**Implementing an Exact Estimation Approach of the Base Stock for the Periodic Review Policy Based on Fill Rate**

**Eugenia BABILONI, Ester GUIJARRO, Manuel CARDÓS**

Universidad Politècnica de València

Camino de Vera s/n, Valencia, Spain

**Abstract**: Recent developments on the fill rate estimation in periodic review stock policies provide new techniques for setting up the stock policy parameters including exact procedures. However, the implementation of this method in real environments may be a difficult task due to issues such as the availability of demand data, the identification of an appropriate demand distribution pattern and the target fill rate established for each item according to a global target fill rate. This paper focuses on the modifications needed in the information management system of a company in order to implement an exact estimation procedure of the stock policies taking advantage of available data.

**Keywords**: Stock control, fill rate, data model, DSS.

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**CITE THIS PAPER AS**:

Eugenia BABILONI, Ester GUIJARRO, Manuel CARDÓS, **Implementing an Exact Estimation Approach of the Base Stock for the Periodic Review Policy Based on Fill Rate**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 22 (3), pp. 289-296, 2013.

**Introduction**

In inventory management two key questions should be answered: (1) when to place a replenishment order; and (2) how much should be order at each review period. When the inventory is managed by a base stock (*R, S*) policy, answering the first question implies to determine the review period *R*, whereas the second question depends on the base stock level *S*. In practical environments, the review period is normally predetermined, so the real problem for managers consists in deter-mining the optimal S such that total inventory costs are minimized or some target customer service level is fulfilled.

Although the majority of the literature has focused on the criterion of minimization of costs, in real environments these costs are difficult to know and estimate, particularly costs incurred by not having stock to attempt the demand [1], [2]. For this reason, practitioners apply the service level criterion to establish the base stock level *S*, being the volume fill rate (denoted just fill rate or *ß* further on) the customer service measure most widely used. Furthermore, when managing inventories it is required to know how to proceed when an item is out of stock and a customer order arrives. This paper assumes that unsatisfied demand from the on hand stock is backordered.

The fill rate is defined as the fraction of demand that is immediately fulfilled from on hand stock. However, in order to avoid complex mathematical formulation, the most common approach to estimate it consists in computing the number of units short instead of computing directly the fulfilled demand per replenishment cycle.

This approach (denoted by *ß _{Trad}* further on) calculates the complement of the quotient between the expected unfulfilled demand per replenishment cycle (also known as expected shortage) and the total expected demand per replenishment cycle, i.e.:

(1)

In the literature, we find quite a number of works suggesting methods to estimate expression (1) in different contexts. However, few of those consider the periodic review systems, although these are arguably most realistic. One limitation of the available methods devoted to estimating *ß _{Trad }*in the (

*R, S*) system for the backordering case is that they usually assume continuous demand patterns, generally normally distributed. In this sense, [1], [3], [4] and [5] suggest methods to estimate it when demand is normally distributed whereas [6], [7] and [8] when demand follows any continuous distribution. The continuous assumption is the most common for modelling the demand since mathematical models are simplified. Unfortunately, in real environments the demand rarely is continuous, being the discrete demand the most common in practice [9].

Another approach to compute the fill rate consists in estimating directly the fraction of the fulfilled demand per replenishment cycle instead of determining the expected shortage, as follows:

[10] and [11] show that expression (1) and expression (2) are not equivalent. To the best of our knowledge, only [11] present exact estimation procedures for both expression (1) and (2) to compute the fill rate for periodic review systems under discrete demand context and backordering case. However the complexity of such methods makes difficult its implementation by practitioners.

On the other hand, applying estimation procedures designed for continuous demand patterns to discrete demands leads to significant deviations that cannot be neglected as shown by [12].

Therefore, the exact estimation of the base stock for the periodic review policy based on the fill rate cannot be avoided but in practice it is feasible if and only if it is included as part of the inventory control module of the management information system. However the implementation of an exact estimation procedure is much more than just applying complex formulas but it also requires taking advantage of available data. The main purpose of this paper is to describe how the information management system of a company can be modified to cope with the main difficulties that arise when an exact approach is applied and how a Decision Support System (*DSS*) can help stock managers.

The rest of this paper is organized as follows. Section 2 points out some important technical problems related to the application of an exact estimation procedure in practice. However, in order to implement this kind of procedures the transactional system of the company should be modified (see Section 3) and a Decision Support System is also needed to help adopting the most appropriate decision on the stock management (see Section 4). Finally, Section 5 includes a summary of the main results provided in this paper. Appendix A includes the description and notation of a periodic review stock policy and Appendix B summarizes the derivation of *ß _{Trad}* and

*ß*in a discrete demand context and its estimation procedures.

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