Steven Nahmias

Perishable Inventory Theory: A Review

This paper reviews the relevant literature on the problem of determining suitable ordering policies for both fixed life perishable inventory, and inventory subject to continuous exponential decay. We consider both deterministic and stochastic demand for single and multiple products. Both optimal and suboptimal order policies are discussed. In addition, a brief review of the application of these models to blood bank management is included. The review concludes with a discussion of some of the interesting open research questions in the area.

Modeling Supply Chain Contracts: A Review

In this review, we summarize model-based research on contracts in the supply chain setting and provide a taxonomy for work in this area. During our discussions it became clear that the field has developed in many directions at once. Furthermore, as we surveyed the Uterature, it was not obvious what constitutes a contract in this context. While the nomenclature “supply chain management” is relatively new, many of the problems that are addressed are not. In particular, mathematical models for optimizing inventory control have a long history as a significant part of the mainstream of operations research and operations management. Inventory modeling, per se, dates to the early part of the century and the ideas of a Westinghouse engineer named Ford Harris (1915). A natural issue to address first is what is meant by supply chain management (SCM) research and how it relates to the vast body of work constituting classical inventory theory.

Optimal Ordering Policies for Perishable Inventory-II

This paper deals with the problem of computing optimal ordering policies for a single product with a lifetime of exactly m periods. Costs are charged against ordering, holding, shortages, and out-dating. To take explicit account of the perishability, we substitute a cost to be incurred at the time of outdating. The functional equation approach of dynamic programming is used to extend this model to the multiperiod case, and the structure of optimal ordering policies is analyzed.

Simple Approximations for a Variety of Dynamic Leadtime Lost-Sales Inventory Models

This paper treats three realistic versions of the classical leadtime lost-sales inventory problem: inclusion of a set-up for ordering, partial backordering, and the random leadtime lost-sales inventory problem, as well as all possible combinations of these three models. The goal of the study is to construct simple myopic approximations for these various models. We report extensive computations that compare the approximations derived with the optimal policies for leadtimes of one and two periods. In addition, we develop a simulator that determines the best S or (s, S) policies by a Fibonacci search of the appropriate response surface and compares the performance of the approximations to these policies for maximum leadtimes of five, ten, and twenty periods for a variety of configurations of the cost structure, demand distribution, and leadtime distribution.

Optimal Policy for a Two-Stage Assembly System under Random Demand

This paper considers an inventory system in which an end product is assembled from two components, each of which is ordered from an external supplier. Only the end product has final demand, which is assumed to be random. Using the functional equation approach of dynamic programming, we characterize the forms of both the optimal order policy for the components and the optimal assembly policy of the end product for a multiperiod problem of arbitrary, but finite length. We believe that this work represents the only exact analysis of an MRP-type assembly system when demand is stochastic.

Demand estimation in lost sales inventory systems

This article considers the problem of estimating parameters of the demand distribution in lost sales inventory systems. In periods when lost sales occur demand is not observed; one knows only that demand is larger than sales. We assume that demands form a sequence of IID normal random variables, which could be a residual demand process after filtering out seasonality and promotional nonstationarities. We examine three estimators for the mean and standard deviation: maximum likelihood estimator, BLUE (best linear unbiased estimator), and a new estimator derived here. Extensive simulations are reported to compare the performance of the estimators for small and large samples and a variety of parameter settings. In addition, I show how all three estimators can be incorporated into sequential updating routines.

Perishable Inventory Systems

Chapter 1. Preliminaries.- Chapter 2. The Basic Multiperiod Dynamic Model.- Chapter 3. Extensions of the Basic Multiperiod Dynamic Model.- Chapter 4. Continuous Review Perishable Inventory Models.- Chapter 5. Approximate Order Policies.- Chapter 6. Inventory Depletion Management.- Chapter 7. Deterministic Models.- Chapter 8. Decaying Inventories.- Chapter 9. Queues with Impatient Customers.- Chapter 10. Blood Bank Inventory Control.- Afterward.

Adjustment Strategies for a Fixed Delivery Contract

We consider a long term contractual agreement between buyer and seller in whichQ units are delivered to the buyer at regular time intervals. It must be true that the delivery quantity,Q, is less than the mean demand per period. In order to manage the inventory, the buyer has the option of adjusting the delivery quantity upwards just prior to a delivery, but must pay a premium to do so. Demand is assumed random, and we model the system in a continuous review setting. We show that the equations one must solve to find optimal adjustment strategies are intractable. A diffusion approximation is developed which when coupled with the solution to an even simpler deterministic version of the problem yields very simple but effective approximations. Extensive computations are included to compare the performance of the optimal and approximate policies. We also empirically derive a formula for computingQ whose accuracy is established computationally. We prove that the fixed delivery contract results in lower variance of orders to the seller. We also include a computational study to find the unit cost discount that equalizes the expected costs for the fixed delivery contract and the base stock contract for a large parameter set.

The Fixed-Charge Perishable Inventory Problem

This paper deals with the problem of determing both optimal and approximate ordering policies for a fixed-life perishable commodity when there is a fixed charge or set-up cost for placing an order. The structure of the optimal policy can be inferred from the analysis of a one-period model under the assumption that the outdating is paid for when the order arrives rather than when the outdating actually occurs. An approximate model is constructed for which s, S policies are optimal by approximating the one expected single-period cost and the transfer function. Two different bounds on the expected out-dating per period are used to obtain two different s, S approximations. Computations are performed for a discrete version of the problem with three different demand distributions and 24 configurations of the cost structure to compare the effectiveness of both approximations with the optimal policy.

Higher-Order Approximations for the Perishable-Inventory Problem

Computation of an optimal policy for ordering a perishable commodity with a fixed lifetime of m periods requires the solution of a dynamic program whose state variable has dimension m − 1. Unless m is small, the computations quickly become unreasonable. By bounding the expected outdating function and using an approximate transfer function, we obtain an approximation of the original problem that can be computed as if the lifetime were r < m periods. When r = 2, it is demonstrated under reasonable assumptions that the appropriate function to be minimized each period is quasi-convex. We include computations that compare this approximation to both the optimal policy and a critical number approximation.