The Multidimensional Knapsack Problem: Structure and Algorithms (original) (raw)
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Exact, Metaheuristic, and Hybrid Approaches to Multidimensional Knapsack Problems
2009
In this chapter we review our recent work on applying hybrid collaborative techniques that integrate branch and bound (B&B) and memetic algorithms (MAs) in order to design effective heuristics for the multidimensional knapsack problem (MKP). To this end, let us recall that branch and bound (B&B)[102] is an exact algorithm for finding optimal solutions to combinatorial problems that works basically by producing convergent lower and upper bounds for the optimal solution using an implicit enumeration scheme.
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.
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.
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.
Improved results on the 0-1 multidimensional knapsack problem
European Journal of Operational Research, 2005
Geometric Constraint and Cutting planes have been successfully used to solve the 0-1 multidimensional knapsack problem. Our algorithm combines Linear Programming with an efficient tabu search. It gives best results when compared with other algorithms on benchmarks issued from the OR-LIBRARY I BRARY. Embedding this algorithm in a variables fixing heuristic still improves our previous results. Furthermore difficult sub problems with about 100 variables issued from the 500 original ones could be generated. These small sub problems are always very hard to solve.
International journal of circuits, systems and signal processing, 2021
The 0-1 Multidimensional Knapsack Problem (MKP) is a NP-Hard problem that has important applications in business and industry. Approximate solution approaches for the MKP in the literature typically provide no guarantee on how close generated solutions are to the optimum. This article demonstrates how general-purpose integer programming software (Gurobi) is iteratively used to generate solutions for the 270 MKP test problems in Beasley's OR-Library such that, on average, the solutions are guaranteed to be within 0.094% of the optimums and execute in 88 seconds on a standard PC. This methodology, called the simple sequential increasing tolerance (SSIT) matheuristic, uses a sequence of increasing tolerances in Gurobi to generate a solution that is guaranteed to be close to the optimum in a short time. This solution strategy generates bounded solutions in a timely manner without requiring the coding of a problem-specific algorithm. The SSIT results (although guaranteed within 0.094% of the optimums) when compared to known optimums deviated only 0.006% from the optimums-far better than any published results for these 270 MKP test instances.
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.
The multiobjective multidimensional knapsack problem: a survey and a new approach
2012
The knapsack problem (KP) and its multidimensional version (MKP) are basic problems in combinatorial optimization. In this paper we consider their multiobjective extension (MOKP and MOMKP), for which the aim is to obtain or to approximate the set of efficient solutions. In a first step, we classify and describe briefly the existing works, that are essentially based on the use of metaheuristics. In a second step, we propose the adaptation of the twophase Pareto local search (2PPLS) to the resolution of the MOMKP. With this aim, we use a very-large scale neighborhood (VLSN) in the second phase of the method, that is the Pareto local search. We compare our results to state-of-the-art results and we show that we obtain results never reached before by heuristics, for the biobjective instances. Finally we consider the extension to three-objective instances.
An Approximate Dynamic Programming Approach to Multidimensional Knapsack Problems
Management Science, 2002
W e present an Approximate Dynamic Programming (ADP) approach for the multidimensional knapsack problem (MKP). We approximate the value function (a) using parametric and nonparametric methods and (b) using a base-heuristic. We propose a new heuristic which adaptively rounds the solution of the linear programming relaxation. Our computational study suggests: (a) the new heuristic produces high quality solutions fast and robustly, (b) state of the art commercial packages like CPLEX require significantly larger computational time to achieve the same quality of solutions, (c) the ADP approach using the new heuristic competes successfully with alternative heuristic methods such as genetic algorithms, (d) the ADP approach based on parametric and nonparametric approximations, while producing reasonable solutions, is not competitive. Overall, this research illustrates that the base-heuristic approach is a promising computational approach for MKPs worthy of further investigation.
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.