Routing two stacking cranes with predetermined container sequences

The scheduling of gantry cranes with respect to mutual interference has received considerable attention in recent years. We consider a subproblem which arises when each crane has a sequence of tasks to be assigned. The problem is concerned with resolving the interference between two cranes by determining which crane avoids the other in order to let it complete its next task first. We provide a fairly general problem framework accounting for different crane systems and various side constraints. We assume a cost function for each task that determines the cost of completing the task at a specific point in time. We then distinguish between the objectives to minimize both the total cost and the maximum cost among tasks. A general dynamic programming framework is provided which allows us to solve all problem versions in pseudo-polynomial time. Furthermore, we show that while the general problem aiming for minimum total cost is binary NP-hard, the general problem aiming for minimum maximum cost can be solved in polynomial time. Finally, we address two important special cases of the former, and we show that they can be solved in polynomial time as well.