Measuring performance by integrating k-medoids with DEA: Mongolian case

Baya aa, Ba chimeg; Ta noczi, Tibo ; Feny es, Ve onika A icle Measu ing pe o mance by in eg a ing k-medoids wi h DEA: Mongolian case Jou nal o Business Economics and Managemen (JBEM) P o ided in Coope a ion wi h: Vilnius Gediminas Technical Uni e si y (VILNIUS TECH) Sugges ed Ci a ion: Baya aa, Ba chimeg; Ta noczi, Tibo ; Feny es, Ve onika (2019) : Measu ing pe o mance by in eg a ing k-medoids wi h DEA: Mongolian case, Jou nal o Business Economics and Managemen (JBEM), ISSN 2029-4433, Vilnius Gediminas Technical Uni e si y, Vilnius, Vol. 20, Iss. 6, pp. 1238-1257, h ps://doi.o g/10.3846/jbem.2019.11237 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/317374 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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Published by VGTU P ess This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (h p://c ea i ecommons. o g/licenses/by/4.0/), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal au ho and sou ce a e c edi ed. *Co esponding au ho . E-mail: baya aa.ba [email protected] Jou nal o Business Economics and Managemen ISSN 1611-1699 / eISSN 2029-4433 2019 Volume 20 Issue 6: 1238–1257 h ps://doi.o g/10.3846/jbem.2019.11237 MEASURING PERFORMANCE BY INTEGRATING K-MEDOIDS WITH DEA: MONGOLIAN CASE Ba chimeg BAYARAA*, Tibo TARNOCZI , Ve onika FENYVES Ins i u e o Accoun ing and Finance, Facul y o Economics and Business, Ká oly Ih ig Doc o al School o Managemen and Business, Uni e si y o Deb ecen, Deb ecen, Hunga y Recei ed 30 Janua y 2019; accep ed 18 July 2019 Abs ac . Pe o mance measu emen encou ages Decision Making Uni s (DMUs) o imp o e hei le el o pe o mance by compa ing hei cu en inancial posi ions wi h ha o hei pee s. Da a En elopmen Analysis (DEA) is a widely used app oach o pe o mance measu emen , hough i is suscep ible when he da a is he e ogeneous. The main objec i e o his s udy is o examine he pe o mance o Mongolian lis ed companies by combining DEA and a k-medoid clus e ing me hod. Clus e ing acili a es he cha ac e iza ion and pa e ns o da a and iden i ica ion o homogenous g oups. This s udy applies he in eg a ion o k-medoids and pe o mance measu emen . The e- sea ch used 89 Mongolian companies’ inancial s a emen s om 2012 o 2015 – ob ained om he Mongolian S ock Exchange websi e. The companies a e g ouped by k-medoids clus e ing, and e - iciency o each clus e is e alua ed by DEA. Acco ding o he silhoue e me hod, he companies a e classi ied in o wo clus e s which a e conside ed i s clus e as small and medium-sized (80), and second clus e as big (9) companies. Bo h clus e s a e analyzed and compa ed by inancial a ios. The mean e iciency sco e o big companies’ is much highe han ha o small and medium-sized companies. In eg a ed esul s show ha clus e -speci ic e iciency p o ides be e pe o mance han p e-clus e ing e iciency esul s. Keywo ds: inancial pe o mance, k-medoids clus e ing, da a en elopmen analysis, inpu e - iciency, a iable e u n o scale, decision making uni . JEL Classi ica ion: C38, C14, L25. In oduc ion The basis o all ypes o analysis is da a. I is possible o ace a ious da a in e e yday li e, wi h o wi hou any p io knowledge. Howe e , u he analysis canno be made wi hou knowing he pa e n and he cha ac e is ics o da a. Da a classi ica ion – by g ouping o clus e ing – is one means o p ima y analysis. Clus e s ha e been de e mined in many ways, ye he e is no single de e mina ion which is globally accep ed. Clus e analysis is used o unde s and- Jou nal o