3D VISUALIZATION OF A CASE-BASED DISTANCE MODEL

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1 8 D VISUALIZATION OF A CASE-BASED DISTANCE MODEL Péte Volf Zoltán Kovács István Szalkai Univesity of Pannonia Veszpém Hungay volfpete@gtk.uni-pannon.hu kovacsz@gtk.uni-pannon.hu szalkai@almos.uni-pannon.hu Abstact Regading to the conveyance of mateials the assignment of stock-keeping units (SKUs is one of the most impotant tasks in a waehouse. The units in the inteest of thei easie handling can be soted into goups. To classifying SKUs the most feuently used method is the ABC analysis. The taditional appoach implements a simplified scheme which is mostly based on annual dolla usage. Classifying units accoding to one chaacteistic can lead to ushed esult because the assignment of stock-keeping units might be influenced by othe factos like numbe of odes weight o lead time. In ecent yeas a numbe of decision models have been developed in ode to make it possible to conside moe than one citeion in the same time. In geneal in the case of epesenting a model the intoduction of the algoithm is moe pefeable than the visualization of data. Howeve the visualization of esults can elieve the compehension of the given poblem and the way how the model woks even in elation to soting. The aim of this eseach is to intoduce a plotting method which visualizes the esults coming fom the Case-based distance model developed by Chen Kilgou and Hipel. This method stengthens the classification by poviding sepaating nonlinea sufaces. Key wods: case-based distance model inventoy management visualization. Intoduction Many obstacles can emege duing decision making and suppot. It can feuently occu that besides the lage amount of data the decision has to be made by consideing moe than one factos o chaacteistics. Fo instance the soting poblem in inventoy management whee thousands of stock-keeping-units have to be soted into goups based on thei chaacteistics. Concening this poblem numeous methods have been developed to simplify analyze and elieve the multiciteial decision making situation in ecent yeas. Visualization techniues can also help the decision suppot pocedue by making the sot of poblem moe undestandable. The taditional ABC analysis is the most feuently used appoach to classify SKUs. The taditional appoach implements a geneal simplified scheme which is mostly based on annual dolla usage. Besides its simplicity one of its advantages is the ease of epesentation of the esults in the Paeto-chat (Hay Mann De Hodgins Hulbet & Lacke 00 which is based on the famous Paeto-obsevation (Mashall 007. The most impotant lack of this method is that it cannot take moe than one citeion into account at the same time. In ecent yeas a numbe of multi-citeia decision models have been developed in ode to compensate this defect of the taditional appoach using methodologies such as the weighted linea optimization (Ramanthan 006 the genetic algoithm (Guvemi & Eel 998 the analytic hieachy pocess (AHP (Patovi & Buton 99 the fuzzy set theoy (Chu Liang & Liao 008 the fuzzy-ahp (Caki & Canbolat 008 etc. On one hand these models can seve as management tools fo solving complex decision

2 situations (Vetschea Chen Hipel & Kilgou 00 on the othe hand the expansion of the numbe of dimensions might cause confusion in intepetation of the esults. Not undestanding o misundestanding the decision context is one of the main (Ma 0 that can lead to mistakes. In geneal the visualization of data can elieve the compehension of the given poblem and its envionment suounding it. Peviously Tufte (98 emphasized the impotance of gaphical epesentation analyzing the spead of cholea epidemic. He stated that the gaphical analysis was moe efficient than the calculation. Meye (99 examined the ole of visual data in the concept of oganizational eseach and suggested eseach uestions whee visual data can be moe meaningful to vebal one. Condon Golden & Wasil (00 poposed a methodology based on Sammon map fo AHP. They analyzed the goup decision pocedue and elative udgment of the membes. They claimed that using gaphical assistant can aise the obectivity of collective decision making. Adle & Raveh (008 intoduced a gaphical method fo data envelopment analysis (DEA using Co-plot which can help to identify efficient units and outlies. Ma (0 made up a sceening methodology suppoted by gaphical visualization based on the concept of multidimensional scaling (MDS. On the field of inventoy management concening the analyses of SKUs one of the main is that the lage numbe of units can lead to impacticable data set that can make the employ of the given model fo decision makes (DMs moe complicated and can hinde the appopiate intepetation of the esults. In this sense the Case-based distance model developed by Chen Kilgou & Hipel (008 excels fom othe models consideing the stuctue of the model because it involves an untapped oppotunity in the view of intepetation of esults. Namely the pocess of detemining the paametes of sufaces sepaating each goup of SKUs is built in the model and visualizing these sufaces can help to discove the magnitude of goups and the elationships between them. The main pupose of the eseach is to give a gaphical suppot to Multiciteial ABC Analysis based on Case-based distance model. Hence fist of all the pocess of the initial dataset geneation will be discussed. It is followed by an intoduction into the visualization of the model which is needed to explain the main findings of the eseach. A 000-point illustative example seves as demonstation of the plotting method step by step. 8 Methodology of Reseach The main aim of the visualization is to demonstate how the Case-based distance model woks and to pesent its esults. The necessay dataset was geneated on andom basis in a contolled model. In this sense the contol model means that the intevals wee ceated in ode to epesent the most analised citeia like unit value weight and lead time. On the othe hand the thee citeia ae easonable because of the epesentation of the model in thee dimensions. Data Analysis The main goal of the Case-based distance model is to sot the stock keeping units into goups based on chaacteistics of the units as the set Q of citeia (Chen Kilgou & Hipel 008. Let T be the set of the units ( i based on Q =... m. i A being analyzed whee A T T g = A B C g min The minimum ( c and the imum ( c values taken up in positions A and

