Design of a near-optimal generalized ABC classificationfor a multi-item inventory control problemvan Wingerden, E.; Tan, T.; van Houtum, G.J.J.A.N.

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Design of a near-optimal generalized ABC classification for a multi-item inventory control problem

E. van Wingerden, T. Tan, G.J. Van Houtum,

Beta Working Paper series 494

BETA publicatie WP 494 (working paper) ISBN ISSN NUR

Eindhoven February 2016

Design of a near-optimal generalized ABC classification for a

multi-item inventory control problem

E. van Wingerden, T. Tan∗, G.J. Van Houtum,

Eindhoven University of Technology, School of Industrial Engineering, the Netherlands

Abstract

In this paper we consider a multi-item, single-location inventory control problem. We lookat how to design a generalized ABC classification scheme to minimize the inventory investmentcosts while satisfying an aggregate fill rate constraint by considering four different aspects ofa classification: the classification criteria, the number of classes, the class sizes, and the targetfill rate per class. Using multiple alternatives per aspect and considering all combinations,we determine the best combination of those four aspects. The results are measured againstthe system approach, which specifies a separate target fill rate for each SKU and therefore isoptimal but more difficult to manage. We find near-optimal results by applying our approachon empirical data. The most important insight that we generate is that the best class sizes -which need to be carefully decided - depend on the target aggregate fill rate whereas commonused rule of thumbs to determine class sizes may lead to large inventory investment costs.

Keywords: Inventory control, classification, SKUs, system-oriented service constraints, greedy heuristics

1. Introduction

A proper management of the inventory is of great importance for many companies as they have

service contracts or want to provide a certain service to their customers, often measured by the

aggregate fill rate (see Thonemann et al., 2002). For many practical cases, the number of stock

keeping units (SKUs) is too large to apply SKU-specific inventory control methods (Buxey, 2006).

Although some companies implement SKU-specific inventory control methods, there are many

companies for which this is not possible due to limited functionality of their ERP system or due to

a lack of knowledge about this topic within the company. As a practical solution, many companies

follow an approach known as an ABC classification (see e.g. Ramanathan, 2006; Hatefi and Torabi,

2015; Flores and Whybark, 1987; Chu et al., 2008; Eilon and Mallya, 1985; Liu et al., 2015): They

often categorize their inventory into three classes and manage the SKUs within a class in the same

way, Class A being the most important category receiving the closest attention and Class C being

the least important one. In this way they are basically able to differentiate between different SKUs

while still having a simple inventory control policy. Nevertheless, the classification of SKUs is done

by rules of thumb: E.g. 20% of SKUs that have the highest annual dollar volume are grouped

∗Corresponding author, Tel.: +31 40 247 3950, E-mail: T.Tan@tue.nl

1

under Class A. Such oversimplification is prone to significant sub optimization. In this paper

we investigate whether this can be improved by a -still simple- ”generalized” ABC classification

approach, in which the classification is made more carefully. In particular, we attempt to provide

an answer to our main research question by making use of two different spare parts data sets of a

maintenance company: How should one design a generalized ABC classification in order to obtain

the lowest inventory investment costs while ensuring a target aggregate fill rate?

When looking at various ABC classifications, there are four aspects that together determine

the performance of the generalized ABC classification. First of all, as different SKUs have different

characteristics, and can be of different importance to the company, SKUs can be ranked based on one

or more criteria. There have been many authors that propose different criteria to rank the SKUs.

The most common criterion is the annual dollar volume (see e.g. Silver et al., 1998; Stevenson,

2007; Nahmias, 1997). However, this criterion does not set the relation with the resulting inventory

costs. As inventory costs can be a large fraction of the total costs, Ramanathan (2006), Bacchetti

and Saccani (2012), Bacchetti et al. (2013), Zhou and Fan (2007), Teunter et al. (2010), Ng (2005),

Babai et al. (2015), Hollier and Vrat (1978), Zhang et al. (2001) and Ernst and Cohen (1990)

propose classifications for SKUs where inventory costs are taken into consideration. Multicriteria

classifications, where SKUs are classified into three classes based on multiple criteria, have also been

proposed (see e.g. Flores et al., 1992). Moreover, in practice we commonly observe two-dimensional

classifications where each SKU is classified twice based on each criterion and the final classification

is obtained by taking the combination of the previous two classifications. In this way, often nine

classes are being used (see e.g. Duchessi et al., 1988; Flores and Whybark, 1987). See Figure 1 for

an example of a two-dimensional classification.

Figure 1: Example of Price and Demand classification

The second aspect we consider is the number of classes used to classify the SKUs. Although

most of the previously mentioned literature considers three classes only, it is possible to extend this

to a larger number of classes. A larger number of classes would lead to a reduction of inventory

investment costs and increase in performance as three classes may not capture the differences

between SKUs sufficiently well (Lee, 2002). However, an increase of the number of classes also

2

makes it more difficult to manage the SKUs. Based on empirical data, we show that having three

classes already gives significant improvement compared to two classes, and that four classes is a

good choice to balance the costs and difficulty of implementing. A further increase in the number

of classes only results in limited cost decreases.

Thirdly, one needs to decide on the class sizes and thus in which class each SKU needs to be

placed. For ABC classifications with three classes, a common rule of thumb (see e.g. Jacobs et al.,

2009; Slack et al., 2007; Stevenson, 2007; Nahmias, 1997) is to have about 20% of the SKUs in class

A, 30% in class B and 50% in class C. However, when the classes are managed by setting different

service levels per class, there are no clear guidelines on how to rank items. There are authors that

argue that class A SKUs should have the highest service target to avoid frequent backlogs (see

e.g. Armstrong, 1985; Stock and Lambert, 2001), whereas other authors claim that class C SKUs

should have the highest service as dealing with stockouts is not worth the effort (see e.g. Knod

and Schonberger, 2001). Nevertheless, these suggestions are not considering the relative size of

each class and the link with the desired service or classification criteria used. To the best of our

knowledge, besides this rule of thumb, there is no literature on this topic. We show that for the

class consisting of the most expensive SKUs, which gets the lowest service level due to their large

impact on the inventory investment costs, the best class size depends on the aggregate fill rate

target. Large cost differences of around 290% are possible by only varying the size of each class.

Finally, one needs to decide on how to manage each class of parts. In this paper this is done by

setting a target fill rate for each class. Although most literature only considers the classification

criteria used to classify the SKUs and does not relate to the resulting inventory costs, there are

a few authors that do consider the resulting inventory costs (see e.g. Teunter et al., 2010; Babai

et al., 2015). The first authors to combine target setting with a classification are Teunter et al.

(2010). However, they stock all SKUs as they only consider strictly positive target fill rates. As we

are interested in the overall performance, it is not always desired to stock all SKUs (meaning that

some SKUs might have zero fill rate), especially when there are large cost differences between the

SKUs (Sherbrooke, 1968).

As these four aspects cannot be considered independently of the others (e.g. the target service

level of a class depends on how many classes there are), we examine each combination of these four

aspects, which to the best of our knowledge has not been done before. We consider four possible

classification criteria, vary the number of classes, use enumeration over possible class sizes, and

set the optimal target fill rate per class given all other aspects. This approach is applied on two

different data sets consisting of real life spare parts data of a maintenance company responsible for

the maintenance of ground support equipment. We use this data to compare the resulting inventory

investment costs to the result of a system approach, which stands for having a separate class for

each SKU (see e.g Sherbrooke, 1968; Van Houtum and Kranenburg, 2015).

The main contribution of this paper is that we propose a generalized ABC classification scheme

that takes the above four aspects jointly into account, and provide a structured approach to design a

classification to minimize inventory investment costs while meeting the aggregate fill rate constraint.

Using real life spare parts data and by comparing multiple classification criteria, we show that when

3

these four aspects are combined, the resulting inventory investment costs can be within one percent

of the result of a system approach, which is shown to give (near) optimal results at the expense of

more complicated inventory control. This approach can also be applied to other criteria, demand

distributions, and inventory policies. Moreover, we show that having the right class sizes, which to

the best of our knowledge has not been discussed in the literature before, can have a big impact

on the performance. We compared the best class sizes with class sizes determined by the common

rule of thumb where about 20% of the SKUs are in class A, 30% in class B and 50% in class C (cf.

Teunter et al., 2010). Cost differences up to 290% are possible and more likely to occur for higher

aggregate fill rate targets. Because we use enumeration to determine the optimal target per class,

our approach can become time consuming when the number of classes increases. Therefore, we also

propose an algorithm that significantly decreases the calculation time for our classification scheme,

whereas the cost increase on average is below 2%.

The remainder of the paper is organized as follows: In Section 2 we introduce the model de-

scription. In Section 3 we give our approach on how to design a generalized ABC classification

scheme that considers all of the four aspects jointly. In Section 4 we first explain our real life data

set of spare parts, which we use for our numerical experiment. We then present the optimality gap

of the alternatives that we consider and we conduct some analysis to obtain more insights regarding

how to set the right class sizes and propose an algorithm to set the target fill rates per class in

a fast way. We conclude our paper and give some recommendations for future research in Section 5.

2. Model

We consider a single warehouse in which several SKUs are kept on stock. Whenever a SKU is

requested, it is immediately delivered from stock on hand, and backordered and delivered as soon

as possible when there is no stock on hand. The set of SKUs is denoted by I; the number of SKUs

is |I| ∈ N := {1, 2, . . .}. For notational convenience, the SKUs are numbered as 1, . . . , |I|. For

each SKU the stock is controlled by a continuous-review basestock policy, with basestock level Si

for SKU i. For each SKU i ∈ I, we assume that demand follows an independent Poisson process

with a constant rate λi (≥ 0), which is particularly plausible in the maintenance and spare parts

environment, but also in some retail environments. The total demand rate for all SKUs is denoted

by Λ =∑

i∈I λi. The replenishment lead time Li (> 0) is assumed to be constant. Each SKU has

a price of ci, and the total inventory investment costs are given by

C(S) =∑i∈I

ciSi, (1)

where S = (S1, . . . , S|I|) denote the vector consisting of all basestock levels. The fill rate for item

i, in steady state, is denoted by βi(Si). The (aggregate) fill rate in steady state is as follows:

β(S) =∑i∈I

λiΛβi(Si). (2)

4

Note that the calculation for the fill rate is as follows (see also Van Houtum and Kranenburg, 2015)

βi(Si) =

Si−1∑x=0

P {Xi = x} ,

P {Xi = x} =(λiLi)

x

x!e−λiLi , x ∈ N0.

The target aggregate fill rate for β(S) is given by βo. The objective is to minimize the total costs

subject to an aggregate fill rate constraint. In mathematical terms the optimization problem is as

follows:(P) min C(S)

subject to β (S) ≥ βo

Si ∈ N0, ∀ i ∈ I

As managing each SKU individually can be very complex, especially for a large number of

SKUs, companies often resort to the use of ABC classifications. In the next section, we show how

we design a generalized ABC classification to obtain minimum inventory investment costs against

the aggregate fill rate target.

Although we made some assumptions on the demand process and the inventory control policy

our approach can be easily applied with other demand processes and/or inventory control policies

as this merely changes the evaluation of the fill rate and costs. The same steps of our approach

can still be applied to design the ABC classification.

3. Design aspects of a generalized ABC classification

In this section we propose a design of a generalized ABC classification scheme. As the four different

aspects cannot be considered separately, we consider each combination and their resulting inventory

investment costs. We first describe the different classification criteria we consider in this research.

Then we explain the number of classes we consider for each of the four different classification

criteria. Thirdly, we explain how we set the class sizes. Finally, we describe how we set the target

service level for each class. Willen we hier verwijzing naar de secties toevoegen?

3.1 Classification criteria

When considering a classification for SKUs, we first need to define the classification criteria that

enable us to rank the SKUs. Once we have ranked the SKUs we can take the other three aspects

into consideration. There has been extensive studies on how to classify SKUs, and recently there

were also authors that relate the classification with the inventory control (see e.g. Babai et al.,

2015; Teunter et al., 2010; Ng, 2005). The annual dollar volume - i.e. annual demand times price

- (ADV) criterion, is the traditional criterion for ABC classifications and found in many textbooks

(see e.g. Silver et al., 1998; Nahmias, 1997). However, if we follow the idea of a system approach

(Sherbrooke, 1986), parts with a high price end up with a lower service level, whereas parts with a

5

high demand end up with a higher service level. Therefore, ADV is a poor construction in terms

of inventory investment costs as demand and price work in opposite directions as under the system

approach. This was also observed by Teunter et al. (2010) and they proposed a classification based

on Demand/Price (D/P), which captures the idea of a higher service for cheap and fast moving

SKUs. They found that this classification criterion was the best performing criterion among the

criteria they considered.

In practice, two-dimensional classifications based on price and demand are often seen to manage

the inventory (see e.g Duchessi et al., 1988; Flores and Whybark, 1987), but they have not been

compared before. This two-dimensional classification classifies SKUs based on both price and

demand and therefore does not have similar problems as the ADV criterion. We compare this

two-dimensional classification to the previous mentioned methods. Note that we also compared

some other classification criteria as a ranking based on demand, leadtime demand, and price only,

but we omit them in our exposition as they did not give better results. Therefore, we only present

the results and findings of the following three criteria:

1. Annual Dollar volume (ADV) (one-dimensional)

This single criterion classification is the most common criterion. The demand rate of each

SKU is multiplied with the price of each SKU and then ordered based on this annual dollar

volume. The SKUs are ranked in descending order, where SKUs with the largest ADV are

considered class A SKUs.

2. Demand/Price (D/P) (one-dimensional)

This classification is proposed by Teunter et al. (2010), and shown to give the best performance

related to inventory costs. The SKUs are classified based on their demand rate divided by their

price. They are then ranked in descending order, where the top group of SKUs, consisting

of fast moving and cheap SKUs, are considered class A SKUs which should get the highest

service level.

3. Price and Demand (P&D) (two-dimensional)

This classification is often found in practice: Each SKU is classified based on both price

and demand. In practice, often nine classes are used, see Figure 1 for an example of this

classification.

3.2 Number of classes

When considering ABC classifications, often three classes per dimension are considered: Class A,

class B, and class C. However, it is also possible to consider different numbers of classes when

classifying SKUs. As less classes will most likely end up with higher investment costs, it makes the

problem less complex. Increasing the number will most likely give better results because there are

more options available, but this will increase the complexity of the problem. Therefore, there is a

trade-off between performance and manageability. Graham (1987) and Silver et al. (1998) mention

that up to six classes can be used as more classes increases the complexity. Using six classes,

6

Teunter et al. (2010) showed that good results can already be obtained. For the two-dimensional

classification, a matrix approach is common with 9 classes Duchessi et al. (1988). Having a 4 by 4

matrix would lead to 16 classes which becomes too difficult and costly to manage.

In our numerical experiment we vary the number of classes between 2 and 6 for the one-

dimensional classifications. For the two-dimensional classification we consider a 2 by 2 matrix and

a 3 by 3 matrix. Moreover, we add the results of the system approach, where each SKU is put in

a separate class. By considering these different class sizes, we obtain in total 14 combinations of

ABC classification criteria and number of classes. Given the number of classes, we introduce a set

of classes K where classes are numbered (1, . . . , |K|).

3.3 Class sizes

For each of the above mentioned combinations, we need to decide on the size of each class, which in

turn determines the class each SKU will be placed in. There has not been much research about the

best class sizes for ABC classifications. Typically a rule of thumb for the ADV classification is used

(see e.g. Jacobs et al., 2009), in which 20% of the SKUs are class A, 30% class B, and 50% class

C. Later on, Teunter et al. (2010) also used this rule of thumb for the D/P classification. One of

our goals in this paper is to find out whether different class sizes have an impact on the inventory

investment costs.

Therefore, we use treshold values to determine which class each SKU is placed into. For example,

in the case we have three classes and SKUs are ranked based on their price and the price of a SKU is

larger than all treshold values, it will be placed in class C (unlike the logic of the ADV classification).

If the price is smaller than the smallest treshold value it will be placed in class A. When there are

more than three classes, or multiple dimensions, there are also more treshold values but the logic

remains the same.

We then use enumeration over the treshold values, which we set equal to the x−th percentile.

With x ∈ {∆, 2∆, . . . , d100/∆e∆}, for each treshold value. Based on the treshold values, we

determine ki, i ∈ I, representing the class to which SKU i belongs.

Note that the treshold values are sorted in ascending order as they must be increasing in size,

and that if two treshold values are equal, this is the same as having fewer classes.

3.4 Target setting

Given the set of classes K and the class to which each SKU i ∈ I belongs, noted by ki, , we need

to decide on how to manage each class of SKUs. It is common to set different service targets for

each class, θk, to manage each class (Teunter et al., 2010). Although there are also other ways to

manage the inventory of each SKU in a class, for instance by choosing Si such that the distance to

the service level target of that SKU’s class, θki , is minimized, we follow the more common approach

of Teunter et al. (2010), where the fill rate has to be greater than or equal to the target for each

SKU in that class. We thus need to decide on θk, k ∈ K, the target fill rate per class. We set the

value of Si, i ∈ I, equal to the lowest integer that satisfies this fill rate target, θki , for SKU i.

7

The corresponding problem (C) is then as follows:

(C) min C(S)

subject to β (S) ≥ βo

Si = min{x|βi (x) ≥ θki , x ∈ N0}, ∀ i ∈ Iθk ∈ [0, 1) , ∀ k ∈ K

Different than what Teunter et al. (2010) does, we allow to have zero stock of SKUs. Sherbrooke

(1968) already showed that when we are mainly concerned with the overall performance it is

sometimes better to take more risk for the very expensive SKUs because their inventory investment

costs are very high, and reduce the risk on many other cheap SKUs.

To obtain the solution that minimizes the inventory investment costs of problem (C), we first

look at each class separately, as the classes are independent due to the fact that the class to which

each SKU belongs is known at this stage. Let

Sk = {Si}i∈I, ki=k,

be the vector of basestock levels for each SKU i in class k, and

Λk =∑

i∈I, ki=kλi,

be the total demand rate for class k. The aggregate fill rate for class k, βk(S), and the inventory

investment costs for class k, Ck(Sk), are calculated as follows:

βk(Sk) =∑

i∈I, ki=k

λiβi(Si)

Λk

Ck(Sk) =∑

i∈I, ki=kciSi.

