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Controlling inventory by combining ABC analysis and fuzzy classification

Ching-Wu Chu *, Gin-Shuh Liang, Chien-Tseng Liao

Department of Shipping and Transportation Management, National Taiwan Ocean University, Pei Ning Road, Keelung 202, Taiwan, ROC

a r t i c l e i n f o

Article history:

Received 7 October 2005

Received in revised form 20 February 2008

Accepted 7 March 2008

Available online 20 March 2008

Keywords:

ABC classification

Multi-criteria inventory control

Fuzzy classification

a b s t r a c t

The objective of inventory management is to make decisions regarding the appropriate

level of inventory. In practice, all inventories cannot be controlled with equal attention.The most widespread used inventory system is the ABC classification system, but the lim-

itation of the ABC control system is that only one criterion is considered.

The purpose of this paper is to propose a new inventory control approach called ABCfuzzy

classification (ABCFC), which can handle variables with either nominal or non-nominal

attribute, incorporate managers experience, judgment into inventory classification, and

can be implemented easily.

Our ABCFC approach is implemented based on the data of the Keelung Port. The results of

our study show that 59 items are identified as very important group, 69 items as important

group, and the remaining 64 items as unimportant group. By comparing the results of ABC

FC with the original data, we find that our ABCFC analysis shows a high accuracy of clas-

sification. Some concluding remarks and suggestions for inventory control are also

provided.

2008 Elsevier Ltd. All rights reserved.

1. Introduction

The inventory control has been a very classical OR problem. An extremely large number of models have been developed to

solve inventory problems. Each model uses a particular set of hypotheses. In practice, organizations have hundreds of differ-

ent types of materials and spare parts, so it is easy to loss sight of effectively managing materials. ABC analysis is one of the

most widely used techniques in organizations. ABC classification allows an organization to separate stock keeping units into

three groups: A very important, B important, and C least important. The amount of time, effort, and resources spent on

inventory control should be in the relative importance of each item.

The classification of items into A, B, C groups has generally been based on just one criterion. For inventory items, the cri-

terion is often the annual dollar usage of the item. However, there may be other criteria that represent other important con-

siderations for management. The criticality of a stock-out of the item, the rate of obsolescence, the scarcity, substitutability,and order size requirement of the item and the lead time of supply, are all examples of such considerations. Thus, it has been

generally recognized that the traditional ABC analysis may not be able to provide a good classification of inventory items in

practice (Guvenir & Erel, 1998; Huiskonen, 2001; Partovi & Anandarajan, 2002 ).

There are many instances when other criteria become important in deciding the importance of an inventory item. This

problem becomes a multi-criteria inventory classification that has been studied by some researchers in the past. In general,

complex computational tools or procedures are needed for multi-criteria ABC classification. The concept of fuzzy theory has

received considerable attention recently and it is often used in handling the fuzziness and uncertainty of data or information.

Fuzzy classification is a technique that uses the available information in a set of independent attributes to predict the value

0360-8352/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.cie.2008.03.006

* Corresponding author. Tel.: +886 2 24622192x3407; fax: +886 2 24631903.

E-mail address: [email protected] (C.-W. Chu).

Computers & Industrial Engineering 55 (2008) 841851

Contents lists available at ScienceDirect

Computers & Industrial Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c a i e
mailto:[email protected]://www.sciencedirect.com/science/journal/03608352http://www.elsevier.com/locate/caiehttp://www.elsevier.com/locate/caiehttp://www.sciencedirect.com/science/journal/03608352mailto:[email protected]

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of a discrete or categorical dependent attribute. The purpose of this paper is to propose a new inventory control approach

called ABCfuzzy classification (ABCFC), which can incorporate managers experience, knowledge, and judgment into

inventory classification and can be implemented easily.

This paper is organized as follows. Section 2 reviews the related research. This is followed by the research methodology in

Section 3. Section 4 presents the empirical results. Finally, some conclusions and suggestions for future research are provided

in Section 5.

2. Related research

Classification has emerged as an important decision making tool in business. Applications of classification technique can

be found in prediction of stock market behavior, credit scoring, classifying inventory items (Flores, Olson, & Dorai, 1992), and

prediction of various events such as credit card usage.

