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Register a new userProjected Inventory Level Policies for Lost Sales Inventory Systems: Asymptotic Optimality in Two Regimes
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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.
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Recent literature established that neural networks can represent good policies across a range of stochastic dynamic models in supply chain and logistics. We incorporate variance reduction techniques in a newly proposed algorithm, to overcome limitations of the model-free algorithms typically employed to learn such neural network policies. For the classical lost sales inventory model, the algorithm learns neural network policies that are 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.
We study periodic review stochastic inventory control in the data-driven setting, in which the retailer makes ordering decisions based only on historical demand observations without any knowledge of the probability distribution of the demand. Since an $(s, S)$-policy is optimal when the demand distribution is known, we investigate the statistical properties of the data-driven $(s, S)$-policy obtained by recursively computing the empirical cost-to-go functions. This policy is inherently challenging to analyze because the recursion induces propagation of the estimation error backwards in time. In this work, we establish the asymptotic properties of this data-driven policy by fully accounting for the error propagation. First, we rigorously show the consistency of the estimated parameters by filling in some gaps (due to unaccounted error propagation) in the existing studies. On the other hand, empirical process theory cannot be directly applied to show asymptotic normality. To explain, the empirical cost-to-go functions for the estimated parameters are not i.i.d. sums, again due to the error propagation. Our main methodological innovation comes from an asymptotic representation for multi-sample $U$-processes in terms of i.i.d. sums. This representation enables us to apply empirical process theory to derive the influence functions of the estimated parameters and establish joint asymptotic normality. Based on these results, we also propose an entirely data-driven estimator of the optimal expected cost and we derive its asymptotic distribution. We demonstrate some useful applications of our asymptotic results, including sample size determination, as well as interval estimation and hypothesis testing on vital parameters of the inventory problem. The results from our numerical simulations conform to our theoretical analysis.
Many retailers today employ inventory management systems based on Re-Order Point Policies, most of which rely on the assumption that all decreases in product inventory levels result from product sales. Unfortunately, it usually happens that small but random quantities of the product get lost, stolen or broken without record as time passes, e.g., as a consequence of shoplifting. This is usual for retailers handling large varieties of inexpensive products, e.g., grocery stores. In turn, over time these discrepancies lead to stock freezing problems, i.e., situations where the system believes the stock is above the re-order point but the actual stock is at zero, and so no replenishments or sales occur. Motivated by these issues, we model the interaction between sales, losses, replenishments and inventory levels as a Dynamic Bayesian Network (DBN), where the inventory levels are unobserved (i.e., hidden) variables we wish to estimate. We present an Expectation-Maximization (EM) algorithm to estimate the parameters of the sale and loss distributions, which relies on solving a one-dimensional dynamic program for the E-step and on solving two separate one-dimensional nonlinear programs for the M-step.
The availability of databases electronically encoding curated regulatory networks and of high-throughput technologies and methods to discover regulatory interactions provides an invaluable source of data to understand the principles underpinning the organization and evolution of these networks responsible for cellular regulation. Nevertheless, data on these sources never goes beyond the regulon level despite the fact that regulatory networks are complex hierarchical-modular structures still challenging our understanding. This brings the necessity for an inventory of systems across a large range of organisms, a key step to rendering feasible comparative systems biology approaches. In this work, we take the first step towards a global understanding of the regulatory networks organization by making a cartography of the functional architectures of diverse bacteria. Abasy (Across-bacteria systems) Atlas provides a comprehensive inventory of annotated functional systems, global network properties, and systems-level elements (global regulators, modular genes shaping functional systems, basal machinery genes, and intermodular genes) predicted by the natural decomposition approach for reconstructed and meta-curated regulatory networks across a large range of bacteria, including pathogenically and biotechnologically relevant organisms. The meta-curation of regulatory datasets provides the most complete and reliable set of regulatory interactions currently available. Abasy Atlas contains systems and system-level elements for 50 regulatory networks comprising 78,649 regulatory interactions covering 42 bacteria in nine taxa, containing 3,708 regulons and 1,776 systems. All this brings together a large corpus of data that will surely inspire studies to generate hypothesis regarding the principles governing the evolution and organization of systems and the functional architectures controlling them.
In this paper, we draw attention to a problem that is often overlooked or ignored by companies practicing hypothesis testing (A/B testing) in online environments. We show that conducting experiments on limited inventory that is shared between variants in the experiment can lead to high false positive rates since the core assumption of independence between the groups is violated. We provide a detailed analysis of the problem in a simplified setting whose parameters are informed by realistic scenarios. The setting we consider is a $2$-dimensional random walk in a semi-infinite strip. It is rich enough to take a finite inventory into account, but is at the same time simple enough to allow for a closed form of the false-positive probability. We prove that high false-positive rates can occur, and develop tools that are suitable to help design adequate tests in follow-up work. Our results also show that high false-negative rates may occur. The proofs rely on a functional limit theorem for the $2$-dimensional random walk in a semi-infinite strip.
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