Abstract
We address a general fault-tolerant version of the k-median problem on a network. Unlike the original k-median, the objective is to find k nodes (medians or facilities) of a network, assign each non-median node (customer) to \(r_j\) distinct medians, and each median nodes to \(r_j-1\) other medians so as to minimize the overall assignment cost. The problem can be considered in context of the so-called reliable facility location, where facilities once located may be subject to failures. Hedging against possible disruptions, each customer is assigned to multiple distinct facilities. We propose a fast and effective heuristic rested upon consecutive searching for lower and upper bounds for the optimal value. The procedure for finding lower bounds is based on a Lagrangian relaxation and a specialized effective subgradient algorithm for solving the corresponding dual problem. The information on dual variables is then used by a core heuristic in order to determine a set of primal variables to be fixed. The effectiveness and efficiency of our approach are demonstrated in a computational experiment on large-scale problem instances taken from TSPLIB. We show that the proposed algorithm is able to fast find near-optimal solutions to problem instances with almost 625 million decision variables (on networks with up to 24978 vertices).
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Aboolian R, Cui T, Shen ZJM (2013) An efficient approach for solving reliable facility location models. INFORMS J Comput 25(4):720–729. https://doi.org/10.1287/ijoc.1120.0534
Albareda-Sambola M, Hinojosa Y, Puerto J (2015) The reliable \(p\)-median problem with at-facility service. Eur J Oper Res 245(3):656–666
Alcaraz J, Landete M, Monge JF (2012) Design and analysis of hybrid metaheuristics for the reliability \(p\)-median problem. Eur J Oper Res 222(1):54–64. https://doi.org/10.1016/j.ejor.2012.04.016
Anthony B, Goyal V, Gupta A, Nagarajan V (2010) A plant location guide for the unsure: approximation algorithms for min–max location problems. Math Oper Res 35(1):79–101. https://doi.org/10.1287/moor.1090.0428
Avella P, Sforza A (1999) Logical reduction tests for the \(p\)-problem. Ann Oper Res 86:105–115
Avella P, Sassano A, Vasilyev I (2007) Computational study of large-scale \(p\)-median problems. Math Program 109(1):89–114
Avella P, Boccia M, Sforza A, Vasilyev I (2008) An effective heuristic for large-scale capacitated facility location problems. J Heuristics 15(6):597–615
Avella P, Boccia M, Salerno S, Vasilyev I (2012) An aggregation heuristic for large scale \(p\)-median problem. Comput Oper Res 39(7):1625–1632
Beasley JE (1993) Lagrangean heuristics for location problems. Eur J Oper Res 65(3):383–399
Berman O, Krass D, Menezes MBC (2007) Facility reliability issues in network \(p\)-median problems: strategic centralization and co-location effects. Oper Res 55(2):332–350. https://doi.org/10.1287/opre.1060.0348
Berman O, Krass D, Menezes MBC (2009) Locating facilities in the presence of disruptions and incomplete information. Decis Sci 40(4):845–868. https://doi.org/10.1111/j.1540-5915.2009.00253.x
Bhattacharya S, Chalermsook P, Mehlhorn K, Neumann A (2014) New approximability results for the robust \(k\)-median problem. In: Ravi R, Gørtz IL (eds) Algorithm theory—SWAT 2014: 14th Scandinavian symposium and workshops, Copenhagen, Denmark, July 2–4, 2014. Proceedings. Springer, Cham, pp 50–61. https://doi.org/10.1007/978-3-319-08404-6_5
Byrka J, Srinivasan A, Swamy C (2010) Fault-tolerant facility location: a randomized dependent lp-rounding algorithm. In: Eisenbrand F, Shepherd FB (eds) Integer programming and combinatorial optimization: 14th international conference, IPCO 2010, Lausanne, Switzerland, June 9–11, 2010. Proceedings. Springer, Berlin, pp 244–257. https://doi.org/10.1007/978-3-642-13036-6_19
Caprara A, Fischetti M, Toth P (1999) A heuristic method for the set covering problem. Oper Res 47(5):730–743
Carrizosa E, Ushakov A, Vasilyev I (2012) A computational study of a nonlinear minsum facility location problem. Comput Oper Res 39(11):2625–2633
Chechik S, Peleg D (2014) Robust fault tolerant uncapacitated facility location. Theor Comput Sci 543(Supplement C):9–23. https://doi.org/10.1016/j.tcs.2014.05.013
Cui T, Ouyang Y, Shen ZJM (2010) Reliable facility location design under the risk of disruptions. Oper Res 58(4–part–1):998–1011. https://doi.org/10.1287/opre.1090.0801
Ding H, Xu J (2015) A unified framework for clustering constrained data without locality property. In: Indyk P (ed) Proceedings the 26th annual ACM-SIAM symposium on discrete algorithms, SODA’15. SIAM, Philadelphia, pp 1471–1490. https://doi.org/10.1137/1.9781611973730.97
Drezner Z (1987) Heuristic solution methods for two location problems with unreliable facilities. J Oper Res Soc 38(6):509–514. https://doi.org/10.1057/jors.1987.88
Fisher ML (1981) The lagrangian relaxation method for solving integer programming problems. Manag Sci 27(1):1–18
García S, Labbé M, Marín A (2011) Solving large \(p\)-median problems with a radius formulation. INFORMS J Comput 23(4):546–556
Geoffrion AM (1974) Lagrangean relaxation for integer programming. In: Balinski ML (ed) Approaches to integer programming, mathematical programming studies, vol 2. Springer, Berlin, pp 82–114
