INFORMATYKA EKONOMICZNA

INFORMATYKA EKONOMICZNA BUSINESS INFORMATICS 1(35) 2015 Wydawnctwo Unwersytetu Ekonomcznego we Wrocławu Wrocław 2015

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Sps treśc Wstęp... 7 Andrzej Chlusk, Leszek Zora: The applcaton of bg data n the management of healthcare organzatons. A revew of selected practcal solutons. 9 Andrzej Dudek, Bartłomej Jefmańsk: The fuzzy TOPSIS method and ts mplementaton n the R programme... 19 Mara Mach-Król: Wedza ontologe w temporalnych systemach ntelgentnych... 28 Marusz Nowak, Edward Gąsorek, Jacek Radomańsk: The applcaton of nformaton technology n collectng data on tenns performance... 36 Paweł Sarka: System nformacyjny banku ntegracja procesów zarządzana ryzykem kredytowym... 53 Bartosz Wachnk: Analza kosztów transakcyjnych w nformatycznych przedsęwzęcach wdrożenowych realzowanych poprzez outsourcng... 70 Paweł Zemba, Jarosław Jankowsk, Waldemar Wolsk: Dobór języka reprezentacj wedzy w ontologach dzedznowych... 84 Summares Andrzej Chlusk, Leszek Zora: Zastosowane bg data w zarządzanu organzacjam służby zdrowa. Przegląd wybranych rozwązań... 9 Andrzej Dudek, Bartłomej Jefmańsk: Rozmyta metoda TOPSIS jej mplementacja w programe R... 19 Mara Mach-Król: Knowledge and ontologes n temporal ntellgent systems... 28 Marusz Nowak, Edward Gąsorek, Jacek Radomańsk: Wykorzystane nformatyk w procese gromadzena danych z meczów tensowych... 36 Paweł Sarka: Bankng nformaton system. An ntegraton of credt rsk management processes... 53 Bartosz Wachnk: The analyss of transacton costs n t mplementaton projects wth the use of outsourcng... 70 Paweł Zemba, Jarosław Jankowsk, Waldemar Wolsk: Selecton of knowledge representaton language n the doman ontologes... 84

INFORMATYKA EKONOMICZNA BUSINESS INFORMATICS 1(35) 2015 ISSN 1507-3866 e-issn 2450-0003 Andrzej Dudek, Bartłomej Jefmańsk Wrocław Unversty of Economcs e-mals: andrzej.dudek@ue.wroc.pl, bartlomej.jefmansk@ue.wroc.pl THE FUZZY TOPSIS METHOD AND ITS IMPLEMENTATION IN THE R PROGRAMME ROZMYTA METODA TOPSIS I JEJ IMPLEMENTACJA W PROGRAMIE R DOI: 10.15611/e.2015.1.02 Summary: The TOPSIS method (Technque for Order Preference by Smlarty Ideal Soluton) suggested by Hwang and Yoon [1981], belongs to the group of pattern lnear orderng methods of multdmensonal objects. A characterstc feature of ths method s a way to evaluate a synthetc crteron s values, whch takes nto consderaton the dstance of an evaluated object from a postve-deal soluton as well as from a negatve-deal soluton. The fuzzy TOPSIS method enables the lnear orderng of objects descrbed through lngustc varables, whose values are expressed n the form of trangular fuzzy numbers. In ths artcle, a way of synthetc measurement estmaton n envronment R was presented, accordng to the assumptons of the fuzzy TOPSIS method proposed by Chen [2000]. Scrpts, whch are ncluded n the artcle make the accomplshment of ths partcular method s stages possble. Keywords: lnear orderng, fuzzy TOPSIS, fuzzy number, lngustc varable, R programme. Streszczene: Metoda TOPSIS (Technque for Order Preference by Smlarty Ideal Soluton), zaproponowana przez Hwanga Yoona (1981 r.), należy do wzorcowych metod porządkowana lnowego obektów welowymarowych. Jej cechą charakterystyczną jest sposób oblczana wartośc kryterum syntetycznego, które uwzględna odległość ocenanego obektu zarówno od wzorca, jak antywzorca rozwoju. Rozmyta metoda TOPSIS umożlwa porządkowane lnowe obektów opsanych za pomocą zmennych lngwstycznych, których wartośc wyrażone są w postac trójkątnych lczb rozmytych. W artykule zaprezentowany został sposób estymacj mary syntetycznej w środowsku R zgodne z założenam rozmytej metody TOPSIS zaproponowanej przez Chena (2000 r.). Zameszczone w artykule skrypty umożlwają realzację poszczególnych etapów metody. Słowa kluczowe: porządkowane lnowe, rozmyta metoda TOPSIS, lczby rozmyte, rozmyte wag, modele IRT.

