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A two-stage algorithm with valid inequalities for the split delivery vehicle routing problem
ARTICLE IN PRESS0925-5273/$ - se
doi:10.1016/j.ijp
Correspondi
E-mail addreInt. J. Production Economics 105 (2007) 228–242
/locate/ijpeA two-stage algorithm with valid inequalities for
the split delivery vehicle routing problem
Mingzhou Jin, Kai Liu, Royce O. Bowden
Department of Industrial and Systems Engineering, Mississippi State University, P.O. Box 9542, Mississippi State, MS, 39762, USA
Received 1 November 2004; accepted 1 April 2006
Available online 3 July 2006Abstract
This paper proposes a two-stage (TS) algorithm with valid inequalities (TSVI) to optimally solve the split delivery
vehicle routing problem (SDVRP). The first stage in the TSVI creates clusters that cover all demand and establishes a lower
bound. The second stage calculates the minimal distance traveled for each cluster by solving the corresponding traveling
salesman problem (TSP). The sum of the minimal distance traveled over all clusters yields an upper bound. The minimal
distance of each TSP is added as cuts (constraints) into the first stage for the vehicles visiting all the points covered by the
TSP problem. This iterative procedure stops with an optimal solution when no new clusters are created in the first stage.
The TS scheme provides opportunities to develop strong valid inequalities for the first stage to improve the overall
computational speed. Seven valid inequalities are developed. One of the valid inequalities is created to address the
symmetric nature of a solution caused by the index combinations of vehicles. Another strong valid inequality is created to
provide a lower bound on the distance traveled for each subset of demand points. Conditions under which a delivery to a
demand point is not split in an optimal solution are studied in order to create additional valid inequalities. The numerical
experiments show that the TSVI significantly outperforms other exact solution approaches provided in the literature for
the SDVRP.
r 2006 Elsevier B.V. All rights reserved.
Keywords: Sp
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