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A column generation scheme for the fixed fleet heterogeneous vehicle routing problem
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0003-2094-7376
2005 (English)Report (Other academic)
Abstract [en]

We present an optimizing column generation procedure for solving the vehicle routing problem with a fixed heterogeneous fleet of vehicles; if the method is used in a truncated fashion it turns into a heuristic. The method is based on a new mathematical formulation, which includes a new type of valid inequalities, strengthened by the use of Chvátal-Gomory rounding, and a Lagrangian dualization of this formulation. The dual problem is attacked by subgradient optimization and a near-optimal dual solution obtained is used for enumerating routes that are promising candidates for being used in an optimal solution. These routes are collected in a set partitioning problem, which is finally solved, and an upper bound to the optimal objective value is obtained. The method is evaluated on a set of small-scale test instances. The valid inequalities improves the lower bound significantly: the improvement depends on the ratio between total customer demand and total vehicle capacity. The qualities of the upper bounds varies quite much among the instances.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet , 2005.
Series
LiTH-MAT-R, 12
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-29699Local ID: 15092OAI: oai:DiVA.org:liu-29699DiVA: diva2:250516
Available from: 2009-10-09 Created: 2009-10-09 Last updated: 2013-08-30
In thesis
1. Accelerating column generation schemes: applications to routing problems
Open this publication in new window or tab >>Accelerating column generation schemes: applications to routing problems
2005 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Many integer optimization problems of great practical importance are today attacked with column generation. Merits of column generation is that it enables the use of compact and flexible formulations of many complex optimization problems, and that it often gives rise to good (strong) formulations. A potential drawback with column generation is the well-known tailing-off phenomenon, that is, that the objective value is improved rapidly in early iterations but very slowly in the late iterations.

We study various techniques for accelerating column generation methods for (integer) linear programs. We evaluate the use of stabilized column generation on a Traveling Salesman Subtour Problem, TSSP, a problem which is closely related to the Prize Collecting Traveling Salesman Problem. We further study how subgradient optimization can be used with the purpose of predicting optimal columns (and, optionally, non-binding restrictions). This technique is tested on the TSSP and the multicommodity network flow problem.

Another idea evaluated in this thesis is the use of over-generation of columns in a column generation scheme for an integer programming problem, in this case a vehicle routing problem with a heterogeneous fleet of vehicles.

The thesis also includes a note on a class of relatives to the Held and Karp 1–tree problem. It is shown that two subclasses have polynomial time-complexity. Further, we note a mistake in an earlier work and provide a counter-example to the erroneous result.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet, 2005. 34 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 978
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-30273 (URN)15790 (Local ID)91-85457-50-7 (ISBN)15790 (Archive number)15790 (OAI)
Public defence
2005-11-28, Glashuset, Hus B, Campus Valla, Linköping, 10:15 (Swedish)
Opponent
Available from: 2009-10-09 Created: 2009-10-09 Last updated: 2012-12-06

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Westerlund, AndreasGöthe-Lundgren, MaudLarsson, Torbjörn

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