Efficient lot-sizing under a differential transportation cost structure for serially distributed warehouses

A major cost element in the logistics of distributed warehousing is transportation cost. In most practical systems, the transportation costs are volume-dependent. The unit transportation costs are usually determined differentially among intervals of shipment volumes. While the unit cost is constant over an interval, it follows a stepwise declining pattern from an interval to the next higher interval of shipment volumes. This structure is analogous to that of quantity-discounted inventory systems. We consider a single-product, serial warehousing system operating under the differential transportation costs and the traditional holding and ordering costs. External demand occurs at the final stage at a constant continuous rate. Demand must be met on time over an infinite horizon under continuous review. The objective is to determine the ordering lot size for each warehouse such that the long-run average cost is minimized. We model the above problem and establish its structural properties. We first consider two-level differential transportation cost structures and develop efficient algorithms to obtain a best integer ratio and optimal powers-of-two policies. Next, we consider multilevel cost structures and develop a hypercube characterization of the solution space. Efficient algorithms to yield a best integer ratio and powers-of-two policies using the hypercube model and a relaxation approach using the two-level model are developed. Extensive computational studies with the algorithms reveal that the proposed modeling and algorithmic approach is both viable and efficient in solving practical distribution problems. Conclusions and direction for future research are presented.