ABC Classification: Service Levels and Inventory Costs

ABC inventory classifications are widely used in practice, with demand value and demand volume as the most common ranking criteria. The standard approach in ABC applications is to set the same service level for all stock keeping units (SKUs) in a class. In this paper, we show (for three large real life datasets) that the application of both demand value and demand volume as ABC ranking criteria, with fixed service levels per class, leads to solutions that are far from cost optimal. An alternative criterion proposed by Zhang et al. performs much better, but is still considerably outperformed by a new criterion proposed in this paper. The new criterion is also more general in that it can take criticality of SKUs into account. Managerial insights are obtained into what class should have the highest/lowest service level, a topic that has been disputed in the literature.


Introduction
ABC inventory classification systems are widely used by business firms to streamline the organization and management of inventories consisting of very large numbers of distinct items, referred to as stock-keeping units (SKUs).APICS (Blackstone and Cox 2008) defines ABC classification as follows.''The classification of a group of items in decreasing order of annual dollar volume (price multiplied by projected volume) or other criteria.This array is then split into three classes, called A, B and C.''More flexible classifications with more than three classes and/or multiple criteria have also been proposed and these will be reviewed in section 2. We remark that, in what follows, we will use the term demand value instead of annual dollar volume and demand volume for the simpler annual demand volume criterion.
The most important reason for applying an ABC classification is that, in most practical cases, the number of different SKUs is too large to implement SKU-specific inventory control methods (Ernst and Cohen 1990).Retailers typically deal with thousands of different SKUs (Buxey 2006) and larger (service) organizations can have tens or hundreds of thousand SKUs.
For inventory control, service levels constitute arguably the most important performance measures.We have encountered a number of companies that use the ABC classification to set service levels, by assigning the same service level to each SKU in a particular class.This is in line with findings from Lee (2002) from NONSTOP solutions (a provider of demandchain optimization services) and Pflitsch (2008) from SLIMSTOCK (a provider of forecasting and inventory management software, including ''Slimstock ABC'' for inventory classification).Both confirm from their extensive experience of implementing inventory control software that the standard approach is to fix service levels per class.
If a company decides to use fixed service levels per class, then an obvious key question is what those service levels should be.However, the literature does not provide clear guidelines.In fact, it is not even clear which class should get the highest/lowest service level.On the one hand, authors have argued that A items are the most critical for a firm in determining the profit and should therefore have the highest service levels in order to avoid frequent backlogs (Armstrong 1985;Stock and Lambert 2001).On the other hand, it has been claimed that dealing with stockouts is not worth the effort for C items and they should therefore get the highest service level (Knod and Schonberger 2001).See Viswanathan and Bhatnagar (2005) for further discussion.
An explanation for this mixed bag of results is that the traditional criteria of demand value and demand volume for ABC classifications have not been developed from an inventory cost perspective.In this paper, we will take such a perspective to develop an alternative criterion based on the objective to minimize total inventory cost whilst achieving a certain required average fill rate (over all SKUs).See section 3 for details.This cost criterion ranks SKUs based on the value of bD hQ , where b is the criticality measured by the shortage cost, D is the demand volume, h is the unit holding cost, and Q is the order size.
Note that the cost criterion takes criticality of SKUs into account through the shortage cost b.As will be discussed in the next section, the importance of criticality has been addressed by many previous authors, especially in the context of spare parts management.However, the two common ranking criteria do not take it into account and neither does an alternative criterion proposed by Zhang et al. (2001).
A comparison of the new cost criterion and the most commonly applied demand value criterion leads to an important insight into the ''sub-optimality'' of the latter with respect to cost-service efficiency.Although both criteria rank an SKU higher if the demand volume is larger, the cost criterion ranks an SKU higher if the holding cost is lower whereas the demand value criterion does the opposite (assuming that a higher price implies a higher holding cost, as is common).This also explains why specialists in commercial ABC applications (Pflitsch 2008) have found that the demand volume criterion is more effective than the most commonly used demand value criterion in reducing inventory costs while maximizing the service level.
Our analysis leading to the cost criterion will also show that it is best used with a fixed cycle service level (1-stockout probability) per class rather than fixed fill rates per class.This has an important practical advantage, because the transformation to safety stocks and reorder points for SKUs is much easier from cycle service levels.
We will empirically compare the two traditional criteria and the one proposed by Zhang et al. (2001) to the new cost criterion, using three large real-life datasets, in a numerical experiment where the target average fill rate and the number of classes are varied.The results will show that the new cost criterion significantly and consistently outperforms all other methods.
The remainder of the paper is organized as follows.Section 2 shortly reviews the relevant literature on ABC classification.In section 3, we propose the new cost criterion based on inventory analysis.In section 4, we compare the cost criterion to other criteria in an extensive numerical experiment.Finally, in section 5, we provide our conclusions, managerial insights and directions for further research.

