abs multicriteria.pdf

Hindawi Publishing CorporationJournal of Industrial EngineeringVolume 2013 Article ID 827274 7 pageshttpdxdoiorg1011552013827274

Research ArticleMultiple Criteria ABC Analysis with FCMClustering

Gulsen Aydin Keskin1 and Coskun Ozkan2

1 Department of Industrial Engineering Faculty of Engineering Kocaeli University Umuttepe Campus 41380 Kocaeli Turkey2Department of Industrial Engineering Faculty of Engineering Yildiz Technical University 34349 Istanbul Turkey

Correspondence should be addressed to Gulsen Aydin Keskin gaydinkocaeliedutr

Received 13 August 2012 Revised 29 October 2012 Accepted 12 November 2012

Academic Editor Josefa Mula

Copyright copy 2013 G Aydin Keskin and C Ozkan is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

e number of stock keeping units (SKUs) possessed by organizations can easily reach quite a few An inventory managementpolicy for each individual SKU is not economical to design ABC analysis is one of the conventionally used approaches to classifySKUs In the classical method the SKUs are ranked with respect to the descending order of the annual dollar usage which is theproduct of unit price and annual demande few of the SKUs that have the highest annual dollar usage are in group A and shouldbe taken into account mostly the SKUs with the least annual dollar usage are in group C and should be taken into account leastthe remaining SKUs are in group B In this study we proposed fuzzy c-means (FCM) clustering to a multicriteria ABC analysisproblem to help managers to make better decision under fuzzy circumstancse e obtained results show that the FCM is a quitesimple and an easily adaptable method to inventory management

1 Introduction

Inventory control is a well-known problem in operationsresearch Several models have been developed to solveinventory problems In business companies have hundredsof different types of materials erefore it is easy to losecontrol of managing the materials Inventory classicationusing ABC analysis is one of the most widely used techniquesin organizations ABC classication allows an organizationto separate stock keeping units (SKUs) into three groups Athe most important B important and C the least importante purpose of classifying items into groups is to establishappropriate levels of control over each item [1 2] e majoradvantage ofABCanalysis is the easiness of use and simplicityto understand e items are classied according to theannual use value which is the product of annual demand andthe average unit price [3]

e classication of items into A B and C groups hasgenerally been implemented according to one criterion Forinventory items the criterion is frequently the annual dollarusage of the item However it has been generally recognizedthat the traditional ABC analysis has a serious drawbackthat may inhibit the effectiveness of the procedure in some

situations Using one criterion only may create problems ofsignicant nancial loss For example class C items with longlead time or class A items prone to obsolescence may incurnancial losses due to a possible interruption of productionandor huge inventory levelserefore it has been proposedthat multicriteria ABC classication such as lead timecriticality of a stockout of the item the rate of obsolescencethe scarcity substitutability and order size requirement of theitem can provide a more comprehensive managerial controland to take other important criteria into consideration [2 45]

Complex computational tools are needed for traditionalmulticriteria ABC classication One of them is the matrix-based methodology At this methodology a joint criteriamatrix is developed in the case of two criteria Howeverthe methodology is relatively difficult to use when morecriteria have to be considered [6] e other technique usedto weigh the criteria is the analytic hierarchy process (AHP)[1 2 7] e main idea of AHP is to derive a single scalarmeasure of importance of inventory items by subjectivelyrating the criteria andor the inventory itemse singlemostimportant issue associated with AHP-based studies is thesubjectivity involved in the analysis [3]

2 Journal of Industrial Engineering

In this paper we are concerned with the multicriteriaclassication method aimed to facilitate inventory manage-ment e classication method is proposed through a casestudy of an automative company Being distinct from thetechniques used for multicriteria ABC analysis the criteriaare not weighed in this method erefore this method doesnot involve subjectivity In addition unlike other methodsour method not only determines the matching to any of theclusters but also gives the membership degrees to all clustersese are our contributions to the literature

e aim of this study is to develop a decision-makingapproach to the ABC analysis method which is a multicri-teria decision-making problem e remainder of the paperis organized as follows ABC analysis is described in Section2 In Section 3 fuzzy c-means (FCM) clustering is explainedIn Section 4 the FCM is applied to a real life problem Finallyconclusions are drawn in Section 5

2 ABC Analysis

e conventional ABC classication based on Pareto prin-ciple developed at General Electric during the 1950s is themost popularmethod to classify the items in inventory [4] Inthis classication inventory items are ordered descendinglywith respect to their annual dollar usage valuese relativelysmall number of items at the top of the list (approx 10)controlling the majority of the total annual dollar usageconstitute class A the majority of the items at the bottom ofthe list (approx 60) controlling a relatively small portionof the total annual dollar usage constitute class C and theitems between the two classes constitute class B (approx30)is classicationmay not always be accurate but it hasbeen found too close to the existing occurrence in companieswith remarkable accuracy Class A inventory items requirecautious inventory control because they represent a largepercentage of the total dollar value of the inventory isstate requires certain demand forecasts and detailed recordkeeping Class C inventory items should receive a exiblecontrol Class B items should have a control effort that liesbetween A and C categories [1 2]

