Solving the Single-Sink, Fixed-Charge, Multiple-Choice Transportation Problem by Dynamic Programming

Authors:
Tue R. L. Christensen;Kim Allan Andersen;Andreas Klose
Affiliations:
Department of Economics and Business, Aarhus University, 8210 Aarhus V, Denmark;Department of Economics and Business, Aarhus University, 8210 Aarhus V, Denmark;Department of Mathematics, Aarhus University, 8000 Aarhus C, Denmark
Venue:
Transportation Science
Year:
2013

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Abstract

This paper considers a minimum-cost network flow problem in a bipartite graph with a single sink. The transportation costs exhibit a staircase cost structure because such types of transportation cost functions are often found in practice. We present a dynamic programming algorithm for solving this so-called single-sink, fixed-charge, multiple-choice transportation problem exactly. The method exploits heuristics and lower bounds to peg binary variables, improve bounds on flow variables, and reduce the state-space variable. In this way, the dynamic programming method is able to solve large instances with up to 10,000 nodes and 10 different transportation modes in a few seconds, much less time than required by a widely used mixed-integer programming solver and other methods proposed in the literature for this problem.