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On branching heuristics for the bi-objective 0/1 unidimensional knapsack problem

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Abstract

This paper focuses on branching strategies that are involved in branch and bound algorithms when solving multi-objective optimization problems. The choice of the branching variable at each node of the search tree constitutes indeed an important component of these algorithms. In this work we focus on multi-objective knapsack problems. In the literature, branching heuristics used for these problems are static, i.e., the order on the variables is determined prior to the execution. This study investigates the benefit of defining more sophisticated branching strategies. We first analyze and compare a representative set of classic branching heuristics and conclude that none can be identified as the best overall heuristic. Using an oracle, we highlight that combining branching heuristics within the same branch and bound algorithm leads to considerably reduced search trees but induces high computational costs. Based on learning adaptive techniques, we propose then dynamic adaptive branching strategies that are able to select the suitable heuristic to apply at each node of the search tree. Experiments are conducted on the bi-objective 0/1 unidimensional knapsack problem.

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Notes

  1. The reduced oracle method using \(c=22\) heuristics is the oracle method, at the exception that the equalities on the quality measure are broken by giving the advantage to the branching heuristic with the best rank in the reduced oracle method.

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Acknowledgements

This work is supported by the following projects: ANR-09-BLAN-0361 “GUaranteed Efficiency for PAReto optimal solutions Determination (GUEPARD)”, the project LigeRO, and the project ANR/DFG-14-CE35-0034-01 “Exact Efficient Solution of Mixed Integer Programming Problems with Multiple Objective Functions (vOpt)”.

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Authors and Affiliations

  1. LIMOS UMR 6158, Mines Saint-Étienne, 158 cours Fauriel CS 62362, 42023, Saint-Étienne Cedex 2, France

    Audrey Cerqueus

  2. IRCCyN UMR CNRS 6597, Université de Nantes, 2 Rue de la Houssinière BP 92208, 44322, Nantes Cedex 03, France

    Xavier Gandibleux & Anthony Przybylski

  3. LERIA, Université d’Angers, 2 Boulevard Lavoisier, 49045, Angers Cedex 01, France

    Frédéric Saubion

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  1. Audrey Cerqueus
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  2. Xavier Gandibleux
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  3. Anthony Przybylski
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  4. Frédéric Saubion
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Correspondence to Audrey Cerqueus.

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Cerqueus, A., Gandibleux, X., Przybylski, A. et al. On branching heuristics for the bi-objective 0/1 unidimensional knapsack problem. J Heuristics 23, 285–319 (2017). https://doi.org/10.1007/s10732-017-9346-9

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