Because of its easy-to-implement nature applicabilityto numerous situations empirically observed benets andremarkable effectiveness in many inventory systems thisapproach is still popularly used in practice However themethod has a serious drawback that may inhibit the effec-tiveness of the procedure in some situations e criterionused in the conventional ABC classication is the annualdollar usage so using one criterion may create problems ofsignicant nancial loss [4 8] us the inventory controlproblem becomes a multicriteria inventory classication thathas been studied by some researchers In the literature someof the considered criteria include inventory cost criticalitylead time commonality obsolescence substitutability num-ber of requests for the item in a year scarcity durabilitysubstitutability repairability order size requirement stock-ability demand distribution and stockout penalty cost [2 3]

21 Related Research In recent years several new mul-ticriteria approaches to inventory classication have been

introduced but all have some drawbacks Classical ABCclassication approach on two criteria is a step forward inmulticriteria ABC classication However the methodologyis relatively difficult to use when more criteria have to beconsidered ere is no obvious way to extend the procedureto more than two criteria [2] e rst paper about ABCanalysis was presented by Flores and Whybark in 1986 [9]Affected from this study Chen et al [8] have presented a case-based multicriteria ABC analysis by accounting additionalcriteria such as lead time and criticality of SKUs to providemanagerial exibility

In 1990 amethodology based on statistical clusteringwasproposed by Ernst and Cohen But this approach requiressubstantial data the use of factor analysis and a clusteringprocedure which may render it impractical in typical stock-room environments Furthermore the clusters themselvesmust be reevaluated in order to classify new stock items sothere is a chance that previously classied stock may end upbeing classied differently every time new items are addedand this may disturb the inventory control procedure Inshort the model may be too sophisticated for the averagemanager [1]

e AHP process has also been proposed by severalauthors for ABC classication [6] Cakir and Canbolat [7]have proposed AHP with integrating fuzzy technique tosolve the multicriteria inventory classication problem eadvantage of the AHP is that it can incorporate many criteriaand its ease of use on amassive accounting andmeasurementsystemOne of the important drawbacks of themethod is thata signicant amount of subjectivity is involved in pairwisecomparisons of criteria rating levels and assigning a ratinglevel and associated weights [1 2]

Articial intelligence is another method for multicriteriainventory classication Guvenir and Erel [4] have presenteda new approach using AHP technique and genetic algorithmto multicriteria classication Articial neural network isanother technique which is applicable to the classicationprocess Partovi and Anandarajan [1] have proposed anarticial neural network (ANN) by utilizing two learningmethods in the ANN named back-propagation and geneticalgorithms for ABC classication of SKUs Ramanathan [3]has proposed a simple weighted linear optimization modelto address the multicriteria inventory classication prob-lem which is similar to data envelopment analysis (DEA)Zhou and Fan [10] have presented an extended version ofthe Ramanathanrsquos model by incorporating some balancingfeatures for multicriteria ABC inventory classication eclassication results are compared with those of Bayes andother fuzzy classiers It is shown that the proposed methodis superior to them Chu et al [2] have proposed a newinventory control approach by integrating ABC and fuzzyclassication

Different from what is mentioned previously Ng [11] hasproposed an alternative weight linear optimization modelfor multicriteria inventory classication by using the ABCprinciple Hadi-Vencheh [12] has presented an extendedversion of the Ng-model which is a simple classier formulticriteria ABC analysis Al Kattan and Bin Adi [13] have

Journal of Industrial Engineering 3

T 1 e data set of all items at the company

SKUs Average unit cost Annual dollar usage Critical factor Lead timeS1 4724 48240 100 300S2 3564 38900 100 700S3 9825 35750 100 200S4 1558 27260 001 200S5 3487 46280 050 300S6 5687 34970 001 400S7 3425 28600 050 500S8 4240 26750 001 300S9 5582 46732 100 200S10 2425 38260 050 100S11 5425 9956 100 500S12 2357 18725 050 400S13 4750 10150 100 800S14 6534 65745 050 300S15 5764 54725 100 200S16 10250 56475 050 100S17 2357 45825 001 500S18 4021 18260 050 700S19 1754 35645 001 200S20 3415 26425 050 300S21 1135 27846 100 500S22 7000 10567 050 300S23 5121 47216 100 700S24 6854 26684 050 800S25 5117 18918 001 100S26 1258 35015 001 100S27 3759 45856 001 100S28 2425 28416 001 500S29 10350 18754 050 300S30 2000 25618 100 100S31 7518 35895 050 500S32 1825 45210 100 300S33 2480 15415 050 200S34 2627 12748 001 400S35 1520 17535 001 200S36 1250 25014 100 200S37 1254 18770 050 800S38 8500 14825 050 400S39 6430 25648 001 400S40 2060 45710 001 300S41 4620 15415 100 100S42 1480 8716 001 200S43 2655 9658 050 100S44 1678 10211 001 200S45 6420 13260 100 500

proposed a method to reduce total inventory cost by thecombination of ABC and 123 analysis

