Abstract
The Multidimensional Knapsack/Covering Problem (KCP) is a 0–1 Integer Programming Problem containing both knapsack and weighted covering constraints, subsuming the well-known Multidimensional Knapsack Problem (MKP) and the Generalized (weighted) Covering Problem. We propose an Alternating Control Tree Search (ACT) method for these problems that iteratively transfers control between the following three components: (1) ACT-1, a process that solves an LP relaxation of the current form of the KCP. (2) ACT-2, a method that partitions the variables according to 0, 1, and fractional values to create sub-problems that can be solved with relatively high efficiency. (3) ACT-3, an updating procedure that adjoins inequalities to produce successively more constrained versions of KCP, and in conjunction with the solution processes of ACT-1 and ACT-2, ensures finite convergence to optimality. The ACT method can also be used as a heuristic approach using early termination rules. Computational results show that the ACT-framework successfully enhances the performance of three widely different heuristics for the KCP. Our ACT-method involving scatter search performs better than any other known method on a large set of KCP-instances from the literature. The ACT-based methods are also found to be highly effective on the MKP.
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References
Achterberg, T.: Conflict analysis in mixed integer programming. Discrete Optim. 4(1), 4–20 (2007)
Arntzen, H., Hvattum, L.M., Løkketangen, A.: Adaptive memory search for multidemand multidimensional knapsack problems. Comput. Oper. Res. 33, 2508–2525 (2006)
Beaujon, G.J., Marin, S.P., McDonald, G.C.: Balancing and optimizing a portfolio of r&d projects. Nav. Res. Logist. 48, 18–40 (2001)
Cappanera, P.: Discrete facility location and routing of obnoxious facilities. PhD thesis, University of Milano (1999)
Cappanera, P., Trubian, M.: A local-search-based heuristic for the demand-constrained multidimensional knapsack problem. INFORMS J. Comput. 17, 82–98 (2005)
Chu, P.C., Beasley, J.E.: A genetic algorithm for the multidimensional knapsack problem. J. Heuristics 4, 63–86 (1998)
Danna, E., Rothberg, E., Pape, C.L.: Exploring relaxation induced neighborhoods to improve MIP solutions. Math. Program. 102(1), 71–90 (2005)
Fischetti, M., Lodi, A.: Local branching. Math. Program. 98(1), 23–47 (2003)
Gavish, B., Pirkul, H.: Allocation of databases and processors in a distributed computing system. In: Akoka, J. (ed.) Management of Distributed Data Processing, pp. 215–231. North-Holland, Amsterdam (1982)
Gilmore, P.C., Gomory, R.E.: The theory and computation of knapsack functions. Oper. Res. 14, 1045–1075 (1966)
Glover, F.: Heuristics for integer programming using surrogate constraints. Decis. Sci. 8(1), 156–166 (1977)
Glover, F., Laguna, M.: Tabu Search. Kluwer Academic, Boston (1997)
Gomes, C., Sellmann, M.: Streamlined constraint reasoning. In: Proceedings, CPAIOR 2004 (2004)
Gomes, C., van Hoeve, W., Leahu, L.: The power of semidefinite programming relaxations for MAX-SAT. In: Proceedings, CPAIOR 2006, pp. 104–118 (2006)
Hanafi, S., Wilbaut, C.: Improved convergent heuristic for 0–1 mixed integer programming. Research Report, University of Valenciennes (2006)
Holmberg, K., Yuan, D.: A Lagrangian heuristic based branch-and-bound approach for the capacitated network design problem. Oper. Res. 48(3), 461–481 (2000)
Hvattum, L.M., Løkketangen, A.: Experiments using scatter search for the multidemand multidimensional knapsack problem. In: Doerner, K.F., et al. (eds.) Metaheuristics: Progress in Complex Systems Optimization. Operations Research/Computer Science Interfaces, vol. 39, pp. 3–24. Springer, Berlin (2007)
Laguna, M., Martí, R.: Scatter Search: Methodology and Implementations in C. Kluwer Academic, Dordrecht (2003)
Lorie, J., Savage, L.: Three problems in capital rationing. J. Bus. 28, 229–239 (1955)
Manne, A., Markowitz, H.: On the solution of discrete programming problems. Econometrica 25, 85–110 (1957)
Plastria, F.: Static competitive facility location: an overview of optimization approaches. Eur. J. Oper. Res. 129, 461–470 (2001)
Puchinger, J., Raidl, G.R.: Combining metaheuristics and exact algorithms in combinatorial optimization: A survey and classification. In: Mira, J., Álvarez, J.R. (eds.) Proceedings of the First International Work-Conference on the Interplay Between Natural and Artificial Computation, Part II. Lecture Notes in Computer Science, vol. 3562, pp. 41–53. Springer, Berlin (2005)
Romero-Morales, D., Carrizosa, E., Conde, E.: Semi-obnoxious location models: a global optimization approach. Eur. J. Oper. Res. 102, 295–301 (1997)
Savelsbergh, M.W.P.: Preprocessing and probing for mixed integer programming problems. ORSA J. Comput. 6, 445–454 (1994)
Sellmann, M., Kliewer, G., Koberstein, A.: Lagrangian cardinality cuts and variable fixing for capacitated network. In: Proceedings of the Tenth Annual European Symposium on Algorithms, pp. 845–858 (2002)
Shih, W.: A branch and bound method for the multiconstraint zero-one knapsack problem. J. Oper. Res. Soc. 30, 369–378 (1979)
Soyster, A.L., Lev, B., Slivka, W.: Zero-one programming with many variables and few constraints. Eur. J. Oper. Res. 2, 195–201 (1978)
Vasquez, M., Hao, J.-K.: A hybrid approach for the 0–1 multidimensional knapsack problem. In: Proceedings of the International Joint Conference on Artificial Intelligence 2001, pp. 328–333 (2001)
Vasquez, M., Vimont, Y.: Improved results on the 0–1 multidimensional knapsack problem. Eur. J. Oper. Res. 165, 70–81 (2005)
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Hvattum, L.M., Arntzen, H., Løkketangen, A. et al. Alternating control tree search for knapsack/covering problems. J Heuristics 16, 239–258 (2010). https://doi.org/10.1007/s10732-008-9100-4
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DOI: https://doi.org/10.1007/s10732-008-9100-4