An exact algebraic ϵ-constraint method for bi-objective linear integer programming based on test sets

•We present a new exact algorithm for bi-objective linear integer problems.•Based on the epsilon-constraint method and algebraic test sets for integer problems.•The Pareto frontier is obtained solving exactly one problem for each point.•Integer problems are solved via reduction with test set, not us...

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Bibliographic Details
Published in:European journal of operational research Vol. 282; no. 2; pp. 453 - 463
Main Authors: Hartillo-Hermoso, María Isabel, Jiménez-Tafur, Haydee, Ucha-Enríquez, José María
Format: Journal Article
Language:English
Published: Elsevier B.V 16.04.2020
Subjects:
Algebraic test sets
Multiple objective programming
Non-dominated set
Pareto set
Unbounded Knapsack Problem
ϵ-constraint method
Non-dominated set
Pareto set
ϵ-constraint method
Unbounded Knapsack Problem
Algebraic test sets
Multiple objective programming
ISSN:0377-2217, 1872-6860
Online Access:Get full text
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Summary:•We present a new exact algorithm for bi-objective linear integer problems.•Based on the epsilon-constraint method and algebraic test sets for integer problems.•The Pareto frontier is obtained solving exactly one problem for each point.•Integer problems are solved via reduction with test set, not using an optimizer.•We obtain efficient points directly, avoiding weakly efficient ones. A new exact algorithm for bi-objective linear integer problems is presented, based on the classic ϵ-constraint method and algebraic test sets for single-objective linear integer problems. Our method provides the complete Pareto frontier N of non-dominated points and, for this purpose, it considers exactly |N| single-objective problems by using reduction with test sets instead of solving with an optimizer. Although we use Gröbner bases for the computation of test sets, which may provoke a bottleneck in principle, the computational results are shown to be promising, especially for unbounded knapsack problems, for which any usual branch-and-cut strategy could be much more expensive. Nevertheless, this algorithm can be considered as a potentially faster alternative to IP-based methods when test sets are available.
ISSN:0377-2217
1872-6860
DOI:10.1016/j.ejor.2019.09.032