Retsef Levi

Approximation Algorithms for Stochastic Inventory Control Models

We consider two classical stochastic inventory control models, the periodic-review stochastic inventory control problem and the stochastic lot-sizing problem. The goal is to coordinate a sequence of orders of a single commodity, aiming to supply stochastic demands over a discrete, finite horizon with minimum expected overall ordering, holding, and backlogging costs. In this paper, we address the important problem of finding computationally efficient and provably good inventory control policies for these models in the presence of correlated, nonstationary (time-dependent), and evolving stochastic demands. This problem arises in many domains and has many practical applications in supply chain management. Our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel cost-balancing techniques. Specifically, in each period, we balance the expected cost of overordering (i.e., costs incurred by excess inventory) against the expected cost of underordering (i.e., costs incurred by not satisfying demand on time). This leads to what we believe to be the first computationally efficient policies with constant worst-case performance guarantees for a general class of important stochastic inventory control models. That is, there exists a constant C such that, for any instance of the problem, the expected cost of the policy is at most C times the expected cost of an optimal policy. In particular, we provide a worst-case guarantee of two for the periodic-review stochastic inventory control problem and a worst-case guarantee of three for the stochastic lot-sizing problem. Our results are valid for all of the currently known approaches in the literature to model correlation and nonstationarity of demands over time.

Provably Near-Optimal Sampling-Based Policies for Stochastic Inventory Control Models

In this paper, we consider two fundamental inventory models, the single-period newsvendor problem and its multiperiod extension, but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions. Under the assumption that the demand distributions are given explicitly, these models are well studied and relatively straightforward to solve. However, in most real-life scenarios, the true demand distributions are not available, or they are too complex to work with. Thus, a sampling-driven algorithmic framework is very attractive, both in practice and in theory. We shall describe how to compute sampling-based policies, that is, policies that are computed based only on observed samples of the demands without any access to, or assumptions on, the true demand distributions. Moreover, we establish bounds on the number of samples required to guarantee that, with high probability, the expected cost of the sampling-based policies is arbitrarily close (i.e., with arbitrarily small relative error) compared to the expected cost of the optimal policies, which have full access to the demand distributions. The bounds that we develop are general, easy to compute, and do not depend at all on the specific demand distributions.

Approximation Algorithms for Capacitated Stochastic Inventory Control Models

We develop the first algorithmic approach to compute provably good ordering policies for a multiperiod, capacitated, stochastic inventory system facing stochastic nonstationary and correlated demands that evolve over time. Our approach is computationally efficient and guaranteed to produce a policy with total expected cost no more than twice that of an optimal policy. As part of our computational approach, we propose a novel scheme to account for backlogging costs in a capacitated, multiperiod environment. Our cost-accounting scheme, called the forced marginal backlogging cost-accounting scheme , is significantly different from the period-by-period accounting approach to backlogging costs used in dynamic programming; it captures the long-term impact of a decision on system performance in the presence of capacity constraints. In the likely event that the per-unit order costs are large compared to the holding and backlogging costs, a transformation of cost parameters yields a significantly improved guarantee. We also introduce new semimyopic policies based on our new cost-accounting scheme to derive bounds on the optimal base-stock levels. We show that these bounds can be used to effectively improve any policy. Finally, empirical evidence is presented that indicates that the typical performance of this approach is significantly stronger than these worst-case guarantees.

A Constant Approximation Algorithm for the One-Warehouse Multiretailer Problem

Deterministic inventory theory provides streamlined optimization models that attempt to capture tradeoffs in managing the flow of goods through a supply chain. We will consider a well-studied inventory model, called the one-warehouse multi-retailer problem (OWMR). and give the first approximation algorithm with constant performance guarantee; more specifically, we give a 2.398-approximation algorithm. Our results are based on an LP-rounding approach, and hence not only provide good algorithmic results, but show strong integrality gaps for these linear programs. Furthermore, we extend this result to obtain a constant performance guarantee for a capacitated variant of this model.

Adaptive Data-Driven Inventory Control with Censored Demand Based on Kaplan-Meier Estimator

Using the well-known product-limit form of the Kaplan-Meier estimator from statistics, we propose a new class of nonparametric adaptive data-driven policies for stochastic inventory control problems. We focus on the distribution-free newsvendor model with censored demands. The assumption is that the demand distribution is not known and there are only sales data available. We study the theoretical performance of the new policies and show that for discrete demand distributions they converge almost surely to the set of optimal solutions. Computational experiments suggest that the new policies converge for general demand distributions, not necessarily discrete, and demonstrate that they are significantly more robust than previously known policies. As a by-product of the theoretical analysis, we obtain new results on the asymptotic consistency of the Kaplan-Meier estimator for discrete random variables that extend existing work in statistics. To the best of our knowledge, this is the first application of the Kaplan-Meier estimator within an adaptive optimization algorithm, in particular, the first application to stochastic inventory control models. We believe that this work will lead to additional applications in other domains.

