Abstract
We propose a Lagrangian heuristic for facility location problems with concave cost functions and apply it to solve the plant location and technology acquisition problem. The problem is decomposed into a mixed integer subproblem and a set of trivial single-variable concave minimization subproblems. We are able to give a closed-form expression for the optimal Lagrangian multipliers such that the Lagrangian bound is obtained in a single iteration. Since the solution of the first subproblem is feasible to the original problem, a feasible solution and an upper bound are readily available. The Lagrangian heuristic can be embedded in a branch-and-bound scheme to close the optimality gap. Computational results show that the approach is capable of reaching high quality solutions efficiently. The proposed approach can be tailored to solve many concave-cost facility location problems.
Similar content being viewed by others
References
Beale, E.M., Forrest, J.J.: Global optimization using special ordered sets. Math. Program. 10(1), 52–69 (1976)
Beasley, J.E.: Or-library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)
Berman, O., Krass, D., Tajbakhsh, M.M.: A coordinated location-inventory model. Eur. J. Oper. Res. 217(3), 500–508 (2012)
Carrizosa, E., Ushakov, A., Vasilyev, I.: A computational study of a nonlinear minsum facility location problem. Comput. Oper. Res. 39(11), 2625–2633 (2012)
Dasci, A., Verter, V.: The plant location and technology acquisition problem. IIE Trans. 33(11), 963–974 (2001)
Daskin, M.S., Coullard, C.R., Shen, Z.M.M.: An inventory-location model: formulation, solution algorithm and computational results. Ann. Oper. Res. 110(1), 83–106 (2002)
Dupont, L.: Branch and bound algorithm for a facility location problem with concave site dependent costs. Int. J. Prod. Econ. 112(1), 245–254 (2008)
Elhedhli, S., Merrick, R.: Green supply chain network design to reduce carbon emissions. Transp. Res. Part D Transp. Environ. 17(5), 370–379 (2012)
Eppen, G.: Effects of centralization on expected costs in a multi-location newsboy problem. Manag. Sci. 25(5), 498–501 (1979)
Falk, J.E., Soland, R.M.: An algotithm for separable nonconvex programming problems. Manag. Sci. 15(9), 550–569 (1969)
Floudas, C.A.: Detreministic Global Optimization: Theory, Methods and Applications. Springer, Secaucus (2000)
Frangioni, A.: About Lagrangian methods in integer optimization. Ann. Oper. Res. 139(1), 163–193 (2005)
Guignard, M., Kim, S.: Lagrangean decomposition: a model yielding stronger lagrangean bounds. Math. Program. 39, 215–228 (1987)
Guisewite, G., Pardalos, P.: Minimum concave-cost network flow problems: applications, complexity, and algorithms. Ann. Oper. Res. 25(1), 75–99 (1990)
Hajiaghayi, M.T., Mahdian, M., Mirrokni, V.S.: The facility location problem with general cost functions. Networks 42(1), 42–47 (2003)
Hale, T.: Trevor Hale’s Location Science References (2009). http://gator.uhd.edu/~halet/
Harkness, J., ReVelle, C.: Facility location with increasing production costs. Eur. J. Oper. Res. 145(1), 1–13 (2003)
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, vol. 2. Springer-Verlag, Berlin (1992)
Kelley, J.E.J.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8(4), 703–712 (1960)
Kuno, T., Utsunomiya, T.: A Lagrangian based branch-and-bound algorithm for production-transportation problems. J. Glob. Optim. 18(1), 59–73 (2000)
Lu, D., Gzara, F., Elhedhli, S.: Facility location with economies and diseconomies of scale: models and column generation heuristics. IIE Trans. 46(6), 585–600 (2014)
Özsen, L., Coullard, C.R., Daskin, M.S.: Capacitated warehouse location model with risk pooling. Nav. Res. Logist. (NRL) 55(4), 295–312 (2008)
Romeijn, H.E., Sharkey, T.C., Shen, Z.M., Zhang, J.: Integrating facility location and production planning decisions. Networks 55(2), 78–89 (2010)
Shu, J., Wu, T., Zhang, K.: Warehouse location and two-echelon inventory management with concave operating cost. Int. J. Prod. Res. 53(9), 2718–2729 (2015)
Soland, R.M.: Optimal facility location with concave costs. Oper. Res. 22(2), 373–382 (1974)
Taha, H.A.: Concave minimization over a convex polyhedron. Nav. Res. Logist. Q. 20(3), 533–548 (1973)
Tuy, H.: Concave programming under linear constraints. Sov. Math. 5, 1437–1440 (1964)
Verter, V.: Uncapacitated and capacitated facility location problems. In: Eiselt, H.A., Marianov, V. (eds.) Foundations of Location Analysis. Springer, US (2011)
Wu, L.Y., Zhang, X.S., Zhang, J.L.: Capacitated facility location problem with general setup cost. Comput. Oper. Res. 33(5), 1226–1241 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Saif, A., Elhedhli, S. A Lagrangian heuristic for concave cost facility location problems: the plant location and technology acquisition problem. Optim Lett 10, 1087–1100 (2016). https://doi.org/10.1007/s11590-016-0998-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-016-0998-4