Approach. IIE Trans. 36, 601–609 (2004a)

    Article  Google Scholar 

  • Francis, R.L., Lowe, T.J., Tamir, A., Emir-Farinas, H.: Aggregation Decomposition and Aggregation Guidelines for a Class of Minimax and Covering Location Models. Geogr. Anal. 36, 332–349 (2004b)

    Article  MATH  Google Scholar 

  • Francis, R.L., Rayco, M.B.: Asymptotically optimal aggregation for some unweighted \(p\)-center problems with rectilinear distances. Stud. Locat. Anal. 10, 25–36 (1996)

    MathSciNet  MATH  Google Scholar 

  • Francis, R.L., Lowe, T.J., Rayco, M.B., Tamir, A.: Aggregation error for location models: survey and analysis. Ann. Oper. Res. 167, 171–208 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Gamal, M.D.H., Salhi, S.: Constructive heuristics for the uncapacitated continuous location-allocation problem. J. Oper. Res. Soc. 52, 821–829 (2001)

    Article  MATH  Google Scholar 

  • Hakimi, S.L.: Optimum locations of switching centers and the absolute centers and medians of a graph. Oper. Res. 12, 450–459 (1964)

    Article  MATH  Google Scholar 

  • Hansen, P., Brimberg, J., Urosevic, D., Mladenovic, N.: Solving large p-median clustering problems by primal-dual variable neighborhood search. Data Min. Knowl. Discov. 19, 351–375 (2009)

    Article  MathSciNet  Google Scholar 

  • Hansen, P., Mladenovic, N.: Variable neighbourhood search for the p-median. Locat. Sci. 5, 207–225 (1997)

    Article  MATH  Google Scholar 

  • Hansen, P., Mladenovic, N., Perez, J.A.M.: Variable neighbourhood search: methods and applications. Ann. Oper. Res. 175, 367–407 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Hillsman, E.L., Rhoda, R.: Errors in measuring distances from populations to service centers. Ann. Reg. Sci. 12, 74–88 (1978)

    Article  Google Scholar 

  • Ilhan, T., Pinar, M.: An efficient exact algorithm for the vertex \(p\)-center problem (2001). http://www.ie.bilkent.edu.tr/~mustafap/pubs/

  • Irawan, C.A., Salhi, S.: Solving large p-median problems by a multistage hybrid approach using demand points aggregation and variable neighbourhood search. J. Glob. Optim. (2013). doi:10.1007/s10898-013-0080-z

  • Irawan, C.A., Salhi, S., Scaparra, M.P.: An adaptive multiphase approach for large unconditional and conditional p-median problems. Eur. J. Oper. Res. 237, 590–605 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Jaeger, M., Kariv, O.: Algorithms for finding \(p\)-centers on a weighted tree (for relatively small \(p)\). Networks 15, 381–389 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  • Kariv, O., Hakimi, S.L.: An algorithmic approach for the vertex \(p-\)center problem. Part I: The \(p-\)centers. SIAM J. Appl. Math. 37, 513–538 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  • Kaveh, A., Nasr, H.: Solving the conditional and unconditional \(p\)-center problem with modified harmony search: a real case study. Sci. Iran. A. 18, 867–877 (2011)

    Article  Google Scholar 

  • Liu, S., Liu, Y., Ni, L. M., Fan, J., Li, M.: Towards Mobility-based Clustering. In: Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 919–928 (2010)

  • Lu, C.: Robust weighted vertex \(p\)-center model considering uncertain data: An application to emergency management. Eur. J. Oper. Res. 230, 113–121 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, C., Sheu, J.: Robust vertex \(p\)-center model for locating urgent relief distribution centers. Comput. Oper. Res. 40, 2128–2137 (2013)

    Article  MathSciNet  Google Scholar 

  • Minieka, E.: The \(m\)-center problem. SIAM Rev. 12, 138–139 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  • Minieka, E.: Conditional centers and medians on a graph. Networks 10, 265–272 (1980)

    Article  MathSciNet  Google Scholar 

  • Mladenovic, N., Labbe, M., Hansen, P.: Solving the \(p-\)center problem with tabu search and variable neighbourhood search. Networks 42, 48–64 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Qu, Q., Liu, S., Yang, B. Jensen, C.S.: Integrating non-spatial preferences into spatial location queries. In: Proceedings of the 26th International Conference on Scientific and Statistical Database Management, Article No. 8 (2014)

  • Rayco, M.B., Francis, R.L., Lowe, T.J.: Error-bound driven demand point aggregation for the rectilinear distance \(p\)-center model. Locat. Sci. 4, 213–235 (1997)

    Article  MATH  Google Scholar 

  • Rayco, M.B., Francis, R.L., Tamir, A.: A \(p\)-center grid-positioning aggregation procedure. Comput. Oper. Res. 26, 1113–1124 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Salhi, S., Al-Khedhairi, A.: Integrating heuristic information into exact methods: the case of the vertex \(p-\)centre problem. J. Oper. Res. Soc. 61, 1619–1631 (2010)

    Article  MATH  Google Scholar 

  • Salhi, S., Sari, M.: A multi-level composite heuristic for the multi-depot vehicle fleet mix problem. Eur. J. Oper. Res. 103, 95–112 (1997)

    Article  MATH  Google Scholar 

  • Shaw, D.X.: A unified limited column generation approach for facility location problems on trees. Ann. Oper. Res. 87, 363–382 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Tansel, B.C., Francis, R.L., Lowe, T.J.: State of the art-location on network: a survey. Part I: the \(p\)-center and \(p\)-median problems. Manag. Sci. 29, 482–497 (1983a)

  • Tansel, B.C., Francis, R.L., Lowe, T.J.: State of the art-location on network: a survey. Part II: exploiting tree network structure. Manag. Sci. 29, 498–511 (1983b)

  • Tansel, B.C., Francis, R.L., Lowe, T.J., Chen, M.: Duality and distance constraints for the nonlinear \(p-\)center and covering problem on a tree network. Oper. Res. 30, 725–744 (1982)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank both referees for their useful suggestions that improved both the content as well as the presentation of the paper. We are also grateful to the Indonesian Government and Department of Industrial Engineering—ITENAS Bandung, Indonesia for the sponsorship of the first author.

Author information

Authors and Affiliations

  1. Centre for Operational Research and Logistics (CORL), Department of Mathematics, University of Portsmouth, Lion Gate Building, Lion Terrace, Portsmouth, PO1 3HF, UK

    Chandra Ade Irawan

  2. Department of Industrial Engineering, Institut Teknologi Nasional, Bandung, 40124, Indonesia

    Chandra Ade Irawan

  3. Centre for Logistics & Heuristic Optimization (CLHO), Kent Business School, University of Kent, Canterbury, Kent, CT2 7PE, UK

    Said Salhi

  4. Department of Information Systems and Decision Sciences, Steven G. Mihaylo College of Business and Economis, California State University-Fullerton, Fullerton, CA, 92834, USA

    Zvi Drezner

Authors
  1. Chandra Ade Irawan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Said Salhi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zvi Drezner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chandra Ade Irawan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Irawan, C.A., Salhi, S. & Drezner, Z. Hybrid meta-heuristics with VNS and exact methods: application to large unconditional and conditional vertex \(p\)-centre problems. J Heuristics 22, 507–537 (2016). https://doi.org/10.1007/s10732-014-9277-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-014-9277-7

Keywords

Search

Navigation