ElsevierGuijarro Tarradellas, E.; Cardós, M.; Babiloni, E. (2012). On the exact calculation of the fill rate in a periodic review inventory policy under discrete demand patterns. European Journal of Operational Research. 218(2): 442-447. doi:10.1016/j.ejor.2011.11.025. On the exact calculation of the fill rate in a periodic review inventory policy under discrete demand patterns GUIJARRO, E., CARDÓS, M., BABILONI, E. Universidad Politécnica de Valencia AbstractThe primary goal of this paper is the development of a generalized method to compute the fill rate for any discrete demand distribution in a periodic review policy. The fill rate is defined as the fraction of demand that is satisfied directly from shelf. In the majority of related work, this service metric is computed by using what is known as the traditional approximation, which calculates the fill rate as the complement of the quotient between the expected unfulfilled demand and the expected demand per replenishment cycle, instead of focusing on the expected fraction of fulfilled demand. This paper shows the systematic underestimation of the fill rate when the traditional approximation is used, and revises both the foundations of the traditional approach and the definition of fill rate itself. As a result, this paper presents the following main contributions: (i) a new exact procedure to compute the traditional approximation for any discrete demand distribution; (ii) a more suitable definition of the fill rate in order to ignore those cycles without demand; and (iii) a new standard procedure to compute the fill rate that outperforms previous approaches, especially when the probability of zero demand is substantial. This paper focuses on the traditional periodic review, order up to level system under any uncorrelated, discrete and stationary demand pattern for the lost sales scenario.Keywords: inventory, fill rate, periodic review, discrete demand, lost sales. 2 Introduction and literature reviewOne of the service measures most commonly used in practice is the volume fill rate, defined as the fraction of demand that is immediately fulfilled from on hand stock (see Silver et al. (1998) or Axsäter (2000) among others). Therefore, in practice it also indicates the size of the backordering demand when this is allowed (Tempelmeier (2007)). The volume fill rate is also known as item fill rate, unit fill rate or just fill rate (denoted by β in the rest of this paper).Over the last sixty years, several works have suggested methods to estimate the fill rate in different contexts. However, to the best of our knowledge, there is not any exact and general method to estimate it under any discrete demand context. This paper focuses on the exact estimation of the fill rate for the periodic review, order up to level (base stock) system. This stock policy is commonly denoted by (R, S) and consists of examining the status of an item every R fixed time periods and launching a replenishment order which raises the inventory position to the order up to level S.Traditionally ...

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This paper focuses on computing on-hand stock levels at the beginning of a replenishment cycle for a lost sales inventory system with periodic reviews and discrete demand. A base-stock policy is used for replenishments. The literature provides an Exact method which requires a huge computational effort, and two closed-form approximate methods that arise from the backordering case, the Non-stockout and the Bijvank&Johansen. In this paper we propose three new and closed-form approaches that explicitly consider the lost sales assumptions:the Adjusted Non-stockout, the Polar Opposite and the 1-Step methods. Existing and proposed methods are evaluated in terms of their accuracy when computing the cycle service level and the fill rate. In this sense, results show that the Bijvank&Johansen and 1-Step methods provide similar performance but present different behaviours in terms of under or over estimating service measures that have different implications on the design of stock policies.

Customer service measures are traditionally used to determine the performance or/and the control parameters of any inventory system. Among them, the fill rate is one of the most widely used in practice and is defined as the fraction of demand that is immediately met from shelf i.e. from the available onhand stock. However, this definition itself set out several problems that lead to consider two different approaches to compute the fill rate: the traditional, which computes the fill rate in terms of units short; and the standard, which directly computes the expected satisfied demand. This paper suggest two expressions, the traditional and the standard, to compute the fill rate in the continuous reorder point, order quantity (s, Q) policy following these approaches. Experimental results shows that the traditional approach is biased since underestimate the real fill rate whereas the standard computes it accurately and therefore both approaches cannot be treated as equivalent. This paper focuses on the lost sales context and discrete distributed demands.

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