Research Background
Classification of SKUs is widely adopted by organizations.The main purpose of classification is to simplify the task of inventory management, by setting stock control methods and service levels per class rather than for each SKU separately.We remark that forecasting procedures can also be class-dependent (Boylan et al. 2008;Syntetos et al. 2005).However, our focus in this paper is restricted to inventory control and service levels.
In some environments, especially in the service/ maintenance industries, the rank of an SKU is determined by its criticality for the functioning of a piece of equipment (Naylor 1996).In most practical situations, however, classification is based on SKU-specific criteria such as the demand value (price multiplied by demand volume) and the demand volume for an SKU.
Obviously, ABC classification was originally named and designed for three classes: A, B, and C.However, the method can easily be extended to more classes, simply by dividing the ranked SKUs into more groups.The number of classes is usually limited to at most six (Graham 1987;Silver et al. 1998).
As discussed in section 1, the traditional classification approach is based on a single criterion.Demand value is the most commonly applied one, followed by demand volume.Another single criterion, discussed in detail later in this section, was proposed by Zhang et al. (2001).Our analysis will also lead to the proposal of a single criterion upon which the classification of SKUs may be based.Obviously, a major advantage of using a single criterion is the simplicity.
However, a number of authors (Buzacott 1999;Chen et al. 2008;Ernst and Cohen 1990;Flores and Whybark 1987;Ng 2007;Partovi and Burton 1992;Ramanathan 2006;Zhou and Fan 2007) have considered the use of multiple criteria, such as the certainty of supply, the rate of obsolescence, the lead time, costs of review and replenishment, design and manufacturing process technology, and substitutability.Accordingly, multi-criteria classifications have been developed.Various multicriteria methodologies have been considered, including weighted linear programming, analytic hierarchy process (AHP), and operations-related groups (ORG).An alternative for using multi-criteria methodologies is to use multiple way classifications, e.g., a two-way classification by purchase cost and demand volume.
Our inventory cost analysis will show that, for achieving cost optimal (or cost-service efficient) solutions, a single criterion is sufficient.However, that criterion does take four system parameters into account: demand volume, holding cost (purchase price), shortage cost (criticality), and average order quantity.Alternatively, a four-way classification could be used but this would not produce gains in terms of inventory cost and would be more difficult to implement.
With one exception, to the best of our knowledge, ABC classification methods have not been proposed or analyzed from an inventory cost perspective.This exception is a study by Zhang et al. (2001), who do propose a classification criterion (given in section 3, after notations are introduced) based on inventory theory.The derivation of this criterion is based on the assumption that service is measured by the probability that an order arrives on time, also known as the cycle service level.
In this paper, we will instead measure service as the fill rate, i.e., the fraction of demands that are satisfied directly from stock on hand.This is the most common way of measuring service (see, e.g., Axsa ¨ter 2006;Silver et al. 1998), and is also used by Zhang et al. (2001) to evaluate their criterion (and other proposed inventory control heuristics).The main advantage of using the fill rate is that it directly reflects the service as experienced by the customers.We will derive a new criterion based on the fill rate objective, and contrast it to the criterion of Zhang et al. (2001) and the traditional demand volume and demand value criteria.

