On the Selection of Customer Service Metrics in Inventory Control

Transcription

1 On the Selection of Customer Service Metrics in Inventory Control Guijarro E 1, Babiloni E, Cardós M, Estellés S Abstract Customer service metrics have been largely studied in literature due to their importance on the performance of the inventory systems. However selecting the most suitable metric for a business environment is a difficult decision for managers since each one has its advantages and disadvantages. To the best of our knowledge, the literature does not offer suggestions on how to select the most suitable metric. This paper reviews and discusses two of the service measures most used in practice, the cycle service level (CSL) and the fill rate (FR), and proposes some guidelines to help practitioners on the selection. We consider different views such as customers expectations, demand data, the behavior of the demand, computational complexity, importance of the item, purpose of the business decisions, and efficiency of the inventory systems. Keywords: Customer Service Metrics, Cycle Service Level, Fill Rate, Guidelines 1 Introduction Customer service levels are a concern of every inventory systems. Generally, the aim of inventory managers is to provide a high level of customer service. However, an organization cannot assure that all demanded items will be available in the proper quantity and moment. For example, when demand is probabilistic, there is a chance of not being able to satisfy part of the demand. Furthermore, other external causes (such as machine failure or late delivery) can affect the availability of items when are demanded. Obviously, poor service levels may result in loss of 1 Ester Guijarro Tarradellas ( esguitar@doe.upv.es) Dpto. de Organización de Empresas. Universitat Politècnica de València. Camino de Vera s/n, Valencia. * This research is part of a project supported by the Universitat Politècnica de València, with reference number PAID-06-11/

2 customers, sales and extra costs of handling backorders. However, attempting to offer a high level of service can also be counterproductive because excessive service levels may result in large inventories and thus in a waste of money (Ronen 1983). For that reason, it is important for managers to have appropriate tools to measure the customer service. Furthermore, customer service metrics are used in inventory control not only to analyze the performance of the policies, but also to establish the parameters of an inventory policy. Then, how to measure the customer service is an important decision for inventory managers. A considerable amount of research has proposed different metrics to compute the customer service [see (Ronen 1983); (Silver et al. 1998) or (Tempelmeier 2000) among others]. The selection of one of them is a difficult decision since each has its strengths and weaknesses and appropriate applications (Fogarty et al. 1991). The purpose of this paper is to propose some guidelines to help practitioners on the selection of the most suitable customer service measure according to the aim of the organization and inventory characteristics. Two of the service measures most used in practical environments are the cycle service level (CSL) and the fill rate (FR). Both CSL and FR have been widely discussed in related literature, but customers service levels are more than adequate, but are poorly defined (Sabath 1978). This paper focuses on them and reviews the classical definition of CSL (Section 2) and FR (Section 3). We discuss that these definitions may provide poor results under some inventory conditions. Furthermore, we review the standard definition of CSL (Cardós et al. 2006) and FR (Guijarro et al. 2012) and highlight a number of advantages and disadvantages of each one. Finally, Section 4 proposes guidelines to help practitioners on the selection of the most appropriate service measure. The conclusions of this paper are presented in Section 5. 2 The Cycle Service Level One of the most usual measures of customer service is CSL. This metric indicates the probability of no stockout per replenishment cycle (Chopra and Meindl 2004), considering a stockout as the situation when the on-hand stock drops to the zero level (Silver et al. 1998). Therefore, CSL can be defined as the fraction of cycles in which the on-hand stock does not drop to the zero level, and can be expressed as CSLClassic 1 P stockout in a replenishment cycle (1) This definition, called classic in this paper (CSL Classic), can be applied to any stock policy and demand pattern. Considering the specific case of periodic review 552

3 stock policy (R, S), CSL is the probability that the stock at the review is not exceeded by the demand during the review period R plus the lead time L, i.e.: Classic RL CSL P D S (2) where D t represents the accumulated demand during t consecutive periods and S represents the order-up-to-level. When demand follows any discrete demand distribution, expression (2) can be rewritten as Classic RL CSL F S (3) Note that this expression can be used both in the backordering case (unfulfilled demand is backordered and met as soon as possible) and in the lost sales case (unfulfilled demand is lost). One of the main advantages of this definition is that it is simple enough to be used by practitioner in real environments. However it presents some limitations. On one hand, it does not take into account demand fulfillment, since it only considers the stock level at the end of the cycle. On the other hand, there are a number of situations where this definition is scarcely useful. For example, under an intermittent or slow movement demand context the probability of no demand when the physical stock is equal to zero is not negligible, so a stockout situation and demand fulfillment can be compatible. Furthermore, this definition leads to consider that the CSL is equal to one if there is no demand during the replenishment cycle. In order to overcome shortcomings explained above (Cardós et al. 2006) propose a more general and appropriate CSL definition, referred as standard CSL (CSL Stnd) in the rest of the paper, as the fraction of cycles in which non zero demand is completely satisfied with physical stock. According with this new definition, CSL can be expressed as CSL P demand during cycle on hand stock demand during cycle 0 (4) Stnd (Cardós et al. 2009) propose two expressions to compute (4) in a periodic review (R, S) when demand is discrete. The first one considers that the unfulfilled demand in a replenishment cycle is backlogged, and then CSL can be computed as CSL Stnd _ Bk R R0 FL S F 1 F 0 (5) where F t( ) is the cumulative distribution function. The second one assumes that the unfulfilled demand is lost. In this case CSL is 553

4 CSL S FROH 0 FR 0 POH (6) 0 1 F 0 Stnd _ LS 0 OH where OH 0 is the on-hand stock at the beginning of the replenishment cycle. The main advantages of the standard definition are: (a) demand is explicitly considered to measure the service level; and (b) it works properly even if there is no demand during the replenishment cycle. However, calculating expressions (5) and (6) may be quite complex, especially in the lost sales scenario. Computing the probability vector of every on-hand stock level at the beginning of the cycle, P OH 0, requires the availability of appropriate tools as sound mathematical background. Furthermore, from a practical point of view, expression (6) may be time-consuming which makes the standard CSL an inappropriate procedure to be used in some business contexts involving thousands of items. R 3 The Fill Rate The second customer service metric reviewed in this paper is FR, defined as the fraction of demand that is completely fulfilled from available stock. The definition and estimation of fill rate have been studied extensively in operation management (OM) literature in the last sixty years. However, this definition may result ambiguous since it does not specify the time period referred to. This ambiguity causes different interpretations of the FR definition. In this sense, (Silver et al. 1998) suggest two possible generalizations: (i) considering FR as the fraction of demand satisfied in each period, and (ii) considering FR as the fraction of demand satisfied within a specified time, i.e. as a long run average. Classical OM literature computes fill rate following the first generalization, i.e. as the fraction of demand per replenishment cycle met from on-hand stock. Common approach to estimate it consists in computing the number of backorders per replenishment cycle, i.e. the demand that is not satisfied, instead of computing directly the fulfilled demand per replenishment cycle [e.g. (Hadley and Whitin 1963); (Silver et al. 1998); (Silver and Bischak 2011)]. This approach, known as the traditional approximation (FR Trad further on) calculates the complement of the quotient between the expected unfulfilled demand per replenishment cycle and the total expected demand per replenishment cycle, i.e. FR Trad E unfulfilled demand per replenishment cycle 1 (7) E total demand per replenishment cycle 554

