Ordering policies for periodic-review inventory systems with quantity-dependent fixed costs

We consider a stochastic periodic-review inventory control system in which the fixed cost depends on the order quantity. In particular, we investigate the optimal ordering policies under three fixed cost structures. The first structure is motivated by transportation and production contracts and considers two fixed costs: if the order size is within a specified limit C, then the fixed cost is K₁; otherwise, it is K₂, where K₁ ≤ K ₂. The second structure contains multiple fixed costs in which the same incremental fixed cost K is incurred for any additional order quantity up to a given identical batch capacity C. In the third structure, in addition to the K incurred as in the previous case, a common fixed cost is charged for any nonzero order size. An example of the former case arises when an order is shipped with a homogeneous fleet of trucks with per-truck fixed costs. A situation in which a fixed administrative cost plus a quantity-dependent trucking cost is incurred for each shipment exemplifies the latter case. For the first cost structure, we separate the analysis according to the conditions (1) K₁ ≤ K₂ ≤ 2K₁ and (2) K₁ ≤ K₂. Under condition (1), we introduce a new concept called C-(K ₁,K₂)-convexity, which enables us to almost completely characterize the optimal ordering policy. Under the general condition (2), we utilize a modified notion to provide a partial characterization of the optimal policy and propose a heuristic policy that performs well under a wide variety of model parameters. For the second cost structure, we show that it is optimal to order an integer multiple of the batch capacity to raise the inventory level to a specified range or band of length C, and then to order an additional full or partial batch size depending on the cost function, with no ordering required above the band. We also characterize a similar optimal policy for the third cost structure. Using different techniques, our study extends or redevelops several existing results in the literature.