Efficient algorithms for solving multiconstraint zero-one knapsack problems to optimality

Gavish, Bezalel; Pirkul, Hasan

doi:10.1007/bf02591863

Cited by 196 publications

(81 citation statements)

References 31 publications

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“…Element i has a weight i w and the knapsack has a weight limit C. If object i is placed into the knapsack, we will obtain a profit i p . The problem is to maximize the total profit under the constraint that the total weights of all chosen objects are at most C. So, the knapsack problem can be formulated as [16,14,17,18]: (3) where i x is a binary variable denoting whether object i is chosen or not. Equation (3) is the constraint that needs to be satisfied and defines the profit of a feasible n-tuple.…”

Section: The Formulation Of the 0/1 Knapsack Problemmentioning

confidence: 99%

Optimization of Educational Systems Using Knapsack Problem

Atashbar¹,

Rahimi²

2011

IJMLC

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Section: The Formulation Of the 0/1 Knapsack Problemmentioning

confidence: 99%

Optimization of Educational Systems Using Knapsack Problem

Atashbar¹,

Rahimi²

2011

IJMLC

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“…The unique knapsack problem (UKP) has been given considerable attention in the literature though it is not, in fact, as difficult as (MKP), more precisely, it can be solved in a pseudo-polynomial time (see [2,3,6,11,12]). We have then tried to transform the original (MKP) into a (UKP) (see also [15,17]). In this purpose, we have used a relaxation technique, that is to say, surrogate relaxation.…”

Section: Introductionmentioning

confidence: 99%

“…Several heuristics have been proposed in order to find out good surrogate multipliers (see in particular [14,15,17]). In practice, it is not important to obtain the optimal multiplier vector, since in the general case we have no guarantee that vðSðu Ã ÞÞ ¼ vðMKPÞ.…”

Section: Introductionmentioning

confidence: 99%

Heuristics for the 0–1 multidimensional knapsack problem

Boyer

Elkihel

Baz

2009

European Journal of Operational Research

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a b s t r a c tTwo 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.

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“…The MDKP has many applications in the fields of capital budgeting, cutting stock, cargo loading, allocating processors and databases in distributed computer systems [8]. Due to its hard complexity (it is an NP-complete combinatorial prob-lem) the size of the instances which can be solved exactly is modest.…”

Section: Introductionmentioning

confidence: 99%

“…Frévillle and Plateau [6] mentioned the limit of 5 constraints and 200 variables for previous works. Exact methods are based on dynamic programming [9,29] or in branchand-bound techniques [26,8,23]. Large size instances can only be managed with approximate methods which establish a balance between the quality of the solutions and the computational time required.…”

Section: Introductionmentioning

confidence: 99%

A Scatter Search Method for the Bi-Criteria Multi-dimensional {0,1}-Knapsack Problem using Surrogate Relaxation

Silva

Clímaco

Figueira

2004

Journal of Mathematical Modelling and Algorithms

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Abstract. This paper presents a scatter search (SS) based method for the bi-criteria multi-dimensional knapsack problem. The method is organized according to the usual structure of SS: (1) diversification, (2) improvement, (3) reference set update, (4) subset generation, and (5) solution combination. Surrogate relaxation is used to convert the multi-constraint problem into a single constraint one, which is used in the diversification method and to evaluate the quality of the solutions. The definition of the appropriate surrogate multiplier vector is also discussed. Tests on several sets of large size instances show that the results are of high quality and an accurate description of the entire set of the non-dominated solutions can be obtained within reasonable computational time.Comparisons with other meta-heuristics are also presented. In the tested instances the obtained set of potentially non-dominated solutions dominates the set found with those meta-heuristics. (2000): 90C27. Mathematics Subject Classification

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Efficient algorithms for solving multiconstraint zero-one knapsack problems to optimality

Cited by 196 publications

References 31 publications

Optimization of Educational Systems Using Knapsack Problem

Optimization of Educational Systems Using Knapsack Problem

Heuristics for the 0–1 multidimensional knapsack problem

A Scatter Search Method for the Bi-Criteria Multi-dimensional {0,1}-Knapsack Problem using Surrogate Relaxation

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