5 Recently, some authors as (Axsäter 2003); (Sobel 2004) or (Zhang and Zhang 2007) have presented a new approach based on the second generalization and they define the fill rate as the long run average fraction of demand satisfied immediately from the on-hand stock, which can be expressed as fulfilled demand in t periods FR lim E t total demand in t periods (8) (Chen et al. 2003) indicate that, applying the renewal theory, estimating the fill rate following the long run definition (8) leads to the traditional approximation (FR Trad) when demand is independent and identically distributed. Then the most widely used method to estimate the fill rate is actually FR Trad, i.e. expression (7). However, as (Guijarro et al. 2012) point out FR Trad is an approximation of the fill rate. The authors find that computing the fill rate as expression (7) leads to significant biases and deviations that cannot be neglected. These deviations are especially important during the cycles without demand because in those cycles FR Trad considers the fill rate equal to 1. However, from a practical point of view, it is useless to consider a service metric when there is no demand to be served. Therefore, cycles that do not show any demand should not be taken into account. For that reason, (Guijarro et al. 2012) propose a new generalized definition of the fill rate (named standard fill rate, FR Stnd in this paper) as the fraction of demand that is fulfilled with the on hand stock during a cycle with positive demand, which is expressed as fulfilled demand per replenishment cycle FRStnd E positive demand during cycle (9) total demand per replenishment cycle When the inventory is managed by a periodic review (R, S) inventory policy, demand follows a discrete distribution and the unfulfilled demand is backordered, expression (9) can be rewritten as S FR NS0 F 0 R NS f 0 R D R 1 1 F 0 1 D 1F 0 (10) FR f S NS Stnd _ Bk L 0 NS0 R DR NS0 R R being f t( ) the probability density function of demand during t consecutive periods and NS 0 the net stock at the beginning of the cycle. In the lost sales case, the standard fill rate can be computed as follows 555

6 S FR i F 0 R i fr j i 0 1 F 0 ji1 j 1F 0 (11) FRStnd _ LS P OH0 i R As can be seen, expression (9) includes explicitly the condition of having positive demand during the cycle; thus it can be used over different demand patterns including intermittent or lumpy demand. However, the computational effort needed is substantial, especially to obtain the whole vector POH 0 in the lost sales scenario. R 4 Guidelines on Selecting the Service Level Metric Despite the metrics reviewed in previous sections have been largely studied, there is not a consensus on their definition and estimation, so the selection of one of them is a difficult task for managers. This section presents guidelines, summarized in Table 1, to help practitioners on the selection based on the following views: Customer expectations. Traditionally, in inventory literature the term customer service refers to what the managers or the company understand about service. Nevertheless, the first question that should be answered by inventory managers is what customers perceive as service including if backlog is allowed or not. For example, frequently when industrial customers are interviewed they indicate that they can wait for some days if there is a stockout, so CSL with backlog may be suitable. Available demand data. Once a customer service is selected, it usually requires some information concerning the demand for the managed item. Unfortunately, in a large number of cases, available data in companies is actually sales data which are not equivalent. Computing the service level using sales data instead of demand can distort the picture especially in the lost sales case because the demand of an item is not recorded when it is in stockout. A strength of the CSL Classic is that it does not require demand data for its calculation since it only considers the stock level at the end of the cycle. The other metrics, however, need to record demand data to identify properly the demand distribution pattern. Demand category. The categorization of demand patterns is an essential tool to choose the best forecasting and stock control policies. (Syntetos et al. 2005) propose a method of categorization of demand patterns based on the squared coefficient of variation (CV 2 ) and the average inter-demand interval (p). The categorization schemes distinguishes four demand categories: smooth (p 1.32 and CV ), intermittent (p>1.32 and CV ), erratic (p 1.32 and CV 2 >0.49) and lumpy (p>1.32 and CV 2 >0.49). The average inter-demand in- 556