3 84 A fom an inteval (Chen Kilgou & Hipel 008. The Euclidean distances taken fom these points detemine the position of the given unit on citeion i.e. the distance fom the exteme values of the inteval. Let the set Z Tg g = A B C be a subset of T g that involves the units which pincipally epesent the set T g and & Hipel 008. c min z T g = A B C is one of these units whee g ( A c ( z c ( A (Chen Kilgou The Case-based distance model is constucted fo n dimensions which adopted on thee citeia becomes capable to be plotted as a thee dimensional system of co-odinates defined by the thee elements of Q whee espectively adius vecto epesents the x the y and the z axis. In the system of co-odinates the adius vectos can be given by the co-odinates of thei end so if the co-odinates of a P point ae and P P P P then the adius vecto heading towads the P point is ; ;. In this sense we find that the coodinates of adius vecto on citeion : - If the uppe limit of the inteval is the point of compaison: ( c ( A c ( z ; 0; 0 ; ( ; c ( A c ( z ; 0 ; 0 ( 0; 0; c ( A c ( z. - If the lowe limit of the inteval is the point of compaison: min ( c ( z c ( A ; 0; 0 ; min ( 0; c ( z c ( A ; 0 ; min ( ; 0; c ( z c ( A 0. ( 0 belonging to P P z T g

4 Péte VOLF Zoltán KOVÁCS István SZALKAI. D Visualization of a Case-Based Distance Model Afte launching the nomalization facto d the distance between the lowe and uppe bounds of the intevals changes to unit i.e. (Chen Kilgou & Hipel Euation d ( A A min ( c ( A c ( A ; = d = = Since the distance between the lowe and uppe bound of inteval is eual to the absolute value of adius vecto on citeion theefoe the Euclidean distances taken fom the points of compaison can be identified as the length of the adius vectos belonging to each unit. So the distance of z T fom the uppe limit of the inteval on the second citeion is: g d = ( A z 0 = ( ( ( c ( ( ( 0; ( ( ; 0 A c z c A c z = d min ( 0 ; c ( A c ( A ; 0 ( c ( ( 0 A c z ( c ( A c( z = = whee. Euation ( 0 = ( min min min ( c c = ( c ( A c ( A = ( ;( c ( A c ( A ; d = = 0 Based on Chen Kilgou & Hipel (008 the weighted aggegation fomula of distances ae:. Euation D ( z = D ( A ; z = w d ( z = 4. Euation D Q Q ( z = D ( z A = w d ( z = espectively. Q Q w w ( ( ( ; ( ; ( x z y z z z ( ( ( ; ( ; ( x z y z z z

5 86 Note that depending on the point of compaison we get two diffeent soting. Based on this fact the visualization has to be sepaated likewise: min - If c ( A i n - If c ( A i n is the point of compaison then we talk about minimum tansfomation the minimum (distoted space; is the point of compaison then we talk about imum tansfomation the imum (distoted space. In this sense the attibute distotion means that the values ceated by the value function ( d z fom the oiginal values ae linea and nonlinea distoted and this value tansfomation affects the sets and the shape of thei sepaating sufaces too. The main aim is to plot the sufaces (ellipsoids bounding the sets in the oiginal space. To achieve this goal the linea and nonlinea tansfomations togethe with thei effects have been discoveed and ae intoduced it in the following section. The softwaes Lingo and MATLAB wee used to calculate the necessay model paametes and to plot the sufaces. Linea and nonlinea tansfomations and thei effects on the visualization Space 0: Let ( p i p i p i P i P i R be the point belonging to a SKU with the co-odinates = whee i =... D and m m m p i p p i i M M M. whee ( m m m and ( M M M ae the minimum and imum values of the oiginal min min min min c o - odinates i.e. ( c ; c ; c c ( c ; c ; c. Space : Linea tansfomation: whee c and i.e. in the case of i =... D

