Column generation (CG) models have several advantages over compact formulations, namely, they provide better LP bounds, may eliminate symmetry, and can hide non-linearities in their subproblems. However, users also encounter drawbacks in the form of slow convergency a.k.a. the tailing-off effect and the oscillation of the dual variables. Among different alternatives for stabilizing the CG process, Ben Amor et al. [Ben Amor, H., Desrosiers, J., and Valério de Carvalho, J. M. (2006). Dual-optimal inequalities for stabilized column generation. Operations Research, 54(3), [454][455][456][457][458][459][460][461][462][463] suggest the use of dual-optimal inequalities (DOIs) in the context of cutting stock and bin packing problems. We generalize their results, provide new classes of (deep) DOIs, and show the applicability to other problems (vector packing, vertex coloring, bin packing with conflicts). We also suggest the dynamic addition of violated dual inequalities in a cutting-plane fashion and the use of dual inequalities that are not necessarily (deep) DOIs. In the latter case, a recovery procedure is needed to restore primal feasibility. Computational results proving the usefulness of the methods are presented.

In the dial-a-ride problem (DARP), user-specified transport requests from origin to destination points have to be served by a fleet of homogeneous vehicles. The problem variant we consider aims at finding a set of minimum-cost routes satisfying constraints on vehicle capacity, time windows, maximum route duration, and maximum user ride times. We propose an adaptive large neighborhood search (ALNS) for its solution. The key novelty of the approach is an exact amortized constant-time algorithm for evaluating the feasibility of request insertions in the repair steps of the ALNS. In addition, we use two optional improvement techniques: a local-search based, intra-route improvement of routes of promising solutions using the Balas-Simonetti neighborhood, and the solution of a set-partitioning model over a subset of all routes generated during the search. With these techniques, the proposed algorithm outperforms the state-of-the-art methods in terms of solution quality. New best solutions are found for several benchmark instances.

A dynamic time window relates to two operations that must be executed within a given time meaning that the difference between the points in time when the two operations are performed is bounded from above. The most prevalent context of dynamic time windows is when precedence is given for the two operations so that it is a priori specified that one operation must take place before the other. A prominent vehicle routing problem with dynamic time windows and precedence is the dial-a-ride problem (DARP), where user-specified transportation requests from origin to destination points must be serviced. The paper presents a new branch-and-cut-and-price solution approach for the DARP, the prototypical vehicle-routing problem with ordinary and dynamic time windows. For the first time (to our knowledge), both ordinary and dynamic time windows are handled in the column-generation subproblem. For the solution, an effective column-generation pricing procedure is derived that allows fast shortest-path computations due to new dominance rules. The new approach is compared with alternative column-generation algorithms that handle dynamic time windows either as constraints of the master program or with less effective labeling procedures. Computational experiments indicate the superiority of the new approach.

This paper considers packing and cutting problems in which a packing/cutting pattern is constrained independently in two or more dimensions. Examples are restrictions with respect to weight, length, and value. We present branch-and-price algorithms to solve these vector packing problems (VPPs) exactly. The underlying column-generation procedure uses an extended master program that is stabilized by (deep) dualoptimal inequalities. While some inequalities are added to the master program right from the beginning (static version), other violated dual-optimal inequalities are added dynamically. The column-generation subproblem is a multidimensional knapsack problem, either binary, bounded, or unbounded depending on the specific master problem formulation. Its fast resolution is decisive for the overall performance of the branch-and-price algorithm. In order to provide a generic but still efficient solution approach for the subproblem, we formulate it as a shortest path problem with resource constraints (SPPRC), yielding the following advantages: (i) Violated dual-optimal inequalities can be identified as a by-product of the SPPRC labeling approach and thus be added dynamically; (ii) branching decisions can be implemented into the subproblem without deteriorating its resolution process; and (iii) larger instances of higher-dimensional VPPs can be tackled with branch-and-price for the first time. Extensive computational results show that our branch-and-price algorithms are capable of solving VPP benchmark instances effectively.

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