Development of core to solve the multidimensional multiple-choice knapsack problem

The multidimensional multiple-choice knapsack problem (MMKP) is an extension of the 0-1 knapsack problem. The core concept has been used to design efficient algorithms for the knapsack problem but the core has not been developed for the MMKP so far. In this paper, we develop an approximate core for the MMKP and utilize it to solve the problem exactly. Computational results show that the algorithm can solve large uncorrelated instances (up to 100 classes and 100 items in each class) and correlated instances with small number of constraints (up to 5) efficiently. In particular, it solves recently published hard instances for the MMKP in less than a second. The algorithm consumes negligible memory, and compared with the best previous exact algorithm for the MMKP performs significantly faster.

[1]  Gwendal Simon,et al.  Hard multidimensional multiple choice knapsack problems, an empirical study , 2010, Comput. Oper. Res..

[2]  Fred W. Glover,et al.  Exploiting nested inequalities and surrogate constraints , 2007, Eur. J. Oper. Res..

[3]  David Pisinger,et al.  Core Problems in Knapsack Algorithms , 1999, Oper. Res..

[4]  David Pisinger.  A minimal algorithm for the Multiple-choice Knapsack Problem , 1995 .

[5]  Abdelkader Sbihi,et al.  A best first search exact algorithm for the Multiple-choice Multidimensional Knapsack Problem , 2007, J. Comb. Optim..

[6]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

[7]  María Auxilio Osorio Lama,et al.  Cutting analysis for MKP , 2004, Proceedings of the Fifth Mexican International Conference in Computer Science, 2004. ENC 2004..

[8]  Michael C. Jäger,et al.  Optimising quality of service for the composition of electronic services , 2007 .

[9]  Mhand Hifi,et al.  A Reactive Local Search-Based Algorithm for the Multiple-Choice Multi-Dimensional Knapsack Problem , 2006, Comput. Optim. Appl..

[10]  Ross J. W. James,et al.  Enumeration Methods for Repeatedly Solving Multidimensional Knapsack Sub-Problems , 2005, IEICE Trans. Inf. Syst..

[11]  Zuren Feng,et al.  An ant colony optimization approach to the multiple-choice multidimensional knapsack problem , 2010, GECCO '10.

[12]  David Pisinger,et al.  A Minimal Algorithm for the 0-1 Knapsack Problem , 1997, Oper. Res..

[13]  Mohammad Sohel Rahman,et al.  Solving the Multidimensional Multiple-choice Knapsack Problem by constructing convex hulls , 2006, Comput. Oper. Res..

[14]  Fred Glover,et al.  Surrogate Constraint Duality in Mathematical Programming , 1975, Oper. Res..

[15]  Eric G. Manning,et al.  Quality adaptation in a multisession multimedia system: model, algorithms, and architecture , 1998 .

[16]  Arnaud Fréville,et al.  An Efficient Preprocessing Procedure for the Multidimensional 0- 1 Knapsack Problem , 1994, Discret. Appl. Math..

[17]  Kin F. Li,et al.  Solving the Knapsack Problem for Adaptive Multimedia Systems , 2002, Stud. Inform. Univ..

[18]  Nikitas J. Dimopoulos,et al.  A new heuristic for solving the multichoice multidimensional knapsack problem , 2005, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

[19]  Egon Balas,et al.  An Algorithm for Large Zero-One Knapsack Problems , 1980, Oper. Res..

[20]  Eric G. Manning,et al.  Heuristic Solutions for the Multiple-Choice Multi-dimension Knapsack Problem , 2001, International Conference on Computational Science.

[21]  David Pisinger,et al.  Algorithms for Knapsack Problems , 1995 .

[22]  Fred W. Glover,et al.  Generating Cuts from Surrogate Constraint Analysis for Zero-One and Multiple Choice Programming , 1997, Comput. Optim. Appl..

[23]  Andrew V. Goldberg,et al.  On finding the exact solution of a zero-one knapsack problem , 1984, STOC '84.

[24]  Günther R. Raidl,et al.  The Core Concept for the Multidimensional Knapsack Problem , 2006, EvoCOP.

[25]  David Pisinger,et al.  An expanding-core algorithm for the exact 0-1 knapsack problem , 1995 .

[26]  René Beier,et al.  Probabilistic analysis of knapsack core algorithms , 2004, SODA '04.

[27]  Mhand Hifi,et al.  A column generation method for the multiple-choice multi-dimensional knapsack problem , 2010, Comput. Optim. Appl..

[28]  Fred W. Glover,et al.  Surrogate Constraints , 1968, Oper. Res..

[29]  Raymond R. Hill,et al.  New greedy heuristics for the Multiple-choice Multi-dimensional Knapsack Problem , 2007 .

[30]  Prabhakant Sinha,et al.  The Multiple-Choice Knapsack Problem , 1979, Oper. Res..

[31]  S. Martello,et al.  Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem , 1999 .