6 Péte VOLF Zoltán KOVÁCS István SZALKAI. D Visualization of a Case-Based Distance Model 5. Euation 87 conseuently the linea distoted co-odinates ae: As a esult we get a unit cube placed in o oigin. Let Space : Minimum wold (nonlinea tansfomation: 6. Euation so in the case of i =... D Note that fo i =... D that is the tansfomed points ae in the same unit cube. Let Space : Maximum wold (nonlinea tansfomation: 7. Euation so in the case of i =... D Note that in the case of i =... D

7 88 that is the tansfomed ae also in the unit cube. It is easy to state that the elation between Space and is: 8. Euation Depending on which space we ae analyzing the position of the points seached ae the followings (Chen Kilgou & Hipel 008: Space : - weight vecto: w ( w w w ; - adii: R and R. B C = i P the paametes These paametes ae given by the optimization model A MCABC expessed by the intoduced tansfomation ( Ξ based on (Chen Kilgou & Hipel 008: 9. Euation whee the conditions ae: 9b. Euation 9c. Euation 9d. Euation 9e. Euation whee 9f. Euation

8 Péte VOLF Zoltán KOVÁCS István SZALKAI. D Visualization of a Case-Based Distance Model Space : - weight vecto: w ( w w w ; - adii: R and R. A B = Paametes ae given in the case of A MCABC by the following model expessed by the intoduced tansfomation ( Η based on (Chen Kilgou & Hipel 008: 0. Euation 89 whee the conditions ae: 0b. euation 0c. Euation 0d. Euation 0e. Euation whee 0f. Euation Plotting method of linea and nonlinea sepaating sufaces d on The nonlinea tansfomation implied by the value function ( z i z = P i = tansfomed the ellipsoids sepaating the set g into planes. In this way the planes sepaating the sets in the distoted spaces can be given by the following fomula: Space :

9 90. Euation whee g = B C. Space :. Euation whee g = A B. Note that due the tansfomations above the assumed elation between Space and cannot be assessed in distoted spaces. The DMs have to etun back to the oiginal space in ode to detemine the inteaction of the two classifications in the same space. To get the thee goups of SKUs and the ellipsoids sepaating them in the oiginal space executing the invese tansfomation is needed. Since a n d ae monotone tansfomations of the ellipsoids sepaating the sets in the following way: Space : i P we can detemine The euation of planes dividing the sets in Space is: which can be efomed using the 6. euation into following fomula: ( p m ( M m ( p m ( M m = R w w expessing p can we find the euation of the ellipsoids:. Euation w ( p m ( M m g ( M m ( p m R w w ( M m ( p m ( M m p = m g w whee g = B C. Space : The euation of planes dividing the sets in Space is:

10 Péte VOLF Zoltán KOVÁCS István SZALKAI. D Visualization of a Case-Based Distance Model which can be efomed using the 7. euation into following fomula: 9 ( M p ( M m ( M p ( M m = R w w expessing p can we find the euation of the ellipsoids: 4. Euation ( M p ( M m g w ( M m ( M p R w w ( M m ( M p ( M m p = M g w whee g = A B. Note that in this case the cente of ellipsoids is the uppe limit of the available values i.e. the imum value. i Since the position of the points P epesenting the units in n dimensional space is detemined by the same citeia theefoe the sets ae able to be plotted in the same space whee thei sections can be identified as the nine sets befoe the eclassification. d z kept the elations between sets that is why applying the 4. euation the efomed shape o f the planes sepaating the sets in Space is The linea and nonlinea tansfomations employed by the value function ( 5. Euation 4/4 suface ( eggs whee g = B C. Respectively the efomed shape of the planes fom the Space can be given in Space by the following fomula: 6. Euation This suface is 4/4 suface ( eggs likewise whee g = A B.