The following is applied for each class k ∈ K, with ε being a small positive value close to zero:

1. Set θk := 0

Si = 0 ∀ ki = k, i ∈ ISk = {Si}i∈I, ki=kCompute Ck(Sk) and βk(Sk)

Ek :={(

Sk, Ck(Sk), βk(Sk), θk)}

2. θk :=min(βi(Si) : i ∈ I, ki = k)

For all i ∈ I, ki = k : If βi(Si) = θk, then Si := Si + 1

Sk = {Si}i∈I, ki=k

3. Compute Ck(Sk) and βk(Sk);

Ek = Ek ∪{(

Sk, Ck(Sk), βk(Sk), θk)}

8

4. If βk(Sk) ≥ 1− ε, then stop, else go to 2

After this procedure, we have a solution vector per class, Ek, consisting of the base-stock levels,

costs, fill rate, and target fill rate. We then enumerate over all different combinations of solutions

per class, θk, and look for the combination that gives the lowest inventory investment costs while

satisfying the aggregate fill rate target.

4. Numerical experiment

In order to find which classification gives the best result for problem (P), we apply our approach

and perform a numerical experiment using empirical data. In Section 4.1 we give an overview of

the data we use for our numerical experiment. We show that the D/P classifications gives near-

optimal results with only a few classes in Section 4.2. Then in Section 4.3 we show the impact of

the class sizes on the performance. Finally we introduce a greedy ABC algorithm that reduces the

calculation time significantly in Section 4.4.

4.1 Data

To apply our approach to design a generalized ABC classification we use data of spare parts for

the maintenance of Ground Support Equipment (GSE) at an airport. The GSE consists of a wide

range of vehicles consisting of passenger cars up to specialised equipment such as de-ice vehicles

used to remove ice off the wings of airplanes. Because of the large variety of GSE, the maintenance

is split up over multiple groups. For our numerical experiment we use two different data sets, where

each data set consists of spare parts usage for a different group of GSE, named set 1 and set 2

from now on. Set 1 consists of 5988 SKUs, whereas set 2 consists of 1092 SKUs. Table 1 gives an

overview of the characteristics of both data sets.

The ratio between slow, medium and fast moving SKUs is quite similar for both data sets. There

Table 1: Data characteristics

Data set 1 Data set 2

Demand Price (e ) Leadtime Demand Price (e ) Leadtime

per year (days) per year (days)

per SKU per SKU

Min 0.10 0.01 1.00 0.02 0.06 1.00

Average 8.53 78.61 5.97 6.97 201.32 4.82

Max 610.50 10,950.00 111.00 566.50 9,500.00 7.00

are many SKUs that are requested only up to three times, and the top 5% of most frequently used

SKUs contribute to over more than 60% of the total demand. For data set 1, over 90% of all SKUs

are delivered within 3 weeks, and for data set 2 about 40% of all SKUs have a lead time of 1 or 2

9

days, based on contracts with suppliers. For all the other SKUs of data set 2 the lead times are set

at 7 days.

4.2 Choosing the best classification criteria

In Table 2 we present our finding regarding the inventory investment costs for the different classifi-

cations for data set 1. To calculate the inventory investment for the system approach we applied the

algorithm as described in Section 2.4.2 of Van Houtum and Kranenburg (2015). Similar findings as

in Table 2 are also obtained for data set 2; see Appendix A. The costs shown are the costs when we

choose the best class sizes and target fill rates. The outcome of the system approach is the optimal

solution where each SKU has its own class and target fill rate. The results regarding the ADV are

in line with the results of Teunter et al. (2010): We clearly see that the ADV classification gives

very poor results. We did not show the findings for the ADV classification for 5 and 6 classes as

they do not give more insights. Even increasing the number of classes in the ADV classification

does not lead to large cost reduction. An explanation for this is that both fast moving SKUs as

well as expensive SKUs can be put in the same class of SKUs according to this criterion. However,

from the inventory investment cost perspective, we prefer to stock less expensive SKUs over expen-

sive SKUs, because expensive SKUs bring higher inventory investment costs. Moreover, it is more

interesting to stock fast moving SKUs over slow moving SKUs because the contribution to the the

aggregate fill rate of fast moving SKUs is larger.

The classification that leads to the lowest inventory investment costs is the D/P classification.

In the system approach, a “biggest bang for the buck” approach is applied (see Section 2.4.2 of

Van Houtum and Kranenburg, 2015); the stock of SKUs that give the largest contribution to the

aggregate fill rate (which is mainly influenced by the demand rate of each SKU) relative to the price

is increased until the aggregate fill rate target is met. The D/P classification captures this trade-off

between the contribution to the aggregate fill rate and costs very well, therefore the SKUs in the

same class will end up with similar fill rate targets as if we would apply a system approach. We find

that by applying the D/P classification with four classes, and setting the best class sizes and target

fill rates, the resulting inventory investment costs are within a few percent of the optimal solution.

The P&D classification is also able to capture this trade-off, although it does not directly capture

this trade-off for each SKU and that explains the slightly higher inventory investment costs.

When we look at the number of classes used for the D/P classifications, we clearly see that

when only three classes are considered, already a significant gain with respect to two classes can

be obtained. Whenever the number of classes is increased, the performance can only increase as

well because the same solution as with less classes is always possible. However, after four classes,

the increase in performance is only limited. As an increase in the number of classes makes it more

difficult and costly to control the inventory policy of each of the four classes, for our data sets having

four classes is sufficient to get most of the gains, while avoiding difficult and costly management of

the SKUs.

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Table 2: Overall inventory investment costs for data set 1

Inventory investment costs Target aggregate fill rate

Classification Number of classes 80% 90% 97%

Annual Dollar Volume

2 e 36,905 e 39,891 e 42,472

3 e 32,926 e 38,128 e 42,346

4 e 32,887 e 38,073 e 42,159

Demand/Price

2 e 1,369 e 3,993 e 14,534

3 e 1,295 e 3,000 e 10,487

4 e 1,255 e 2,945 e 10,138

5 e 1,254 e 2,937 e 10,017

6 e 1,239 e 2,937 e 9,958

Price and Demand4 (2x2) e 1,618 e 3,632 e 12,441

9 (3x3) e 1,424 e 3,255 e 12,084

One class e 39,242 e 40,346 e 43,583

System approach e 1,154 e 2,894 e 9,871

4.3 The impact of class sizes on performance

When considering the size of each class, we would like to gain insights in the importance of choosing

the best sizes. As the D/P criterion gives the best overall results, we consider the impact of the

class sizes for this criterion only. There has not been much literature about the proper class sizes.

