A Similarity-Based Approach for the All-Time Demand Prediction of New Automotive Spare Parts

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1 Proceedngs of the 51 st Hawa Internatonal Conference on System Scences 2018 A Smlarty-Based Approach for the All-Tme Demand Predcton of New Automotve Spare Parts Danel Steuer Karlsruhe Insttute of Technology d.b.steuer@gmal.com Peter Korevaar IBM Germany korevaar@de.bm.com Verena Hutterer IBM Germany verena.hutterer@de.bm.com Hansjörg Fromm Karlsruhe Insttute of Technology hansjoerg.fromm@kt.edu Abstract The all-tme demand predcton of new automotve spare parts s an area that has not receved much attenton n theory and practce n the past. Ths paper presents a new approach for tacklng ths mportant, but complex problem. The approach assumes that new parts wll have smlar demand patterns as comparable parts had n the past. For the study, the demand hstores of spare parts were provded from a German automotve manufacturer. In a frst step, hstorcal fullcycle demand patterns were clustered based on smlarty, and a representatve demand pattern (RDP) was determned for each cluster. In a second step, a classfcaton model was traned on the hstorcal data relatng demand patterns wth selected parts characterstcs. In a thrd step, the classfer was used to predct for each new part to whch cluster t belongs. The RDP of ths cluster s then used to calculate the part s all-tme demand. The developed models have been valdated usng standard qualty measures used n analytcs and wll now be used n practce. 1. Introducton Accordng to Arthur D. Lttle [14], 23% of the overall revenue and 54% of the proft of automotve manufacturers (OEMs) come from aftersales servces and parts. On the other hand, 50-70% of the spare parts nventory s often unused. Ths means that even f the spare parts busness s of so hgh mportance forecastng and nventory plannng methods are stll nadequate and need to be mproved. The Automotve Industry has seen a steady ncrease n complexty over the last decades. The number of models and model varants that the OEMs release to market has grown sgnfcantly. And accordngly, the URI: ISBN: (CC BY-NC-ND 4.0) overall number of spare parts. Yet, the OEMs ensure the avalablty of spare parts for years after end of producton (EOP) of a vehcle. Wth every new model, thousands of new parts are ntroduced for whch the OEM has to predct the total demand of requred spare parts over ther complete lfecycles. Ths s an mportant prerequste for sourcng decsons and prce negotatons wth the parts supplers. Due to ts complexty, lfecycle forecastng methods for spare parts have not receved adequate attenton n theory and practce. Academc research and ndustral practce have largely concentrated on short-term methods for spare parts forecastng and nventory control, whch are already rather sophstcated [3, 4, 23]. Lfecycle forecastng s stll predomnantly based on judgement. Lfecycle forecastng for automotve spare parts bears some ndustry-specfc problems, a mxture of market and technologcal dependences. The major unknowns are: - The market success (sales) of the prmary product (the vehcle) tself - The lfecycle (usage tme) of the prmary product and the tme that the vehcle stays n the servce responsblty of the OEM (before mgratng to the ndependent aftermarket) - The relablty and wear-out characterstcs of parts (unknown especally for new parts), whch determne the number of tmes a part gets replaced durng the lfecycle of a vehcle Each of these unknowns represents a challenge for tself. Altogether, they consttute a complex forecastng problem. As wll be shown later, ths brought us to choose an emprcal approach n order to tackle ths complex problem. Page 1525

2 2. Related Work Related work s found n dfferent areas of research: new product sales forecastng and end-of-lfe forecastng. All of these areas, however, cover only sngle aspects of our problem. New product sales forecastng. Ths area has a long hstory n marketng research. Mas-Machuca et al. [15] gve an overvew of methods that have been appled for new product sales forecastng. They dstngush between (a) judgmental methods, (b) consumer and market research, (c) cause and effect models, and (d) artfcal ntellgence. An example of a combned judgmental /quanttatve model s presented n Goodwn et al. [8] who estmate the parameters of a Bass model by regresson wth hstorcal sales data of analogous products and assume that the parameters wll be applcable for the new product. The Bass model [1] s a combnaton of exponental functons wth two parameters, the coeffcent of nnovaton and the coeffcent of adopton, that allows to model a multtude of typcal