Business Economics and Managemen , 2019, 20(6): 1238–1257 1239 ing ( inding meaning ul g oups o objec s ha sha e common cha ac e is ics) and u ili y ( o abs ac he ep esen a i e objec om among many o he s in he same clus e s) (Wu, 2012). Clus e ing echniques a e di ided in o pa i ional and hie a chical ypes. The mos popu- la and well-known pa i ional clus e echnique is k-means, which is widely employed in esea ch. Al hough k-means is a popula choice among pa i ional clus e s, i is sensi i e o ou lie s. On he con a y, he k-medoids algo i hm is mo e obus and less sensi i e o ou li- e s. Resea ch, which compa ed k-medoids wi h k-means, sugges ed he k-medoid was be e in all aspec s. Fo example, A o a and Va shney (2016) compa ed k-means and k-medoids in hei esea ch. Thei esul s p o ed ha k-medoids is be e han k-means; as execu ion ime, sensi i i y o ou lie s and space complexi y o o e lapping a e all less. Na ayana and Vasu- ma hi (2018) s a ed in hei wo k ha he k-medoids echnique is mo e accu a e and easie o unde s and han k-means clus e ing. Mo eo e , Pa el and Singh (2013) s udied a new app oach o k-means and k-medoids algo i hm and concluded ha k-medoids imp o ed accu acy. Howe e , A bin, Suhaimi, Mokh a , and O hman (2016) e alua ed k-means and k-medoids, and bo h me hods we e ound o be good ha ing mean e o s less han h ee. The K-medoids algo i hm, which was p oposed by Kau man and Rousseeuw (1987), was de eloped and in es iga ed by a ious esea che s om di e en ields. Fo example, Ho- Kieu, Vo-Van, and Nguyen-T ang (2018) e i ied and compa ed he e ec i eness and easibil- i y o he k-medoids me hod and algo i hm wi h a ious o he algo i hms’ h ough a i icial and eal da ase s. The esul s e ealed an ou s anding pe o mance h ough he e alua ion c i e ia. Pa k and Jun (2009) ecommended a simple and as algo i hm o k-medoids clus- e ing. In his wo k, a new algo i hm – which uns like he k-means algo i hm – was p o- posed. Mohammad, Zadegan, Mi zaie, and Sadoughi, (2013) examined anked k-medoids. They in oduced a new k-medoids algo i hm, which can ind all Gaussian-shaped clus e s. Zhang and Couloigne (2005) sugges ed a new k-medoids algo i hm o spa ial clus e ing in la ge applica ions. Gandhi and S i as a a (2014) p esen ed an o e iew o he modi ied k-medoids algo i hm o imp o e scalabili y and e iciency. Sood and Bansal (2013) su eyed he combina ion o he k-medoids algo i hm and he ba algo i hm. Mei and Chen (2011) in es iga ed medoid-based uzzy ela ional clus e ing. Clus e ing i sel is no he inal esul , a he i is a possible da a inpu o u he analysis. The e o e, he aim o he s udy was o imp o e he accu acy o pe o mance analysis by in e- g a ing i wi h clus e ing. I mus be men ioned ha clus e ing me hods ha e been used wi h e iciency analysis be o e. Fo example, Om ani, Sha aa , and Em ouznejad (2018) in eg a ed a uzzy clus e ing coope a i e game wi h DEA model in an applica ion o hospi al e iciency. In hei esea ch, 288 hospi als om 31 p o inces o I an examined. Simila ly, Jahangoshai, Rezaee, Jozmaleki, and Valipou (2018) in eg a ed uzzy C-means, DEA, and an a i icial neu al ne wo k. They ob ained hei da a ( om 2007 o 2012) om he Teh an S ock Ma ke and used inancial a ios as a iables. Thaka e and Bagal (2015) e alua ed he pe o mance o k-means wi h a ious me ics. They concluded ha he pe o mance o he k-means algo- i hm is based on he dis ance me ics as well he da abase used. Kim, Lee, and Kang (2018) in eg a ed eigh in e nal clus e ing e iciency measu es based on DEA and p oposed a new clus e alidi y