Improved results on the 0-1 multidimensional knapsack problem (original) (raw)
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A hybrid approach for the 0-1 multidimensional knapsack problem
2001
We present a hybrid approach for the 0-1 multidimensional knapsack problem. The proposed approach combines linear programming and Tabu Search. The resulting algorithm improves significantly on the best known results of a set of more than 150 benchmark instances. [Toyoda, 1975] Y. Toyoda. A simplified algorithm for obtaining approximate solutions to zero-one programming problem.
Heuristics for the 0–1 multidimensional knapsack problem
European Journal of Operational Research, 2009
Two heuristics for the 0-1 multidimensional knapsack problem (MKP) are presented. The first one uses surrogate relaxation, and the relaxed problem is solved via a modified dynamic-programming algorithm. The heuristics provides a feasible solution for (MKP). The second one combines a limited-branch-and-cutprocedure 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. Computational experiences show that our approaches give better results than the existing heuristics, and thus permit one to obtain a smaller gap between the solution provided and an optimal solution.
Solving the 0-1 Multidimensional Knapsack Problem Using Tabu Search and Visualization
2008
We propose an exact method which combines the resolution search and branch & bound algorithms for solving the 0-1 Multidimensional Knapsack Problem. This algorithm is able to prove large-scale strong correlated instances. The optimal values of the 10 constraint, 500 variable instances of the OR-Library are exposed. These values were previously unknown.
The Multidimensional Knapsack Problem: Structure and Algorithms
2010
We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different Integer Linear Programming (ILP) based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to the LP-relaxation and the original problem and then introduce a new core concept for the MKP, which we study extensively. The empirical analysis is then used to develop new concepts for solving the MKP using ILP-based and memetic algorithms. Different collaborative combinations of the presented methods are discussed and evaluated. Further computational experiments with longer run-times are also performed in order to compare the solutions of our approaches to the best known solutions of another so far leading approach for common MKP benchmark instances. The extensive computational experiments show the effectiveness of the proposed methods, which yield highly competitive results in significantly shorter run-times than previously described approaches.
Proceedings of the International Conference on Evolutionary Computation Theory and Applications, 2011
In this paper, we present a new population-based heuristic for the multidimensional 0-1 knapsack problem (MKP) which is combined with 0-1 linear programming to improve the quality of the final heuristic solution. The MKP is one of the most well known NP-hard problems and has received wide attention from the operational research community during the last four decades. MKP arises in several practical problems such as the capital budgeting problem, cargo loading, cutting stock problem, and computing processors allocation in huge distributed systems. Several different techniques have been proposed to solve this problem. However, according to its NP-hard nature, exact methods are unable to find optimal solutions for larger problem instances. Heuristic methods have become the alternative, and the last generation of them, are being successfully applied to this problem. Hence, in practice, heuristic algorithms to generate nearoptimal solutions for larger problem instances are of special interest. The presented hybrid heuristic approach exploits the fact, that using a state-of-the-art solver a small binary linear programming (BLP) problem can be solved within reasonable time. The computational experiments show that the presented combined approach produces highly competitive results in significantly shorter run-times than the previously described approaches.
Hybrid heuristic algorithm for the multidimensional knapsack problem
2012 IV International Conference "Problems of Cybernetics and Informatics" (PCI), 2012
In this work, a new hybrid heuristic algorithm for the 0/1 multidimensional knapsack problem is proposed. In the algorithm, Lagrange multipliers for every constraint are determined to reduce the problem to single dimension and some initial solutions are obtained with greedy algorithms. Then, these solutions are improved with iterative procedures. In order to test efficiency of the algorithm, computational experiments were done on some library problems in literature. It was observed that the algorithm has high efficiency in terms of solutions and time.
A Local-Search-Based Heuristic for the Demand-Constrained Multidimensional Knapsack Problem
INFORMS Journal on Computing, 2005
W e consider an extension of the 0-1 multidimensional knapsack problem in which there are greater-thanor-equal-to inequalities, called demand constraints, in addition to the standard less-than-or-equal-to constraints. Moreover, the objective function coefficients are not constrained in sign. This problem is worth considering because it is embedded in models of practical application, it has an intriguing combinatorial structure, and it appears to be a challenging problem for commercial ILP solvers. Our approach is based on a nested tabusearch algorithm in which neighborhoods with different structures are exploited. First, a tabu-search procedure is carried out in which mainly the infeasible region is explored. Once feasibility has been established, a second tabu-search procedure, which analyzes only feasible solutions, is applied. The algorithm has been tested on a wide set of instances. Computational results are discussed.
Fast, effective heuristics for the 0-1 multi-dimensional knapsack problem
Computers & Operations Research, 2009
The objective of the multidimensional knapsack problem (MKP) is to find a subset of items with maximum value that satisfies a number of knapsack constraints. Solution methods for MKP, both heuristic and exact, have been researched for several decades. This paper introduces several fast and effective heuristics for MKP that are based on solving the LP relaxation of the problem. Improving procedures are proposed to strengthen the results of these heuristics. Additionally, the heuristics are run with appropriate deterministic or randomly generated constraints imposed on the linear relaxation that allow generating a number of good solutions. All algorithms are tested experimentally on a widely used set of benchmark problem instances to show that they compare favourably with the best-performing heuristics available in the literature.
A multi-level search strategy for the 0-1 Multidimensional Knapsack Problem
Discrete Applied Mathematics, 2010
We propose an exact method based on a multi-level search strategy for solving the 0-1 Multidimensional Knapsack Problem. Our search strategy is primarily based on the reduced costs of the non-basic variables of the LP-relaxation solution. Considering that the variables are sorted in decreasing order of their absolute reduced cost value, the top level branches of the search tree are enumerated following Resolution Search strategy, the middle level branches are enumerated following Branch & Bound strategy and the lower level branches are enumerated according to a simple Depth First Search enumeration strategy. Experimentally, this cooperative scheme is able to solve optimally large-scale strongly correlated 0-1 Multidimensional Knapsack Problem instances. The optimal values of all the 10 constraint, 500 variable instances and some of the 30 constraint, 250 variable instances of the OR-Library were found. These values were previously unknown.