Flores and Whybark (1986) have proposed a matrix-based methodology. A joint criteria matrix is developed in the case of

two criteria. Though this is a step forward in multi-criteria ABC classification, it is difficult to use when more criteria have to

be considered. Ernst and Cohen (1990) have presented a methodology based on statistical clustering. The main advantage of

this approach is that it can accommodate large combinations of attributes. However, this approach requires substantial data,

the use of factor analysis, a clustering procedure, which may render it impractical in typical stockroom environments.

The analytic hierarchy process introduced by Saaty (1980) has been adopted by some authors for ABC classification (Gaj-

pal, Ganesh, & Rajendran, 1994; Partovi & Burton, 1993; Partovi & Hopton, 1994). The advantage of the AHP is that it can

incorporate many criteria and ease of use on a massive accounting and measurement system, but its shortcoming is that

a significant amount of subjectivity is involved in pairwise comparisons of criteria.Artificial intelligence is another method for multi-criteria inventory classification. Guvenir and Erel (1998) have

employed the genetic algorithm to the inventory classification problem. Artificial neural network is another artificial intel-

ligence-based technique, which is applicable to the classification process. Partovi and Anandarajan (2002) have proposed an

artificial neural network to classify SKUs in a pharmaceutical industry. Two learning methods, back propagation and genetic

algorithm, are used in the method. Clearly, these approaches are heuristic and may not provide good results at all environ-

ments. Recently, Ramanathan (2006) has presented a simple classification scheme using weighted linear optimization, which

is similar to data envelopment analysis (DEA). Zhou and Khotanzad (2007) have suggested a method for design of fuzzy-rule-

based classifier using genetic algorithm. The classification results are compared with those of Bayes and other fuzzy classi-

fiers. It is shown that the proposed method is superior to them.

The goal of discriminant analysis is to use the data in the sample to develop a rule or a method for classifying the new

observation into appropriate group based on the observed value on the independent variables. However, discriminant anal-

ysis is based on some assumptions (Johnson & Wichern, 1998). If the real situation deviates from these assumptions, the re-

sults from discriminant analysis will not be accurate and reliable. Lin and Chen (2004) have proposed a method to the fuzzydiscriminant analysis for groups of crisp data. The authors have utilized a genetic algorithm to determine the membership

function of each group by minimizing the classification error.

Rule induction uses induction to determine a relationship between observations, which can be used for predicting one of

the variables. Quinlans algorithm (Quinlan, 1979) is a popular induction algorithm which uses entropy to measure the infor-

mation content of each attribute and then derives rules through a repetitive decomposition process. This process may reduce

the accuracy of the rules. Composite rule induction (Liang, 1992) accesses probabilities for rules and applies different meth-

ods to handle both nominal and non-nominal attributes. Its stable but too complex.

Some other studies (Hu & Tzeng, 2003; Hu, Chen, & Tzeng, 2003) have suggested the fuzzy data mining techniques to deal

with the classification problem. Determining membership functions and minimum fuzzy support in finding fuzzy association

rules are important issues for using those methods (Hu, 2005).

The main difference of this paper from earlier work is as follows: (1) the approach can handle any combination of item

attribute information that is important for managerial purposes (e.g., the criticality of a stock-out, order size requirement of

the item); (2) managers preference for grouping based on operational performance can be accommodated; (3) fuzzy statis-tical discrimination criteria are considered; (4) our ABCFC approach can be easily implemented on the spreadsheet, which is

more accessible to practitioners.

3. Research methodology

3.1. ABC classification system

In the 18th century, Villefredo Pareto, in a study of the distribution of wealth in Milan, found that 20% of the people con-

trolled 80% of the wealth. This logic of the few having the most importance and the many having the little importance has

been broadened to include many situations and is termed the Pareto principle. This is true in inventory systems where a few

items account for the bulk of our investment.

Any inventory system must specify when an order to be placed for an item and how many units to order. In practice, there

are so many items involved, so it is not practical to model and control each item with equal attention. To deal with this prob-

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lem, the ABC classification divides inventory items into three groups based on the annual dollar purchases of an inventory

item.

The three groups used in the ABC system are as follows:

A items (high value items): The 1520% of the items that account for 7580% of the total annual inventory value.

