Advances in shortest path based column generation for integer programming
Engineer, Faramroze Godrej
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Branch-price-and-cut algorithms are among the most successful exact optimization approaches for solving many routing and scheduling problems. This is due, in part, to the availability of extremely efficient and effective dynamic programming algorithms for solving the pricing problem, and the availability of efficient and effective branching schemes and cutting planes that drive integrality. In terms of branch-price-and-cut, two obstacles we face today are (1) being able to solve harder and larger pricing problems, and (2) solving mixed-integer column generation formulations that suffer from relatively weak LP bounds compared to the more traditional 0-1 set partitioning type. As part of the work presented in this thesis, we encounter column generation formulations motivated by real life problems that require overcoming both types of challenges. The first part of this thesis is dedicated to solving the resource constrained shortest path problem (RCSPP) arising in column generation pricing problems for formulations involving extremely large networks and a huge number of local resource constraints. We present a relaxation-based dynamic programming algorithm that alternates between a forward and a backward search. Each search employs bounds derived in the previous search to prune the search, and between consecutive searches, the relaxation is tightened over a set of critical resources and arcs. The second part of this thesis focuses in the fixed charge shortest path problem (FCSPP) in which the amount of resource consumed is itself a continuous bounded variable. By exploiting the structure of optimal solutions to FCSPP, we design and implement a solution approach that relies on solving multiple RCSPPs, and therefore can again make use of extremely efficient and effective dynamic programming algorithms. In the third and final part of this thesis, we present a branch-price-and-cut algorithm for the inventory routing problem (IRP). We extend a class of cuts known for the vehicle routing problem, and develop a new class of cuts specifically for IRP to tighten the formulation. Both the branching schemes and cuts preserve the structure of the pricing problem making them efficiently implementable within a branch-price-and-cut algorithm.