A New ABC Classification Criterion
In practice, inventory managers usually try to minimize cost while maximizing the service level.The most common way (see, e.g., Silver et al. 1998) to define the service level is as the fraction of demand that is satisfied directly from stock on hand; the socalled fill rate.For a multi-SKU inventory system, the average fill rate over all SKUs can be calculated (see, e.g., Thonemann et al. 2002) as the weighted average of the single-SKU fill rates, where the weights are the fractions of demands for the different SKUs.
Such a system with a mixed cost-service objective is difficult to analyze in general.For this reason, inventory theory is dominated by a cost approach rather than a service approach.In a cost approach, instead of including a service restriction, missed demands incur a penalty cost and the objective is to minimize the total cost including inventory and penalty costs.The practical validity of this alternative cost approach lies in the fact that it leads to the same set of costservice efficient solutions as the corresponding service approach (see, e.g., Silver et al. 1998).In this section, we will use a cost approach to derive a new criterion for ranking SKUs.
So, we consider a multi-SKU inventory system where the objective is to minimize the total cost, consisting of inventory holding costs (per SKU and per time unit) and penalty/shortage costs per backordered demand.We do not assume a specific reorder policy, but will show that our analysis and the resulting criterion apply to both reorder point, reorder quantity (R, Q) policies as well as reorder point, order-up-to-level (s, S) policies, and for both continuous and periodic review.These are the most commonly used inventory policies.The criterion can also be used for any type of demand distribution, although we will consider only Normal and Gamma distributed demand in our numerical investigation.
We introduce the following notation.Note that ''(average)'' is added for the order quantity as it may vary for an (s, S) inventory policy.
N: number of SKUs b i : penalty cost (per backordered item) for SKU i CSL i : cycle service level for SKU i D i : demand per unit time for SKU i FR i : fill rate for SKU i h i : inventory holding cost (per item per time unit) for SKU i L i : lead time for SKU i Q i : (average) order quantity for SKU i SS i : safety stock (average inventory level just before an order arrives) for SKU i.
The total cost for all SKUs can be expressed as Here, the three terms represent the safety stock cost, the cycle stock cost and the shortage cost, respectively.As is shown in Appendix A, minimizing the total cost leads to the following approximate newsboy-type optimality condition for each SKU: Note from (2) that the approximately optimal cycle service level can be negative (because of the underlying approximation) for ''extreme'' cases with very low demand as well as backorder costs.However, this does not affect the general applicability of the ABC method that we will develop, as we will only use (2) to rank the SKUs and not to actually set their service levels.
We remark that for normally distributed demand and a continuous review (R, Q) policy, the safety stock corresponding to (2) can be calculated using the wellknown ''safety factor'' rule as where safety factor k i can be calculated using the inverse distribution function for standard normal demand (available in Excel) at CSL i .
Condition (2) has the following intuitive costbalance explanation.Holding an SKU of type i in safety stock for the length Q i D i of a replenishment cycle leads to a holding cost of h i Q i D i per unit of SKU.However, if a cycle ends with a stock-out then each unit of safety stock prevents a backorder at cost b i , and hence the expected backorder cost reduction is b i (1 À CSL i ).So, (2) balances holding and backorder costs optimally.
Condition (2) implies that the cycle service level for an SKU is increasing in the ratio b i D i h i Q i .Based on this result, we propose the following new ABC classification method: rank the SKUs based on the ratio b i D i h i Q i in descending order and fix the cycle service level per class.In other words, an SKU with higher optimal CSL (or higher b i D i h i Q i Þ gets a higher rank.Because this new ranking criterion and classification are based on a cost minimization approach, we refer to them as cost criterion and cost classification. An advantage of the cost criterion, compared with the common criteria and to the alternative proposed by Zhang et al. (2001), is that it takes shortage cost (or criticality) into account.This implies that it can also be used in situations, such as for spare parts management, where criticality is essential.
In common with the demand value and demand volume classifications, the cost method also ranks SKUs higher if the demand rate is larger.In addition, higher holding cost or order quantities induce a decrease in the values of the ratio, which is an intuitively appealing property from an inventory cost point of view.Note that as a higher holding cost typically indicates a higher purchase price, this holding cost effect is opposite to the price effect of the demand value classification.Due to this ''reverse'' use of prices, we expect the demand value classification to perform worse than the demand volume classification (that ignores prices) and much worse than the cost classification.The results from an empirical investigation in section 4 will confirm this.
The criterion of Zhang et al. (2001) . As our cost criterion, this one ranks an SKU higher if the demand rate is larger or if the holding cost is smaller.However, the holding cost is squared and hence its effect more dominant.Furthermore, the order quantity has no effect on rank, but the lead time does.So, aside from the inclusion of criticality, there are other major differences between our criterion and that of Zhang et al. (2001).Section 4 will show that our criterion performs significantly better than that of Zhang et al. (2001), and that both outperform the traditional criteria.
It is interesting to note that, despite the fill rate objective that was translated into a penalty cost per backorder, the optimality condition (2) fixes the cycle service level rather than the fill rate of an SKU.Fixing stockout probabilities per class instead of fill rates has the important advantage of easier implementation.Transforming a fill rate into a reorder point involves the use of complicated inverse loss functions (see, e.g., Axsa ¨ter 2006), whereas the inverse distribution function is sufficient for linking the stockout probability to the reorder point.Indeed, inventory planning tools sometimes do not allow the use of fixed fill rates.SAP R/3, for instance, is restricted to the use of stockout probabilities, although the more advanced planning and optimization tool SAP APO does offer the option to fix fill rates as well.