B items (medium value items): The 3040% of the items that account for approximately 15% of the total annual inventory

value.

C items (low value items): The 4050% of the items that account for 1015% of the total annual inventory value.

These classifications may not always be exact, but they have been found to be close to the actual occurrence in firms with

remarkable accuracy (Swamidass, 2000).

3.2. Fuzzy classification

Fuzzy classification analysis is usually used to classify the training data set (a data set which is used to induce the

membership function) and to predict the testing data (Zhou & Khotanzad, 2007). The training data set contains a num-

ber of examples. An example contains value for a dependent attribute and several attribute values can be either nom-

inal or non-nominal.

In order to use fuzzy classification, we must generate the membership function from the input training data set. Ta-

maki, Kanagawa, and Ohta (1998) have presented a method to obtain the membership functions which satisfy the

restriction as the fuzzy event against to given probability function. Although the idea of proposed method is good,the identification procedure of membership function is complex. Medasani, Kim, and Krishnapuram (1998) have pro-

vided a general overview of several methods for generating membership functions for fuzzy pattern recognition appli-

cations. There are many membership functions, for examples, the triangular fuzzy membership function, trapezoidal

fuzzy membership function, Gaussian fuzzy membership function, Z Spline fuzzy membership function, and S Spline fuz-

zy membership function, can be utilized in fuzzy classification ( Medasani et al., 1998; Zhou & Khotanzad, 2007). Since

the nature of nominal and non-nominal data is different, we first treat the two types of data differently and discuss

them in the following subsections, respectively. Then, fuzzy classification rules are discussed. Finally, a numerical exam-

ple for demonstrating calculation procedures is provided.

3.2.1. Independent nominal attributes

Let Y and X1,X2, . . . ,Xk be the dependent nominal attribute and independent nominal attributes, respectively. The mem-

bership function of independent nominal attribute can be obtained in three steps:

(1) For each Yand X0 (0 = 1, . . . , k), classifying all examples in the input training data set by their dependent attribute val-

ues Cj (j = 1,2, . . ., n) and independent attribute values Vi (i = 1,2, . . ., m), we can obtain the occurrence frequency table

by counting the occurrence frequency (fij) corresponding to the combination of Vi and Cj.

(2) For each row ofTable 1, divide each entry in row i (i = 1,2, . . . , m) of Table 1 by the sum of entire entries in row i. This

will yield a new Table 2 in which the sum of entire entries in each row is equal to 1 i:e:; gij

fijPnk1

fikand

Pnk1gik 1.

(3) For each j, 16j 6 n, the membership function lYCj X0 is defined as follows:

lYCjX0

g1j; if X0 V1

g2j; if X0 V2

.

.

.

gmj; if X0 Vm

8>>>>>>>>>:

Table 1

The occurrence frequency table ofY and X0

X0 Y

C1 C2 . . . Cn

V1 f11 f12 . . . f1nV2 f21 f22 . . . f2n. . . . . . . . . . . . . . .

Vm fm1 fm2. . .

fmn

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3.2.2. Independent non-nominal attributes

For independent non-nominal attributes, sample mean and variance provide valuable information about the population

and hence are used to formulate the membership function.

Without loss of the generality and matching with our research, we assume a three-class classification problem is consid-

ered. The distribution of attribute X0 for classes i (i.e., Y= Ci, i = 1,2,3) has population mean li and variance r2i . Since the pop-

ulation mean and variance are not available, we use sample mean Xi and variance S2i to estimate li and variance r

2i : Suppose

that X1 < X2 < X3:

The membership function of independent non-nominal attributes can be obtained in two steps:

(1) Calculate the cut values XC12 ;XC23 and thresholds values X2L, X3L, X1R, and X2R which are defined as follows:

