A Filter-and-Fan Metaheuristic for the 0-1 Multidimensional Knapsack Problem

This paper proposes a new hybrid tree search algorithm to the Multidimensional Knapsack Problem (MKP) that effectively combines tabu search with a dynamic and adaptive neighborhood search procedure. The authors’ heuristic, based on a filter-and-fan (F&F) procedure, uses a Linear Programming-based Heuristic to generate a starting solution to the F&F process. A tabu search procedure is used to try to enhance the best solution value provided by the F&F method that generates compound moves by a strategically truncated form of tree search. They report the first application of the F&F method to the MKP. Experimental results obtained on a wide set of benchmark problems clearly demonstrate the competitiveness of the proposed method compared to the state-of-the-art heuristic methods. DOI: 10.4018/jamc.2012100103 44 International Journal of Applied Metaheuristic Computing, 3(4), 43-63, October-December 2012 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. optimal or near-optimal solutions (the reader is referred to Fréville, 2004; Fréville & Hanafi, 2005, for a comprehensive annotated bibliography). Some very efficient algorithms (Kellerer et al., 2004; Martello & Toth, 1990) exist when m = 1, but as m increases, exact methods usually fail to provide an optimal solution for even moderate size instances. Hybrid tabu search techniques have been developed for the MKP (Glover & Kochenberger, 1996; Hanafi & Fréville, 1998), and most of the best-known solutions for the set of MKP instances available in the OR-Library (Beasley, 1990) were obtained by Vasquez & Hao (2001) and Vasquez & Vimont (2005). Other recent methods have obtained encouraging results by making a compromise between solution quality and computational effort. For example, Puchinger et al. (2006) proposed an extension of the classic core concept for the MKP (Pisinger, 1995) and also described an extension of the variable neighborhood search metaheuristic (Hansen & Mladenovic, 2001) used with a branch-and-cut algorithm. In addition, Hanafi and Glover (2007) offered an exploitation of nested inequalities and surrogate constraints on the MKP better than that proposed by Osorio et al. (2002), but did not offer computational results. Akcay et al. (2007) proposed a greedy heuristic based on the effective capacity notion defined as the maximum number of copies of an item that can be accepted if the entire knapsack were to be used for that item alone. Also in Boyer et al. (2009) two heuristics are presented. The first one uses surrogate relaxation where the relaxed problem is solved via a modified dynamic-programming algorithm. The heuristic provides a feasible solution. The second one combines a limited-branch-and-cut procedure with the previous approach, and tries to improve the bound obtained by exploring some nodes that have been rejected by the modified dynamic programming algorithm. Finally, Fleszar and Hindi (2009) proposed several fast and effective heuristics that are based on solving the linear programming relaxation of the problem. A wide variety of different techniques have been also used for solving optimally the MKP. Among others, we can mention dynamic programming approaches (Gilmore & Gomory, 1966; Green, 1967; Weigartner & Ness, 1967), tree search algorithms (Shih, 1979; Fayard & Plateau, 1982; Gavish & Pirkul, 1985; Vimont et al., 2008) and hybrid constraint programming and integer linear programming approaches (Oliva et al., 2001). Recently, Boussier et al. (2010) presented an exact method based on a multi-level search strategy. This paper proposes a new hybrid heuristic algorithm to solve the 0-1 MKP that effectively combines a tabu search procedure with a dynamic and adaptive neighborhood search procedure based on filter-and-fan (F&F) method (Glover, 1998; Rego & Glover, 2002). Our main objective is to study the effectiveness of the F&F algorithm in solving the MKP while keeping a relatively straightforward implementation of the approach. Experimental results, obtained on a wide set of benchmark problems; demonstrate the competitiveness of the proposed algorithm compared to the state-of-the-art heuristic methods. The remainder of this paper is organized as follows. Section 2 describes the basic Filter-andfan model and presents the proposed algorithm. Section 3 discusses the experimental results obtained on a wide set of benchmark problems and establishes a comparative analysis with the state-of-the-art heuristic methods. Finally, Section 4 presents the conclusions. 2. FILTER-AND-FAN METHOD The F&F method was introduced by Glover (1998) as a method for refining solutions obtained by scatter search, then it was further extended by Rego and Glover (2002). To create a combined neighborhood search strategies, this method was also proposed as an alternative to ejection chain methods (Rego & Glover, 2010; Glover & Rego, 2006). This approach consists in the integration of the filtration and sequential dispersion of candidate list strategies used in the 19 more pages are available in the full version of this document, which may be purchased using the "Add to Cart" button on the product's webpage: www.igi-global.com/article/filter-fan-metaheuristicmultidimensional-knapsack/74738?camid=4v1 This title is available in InfoSci-Journals, InfoSci-Journal Disciplines Computer Science, Security, and Information Technology. Recommend this product to your librarian: www.igi-global.com/e-resources/libraryrecommendation/?id=2