The vehicle routing problem with time windows consists of delivering goods at minimum cost to a set of customers using an unlimited number of capacitated vehicles assigned to a single depot. Each customer must be visited within a prescribed time window. The most recent successful solution methods for this problem are branch-and-price-and-cut algorithms where the column generation subproblem is an elementary shortest-path problem with resource constraints (ESPPRC). In this paper, we propose new ideas having the potential to improve such a methodology. First, we develop a tabu search heuristic for the ESPPRC that allows, in most iterations, the generation of negative reduced cost columns in a short computation time. Second, to further accelerate the subproblem solution process, we propose to relax the elementarity requirements for a subset of the nodes. This relaxation, however, yields weaker lower bounds. Third, we introduce a generalization of the k-path inequalities and highlight that these generalized inequalities can, in theory, be stronger than the traditional ones. Finally, combining these ideas with the most recent advances published in the literature, we present a wide variety of computational results on the Solomon's 100-customer benchmark instances. In particular, we report solving five previously unsolved instances.

In this article, we present a comparative study of several strategies that can be applied to achieve the so‐called elementary lower bound in vehicle routing problems, that is, the bound obtained when all positive‐valued variables in an optimal solution of the linear relaxation of the set‐partitioning formulation correspond to vehicle routes without cycles. This bound can be achieved by solving the resource‐constrained elementary shortest path problem—an N P ‐hard problem—as the pricing problem in a column generation algorithm, but several other strategies can be used to ultimately produce the same lower bound in less computational effort. State‐of‐the‐art algorithms for vehicle routing problems rely on the quality of this lower bound to either bound the size of the search tree in a branch‐and‐price algorithm or the complexity of an enumeration procedure used to limit the number of variables in the set‐partitioning model. We consider several strategies for imposing elementarity that involve ng‐paths, strong degree constraints, and decremental state‐space relaxation. We compare the performance of these strategies on some selected instances of the vehicle routing problem with time windows. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(1), 88–99. 2015

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The min‐max k‐vehicles windy rural postman problem consists of minimizing the maximal distance traveled by a vehicle to find a set of balanced routes that jointly service all the required edges in a windy graph. This is a very difficult problem, for which a branch‐and‐cut algorithm has already been proposed, providing good results when the number of vehicles is small. In this article, we present a branch‐price‐and‐cut method capable of obtaining optimal solutions for this problem when the number of vehicles is larger for the same set of required edges. Extensive computational results on instances from the literature are presented. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 63(1), 34–45 2014

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