XC12

S1X2 S2X1

S1 S2 ; XC23

S2X3 S3X2

S2 S3

X2L X2 3S2; X3L X3 3S3

X1R X1 3S1; X2R X2 3S2

(2) Find the membership function lYC1 X0 for Y= C1, lYC2 X0 for Y= C2, and lYC3 X0 for Y= C3. If X2R > X2L, then the

membership functions are defined as follows:

lY C1X0

1; X0 < XC12X1RX0

X1RXC12; XC12 6 X0 < X1R

0; X1R 6 X0

8>:

lY C2 X0

0; X0 < X2L or X0 P X2RX0X2L

XC12X2L

; X2L 6 X0 < XC12

1; XC12 6 X0 < XC23X2RX0

X2RXC23; XC23 6 X0 < X2R

8>>>>>>>>>:

lY C3X0

0; X0 < X3LX0X3L

XC23X3L

; X3L 6 X0 < XC23

1; XC23 6 X0

8>:

Fig. 1 represents the graph of membership functions for the case l3 > l2 > l1 and X2R > X2L. With similar reasoning, the mem-

bership function of more than three classification problem can be derived.

3.2.3. The fuzzy classification rule and an illustrated example

The fuzzy classification can be summarized in the following steps:

(1) Decide the dependent attribute Yand independent attributes X0 (0 = 1,2, . . ., k) where the attribute ofYis nominal and

independent attributes X0 (0 = 1,2, . . . , k) can be either nominal or non-nominal.

(2) Find the values of the dependent attribute Yand independent attributes X0, and use C1, C2, . . . , Cn to denote the values of

dependent attribute Y and V1, V2, . . ., Vm to denote the values of independent attributes X0, respectively.

(3) Generate the membership function of independent nominal attribute based on the steps mentioned in Section 3.2.1.

(4) Generate the membership function of independent non-nominal attribute based on the steps mentioned in Section

3.2.2.

(5) Denote a specific inventory item as It. Based on Steps (3) or (4), substituting any value ofVi into membership function,

we can obtain n values lItYC1 Vi; . . . ; lItYCn

Vi.

(6) DefinelYCj It

Pki1

lItYCj

Vi

k, which represents the grade of membershipofIt in class Y= Cj. Fuzzy classification rulecan

be defined as follows: iflYCt It maxflYC1 It;lYC2 It; . . . ; lYCn Itg, then inventory item It is assigned to class Ct.

A set of inventory data shown in Table 3 is used to illustrate the procedure of fuzzy classification.

Table 2

The relative frequency table ofY and X0

X0 Y

C1 C2 . . . Cn

V1 g11 g12 . . . g1nV2 g21 g22 . . . g2n. . . . . . . . . . . . . . .

Vm gm1 gm2 . . . gmn

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(1) Assume that there are 60 inventory items in a company. Y is the criticality of an inventory item which is a nominal

attribute and includes three levels in the classification: 2, very critical; 1, critical; and 0, uncritical, X1 is severity of

the impact of the inventory running out which is also a nominal attribute and includes three levels in the classifica-tion: 2, very severe; 1, severe; and 0, not severe, and X2 is the usage frequency of the inventory item within the plan-

ning period which is a non-nominal attribute.

(2) Let us handle the nominal attributes first. The occurrence frequency table for the nominal attributes X1 and Y can be

counted and summarized in Table 4.

For each row ofTable 4, divide each entry in row i (i = 1,2, 3) ofTable 4 by the sum of entire entries in row i. This will yield

a new Table 5.

Based on the definition in Section 3.2.1, we have the membership functions of lYCj X1; j 0;1; 2 as follows:

lY0X1

0:76; if X1 0

0:12; if X1 1

0:04; if X1 2

8>:

lY1X1

0:18; if X1 0

0:35; if X1 1

0:35; if X1 2

8>:

lY2X1

0:06; if X1 0

0:53; if X1 1

0:61 if X1 2

8>:

(3) Next we will handle the non-nominal attribute X2. Based on the values of Y, we can classify the data of X2 into three

groups and calculate the sample mean and standard deviation of each group as shown in Table 6.

With the information in Table 6 and the formula mentioned in Section 3.2.2, the cut values XC12, XC23 and thresholds val-

ues X2L, X3L, X1R, and X2R can be obtained as follows:

1.0

1.0

X

X

X

X

X2L X3LXC12 X1RXC23 X2R

X3L XC23

X2L XC12 XC23 X2R

XC12 X1R XC23 X2R

Y=C1 Y=C2 Y=C3

The membership function of

Y=C3


Y=C1


Y=C2

1.0

Fig. 1. The membership functions of three-class non-nominal data.

