FORMATION OF PART FAMILY IN RECONFIGURABLE MANUFACTURING SYSTEM USING PRINCIPLE COMPONENT ANALYSIS AND K-MEANS ALGORITHM

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1 als of DM for 01 & Proceedigs of the 3rd Iteratioal DM Symposium, Volume 3, No.1, ISSN ISBN , CDROM versio, Ed. B. Kataliic, Published by DM Iteratioal, Viea, ustria, EU, 01 Make Harmoy betwee Techology ad Nature, ad Your Mid will Fly Free as a Bird als & Proceedigs of DM Iteratioal 01 FORMTION OF PRT FMILY IN RECONFIGURBLE MNUFCTURING SYSTEM USING PRINCIPLE COMPONENT NLYSIS ND K-MENS LGORITHM GUPT, [shutosh]; JIN, P. K. & KUMR, D[iesh] bstract: The Recofigurable Maufacturig Systems (RMS) is the ext step i maufacturig, allowig the productio of ay quatity of highly customised ad complex parts together with the beefits of mass productio. I RMSs, parts are grouped ito families, each of which requires a specific system cofiguratio. Iitially system is cofigured to produce the first family of parts. Oce it is fiished, the system is recofigured i order to produce the secod family, ad so forth. The effectiveess of a RMS depeds o the formatio of the optimum set of part families addressig various recofigurability issues. The aim of this work is to establish a methodology for groupig parts ito families for effective workig of Recofigurable Maufacturig Systems (RMSs). The methodology carried out i three phases. I the first phase, the correlatio matrix is used as similarity coefficiet matrix. I the secod phase, Pricipal Compoet alysis (PC) is applied to fid the eigevalues ad eigevectors o the correlatio similarity matrix. scatter plot aalysis as a cluster aalysis is applied to make parts groups while maximizig correlatio betwee parts. I the third phase, gglomerative Hierarchical K-meas algorithm improved the parts family formatio usig Euclidea distace resultig i a optimum set of part families for recofigurable maufacturig system. Keywords: Recofigurable maufacturig system, Part family formatio, Priciple Compoet alysis, K-meas algorithm, Similarity coefficiet, Multivariate aalysis 1. INTRODUCTION Curret market treds are characterized by globalizatio, ew product requiremets, rapidly chagig demads, ad a cotiuous improvemet of the existig techology for maufacturig activities. Time reductio to itroduce ew products to the market with high quality ad low cost is a ecessity for eterprise survival i this ew sceario. The key factor i this highly competitive eviromet is the ability of the compaies to lauch ew products to the market, with high quality, ad low cost. For achievig this, the maufacturig system must have exact capacity ad fuctioality to yield differet batch sizes of differet product types. Referece [1] were first, defied a RMS as a maufacturig system desiged at the outset for rapid chages i structure as well as i hardware ad software compoets i order to quickly adust productio capacity ad fuctioality withi a part family i respose to sudde chages i market or i regulatory requiremets. I the same way, Refrece [] cosider a RMS as a maufacturig system cofigured to produce a family of products that shares some similarities. s these defiitios state, the formatio of part families is a cetral issue i RMS. The key attribute of part families is that all the compoets withi a family require similar productio systems ad thus RMSs should have the exact capacity ad fuctioality required to maufacture a part family, allowig cost effectiveess [3]. It is suggested that groupig parts ito families i RMSs has a positive effect o the itroductio of ew products i market [4]. Iitially, RMS is cofigured to produce the first selected family ad the it is recofigured to effectively produce the followig part family, ad so forth. Thus, the RMS cofiguratio chages over a umber of part families to complete all the batches. The first issue which cosidered i RMS is the formatio of part families. Literature presets plety of methods to obtai families ad diverse formatio criteria. These methods ad criteria caot be used directly i recofigurable maufacturig because it has its ow sigularities that are to be take ito cosideratio, which differ from other maufacturig paradigms. Referece [5] stated that formatio of part family i RMS has to be based o some groupig criteria. Modularity ad commoality are two importat criteria for groupig cosidered suitable for RMS. For customized products, modularity allows the assembly of simple ad fuctioally idepedet parts. These simple ad fuctioally idepedet parts use stadard parts which are group together to o the basis of operatios similarity. other importat criterio is commoality cocept which is related to the part variety ad is defied as measure of how well the product uses stadard parts. The stadard parts require similar type of operatios ad grouped together to capture the iheret advatages