Distributionally Robust Dynamic Inventory Control: Automatic Recovery under Martingale Demand

Abstract

Problem definition: We study dynamic inventory control for battery-swapping stations that combine central replenishment with automatic local recovery. Central replenishment improves immediate battery availability but reduces the depleted-battery pool available for subsequent recovery, creating an intertemporal trade-off. Managing this trade-off is particularly challenging when historical demand data are limited and future demand is uncertain and temporally dependent. Methodology/results: We develop a time-consistent distributionally robust dynamic program under martingale demand ambiguity, using the latest observed demand as the next conditional mean without requiring a fully specified demand distribution. We establish an optimal order-up-to policy for every finite horizon and identify two critical recovery rates governing replenishment decisions. Under a regular penalty condition, we derive explicit finite-horizon policies and worst-case demand distributions, showing that sufficiently strong automatic recovery makes central replenishment unnecessary in nonterminal periods. Extensions to submartingale and supermartingale demand reveal contrasting behaviors, including eventual full-capacity replenishment under upward ambiguity and recovery-induced positive replenishment under downward ambiguity. Managerial implications: Numerical experiments demonstrate that the robust policy is particularly competitive when demand is highly dispersed and historical information is limited, whereas its advantage diminishes when demand becomes more predictable or the conditional-mean assumption becomes less reliable. Our findings also reveal that greater reliance on local recovery may compromise service availability, highlighting the importance of jointly considering demand information, recovery capability, and service requirements when making replenishment decisions.