11 9 Results of Reseach The esults show that limitation of the n dimensional model into thee dimensions made it possible to visualize the model in opeation. The mathematical backgound of plotting method with the euisite fomulas has been biefly intoduced above. The following illustative example suppots to demonstate the main findings of eseach concening the visualization of Casebased distance model developed by Chen Kilgou & Hipel (008 step-by-step in figues. The illustative example consists of 000 units. Each unit has thee paametes epesented on the thee axes of a thee dimensional coodinate-system unit value (x anges fom 00 to 785; weight (y anges fom 9 59 to 99 and lead time (z anging fom 004 to 4999 assuming that the lead time is a continuos vaiable which indicates the aveage of the egula lead times and in this sense it is not needed to be discete. The intevals of unit value and weight wee geneated based on nomal distibution in the case of lead time a shote inteval was detemined based on unifom distibution. Figue illustates the 000-point dataset in thee dimensional coodinate-system. The distibution of the unit value consists of thee nomal distibutions with diffeent paametes. Hence it is sepaated into thee pats in ode to make the movement of the dataset in the model to be taceable. Figue : Visualization of points P (oiginal co-odinates. i i i i Visualization of points = ( p p p i = in Space 0 Applying the paametes of the esults of A MCABC and A MCABC can be epesented in thee dimensional coodinate-system. The model paametes ae the followings: Space (Minimum wold: - weight vecto: w = ( 0840; 074; adii: R B = and Space (Maximum wold: R C = weight vecto: w = ( 0990; 0740; adii: R A = 0 48 and R B = ; ; each goup

12 Péte VOLF Zoltán KOVÁCS István SZALKAI. D Visualization of a Case-Based Distance Model Note that both in the minimum and imum wolds the cente of the compaison is the o oigin but the ode of sets is convese (see Figue. The convese ode of these sets has been caused by the effect of linea and nonlinea tansfomations (intoduced biefly in 6-7 euations. 9 Figue : Visualization of goups in Space and (making: Black goup A Gey goup B Light gey goup C. Substituting the coodinates and the paametes above into euations and makes it possible to plot the sufaces sepaating each goup fom the othes (see Figue. In both distoted spaces the sepaating sufaces uested fo ae planes. Figue : Visualization of goups and planes sepaating them in Space and Space. The A B and C goups can be plotted in the oiginal spaces afte executing the invese i i i i i tansfomation indicated as Ξ( Τ( P = ( p p p = P i i i i i Η( Τ( P = ( p p p = P and. The bounding sufaces belonging to each goup can be epesented by substituting these efomed coodinates with changeless model paametes into euations -4 (see Figue 4. These sufaces in the oiginal space ae ellipsoids nonlinea sufaces in accodance with the initial assumptions.

13 94 Figue 4: Visualization of goups of SKUs and ellipsoids bounding them based on the minimum and imum wold using the co-odinates of X i - i i i i i - i i i i ( T( P = ( p p p = P and H( T( P = ( p p p = P Due to the invese tansfomation the esults of the two classifications can be epesented in the same oiginal space (see Figue 5.. Figue 5: Visualization of classification based on minimum and imum wolds in one space (Space 0 bottom-view and side-view. Since the implied linea and nonlinea tansfomations ae monotonic that is why the epesentation of the esults of classifications keeps the elations between the goups in distoted spaces as well. The minimum and imum wolds can be completed by using euations 5-6 espectively (see Figue 6.

14 Péte VOLF Zoltán KOVÁCS István SZALKAI. D Visualization of a Case-Based Distance Model 95 Figue 6: The efomed shape of the planes fom Space in Space and fom Space in Space. Figue 6 shows that the shape of the sufaces in one wold is oval ( egg in the othe wold intecepting the planes. Discussion The adage A pictue is woth a thousand wods efes to the idea that a complex poblem can be modeled in the way that manages can oveview easily. The models like pesented above can suppot evey stages of decision making. Based on this manages will be able to make faste the pocess to get to the ight decision. It inceases the esponsiveness of oganizations. Visualization is extemly impotant in the case of Multiciteial decisions when moe factos ae supposed to be taken into consideation at the same time. The plotting method poposed in this pape is a manageial tool to demonstate how the magnitude of goups vaies caused by changing in the selected citeia. Knowing the linea and nonlinea tansfomations and the fomulas of each sufaces the effect of citeia selection to the classification o the effect of switching fom one citeion to anothe keeping the othe citeia stable can be studied which is one of the main contibution of the plotting method. In this sense the visualization can suppot not only the DMs but also the developes of the model to discove the oppotunities in the model and to avoid the occuing. The visualization highlighted what happens to the oiginal data duing the model that can make the explanation of esults much easie. Discoveing the movement and the distotion of the oiginal dataset in each spaces could lead to the obsevation of the eal elation between the two classification appoaches (minimum and imum wolds. Using fomulas (euation -4 gained fom the model allowed to poduce Figue (4 which illustates the position of the categoies in the way as oiginal model developes (Chen Kilgou & Hipel 008 assumed. Futhemoe only the epesentation of the sufaces (ellipsoids in the oiginal space could help to detemine the elationship fistly between the distoted spaces and secondly between the goups. On the othe hand the visualization of the goups and sufaces in a thee dimensional coodinate-system evealed that the ule of eclassification poposed by Floes & Whybak (987 and used as a basis of eclassification in Case-based distance model (Chen Kilgou & Hipel 008 is not unambiguos. Plotting the categoies of SKUs fom the two diffeent appoaches in one space still does not esult in the final classification. The method of egouping of the nine goups has to be econsideed and can fom a subect of futhe eseach.