A common proposal is to have around 20% of the SKUs in class A, 30% in class B and 50% in class

C (Teunter et al., 2010). Based on our numerical experiment, our first key observation is that there

are large inventory investment costs differences when only changing the percentage of demand in

each class. In the best solutions found, there is always at least one class for which SKUs are not

stocked. This is always the class for which the value of D/P is the smallest, thus consisting of

very expensive slow moving SKUs. These SKUs have a very large impact on the total inventory

costs. Given the aggregate fill rate target, we know the fraction of demand that does not have to be

satisfied directly from stock on hand. Whenever the fraction of total demand in this class is larger

than the fraction of total demand that does not have to be satisfied from stock on hand directly,

it is impossible to find a feasible solution without stocking all SKUs in that class. We call this a

“forced presence” of stock for these SKUs. Therefore, we expect that the commonly used rule of

thumb, where the fraction of demand of this class of SKUs is set to 50% (see e.g. Teunter et al.,

2010), may lead to a forced presence of stock for these SKUs and thus a poor solution. Moreover,

we find that the fraction of demand in this class is always close to the maximum allowed fraction

of demand that does not result in a forced presence of the SKUs. Figure 2 shows the fraction of

total demand in this class for the best solution, against the inventory investment costs for different

aggregate fill rate targets when there are three classes. Thus, for higher aggregate fill rate targets,

the total fraction of demand in this class is smaller than for lower targets. In Table 3 we present

11

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

0,00% 5,00% 10,00% 15,00% 20,00% 25,00%

Inve

nto

ry in

vest

me

nt

cost

s

Percentage of demand in class C

80 percentaggregate fillrate

90 percentaggregate fillrate

97 percentaggregate fillrate

Figure 2: Impact of different class sizes on investment costs

our findings when we would use the commonly used rule of thumb, as suggested by Teunter et al.

(2010), to divide the SKUs over the classes. We compare these findings with the findings when we

set the best class sizes. Based on our data, even if the target fill rates are set optimally, huge cost

differences are found, even up to 290%. This difference can be explained by the forced presence of

class C SKUs. However, we should note that the cost differences between SKUs play an important

role in the magnitude of the impact. If there would be no cost difference, it would make less benefit

to optimize the class size. Moreover, if it is required to have strictly positive basestock levels (so

that all SKUs are really kept on stock), we do not have the forced presence of stock for the SKUs

and setting the best class size would have less impact.

4.4 ABC classification target setting

In order to apply our approach, we make use of enumeration to determine the optimal target setting

for each class, as described in Section 3.4. Unfortunately, when the number of classes is increas-

12

Table 3: Inventory investment costs for data set 1 based on D/P classification

Number of classesTarget aggregate fill rate

80% 90% 97%

3 best e 1,295 e 3,000 e 10,487

3 rule of thumb e 3,234 e 3,532 e 39,691

4 best e 1,255 e 2,945 e 10,138

4 rule of thumb e 1,314 e 4,620 e 39,591

ing, the run time is increasing exponentially; for the D/P classification with five classes and an

aggregate fill rate target of 90% the run times are around 45 minutes. Although these calculations

are not done on a regular basis, companies may be interested in sensitivity analysis to see how

the total inventory investment costs change with respect to the aggregate fill rate in order to set

their goals. Therefore, we propose a greedy ABC algorithm to set the target fill rates for each class

based on getting the “biggest bang for the buck”. We increase θk with a small step size, θstep, for

the class where we get the biggest increase in aggregate fill rate per euro invested. This ratio, Γk,

is defined as the difference in aggregate fill rate after increasing θk, divided by the difference in

inventory investment costs. Because we do not have to calculate all solutions for each class and use

enumeration, this results in significantly reduced run times. The greedy ABC algorithm is as follows:

1. Set θk := 0 for all k ∈ KSi = 0 for all i ∈ IS = (S1, S2, . . . , S|I|)

Compute C(S) and β(S)

2. For all k ∈ K:

(a) For all i ∈ I:

If ki = k: vi = min{x|βi (x) ≥ θki + θstep, x ∈ N0

}− Si

else, vi = 0

(b) vk = (v1, v2, . . . , v|I|)

Γk := (β(S + vk))− β(Sk))/(C(S + vk)− C(S))

3. m :=arg max{Γk : k ∈ K}S := S + vk

θk :=min{βi(S) : i ∈ I, ki = k} for all k ∈ K

4. Compute C(S) and β(S);

If βk(S) ≥ βo, then stop, else go to 2

The run-times using this algorithm are around 0.15 seconds when there are two classes up to around

980 seconds for six classes on a Pentium Intel(R) Core(TM) i5-2520M CPU with 4GB memory. We

13

find that savings in calculation time of around factor 20 are obtained for five classes, and increasing

for larger numbers of classes. The increase in costs compared to the optimal target setting, over

all different targets per class with more than three classes, is less than two percent on average.

Thus for a larger number of classes or a larger number of SKUs, the greedy ABC algorithm can

significantly speed up the calculation time of our approach whereas the cost increase is limited,

especially compared to the cost savings when coming from no classification at all.

5. Conclusions and suggestions for further research

In this paper we looked at how to design a generalized ABC classification to minimize the inventory

investment costs by combining four different aspects: Classification criteria, number of classes, class

sizes, target fill rates per class.

We use a structured approach to combine these four aspects, and using empirical data we

show that it is possible to obtain a solution within a few percent of the optimal solution by using

D/P as classification criterion with only four classes. Having proper class sizes turns out to be

of great importance; using the rule of thumb as proposed by Teunter et al. (2010), may lead to

up to around 290% higher inventory investment costs. This difference is explained by the forced

presence of stock of expensive and slow moving SKUs. We propose a greedy ABC algorithm which

significantly reduces run times, whereas the cost increase of using this algorithm is only limited to

a few percent on average.