3 Fuzzy C-Means (FCM) Algorithm

e fuzzy c-means (FCM) algorithm is a useful tool forclustering which partitions a real 119899119899-dimensional datasetinto c fuzzy clusters by minimizing the clustering objective

function to describe an underlying structure within the data[14]

Although the nal version of the FCM algorithm wasintroduced by Bezdek the origins of the algorithm are tracedback to Dunn [15] Aerwards several studies have usedthe FCM algorithm in many engineering elds such as part-machine grouping in cellular manufacturing [16] imagesegmentation [17 18] new product development (NPD)


T 2 Normalized value data set of all items at the company


[19] health sector [20] geology [21] data envelopmentanalysis [22] business [23] and so on

In the case of the well-separated data group the hardclustering approach can be a natural solutionWhen the clus-ters overlap and some of the data partially belong to severalclusters as seen in Figure 1 the fuzzy clustering is a natural

way to describe this situation and the membership degree ofa data object to a cluster is a value between the interval [0 1]

At FC the clusters are dened with respect to clusternumbers (119888119888) and initial membership values of the inputvector e memberships of the clusters are dened with thecorresponding membership values Also at the algorithm


X2

X1

F 1 Illustration of fuzzy clustering [24]

clusters are described by prototypes which represent thecluster centers

It is an iteratively optimal algorithm based on the iterativeminimization of the objective function in (1) as follows

119869119869119898119898 (119880119880119880119880119880) =11989911989910055761005576119896119896=119896

11988811988810055761005576119894119894=119896

11990611990611989811989811989611989611989411989410092651009265119909119909119896119896 minus 119907119907119894119894100926510092652 (1)

In (1) 119899119899 is the total number of data vectors in a given dataset and 119888119888 is the number of clusters 119883119883 = 119883119909119909119896119880 1199091199092119880hellip 119880 119909119909119899119899 sub119877119877119904119904 and 119880119880 = 119883119907119907119896119880 1199071199072119880hellip 119880 119907119907119888119888 sub 119877119877119904119904 are the feature data andcluster centers 119880119880 = (119906119906119896119896119894119894)119899119899119899119888119888 is a fuzzy partition matrix thatis composed of the membership of each feature vector 119909119909119896119896 ineach cluster 119894119894 Here 119906119906119896119896119894119894 should satisfy sum119888119888

119894119894=119896 119906119906119896119896119894119894 = 119896 for 119896119896 =119896119880 2119880hellip 119880 119899119899 and 119906119906119896119896119894119894 ge 0 for all 119894119894 = 119896119880 2119880hellip 119880 119888119888 and 119896119896 = 119896119880 2119880hellip 119880 119899119899e exponent 119898119898 119898 119896 in (1) is a parameter called fuzzier Tominimize (1) the cluster centers 119907119907119894119894 and membership matrix119880119880 need to be calculated with regards to the following iterativeformula

119906119906119896119896119894119894

=

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1007656100765611988811988810055761005576119895119895=1198961007655100765510092651009265119909119909119896119896 minus 1199071199071198941198941009265100926510092661009266119909119909119896119896 minus 11990711990711989511989510092661009266

100767110076712(119898119898minus119896)

10076721007672

minus119896

if 10092661009266119909119909119896119896 minus 11990711990711989511989510092661009266 119898 0119880

119896119880 if 10092651009265119909119909119896119896 minus 11990711990711989411989410092651009265 = 01198800119880 if exist119895119895 119895 119894119894 10092661009266119909119909119896119896 minus 11990711990711989511989510092661009266 = 0119880

for 119870119870 = 119896119880hellip 119880 119899119899119880 119894119894 = 119896119880hellip 119880 119888119888119880(2)

119907119907119894119894 =sum119899119899119896119896=119896 119906119906

119898119898119896119896119894119894 sdot 119909119909119896119896

sum119873119873119894119894=119896 119906119906119898119898119896119896119894119894

119880 119894119894 = 119896119880 2119880hellip 119880 119888119888 (3)

e procedure of the Fuzzy C-Means (FCM) algorithm is asseen hereinaer

Step 1 Input the number of clusters 119888119888 the fuzzier119898119898 and thedistance function sdot

Step 2 Initialize the cluster centers 1199071199070119894119894 (119894119894 = 119896119880 2119880hellip 119880 119888119888)