20 Andrzej Dudek, Bartłomej Jefmańsk 1. Introducton A lot of soco-economc phenomena have a complex character, therefore t s mpossble to descrbe them by a sngle varable. The objects evaluaton from the vew of such phenomena s possble through the constructon of a synthetc varable, aggregatng fragmentary nformaton ncluded n partcular crtera. Assessments of sngle crtera are often expressed n the form of lngustc values, whch are the expresson emergng from a natural language. Statstcal analyss of ths knd of nformaton s possble, among others, through the substtuton of lngustc expresson by trangular fuzzy numbers, whch consttute an exceptonal case of fuzzy sets. For such prepared data, one can apply the fuzzy modfcaton of lnear orderng methods. One of them s a fuzzy TOPSIS method whch fnds the applcaton n, among others, a stuaton when assessments and/or weghts of crtera are expressed n the form of trangular fuzzy numbers. The am of ths artcle s to dscuss a fuzzy TOPSIS method n the ssue of lnear orderng objects and also n a presentaton of some parts of authoral scrpts n envronment R enablng the accomplshment of ths procedure. Scrpts functonalty was llustrated by an example of the fuzzy TOPSIS method applcaton, n the research on the satsfacton of the employees of admnstratve and terrtoral admnstratve offces. 2. Fuzzy TOPSIS method The orderng of objects from the best one to the worst one consderng an assumed synthetc measure, whch s not subjected to a drect measurement, belongs to the task of lnear orderng. The tool of these methods s the synthetc measure of development, consttutng a certan functon, aggregatng fragmentary nformaton ncluded n partcular varables. Accordng to the way of synthetc measures constructon of aggregaton formula development, one can dvde them nto pattern and nonpattern ones. Pattern formulas are based on dfferent type of dstances of assessed objects from a pattern object [Walesak 2006]. Hellwg s method [1968] and the TOPSIS method suggested by Hwang and Yoon [1981], can be ncluded nto the basc pattern lnear orderng methods [Wysock 2010]. The TOPSIS method can be treated as a modfcaton of Hellwg s method. The dfference s n the way of evaluatng the synthetc crteron value. In the TOPSIS method, the formula whch serves to evaluate ths crteron takes nto account, apart from the dstance of the assessed object, also a dstance from nonpattern development. The fuzzy TOPSIS method was proposed by Chen [2000]. The dfference n relaton to the prmary verson of ths method s n expressng assessments and/or crtera s weghts n the form of trangular fuzzy numbers. An example of applyng ths method can be found, among others, n the studes of: Chang and Tseng [2008],

The fuzzy TOPSIS method and ts mplementaton n the R programme 21 Uyun and Rad [2011], Mad and Tap [2011], Yayla et al. [Yayla, Yldz, Özbek 2012], Jannatfar et al. [Jannatfar, Shah, Morad 2012], Erdoğan et al. [2013], Atae [2013], Ka et al. [Ka, Danae, Oroe 2014]. Let us assume that a certan set of objects A = { A = 1,..., n and a set of crtera ~ C = { C j j = 1,..., m, where X = { ~ xj = 1,..., n; j = 1,..., m stand for a set of fuzzy ~ W = w~ j j = 1,..., m a set of fuzzy weghts. The lnear evaluaton crteron and { orderng of objects wth the applcaton of the fuzzy TOPSIS method wth the above outlned assumptons requres the accomplshment of the followng steps [Chen 2000]: Step 1. Calculaton of normalzed fuzzy evaluaton crtera: ~ ~ xj zj =, = 1,..., n; j = 1,..., m. (1) n ~ x 2 = 1 j Step 2. Calculaton of weghted normalzed fuzzy evaluaton crtera: v = wz. (2) j j j Step 3. Appontng postve-deal soluton A development: + A and negatve-deal soluton { 1 2 m { j 1 j 2 { 1 2 m { j 1 j 2 A + + + + = v, v,..., v = (max v j J ),(mn v j J ) = 1,..., n, (3) A = v, v,..., v = (mn v j J ),(max v j J ) = 1,..., n, (4) where J 1 and J 2 are respectvely the beneft crteron and the cost crteron. Step 4. Calculaton for each object of a dstance from postve-deal soluton and negatve-deal soluton d (n the orgnal work t s an Eucldean dstance). Step 5. Calculaton of a synthetc measure: CC + = d d + + d, + d = ( 1,..., n). (5) Measure values (5) are normalzed n an nterval < 0 ; 1>. The smaller the dstance of an object from a postve-deal soluton and the bgger from a negatve-deal soluton, the closer the value of a synthetc measure s to coheson. Step 6. Establshng the objects rankng. The best object has the bggest value of a synthetc measure.