Primal-Dual Algorithms for Deterministic Inventory Problems

We consider several classical models in deterministic inventory theory: the single-item lot-sizing problem, the joint replenishment problem, and the multistage assembly problem. These inventory models have been studied extensively, and play a fundamental role in broader planning issues, such as the management of supply chains. For each of these problems, we wish to balance the cost of maintaining surplus inventory for future demand against the cost of replenishing inventory more frequently. For example, in the joint replenishment problem, demand for several commodities is specified over a discrete finite planning horizon, the cost of maintaining inventory is linear in the number of units held, but the cost incurred for ordering a commodity is independent of the size of the order; furthermore, there is an additional fixed cost incurred each time a nonempty subset of commodities is ordered. The goal is to find a policy that satisfies all demands on time and minimizes the overall holding and ordering cost. We shall give a novel primal-dual framework for designing algorithms for these models that significantly improve known results in several ways: the performance guarantees for the quality of the solutions improve on or match previously known results; the performance guarantees hold under much more general assumptions about the structure of the costs, and the algorithms and their analysis are significantly simpler than previous known results. Finally, our primal-dual framework departs from the structure of previously studied primal-dual approximation algorithms in significant ways, and we believe that our approach may find applications in other settings. More specifically, we provide 2-approximation algorithms for the joint replenishment problem and for the assembly problem, and solve the single-item lot-sizing problem to optimality. The results for the joint replenishment and the lot-sizing problems also hold for their generalizations with back orders allowed. As a byproduct of our work, we prove known and new upper bounds on the integrality gap of some linear-programming (LP) relaxations of the abovementioned problems.

A 2-Approximation Algorithm for Stochastic Inventory Control Models with Lost Sales

In this paper, we describe the first computationally efficient policies for stochastic inventory models with lost sales and replenishment lead times that admit worst-case performance guarantees. In particular, we introduce dual-balancing policies for lost-sales models that are conceptually similar to dual-balancing policies recently introduced for a broad class of inventory models in which demand is backlogged rather than lost. That is, in each period, we balance two opposing costs: the expected marginal holding costs against the expected marginal lost-sales cost. Specifically, we show that the dual-balancing policies for the lost-sales models provide a worst-case performance guarantee of two under relatively general demand structures. In particular, the guarantee holds for independent (not necessarily identically distributed) demands and for models with correlated demands such as the AR(1) model and the multiplicative autoregressive demand model. The policies and the worst-case guarantee extend to models with capacity constraints on the size of the order and stochastic lead times. Our analysis has several novel elements beyond the balancing ideas for backorder models.

A model for understanding the impacts of demand and capacity on waiting time to enter a congested recovery room.

Background:When a recovery room is fully occupied, patients frequently wait in the operating room after emerging from anesthesia. The frequency and duration of such delays depend on operating room case volume, average recovery time, and recovery room capacity. Methods:The authors developed a simple yet nontrivial queueing model to predict the dynamics among the operating and recovery rooms as a function of the number of recovery beds, surgery case volume, recovery time, and other parameters. They hypothesized that the model could predict the observed distribution of patients in recovery and on waitlists, and they used statistical goodness-of-fit methods to test this hypothesis against data from their hospital. Numerical simulations and a survey were used to better understand the applicability of the model assumptions in other hospitals. Results:Statistical tests cannot reject the prediction, and the model assumptions and predictions are in agreement with data. The survey and simulations suggest that the model is likely to be applicable at other hospitals. Small changes in capacity, such as addition of three beds (roughly 10% of capacity) are predicted to reduce waiting for recovery beds by approximately 60%. Conversely, even modest caseload increases could dramatically increase waiting. Conclusions:A key managerial insight is that there is a sensitive relationship among caseload and number of recovery beds and the magnitude of recovery congestion. This is typical in highly utilized systems. The queueing approach is useful because it enables the investigation of future scenarios for which historical data are not directly applicable.

Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs

In the rectangle stabbing problem, we are given a set of axis parallel rectangles and a set of horizontal and vertical lines, and our goal is to find a minimum size subset of lines that intersect all the rectangles. In this article, we study the capacitated version of this problem in which the input includes an integral capacity for each line. The capacity of a line bounds the number of rectangles that the line can cover. We consider two versions of this problem. In the first, one is allowed to use only a single copy of each line (hard capacities), and in the second, one is allowed to use multiple copies of every line, but the multiplicities are counted in the size (or weight) of the solution (soft capacities). We present an exact polynomial-time algorithm for the weighted one dimensional case with hard capacities that can be extended to the one dimensional weighted case with soft capacities. This algorithm is also extended to solve a certain capacitated multi-item lot-sizing inventory problem with joint set-up costs. For the case of d-dimensional rectangle stabbing with soft capacities, we present a 3d-approximation algorithm for the unweighted case. For d-dimensional rectangle stabbing problem with hard capacities, we present a bi-criteria algorithm that computes 4d-approximate solutions that use at most two copies of every line. Finally, we present hardness results for rectangle stabbing when the dimension is part of the input and for a two-dimensional weighted version with hard capacities.

The Data-Driven Newsvendor Problem: New Bounds and Insights

Consider the newsvendor model, but under the assumption that the underlying demand distribution is not known as part of the input. Instead, the only information available is a random, independent sample drawn from the demand distribution. This paper analyzes the sample average approximation SAA approach for the data-driven newsvendor problem. We obtain a new analytical bound on the probability that the relative regret of the SAA solution exceeds a threshold. This bound is significantly tighter than existing bounds, and it matches the empirical accuracy of the SAA solution observed in extensive computational experiments. This bound reveals that the demand distributions weighted mean spread affects the accuracy of the SAA heuristic.