Awi Federgruen

Combined Pricing and Inventory Control Under Uncertainty

This paper addresses the simultaneous determination of pricing and inventory replenishment strategies in the face of demand uncertainty. More specifically, we analyze the following single item, periodic review model. Demands in consecutive periods are independent, but their distributions depend on the items price in accordance with general stochastic demand functions. The price charged in any given period can be specified dynamically as a function of the state of the system. A replenishment order may be placed at the beginning of some or all of the periods. Stockouts are fully backlogged. We address both finite and infinite horizon models, with the objective of maximizing total expected discounted profit or its time average value, assuming that prices can either be adjusted arbitrarily (upward or downward) or that they can only be decreased. We characterize the structure of an optimal combined pricing and inventory strategy for all of the above types of models. We also develop an efficient value iteration method to compute these optimal strategies. Finally, we report on an extensive numerical study that characterizes various qualitative properties of the optimal strategies and corresponding optimal profit values.

A Combined Vehicle Routing and Inventory Allocation Problem

We address the combined problem of allocating a scarce resource among several locations, and planning deliveries using a fleet of vehicles. Demands are random, and holding and shortage costs must be considered in the decision along with transportation costs. We show how to extend some of the available methods for the deterministic vehicle routing problem to this case. Computational results using one such adaptation show that the algorithm is fast enough for practical work, and that substantial cost savings can be achieved with this approach.

An Inventory Model with Limited Production Capacity and Uncertain Demands II. The Discounted-Cost Criterion

This paper considers a single-item, periodic-review inventory model with uncertain demands. We assume a finite production capacity in each period. With stationary data, a convex one-period cost function and a continuous demand distribution, we show under a few additional unrestrictive assumptions that a modified basic-stock policy is optimal under the discounted cost criterion, both for finite and infinite planning horizons. In addition we characterize the optimal base-stock levels in several ways.

Computational Issues in an Infinite-Horizon, Multiechelon Inventory Model

Clark and Scarf Clark, A., H. Scarf. 1960. Optimal policies for a multi-echelon inventory problem. Mgmt. Sci.6 475-490. characterize optimal policies in a two-echelon, two-location inventory model. We extend their result to the infinite-horizon case for both discounted and average costs. The computations required are far easier than for the finite horizon problem. Further simplification is achieved for normal demands. We also consider the more interesting case of multiple locations at the lower echelon. We show that, under certain conditions, this problem can be closely approximated by a model with one such location. A rather simple computation thus yields both a near-optimal policy and a good approximation of the cost of the system.

Finding Optimal (s, S) Policies Is About As Simple As Evaluating a Single Policy

In this paper, a new algorithm for computing optimal ( s , S ) policies is derived based upon a number of new properties of the infinite horizon cost function c ( s , S ) as well as a new upper bound for optimal order-up-to levels S * and a new lower bound for optimal reorder levels s *. The algorithm is simple and easy to understand. Its computational complexity is only 2.4 times that required to evaluate a (specific) single ( s , S ) policy. The algorithm applies to both periodic review and continuous review inventory systems.

An efficient algorithm for computing an optimal ( r,Q ) policy in continuous review stochastic inventory systems

The reorder point/reorder quantity policies, also referred to as r, Q policies, are widely used in industry and extensively studied in the literature. However, for a period of almost 30 years there has been no efficient algorithm for computing optimal control parameters for such policies. In this paper, we present a surprisingly simple and efficient algorithm for the determination of an optimal r*, Q* policy. The computational complexity of the algorithm is linear in Q*. For the most prevalent case of linear holding, backlogging and stockout penalty costs in addition to fixed order costs, the algorithm requires at most 6r* + 13Q* elementary operations additions, comparisons and multiplications, and hence, no more than 13 times the amount of work required to do a single evaluation of the long-run average cost function in the point r*, Q*.

Selecting a Portfolio of Suppliers Under Demand and Supply Risks

We analyze a planning model for a firm or public organization that needs to cover uncertain demand for a given item by procuring supplies from multiple sources. Each source faces a random yield factor with a general probability distribution. The model considers a single demand season. All supplies need to be ordered before the start of the season. The planning problem amounts to selecting which of the given set of suppliers to retain, and how much to order from each, so as to minimize total procurement costs while ensuring that the uncertain demand is met with a given probability. The total procurement costs consist of variable costs that are proportional to the total quantity delivered by the suppliers, and a fixed cost for each participating supplier, incurred irrespective of his supply level. Each potential supplier is characterized by a given fixed cost and a given distribution of his random yield factor. The yield factors at different suppliers are assumed to be independent of the seasons demand, which is described by a general probability distribution. Determining the optimal set of suppliers, the aggregate order and its allocation among the suppliers, on the basis of the exact shortfall probability, is prohibitively difficult. We have therefore developed two approximations for the shortfall probability. Although both approximations are shown to be highly accurate, the first, based on a large-deviations technique (LDT), has the advantage of resulting in a rigorous upper bound for the required total order and associated costs. The second approximation is based on a central limit theorem (CLT) and is shown to be asymptotically accurate, whereas the order quantities determined by this method are asymptotically optimal as the number of suppliers grows. Most importantly, this CLT-based approximation permits many important qualitative insights.

Design for Postponement: A Comprehensive Characterization of Its Benefits Under Unknown Demand Distributions

Recent papers have developed analytical models to explain and quantify the benefits of delayed differentiation and quick response programs. These models assume that while demands in each period are random, they are independent across time and their distribution is perfectly known, i.e., sales forecasts do not need to be updated as time progresses. In this paper, we characterize these benefits in more general settings, where parameters of the demand distributions fail to be known with accuracy or where consecutive demands are correlated. Here it is necessary to revise estimates of the parameters of the demand distributions on the basis of observed demand data. We analyze these systems in a Bayesian framework, assuming that our initial information about the parameters of the demand distributions is characterized via prior distributions. We also characterize the structure of close-to-optimal ordering rules in these systems, for a variety of types of order cost functions.

Two-echelon distribution systems with vehicle routing costs and central inventories

We consider distribution systems with a single depot and many retailers each of which faces external demands for a single item that occurs at a specific deterministic demand rate. All stock enters the systems through the depot where it can be stored and then picked up and distributed to the retailers by a fleet of vehicles, combining deliveries into efficient routes. We extend earlier methods for obtaining low complexity lower bounds and heuristics for systems without central stock. We show under mild probabilistic assumptions that the generated solutions and bounds come asymptotically within a few percentage points of optimality (within the considered class of strategies). A numerical study exhibits the performance of these heuristics and bounds for problems of moderate size.

An allocation and distribution model for perishable products

This paper presents an allocation model for a perishable product, distributed from a regional center to a given set of locations with random demands. We consider the combined problem of allocating the available inventory at the center while deciding how these deliveries should be performed. Two types of delivery patterns are analyzed: the first pattern assumes that all demand points receive individual deliveries; the second pattern subsumes the frequently occurring case in which deliveries are combined in multistop routes traveled by a fleet of vehicles. Computational experience is reported.