22 Andrzej Dudek, Bartłomej Jefmańsk 3. The assessment of employees satsfacton wth an applcaton of the fuzzy TOPSIS method The fuzzy TOPSIS method was appled to lnear orderng of nne selected commune offces of the Zachodnopomorske (West Pomeranan) Vovodshp n respect of the employees satsfacton 1. For the above mentoned purpose, specal scrpts had been worked out, whch can be downloaded from the webste of the Department of Econometrcs and Informaton Technology, Wroclaw Unversty of Economcs 2. The opnons of 611 employees were taken nto account. The detaled characterstcs of the research materal was ntroduced n the study of Błońsk and Jefmańsk [2013]. The crtera were grouped nto dmensons, accordng to the SERVQUAL model: C 1 -C 2 relablty, C 3 -C 7 responsveness, C 8 -C 11 assurance, C 12 -C 16 empathy, C 17 -C 23 tangbles. The names of the crtera were dstngushed n Table 1 below. Table 1. Crtera of employee s satsfacton Symbol C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 C 11 C 12 C 13 C 14 C 15 C 16 C 17 C 18 C 19 C 20 C 21 C 22 C 23 Crtera Tmely handlng of cases between co-workers at the offce Relable handlng of cases between co-workers at the offce (no errors) Desre to help from the other offce staff Cooperaton n handlng of cases by customers wth other offce staff Desre to help from the other offce staff n emergences and crss stuatons Desre to help from the superor Identfyng the employees wth the offce Confdentalty (non-commentng) of customer cases by the offce staff Adjust the level of knowledge and sklls to the poston held Mutual respect and kndness at work Sense of job securty Desre to share nformaton helpful n handlng of customer cases Transmsson of nformaton between employees n a meanngful way Adaptng workng tme to the needs of customers Effcent flow of nformaton between employees and superors Clarty n commands formulated by the superor Decor Functonalty of the workplace (space, lghtng, etc.) Avalablty of workng facltes (fax, telephone, computer, coper) Fnancal motvaton Non-fnancal motvaton Tranng Opportunty for professonal development Source: [Błońsk, Jefmańsk 2013]. 1 The research was part of the task: Customer and Local Government Employees Satsfacton carred out n the framework of the project: Implementaton of management mprovements n local government unts n the area of Zachodnopomorske (Western Pomerana) provnce. Project manager: Prof. T. Lubńska, PhD, Szczecn Unversty; task manager: Prof. Jolanta Wtek, PhD. 2 htp://wgrt.ae.jgora.pl/ad/materaly/r/topss_chan_procedure.r.