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XC12 14:47; XC23 19:08

X2L 9:13; X3L 15:22

X1R 20:33; X2R 23:53

With the definition in Section 3.2.2, we obtain the membership functions of lYCj X2; j 0;1;2 as the following:

Table 3

The data set for fuzzy classification

Item No. Y X1 X2

1 2 2 23

2 2 2 24

3 2 2 21

4 2 2 24

5 2 2 23

6 2 2 257 2 2 26

8 2 2 24

9 2 2 22

10 2 2 20

11 2 2 23

12 2 2 20

13 2 2 19

14 2 2 20

15 2 2 21

16 2 2 22

17 2 1 20

18 2 1 23

19 2 1 21

20 2 1 20

21 2 1 18

22 2 1 2023 2 1 19

24 2 1 21

25 2 1 20

26 2 0 19

27 1 2 18

28 1 2 17

29 1 2 16

30 1 2 14

31 1 2 18

32 1 2 19

33 1 2 21

34 1 2 18

35 1 2 19

36 1 1 14

37 1 1 16

38 1 1 1539 1 1 17

40 1 1 15

41 1 1 18

42 1 0 13

43 1 0 12

44 1 0 14

45 0 2 16

46 0 1 15

47 0 1 14

48 0 0 17

49 0 0 16

50 0 0 12

51 0 0 11

52 0 0 13

53 0 0 10

54 0 0 12

55 0 0 12

56 0 0 9

57 0 0 11

58 0 0 13

59 0 0 10

60 0 0 8

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lY0X2

1; X2 < 14:4720:33X2

5:86; 14:47 6 X2 < 20:33

0; 20:33 6 X2

8>:

lY1X2

0; X2 < 9:13 or X2P

23:53X29:135:34

; 9:13 6 X2 < 14:47

1; 14:47 6 X2 < 19:0823:53X2

4:45; 19:08 6 X2 < 23:53

8>>>>>:

lY2X2

0; X2 < 15:22X215:22

3:86; 15:22 6 X2 < 19:08

1; 19:08 6 X2

8>:

(4) Calculate the membership function values and classify inventory items. After the membership functions are con-

structed, we can obtain the membership function values based on Steps (2) and (3) and classify an inventory item

according to Step (6) mentioned in Section 3.2.3. Let us take the first inventory item as an example. Y= 2, X1 = 2,

and X2

= 23. Given X1

= 2, from the membership functions in Step (2), we can find lY=0

(X1

) = 0.04, lY=1

(X1

) = 0.35,

and lY=2 (X1) = 0.61. Substituting X2 = 23 into the membership functions mentioned in Step (3), we can calculate

lY=0 (X2) = 0, lY=1(X2) = 0.12, and lY=2 (X2) = 1. Based on the definition in Step (6) we have the grade of membership

in each class, lY=0(I1) = 0.02, lY=1 (I1) = 0.235, and lY=2 (I1) = 0.805. Since 0.805 > 0.235 > 0.002, according to fuzzy clas-

sification rule, we classify this inventory into group Y= 2, that is this item is very critical. The results of fuzzy classi-

fication are shown in Table 7.

The last column,bY, in Table 7 stands for prediction value ofY(i.e., classified group of this inventory item). Under the col-umn of bY, * stands for misclassified inventory item. There are 11 misclassified inventory items in our classification. By com-paring bY with Y, we can find that the accuracy is about 82% 6011

60 0:82

.

If this classification accuracy is acceptable to the manger, the membership functions can be used to classify the new

inventory items directly without reconstructing the membership functions again. This is one of the major advantages of

our ABCFC approach. For example, a new inventory item is added to the warehouse. Based on the managers experience

and judgment, the impact of the inventory running out is severe, i.e., X1 = 1, and the usage frequency of the inventory itemis estimated about 18, i.e., X2 = 18.