7 terval is directly related to the probability of no demand during a cycle. Then, if there is a high probability of having cycles without demand, which correspond to intermittent and lumpy categories, only the standard definitions of the service measures (CSL Stnd and FR Stnd) work properly. On the other hand, if the probability of zero demand is low, then classical definitions may fit. Computational complexity and accuracy. As Section 2 and 3 explain, both CSL Classic and FR Trad are simple enough expressions to be used in real environments, especially the classical CSL. However, these metrics present poor results in some situations. ABC item classification. The importance of the managed item should be taken into account as well as the specific business environment to decide if the risk of using approximated metrics is acceptable. For example, for A items the aim is usually to avoid unpredictable stockout situations. Then, the complexity of the exact expressions (CSL Stnd and FR Stnd) seems to be justified. However, the aim for C items is to ease its management, so CSL Classic and FR Trad can provide a good performance and low impact on the business performance. Business Purpose. As explained before, CSL Classic, CSL Stnd and FR Stnd are short term metrics whereas FR Trad is a long run metric. The time window considered is connected to the purpose of the decisions. For example, in the purpose is financial, managers need long-range information, being the FR Trad the most appropriate customer measure. On the other hand, if the purpose is operational, the information should be short-term because managers make decisions for the next period. In this case, either CSL Classic, CSL Stnd or FR Stnd may suit. Inventory Systems Efficiency. While CSL is a more restrictive metric (when just one unit is not served, CSL is equal to zero for that period), FR provides more information since it computes the fraction of demand served. Therefore, if the expected service level is high, CSL may lead to overestimate the stock needed when it is used to determine the policy parameters. Table 1 Guidelines for the selection of the service level metric Views CSL classic CSL standard FR traditional FR standard Customer expectations Available demand data just stock Demand category Smooth All Smooth All Computational complexity and accuracy simple accurate simple accurate Item class B or C A B or C A Business purpose Operational Operational Financial Operational Inventory systems efficiency medium target medium target medium and high target medium and high target 557

8 5 Conclusions Despite of the general consensus on the importance of the selected service metric on the performance of inventory systems, the literature does not offer many clues to help practitioners on selecting the most appropriate metric for a business environment. That is the contribution of this paper. It cannot be affirmed that any metric is better than others because each of them focuses on different aspects of the delicate trade-off between the expectations of the customers, availability of demand data vs. sales data, the behaviour of the demand, computational complexity and accuracy of the service level measurement, importance of the item, aim of the business decision focused on financial or operational goals, and efficiency of the inventory system avoiding overestimation of the inventory needed. Therefore, the selection of the service metric has to be adopted considering the impact on customers relationships, operations and business performance aligned with the strategy of the organization. 6 References Axsäter S (2003) Note: Optimal Policies for Serial Inventory Systems Under Fill Rate Constraints. Management Science 49: Cardós M, Babiloni E, Palmer M E, and Albarracín J.M. (2009) Effects on Undershoots and Lost Sales on the Cycle Service Level for Periodic and Continuous Review Policies. Ieee Transactions on Information Forensics and Security Cardós M, Miralles C, and Ros L (2006) An exact calculation of the cycle service level in a generalized periodic review system. J Opl Res Soc 57: Chen J H, Lin D K J, and Thomas D J (2003) On the single item fill rate for a finite horizon. Operations Research Letters 31: Chopra S, Meindl P (2004) Supply Chain Management. Pearson. Prentice Hall, Fogarty D W, Blackstone J H, Hoffman T R (1991) Production and Inventory Management. South-Western Publishing Co., Cincinnati, Ohio Guijarro E, Cardós M, and Babiloni E (2012) On the exact calculation of the fill rate in a periodic review inventory policy under discrete demand patterns. Eur J Oper Res 218: Hadley G, Whitin T (1963) Analysis of Inventory Systems. Prentice-Hall, Englewood Cliffs, NJ Ronen D (1983) Inventory service levels - comparison of measures. International Journal of Operation and Production Management 3:37-45 Sabath R E (1978) How much service do customers really want? Business Horizons 21:26-32 Silver E A, Pyke D F, Peterson R (1998) Inventory Management and Production Planning and Scheduling. Wiley, New York Silver E A, Bischak D P (2011) The exact fill rate in a periodic review base stock system under normally distributed demand. Omega-int J Manage S 39: Sobel M J (2004) Fill rates of single-stage and multistage supply systems. Manufacturing and Service Operations Management 6:41-52 Syntetos A A, Boylan J E, and Croston J D (2005) On the categorization of demand patterns. J Opl Res Soc 56:

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