Empirical Data
The empirical investigation is based on three real life datasets.Dataset 1 is from a warehouse supplying spare parts globally for the installed base of machines that are used in the textile industry.Dataset 2 is from a retailer that sells bike and car parts and accessories.Dataset 3 is from a retailer that sells do-it-yourself products.Table 1 shows the size of these datasets at well as key statistics for demand, lead time, and purchase price.
To apply and evaluate the different classification schemes, we also need information on the order quantities, demand variations, and holding costs.All datasets contain information on the order quantity.Datasets 1 and 2 also contain a 24-months demand history for each SKU that was used to calculate the Note that the lead time is constant and small across all SKUs for Dataset 2, which is because all orders are delivered from a large warehouse within the same country in two days.
standard deviation of demand per month, which was then converted to the standard deviation of lead time demand.No demand histories are available for dataset 3, but we were able to estimate standard deviations based on the current reorder points (that are given) by assuming that a safety factor of two was used for all SKUs.
There is no information on the (annual) holding cost rate and therefore it is not possible to multiply purchase prices by that rate to obtain holding costs.However, this is not a problem since the holding cost rate is only a scaling factor and therefore any value will lead to exactly the same relative cost results (per cent cost increase of one method compared with another).Therefore, and also because relative cost results are easier to interpret and compare across datasets, we will present relative instead of absolute cost figures in what follows.