T 3 e cluster memberships of all items aer FCM

SKUs 119862119862119896 1198621198622 119862119862119862S1 0120297 0044415 0835287S2 0482047 0116516 0401437S3 0312767 013446 0552773S4 0045124 0919403 0035473S5 029339 025688 044973S6 0221803 0649321 0128876S7 073305 0125228 0141722S8 0055329 0910642 0034029S9 0127868 0055433 0816699S10 0225196 0380499 0394306S11 0565756 0102472 0331772S12 0527109 027132 0201571S13 0567224 0143353 0289423S14 0299279 0241065 0459655S15 0175279 0087093 0737628S16 0305682 0261788 043253S17 023184 0611386 0156774S18 0696526 0149756 0153718S19 0062379 0884495 0053126S20 0435864 0284404 0279732S21 0414061 0128463 0457476S22 0524111 0227003 0248886S23 0444989 0119633 0435377S24 0620943 016766 0211397S25 0142157 0747277 0110566S26 0098581 080764 0093779S27 0143794 0708718 0147487S28 018746 0714468 0098072S29 0449008 0232609 0318382S30 0244025 0145713 0610262S31 0604867 0142195 0252938S32 0202419 0094825 0702756S33 0326855 0407832 0265313S34 015908 0758182 0082737S35 0076982 0866962 0056056S36 0273608 0143173 0583219S37 0527114 0247646 0225239S38 0567682 0188999 0243319S39 0262399 0601451 0136149S40 0132168 0756197 0111636S41 0292748 0140513 0566739S42 0137553 076635 0096096S43 0299306 0421279 0279415S44 0123892 0789745 0086362S45 0559776 0096093 0344132

Step 3 Compute 119906119906119896119896119894119894 (119896119896 = 119896119880 2119880hellip119899119899119896 119894119894 = 119896119880 2119880hellip 119880 119888119888) by using(2)

Step 4 Compute 119907119907119896119894119894 (119894119894 = 119896119880 2119880hellip 119880 119888119888) by using (3)


T 4 Cluster members and labels

Labels Cluster members Mean of each clusterA S2 S7 S11 S12 S13 S18 S20 S22 S23 S24 S29 S31 S37 S38 S45 200B S1 S3 S5 S9 S10 S14 S15 S16 S21 S30 S32 S36 S41 196C S4 S6 S8 S17 S19 S25 S26 S27 S28 S33 S34 S35 S39 S40 S42 S43 S44 078

Step 5 Ifmax1le119894119894le119894119894(1199071199070119894119894 minus 119907119907

1119894119894 119907119907

1119894119894 ) le 120576120576 then go to Step 6 else

let 1199071199070119894119894 = 1199071199071119894119894 (119894119894 = 1119894 119894119894119894 119894 119894119894) and go to Step 3

Step 6 Output the clustering results cluster centers 1199071199071119894119894 (119894119894 =1119894 119894119894119894 119894 119894119894) membership matrix119880119880 and in some applicationsthe elements of each cluster 119894119894 that is all the 119909119909119896119896 such that 119906119906119896119896119894119894 gt119906119906119896119896119896119896 for all 119896119896 119895 119896119896

Step 7 Stop [25]

MATLAB R2010a is used to perform FCM ABC analysismethodologymentioned previously In the following sectiona real life problem has been introduced and the applicationof the FCM algorithm for ABC analysis has been explained

4 Numerical Example

In this study an alternative ABC analysis based on FCMis proposed for an automotive company In the proposedmethod the SKUs are clusteredwith FCMus the items areseparated to different clusters according to their attributese major advantage of FCM is not only attaching dataobjects to decisively one cluster but also determining themembership degrees to all other formed clusters e SKUwith high membership degree indicates the characteristic ofcluster well more than the other members

e empirical investigation for this study was carriedout by using a real life data obtained from an automotivecompany located in Turkey A data set containing 45 items asseen in Table 1was used to design the frame of the study Eachdata represented the classication of items with regard tofour attributes average unit cost annual dollar usage criticalfactor and lead time ese attributes were dened basedon the literature review and the opinions of the companymanagers

As it is seen in Table 1 the variables of the average unitcost and the annual dollar usage are much higher than thevariables of the critical factor and the lead time ereforethis study includes preprocessing In the preprocessing partwe normalize the data set to state all criteria in the same way

Normalized data are calculated according to the followingequation

119873119873119873119873119894119894119894119896119896 =119873119873 10076491007649119894119894119894 11989611989610076651007665 minus min 1007649100764911989611989610076651007665max 1007649100764911989611989610076651007665 minus min 1007649100764911989611989610076651007665

119894 (4)

where 119873119873 is the input value119873119873119873119873 is the normalized value 119894119894 is thenumber of items and 119896119896 represents the criterion number [26]us Table 2 is obtained