15 96 Conclusions In ecent yeas a numbe of models fo Multi-citeia ABC analysis have been developed in ode to make it possible to take moe than one citeion into account at the same time. But in geneal in the case of epesenting a model the intoduction of the algoithm is moe pefeable than the visualization of data. Howeve the visualization of esults can elieve the compehension of the given poblem and the way how the model woks even in elation to soting based on Case-based distance model. The difficulty with intepetation of the esults of classification is that the oiginal data have been tansfomed by the value function employed in the model which affected the position of the goups and the shape of the sufaces bounding them. Duing the analysis of the mathematical backgound we contibute to the Case-based distance model with a gaphical extension which is a poposal to the epesentation of the final classification of the goups of SKUs in one space. Refeences Adle N. & Raveh A. (008. Pesenting DEA gaphically. Omega Vol. 6 N Caki O. & Canbolat M.S. (008. A web-based decision suppot system fo multi-citeia inventoy classification using fuzzy AHP methodology. Expet Systems with Applications Chen Y. Li K. W. Kilgou D. M. & Hipel K. W. (008. A case-based distance model fo multiple citeia ABC analysis. Computes & Opeations Reseach Chu Ch.W. Liang G. S. & Liao Ch. T. (008. Contolling inventoy by combining ABC analysis and fuzzy classification. Computes & Industial Engineeing Condon E. Golden B. & Wasil E. (00. Visualizing goup decisions in analythic hieachy pocess. Computes & Opeations Reseach Floes B. E. & Whybak D. C. (987. Implementing multiple citeia ABC analysis. Jounal of Opeations Management 7 ( Hay M. J. Mann P. S. De Hodgins O. C. Hulbet R. L. & Lacke C. J. (00. Pactiotione s Guide to Statistics and Lean Six Sigma fo Pocess Impovements. Hoboken Jew Jesey John Wiley & Sons. Guvemi H. A. & Eel E. (998. Multiciteia inventoy classification using a genetic algoithm. Euopean Jounal of Opeational Reseach Ma L. C. (0. Sceening altenatives gaphically by an extended case-based distance appoach. Omega Vol. 40 N Mashall A. J. (007. Vilfedo Paeto s sociology a famewok fo political psychology. Ashgate UK Aldeshot. Meye A. D. (99. Visual Data in Oganizational Reseach. Oganization Reseach Vol. No Patovi F. Y. & Buton J. (99. Using the analytic hieachy pocess fo ABC analysis. Intenational Jounal of Poduction and Opeations Management Ramanathan R. (006. ABC inventoy classification with multiple-citeia using weighted linea optimization. Computes & Opeations Reseach Tufte E. R. (98. Visual Display of Quantitative Infomation. Chesie CT Gaphics Pess. Vetschea R. Chen Y. Hipel K. W. & Kilgou D. M. (00. Robustness and infomation levels in case-based multiple citeia soting. Euopean Jounal of Opeational Reseach

16 Péte VOLF Zoltán KOVÁCS István SZALKAI. D Visualization of a Case-Based Distance Model Advised by Zsolt T. Kosztyan Univesity of Pannonia Hungay 97 Received: Septembe 5 0 Accepted: Octobe 0 0 Péte Volf Zoltán Kovács István Szalkai Ph. D. Candidate Univesity of Pannonia H-800 Egyetem Steet 0 Veszpém Hungay. volfpete@gtk.uni-pannon.hu Website: P%C%A9te_adatlapa Pofesso Univesity of Pannonia H-800 Egyetem Steet 0 Veszpém Hungay. kovacsz@gtk.uni-pannon.hu Website: Kov%C%Acs_Zolt%C%An_adatlapa Assistant Pofesso Univesity of Pannonia H-800 Egyetem Steet 0 Veszpém Hungay. szalkai@almos.uni-pannon.hu Website: