A Bayesian approach to a dynamic inventory model under an unknown demand distribution

Abstract

In this paper, the Bayesian approach to demand estimation is outlined for the cases of stationary as well as non-stationary demand. The optimal policy is derived for an inventory model that allows stock disposal, and is shown to be the solution of a dynamic programming backward recursion. Then, a method is given to search for the optimal order level around the myopic order level. Finally, a numerical study is performed to make a profit comparison between the Bayesian and non-Bayesian approaches, when the demand follows a stationary lognormal distribution. A profit comparison is also made between the stationary and non-stationary Bayesian approaches to observe whether the Bayesian approach incorporates non-stationarity in the demand. And, it is observed whether stock disposal reduces the losses due to ignoring non-stationarity in the demand.

Scope and purpose

In the context of inventory models, one of the crucial factors to determine an optimal inventory policy, is the accurate forecasting or estimation of the demand for items in the inventory. The assumption of a constant demand is seriously questioned in recent times, since in reality the demand is generally uncertain and may even vary with time. For instance, the demand for new products, spare parts, or style goods, is likely to fluctuate widely, the average demand is quite likely to be low, and may exhibit a trend. In such situations, the Bayesian approach is a very useful tool for demand estimation, which is applicable even when past observations are scarce. In this paper, we use this approach to estimate the demand for an item, and obtain the expressions for finding the optimal inventory policies. We give a simpler method to find the optimal inventory policy, since the procedure to obtain the optimal inventory policy in the Bayesian framework, is quite tedious especially for long planning horizons, and in cases where the future demand becomes unpredictable. To widen the application of the method, we have given a general procedure which is not restricted to any particular probability distribution for the demand. We compare the Bayesian approach with the corresponding non-Bayesian approach, in terms of the optimum expected profits, when the demand follows a lognormal distribution. We also investigate how well the Bayesian approach incorporates non-stationarity in the demand.

Introduction

The problem of demand estimation, is an important aspect in the analysis of probabilistic inventory systems. It is generally assumed that the demand distribution has known parameters and is static throughout the planning horizon. In practice, the parameters have to be fixed subjectively, or statistically estimated using past demand information. But it is almost impossible to specify exactly the true values of the parameters, especially in the absence of abundant demand information, as in the case of demand for new products. Moreover, sometimes, due to many reasons, demand may exhibit a trend. And the optimal solutions are very sensitive to the changes in the demand rate. To incorporate this or otherwise, it is more appropriate to assume randomness in the parameters as well. For this purpose, a prior distribution is considered for the unknown parameters of the demand distribution, based on past experience or intuition. This distribution can be updated as and when fresh demand occurs. One of the best systematic methods for incorporating current demand information and updating the demand distribution, is known to be the Bayesian approach.

The Bayesian approach can be applied to inventory systems with either a finite or an infinite planning horizon. Items like computers and related products or even motor vehicles, are being continuously updated and new versions are introduced in the market. Inventory of such items generally have finite planning horizons with fluctuating demands, and the Bayesian set-up could be appropriate. Brown and Rogers [1] and Eppen and Iyer [2] have considered specific finite horizon problems under the Bayesian framework. The Bayesian approach can also be applied to infinite horizon problems in the initial stages, until the demand for the product stabilizes or enough data accumulates for using other estimation procedures. In this age of information technology, obtaining data from time to time for updating the information about uncertain quantities like demand, deterioration, or supply, is not a problem. Hence, with the easy availability of such information, the Bayesian approach is expected to give better results.

The Bayesian approach to inventory modeling, has been investigated earlier for specific cases. The optimal policy has been characterized when the demand distribution belongs to a particular case of the one-parameter exponential or range family of distributions, by Scarf [3] and Karlin [4]. The dynamic programming procedure for finding the optimal solutions for an inventory model with an unknown Poisson demand, has been outlined by Brown and Rogers [1] and Zacks [5]. For particular demand distributions, the procedure to find the optimal policy can be slightly simplified as shown by Azoury [6]. In the same situation, it can be approximated by a myopic policy without significant losses, as shown by Lovejoy [7]. Bounds for the optimal order level in the lost sales inventory problem, have been given by Morton and Pentico [8]. The Bayesian approach has been compared with the non-Bayesian approach in specific situations by Azoury and Miller [9], Kaplan [10] and Hill [11]. Bayesian inventory models with time-dependent demand have been investigated by Popovic [12] and Eppen and Iyer [2].

Our aim is to investigate the utility of the Bayesian approach without restricting the demand distribution to any particular family of distributions. Also, previous numerical studies were restricted to a planning horizon of at the most two periods, in which case the optimal policy can be directly obtained by backward recursion. However, the problem becomes quite difficult to solve for longer planning horizons. Hence, we propose a simpler method to obtain the optimal policy, which can be easily implemented even for long planning horizons. We compare the policy so obtained with the corresponding non-Bayesian policy, for various cost and demand parameters. We also consider simple non-stationary demand structures and investigate whether the Bayesian approach implicitly accounts for such non-stationarity.

The rest of the paper is organized as follows. In Section 2, the Bayesian approach is outlined for any given demand distribution, and is extended to the case where there is non-stationarity in the mean demand. In Section 3, the relevant inventory model is described, and a method to find the optimal policy is proposed. The algorithm to find the optimal policy, is outlined in Section 4. Section 5 illustrates the Bayesian approach for a lognormal demand distribution. In Section 6, the numerical study performed to compare the Bayesian and non-Bayesian approaches, as well as the stationary and non-stationary Bayesian approaches, is described. A general summary and conclusions are given in Section 7.