Aer executing the FCMcode inMatlab themembershipdegrees of the SKUs to the clusters are obtained Table 3

presents the membership degrees of each item to all clustersFor instance item S1 belongs to clusters C1 C2 and C3with membership degrees 0120297 0044415 and 0835287respectively See Tables 1 2 and 3

Aer classifying thematerials by these attributes the nextstep is labeling themasA B andC as seen inTable 4 Labelingof the classes is calculated in three steps At the rst stepnormalized criteria values are summed up for each SKU Sec-ondly the SKU criteria values in the same cluster are summedup Finally an average value is obtained by dividing the totalvalue of each cluster by the number of SKUse cluster withthe highest mean is labeled as A the second cluster is labeledas B and the lowest is labeled as C

As it is seen in Tables 3 and 4 all items are attached toclusters with regard to the highest membership degrees Forinstance S2 is attached to the cluster A with 0482047 and S1is attached to the cluster Bwith 0835287membership degreeIn addition to this when we examined S10 we noticed thatas a close membership degree this item belongs to clusterC with a membership degree of 0380499 and also belongsto cluster B with a membership degree of 0394306 Here itshould be considered that this item could take part at clusterC depending on decision maker and conditions (ie leadtime cost frequency of use etc) Another subject to bediscussed is about items S2 and S7 e membership degreeof S2 to cluster A is 0482047 and S7 is 073305 Under thiscircumstance it can be said that S7 represents cluster A betterthan S2

As seen from Table 4 the method clusters items 2 7 1112 13 18 20 22 23 24 29 31 37 38 and 45 as cluster Athat required the highest attention erefore the companyshould have to pay more attention to this type of SKUs

5 Discussion and Conclusion

Several methods are used for inventory classication Classi-cation has emerged as an important decision-making tool inbusiness life such as credit scoring and classifying inventoryitems [2]

At conventional inventory classication the classicationanalysis is based on a single criterion the annual usage costwhich is simply the product of annual usage and dollarvalue per unit item Although it is very straightforward andnancially viable to apply such an analysis the method hassome quirks Decision makers relying on this traditionalanalysis ignore some very crucial qualitative criteria To over-come the limitations of the traditional classication analysismany researchers concentrated on incorporating multicrite-ria judgments into the inventory classication procedure [7]such as average unit cost criticality and lead time


In this paper FCM is used as an effective solutionmethodfor ABC analysis for the rst time is solution methodeliminates the criticism related to the subjectivity involvedin weighing the criteria at the classical multicriteria ABCanalysis FCMalgorithmnot only selects the inventory(ies) tocare about mostly but also clusters all of the items accordingto their membership degrees Consequently FCM can beeasily and eciently applied to all sectors and all rms due toits eibility nder these circumstances the FCM algorithmapplied in this paper is a valuable alternative among the othermodeling methods to any ABC analysis problem

References

[1] F Y Partovi and M Anandarajan ldquoClassifying inventory usingan articial neural network approachrdquo Computers and Indus-trial Engineering vol 41 no 4 pp 389ndash404 2001

[2] C-W Chu G S Liang and C T Liao ldquoControlling inventoryby combiningABC analysis and fuzzy classicationrdquoComputersand Industrial Engineering vol 55 no 4 pp 841ndash851 2008

[3] R Ramanathan ldquoABC inventory classication with multiple-criteria using weighted linear optimizationrdquo Computers andOperations Research vol 33 no 3 pp 695ndash700 2006

[4] H A Guvenir and E Erel ldquoMulticriteria inventory classica-tion using a genetic algorithmrdquo European Journal of OperationalResearch vol 105 no 1 pp 29ndash37 1998

[5] M C Yu ldquoMulti-criteria ABC analysis using articial-intelli-gence-based classication techniquesrdquo Expert Systems withApplications vol 38 no 4 pp 3416ndash3421 2011

[6] B E Flores D L Olson and V K Dorai ldquoManagement ofmulticriteria inventory classicationrdquo Mathematical and Com-puter Modelling vol 16 no 12 pp 71ndash82 1992

[7] O Cakir and M S Canbolat ldquoA web-based decision supportsystem for multi-criteria inventory classication using fuzzyAHP methodologyrdquo Expert Systems with Applications vol 35no 3 pp 1367ndash1378 2008

[8] Y Chen K W Li D Marc Kilgour and K W Hipel ldquoAcase-based distance model for multiple criteria ABC analysisrdquoComputers and Operations Research vol 35 no 3 pp 776ndash7962008

[9] B E Flores andDCWhybark ldquoImplementingmultiple criteriaABC analysisrdquo Journal of Operations Management vol 7 no 1-2 pp 79ndash85 1987

[10] P Zhou and L Fan ldquoA note on multi-criteria ABC inventoryclassication using weighted linear optimizationrdquo EuropeanJournal of Operational Research vol 182 no 3 pp 1488ndash14912007