Experimental Setting
All three companies from which the datasets were obtained apply a continuous review (R, Q) inventory policy, which we also do in our numerical investigation.As is common in the literature (for safety stock calculations), order quantities (Q) are assumed to be fixed as they often result from constraints on, e.g., minimal order or packet size.However, by using different classes and corresponding optimal class service levels (of which the calculation is discussed later), the various ABC criteria lead to different order levels (R).That is, they distribute the safety stock in different ways over all SKUs.Because firms typically do not plan for orders to come in late, we restrict the focus on positive safety stocks by not allowing the cycle service level (i.e., the probability that an order arrives on time) of any class to be less than 50%.
We use the Normal distribution as well as the Gamma distribution to model lead time demand.The Normal distribution is the most widely applied distribution for inventory control.However, it may not be suitable for SKUs with slow moving demand, since such SKUs typically have large coefficients of variation, resulting in a considerable probability that demand is negative when using the Normal distribution (with the correct mean and variance).Therefore, we also consider the nonnegative and skewed Gamma distribution, which has been suggested by many others (e.g., Aviv and Federgruen 2001;Bagchi et al. 1986;Das 1976) as a suitable alternative to the Normal distribution for modelling lead time demand.
We compare four different ranking criteria: demand value, demand volume, Zhang et al., and cost.Each criterion is applied with three and six classes.With three classes, the sizes are determined using the commonly applied rule that classes A, B, and C contains about 20%, 30%, and 50% of all SKUs (see, e.g., Jacobs et al. 2009;Slack et al. 2007;Stevenson 2007-the recommended percentages do vary somewhat).As this implies that class sizes increase with a reasonably constant factor, we applied this logic (and round to integer numbers) to obtain class sizes of 4% (A), 7% (B), 10% (C), 16% (D), 25% (E), and 38% (F) when there are six classes.We remark that we also experimented with different (methods for setting) class sizes, but that did not lead to (consistently) better results or different results on the comparative performance of the four criteria.
For each criterion and number of classes, we (use the Solver tool in Excel to) find the cycle service levels per class that minimize the total inventory cost for all SKUs, whilst ensuring a certain overall (i.e., over all SKUs) target fill rate.Conversion of cycle service levels to order levels (R) is straightforward by using the inverse distribution function.Calculation of the fill rate for an SKU at a given service level can be done using the results of Appendices A and B (for Normal and Gamma demand).The overall fill rate is the average over all SKUs weighted by their annual demand rates.The specific target fill rates that we consider are 95% and 99%.
Besides calculating the minimum cost solution for each of the criteria and either three or six classes, we also calculated the optimal solution without a restriction on the number of classes.Although this solution is obviously not implementable in an ABC framework, it does provide a useful lower bound on the cost that can be used as a benchmark.For ease of presentation, we will refer to this solution as the optimal solution in section 4.3.

Results
Since order quantities are constant and not influenced by the classification criterion, cycle inventory costs are the same for each criterion.Therefore, we will present figures for the relative safety stock cost only.
It turns out that the results for Normal and Gamma demand are very similar.Therefore, we first discuss the results for Normal demand extensively, followed by a shorter discussion on Gamma demand.
Tables 2-4 summarize the relative cost performance, measured by the percentage increase in the safety stock cost compared with the optimal solution (without a restriction on the number of classes), for Normal demand.Detailed results with optimal cycle service levels and corresponding fill rates for all classes are provided in Appendix C for Dataset 1. Similar results were obtained for the other two datasets.Note that for Dataset 2 (Table 3) only the results for a 99% target fill rate are presented.The reason for this is that due to the small lead time of 2 days for all SKUs (see also section 4.1), a 95% (or in fact even a 97%) fill rate is achieved with zero safety stock (i.e., a cycle service level of 50%) for all SKUs.
Although the exact figures vary per dataset, the qualitative results are the same.Firstly, the new cost criterion is the clear winner.The criterion from Zhang et al. (2001) also performs much better than the two traditional criteria.Indeed, the performance of the traditional criteria can be described as dramatic, with the worst performance for the most commonly used demand value criterion.Please recall from section 3 that we expected the demand value criterion to have the poorest performance, as it ranks more expensive SKUs higher whereas, from a cost perspective, they should be ranked lower.
Another result for the cost criterion is that there is a considerable cost reduction that results from using six instead of three classes.Doing so reduces the relative safety stock cost (compared with the optimal solution) from 31% to 10%.The small latter figure also indicates that six classes are sufficient to obtain a nearly optimal solution.
Recall from section 1 that there is no agreement in the literature on which class should get the highest service level when the two traditional ranking criteria are used.The detailed results in Appendix C and similar results for the other two datasets offer an explanation for this lack of agreement, since neither the demand volume nor the demand value criterion have a consistent positive or negative ''trend'' in the cycle service levels or fill rates over the classes.The new cost criterion ''by design'' does show the cycle service level to be highest for class A, then B, and so on.Note, however, that this does not necessarily imply the same pattern for the fill rates, as those depend on order quantities as well.
As mentioned at the start of this section, the results for Gamma demand are very similar.To illustrate this, we present the summarized results for Dataset 1 in Table 5.
The cost criterion performs best for Gamma demand as well, and six classes are sufficient for obtaining a solution that is very close to optimal.All other findings for Normal demand apply to Gamma demand as well.

Conclusion and Extensions
In this paper, we have studied ABC classifications where, as is the standard approach in practice, service levels are fixed per class.Based on well-known results from the inventory theory, we proposed a new cost criterion for ranking SKUs: based on b i D i h i Q i , where b i is the shortage cost (i.e., the criticality), D i is the demand rate, h i is the inventory holding cost, and Q i is the order quantity for SKU i.
The cost criterion can be applied using the following simple steps.
1. Rank all SKUs in descending order of b i D i h i Q i .2. Divide the SKUs into classes A, B, and so on.Our results showed that using increasing class sizes of 20%, 30% and 50% for three classes (as is usual) and 4%, 7%, 10%, 16%, 25% and 38% for six classes works well.3. Fix the cycle service level for each class, where A should have the highest service level, followed by B, and so on.
An important advantage of the cost criterion over those discussed in the literature is that criticality is taken into account.Moreover, since criticality is com- bined with other relevant parameters into a single criterion, more complex multi-criteria methods can be avoided.
In an extensive numerical experiment using three real life datasets, we compared the cost criterion with the traditional demand value and demand volume criteria and to another criterion proposed by Zhang et al. (2001).As these other methods do not take criticality into account, this was not considered in the experiment.The cost criterion consistently outperformed all other methods across datasets, target service levels (95% and 99%), and types of demand distribution (Normal and Gamma) in minimizing the safety stock cost.Zhang et al.'s criterion also performed reasonably well, and certainly much better than the two traditional criteria.Both the demand value and the demand volume criteria more than doubled the safety stock cost in all cases considered, with the most common demand value criterion providing the worst performance.
Based on these results, we recommend the use of the cost criterion in combination with fixed cycle service levels for each class.One should keep in mind that our focus was purely on minimizing inventory costs whilst maximizing service.If classification is also used for other reasons, e.g., to determine the types of inventory policies used, the frequency of stock counts or the frequency of orders to be placed, then other performance measures may come into play.Nevertheless, we feel that the size of the potential savings warrants consideration of the newly proposed classification method in these situations as well.This leads to an interesting direction for further research, namely to evaluate the cost classification method in such settings.
Another avenue for future research is to develop and test rules of thumb for how to set the cycle service levels for the various classes when applying the cost criterion.Simple rules may be preferred in practice over finding the optimal cycle service levels as was done (in Excel) for our numerical experiment.Based on (2), a logical rule is to set the stockout probability (1-cycle service level) for each class proportional to the reciprocal of the average value of the criterion for that class.However, extensive testing is needed.

Appendix A. Inventory Cost Analysis
In this Appendix, we will derive (2) as the approximate optimality condition for the reorder level.We start by doing so for the continuous review (R, Q) policy, and then consider the continuous review (s, S) policy and finally the periodic review (s, S) policy.