market penetraton curves. The underlyng assumptons of Bass make the model partcularly suted for frequently purchased consumer products. Snce the demand of spare parts s sporadc and extends over a long perod of tme, these models are of lmted nterest for our case. Thomassey and Hapette [21] descrbe a forecastng approach for new apparel tems very smlar to the approach we have chosen. The sales tme seres of hstorcal tems are clustered usng self-organzng maps (SOM). The clusters are representng prototypes for typcal sales hstores. A classfcaton procedure based on probablstc neural networks establshes a relaton between the prototypes and ndvdual tem characterstcs. For each new tem (wthout any sales hstory), the classfcaton procedure selects a prototype accordng to the new tem s characterstcs and apples the prototype s sales curve to predct the sales future of the new tem. The approach of Thomassey and Hapette was not lmted by the number of predefned functons, but was appled n a fast-sellng fashon envronment. Therefore, a smple transferablty to spare parts could not be assumed. End-of-lfe forecastng. End-of-lfe forecastng dffers from new product forecastng by the exstence of a partal demand hstory [7, 22]. Jonas et al. [12] descrbe an approach that assumes that electronc spare parts of the same commodty exhbt smlar demand patterns n the future, as they dd n the past. They ft a connected lne segment model (CLSM) consstng of three dfferent phases to hstorcal demand tme seres, normalze t, and obtan typcal demand models (TDM) by clusterng all normalzed CLSMs. For demand predcton, they use a fuzzy smlarty approach to ft the partal demand hstory of an already actve part wth the approprate TDM. Other approaches usng partal demand hstores are presented by Mexell and Wu [16] and by Bensng [2]. It s clear that for the predcton of new parts wthout demand hstory, end-of-lfe forecastng cannot be drectly used, even f some of the appled technques are very relevant. 3. Our Approach Our approach s largely determned by the requrements that (a) no demand hstory s avalable for the new parts to be predcted, (b) no predefned demand shape shall be assumed for the predcton, and (c) no expert knowledge s needed n the process. These requrements could be fulflled n a three-step approach consstng of the steps clusterng, classfcaton, and predcton (Fgure 1). In step 1, a large amount of full-lfecycle demand patterns s frst normalzed and then clustered accordng to ther shapes. A representatve demand pattern (RDP) s selected to represent a cluster as the bass for the later predcton. In step 2, a classfcaton model s traned on the same hstorcal data and relates certan characterstcs of the parts wth the dentfed clusters. In step 3, the classfcaton model s used to predct the approprate cluster for a new part based on ts characterstcs. The representatve demand pattern (RDP) of the predcted cluster s used to make a demand forecast for the new part. Fgure 1. Three-step approach for the smlarty-based all-tme demand predcton of new spare parts Page 1526

3 3.1. Data Source The data was already collected and ts qualty ensured before ths research started. It conssts of spare parts master data, transacton data and replacement nformaton. The monthly and yearly demand data s avalable from 1988 untl today. The data sources were already prepared for modelng n general before ths research started and a set of records wth at least 18 consecutve years of demand hstory wthn the gven tme perod were ntally dentfed as sutable parts from the prepared data set. Each of these records stands for one spare part, contanng 243 dfferent data felds and s unquely dentfable by ts key attrbute, the spare part number. The other felds carry nformaton regardng the hstorcal demand and characterstcs of each spare part. Moreover, smoothed demand values of the ndvdual parts were made avalable. Those demand values were created by a so called cubc splne ft of the hstorcal aggregated yearly demand. Cubc splne fttng s a well-known method for non-lnear regresson modellng. A cubc splne s constructed of pecewse thrd-order polynomals ax³ + bx² + cx + d wth approprate coeffcents a, b, c and d values [6, 9]. Two of the four coeffcents of each pecewse polynomal are used to lnk the peces such that they ft seamlessly and are dfferentable at the fttng ponts. The remanng coeffcents are used to ft the full curve to the data such that a least square ft s made and the sum of the smoothed demand values equals the orgnal demand values Data Preparaton Not every spare part n the gven data set fulflls the crtera of a full demand lfecycle meanng that the start and the end of the lfecycle should be from 1988 untl today. The earlest start of producton (SOP) of one spare part was for example 1946 and the most recent one 2001, meanng that the parts are n dfferent phases of ther product lfecycle and we cannot see the whole lfecycle snce demand data s only avalable for the years of 1988 untl today, resultng n demand data that only represents a fracton of ther lfecycle. Therefore, a selecton of full-lfecycle parts out of all spare parts s necessary. Snce the model attempts to analyze and predct the all-tme demand of spare parts, only parts wth full demand lfecycles can be ncluded for learnng. To ensure full demand lfecycles the followng crtera s mplemented, resultng n parts that can be ncluded nto the model: - Number of hstorcal demand years > 18 (together wth the end of producton (EOP) date, a long enough demand hstory s ensured) - EOP <= 2009 In a second preparaton step, the demand data of every part s shfted to the same start year n order to be able to compare the smlarty of ther demand lfecycle patterns. To correctly assess the smlarty of demand patterns of dfferent parts, the patterns need to have the same length and start at the same tme otherwse the demand patterns are not comparable. Snce the parts have dfferent SOP and EOS (end of servce) dates, an adjustment s necessary. To realze ths adjustment, the demand or respectvely the part s SOP date s shfted n tme, such as f all parts would have the same SOP date. The thrd step n the data preparaton process for clusterng s the normalzaton of the demand data. After shftng the demand data to the same startng pont, the demand data D of each part needs to be normalzed n order to compare the demand patterns of the parts. The reason for normalzaton s to make sure that parts wth the same shape of the demand pattern are clustered together even though some of them mght have a hgh overall demand and some have a lower overall demand. Normalzaton s done by dvdng the actual demand at the tme t by the sum of the demand of the full lfecycle for each ndvdual part. The last step wthn the data preparaton for clusterng s the creaton of the dstance matrx or dssmlarty matrx for the comparson of the smlarty between each part n order to cluster them by smlarty. Snce the clusterng s based on the smlarty of demand patterns, the dstance between every ndvdual demand pattern to all other patterns needs to be calculated. In consderaton of the fact that the spare part demand has already been shfted and normalzed, the dstance between two demand patterns s calculated by capturng the dstance between the ndvdual demand data ponts over tme. Due to normalzaton, the total demand over tme for each part equals 1, whch means for the measurement of dstance of the ndvdual parts that t can be bascally descrbed as the dstance between hstograms. When t comes to smlarty or dstance measurement of hstograms, several dstance measures exst. Wthn ths research, the ch-square (χ2) dstance s used snce t works well for the gven scenaro [18]. The χ2 hstogram dstance-measure orgnates from the χ2 test-statstc where t s used to test the ft between a dstrbuton functon and observed frequences [20]. The χ2 dstance, between two parts s represented by the followng formula: - SOP = /- 3years Page 1527

4 where T t1 ( xt ( x t 2 yt ) y ) xt = normalzed demand of part x at tme t yt = normalzed demand of part y at tme t T = total tmespan of avalable demand data t The values of the χ2 dstance range between [0,1], where a dstance of 0 equals a hgh smlarty and 1 equals low smlarty. Thus, a quadratc dstance matrx s created, whch s the bass for the creaton of the clusters Clusterng From the avalable parts, exhbted a full demand lfecycle and could therefore be ncluded n the clusterng process. A revew of clusterng technques resulted n the selecton of k-medods clusterng, snce t s resstant to outlers and uses actual parts as centrods for the clusters. The avalable algorthms are PAM (parttonng around medods) and CLARA (CLusterng LARge Applcatons). We started wth PAM and reserved CLARA for cases wth run tme capacty constrants. As an teratve algorthm, PAM requres a set of parts that are handled as ntal cluster centers [11]. For ths purpose, we selected parts that have many smlar parts and hence are lkely to be good representatve demand patterns (RDP). Based on the whole data set, a sub-set of parts wth a hgh count of hghly smlar parts (χ2 dstance < 0.01) s created and then a sample of k (number of clusters) parts s randomly chosen as the ntal cluster centers. The threshold for the parts belongng to the ntal set s chosen to be the followng: #(parts wth χ2 