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We describe a specialized dynamic programming algorithm for factory crane schedul- ing. An innovative data structure controls the memory requirements of the state space and enables solution of problems of realistic size. The algorithm finds optimal space- time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must not interfere with each other, although one may yield to the other. The state space representa- tion permits a wide variety of constraints and objective functions. It is stored in a compressed data structure that uses a cartesian product of intervals of states and an array of two-dimensional circular queues. We also show that only certain types of trajectories need be considered. The algorithm found optimal solutions in less than a minute for medium-sized instances of the problem (160 tasks, spanning four hours). It can also be used to create benchmarks for tuning heuristic algorithms that solve larger instances.

We study the problem of finding optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is a assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must be operated so as not to interfere with each other, although one crane may need to yield to another. The objective is generally to follow a production schedule as closely as possible. We show that only certain types of trajectories need be considered to obtain an optimal solution. This simplifies the operation of the cranes and enhances safety, because the cranes move according to predictable patterns. We present a specialized dynamic programming algorithm that solves the problem. To control the state space size we use an innovative state space representation based on a cartesian product of intervals of states and an array of two-dimensional circular queues. The algorithm solves medium-sized instances of the problem to optimality and can be used to create benchmarks for tuning heuristic algorithms that solve larger instances.