We study the Panama Canal as a bidirectional hybrid flow shop with parallel mechanical stages, shared navigational stages, and vessels as jobs. The canal uses freshwater to enable interoceanic transit. Thus, resource conservation is a critical sustainability challenge for the PC. Although water conservation modes exist, the system has an inherent trade-off between vessel transit time and freshwater savings. An integrated model can better balance these conflicting objectives: makespan and freshwater usage, achieving meaningful resource savings through seasonal scheduling policies and job coordination. This work makes three main contributions. First, it shows that the saving mode requirements, where groups of jobs (macrojob) must be tightly coordinated, enable a partition-based representation of the job set. Second, our research demonstrates that for a macrojob, the processing cost admits an additive lag-and-propagation structure across hybrid stages, allowing sequences of hybrid machines to be modeled as a single equivalent supermachine. Third, this paper presents a Dantzig-Wolfe decomposition method where each column corresponds to a feasible macrojob. The master problem selects among these macrojobs, while the pricing procedure generates candidate macrojobs by recursively exploiting the lag-and-propagation structure. As a result, this avoids embedding a costly hybrid-scheduling subproblem in column generation. Finally, a bidirectional flow shop macro scheduler evaluates both objectives subject to all operational constraints. This separation of representation and evaluation provides a scalable framework for a resource-constrained scheduling problem, that would be hard to solve using a single formulation.

Keywords

Hybrid Flow Shop scheduling, Sustainable scheduling, Dantzig-Wolfe decomposition, Macrojob modeling