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 R 2 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.