 # A Gentle Introduction to the Bayesian Paradigm for Some Inventory Models

Vinti Dhaka (Banasthali University, India), Chandra K. Jaggi (University of Delhi, India), Sarla Pareek (Banasthali University, India) and Piyush Kant Rai (Banasthali University, India)
DOI: 10.4018/978-1-4666-9888-8.ch016

## Abstract

The recent era describes the demand of inventory systems which are governed through random cause effect phenomenon prevailing the strongest use of random models in the concerned area. Bayesian probability model serve the demands of present need in such inventory systems. The present study deals the use of basic Bayesian theory in the development of some of the inventory models, for e.g.: The inventory model for deteriorating items; Designing of the classical (s, Q) models, etc. Here the motivation of use of Bayes theory is to test the efficacy of optimal design of above said models when demand is supposed to be random having some basic probability distributions. In this regard we discuss the inventory model for deteriorating items and the (s, Q) model and their mathematical solution under Bayesian approach.
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## Purpose For The Model

As in the inventory models, one of the crucial factors to determine an optimal inventory policy is the accurate estimation of the demand for items in the inventory. The assumption of a stable demand and deterioration are critically questioned in recent times. Since in reality the demand is generally uncertain and may vary with time. Such as, the demand for the new product, spare parts or style goods, is likely to fluctuate widely, and the average demand is quite likely to be low, and may exhibit an indentified trend. In such situations, the Bayesian approach is a very useful tool for estimating the demand, and this tool is applicable whenever the past observations are not available. In this article, authors use this approach for estimating the demand for an item, and also obtain the expressions for finding the optimal inventory policies.

### Implementation for the Utility of Bayesian Approach for the Proposed Model

Bayesian techniques is most powerful method for modeling uncertain behavior of any random phenomenon in the light of posterior distribution using prior degree of belief i.e. it is a method which is logically correct more accurate an easy to communicate & understand.

With the help of different types of proper & improper prior one can check the efficiency of the methodology in the area of random deterioration under inventory. Moreover it is advanced methodology of drawing inference under several random parameters (high dimensional data) of inventory theory. In addition to applying any model it is also select the appropriate model (probability models) and test their efficacy under varying either parameters or prior information and loss functions.

### Drawbacks of the Proposed Model under Bayesian Approach

Although it is advance tool of measuring the randomness in the environment, yet there are some limitations of the model too. If multiparametric cases arrive then it is very difficult to estimate a posterior probability of the distributions theoretically. Also, if there is no such strong prior information for the parameters then the method provides complex mode for researchers to describe the bayes estimators for the parameters. Since, the posterior distribution as well as the Bayes estimators involves multiple integrals which required some simulation techniques to that will not be realistic situations always.

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## Introduction

The problem of estimating the demand is an important aspect in the analysis of probabilistic inventory systems. Generally it is assumed that the demand distribution has known parameters and is also 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 up dated 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 (1973) and Eppen and Iyer (1997) 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.

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