of GT such as reduced setup times, reduced i-process ivetories, improved product quality, shorter lead time, reduced tool requiremets, improved productivity, ad better overall cotrol of operatios. Dedicated Maufacturig System focuses o the ecoomic productio of oe specific part type oly. The first maufacturig system focused o cost-effective maufacturig of several part types simultaeously was Cellular Maufacturig System (CMS). The developmet of CMSs has bee closely liked to groupig of parts ito families. I CMS, the part family formatio is achieved usig followig techiques: descriptive procedures, mathematical programmig approaches, ad artificial itelligece methods.the hierarchical clusterig agglomerative methods group together similar elemets (products) i clusters based o their attribute similarities. The coefficiets that measure similarity betwee two parts are calculated from the icidece matrix. fter that, a dedrogram shows the similarity degree to group parts. They used similarity or dissimilarity coefficiets amog parts to obtai the groups. Hierarchical clusterig algorithms yield a dedrogram represetig the ested family of parts ad similarity levels at which families chage. No

2 hierarchical clusterig algorithms, o the other had, obtai a sigle partitio of the data istead of a clusterig structure. Most of the existig Cell Formatio methods suffer from oe or more drawbacks. Their maor commo drawbacks are the iflexibility ad the limited idustrial applicatio due to the o-availability of software programs supportig them. Referece [5] also focused o groupig the parts by modifyig existig CMS methods istead of developig the ew method i order to stad the requiremets of RMS. So, ew part groups formatio approaches that overcome these limitatios are clearly eeded. This paper proposes a ew approach based o ew similarity coefficiet method for the part family formatio i RMS eviromet.to this effect, a ew method has bee proposed usig correlatio as a similarity coefficiet. This ew similarity coefficiet is based o the correlatio betwee operatio sequeces which are required to produce the parts. Furthermore, Pricipal Compoet alysis (PC) is used to cluster the parts followed by agglomerative hierarchical K- meas for idetifyig part groups o the basis of operatioal sequece similarity. Pricipal Compoet alysis (PC) is the best kow ad oldest techique i multivariate aalysis [6]. Referece [7] was first to itroduce it to recast liear regressio aalysis ito a ew form. PC is frequetly used for the data set with some itrisic complexity [8, 9]. Referece [10] used PC for cocurret part machie group formatio problem i CMS. It is a quatitatively rigorous method for achievig the simplificatio. The method geerates a ew set of variables, called Pricipal Compoets (PCs). Each PC is a liear combiatio of the origial variables. ll the PCs are orthogoal to each other, so there is o redudat iformatio. The umber of PCs extracted i a PC is equal to the umber of observed variables beig aalyzed. However, i most aalyses, oly the first few compoets accout for meaigful amout of variace, ad hece those first few compoets are retaied, iterpreted, ad used i subsequet aalysis ad rest are eglected. Whe the aalysis is complete, the resultig compoets display varyig degrees of correlatio with the observed variables, but are completely ucorrelated with oe aother. The K-meas algorithm is a clusterig techique. The K-meas algorithm radomly selects K data poits as iitial cluster cetroids. cetroid is a artificial poit i the space which represets a average locatio of the particular cluster. K clusters are formed by assigig each data poit to its earest cetroid. New virtual cetroids are the calculated for each cluster. These processes are iterated util a predefied umber of iteratio is reached or the clusters did ot chage aymore. The maor problem with the K-meas algorithm is that its iitial startig poits are geerated radomly ad does ot guaratee the uique clusterig results [11]. lso, due to the o-hierarchical ature of the algorithm, a hierarchical relatioship betwee the clusters is eeded. This hierarchical relatioship is effective to visualize ad aalyze the large data sets. The hierarchical techique is classified ito agglomerative method ad divisive method. The divisive method is the top dow approach i which iitially all the obects are icluded i a sigle cluster. The, the sigle cluster is divided ito sub-clusters util each obect costitutes a cluster. agglomerative method is the bottom up approach i which each obect is assumed as a separate cluster ad the they are clustered i successio util a sigle cluster which cosists of the etire obect set is formed. So, a gglomerative Hierarchical K-meas Clusterig lgorithm (HKC) is used for the part family formatio. The outlie of the paper is as follows: The proposed methodology is