index. Amin, Wan-Ismail, Abdul-Rasid, and Selemani (2014) analyzed some issues con on he DEA clus e ing algo i hm. Kian a Ahadzadeh Namin, Alam Tab iz, Na- 1240 B. Baya aa e al. Measu ing pe o mance by in eg a ing k-medoids wi h DEA: Mongolian case ja i, and Hosseinzadeh Lo i (2017) examined a hyb id clus e and DEA. They concluded ha he e was a pe cep ible di e ence be ween he e iciency o DMUs wi h he uppe bound and wi h he lowe bound. Po, Guh, and Yang (2009) p esen ed a new clus e ing app oach using DEA. Bi, Song, and Wu (2014) p oposed slack-based measu e-based clus e ing me hod o classi y he en i onmen al pe o mance o Chinese indus y. Dai and Kuosmanen (2014) p oposed benchma king using he clus e ing me hod. They concluded clus e -speci ic e i- ciency anking p o ides mo e e icien and meaning ul benchma king han he con en ional app oach. Mo eo e , some esea che s in eg a ed clus e ing wi h a pa ame ic me hod such as, ‘Bayesian clus e ing in s ochas ic on ie analysis’ by G i in (2011). Acco ding o he Mongolian S ock Exchange’s (MSE) esea ch, 475 companies we e eg- is e ed o he las 10 yea s; howe e , 202 companies we e delis ed by Ma ch o 2019. This shows he need o he lis ed companies o be e alua ed p ope ly – by hei pe o mance – and make imp o emen s based on bes p ac ices. F om he esea che s’ poin o iew, he e is no published esea ch used in inancial s a emen s’ pa ame e s o k-medoids. Also, he da a analyzed in his pape has high a iabili y. The e o e, he algo i hm o k-medoids is chosen ins ead o k-means; which is less sensi i e o noise and ou lie s han k-means. Mo eo e , he e is no published esea ch using k-medoids o Mongolian companies, so his s udy ap- plies k-medoids which is a compa ably new clus e ing me hod. The main goal is o measu e he pe o mance o Mongolian lis ed companies. To do so, he ollowing s eps a e made: – Fundamen al s a is ical analysis o decide whe he clus e ing is app op ia e o he da a; – De e mina ion o he op imal numbe o clus e s, using he silhoue e me hod; – Iden i ica ion o clus e s by K-medoids; – Analysis o each clus e by a io analysis; – E alua ion o he pe o mance o each clus e , and compa ison be ween he pe o - mance wi h ha o p e-clus e ing pe o mance. The emainde o his pape is o ganized in he ollowing way. A e he in oduc ion, he i s sec ion e iews he li e a u e on pe o mance measu emen , he DEA me hod, clus- e ing, and he k-medoids algo i hm. The second sec ion p esen s da a se s and a iables used du ing he analysis. Sec ion h ee consis s o empi ical esul s and he conclusions a e p esen ed in sec ion ou . 1. Li e a u e e iew 1.1. Pe o mance measu emen Co po a e pe o mance is he measu emen o wha has been achie ed by a company. Mas i (2013) no es, ‘Pe o mance measu emen sys em is used by an o ganiza ion no jus o de- e mine whe he i s objec i es ha e been me bu also as a means o compa ing hei pe o - mance wi h ha o o he DMUs (decision-making uni s)’. DMUs can be i ms, o ganiza ions, di isions, indus ies, p ojec s o indi iduals. Pe o mance can be di ided in o wo pa s: e iciency and e ec i eness, which a e o en con used. Neely, G ego y, and Pla s (1995) desc ibed e ec i eness as he ex en o which Jou nal o Business Economics and Managemen , 2019, 20(6): 1238–1257 1241 cus ome s’ equi emen s a e me , while e iciency is a measu emen o how economically he i m’s esou ces a e u ilized when p o iding a gi en le el o cus ome sa is ac ion. In con as , Coope , Sei o d, and Tone (2006) desc ibed e ec i eness as goal achie emen and e iciency as he e alua ions o he esou ces used. The scope