Abstract
We propose a two-phase heuristic for the generalized assignment problem (GAP). The first phase—a generic variable-fixing method—heuristically eliminates up to 98% of the variables without sacrificing the solution quality. The second phase takes as input the small reduced GAP obtained during the first phase and applies a very large scale neighborhood search. The definition of the successive exponential size neighborhoods is guided by the subgradient method applied to the Lagrangian relaxation of the knapsack constraints via the reduced costs. Searching the proposed neighborhood is NP-hard and amounts to solving a monotone binary program (BP) with m constraints and p variables, where m and p are respectively the number of agents and tasks of the reduced GAP (monotone BPs are BPs with two nonzero coefficients of opposite sign per column). To the best of our knowledge, this is the first time the above ideas are exposed. Extensive testing on large scale GAP instances is presented and previously unknown better values for eight instances are obtained. Comparison to well-established methods shows that this new approach is competitive and constitutes a substantial addition to the arsenal of tools for solving GAP.
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References
Avella P, Boccia M, Vasilyev I (2008) A computational study of exact knapsack separation for the generalized assignment problem. Comput Optim Appl 45(3):543–555. https://doi.org/10.1007/s10589-008-9183-8
Barbas J, Marín Á (2004) Maximal covering code multiplexing access telecommunication networks. Eur J Oper Res 159(1):219–238. https://doi.org/10.1016/s0377-2217(03)00381-3
Campbell GM, Diaby M (2002) Development and evaluation of an assignment heuristic for allocating cross-trained workers. Eur J Oper Res 138(1):9–20. https://doi.org/10.1016/s0377-2217(01)00107-2
Campbell JF, Langevin A (1995) The snow disposal assignment problem. J Oper Res Soc 46(8):919–929. https://doi.org/10.1057/jors.1995.131
Cattrysse D, Degraeve Z, Tistaert J (1998) Solving the generalised assignment problem using polyhedral results. Eur J Oper Res 108(3):618–628. https://doi.org/10.1016/s0377-2217(97)00054-4
Chekuri C, Khanna S (2005) A polynomial time approximation scheme for the multiple knapsack problem. SIAM J Comput 35(3):713–728. https://doi.org/10.1137/S0097539700382820
Chu PC, Beasley JE (1997) A genetic algorithm for the generalised assignment problem. Comput Oper Res 24(1):17–23. https://doi.org/10.1016/s0305-0548(96)00032-9
Cohen R, Katzir L, Raz D (2006) An efficient approximation for the generalized assignment problem. Inform Process Lett 100(4):162–166. https://doi.org/10.1016/j.ipl.2006.06.003
Cromley RG, Hanink DM (1999) Coupling land use allocation models with raster GIS. J Geogr Syst 1(2):137–153. https://doi.org/10.1007/s101090050009
Díaz JA, Fernández E (2001) A tabu search heuristic for the generalized assignment problem. Eur J Oper Res 132(1):22–38. https://doi.org/10.1016/s0377-2217(00)00108-9
Dobson G, Nambimadom RS (2001) The batch loading and scheduling problem. Oper Res 49(1):52–65. https://doi.org/10.1287/opre.49.1.52.11189
Foulds LR, Wilson JM (1997) A variation of the generalized assignment problem arising in the New Zealand dairy industry. Ann Oper Res 69:105–114. https://doi.org/10.1023/a:1018968625626
François V, Arda Y, Crama Y, Laporte G (2016) Large neighborhood search for multi-trip vehicle routing. Eur J Oper Res 255(2):422–441. https://doi.org/10.1016/j.ejor.2016.04.065
Grangier P, Gendreau M, Lehuédé F, Rousseau LM (2016) An adaptive large neighborhood search for the two-echelon multiple-trip vehicle routing problem with satellite synchronization. Eur J Oper Res 254(1):80–91. https://doi.org/10.1016/j.ejor.2016.03.040
Haddadi S (1999) Lagrangian decomposition based heuristic for the generalized assignment problem. INFOR 37(4):392–402. https://doi.org/10.1080/03155986.1999.11732392
Haddadi S, Ouzia H (2004) Effective algorithm and heuristic for the generalized assignment problem. Eur J Oper Res 153(1):184–190. https://doi.org/10.1016/s0377-2217(02)00710-5
Haddadi S, Cheraitia M, Salhi A (2018) A two-phase heuristic for set covering. Int J Math Oper Res 13(1):61–78. https://doi.org/10.1504/IJMOR.2018.092962
Higgins AJ (1999) Optimizing cane supply decisions within a sugar mill region. J Sched 2(5):229–244. 10.1002/(SICI)1099-1425(199909/10)2:5\(<\)229::AID-JOS29\(>\)3.0.CO;2-L
Higgins AJ (2001) A dynamic tabu search for large-scale generalised assignment problems. Comput Oper Res 28(10):1039–1048. https://doi.org/10.1016/s0305-0548(00)00024-1
Hochbaum DS (2004) Monotonizing linear programs with up to two nonzeroes per column. Oper Res Lett 32(1):49–58. https://doi.org/10.1016/s0167-6377(03)00074-9
Mitrović-Minić S, Punnen AP (2008) Very large-scale variable neighborhood search for the generalized assignment problem. J Interdiscipl Math 11(5):653–670. https://doi.org/10.1080/09720502.2008.10700590
Monfared MAS, Etemadi M (2006) The impact of energy function structure on solving generalized assignment problem using Hopfield neural network. Eur J Oper Res 168(2):645–654. https://doi.org/10.1016/j.ejor.2004.06.015
Nauss RM (2003) Solving the generalized assignment problem: an optimizing and heuristic approach. INFORMS J Comput 15(3):249–266. https://doi.org/10.1287/ijoc.15.3.249.16075
Nowakovski J, Schwärzler W, Triesch E (1999) Using the generalized assignment problem in scheduling the ROSAT space telescope. Eur J Oper Res 112(3):531–541. https://doi.org/10.1016/s0377-2217(97)00408-6
Nutov Z, Beniaminy I, Yuster R (2006) A \((1{-}1/\epsilon )\)-approximation algorithm for the generalized assignment problem. Oper Res Lett 34(3):283–288. https://doi.org/10.1016/j.orl.2005.05.006
Öncan T (2007) A survey of the generalized assignment problem and its applications. INFOR 45(3):123–141. https://doi.org/10.3138/infor.45.3.123
Özbakir L, Baykasoğlu A, Tapkan P (2010) Bees algorithm for generalized assignment problem. Appl Math Comput 215(11):3782–3795. https://doi.org/10.1016/j.amc.2009.11.018
Pigatti A, de Aragão MP, Uchoa E (2005) Stabilized branch-and-cut-and-price for the generalized assignment problem. Electron Notes Discrete Math 19:389–395. https://doi.org/10.1016/j.endm.2005.05.052
Pisinger D, Ropke S (2010) Large neighborhood search. In: Handbook of metaheuristics. Springer Nature, pp 399–419. https://doi.org/10.1007/978-1-4419-1665-5_13
Posta M, Ferland JA, Michelon P (2011) An exact method with variable fixing for solving the generalized assignment problem. Comput Optim Appl 52(3):629–644. https://doi.org/10.1007/s10589-011-9432-0
Privault C, Herault L (1998) Solving a realworld assignment problem with a metaheuristic. J Heuristics 4(4):383–398. https://doi.org/10.1023/a:1009618009594
Ruland KS (1999) A model for aeromedical routing and scheduling. Int Trans Oper Res 6(1):57–73. https://doi.org/10.1111/j.1475-3995.1999.tb00143.x
Savelsbergh M (1997) A branch-and-price algorithm for the generalized assignment problem. Oper Res 45(6):831–841. https://doi.org/10.1287/opre.45.6.831
Wilson JM (1997) A genetic algorithm for the generalised assignment problem. J Oper Res Soc 48(8):804–809. https://doi.org/10.1057/palgrave.jors.2600431
Woodcock AJ, Wilson JM (2010) A hybrid tabu search/branch & bound approach to solving the generalized assignment problem. Eur J Oper Res 207(2):566–578. https://doi.org/10.1016/j.ejor.2010.05.007
Yagiura M, Yamaguchi T, Ibaraki T (1998) A variable depth search algorithm with branching search for the generalized assignment problem. Optim Method Softw 10(2):419–441. https://doi.org/10.1080/10556789808805722
Yagiura M, Ibaraki T, Glover F (2004) An ejection chain approach for the generalized assignment problem. INFORMS J Comput 16(2):133–151. https://doi.org/10.1287/ijoc.1030.0036
Yagiura M, Ibaraki T, Glover F (2006) A path relinking approach with ejection chains for the generalized assignment problem. Eur J Oper Res 169(2):548–569. https://doi.org/10.1016/j.ejor.2004.08.015
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Many thanks to the referees for their fruitful comments and suggestions. One of them highly contributes to improving the presentation and content of the article in several points.
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Haddadi, S. Variable-fixing then subgradient optimization guided very large scale neighborhood search for the generalized assignment problem. 4OR-Q J Oper Res 17, 261–295 (2019). https://doi.org/10.1007/s10288-018-0389-z
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DOI: https://doi.org/10.1007/s10288-018-0389-z