dstance < 0.01) > 20 * k Ths means that only parts whch have more than k 20 parts, whch have a χ2- dstance smaller than χ2 dstance = 0.01 to the respectve part, are elgble for the ntal set. The threshold for 20 k was set n order to keep the number of possbly elgble parts low and therefore to keep the computng tme for PAM short. The next step s the decson on the number of clusters k, whch n PAM has to be made n advance. It s well known that the choce of k s one of the most dffcult problems of cluster analyss, for whch no unque soluton exsts [18]. For our case, we selected two qualty measures, the Dunn ndex and the slhouette wdth, as decson support crtera. One of the most frequently cted qualty crterons for the dentfcaton of the optmal number of clusters s the Dunn Index [5]. The Dunn ndex ams at dentfyng clusters wth hgh nter-cluster dstance and low ntra-cluster dstance. The Dunn ndex for K clusters C wth = 1,..., K s defned by DU k mn mn f (, j) j where d( C, C j ) f (, j) max dam( C m and d(c,cj) s the dssmlarty/dstance between cluster C and Cj and s defned by d( C, C ) max Dstance( x, y) j xc yc j and dam(cm) s the ntra-cluster functon or dameter of the cluster defned by the equaton dam( C) max Dstance( x, y) x, yc In short, the hgher the values of the Dunn Index, the more compact and well separated are the clusters. The slhouette wdth s based on a paper by Rousseeuw [19] whch descrbes that clusters can be represented by ther so-called slhouette s(), whch s a comparson of ts tghtness and separaton. Ths crteron sgnals whch objects are postoned well wthn ther cluster, and whch ones are just somewhere n between clusters. To assess the relatve qualty of the clusters and to get an overvew of how the data s confgured, the entre clusterng can be dsplayed by combnng the slhouettes of all objects nto a sngle plot. The average slhouette, called slhouette wdth, provdes an evaluaton of clusterng valdty, and mght be used to select an approprate number of clusters. The slhouette s() tself s represented by the followng formula b( ) a( ) s( ) max a( ), b( ) where b() = avg. dssmlarty of and nearest neghbor cluster, and a() = avg. dssmlarty of I and the other objects n the cluster The slhouette s() can range from [-1, 1], where +1 and -1 mean that the object belongs to an adequate or m ) Page 1528

5 nadequate cluster, respectvely. If the slhouette value of the object belongng to the cluster A s close to zero, t means that the object can also be n the nearest neghbor cluster to A. The objectve functon, as lke mentoned above, s the average of the slhouette s() over the number of objects to be classfed, and the best clusterng s reached when ths functon s maxmzed [19]. We conducted experments to fnd the optmal number k of clusters accordng to the qualty crtera. The frst fndng was that outler parts exsted whch had rare demand patterns and dstorted the shape of the ndvdual clusters sgnfcantly. These outlers had to be removed before a smooth clusterng could be conducted. All further nvestgatons were now based on parts wthout outlers. Fgure 2 shows the development of the slhouette wdth for an ncreasng number of clusters. We see a relatvely flat optmum n the range of clusters. Fgure 3: Dunn ndex for dfferent numbers of clusters k A total number of k = 8 clusters, turned out to be the optmal number of clusters, snce ths number results n the hghest cluster qualty possble and at the same tme represents as many dstnctve demand patterns wth a mnmal number of repeatng RDPs (Fgure 5). The red curves are the cluster centers, and the green curves are the cluster means, whch were used as the representatve demand pattern (RDP) for that cluster. Parts are farly equally dstrbuted across the 8 clusters. The proporton vares between ca. 8% and 18% of all parts per cluster. The largest clusters are cluster 2 wth 15,75%, cluster 3 wth 17.92% and cluster 6 wth 15,52% of all parts. Cluster 1 wth 8,52% of all parts s the smallest cluster. The full parts dstrbuton and the average χ2 dstance per cluster s shown n Fgure 4. Fgure 2: Slhouette wdth for dfferent numbers of clusters k The Dunn ndex showed an overall declne, but had some local optma n the range of 5 12 (Fgure 3). We carefully analyzed all clusters for dfferent k n the range suggested by the slhouette wdth and the Dunn ndex. For ths purpose, the shape of all demand lfecycle curves belongng to one cluster, together wth the cluster centers and means, were vsualzed n graphs as shown n Fgure 4. Partcularly, we looked at the smoothness and meanngfulness of the clusters as they have been proposed by the clusterng algorthm for dfferent numbers k, and for the dstrbuton of cluster szes. Fgure 4: Absolute and relatve cluster szes (percentages of all parts) and average χ2 dstance for each cluster for the optmal number of k=8 clusters Page 1529