preseted i Sectio. fterwards, oe umerical example gives illustratio of the proposed methodology i phased maer i Sectio 3. Sectio 4 discusses the results. Lastly, coclusios are draw i sectio 5.. METHODOLOGY I this sectio, a ovel method based o PC ad HKC is developed for part family formatio. The proposed methodology cosists of three phases as show i Figure 1. The obective of the methodology is to cluster the parts ito k part families based o operatios sequece similarity. PHSE 1 Similarity Coefficiet Matrix PHSE Cluster alysis for Correlatio PHSE 3 lgorithm for Part family formatio Fig.1.Proposed methodology.1 Similarity coefficiet matrix Iitial Icidece Matrix Stadardizatio of the iitial data Costructio of the Correlatio matrix S pplicatio of the PC alysis based o the first two Priciple compoets Prelimiary Solutio: Groupig of parts pply K- meas algorithm i PC based Prelimiary solutio space gglomerative Hierarchical clusterig of parts Fial Solutio:Formatio of part family The first phase starts with buildig a similarity coefficiet matrix. The iitial part-operatio icidece matrix as show i equatio (1) is a biary matrix i which rows represet the operatios ad colums stad for parts. This matrix looks like the traspose of the classical part-machie icidece matrix. a a a a a a m 1 m (1) Where a i =1 if part requires operatio i ad a i = 0 otherwise

3 Let P is a biary row vector of matrix, such that P a1, a,, a. Iitial matrix is further stadardized by usig a suitable method of stadardizatio [1]. I this work, the geeral stadardizatio is used ad applied to the iitial icidece matrix. The stadardizatio process is expressed as follows: S P B P E Where, E is the average of row vector 1 E k 1 ( ak E ) k1 E a k E 1 T S B B 1 1 ad S b b ii i ik k k 1 P ad the row vector of the stadardized matrix B. expressed as: () B P is E ca be Here a k is the elemet of iitial icidece matrix; is the umber of elemets i a row vector. Similarly, To simplify the equatio 4 further, Huyghes Koig theorem is applied to yield Oce the stadardized matrix is formed, the proposed similarity coefficiet is based o the simple correlatio matrix of the stadard icidece matrix. The correlatio matrix S is defied as follows: (3) (4) (5) (6) S is the square matrix i which elemets are give by:. Cluster aalysis for correlatio I the secod phase of the proposed approach, the part family formatio is doe o the basis of operatios similarity usig Priciple Compoet alysis. Priciple Compoet alysis is a dimesio reductio techique which attempts to model the total variace of the origial data set, via ew ucorrelated variables called Pricipal Compoets. PC cosists of determiig a small umber of pricipal compoets that recover as much variability i the data as possible. These compoets are liear combiatios of the origial variables ad accout for the total variace of the origial data. The first priciple compoet is a sigle axis i space. Whe each (7) observatio is proected o that axis, resultat is a ew variable. The variace of this variable is maximum amog all possible choices of the first axis. The secod priciple compoet is aother axis i space, perpedicular to the first oe. Proectig the observatios o this axis geerates aother ew variable. The variace of this ew variable is agai maximum amog all possible choices of this secod axis. The full set of pricipal compoets is as large as the origial set of variables. However, the sum of the variaces of the first few pricipal compoets is usually 80% or more of the total variace of the origial data [13]. The first compoet extracted i pricipal compoet aalysis accouts for a maximal amout of total variace i the observed variables. Uder typical coditios, this meas that the first compoet is correlated with at least some of the observed variables. The secod compoet extracted is havig two importat characteristics. First, this compoet accouts for a maximal amout of variace i the data set that was ot accouted for by the first compoet. This meas that the secod compoet is correlated with some of the observed variables that did ot display strog correlatios with first compoet. The secod characteristic of the secod compoet is that it is ucorrelated with the first compoet. This meas that the correlatio betwee first ad secod priciple compoets is zero [14]. The remaiig priciple compoets that are extracted i the aalysis display the same two characteristics metioed above. That is, each priciple compoet accouts for a maximal amout of variace i the observed variables that was ot accouted for by the precedig compoets, ad is ucorrelated with all of the precedig compoets. Pricipal compoets aalysis proceeds i this fashio, with each ew compoet accoutig for progressively smaller ad smaller amouts of variace. Thus, the study of pricipal compoets is cosidered as puttig ito terms the usual developmets of eigevalues ad eigevectors for positive semi-defiite matrices. The eigevector equatio where the terms 1 m are real, o-egative roots of the determiat polyomial of degree P is give as: det ( S ) 0 ; i 1, m i Let {F 1, F,, F m } be correspodig eigevectors. Whe PC was performed o the mea cetered data, a model with the first ad the secod pricipal compoets was usually obtaied. This model explais the procedure to determie the priciple compoet i the data. 1 1 Where, PC = (9) m m k1 I this applicatio of PC, the obective is to cluster parts ito families. s part-operatio matrix is biary i ature, two pricipal compoets are eough to aalyse correlatio betwee elemets (i.e. parts). k (8)

4 .3 lgorithm for part family formatio The obective of the third phase is to assig parts ito families after the prelimiary groupig as doe i the secod phase of the proposed approach. gglomerative Hierarchical K-meas Clusterig lgorithm is used for this purpose. The algorithm for assigig the parts (P i ) ito parts families is give below: For each part k=1 to P i do Step 1:Take poits (Iitial seeds) i -dimesioal plae, where is the umber of variables. (Iitial seeds are take based o score values (coordiate) of each part o first two priciple compoets. Each seed represets a part which cotais associated part operatios.) Step :Compute Euclidea distace for each part. Now, applyig the equatios, 3 ad 5 to the part operatio icidece matrix give above, yield stadardized matrix B, which represets Stadard Sequece Part Operatio Matrix (SSPOM) i the case. The followig procedure is adopted to determie various elemets of SSPOM. P 1 P P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 OP OP OP OP OP OP OP Tab.. Part Operatio Icidece Matrix: POIM (P k, P i ) = ( x x ) ( y y ) (10) k i k i Say, for part P 1, E 1 =/7 = 0.86 Where, x i ad y i are the co-ordiates of part P i o two pricipal compoets axis. Step 3:Sice the obective is to group parts with miimum distace, part uder cosideratio (say P i ) is assiged to a family (say P k ) o the basis of least smallest distace to the part family P k. Step 4:Draw the dedrogram of the sequece of family formatio. The iteratio cotiues util all parts (P i ) are assiged to part families. 3. NUMERICL ILLUSTRTION example case has bee take to demostrate the proposed methodology where seve operatios (OPs) are required to maufacture the eleve parts i recofigurable maufacturig system. Parts are labeled as P 1 to P 11 ad operatios as OP 1 to OP 7. Table 1 shows part s operatioal sequece requiremets. Part Number Required Operatios Sequece P 1 OP 1, OP P OP, OP 3 P 3 OP 1, OP 5, OP 6 P 4 OP 4, OP 5, OP 6,OP 8 P 5 OP 4, OP 5, OP 6 P 6 OP, OP 3 P 7 OP 1, OP 6 P 8 OP 6, OP 7 P 9 OP 3 P 10 OP 5,OP 6, OP 7 P 11 OP 1, OP 5 Tab.1. Parts ad Operatios sequece data ccordigly a Part Operatio Icidece Matrix (POIM) is costructed as show i Table (0.86) = 0.45 The member coefficiet betwee P 1 ad OP 1 (i.e. b 11 ) is calculated as follows: Similarly, b b ad b The same procedure is repeated for other elemets. Fially, SSPOM is obtaied as show i Table 3. From the above SSPOM ad o the basis of proposed similarity coefficiet (as give i the eq. 7), the correlatio matrix (S) is obtaied as show i Table 4. Now cluster aalysis is performed based o PC method ad by usig eq. 8. The computed eigevalues for the correlatio matrix (S) ad their associated variace, ad cumulative variace are listed, sorted i a descedig order as show i Table 5. The two priciple compoets are havig the maximum variace (i.e. almost 80%) ad are sufficietly eough to represet the all parts operatios data. Hece for the further aalysis two priciple compoets are take. P1 P P3 P4 P5 P6 P7 P8 P9 P10 P11 OP OP OP OP OP OP OP Tab.3. Stadard Sequece Part Operatio Matrix

5 Scores o PC (8.11%) P 1 P P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P P P P P P P P P P P Tab.4. Correlatio matris S No of pricipal compoets Eige value % variace Cumulative % variace 1 1.5e e Tab.5. Priciple compoets, Eige value ad Percetage Variace Scatter Plot 1.5 P10 1 P8 0.5 P4 P5 0 P9-0.5 P P6 P11 P1-1 P3 P Scores o PC 1 (50.04%) Fig.. Graphical represetatio of Scatter Plot Further, the graphical aalysis is performed by a two dimesioal scatter plot where each part is represeted by a dot ad the two axes of the scatter plot are the two priciple compoets. This scatter plot idicates the relatioship betwee parts as show i Figure. There is high correlatio betwee parts which are closely placed ad thereby are strogly associated with each other such as (P ad P 6 ) ad (P 4 ad P 5 ). The various correlatio values are show i Table 4. O the basis of correlatio results, the followig priciple situatios are recovered from the scatter plot: i. Two eighborig parts havig low distace measure belog to the same group such as P ad P 6 ; P 4 ad P 5. ii. Part group (P 4 ad P 5 ) which is almost