Continuous Review (R, Q) Policy
The continuous review (R, Q) policy triggers an order of size Q when the inventory level drops to R, assuming continuous demand.With this definition, we can express (see section 5.10 of Axsa ¨ter 2006) the fill rate as and the safety stock as Using these expressions, the average cost per time unit (1) can be rewritten as To avoid confusion of (A1) with cost expression (5.69) in section 5.10 of Axsa ¨ter (2006), we remark that we use notation m instead of m 0 for the expected lead time demand, and that our loss function G(.) can apply to any distribution and not just standard normal demand.The latter also explains the standard deviation terms in expression (5.69) of Axsa ¨ter (2006).
It is common in the literature (see section 5.10 of Axsa ¨ter 2006) to replace the exact cost by a simpler approximate cost expression.If there are relatively few backorders (as is typical in practice where target service levels are usually at least 95%) and if Q is relatively large (as is again typical to avoid extensive ordering costs and because of minimum required order quantities), then the following expression is an accurate approximation of (A1): Note that this cost approximation is easy to interpret.The first part of (A2) is the approximate safety stock cost.The second part represents the average cycle holding cost from ordering Q items at a time.The average number of orders placed per time unit (year) is D/Q and the expected backorder when an order arrives is approximately G(R), leading to the third part.
It is easy to verify that C is convex in R and hence the optimum reorder point is obtained by setting the derivative to zero, which gives or equivalently (see also section 5.10 of Axsa ¨ter 2006 for normally distributed demand) Since F(R) is equal to the cycle service level, this condition is equivalent to (2).

Continuous Review (s, S)
The (s, S) policy triggers an order whenever the inventory position drops to or below s, replenishing the inventory position to S. Since the (s, S) policy is equivalent to an (R, Q) policy with R 5 s and Q 5 S À s for continuous demand, we only need to consider discrete demand here.
In fact, the key difference between discrete and continuous demand is that the inventory position can ''undershoot'' the reorder level.If this undershoot is included in the lead time demand, then the above analysis for (R, Q) policies still holds.This can be done by adding the expected undershoot to m, and by redefining f(.) and F(.) so that they correspond to the distribution of lead time demand plus undershoot.

Periodic Review (s, S)
The periodic review policy only reviews stock at the beginning of each review period (of T time units).Therefore, as for the continuous review (s, S) policy under discrete demand, the inventory position may drop below the reorder point s.This undershoot can be incorporated in the same way as discussed for the continuous review (s, S) policy.
We note that, in the literature (see, e.g., Axsa ¨ter 2006), it is often assumed that all demand in a review period takes place directly after the replenishment decision.Under this assumption, the relevant undershoot distribution corresponds to demand over the ''full'' T time units, so that the distribution for lead time demand plus undershoot is that over L1T time units.

Appendix B. Calculating the Loss Function for Normal and Gamma Demand Distributions
As in Appendix A, we define the loss function G(.) as In this appendix we derive expressions for G(x) that are used in our numerical experiment, starting with normally distributed demand and followed by gamma distributed demand.

Normal Demand
Let us denote the Normal probability density and distribution functions by j(.) and F(.), respectively.As shown in section 5.3.4 of Axsa ¨ter ( 2006), the loss function for a normal distribution with mean m and s can be rewritten as This can easily be calculated in Excel.As for normal demand, it should be straightforward to apply this expression in Excel as the Gamma distribution function is available.However, we discovered bugs in this specific Excel function (for small means and large variances) and had to resort to FOR-TRAN for these calculations.

Gamma Demand
We introduce the following notation D: Demand (per time unit) L: Lead time (in time units) m: Expected lead time demand f(.): Probability density function of lead time demand F(.): Probability distribution function of lead time demand We further define the loss function G(.) Let us denote the Gamma probability density and distribution functions with parameters b and c by f c, b (x) and F c, b (x), respectively.The probability density function ¼ bc À R þ RF c;b ðRÞ À bcF cþ1;b ðRÞ: Appendix C. Detailed Results for Dataset 1 and Normal Demand

Table 1
Descriptive Statistics for the Three Datasets

Table 2
Summarized Results for Dataset 1 and Normal Demand Table 4 Summarized Results for Dataset 3 and Normal Demand

Table C1 Detailed
Results for 3 Classes and a 95% Target Average Fill Rate Table C2 Detailed Results for 6 Classes and a 95% Target Average Fill Rate Table C3 Detailed Results for 3 Classes and a 99% Target Average Fill Rate Table C4 Detailed Results for 6 Classes and a 99% Target Average Fill Rate