[11] W L Ng ldquoA simple classier formultiple criteria ABC analysisrdquoEuropean Journal of Operational Research vol 177 no 1 pp344ndash353 2007

[12] A Hadi-Vencheh ldquoAn improvement to multiple criteriaABC inventory classicationrdquo European Journal of OperationalResearch vol 201 no 3 pp 962ndash965 2010

[13] I Al Kattan and A Bin Adi ldquoMulti-criteria decision making ontotal inventory cost and technical readinessrdquo International Jour-nal Interaction Des Manufacturing vol 2 no 3 pp 137ndash1502008

[14] D Li H Gu and L Zhang ldquoA fuzzy c-means clusteringalgorithm based on nearest-neighbor intervals for incompletedatardquo Expert Systems with Applications vol 37 no 10 pp6942ndash6947 2010

[15] G E Tsekouras and H Sarimveis ldquoA new approach for meas-uring the validity of the fuzzy c-means algorithmrdquo Advances inEngineering Soware vol 35 no 8-9 pp 567ndash575 2004

[16] S Lozano D Dobado J Larraeta and L Onieva ldquoModiedfuzzy C-means algorithm for cellular manufacturingrdquo FuzzySets and Systems vol 126 no 1 pp 23ndash32 2002

[17] J Kang LMin Q Luan Li and J Liu ldquoNovelmodied fuzzyc-means algorithm with applicationsrdquo Digital Signal Processingvol 19 no 2 pp 309ndash319 2009

[18] Z Yang F L Chung and W Shitong ldquoRobust fuzzy clus-tering-based image segmentationrdquo Applied So ComputingJournal vol 9 no 1 pp 80ndash84 2009

[19] W Yan C H Chen and M D Shieh ldquoProduct concept gen-eration and selection using sorting technique and fuzzy c-means algorithmrdquo Computers and Industrial Engineering vol50 no 3 pp 273ndash285 2006

[20] Z Hou W Qian S Huang Q Hu and W L NowinskildquoRegularized fuzzy c-meansmethod for brain tissue clusteringrdquoPattern Recognition Letters vol 28 no 13 pp 1788ndash1794 2007

[21] A B Goktepe S Altun and A Sezer ldquoSoil clustering by fuzzyc-means algorithmrdquo Advances in Engineering Soware vol 36no 10 pp 691ndash698 2005

[22] A Azadeh M Anvari B Ziaei and K Sadeghi ldquoAn integratedfuzzyDEA-fuzzyC-means-simulation for optimization of oper-ator allocation in cellularmanufacturing systemsrdquo InternationalJournal of AdvancedManufacturing Technology vol 46 no 1ndash4pp 361ndash375 2010

[23] I Ozkan I B Tuumlrkşen and N Canpolat ldquoA currency crisis andits perception with fuzzy C-meansrdquo Information Sciences vol178 no 8 pp 1923ndash1934 2008

[24] S A Mingoti and J O Lima ldquoComparing SOM neural networkwith Fuzzy c-means K-means and traditional hierarchical clus-tering algorithmsrdquo European Journal of Operational Researchvol 174 no 3 pp 1742ndash1759 2006

[25] H Sun S Wang and Q Jiang ldquoFCM-based model selectionalgorithms for determining the number of clustersrdquo PatternRecognition vol 37 no 10 pp 2027ndash2037 2004

[26] G A Keskin S Ilhan and C Oumlzkan ldquoe Fuzzy ART algo-rithm a categorization method for supplier evaluation andselectionrdquo Expert Systems with Applications vol 37 no 2 pp1235ndash1240 2010

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In this paper we are concerned with the multicriteriaclassication method aimed to facilitate inventory manage-ment e classication method is proposed through a casestudy of an automative company Being distinct from thetechniques used for multicriteria ABC analysis the criteriaare not weighed in this method erefore this method doesnot involve subjectivity In addition unlike other methodsour method not only determines the matching to any of theclusters but also gives the membership degrees to all clustersese are our contributions to the literature

e aim of this study is to develop a decision-makingapproach to the ABC analysis method which is a multicri-teria decision-making problem e remainder of the paperis organized as follows ABC analysis is described in Section2 In Section 3 fuzzy c-means (FCM) clustering is explainedIn Section 4 the FCM is applied to a real life problem Finallyconclusions are drawn in Section 5

2 ABC Analysis

e conventional ABC classication based on Pareto prin-ciple developed at General Electric during the 1950s is themost popularmethod to classify the items in inventory [4] Inthis classication inventory items are ordered descendinglywith respect to their annual dollar usage valuese relativelysmall number of items at the top of the list (approx 10)controlling the majority of the total annual dollar usageconstitute class A the majority of the items at the bottom ofthe list (approx 60) controlling a relatively small portionof the total annual dollar usage constitute class C and theitems between the two classes constitute class B (approx30)is classicationmay not always be accurate but it hasbeen found too close to the existing occurrence in companieswith remarkable accuracy Class A inventory items requirecautious inventory control because they represent a largepercentage of the total dollar value of the inventory isstate requires certain demand forecasts and detailed recordkeeping Class C inventory items should receive a exiblecontrol Class B items should have a control effort that liesbetween A and C categories [1 2]

