An extended formulation of moldable task scheduling problem and its application to quay crane assignments

In this paper, we study an extended formulation of moldable task scheduling problem (MTSP) motivated by the assignments of quay cranes to vessels. In container terminals, handling time of a vessel depends on the number of quay cranes assigned to that vessel. This characteristic allows us to model quay crane assignment problem (QCAP) as a variant of MTSP. By considering the modeling requirements of various properties of QCAP, we develop an extended formulation of MTSP with specific task to machine assignments. Even though this formulation brings modeling flexibility, it can only be solved for small instances because of its size. For this reason, we provide a generic solution algorithm based on a logic based Benders decomposition by utilizing the extended formulation. There are various characteristics of QCAP observed in different terminals. Accordingly, we implement the proposed decomposition algorithm for contiguous assignments of QCs, uniform QCs as well as the availability of QCs. Computational experiments show that the proposed algorithm is able to solve instances of considerable sizes to optimality and provides a modeling flexibility that allows implementation to different terminal settings.