The fuzzy TOPSIS method and ts mplementaton n the R programme 23 In the crtera s assessment, a ratng scale was appled wth the followng tems: 1 very low, 2 low, 3 medum, 4 hgh, 5 very hgh. The ranges of trangular fuzzy numbers representng partcular tems were determned n accordance wth the method usng the Partal Credt Model belongng to the famly of IRT models [Jefmańsk 2015]. In Table 2, the parameters values of fuzzy numbers were dstngushed, they are descrbed as: a left spread of fuzzy number, b the centre of the fuzzy number area, for whch the values of functon s membershp equals value 1, c rght spread of fuzzy number. Table 2. Parameters of trangular fuzzy numbers for partcular tems Categores Crtera very low low medum hgh very hgh A B c a b c a b c a b c a b c C 1-10 -10-3.07-3.07-1,90-0,72-0,72 2,27 3,82 3,82 5,91 7,99 7,99 10 10 C 2-10 -10-4.41-4.41-2,63-0,84-0,84 2,96 5,08 5,08 6,96 8,84 8,84 10 10 C 3-10 -10-2,19-2,19-1,18-0,17-0,17 1,52 2,87 2,87 4,06 5,24 5,24 10 10 C 4-10 -10-4,09-4,09-2,48-0,87-0,87 1,84 2,8 2,8 4,29 5,78 5,78 10 10 C 5-10 -10-1,85-1,85-1,10-0,35-0,35 1,35 2,35 2,35 3,55 4,75 4,75 10 10 C 6-10 -10-1,29-1,29-0,89-0,48-0,48 1,06 1,63 1,63 2,83 4,02 4,02 10 10 C 7-10 -10-1,76-1,76-0,99-0,22-0,22 1,65 3,07 3,07 4,42 5,76 5,76 10 10 C 8-10 -10-1,67-1,67-0,92-0,16-0,16 1,01 1,85 1,85 3,12 4,38 4,38 10 10 C 9-10 -10-2,11-2,11-1,62-1,13-1,13 1,41 1,68 1,68 3,69 5,69 5,69 10 10 C 10-10 -10-1,28-1,28-1,11-0,93-0,93 1,39 1,85 1,85 2,92 3,99 3,99 10 10 C 11-10 -10-1,6-1,6-1,04-0,48-0,48 1,41 2,33 2,33 3,83 5,32 5,32 10 10 C 12-10 -10-3,08-3,08-2,02-0,95-0,95 1,69 2,43 2,43 4,15 5,87 5,87 10 10 C 13-10 -10-1,76-1,76-1,37-0,98-0,98 1,79 2,59 2,59 4,17 5,75 5,75 10 10 C 14-10 -10-1,36-1,36-1,24-1,12-1,12 1,52 1,92 1,92 3,81 5,7 5,7 10 10 C 15-10 -10-1,6-1,6-1,00-0,4-0,4 1,46 2,52 2,52 4,12 5,72 5,72 10 10 C 16-10 -10-1,77-1,77-1,20-0,63-0,63 1,23 1,82 1,82 3,35 4,88 4,88 10 10 C 17-10 -10-1,29-1,29-0,87-0,44-0,44 1,15 1,86 1,86 2,57 3,27 3,27 10 10 C 18-10 -10-1,57-1,57-1,04-0,5-0,5 0,96 1,42 1,42 2,4 3,38 3,38 10 10 C 19-10 -10-2,17-2,17-1,86-1,55-1,55 0,87 0,18 0,18 1,32 2,46 2,46 10 10 C 20-10 -10-1 -1-0,62-0,23-0,23 1,4 2,57 2,57 3,5 4,42 4,42 10 10 C 21-10 -10-1,23-1,23-0,64-0,05-0,05 1,16 2,26 2,26 3,25 4,23 4,23 10 10 C 22-10 -10-1,03-1,03-0,86-0,69-0,69 1,13 1,57 1,57 2,9 4,22 4,22 10 10 C 23-10 -10-1,05-1,05-0,78-0,5-0,5 1,15 1,8 1,8 2,86 3,92 3,92 10 10 Source: own computatons. For each of the objects, the average assessment was calculated from the crteron n accordance wth the prncples of fuzzy numbers arthmetc. The average crtera s

