Dynamic Programming-Based Column Generation on Time-Expanded Networks: Application to the Dial-a-Flight Problem

Authors:
Faramroze G. Engineer;George L. Nemhauser;Martin W. P. Savelsbergh
Affiliations:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, New South Wales 2308, Australia;H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30308;H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30308
Venue:
INFORMS Journal on Computing
Year:
2011

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Abstract

We present a relaxation-based dynamic programming algorithm for solving resource-constrained shortest-path problems (RCSPPs) in the context of column generation for the dial-a-flight problem. The resulting network formulation and pricing problem require solving RCSPPs on extremely large time-expanded networks having a huge number of local resource constraints, i.e., constraints that apply to small subnetworks. The relaxation-based dynamic programming algorithm alternates between a forward and a backward search. Each search employs bounds derived in the previous search to prune the search space. Between consecutive searches, the relaxation is tightened using a set of critical resources and a set of critical arcs over which these resources are consumed. As a result, a relatively small state space is maintained, and many paths can be pruned while guaranteeing that an optimal path is ultimately found.