Because of its easy-to-implement nature applicabilityto numerous situations empirically observed benets andremarkable effectiveness in many inventory systems thisapproach is still popularly used in practice However themethod has a serious drawback that may inhibit the effec-tiveness of the procedure in some situations e criterionused in the conventional ABC classication is the annualdollar usage so using one criterion may create problems ofsignicant nancial loss [4 8] us the inventory controlproblem becomes a multicriteria inventory classication thathas been studied by some researchers In the literature someof the considered criteria include inventory cost criticalitylead time commonality obsolescence substitutability num-ber of requests for the item in a year scarcity durabilitysubstitutability repairability order size requirement stock-ability demand distribution and stockout penalty cost [2 3]

21 Related Research In recent years several new mul-ticriteria approaches to inventory classication have been

introduced but all have some drawbacks Classical ABCclassication approach on two criteria is a step forward inmulticriteria ABC classication However the methodologyis relatively difficult to use when more criteria have to beconsidered ere is no obvious way to extend the procedureto more than two criteria [2] e rst paper about ABCanalysis was presented by Flores and Whybark in 1986 [9]Affected from this study Chen et al [8] have presented a case-based multicriteria ABC analysis by accounting additionalcriteria such as lead time and criticality of SKUs to providemanagerial exibility

In 1990 amethodology based on statistical clusteringwasproposed by Ernst and Cohen But this approach requiressubstantial data the use of factor analysis and a clusteringprocedure which may render it impractical in typical stock-room environments Furthermore the clusters themselvesmust be reevaluated in order to classify new stock items sothere is a chance that previously classied stock may end upbeing classied differently every time new items are addedand this may disturb the inventory control procedure Inshort the model may be too sophisticated for the averagemanager [1]

e AHP process has also been proposed by severalauthors for ABC classication [6] Cakir and Canbolat [7]have proposed AHP with integrating fuzzy technique tosolve the multicriteria inventory classication problem eadvantage of the AHP is that it can incorporate many criteriaand its ease of use on amassive accounting andmeasurementsystemOne of the important drawbacks of themethod is thata signicant amount of subjectivity is involved in pairwisecomparisons of criteria rating levels and assigning a ratinglevel and associated weights [1 2]

Articial intelligence is another method for multicriteriainventory classication Guvenir and Erel [4] have presenteda new approach using AHP technique and genetic algorithmto multicriteria classication Articial neural network isanother technique which is applicable to the classicationprocess Partovi and Anandarajan [1] have proposed anarticial neural network (ANN) by utilizing two learningmethods in the ANN named back-propagation and geneticalgorithms for ABC classication of SKUs Ramanathan [3]has proposed a simple weighted linear optimization modelto address the multicriteria inventory classication prob-lem which is similar to data envelopment analysis (DEA)Zhou and Fan [10] have presented an extended version ofthe Ramanathanrsquos model by incorporating some balancingfeatures for multicriteria ABC inventory classication eclassication results are compared with those of Bayes andother fuzzy classiers It is shown that the proposed methodis superior to them Chu et al [2] have proposed a newinventory control approach by integrating ABC and fuzzyclassication

Different from what is mentioned previously Ng [11] hasproposed an alternative weight linear optimization modelfor multicriteria inventory classication by using the ABCprinciple Hadi-Vencheh [12] has presented an extendedversion of the Ng-model which is a simple classier formulticriteria ABC analysis Al Kattan and Bin Adi [13] have

















X2

X1




119869119869119898119898 (119880119880119880119880119880) =11989911989910055761005576119896119896=119896

11988811988810055761005576119894119894=119896

11990611990611989811989811989611989611989411989410092651009265119909119909119896119896 minus 119907119907119894119894100926510092652 (1)



119906119906119896119896119894119894

=

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1007656100765611988811988810055761005576119895119895=1198961007655100765510092651009265119909119909119896119896 minus 1199071199071198941198941009265100926510092661009266119909119909119896119896 minus 11990711990711989511989510092661009266

100767110076712(119898119898minus119896)

10076721007672

minus119896

if 10092661009266119909119909119896119896 minus 11990711990711989511989510092661009266 119898 0119880



119907119907119894119894 =sum119899119899119896119896=119896 119906119906

119898119898119896119896119894119894 sdot 119909119909119896119896

sum119873119873119894119894=119896 119906119906119898119898119896119896119894119894

