Two-stage column generation and applications in container terminal management

Seaport container terminals are source of many interesting large-scale optimization problems, which arise in the management of operations at several decision levels. In this paper we firstly illustrate a recently proposed framework, called two-stage column generation (Salani and Vacca, 2008), which is suitable for a particular class of large-scale problems, where the structure of the original formulation, namely the presence of a combinatorial number of decision variables, does not allow for straightforward reformulation. The main idea of their approach is to start from a meaningful subset of original variables, for which the problem remains feasible, apply the DW decomposition to this subset of variables, solve the resulting reformulation with standard column generation and then iteratively perform the explicit pricing on original compact-formulation variables, retracing back the reformulation and using complementary-slackness conditions. The two-stage column generation framework is suitable to be applied to the Tactical Berth Allocation Problem (TBAP) with Quay Crane Assignment, an integrated decision problem which occurs in the management of the quayside resources in container terminals. A compact formulation of TBAP has been recently presented by Giallombardo et al. (2008); in this paper, we propose a Dantzig-Wolfe reformulation of the problem and we show that standard column generation cannot cope with the pricing induced by such reformulation, because of the particular structure of the problem. It makes sense therefore to use two-stage column generation and we illustrate the application of the method to TBAP. Conclusions and future research directions are discussed.