to the other part group (P 6 ad P ; P 9 ) are egatively correlated ad thus caot belog to the same group. iii. Two parts which are placed almost 90 0 to each other such as P 9 ad P 8 are idepedet ad thus caot belog to the same group. It is clearly see i the Figure. The co-ordiate (score) of each part o first two priciple compoets is obtaied from scatter plot as show i Table 6. Part Number First Priciple Compoet (50.04%) Secod Priciple Compoet (8.11%) P P P P P P P P P P P Tab.6. Score (Co-ordiate) of parts The third phase of the methodology is to cluster parts, obtaied from the prelimiary solutio space i secod phase, ito family by usig HKC metioed i the methodology. The algorithm is iitialized by takig score values (coordiates) of parts from scatter plot as a startig poit. The algorithm starts with the iitial solutio i.e. the umber of parts ad the computes ad stores the euclidea distace betwee each part usig equatio 10. The parts havig miimum distace are grouped first ad these grouped parts are removed from the subsequet iteratio. This process cotiues util all the parts are grouped together. The algorithm fially provides output i the form of dedrogram (Figure 3)

6 5. CONCLUSION Fig.3. Hierarchical Clusterig of parts 4. RESULT The resultig dedrogram is based o the distaces to K-meas earest group. I K-meas algorithm, the measured distace is kow as dissimilarity measure. Hece, the magitude of the distace to earest group of parts represets the dissimilarity associated with the part. The ratio of the distace to the earest group to the maximum distace measured i the dedrogram gives the percetage of dissimilarity of the particular parts group. I Figure 3, parts P ad part P 6 are foud at zero distace. So their percetage of dissimilarity is zero. It meas that they are 100% similar i operatios. The distace betwee parts P 4 ad P 5 is ad their associated percetage of dissimilarity is calculated to 6.0 % ad level of similarity is 94%. The same procedure is repeated to calculate the other similarity level for remaiig cluster of parts. The obtaied results are summarized i Table 7. Distace betwee parts group Simi larity Leve l (%) Formed part families Numb er of famili es P,6, P 1,P 3, P 4,P 5,P 7,P 8,P 9,P 10,P P,6, P 1,P 3, 09 P 4,5,P 7,P 8,P 9,P 10,P P,6, P 1,P 3,11, P 4,5,P 7,P 8,P 9,P P,6,9,P 1,P 3,11, P 4,5,P 7,P 8, P P,6,9,P 1,P 3,11,7, P 4,5, P 8, P P,6,9,P 1,P 3,11,7, P 4,5, P 8, P,6,9,P 1,3,11,7, P 4,5, P 8, P,6,9,P 1,3,11,7, P 4,5,8, P,6,9,P 1,3,11,7,4,5,8, P,6,9,1,3,11,7,4,5,8,10 01 Tab.7. Percetage of similarity ad Formed part family O the basis of the formed part families the system plaer first cofigures the maufacturig system to produce the first part family. Oce it is fiished, the system is recofigured to produce the secod part family ad so forth. Each system recofiguratio adds cost to the productio of the parts. Hece there is a eed to arrive at a suitable umber of part families by selectig a appropriate value of percetage of similarity as a cut off to achieve miimum cost solutio of the problem This work has preseted a ovel methodology for groupig parts ito families o the basis of operatio sequece similarity which is a cetral issue i the desig of recofigurable maufacturig systems. correlatio aalysis model is formulated to group the parts ad operatios sequece ad Correlatio matrix is used as the similarity coefficiet matrix. Fially, Priciple Compoet alysis ad gglomerative Hierarchical K- meas algorithm is applied to fid the level of similarity i parts. The obtaied part families are based o compactess of the family formatio o the basis of operatioal sequece similarity. Furthermore it uses PC, which is available i may commercial software packages. lthough, PC is used to reduce the dimesio of data but sometimes high dimesioal data may have may redudat or irrelevat features. These redudat features are of o help for clusterig ad may create osie. For this, a oise reductio techique ca be applied to remove the oise. However the proposed work ca be further exteded to accommodate other factors such as, productio volume, alterative operatio sequeces ad alterative routigs. 6. REFERENCES [1] Kore,Y.,Jovae,F.,Heisel,U.,Moriwaki,T.,Pritschow,G.,Ulsoy,. G.&VaBrussel,H.(1999). Recofigurable Maufacturig Systems. CIRP als, Vol. 48, No., pp [] Xiaobo, Z., Jiacai, W., & Zhebi, L.(000). stochastic model of a recofigurable maufacturig system, Part 1: a framework. Iteratioal Joural of Productio Research, Vol.38, No.10, [3] Lokesh,.K. & Jai, P.K.(011). model ad optimizatio approach for recofigurable maufacturig system cofiguratio desig. Iteratioal Joural of Productio Research, Vol.50, No.1, ISSN [4] bdi, M.R. & Labib,.W.(004). Groupig ad selectig products: the desig key of Recofigurable Maufacturig Systems (RMSs). 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