Gomes T, Tapolcai J, Esposito C, Hutchison D, Kuipers F, Rak J, de Sousa A, Iossifides A, Travanca R, André J, Jorge L, Martins L, Ugalde PO, Pašić A, Pezaros D, Jouet S, Secci S, Tornatore M (2016) A survey of strategies for communication networks to protect against large-scale natural disasters. In: Proceedings of 8th international workshop on resilient networks design and modeling (RNDM). Gdańsk University of Technology, Gdańsk, pp 11–22. https://doi.org/10.1109/RNDM.2016.7608263
Govindan K, Fattahi M, Keyvanshokooh E (2017) Supply chain network design under uncertainty: a comprehensive review and future research directions. Eur J Oper Res 263(1):108–141. https://doi.org/10.1016/j.ejor.2017.04.009
Guha S, Meyerson A, Munagala K (2003) A constant factor approximation algorithm for the fault-tolerant facility location problem. J Algorithms 48(2):429–440. https://doi.org/10.1016/S0196-6774(03)00056-7
Hajiaghayi M, Hu W, Li J, Li S, Saha B (2016) A constant factor approximation algorithm for fault-tolerant \(k\)-median. ACM Trans Algorithms 12(3):36:1–36:19. https://doi.org/10.1145/2854153
Hakimi SL (1965) Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper Res 13(3):462–475
Hansen P, Brimberg J, Urosević D, Mladenović N (2009) Solving large \(p\)-median clustering problems by primal-dual variable neighborhood search. Data Min Knowl Discov 19(3):351–375
Irawan CA, Salhi S, Scaparra MP (2014) An adaptive multiphase approach for large unconditional and conditional p-median problems. Eur J Oper Res 237(2):590–605
Jain K, Vazirani VV (2001) Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. J ACM 48(2):274–296
Jain K, Vazirani VV (2004) An approximation algorithm for the fault tolerant metric facility location problem. Algorithmica 38(3):433–439. https://doi.org/10.1007/s00453-003-1070-1
Kariv O, Hakimi S (1979) An algorithmic approach to network location problems. ii: the \(p\)-medians. SIAM J Appl Math 37(3):539–560
Li Q, Zeng B, Savachkin A (2013) Reliable facility location design under disruptions. Comput Oper Res 40(4):901–909. https://doi.org/10.1016/j.cor.2012.11.012
Lim M, Daskin M, Bassamboo A, Chopra S (2010) A facility reliability problem: formulation, properties, and algorithm. Nav Res Logist 57(1):58–70. https://doi.org/10.1002/nav.20385
Lu M, Ran L, Shen ZJM (2015) Reliable facility location design under uncertain correlated disruptions. Manuf Serv Oper Manag 17(4):445–455. https://doi.org/10.1287/msom.2015.0541
Masone A, Sterle C, Vasilyev I, Ushakov A (2018) A three-stage p-median based exact method for the optimal diversity management problem. Networks. https://doi.org/10.1002/net.21821
Mulvey JM, Crowder HP (1979) Cluster analysis: an application of lagrangian relaxation. Manag Sci 25(4):329–340
Rybicki B, Byrka J (2015) Improved approximation algorithm for fault-tolerant facility placement. In: Bampis E, Svensson O (eds) Approximation and online algorithms: 12th international workshop, WAOA 2014, Wrocław, Poland, Sept 11–12, 2014, Revised selected papers. Springer, Cham, pp 59–70. https://doi.org/10.1007/978-3-319-18263-6_6
Snyder LV, Daskin MS (2005) Reliability models for facility location: the expected failure cost case. Transp Sci 39(3):400–416
Snyder LV, Atan Z, Peng P, Rong Y, Schmitt AJ, Sinsoysal B (2016) Or/ms models for supply chain disruptions: a review. IIE Trans 48(2):89–109. https://doi.org/10.1080/0740817X.2015.1067735
Swamy C, Shmoys DB (2008) Fault-tolerant facility location. ACM Trans Algorithms 4(4):51:1–51:27. https://doi.org/10.1145/1383369.1383382
Vasilyev I, Ushakov A (2017) A shared memory parallel heuristic algorithm for the large-scale \(p\)-median problem. In: Sforza A, Sterle C (eds) Optimization and decision science: methodologies and applications: ODS, Sorrento, Italy, Sept 4–7, 2017. Springer, Cham, pp 295–302. https://doi.org/10.1007/978-3-319-67308-0-30
Yan L, Chrobak M (2015) Lp-rounding algorithms for the fault-tolerant facility placement problem. J Discrete Algorithms 33(Supplement C):93–114. https://doi.org/10.1016/j.jda.2015.03.004
Yu G, Haskell WB, Liu Y (2017) Resilient facility location against the risk of disruptions. Transp Res B Methodol 104(Supplement C):82–105. https://doi.org/10.1016/j.trb.2017.06.014
Yun L, Qin Y, Fan H, Ji C, Li X, Jia L (2015) A reliability model for facility location design under imperfect information. Transp Res B Methodol 81(Part 2):596–615. https://doi.org/10.1016/j.trb.2014.10.010
Zhang Y, Snyder LV, Qi M, Miao L (2016) A heterogeneous reliable location model with risk pooling under supply disruptions. Transp Res B Methodol 83(Supplement C):151–178. https://doi.org/10.1016/j.trb.2015.11.009
Acknowledgements
This study was funded by Russian Foundation of Basic Research, Project No. 18-07-01037.
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All the authors (I. Vasilyev, A. V. Ushakov, N. Maltugueva and A. Sfroza) declare that they have no conflict of interest.
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Communicated by P. Beraldi, M.Boccia, C. Sterle.
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Vasilyev, I., Ushakov, A.V., Maltugueva, N. et al. An effective heuristic for large-scale fault-tolerant k-median problem. Soft Comput 23, 2959–2967 (2019). https://doi.org/10.1007/s00500-018-3562-6
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DOI: https://doi.org/10.1007/s00500-018-3562-6