Annals of Operations Research

, Volume 272, Issue 1–2, pp 41–67 | Cite as

Formulation and solution of a two-stage capacitated facility location problem with multilevel capacities

  • Chandra Ade IrawanEmail author
  • Dylan Jones
S.I.: Advances in Theoretical and Applied Combinatorial Optimization


In this paper, the multi-product facility location problem in a two-stage supply chain is investigated. In this problem, the locations of depots (distribution centres) need to be determined along with their corresponding capacities. Moreover, the product flows from the plants to depots and onto customers must also be optimised. Here, plants have a production limit whereas potential depots have several possible capacity levels to choose from, which are defined as multilevel capacities. Plants must serve customer demands via depots. Two integer linear programming (ILP) models are introduced to solve the problem in order to minimise the fixed costs of opening depots and transportation costs. In the first model, the depot capacity is based on the maximum number of each product that can be stored whereas in the second one, the capacity is determined by the size (volume) of the depot. For large problems, the models are very difficult to solve using an exact method. Therefore, a matheuristic approach based on an aggregation approach and an exact method (ILP) is proposed in order to solve such problems. The methods are assessed using randomly generated data sets and existing data sets taken from the literature. The solutions obtained from the computational study confirm the effectiveness of the proposed matheuristic approach which outperforms the exact method. In addition, a case study arising from the wind energy sector in the UK is presented.


Facility location Matheuristic ILP 



The research leading to these results has received funding from the European Union Seventh Framework Programme under the agreement SCP2-GA-2013-614020 (LEANWIND: Logistic Efficiencies And Naval architecture for Wind Installations with Novel Developments).


