Abstract
In this paper a multi-stage facility location problem with staircase costs and splitting of commodities is introduced and formulated as a mixed integer program. The problem is motivated by an application in the context of a reverse supply chain for end-of-life vehicles. We propose a two-phase heuristic solution approach: The greedy construction heuristic utilizes the solution obtained by the LP-relaxation of the problem. In the improvement heuristic we combine ADD, DROP and SWAP neighborhoods with a diversification strategy to a Variable Neighborhood Descent (VND) and to a Variable Neighborhood Search (VNS) approach. Computational results show that this approach is able reduce the duality gap provided by state-of-the-art MIP solver CPLEX for small and medium-sized instances and is also able to provide high-quality solutions for large-scale instances with up to 2,900 candidate facilities. The building blocks of the solution approach can easily be rearranged in order to solve other facility location problems.
Similar content being viewed by others
References
Aardal K, Labbé M, Leung J and Queyranne M (1996). On the two-level uncapacitated facility location problem. INFORMS J Comput 8: 289–301
Aikens CH (1985). Facility location models for distribution planning. Eur J Oper Res 22: 263–279
Avanthay C, Hertz A and Zufferey N (2003). A variable neighborhood search for graph coloring. Eur J Oper Res 151: 379–388
Balinski M (1965). Integer programming: methods, uses, computation. Manage Sci 12: 254–313
Brandreau ML and Chiu SS (1989). An overview of representative problems in location research. Manage Sci 35(6): 645–674
Croxton KL, Gendron B and Magnanti TL (2003). A comparison of mixed-integer programming models for nonconvex piecewise linear cost minimization problems. Manage Sci 49(9): 1268–1273
Daskin MS (1995). Network and discrete location—models, algorithms and applications. Wiley, New York
Domschke W and Drexl A (1985). Add-heuristics’ starting procedures for capacitated plant location models. Eur J Oper Res 21: 47–53
Domschke W and Drexl A (1996). Logistik: standorte. Oldenbourg, München
Drezner Z and Hamacher HW (2002). Facility location: application and theory. Springer, Berlin
Drezner Z and Hamacher HW (1995). Facility location: a survey of applications and methods. Springer, Berlin
Erlenkotter D (1978). A dual-based procedure for uncapacitated facility location. Oper Res 26: 992–1009
Feldman E, Lehrer FA and Ray TL (1966). Warehouse location under continuous economies of scale. Manag Sci 12(9): 670–684
Hansen P and Mladenovic N (1997). Variable neighborhood search for the p-median. Location Sci 5(4): 207–226
Hansen P and Mladenovic N (2001). Variable neighborhood search: Principles and applications. Eur J Oper Res 130(1): 449–467
Hansen P, Mladenovic N (2003) Variable neighborhood search. In: Glover F, Kochenberger G (eds) Handbook of metaheuristics. Kluwer, Dordrecht, pp 145–184
Hansen P, Mladenovic N and Urosevic D (2004). Variable neighborhood search for the maximum clique. Discrete Appl. Math. 145: 117–125
Holmberg K (1994). Solving the staircase cost facility location problem with decomposition and piecewise linearization. Eur J Oper Res 75: 41–61
Holmberg K and Ling J (1997). A lagrangian heuristic for the facility location problem with staircase costs. Eur J Oper Res 97: 63–74
Jacobsen SK (1983). Heuristics for the capacitated plant location model. Eur J Oper Res 12: 253–261
Kaufman L, vanden Eede M and Hansen P (1977). A plant and warehouse location problem. Oper Res Q 28(3): 547–554
Khumawala BM (1972). An efficient branch and bound algorithm for the warehouse location problem. Manage Sci 18: B718–B731
Khumawala BM (1973). An efficient heuristic procedure for the uncapacitated warehouse location problem. Naval Res Logis Q 20: 109–121
Khumawala BM (1974). An efficient heuristic procedure for the capacitated warehouse location problem. Naval Res Logis Q 21: 609–623
Klose A and Drexl A (2004). Facility location models for distribution system design. Eur J Oper Res 162: 4–29
Kuehn AA and Hamburger MJ (1963). A heuristic program for locating warehouses. Manag Sci 9(4): 643–666
Melo MT, Nickel S and Saldanha da Gama F (2003). Large-scale models for dynamic multi-commodity capacitated facility location. Berichte des Fraunhofer ITWM 58: 1–40
Mladenovic N and Hansen P (1997). Variable neighborhood search. Comput Oper Res 24(11): 1097–1100
Owen SH and Daskin MS (1998). Strategic facility location: a review. Eur J Oper Res 111: 423–447
Revelle CS and Laporte G (1996). The plant location problem: new models and research prospects. Oper Res 44: 864–874
Ribeiro CC and Souza MC (2002). Variable neighborhood search for the degree-constrained minimum spanning tree problem. Discrete Appl Math 118: 43–54
Schleiffer R, Wollenweber J, Sebastian H-J, Golm F, Kapoustina N (2004) Application of genetic algorithms for the design of large-scale reverse logistic networks in europe’s automotive industry. Proceeding of the 37th Hawaii International Conference on System Sciences, pp 1–10
Shmoys DB, Tardos É, Aardal K (1997) Approximation algorithms for facility location problems. In: Proceeding of the 8th ACM-SIAM Symposium on Discrete Algorithms, pp 619–628
Tcha D and Lee B (1984). A branch-and-bound algorithm for the multi-level uncapacitated facility location problem. Eur J Oper Res 18: 35–43
Erlenkotter D and Roy T (1982). A dual based procedure for dynamic facility location. Manag Sci 28: 1091–1105
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wollenweber, J. A multi-stage facility location problem with staircase costs and splitting of commodities: model, heuristic approach and application. OR Spectrum 30, 655–673 (2008). https://doi.org/10.1007/s00291-007-0114-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00291-007-0114-3