o his s udy is o e alua e e iciency, no e ec i eness. E iciency is he a io calcula ed om inpu esou ces and ou pu esul s, o e alua e whe he he use o inpu esou ces is e ec i ely employed o he ou come o no (Azadeh, Ghade i, Mi an, Eb ahimipou , & Suzuki, 2007; Ueasin, Liao, & Wongchai, 2015). The ob- jec i e o e iciency measu emen is o de ec weak a eas so ha app op ia e e o s can be de o ed o imp o e pe o mance. E iciency (cos e iciency o o e all e iciency) has wo componen s: echnical e iciency (abili y o a oid was e by p oducing as much ou pu , as inpu usage allows), and alloca i e e - iciency (combining inpu and ou pu in an op imal p opo ion based on p ices) (Munisamy- Do aisamy, 2004). O e all e iciency (cos e iciency) means he i m mus be able o choose he igh mix o inpu s and use hem in a echnically e icien manne (Boge o & O o, 2011) OE = TE × AE, (1) whe e OE is o e all e iciency, TE is he echnical e iciency, AE is he alloca i e e iciency. Since alloca i e e iciency equi es p ice in o ma ion, his esea ch conce ns echnical e iciency only. Technical e iciency signi ies a le el o pe o mance ha desc ibes a p ocess which uses he lowes amoun o inpu s o c ea e he g ea es amoun o ou pu s. I is no e- wo hy ha echnical e iciencies can be gained by sac i icing quali y, since highe quali y can be a ained by educing p oduc i i y and inc easing cos s (Sudi , 1996). In he simples case – whe e a p ocess o uni has a single inpu and a single ou pu – echnical e iciency is de ined as: =y TE x , (2) whe e x is inpu ec o , y is he ou pu ec o . Typically, DMUs use mul iple inpu s and ou pu s (Bousso iane, Dyson, & Thanassoulis, 1991) and in ha case, e iciency is de e mined as: TE= =Weigh ed sum o ou pu s . Weigh ed sum o inpu s TE (3) E icien companies ake a sco e o 1, so he e iciency sco e which is close o 1 shows be e pe o mance. A e calcula ing he e iciency sco e, ine iciency can be easily de e - mined by sub ac ing he e iciency sco e om one. The smalle he ine iciency is, he be e he pe o mance is (Boge o & O o, 2011). 1.2. Da a en elopmen analysis E iciency measu emen me hods can be di ided in o h ee main ca ego ies: a io indica o s, pa ame ic and nonpa ame ic me hods (Vinco á, 2005). A signi ican di e ence be ween he pa ame ic and he non-pa ame ic app oaches is he es ima ion me hod. DEA is a non- pa ame ic app oach o weigh he inpu s/ou pu s and o measu e he ela i e e iciency o 1242 B. Baya aa e al. Measu ing pe o mance by in eg a ing k-medoids wi h DEA: Mongolian case DMUs (Ablanedo-Rosas, Gao, Zheng, Alidaee, & Wang, 2010). The idea o DEA was i s in- oduced by Fa ell (1957) and de eloped by Cha nes, Coope , and Rhodes (1978). The gene al idea o DEA is conside ing DMUs o ha e he same echnology se . The echnology se is he se o ou pu s ha can be p oduced by using a ailable inpu s which akes ze o o posi i e numbe s as inpu and ou pu a iables (non-nega i e). DEA de e mines wo o ien a ions: inpu e iciency and ou pu e iciency. Inpu e iciency is app op ia e when one is in e es ed in minimizing inpu s, and ou pu e iciency when one is in e es ed in maximizing ou pu . Inpu e iciency: ( ) ( ) + = = ∈ ∈= 0 00 * 00 * ( , ; min{ | , }E Exy T ER Exy T , (4) whe e (x, y) means easibili y o he ec o , and T* he smalles se which is consis en wi h he da a. Inpu e iciency akes a alue be ween 0.0 and 1.0. Fo example, a alue o 0.6 ob ained by he inpu -o ien ed me hod shows he possibili y o p oduce he same ou pu when he inpu s a e dec eased by 40%. The on ie scale o DEA consis s o cons an e u n o scale (CRS) and a iable e u n o scale (VRS). VRS consis s o inc easing (IRS) and dec easing e u n o scale (DRS) (Feny es, Ta nóczi, & Zsidó, 2015). Choosing be ween DRS and IRS depends on he i m’s indus y. In his esea ch, he inpu e iciency VRS model by R s a is ical p og am is used. Dec easing Re u n o Scale: ( ) ( ) ∈ ≤λ≤ ⇒λ ∈=, , 0 1 , xy T xy T . (5) Inc easing Re u ns o Scale: ( ) ( ) ∈ λ ≤ ⇒ λ ∈=, , 1 , xy T xy T . (6) Al hough CRS is he basic DEA model, i is app op ia e when DMUs eely p oduce unde hei op imal size. Bu he he e ogenei y o he da a shows ha he e is no a pe ec compe i ion among he DMUs. The e o e, VRS model is chosen which is mo e ealis ic in his s udy. When a da ase con ains wide- anging companies, he impo ance o he pe o - mance measu emen maybe ques ionable, he e o e, pe o mance measu emen is in eg a ed wi h clus e ing. 1.3. Clus e ing Clus e ing plays an essen ial ole in helping people o analyze, desc ibe and u ilize he alu- able in o ma ion hidden in he g oups (Wu, 2012). Clus e analysis is one o he da a mining me hods o disco e ing knowledge in mul idimensional da a. The p ima y goal o clus e analysis is o iden i y pa e n o g oups o objec s wi hin a da a se which ha e high simila i y. Clus e ing echniques a e di ided in o he pa i ional and hie a chical. Pa i ional clus e ing me hods di ec ly di ide da a poin s in o some p e-speci ied numbe o clus e s wi hou he hie a chical s uc u e. In con as , hie a chical clus e ing g oups he da a wi h a sequence o nes ed pa i ions, ei he om single on clus e s o a clus e including all o he indi iduals; o ice e sa (Xu & Wunsch, 2008). Pa i ioning me hods eloca e ins ances by mo ing hem om one clus e o ano he , s a ing om an ini ial pa i ioning (Rokach & Maimon, 2010). Jou nal o Business Economics and Managemen , 2019, 20(6): 1238–1257 1243 Based on he way o app oach he cen e , clus e analyses a e classi ied as: ha d (c isp) clus e ing and so ( uzzy) clus e ing (Ho-Kieu e al., 2018). The mos common and well- known ha d clus e ing is k-means. The k-means is a simple and as clus e ing me hod. Mo eo e , a k-means algo i hm has he excellen abili y o handle a la ge numbe o in es- iga ed da a. The k-means algo i hm applies a s anda d dis ance measu e o mula, o de e - mine he simila i y o he da a epe i i ely, o ob ain he high in e -clus e dis ance among clus e s (A bin e al., 2016). K-means clus e ing i e a i ely inds he k cen oids and assigns e e y objec o he nea es cen oid (Pa k & Jun, 2009). The cen oids a e upda ed by aking he a e age o all da a. The e o e, i he e a e ou lie s in da a, he cen oids will be pushed o he ou lie s. Ex emely high alues migh subs an ially dis o he dis ibu ion o da a, which is he d awback o k-means. In con as o k-means, k-medoids uses he mos cen ally loca ed objec in a clus e – ins ead o cen e mass – which helps o o e come he k-means’ d awback. 1.4. K-medoids algo i hm K-medoids algo i hm is compu a ionally ha de han ha o he k-means due o compu ing he medoids using he equency o occu ences. Clus e ing endency, which shows whe he he clus e ing is app op ia e o he da a, mus be assessed, be o e employing a clus e ing algo i hm. A e wa ds, he numbe o clus e s and algo i hms mus be de e mined. Finally, clus e alida ion (goodness o clus e ing esul s) should be done. Acco ding o Kassamba a (2017), he mos common algo i hm o k-medoids clus e ing is he Pa i ioning A ound Medoids (PAM) and is as ollows: – Ini ialize. Randomly selec k (numbe o he clus e ) o he n da a poin s as he ‘me- diods’. Like k-means, k-medoids equi es a p e-se numbe o clus e (k). The e is no