Non-approximability of the single crane container transhipment problem

The makespan of operations at container terminals is crucial for the lead time of cargo and consequently the reduction of transportation costs. Therefore, an efficient transhipment and short storage of containers are demanded. Our paper refers to the consolidation process of trains in a container transhipment terminal as well as to the intermediate storage of containers in seaports in order to accelerate the loading and unloading of the vessels. It can also be encountered in automated storage/retrieval systems. Each of these (container) storage and retrieval moves corresponds to a crane operation, carrying a load from its pickup to its drop-off position. The problem is to find a permutation of the loaded crane moves that minimises the total empty crane travel time, which is the sum of times the crane needs to get from the last drop-off point of a load to the next pickup point of a load. We address the problem as an extension of an asymmetric travelling salesman problem (ATSP), assuming that n ordered pairs of points in the two-dimensional Euclidean space need to be traversed. Each point corresponds to a crane operation carrying a load from its pickup to its drop-off position. Despite that the problem seems to be easier than the ATSP, because a simple constant factor approximation exists, which was for a long time an open question for the ATSP, we are the first to prove that there is no polynomial-time approximation algorithm with an approximation guarantee less than $1+{0.23}/{n} $1+0.23/n unless $\textsc{P}=\textsc{NP} $P=NP.