For further research it may be interesting to find simple rules to determine proper class sizes

instead of enumeration, which may further reduce the run time of our approach. Moreover, we used

a basestock policy for each class to manage the inventory, whereas it might be interesting to apply

different policies for different classes. Especially if transport costs are involved for each order. By

including these costs and batching for one or more of the classes, it would be interesting to see

whether the D/P classification would still perform best. Note that although we did not investigate

this problem, our methodology can still be applied as the difference lies only in the evaluation of

the performance and costs. Another interesting point for further research is the target setting.

We assumed that each SKU at least has to meet the target for its class. However, it might be

interesting to see how the performance of the different classifications would change if we would set

the basestock levels such that the difference with the target is minimized instead. This may also

reduce the impact of forced presence of stock.

Acknowledgements

The authors gratefully acknowledge the support of the Netherlands Organisation for Scientific

Research.

14

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16

A. Inventory investment costs for data set 2

In this Section, we show the inventory investment costs as a result of our approach for data set 2,

for the different classification criteria. See Table 4 for the results.

Table 4: Overall inventory investment costs for data set 2

Inventory investment costs Target aggregate fillrate

Classification Number of classes 80% 90% 97%

Annual Dollar Volume

2 e 69,566 e 76,777 e 84,606

3 e 66,796 e 75,270 e 83,595

4 e 49,387 e 66,336 e 82,266

Demand/Price

2 e 1,457 e 5,393 e 27,082

3 e 1,062 e 3,684 e 17,152

4 e 1,062 e 3,499 e 16,576

5 e 1,062 e 3,475 e 16,489

6 e 1,062 e 3,473 e 16,479

Price and Demand4 (2x2) e 1,538 e 4,776 e 22,058

9 (3x3) e 1,266 e 4,243 e 19,493

One class e 76,052 e 76,810 e 84,612

System approach e 947 e 3,469 e 16,205

17

B. Percentage of total demand per class for data set 2

In Figure 3 the percentage of total demand per class for data set 2 is given in the case of four classes

with D/P as classification criteria. Class A consists of the SKUs which have the biggest value for

D/P.

Figure 3: Percentage of demand in each class for best settings with four classes

18

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Heidi Romero, Remco Dijkman, Paul Grefen, Arjan van Weele

378 2012 A Generic Material Flow Control Model for Two Different Industries

S.W.A. Haneyah, J.M.J. Schutten, P.C. Schuur, W.H.M. Zijm

Nr. Year Title Author(s)377 2012 Dynamic demand fulfillment in spare parts networks with

multiple customer classesH.G.H. Tiemessen, M. Fleischmann, G.J. van Houtum, J.A.E.E. van Nunen, E. Pratsini

376 2012 Paper has been replaced by wp 417 K. Fikse, S.W.A. Haneyah, J.M.J. Schutten

375 2012 Strategies for dynamic appointment making by container terminals

Albert Douma, Martijn Mes

374 2012 MyPHRMachines: Lifelong Personal Health Records in the Cloud

Pieter van Gorp, Marco Comuzzi

373 2012 Service differentiation in spare parts supply through dedicated stocks

E.M. Alvarez, M.C. van der Heijden, W.H.M. Zijm

372 2012 Spare parts inventory pooling: how to share the benefits? Frank Karsten, Rob Basten

371 2012 Condition based spare parts supply X. Lin, R.J.I. Basten, A.A. Kranenburg, G.J. van Houtum

370 2012 Using Simulation to Assess the Opportunities of Dynamic Waste Collection

Martijn Mes

369 2012 Aggregate overhaul and supply chain planning for rotables J. Arts, S.D. Flapper, K. Vernooij

368 2012 Operating Room Rescheduling J.T. van Essen, J.L. Hurink, W. Hartholt, B.J. van den Akker

367 2011 Switching Transport Modes to Meet Voluntary Carbon Emission Targets

Kristel M.R. Hoen, Tarkan Tan, Jan C. Fransoo, Geert-Jan van Houtum

366 2011 On two-echelon inventory systems with Poisson demand and lost sales

Elisa Alvarez, Matthieu van der Heijden

365 2011 Minimizing the Waiting Time for Emergency Surgery J.T. van Essen, E.W. Hans, J.L. Hurink, A. Oversberg

364 2012 Vehicle Routing Problem with Stochastic Travel Times Including Soft Time Windows and Service Costs

Duygu Tas, Nico Dellaert, Tom van Woensel, Ton de Kok

363 2011 A New Approximate Evaluation Method for Two-Echelon Inventory Systems with Emergency Shipments

Erhun Özkan, Geert-Jan van Houtum, Yasemin Serin

362 2011 Approximating Multi-Objective Time-Dependent Optimization Problems

Said Dabia, El-Ghazali Talbi, Tom Van Woensel, Ton de Kok

361 2011 Branch and Cut and Price for the Time Dependent Vehicle Routing Problem with Time Windows

Said Dabia, Stefan Röpke, Tom Van Woensel, Ton de Kok

360 2011 Analysis of an Assemble-to-Order System with Different Review Periods

A.G. Karaarslan, G.P. Kiesmüller, A.G. de Kok

359 2011 Interval Availability Analysis of a Two-Echelon, Multi-Item System

Ahmad Al Hanbali, Matthieu van der Heijden

358 2011 Carbon-Optimal and Carbon-Neutral Supply Chains Felipe Caro, Charles J. Corbett, Tarkan Tan, Rob Zuidwijk

357 2011 Generic Planning and Control of Automated Material Handling Systems: Practical Requirements Versus Existing Theory

Sameh Haneyah, Henk Zijm, Marco Schutten, Peter Schuur

356 2011 Last time buy decisions for products sold under warranty Matthieu van der Heijden, Bermawi Iskandar

355 2011 Spatial concentration and location dynamics in logistics: the case of a Dutch province

Frank P. van den Heuvel, Peter W. de Langen, Karel H. van Donselaar, Jan C. Fransoo

354 2011 Identification of Employment Concentration Areas Frank P. van den Heuvel, Peter W. de Langen, Karel H. van Donselaar, Jan C. Fransoo

353 2011 BPMN 2.0 Execution Semantics Formalized as Graph Rewrite Rules: extended version