24 Andrzej Dudek, Bartłomej Jefmańsk assessments wll also have the form of trangular fuzzy numbers. The suggested scrpt requres readng the data below n the followng way: v1 v2 v3 v4 [1,] 1.820000 2.100000 1.3653846 0.8215385 [2,] 3.896154 4.292692 3.1765385 3.0396154 [3,] 5.326923 5.916923 4.1769231 4.2069231 [4,] 1.142222 1.438889 1.0038889 0.4669444 [5,] 3.549306 4.175000 2.6065278 2.6956944 [6,] 5.164722 6.053333 3.7841667 3.8683333 [7,] -0.747500-0.786875-0.4981250-1.2415625 [8,] 1.517344 1.977812 0.8482812 0.8232812 [9,] 3.450312 4.199375 2.5021875 2.3331250 [10,] 1.830417 2.464583 1.5191667 1.1366667 [11,] 4.216875 5.060625 2.8862500 3.0872917 [12,] 6.063333 7.026667 4.1258333 4.3854167 [13,] 1.737018 2.298947 1.4364912 1.0166667 [14,] 4.090263 4.867018 2.8579825 3.0106140 [15,] 5.877719 6.795789 4.0707018 4.2894737 [16,] 2.558889 3.435556 2.0255556 1.7805556 [17,] 4.895278 5.848889 3.3508333 3.6080556 [18,] 6.831667 7.795556 4.5816667 4.9522222 [19,] 0.978500 1.349500 0.9450000 0.4370000 [20,] 3.515750 4.280750 2.3990000 2.6012500 [21,] 5.261000 6.288000 3.6660000 3.8085000 [22,] 0.255625 0.584375 0.2293750-0.2656250 [23,] 2.605938 3.300937 1.5818750 1.7462500 [24,] 4.669375 5.526875 3.2525000 3.2575000 [25,] 2.945625 3.835000 2.2581250 2.0687500 [26,] 5.252187 6.150000 3.7928125 4.0331250 [27,] 7.073125 7.972500 4.9450000 5.2987500 The extract of the read by scrpt data fle ncludes the values of the four frst crtera for the nne objects whch were analysed. The frst three lnes nclude parameters a, b and c of the fuzzy assessments of the frst four crtera characterzng the frst object. The next threesomes of lnes nclude the parameters of the fuzzy assessments for further objects. It has to be emphassed that a scrpt assumes that all crtera nfluence stmulatngly a synthetc crteron. Therefore, n cases when n the crtera s set there are a cost crteron, they should be prevously transformed nto beneft crteron. Formulas enablng such a treatment for trangular fuzzy numbers were ntroduced among others n the study of Wysock [2010]. The fuzzy TOPSIS method assumes n the frst step of procedure, the normalzaton of fuzzy numbers accordng to the formula of lnear scale transformaton. An extract of the code responsble for the normalzaton of trangular fuzzy numbers s ntroduced below: normalzaton.chen<-functon(fuzzydata,type="n0"){ toreturn<-fuzzydata;

The fuzzy TOPSIS method and ts mplementaton n the R programme 25 for(j n 1:dm(fuzzyData)[2]){ f(type=="n0"){ toreturn[,j,]<-fuzzydata[,j,]/max(fuzzydata[,j,3]); else{ toreturn[,j,]<-fuzzydata[,j,]/mn(fuzzydata[,j,1]); toreturn; Accordng to the second step of the fuzzy TOPSIS method s procedure, the weghts of partcular crtera can be expressed n the form of trangular fuzzy numbers. For ths purpose, one should gve n the scrpt the coordnates of a vector ntroduced n the followng way: the frst three ones consttute the followng parameters a, b and c of the trangular fuzzy number representng the weght of the frst crteron. In ths example the same system of weghts has been assumed for all varables, therefore the parameters values of fuzzy numbers representng weghts are the same and come to 1. Because the example takes nto account 23 crtera, a weght vector should nclude 69 elements: weghtsraw<-c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1); Further stages of the fuzzy TOPSIS method are responsble for determnng patterns and non-patterns, the calculaton of Eucldean dstances of each object from a pattern and non-pattern, and a calculaton for each object synthetc crteron s value. The dstngushed actvtes are llustrated n the extract of a code below: for( n 1:n){ for(j n 1:m){ for(k n 1:3){ data1[,j,k]<-dataraw[k+(j-1)*3+(-1)*m*3] weghts[j,k]<-weghtsraw[k+(j-1)*3]; data3<-data2<-normalzaton.chen(data1) for( n 1:n){ data3[,,]<-data2[,,]*weghts; dstanceplus<-rep(0,n); dstanceant<-rep(0,n); for( n 1:n){ for(j n 1:m){ dstanceplus[]<-dstanceplus[]+sqrt(1/3* ((data3[,j,1]- 1)^2+(data3[,j,2]-1)^2+(data3[,j,3]-1)^2)); dstanceant[]<-dstanceant[]+sqrt(1/3*

