Projected Inventory Level Policies for Lost Sales Inventory Systems: Asymptotic Optimality in Two Regimes

We consider the canonical periodic review lost sales inventory system with positive lead-times and stochastic i.i.d. demand under the average cost criterion. We introduce a new policy that places orders such that the expected inventory level at the time of arrival of an order is at a fixed level and call it the Projected Inventory Level (PIL) policy. We prove that this policy has a cost-rate superior to the equivalent system where excess demand is back-ordered instead of lost and is therefore asymptotically optimal as the cost of losing a sale approaches infinity under mild distributional assumptions. We further show that this policy dominates the constant order policy for any finite lead-time and is therefore asymptotically optimal as the lead-time approaches infinity for the case of exponentially distributed demand per period. Numerical results show this policy also performs superior relative to other policies

Deep controlled learning of dynamic policies with an application to lost-sales inventory control

Recent literature established that neural networks can represent good policies across a range of stochastic dynamic models in supply chain and logistics. We propose a new algorithm that incorporates variance reduction techniques, to overcome limitations of algorithms typically employed in literature to learn such neural network policies. For the classical lost sales inventory model, the algorithm learns neural network policies that are vastly superior to those learned using model-free algorithms, while outperforming the best heuristic benchmarks by an order of magnitude. The algorithm is an interesting candidate to apply to other stochastic dynamic problems in supply chain and logistics, because the ideas in its development are generic

Base-stock policies for lost-sales models: Aggregation and asymptotics

This paper considers the optimization of the base-stock level for the classical periodic review lost-sales inventory system. The optimal policy for this system is not fully understood and computationally expensive to obtain. Base-stock policies for this system are asymptotically optimal as lost-sales costs approach infinity, easy to implement and prevalent in practice. Unfortunately, the state space needed to evaluate a base-stock policy exactly grows exponentially in both the lead time and the base-stock level. We show that the dynamics of this system can be aggregated into a one-dimensional state space description that grows linearly in the base-stock level only by taking a non-traditional view of the dynamics. We provide asymptotics for the transition probabilities within this single dimensional state space and show that these asymptotics have good convergence properties that are independent of the lead time under mild conditions on the demand distribution. Furthermore, we show that these asymptotics satisfy a certain ow conservation property. These results lead to a new and computationally efficient heuristic to set base-stock levels in lost-sales systems. In a numerical study we demonstrate that this approach performs better than existing heuristics with an average gap with the best base-stock policy of 0.01% across a large test-bed

Scheduling a Make-To-Stock Queue: Index Policies and Hedging Points

A single machine produces several different classes of items in a make-to-stock mode. We consider the problem of scheduling the machine to regulate finished goods inventory, minimizing holding and backorder or holding and lost sales costs. Demands are Poisson, service times are exponentially distributed, and there are no delays or costs associated with switching products. A scheduling policy dictates whether the machine is idle or busy, and specifies the job class to serve in the latter case. Since the optimal solution can only be numerically computed for problems with several products, our goal is to develop effective policies that are computationally tractable for a large number of products. We develop index policies to decide which class to serve, including Whittle's "restless bandit" index, which possesses a certain asymptotic optimality. Several idleness policies, which are characterized by hedging points, are derived, and the best policy is obtained from a heavy traffic diffusion approximation. Nine sample problems are considered in a numerical study, and the average suboptimality of the best policy is less than 3%