Dynamic programming algorithms for the bi-objective integer knapsack problem

•A property of the traditional dynamic programming algorithm is identified.•The first algorithm is developed by directly using the property.•The second algorithm is developed by using the property in conjunction with the bound sets.•Insufficiency of linear relaxation solutions to estimate an upper b...

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Bibliographic Details
Published in:European journal of operational research Vol. 236; no. 1; pp. 85 - 99
Main Authors: Rong, Aiying, Figueira, José Rui
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.07.2014
Elsevier Sequoia S.A
Subjects:
Algorithms
Bound sets
Dynamic programming
Integer knapsack problems
Integer programming
Integers
Knapsack problem
Mathematical models
Multi-objective optimization
Numerical analysis
Pareto optimality
Pareto optimum
Studies
Tightening
Upper bounds
Multi-objective optimization
Dynamic programming
Integer knapsack problems
Bound sets
ISSN:0377-2217, 1872-6860
Online Access:Get full text
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Summary:•A property of the traditional dynamic programming algorithm is identified.•The first algorithm is developed by directly using the property.•The second algorithm is developed by using the property in conjunction with the bound sets.•Insufficiency of linear relaxation solutions to estimate an upper bound set is identified.•An extended upper set is proposed on the basis of the set of the linear relaxation solutions. This paper presents two new dynamic programming (DP) algorithms to find the exact Pareto frontier for the bi-objective integer knapsack problem. First, a property of the traditional DP algorithm for the multi-objective integer knapsack problem is identified. The first algorithm is developed by directly using the property. The second algorithm is a hybrid DP approach using the concept of the bound sets. The property is used in conjunction with the bound sets. Next, the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set. It means that the upper bound set is underestimated. Then, an extended upper bound set is proposed on the basis of the set of linear relaxation solutions. The efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set. The numerical results obtained from different types of bi-objective instances show the effectiveness of the proposed approach.
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ISSN:0377-2217
1872-6860
DOI:10.1016/j.ejor.2013.11.032