Discrete-Time MDP Modeling for Multi-Item Capacitated Lot Sizing with Stochastic Demand Timing

This paper studies a finite-horizon multi-item capacitated lot-sizing problem in which demand quantities are deterministic, while demand-arrival periods are stochastic. Each demand occurs once within a known time window and must be satisfied no later than its deadline. The proposed model makes production and allocation decisions at the demand level, allowing it to represent capacity competition, demand-specific backlog, and allocation-dependent inventory dynamics. The stochastic problem is formulated as a discrete-time Markov decision process (DTMDP), including the state space, feasible actions, transition kernel, and one-period cost function. To isolate the computational effect of stochastic timing, each stochastic instance is first compared with a deterministic counterpart in which each arrival distribution is replaced by its most likely arrival period. This comparison shows that stochastic timing substantially increases the number of states, the number of transitions, solution time, and memory pressure. A genetic algorithm (GA) is then proposed for the stochastic-timing problem. The GA searches over feasible state-feedback policies and evaluates each policy exactly under the DTMDP transition model. Computational experiments on 330 benchmark instances show that the GA remains close to the exact stochastic solution whenever the latter is available, with an average optimality gap of about $3.44\%$. On the difficult benchmark instances, comprising 90 test cases, the GA remains below the $5\%$ optimality-gap threshold and achieves an average optimization speedup of $6.89 \pm 1.41$ at the $95\%$ confidence level. For instances that cannot be solved exactly on the available hardware, an empirical Bellman-time regression is used to estimate the missing exact resolution time and extrapolate the expected GA speedup.

A Policy Decomposition Framework for Dynamic Order Fulfillment Operations

Modern supply chains span diverse operational environments, ranging from e-commerce distribution networks to customized production-to-order manufacturing lines. Across these settings, operational efficiency depends on coordinating two highly interdependent stages: order preparation and downstream delivery. Although these stages are traditionally managed in isolation, real-world fulfillment systems must satisfy stringent delivery expectations under dynamic stochastic order arrivals. To bridge this gap, we introduce the Dynamic Order Fulfillment Problem (DOFP), a new problem class unifying logistical challenges previously studied separately. We model DOFP as a Markov decision process whose state and decision spaces are partitioned into preparation and delivery sub-spaces, linked by synchronization constraints. While recent approaches attempt to optimize both fulfillment stages simultaneously over myopic rolling horizons, our framework isolates and optimizes the downstream delivery policy, treating preparation strictly as a state-level constraint filter. To solve this, we develop the Decomposition-Driven Framework with Value Function Approximation (DDF-VFA), which utilizes a novel policy-level decomposition. This design partitions the search into a delivery-stage master problem and a preparation-stage compatibility subproblem, iteratively refined via feedback loops. DDF-VFA executes this strategy by combining a large-neighborhood search over partial delivery decisions with a neural-network value function approximation for the cost-to-go. Numerical illustrations on two example variants using real-world datasets show that DDF-VFA consistently outperforms benchmarks that optimize the two stages independently or jointly without decomposition. Finally, the framework naturally scales to accommodate additional real-world complexities such as batched or multi-stage preparation.