26 Andrzej Dudek, Bartłomej Jefmańsk ((data3[,j,1])^2+(data3[,j,2])^2+(data3[,j,3])^2)); The last code lne of the studed scrpt: prnt(dstanceant/(dstanceplus+dstanceant)); allows to show the values of synthetc measure for each of the objects: [1] 0.5557600 0.4643094 0.2476754 0.5169039 0.5113564 0.5944460 [7] 0.4320480 0.3391906 0.6540052 The hghest level of employees satsfacton was observed n the last form of the analysed offce. The values of synthetc measure for ths object came to, approxmately 0,65. The lowest value of synthetc measure was (0,25), whch n other words means that the lowest level of employees satsfacton was noted for the thr offce n lne. 4. Concluson The lnear orderng of objects descrbed through lngustc varables s possble through expressng the values of these varables n the form of fuzzy numbers, and then the applcaton of a certan method of lnear orderng. One such method often appled n the subject lterature s the fuzzy TOPSIS method. In ths artcle, the extracts of the studed scrpt were presented, for a classcal verson of ths method suggested by Chen [2000]. It assumes that the crtera of objects assessment, as well as the weght ordered for each of the crtera, are expressed n the form of trangular fuzzy numbers. Lterature Atae E., 2013, Applcaton of TOPSIS and fuzzy TOPSIS methods for plant layout desgn, World Appled Scences Journal, vol. 24, ss. 7, pp. 908-913. Błońsk K., Jefmańsk B., 2013, Determnants of satsfacton of the employees of local government unts, Economcs & Socology, vol. 6, no. 2, pp. 158-170. Chang S.-H., Tseng H.-E., 2008, Fuzzy TOPSIS decson method for confguraton management, Internatonal Journal of Industral Engneerng, vol. 15, ss. 3, pp. 304-313. Chen C.-T., 2000, Extensons of the TOPSIS for group decson-makng under fuzzy envronment, Fuzzy Sets and Systems, no. 114, pp. 1-9. Erdoğan M., Blşk Ö.N., Kaya İ., Baraçh H., 2013, A customer satsfacton model based on fuzzy TOPSIS and SERVQUAL methods, Lecture Notes n Management Scence, vol. 5, pp. 74-83. Hwang C.L., Yoon K., 1981, Multple Attrbute Decson Makng. Methods and Applcatons, Sprnger, Berln.

The fuzzy TOPSIS method and ts mplementaton n the R programme 27 Hellwg Z., 1968, Zastosowane metody taksonomcznej do typologcznego podzału krajów ze względu na pozom ch rozwoju oraz zasoby strukturę wykwalfkowanych kadr, Przegląd Statystyczny, nr 15, pp. 307-327. Jannatfar H., Shah M.K.K., Morad J.S., 2012, Assessng ntellectual captal management by fuzzy TOPSIS, Decson Scence Letters, vol. 2, ss. 6, pp. 1991-2000. Jefmańsk B., 2015, Zastosowane model IRT w konstrukcj rozmytego system wag dla zmennych w zagadnenu porządkowana lnowego na przykładze metody TOPSIS, Prace Naukowe Unwersytetu Ekonomcznego we Wrocławu (w druku). Ka S.H., Danae A., Oroe M., 2014, An applcaton of fuzzy TOPSIS on rankng products: A case study of faucet devces, Decson Scence Letters, vol. 3, ss. 1, pp. 43-48. Mad E.N., Tap A.O.M., 2011, Fuzzy TOPSIS Method n the Selecton of Investment Boards by Incorporatng Operatonal Rsks, Proceedngs of the World Congress on Engneerng, vol. 1, pp. 291-295. Uyun S., Rad I., 2011, A Fuzzy TOPSIS multple-attrbute decson makng for scholarshp selecton, Telkomnka, vol. 9, no. 1, pp. 37-46. Walesak M., 2006, Uogólnona mara odległośc w statystycznej analze welowymarowej, Wydawnctwo Akadem Ekonomcznej we Wrocławu, Wrocław. Wysock F., 2010, Metody taksonomczne w rozpoznawanu typów ekonomcznych rolnctwa obszarów wejskch, Wydawnctwo Unwersytetu Przyrodnczego w Poznanu, Poznań. Yayla A.Y., Yldz A., Özbek A., 2012, Fuzzy TOPSIS method n suppler selecton and applcaton n the garment ndustry, FIBRES & TEXTILES n Eastern Europe, vol. 20, no. 4, pp. 20-23.