6 3.4. Classfcaton The objectve of ths classfcaton s to assocate a new spare part wth one of the prevously dentfed clusters, or representatve demand patterns (RDP). As descrbed above, there are 243 ndvdual features per part whch reflect parts and demand characterstcs. In a frst step, we dentfed the features that were most nformatve n the ntroducton phase of a new part. The resultng features were afterwards tested for ther relevance by several expermental classfcatons and wth the help of the IBM SPSS Modeler feature selecton node. 7 features turned out to be sgnfcant. In a seres of experments, dfferent classfcaton algorthms were tested: Decson Tree (C 5.0), Fgure 5. Optmal clusterng output for k=8 ncludng the cluster centers (darker) and cluster means (lghter). The latter are used as representatve demand patterns (RDPs). Artfcal Neural Network (ANN), k-nearest Neghbor (knn), Naïve Bayes, and Support Vector Machne (SVM). Each classfer was traned based on the 5-fold cross valdaton on stratfed sample data. The Support Vector Machne (SVM) reaches the hghest classfcaton accuracy of 68.4%, Naïve Bayes s second wth a classfcaton accuracy of 59.4% and the C 5.0 Decson Tree reaches an accuracy of 55.6%, whch ranks t thrd. Fgure 6 shows precson and recall for the best classfer SVM for each ndvdual cluster and over all clusters. Page 1530

7 K False as K True as K K as False Precson of K Recall of K Accuracy ,81% 72,19% ,85% 66,37% ,31% 66,04% ,14% 71,15% ,11% 73,93% ,33% 63,03% ,24% 75,21% ,25% 69,27% Total 68,00% 69,65% 68,43% Fgure 6. Performance of SVM Classfer 3.5. Demand Predcton After assessng how the parts can be clustered and whch classfcaton method creates the best predcton outcome, a model for the actual demand value needs to be mplemented. The bass for ths forecast are the normalzed RDPs of each cluster that were dentfed durng the clusterng step. Snce the RDPs are normalzed demand patterns they must be scaled up n order to determne the real demand for the ndvdual part. Scalng up the demand wth the help of the RDPs requres at least the nformaton about the frst year of demand of the ndvdual parts (named P1). As shown n the followng formula, the all-tme demand of a new spare part n s calculated by accumulatng the normalzed demand of n s respectve RDP up to year one and then dvde the year one demand by ths accumulated normalzed demand value. all tme demand ( n ) 1 t 0 P1( n ) RDP( n ) As the frst year of demand s not known for new parts, t must be derved respectvely. Ths can be done by usng the planned number of produced vehcles n the frst year, named NV, that contan P1, and the falure rate alpha of the new parts. Dependng on the avalablty of data, the followng alternatves for calculatng P1 are avalable: 1) Falure rate alpha s known, planned number of vehcles n frst year NV s known: P1 = alpha NV / 2. 2) Falure rate alpha s known, planned number of vehcles n frst year NV s unknown: Estmate NV and use the formula under 1). If no such estmaton s possble, P1 cannot be estmated (or, n other words: any estmaton of P1 would mply an estmaton of NV) 3) Falure rate alpha s unknown, planned number of vehcles n frst year NV s known: Estmate alpha from hstorcal smlar parts (same door, same engne part,...) and use the formula under 1). If no such estmaton s possble, P1 cannot be estmated (or, n other words: any estmaton of P1 would mply an estmaton of alpha) 4) Falure rate alpha s unknown, planned number of vehcles n frst year P1 s unknown: Estmate P1 from hstorcal smlar parts (same door, same engne part etc.). Ths s not an accurate method, snce t assumes that NV remans unchanged In addton to the applcaton for new parts, ths model can also be very useful for parts wth only a few years of hstorcal demand. As mentoned n chapter 3, n the early stages of a product lfe cycle, exstng predcton models based on hstorcal demand do not provde good forecastng results when t comes to all-tme demand predcton. For these parts, the New Parts Model presented n ths paper could provde a stable and trustworthy alternatve Implementaton and Evaluaton Smple tasks lke the normalzaton of demand or the creaton of the dstance matrx were developed n PERL. The shftng of the demand to the same hstorc start year, the clusterng of normalzed and shfted demand patterns and the vsualzaton of the clusters were developed usng R. Feature selecton, clusterng, and classfcaton was done usng IBM SPSS Modeler. As already mentoned, the evaluaton of the algorthms was done step by step. The results were promsng and n a range of accuracy that s suffcent for the demand plannng of new parts. A full evaluaton of the all-tme demand predcton can only be made as more and more parts complete ther lfecycles. 