Pieter van Gorp, Remco Dijkman

352 2011 Resource pooling and cost allocation among independent service providers

Frank Karsten, Marco Slikker, Geert-Jan van Houtum

Nr. Year Title Author(s)351 2011 A Framework for Business Innovation Directions E. Lüftenegger, S. Angelov, P. Grefen

350 2011 The Road to a Business Process Architecture: An Overview of Approaches and their Use

Remco Dijkman, Irene Vanderfeesten, Hajo A. Reijers

349 2011 Effect of carbon emission regulations on transport mode selection under stochastic demand

K.M.R. Hoen, T. Tan, J.C. Fransoo, G.J. van Houtum

348 2011 An improved MIP-based combinatorial approach for a multi-skill workforce scheduling problem

Murat Firat, Cor Hurkens

347 2011 An approximate approach for the joint problem of level of repair analysis and spare parts stocking

R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

346 2011 Joint optimization of level of repair analysis and spare parts stocks

R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

345 2011 Inventory control with manufacturing lead time flexibility Ton G. de Kok

344 2011 Analysis of resource pooling games via a new extension of the Erlang loss function

Frank Karsten, Marco Slikker, Geert-Jan van Houtum

343 2011 Vehicle refueling with limited resources Murat Firat, C.A.J. Hurkens, Gerhard J. Woeginger

342 2011 Optimal Inventory Policies with Non-stationary Supply Disruptions and Advance Supply Information

Bilge Atasoy, Refik Güllü, Tarkan Tan

341 2011 Redundancy Optimization for Critical Components in High-Availability Capital Goods

Kurtulus Baris Öner, Alan Scheller-Wolf, Geert-Jan van Houtum

340 2011 Making Decision Process Knowledge Explicit Using the Product Data Model

Razvan Petrusel, Irene Vanderfeesten, Cristina Claudia Dolean, Daniel Mican

339 2010 Analysis of a two-echelon inventory system with two supply modes

Joachim Arts, Gudrun Kiesmüller

338 2010 Analysis of the dial-a-ride problem of Hunsaker and Savelsbergh

Murat Firat, Gerhard J. Woeginger

335 2010 Attaining stability in multi-skill workforce scheduling Murat Firat, Cor Hurkens334 2010 Flexible Heuristics Miner (FHM) A.J.M.M. Weijters, J.T.S. Ribeiro333 2010 An exact approach for relating recovering surgical patient

workload to the master surgical scheduleP.T. Vanberkel, R.J. Boucherie, E.W. Hans, J.L. Hurink, W.A.M. van Lent, W.H. van Harten

332 2010 Efficiency evaluation for pooling resources in health care Peter T. Vanberkel, Richard J. Boucherie, Erwin W. Hans, Johann L. Hurink, Nelly Litvak

331 2010 The Effect of Workload Constraints in Mathematical Programming Models for Production Planning

M.M. Jansen, A.G. de Kok, I.J.B.F. Adan

330 2010 Using pipeline information in a multi-echelon spare parts inventory system

Christian Howard, Ingrid Reijnen, Johan Marklund, Tarkan Tan

329 2010 Reducing costs of repairable spare parts supply systems via dynamic scheduling

H.G.H. Tiemessen, G.J. van Houtum

328 2010 Identification of Employment Concentration and Specialization Areas: Theory and Application

Frank P. van den Heuvel, Peter W. de Langen, Karel H. van Donselaar, Jan C. Fransoo

327 2010 A combinatorial approach to multi-skill workforce scheduling M. Firat, C. Hurkens

326 2010 Stability in multi-skill workforce scheduling M. Firat, C. Hurkens, A. Laugier325 2010 Maintenance spare parts planning and control: A framework

for control and agenda for future researchM.A. Driessen, J.J. Arts, G.J. van Houtum, W.D. Rustenburg, B. Huisman

324 2010 Near-optimal heuristics to set base stock levels in a two-echelon distribution network

R.J.I. Basten, G.J. van Houtum

323 2010 Inventory reduction in spare part networks by selective throughput time reduction

M.C. van der Heijden, E.M. Alvarez, J.M.J. Schutten

Nr. Year Title Author(s)322 2010 The selective use of emergency shipments for service-

contract differentiationE.M. Alvarez, M.C. van der Heijden, W.H.M. Zijm

321 2010 Heuristics for Multi-Item Two-Echelon Spare Parts Inventory Control Problem with Batch Ordering in the Central Warehouse

Engin Topan, Z. Pelin Bayindir, Tarkan Tan

320 2010 Preventing or escaping the suppression mechanism: intervention conditions

Bob Walrave, Kim E. van Oorschot, A. Georges L. Romme

319 2010 Hospital admission planning to optimize major resources utilization under uncertainty

Nico Dellaert, Jully Jeunet

318 2010 Minimal Protocol Adaptors for Interacting Services R. Seguel, R. Eshuis, P. Grefen317 2010 Teaching Retail Operations in Business and Engineering

SchoolsTom Van Woensel, Marshall L. Fisher, Jan C. Fransoo

316 2010 Design for Availability: Creating Value for Manufacturers and Customers

Lydie P.M. Smets, Geert-Jan van Houtum, Fred Langerak

315 2010 Transforming Process Models: executable rewrite rules versus a formalized Java program

Pieter van Gorp, Rik Eshuis

314 2010 Working paper 314 is no longer available ----313 2010 A Dynamic Programming Approach to Multi-Objective Time-

Dependent Capacitated Single Vehicle Routing Problems with Time Windows

S. Dabia, T. van Woensel, A.G. de Kok

312 2010 Tales of a So(u)rcerer: Optimal Sourcing Decisions Under Alternative Capacitated Suppliers and General Cost Structures

Osman Alp, Tarkan Tan

311 2010 In-store replenishment procedures for perishable inventory in a retail environment with handling costs and storage constraints

R.A.C.M. Broekmeulen, C.H.M. Bakx

310 2010 The state of the art of innovation-driven business models in the financial services industry

E. Lüftenegger, S. Angelov, E. van der Linden, P. Grefen

309 2010 Design of Complex Architectures Using a Three Dimension Approach: the CrossWork Case

