Optimal layout of transshipment facility locations on an infinite homogeneous plane

This paper studies optimal spatial layout of transshipment facilities and the corresponding service regions on an infinite homogeneous plane R2 that minimize the total cost for facility set-up, outbound delivery and inbound replenishment transportation. The problem has strong implications in the context of freight logistics and transit system design. This paper first focuses on a Euclidean plane and shows that a tight upper bound can be achieved by a type of elongated cyclic hexagons, while a cost lower bound based on relaxation and idealization is also obtained. The gap between the analytical upper and lower bounds is within 0.3%. This paper then shows that a similar elongated non-cyclic hexagon shape, with proper orientation, is actually optimal for service regions on a rectilinear metric plane. Numerical experiments are conducted to verify the analytical findings and to draw further insights.