119880 119894119894 = 119896119880 2119880hellip 119880 119888119888 (3)












1119894119894 119907119907


let 1199071199070119894119894 = 1199071199071119894119894 (119894119894 = 1119894 119894119894119894 119894 119894119894) and go to Step 3


Step 7 Stop [25]


4 Numerical Example






119894 (4)












References




























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of








Journal of










Volume 2014

RoboticsJournal of


















X2

X1




119869119869119898119898 (119880119880119880119880119880) =11989911989910055761005576119896119896=119896

11988811988810055761005576119894119894=119896

11990611990611989811989811989611989611989411989410092651009265119909119909119896119896 minus 119907119907119894119894100926510092652 (1)



119906119906119896119896119894119894

=

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1007656100765611988811988810055761005576119895119895=1198961007655100765510092651009265119909119909119896119896 minus 1199071199071198941198941009265100926510092661009266119909119909119896119896 minus 11990711990711989511989510092661009266

100767110076712(119898119898minus119896)

10076721007672

minus119896

if 10092661009266119909119909119896119896 minus 11990711990711989511989510092661009266 119898 0119880



119907119907119894119894 =sum119899119899119896119896=119896 119906119906

119898119898119896119896119894119894 sdot 119909119909119896119896

sum119873119873119894119894=119896 119906119906119898119898119896119896119894119894

119880 119894119894 = 119896119880 2119880hellip 119880 119888119888 (3)












1119894119894 119907119907


let 1199071199070119894119894 = 1199071199071119894119894 (119894119894 = 1119894 119894119894119894 119894 119894119894) and go to Step 3


Step 7 Stop [25]


4 Numerical Example






119894 (4)












References




























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of








Journal of










Volume 2014

RoboticsJournal of










X2

X1




119869119869119898119898 (119880119880119880119880119880) =11989911989910055761005576119896119896=119896

11988811988810055761005576119894119894=119896

11990611990611989811989811989611989611989411989410092651009265119909119909119896119896 minus 119907119907119894119894100926510092652 (1)



119906119906119896119896119894119894

=

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1007656100765611988811988810055761005576119895119895=1198961007655100765510092651009265119909119909119896119896 minus 1199071199071198941198941009265100926510092661009266119909119909119896119896 minus 11990711990711989511989510092661009266

100767110076712(119898119898minus119896)

10076721007672

minus119896

if 10092661009266119909119909119896119896 minus 11990711990711989511989510092661009266 119898 0119880



119907119907119894119894 =sum119899119899119896119896=119896 119906119906

119898119898119896119896119894119894 sdot 119909119909119896119896

sum119873119873119894119894=119896 119906119906119898119898119896119896119894119894

119880 119894119894 = 119896119880 2119880hellip 119880 119888119888 (3)












1119894119894 119907119907


let 1199071199070119894119894 = 1199071199071119894119894 (119894119894 = 1119894 119894119894119894 119894 119894119894) and go to Step 3


Step 7 Stop [25]


4 Numerical Example






119894 (4)












References




























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of








Journal of










Volume 2014

RoboticsJournal of



X2

X1




119869119869119898119898 (119880119880119880119880119880) =11989911989910055761005576119896119896=119896

11988811988810055761005576119894119894=119896

11990611990611989811989811989611989611989411989410092651009265119909119909119896119896 minus 119907119907119894119894100926510092652 (1)



119906119906119896119896119894119894

=

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1007656100765611988811988810055761005576119895119895=1198961007655100765510092651009265119909119909119896119896 minus 1199071199071198941198941009265100926510092661009266119909119909119896119896 minus 11990711990711989511989510092661009266

100767110076712(119898119898minus119896)

10076721007672

minus119896

if 10092661009266119909119909119896119896 minus 11990711990711989511989510092661009266 119898 0119880



119907119907119894119894 =sum119899119899119896119896=119896 119906119906

119898119898119896119896119894119894 sdot 119909119909119896119896

sum119873119873119894119894=119896 119906119906119898119898119896119896119894119894

119880 119894119894 = 119896119880 2119880hellip 119880 119888119888 (3)












1119894119894 119907119907


let 1199071199070119894119894 = 1199071199071119894119894 (119894119894 = 1119894 119894119894119894 119894 119894119894) and go to Step 3


Step 7 Stop [25]


4 Numerical Example






119894 (4)












References




























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of








Journal of










Volume 2014

RoboticsJournal of






1119894119894 119907119907


let 1199071199070119894119894 = 1199071199071119894119894 (119894119894 = 1119894 119894119894119894 119894 119894119894) and go to Step 3


Step 7 Stop [25]


4 Numerical Example






119894 (4)












References




























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of








Journal of










Volume 2014

RoboticsJournal of




References




























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of








Journal of










Volume 2014

RoboticsJournal of



VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of








Journal of










Volume 2014

RoboticsJournal of


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