Table 4

The occurrence frequency table ofY and X1

X1 Y

2 1 0

2 16 9 1

1 9 6 2

0 1 3 13

Table 5

The relative frequency table ofY and X1

X1 Y

2 1 0

2 0.61 0.35 0.04

1 0.53 0.35 0.12

0 0.06 0.18 0.76

Table 6

The sample means and standard deviations

The usage frequency Y Mean Standard deviation

X2 0 12.44 2.63

1 16.33 2.4

2 21.46 2.08

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Following the same procedures demonstrated above, we can obtain lY=0 (X1) = 0.12, lY=1 (X1) = 0.35, lY=2 (X1) =

0.53, lY=0 (X2) = 0.4, lY=1 (X2) = 1, lY=2 (X2) = 0.72, lY=0 (Inew) = 0.26, lY=1 (Inew) = 0.675, and lY=2 (Inew) = 0.625. Since

0.675 > 0.625 > 0.26, according to fuzzy classification rule, we classify this new inventory into group Y= 1, that is this item

is critical.

Table 7

The results of fuzzy classification

Item lY=0 (X1) lY=1 (X1) lY=2 (X1) lY=0 (X2) l Y=1 (X2) lY=2 (X2) l Y=0 (It) lY=1 (It) lY=2 (It) bY1 0.04 0.35 0.61 0 0.12 1 0.02 0.235 0.805 2

2 0.04 0.35 0.61 0 0 1 0.02 0.175 0.805 2

3 0.04 0.35 0.61 0 0.57 1 0.02 0.46 0.805 2

4 0.04 0.35 0.61 0 0 1 0.02 0.175 0.805 2

5 0.04 0.35 0.61 0 0.12 1 0.02 0.235 0.805 2

6 0.04 0.35 0.61 0 0 1 0.02 0.175 0.805 27 0.04 0.35 0.61 0 0 1 0.02 0.175 0.805 2

8 0.04 0.35 0.61 0 0 1 0.02 0.175 0.805 2

9 0.04 0.35 0.61 0 0.34 1 0.02 0.345 0.805 2

10 0.04 0.35 0.61 0.05 0.79 1 0.045 0.57 0.805 2

11 0.04 0.35 0.61 0 0.12 1 0.02 0.235 0.805 2

12 0.04 0.35 0.61 0.05 0.79 1 0.045 0.57 0.805 2

13 0.04 0.35 0.61 0.23 1 0.98 0.135 0.675 0.795 2

14 0.04 0.35 0.61 0.05 0.79 1 0.045 0.57 0.805 2

15 0.04 0.35 0.61 0 0.57 1 0.02 0.46 0.805 2

16 0.04 0.35 0.61 0 0.34 1 0.02 0.345 0.805 2

17 0.12 0.35 0.53 0.05 0.79 1 0.085 0.57 0.765 2

18 0.12 0.35 0.53 0 0.12 1 0.06 0.235 0.765 2

19 0.12 0.35 0.53 0 0.57 1 0.06 0.46 0.765 2

20 0.12 0.35 0.53 0.05 0.79 1 0.085 0.57 0.765 2

21 0.12 0.35 0.53 0.4 1 0.72 0.26 0.675 0.625 1*

22 0.12 0.35 0.53 0.05 0.79 1 0.085 0.57 0.765 223 0.12 0.35 0.53 0.23 1 0.98 0.175 0.675 0.755 2

24 0.12 0.35 0.53 0 0.57 1 0.06 0.46 0.765 2

25 0.12 0.35 0.53 0.05 1 1 0.085 0.57 0.765 2

26 0.76 0.18 0.06 0.23 1 0.98 0.495 0.59 0.52 1*

27 0.04 0.35 0.61 0.4 1 0.72 0.22 0.675 0.665 1

28 0.04 0.35 0.61 0.57 1 0.46 0.305 0.675 0.535 1

29 0.04 0.35 0.61 0.74 1 0.2 0.39 0.675 0.405 1

30 0.04 0.35 0.61 1 0.91 1 0.52 0.63 0.305 1

31 0.04 0.35 0.61 0.4 1 0.72 0.22 0.675 0.665 1

32 0.04 0.35 0.61 0.23 1 0.98 0.135 0.675 0.795 2*

33 0.04 0.35 0.61 0 0.57 1 0.02 0.46 0.805 2*

34 0.04 0.35 0.61 0.4 1 0.72 0.22 0.675 0.665 1

35 0.04 0.35 0.61 0.23 1 0.98 0.185 0.675 0.795 2*

36 0.12 0.35 0.53 1 0.91 0 0.56 0.63 0.265 1

37 0.12 0.35 0.53 0.74 1 0.2 0.43 0.675 0.365 1

38 0.12 0.35 0.53 0.91 1 0 0.515 0.675 0.265 139 0.12 0.35 0.53 0.57 1 0.46 0.345 0.675 0.495 1

