MIP approaches for the integrated berth allocation and quay crane assignment and scheduling problem ∗ Agostinho Agra aagra@ua.pt Department of Mathematics and CIDMA, University of Aveiro Maryse Oliveira maryse.oliveira@ua.pt Department of Mathematics, University of Aveiro September 2, 2017 Abstract In this paper we consider an integrated berth allocation and quay crane assignment and scheduling problem motivated by a real case where a heterogeneous set of cranes is considered. A first mathematical model based on the relative position formulation (RPF) for the berth allocation aspects is presented. Then, a new model is introduced to avoid the big-M constraints included in the RPF. This model results from a discretization of the time and space variables. For the new discretized model several enhancements, such as valid inequalities, are introduced. In order to derive good feasible solutions, a rolling horizon heuristic (RHH) is presented. A branch and cut approach that uses the enhanced discretized model and incorporates the upper bounds provided by the RHH solution is proposed. Computational tests are reported to show (i) the quality of the linear relaxation of the enhanced models; (ii) the effectiveness of the exact approach to solve to optimality a set of real instances; and (iii) the scalability of the RHH based on the enhanced mathematical model which is able to provide good feasible solutions for large size instances. keywords: Logistics; Berth allocation; Quay crane scheduling; mixed integer formulations; branch and cut; rolling horizon heuristic. 1 Introduction Maritime transportation is a major mode of transportation. Around 80% of global trade by volume and over 70% of global trade by value are carried by sea and are handled by ports worldwide [37]. Port activities involve several interrelated decisions, as berth assignment of vessels, quay crane assignment and scheduling, cargo placement, etc. During peak periods of activity delays may occur leading to large waiting costs. The request for operational efficiency at ports has motivated the increase of research during the last decades on such optimization problems, which is even more ∗ Published in European Journal of Operations Research, 264 (1), 138– 148 2018 1