  1. Abdmouleh, Z., Alammari, R. A. M., & Gastli, A. (2015). Review of policies encouraging renewable energy integration and best practices. Renewable and Sustainable Energy Reviews, 45, 249–262.CrossRefGoogle Scholar
  2. Besnard, F., Fischer, K., & Bertling, L. (2013). A model for the optimization of the maintenance support organization for offshore wind farms. IEEE Transactions on Sustainable Energy, 4(2), 443–50.CrossRefGoogle Scholar
  3. Camacho-Vallejo, J., Muñoz-Sánchez, R., & González-Velarde, J. L. (2015). A heuristic algorithm for a supply chain’s production–distribution planning. Computers and Operations Research, 61, 110–121.CrossRefGoogle Scholar
  4. Correia, I., & Captivo, M. E. (2003). A lagrangean heuristic for a modular capacitated location problem. Annals of Operations Research, 122, 141–161.CrossRefGoogle Scholar
  5. De Regt, D. N. (2012). Economic feasibility of offshore service locations for maintenance of offshore wind farms on the Dutch part of the North Sea. MSc thesis. Delft, The Netherlands: Delft Institute of Applied Mathematics.Google Scholar
  6. Eskandarpour, M., Dejax, P., & Péton, O. (2017). A large neighborhood search heuristic for supply chain network design. Computers and Operations Research, 80, 23–37.CrossRefGoogle Scholar
  7. Francis, R. L., Lowe, T. J., Rayco, M. B., & Tamir, A. (2009). Aggregation error for location models: Survey and analysis. Annals of Operations Research, 167, 171–208.CrossRefGoogle Scholar
  8. Gao, L. L., & Robinson, E. P. (1992). A dual-based optimization procedure for the two-echelon uncapacitated facility location problem. Naval Research Logistics, 39(2), 191–212.CrossRefGoogle Scholar
  9. Hinojosa, Y., Puerto, J., & Fernandez, F. R. (2000). A multiperiod two-echelon multicommodity capacitated plant location problem. European Journal of Operational Research, 123(2), 271–291.CrossRefGoogle Scholar
  10. Irawan, C. A., Jones, D., & Ouelhadj, D. (2015). Bi-objective optimisation model for installation scheduling in offshore wind farms. Computers and Operations Research.
  11. Irawan, C. A., Ouelhadj, D., Jones, D., Stålhane, M., & Sperstad, I. B. (2017a). Optimisation of maintenance routing and scheduling for offshore wind farms. European Journal of Operational Research, 256, 76–89.CrossRefGoogle Scholar
  12. Irawan, C. A., & Salhi, S. (2015a). Solving large p-median problems by a multistage hybrid approach using demand points aggregation and variable neighbourhood search. Journal of Global Optimization, 63, 537–554.CrossRefGoogle Scholar
  13. Irawan, C. A., & Salhi, S. (2015b). Aggregation and non-aggregation techniques for large facility location problems—A survey. Yugoslav Journal of Operations Research, 25, 1–11.CrossRefGoogle Scholar
  14. Irawan, C. A., Salhi, S., & Drezner, Z. (2016). Hybrid meta-heuristics with VNS and exact methods: Application to large unconditional and conditional vertex p-centre problems. Journal of Heuristics, 22, 507–537.CrossRefGoogle Scholar
  15. Irawan, C. A., Salhi, S., Luis, M., & Azizi, N. (2017b). The continuous single source location problem with capacity and zone-dependent fixed cost: Models and solution approaches. European Journal of Operational Research.
  16. Irawan, C. A., Salhi, S., & Scaparra, M. P. (2014). An adaptive multiphase approach for large unconditional and conditional \(p\)-median problems. European Journal of Operational Research, 237, 590–605.CrossRefGoogle Scholar
  17. Jones, D. F., & Wall, G. (2016). An extended goal programming model for site selection in the offshore wind farm sector. Annals of Operations Research, 245(1), 121–135.CrossRefGoogle Scholar
  18. Keskin, B. B., & Üster, H. (2007a). A scatter search-based heuristic to locate capacitated transshipment points. Computers and Operations Research, 34(10), 3112–3125.CrossRefGoogle Scholar
  19. Keskin, B. B., & Üster, H. (2007b). Meta-heuristic approaches with memory and evolution for a multi-product production/distribution system design problem. European Journal of Operational Research, 182(2), 663–682.CrossRefGoogle Scholar
  20. Klose, A. (1999). An LP-based heuristic for two-stage capacitated facility location problems. Journal of Operational Research Society, 50(2), 157–166.CrossRefGoogle Scholar
  21. Klose, A. (2000). A Lagrangean relax-and-cut approach for the two-stage capacitated facility location problem. European Journal of Operational Research, 126(2), 408–421.CrossRefGoogle Scholar
  22. Krarup, J., & Pruzan, P. M. (1983). The simple plant location problem: Survey and synthesis. European Journal of Operational Research, 12, 36–81.CrossRefGoogle Scholar
  23. Li, J., Chu, F., Prins, C., & Zhu, Z. (2014). Lower and upper bounds for a two-stage capacitated facility location problem with handling costs. European Journal of Operational Research, 236, 957–967.CrossRefGoogle Scholar
  24. Li, J., Prins, C., & Chu, F. (2012). A scatter search for a multi-type transshipment point location problem with multicommodity flow. Journal of Intelligent Manufacturing, 23, 1103–1117.CrossRefGoogle Scholar
  25. Li, J., Ren, C., Dong, J., He, M., Wang, Q., Chen, F., et al. (2011). A Lagrangean-based heuristic for a two-stage facility location problem with handling costs. In IEEE international conference on service operations, logistics, and informatics (SOLI) (pp. 319–324).Google Scholar
  26. Lindqvist, M., & Lundin, J. (2010). Spare part logistics and optimization for wind turbines: Methods for cost-effective supply and storage. MS thesis. Uppsala, Sweden: Department of Information Technology, Division of Systems and Control, Uppsala University.Google Scholar
  27. Marín, A., & Pelegrín, B. (1999). Applying Lagrangian relaxation to the resolution of two-stage location problems. Annals of Operations Research, 86, 179–198.CrossRefGoogle Scholar
  28. Mišković, S., & Stanimirović, Z. (2016). Solving the robust two-stage capacitated facility location problem with uncertain transportation costs. Optimization Letters.
  29. Rodríguez, S. V., Plà, L. M., & Faulin, J. (2014). New opportunities in operations research to improve pork supply chain efficiency. Annals of Operations Research, 219, 5–23.CrossRefGoogle Scholar
  30. Stefanello, F., de Araújo, O. C. B., & Müllerc, F. M. (2015). Matheuristics for the capacitated p-median problem. International Transaction in Operational Research, 22, 149–167.CrossRefGoogle Scholar
  31. Tracht, K., Westerholt, J., & Schuh, P. (2013). Spare parts planning for offshore wind turbines subject to restrictive maintenance conditions. Procedia CIRP, 7, 563–568.CrossRefGoogle Scholar
  32. Tragantalerngsak, S., Holt, J., & Rönnqvist, M. (1997). Lagrangean heuristics for the two-echelon, single-source, capacitated facility location problem. European Journal of Operational Research, 102(3), 611–625.CrossRefGoogle Scholar
  33. Tragantalerngsak, S., Holt, J., & Rönnqvist, M. (2000). An exact method for the two-echelon, single-source, capacitated facility location problem. European Journal of Operational Research, 123(3), 473–489.CrossRefGoogle Scholar
  34. Whitaker, R. A. (1983). A fast algorithm for the greedy interchange for large-scale clustering and median location problems. INFOR, 21, 95–108.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Nottingham University Business School ChinaUniversity of Nottingham Ningbo ChinaNingboChina
  2. 2.Department of Mathematics, Centre for Operational Research and LogisticsUniversity of PortsmouthPortsmouthUK

Personalised recommendations