40 0.12 0.35 0.53 0.91 1 0 0.515 0.675 0.265 1

41 0.12 0.35 0.53 0.4 1 0.46 0.26 0.675 0.625 1

42 0.76 0.18 0.06 1 1 0 0.88 0.45 0.03 0*

43 0.76 0.18 0.06 1 1 0.72 0.88 0.36 0.03 0*

44 0.76 0.18 0.61 1 1 0 0.88 0.545 0.03 0*

45 0.04 0.35 0.53 0.74 1 0.2 0.195 0.675 0.405 1*

46 0.12 0.35 0.53 0.91 0.91 0 0.515 0.675 0.265 1*

47 0.12 0.35 0.06 1 1 0 0.56 0.63 0.265 1*

48 0.76 0.18 0.06 0.57 1 0.46 0.665 0.59 0.26 0

49 0.76 0.18 0.06 0.74 1 0.2 0.75 0.59 0.13 0

50 0.76 0.18 0.06 1 0.54 0 0.88 0.36 0.03 0

51 0.76 0.18 0.06 1 0.35 0 0.88 0.265 0.03 0

52 0.76 0.18 0.06 1 0.72 0 0.88 0.45 0.03 0

53 0.76 0.18 0.06 1 0.16 0 0.88 0.17 0.03 0

54 0.76 0.18 0.06 1 0.54 0 0.88 0.36 0.03 0

55 0.76 0.18 0.06 1 0.54 0 0.88 0.36 0.03 0

56 0.76 0.18 0.06 1 0 0 0.88 0.09 0.03 0

57 0.76 0.18 0.06 1 0.35 0 0.88 0.265 0.03 0

58 0.76 0.18 0.06 1 0.72 0 0.88 0.45 0.03 0

59 0.76 0.18 0.06 1 0.16 0 0.88 0.17 0.03 0

60 0.76 0.18 0.06 1 0 0 0.88 0.09 0.03 0

bY stands for prediction value of Y (i.e., classified group of this item).* stands for misclassified items.


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3.2.4. The ABCfuzzy classification rule

From related research in Section 2, we know that the major shortcoming of ABC is that it has been based on just one cri-

terion. Hence, many researchers have devoted themselves to developing multi-criteria ABC analysis to control inventories.

Our approach combines the traditional ABC with fuzzy classification to classify inventory items and the solution procedure of

ABCFC consists of the following steps:

(1) Design the criticality function of inventory items,

Y fX1;X2; where

Yis the criticality of an inventory item including three levels in the classification: very critical, critical, and uncritical,X1 is severity of the impact of the inventory running out including three levels in the classification: very severe, severe,

and not severe, and X2 is the usage frequency of the inventory item within the planning period. For illustration pur-

pose, there are two independent variables in our case. The number of independent variables is not necessary restricted

to two.

(2) Classify all inventory items based on traditional ABC analysis. We can obtain three groups of inventory items: A group,

B group, and C group and denote each group with A1, A2, and A3, respectively.

(3) Use fuzzy classification to classify A1, A2, and A3 groups, respectively. All the inventory items in each group, A1, A2, and

A3, can be further divided into three subgroups based on their criticality: very critical, critical, and uncritical. Table 8

shows all nine subgroups of inventory items from ABC and fuzzy classification.

With ABCFC analysis, there are nine classified groups that could each require different management policies. To reduce

the combinations to a manageable number, which is similar to traditional ABC analysis, we further combine nine classified

groups into three combined groups as follows:

very important group fA1B1;A2B1;A1B2g

important group fA3B1;A2B2;A1B3g

unimportant group fA3B2;A2B3;A3B3g

4. Empirical results

The empirical investigation was carried out during a period of 12 months and one of the researchers is part of the orga-

nization, Keelung Port, located in Northeastern Taiwan. Data were collected by interviewing managers of spare parts, study-

ing documents, and analyzing numerical data.

