Optimal Layout of Transshipment Facilities on An Infinite Homogeneous Plane

TL;DR

This paper investigates the optimal spatial layout of transshipment facilities on an infinite plane, proving known conjectures, deriving bounds for costs, and identifying hexagon-shaped service regions as optimal under various transportation cost conditions.

Contribution

It provides a new proof for Gersho's conjecture, extends the shape analysis to rectilinear metrics, and offers tight bounds on transportation costs for facility layout optimization.

Findings

Optimal service regions are regular hexagons when inbound costs are ignored.

Elongated cyclic hexagons approximate optimal regions when inbound costs are significant.

Analytical bounds on total transportation costs are within 0.3% of each other.

Abstract

This paper studies optimal spatial layout of transshipment facilities and the corresponding service regions on an infinite homogeneous plane $\mathbb{R}^2$ 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 presents a new proof for the known Gersho's conjecture, which states that the optimal shape of each service region should be a regular hexagon if the inbound transportation cost is ignored. When inbound transportation cost becomes non-negligible, however, we show 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 is actually optimal for service regions on a rectilinear metric plane. Numerical experiments and sensitivity analyses are conducted to verify the analytical findings and to draw managerial insights.