R. Seguel, P. Grefen, R. Eshuis

308 2010 Effect of carbon emission regulations on transport mode selection in supply chains

K.M.R. Hoen, T. Tan, J.C. Fransoo, G.J. van Houtum

307 2010 Interaction between intelligent agent strategies for real-time transportation planning

Martijn Mes, Matthieu van der Heijden, Peter Schuur

306 2010 Internal Slackening Scoring Methods Marco Slikker, Peter Borm, René van den Brink

305 2010 Vehicle Routing with Traffic Congestion and Drivers' Driving and Working Rules

A.L. Kok, E.W. Hans, J.M.J. Schutten, W.H.M. Zijm

304 2010 Practical extensions to the level of repair analysis R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

303 2010 Ocean Container Transport: An Underestimated and Critical Link in Global Supply Chain Performance

Jan C. Fransoo, Chung-Yee Lee

302 2010 Capacity reservation and utilization for a manufacturer with uncertain capacity and demand

Y. Boulaksil; J.C. Fransoo; T. Tan

300 2009 Spare parts inventory pooling games F.J.P. Karsten; M. Slikker; G.J. van Houtum

299 2009 Capacity flexibility allocation in an outsourced supply chain with reservation

Y. Boulaksil, M. Grunow, J.C. Fransoo

298 2010 An optimal approach for the joint problem of level of repair analysis and spare parts stocking

R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

297 2009 Responding to the Lehman Wave: Sales Forecasting and Supply Management during the Credit Crisis

Robert Peels, Maximiliano Udenio, Jan C. Fransoo, Marcel Wolfs, Tom Hendrikx

296 2009 An exact approach for relating recovering surgical patient workload to the master surgical schedule

Peter T. Vanberkel, Richard J. Boucherie, Erwin W. Hans, Johann L. Hurink, Wineke A.M. van Lent, Wim H. van Harten

Nr. Year Title Author(s)295 2009 An iterative method for the simultaneous optimization of

repair decisions and spare parts stocksR.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

294 2009 Fujaba hits the Wall(-e) Pieter van Gorp, Ruben Jubeh, Bernhard Grusie, Anne Keller

293 2009 Implementation of a Healthcare Process in Four Different Workflow Systems

R.S. Mans, W.M.P. van der Aalst, N.C. Russell, P.J.M. Bakker

292 2009 Business Process Model Repositories - Framework and Survey

Zhiqiang Yan, Remco Dijkman, Paul Grefen

291 2009 Efficient Optimization of the Dual-Index Policy Using Markov Chains

Joachim Arts, Marcel van Vuuren, Gudrun Kiesmuller

290 2009 Hierarchical Knowledge-Gradient for Sequential Sampling Martijn R.K. Mes; Warren B. Powell; Peter I. Frazier

289 2009 Analyzing combined vehicle routing and break scheduling from a distributed decision making perspective

C.M. Meyer; A.L. Kok; H. Kopfer; J.M.J. Schutten

288 2010 Lead time anticipation in Supply Chain Operations Planning Michiel Jansen; Ton G. de Kok; Jan C. Fransoo

287 2009 Inventory Models with Lateral Transshipments: A Review Colin Paterson; Gudrun Kiesmuller; Ruud Teunter; Kevin Glazebrook

286 2009 Efficiency evaluation for pooling resources in health care P.T. Vanberkel; R.J. Boucherie; E.W. Hans; J.L. Hurink; N. Litvak

285 2009 A Survey of Health Care Models that Encompass Multiple Departments

P.T. Vanberkel; R.J. Boucherie; E.W. Hans; J.L. Hurink; N. Litvak

284 2009 Supporting Process Control in Business Collaborations S. Angelov; K. Vidyasankar; J. Vonk; P. Grefen

283 2009 Inventory Control with Partial Batch Ordering O. Alp; W.T. Huh; T. Tan282 2009 Translating Safe Petri Nets to Statecharts in a Structure-

Preserving WayR. Eshuis

281 2009 The link between product data model and process model J.J.C.L. Vogelaar; H.A. Reijers

280 2009 Inventory planning for spare parts networks with delivery time requirements

I.C. Reijnen; T. Tan; G.J. van Houtum

279 2009 Co-Evolution of Demand and Supply under Competition B. Vermeulen; A.G. de Kok

278 2010 Toward Meso-level Product-Market Network Indices for Strategic Product Selection and (Re)Design Guidelines over the Product Life-Cycle

B. Vermeulen, A.G. de Kok

277 2009 An Efficient Method to Construct Minimal Protocol Adaptors R. Seguel, R. Eshuis, P. Grefen

276 2009 Coordinating Supply Chains: a Bilevel Programming Approach Ton G. de Kok, Gabriella Muratore

275 2009 Inventory redistribution for fashion products under demand parameter update

G.P. Kiesmuller, S. Minner

274 2009 Comparing Markov chains: Combining aggregation and precedence relations applied to sets of states

A. Busic, I.M.H. Vliegen, A. Scheller-Wolf

273 2009 Separate tools or tool kits: an exploratory study of engineers' preferences

I.M.H. Vliegen, P.A.M. Kleingeld, G.J. van Houtum

272 2009 An Exact Solution Procedure for Multi-Item Two-Echelon Spare Parts Inventory Control Problem with Batch Ordering

271 2009 Distributed Decision Making in Combined Vehicle Routing and Break Scheduling

C.M. Meyer, H. Kopfer, A.L. Kok, M. Schutten

270 2009 Dynamic Programming Algorithm for the Vehicle Routing Problem with Time Windows and EC Social Legislation

A.L. Kok, C.M. Meyer, H. Kopfer, J.M.J. Schutten

Nr. Year Title Author(s)269 2009 Similarity of Business Process Models: Metics and Evaluation Remco Dijkman, Marlon Dumas,

Boudewijn van Dongen, Reina Kaarik, Jan Mendling

267 2009 Vehicle routing under time-dependent travel times: the impact of congestion avoidance

A.L. Kok, E.W. Hans, J.M.J. Schutten

266 2009 Restricted dynamic programming: a flexible framework for solving realistic VRPs

J. Gromicho; J.J. van Hoorn; A.L. Kok; J.M.J. Schutten;

Working Papers published before 2009 see: http://beta.ieis.tue.nl