Weak aggregating algorithm for the distribution-free perishable inventory problem

Abstract

We formulate the multiperiod, distribution-free perishable inventory problem as a problem of prediction with expert advice and apply an online learning method (the Weak Aggregating Algorithm) to solve it. We show that the asymptotic average performance of this method is as good as that of any time-dependent stocking rule in a given parametric class.

Introduction

In the classical perishable inventory (also called “newsvendor”) problem, a decision-maker must choose the starting inventory level for a product in a future selling period when demand for the product is uncertain. No replenishment is possible during the selling period, and the product is perishable, i.e. it loses part or all of its value at the end of the period. The newsvendor aims to achieve the maximum return by balancing the risk of lost sales because of understocking against that of inventory spoilage because of overstocking.

This problem is a practical inventory control problem faced by many firms whose products are perishable (including newspaper publishers), but it is also a component of other important problems; for example, bank cash management, water reservoir management, and airline revenue management. The problem has a simple solution when the probability distribution of demand is known but becomes significantly more challenging when distributional information is limited, inaccurate, or unavailable. We deal with the case in which there is no prior knowledge of the distribution.

In the distribution-free case, if the newsvendor faces only a single decision period, the problem falls into the category of decision analysis under uncertainty typified by the min-max approach of [10]. If there are multiple selling periods, it is possible to gain information about the demand distribution over time and adapt ordering policies accordingly. (We will use the term multi-period to mean multiple selling periods rather than multiple opportunities to order, an interpretation that has been used in some prior research.) Recent examples of work on the multi-period version include both Bayesian (see [8]) and nonparametric approaches (see [11]). The latter paper contains an up-to-date review of the extensive literature in this area. Other articles proposing non-parametric approaches and/or bound for inventory problems include [1] which proposes distribution-free upper and lower bounds for the order quantity and reorder point in a service-constrained non-perishable inventory system. [7], [3] consider the perishable inventory system where the functional form of demand distribution is known and develop an operational statistics approach to find a decision rule that maximizes the performance uniformly for all possible values of the unknown demand parameters. Finally, [6] considers a sampling-based approach in which it is possible to obtain bounds on the number of samples needed to attain a specified accuracy level.

In this article, we propose a novel approach to the distribution-free, multi-period problem that utilizes recent advances in the theory of prediction and learning with expert advice (see Chapter 2 of [2]). This approach leads to an algorithm with performance guarantees under more general assumptions than those previously achieved (existing non-parametric results require, at least, independence of demands over time). The ‘experts’ in this treatment are passive predictors of the best starting inventory levels over time out of the continuum of possible levels suggested by functions in a given parametric class with a bounded finite-dimensional parameter space; thus, we consider an infinite pool of experts. (The case of a finite number of stocking levels, and hence experts, is a straightforward variant of this.) The algorithm progresses, in essence, by forming successive weighted averages of the expert predictions, where the weights are adjusted according to the success of the experts in previous periods.

We make the following contributions:

• We cast the newsvendor problem as online learning with expert advice and show that the Weak Aggregating Algorithm (WAA) of [5] can be applied to the problem.

• We prove that the performance of the algorithm is asymptotically as good as the performance of the best time-dependent strategy in a given parametric class with a bounded parameter space. Thus, in the setting of the newsvendor problem, we obtain stronger results than the existing analysis of the WAA for a finite collection of experts.

• The performance bound holds in the absence of any statistical assumptions about the demand sequence.

In the next section we provide a formal statement of the WAA. The newsvendor problem, the explicit WAA specialized to the newsvendor problem and its analysis are given in Section 3.