# Column generation algorithms for bi-objective combinatorial optimization problems with a min–max objective

Column generation algorithms for bi-objective combinatorial optimization problems with a... Many practical combinatorial optimization problems can be described by integer linear programs having an exponential number of variables, and they are efficiently solved by column generation algorithms. For these problems, column generation is used to compute good dual bounds that can be incorporated in branch-and-price algorithms. Recent research has concentrated on describing lower and upper bounds of bi-objective and general multi-objective problems with sets of points (bound sets). An important issue to address when computing a bound set by column generation is how to efficiently search for columns corresponding to each point of the bound set. In this work, we propose a generalized column generation scheme to compute bound sets for bi-objective combinatorial optimization problems. We present specific implementations of the generalized scheme for the case where one objective is a min–max function by using a variant of the $$\varepsilon$$ ε -constraint method to efficiently model these problems. The proposed strategies are applied to a bi-objective extension of the multi-vehicle covering tour problem, and their relative performances based on different criteria are compared. The results show that good bound sets can be obtained in reasonable times if columns are efficiently managed. The variant of the $$\varepsilon$$ ε -constraint presented is also better than a standard $$\varepsilon$$ ε -constraint method in terms of the quality of the bound sets. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png EURO Journal on Computational Optimization Springer Journals

# Column generation algorithms for bi-objective combinatorial optimization problems with a min–max objective

, Volume 6 (2) – Nov 2, 2017
26 pages

/lp/springer_journal/column-generation-algorithms-for-bi-objective-combinatorial-mZMz982zdf
Publisher
Springer Journals
Copyright © 2017 by Springer-Verlag GmbH and EURO - The Association of European Operational Research Societies
Subject
Business and Management; Operations Research/Decision Theory; Operations Management; Operations Research, Management Science; Optimization
ISSN
2192-4406
eISSN
2192-4414
D.O.I.
10.1007/s13675-017-0090-6
Publisher site
See Article on Publisher Site

### Abstract

Many practical combinatorial optimization problems can be described by integer linear programs having an exponential number of variables, and they are efficiently solved by column generation algorithms. For these problems, column generation is used to compute good dual bounds that can be incorporated in branch-and-price algorithms. Recent research has concentrated on describing lower and upper bounds of bi-objective and general multi-objective problems with sets of points (bound sets). An important issue to address when computing a bound set by column generation is how to efficiently search for columns corresponding to each point of the bound set. In this work, we propose a generalized column generation scheme to compute bound sets for bi-objective combinatorial optimization problems. We present specific implementations of the generalized scheme for the case where one objective is a min–max function by using a variant of the $$\varepsilon$$ ε -constraint method to efficiently model these problems. The proposed strategies are applied to a bi-objective extension of the multi-vehicle covering tour problem, and their relative performances based on different criteria are compared. The results show that good bound sets can be obtained in reasonable times if columns are efficiently managed. The variant of the $$\varepsilon$$ ε -constraint presented is also better than a standard $$\varepsilon$$ ε -constraint method in terms of the quality of the bound sets.

### Journal

EURO Journal on Computational OptimizationSpringer Journals

Published: Nov 2, 2017

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