The data consists of one year demand history of 192 spare parts as well as information including unit price, usage fre-

quency, procurement lead time, current item status, the criticality of an inventory item, and the severity of the impact of

the inventory running out. To avoid the lengthy presentation, we do not provide the detail of the original data, the related

tables and the results of ABCFC in the paper. The original data and fore mentioned results of ABCFC are available from the

authors upon request.

Based on the results of ABCFC, we can count the number of misclassified prediction values and know that our ABCFC

analysis shows an accuracy of 72.4%. Table 9 shows the results of ABCFC analysis in which 59 items are identified as very

Table 8

The matrix of ABC and fuzzy classification

ABC FC

Very critical Critical Uncritical

B1 B2 B3

A1 A1B1 A1B2 A1B3A2 A2B1 A2B2 A2B3A3 A3B1 A3B2 A3B3

Table 9

The classification results of ABCfuzzy classification

ABC FC

B1 B2 B3

A1 34 0 10

A2 25 0 6

A3 59 0 58

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important group, 69 items are identified as important group, and the remaining 64 items are identified as the unimportant

group. Looking at Table 10, we can see that the ABCFC classifies more inventory items as very important group and impor-

tant group. This result seems to make sense since fuzzy classification has taken more criteria into consideration.

Frequently used inventory management policies for ABC inventory control are shown in Table 11 (Silver, Pyke, & Peterson,1998). The spare parts of the Keelung Port are purchased at a monthly and quarterly base, so the management policies sug-

gested for the periodic system in Table 11 satisfy our need. Based on the results of ABCFC, we further examined the data in

very important and important groups. The annual usage frequency of most inventory items is less than 10 and that of a large

portion of inventory items is equal to 1. Hence, (S, S-1) inventory policy has been suggested to control these items. (R, s, S)

and (R, S) inventory control models can be used to control items in very important and important groups with high usage,

but it takes a lot of time and effort to obtain the optimal solution. In stead of using the traditional OR method to solve the

problem, some policies have been suggested to managers as follows:

(1) More frequent counts should be made to improve the accuracy of inventory record; (2) reduce the procurement lead

time to low down the safety stock; (3) increase the accuracy of forecast to cut down unnecessary inventory; (4) the

order quantity and safety stock level should be established for each item depending on both the criticality and eco-

nomics; (5) a specific period for reconsidering the classification of the inventory items is necessary, since it is a chang-

ing world. As to the unimportant group, the two bins method is suggested.

5. Conclusions

In todays business environment, an organization must maintain an appropriate balance between critical stock-outs and

inventory holding costs. Because customer service is not a principal factor for attracting new customers, but it is frequently a

major reason for losing them. Many researchers have devoted to achieving this appropriate balance. In this paper, a new

inventory control approach combing ABC with fuzzy classification has been proposed and illustrated in the presence of nom-

inal and non-nominal attributes.

An illustration is demonstrated and our ABCFC approach is also implemented based on the data of the Keelung Port. The

results of our study show that 59 items are identified as very important group, 69 items as important group, and the remain-

ing 64 items as unimportant group. By comparing the results of ABCFC with the original data, we find that our ABCFC anal-

ysis shows a high accuracy of classification. Some inventory control policies are also suggested.Future research is required for the organization perspective on inventory control to grow in size. The paper has initiated

its development based on the data of the Keelung Port containing only 192 inventory items, but it mainly focused on the

creation of an approach that is most useful in practice. It would be interesting to see if the implementation results of differ-

ent types of organizations with more inventory items will be further improved as well.

Acknowledgements

The authors are grateful to reviewers for their valuable comments. This work was partially supported by the National Sci-

ence Council of Taiwan under Grant NSC 93-2416-H-019-003.

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Table 10

A comparative table of classification results from two different methods

Method Group

A (very important) B (important) C (unimportant)

ABC analysis 44 31 117

ABCfuzzy classification 59 69 64

Table 11

Inventory management policies for different inventory control systems

Classification System

Continuous review Periodic review

Very important (s, S) (R, s, S)

Important (s, Q) (R, S)

Unimportant Easily implemented method, such as two bins method


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