4. Summary After an extensve lterature study on long-term and new product plannng, we decded to choose a smlarty-based approach to tackle the problem of alltme predcton of new automotve spare parts. We have found a few papers that have already used ths specfc approach [2, 12, 16, 21]. However, none of these are usng ther models for predctng new spare parts demand. Instead, they mostly rely on partal demand data to predct demand of parts n use. Further drawbacks of the research so far were the usage of partal demand patterns only for predctng smlarty, the use of common dstrbuton functons n order to strongly smplfy the demand patterns, or the need of expert knowledge to create the fnal outcome of the predcton. Page 1531

8 Our chosen approach provdes a fully qualfed model for the predcton of new spare parts. It conssts of three easly comprehensble steps: clusterng, classfcaton, and demand predcton. The frst two steps could be evaluated and showed good results, so that the overall approach seems to be very well suted for the complex problem of all-tme demand predcton of automotve spare parts. A generalzaton of the approach to other ndustral stuatons seems possble. 5. References [1] Bass, F. M. (1969). A New Product Growth Model for Consumer Durables. Management Scence, 15, [2] Bensng, L. (2012). Lfe cycle predcton for spare parts at IBM spare parts operatons. Unversty of Twente. [3] Boylan, J. E., & Syntetos, A. A. (2008). Forecastng for Inventory Management of Servce Parts. In Sprnger Seres n Relablty Engneerng (Vol. 8, pp ). [4] Croston, J. D. (1972). Forecastng and Stock Control for Intermttent Demands. Journal of the Operatonal Research Socety, 23(3), [5] Dunn, J. C. (1974). Well-Separated Clusters and Optmal Fuzzy Parttons. Journal of Cybernetcs, 4(1), [6] Elers, P. H. C., & Marx, B. D. (1996). Flexble smoothng wth B -splnes and penaltes. Statstcal Scence, 11(2), [7] Fortun, L. (1980). The All-Tme Requrement of Spare Parts for Servce After Sales Theoretcal Analyss and Practcal Results. Internatonal Journal of Operatons & Producton Management, 1(1), [8] Goodwn, P., Dyussekeneva, K., & Meeran, S. (2012). The use of analoges n forecastng the annual sales of new electroncs products. IMA Journal of Management Mathematcs, 24(4), [9] Haste, T. J., & Tbshran, R. (1993). Varyngcoeffcent Models. Journal of the Royal Statstcal Socety, 55(4), [10] Hong, J. S., Koo, H.-Y., Lee, C.-S., & Ahn, J. (2008). Forecastng servce parts demand for a dscontnued product. IIE Transactons, 40(7), [11] Jan, A. K., Murty, M. N., & Flynn, P. J. (1999). Data clusterng: a revew. ACM Computng Surveys, 31(3), [12] Jonas, T., Toth, Z. E., & Domb, J. (2015). A Knowledge Dscovery Based Approach to Long-Term Forecastng of Demand for Electronc Spare Parts. Computatonal Intellgence and Informatcs, [13] Kaufman, L. R., & Rousseeuw, P. P. (1990). Fndng groups n data: An ntroducton to cluster analyss. Hoboken NJ John Wley & Sons Inc. ISO 690. [14] Lttle, A. D. (2008). Automotve After Sales [15] Mas-Machuca, M., Sanz, M., & Martnez-Costa, C. (2014). A revew of forecastng models for new products. Intangble Captal, 10(1), [16] Mexell, M., & Wu, S. (2001). Scenaro analyss of demand n a technology market usng leadng ndcators. Semconductor Manufacturng, IEEE, 14(1), [17] Moore, J. R. (1971). Forecastng and Schedulng for Past-Model Replacement Parts. Management Scence, 18(4- Part-I), B 200 B 213. [18] Pele, O., & Werman, M. (2010). The quadratc-ch hstogram dstance famly. In Lecture Notes n Computer Scence (Vol LNCS, pp ). School of Computer Scence. The Hebrew Unversty of Jerusalem. [19] Rousseeuw, P. J. (1987). Slhouettes: A graphcal ad to the nterpretaton and valdaton of cluster analyss. Journal of Computatonal and Appled Mathematcs, 20(C), [20] Snedecor, G. W. & Cochran, W. G. (1977). Statstcal methods appled to experments n agrculture and bology. Ctaton Classcs, 19, 1. [21] Thomassey, S., & Happette, M. (2007). A neural clusterng and classfcaton system for sales forecastng of new apparel tems. Appled Soft Computng Journal, 7(4), [22] Teunter, R. H., & Fortun, L. (1999). End-of-lfe servce. Internatonal Journal of Producton Economcs, 59(1), [23] Zhu, S., Dekker, R., Van Jaarsveld, W., Renje, R. W., & Konng, A. J. (2017). An mproved method for forecastng spare parts demand usng extreme value theory. European Journal of Operatonal Research, 261(1), Page 1532