inal app oach o de e mining he numbe o clus e s, bu iden i ying an inap- p op ia e numbe o clus e s can lead o meaningless clus e s (which do no exis ). A use ul app oach o assess he op imal numbe o clus e s is he silhoue e me hod (Kassamba a, 2017). – Calcula e he dissimila i y ma ix. A s anda d way o exp ess simila i y is h ough a se o dis ances be ween pai s o objec s (Ha igan, 1989). Da a wi hin he g oup (in a-clus e ) a e simila , while da a be ween he g oups (in e -clus e ) a e di e en , based on he speci ic c i e ia. – Assign e e y objec o he closes medoid. The closes medoid is de ined by using any alid dis ance me ic, mos commonly: Euclidean dis ance ( he oo -sum-o -squa es o di e ences), Manha an dis ance ( he sum o absolu e dis ances) o Minkowski dis ance. Manha an is mo e obus han Euclidean dis ances when da a con ains ou - lie s. I any o he objec s o he clus e dec eases he a e age dissimila i y coe icien , selec he en i y ha educes his coe icien he mos as he medoid o his clus e . Each clus e ing algo i hm c ea es a di e en clus e o he same da a, he e o e, clus e alida ion is essen ial o de e mine whe he he clus e s a e meaning ul o jus a i ac s o he clus e ing algo i hm. – The e a e h ee ca ego ies o alida ion: – Ex e nal; used o selec a sui able clus e ing algo i hm; 1244 B. Baya aa e al. Measu ing pe o mance by in eg a ing k-medoids wi h DEA: Mongolian case – In e nal: measu es he compac ness (wi hin clus e a ia ion), and he connec edness and he sepa a ion (how well-sepa a ed) o he clus e pa i ions; – Rela i e c i e ia (Kassamba a, 2017). 2. Da a and a iables Mongolian companies a e o ganized as public o non-public. Since public companies’ inan- cial epo s a e equi ed o be audi ed, hei da a is mo e eliable ( han ha o non-public companies) and is publicly a ailable. In his esea ch, 89 public companies’ inancial s a emen s ( om 2012 o 2015) we e ob ained om he MSE (Mongolian S ock Exchange) and used as da a. In 2009, he MSE s a ed p o iding downloadable inancial s a emen s; hough only nine companies’ inancial s a emen s we e a ailable a ha ime. In 2010, he numbe o publicly a ailable inancial s a emen s ose d ama ically o 100. Howe e , he o m o inancial s a emen s was changed in 2012, which made i di icul o compa e he inancial s a emen s be o e and a e 2012. Al hough he e a e 334 egis e ed public companies, no all he companies we e sui able sou ces o da a. Some companies’ inancial epo s we e deduc ed om esea ch due o bank up cy, lack o annual epo s, and ze o alues in inancial da a. Only 137 companies ou o 334, epo ed publicly hei inancial s a emen s o 2015. The inancial s a emen s which a e used in he esea ch me he equi emen s o consis ency, compa abili y, and accu acy. In his pape , o al asse s and e enue a e chosen o k-medoids. Da a includes companies in di e en sec o s, including, se ices, mining, manu ac u ing, e c. The majo i y o cos s in he se ice sec o a e ope a ional cos s, while he cos s o goods sold we e g ea e in manu ac- u ing. Simila ly, cu en asse s cons i u e mo e in he se ice sec o and less in he mining sec o . Conside ing he cha ac e is ics o sec o s, e enue and a e - ax p o i a e chosen as ou pu a iables; while non-cu en asse s, cu en asse s, cos o goods sold and ope a ional cos s a e inpu a iables in he DEA. The desc ip i e s a is ics o each yea a e a ached in Appendix 1. As we can see om Appendix 1, he coe icien s o a ia ion a e ema kably high (> 200%), especially o he las a iable (a e - ax p o i > 400%). The high alues o he coe icien s o a ia ion indica e ha he a iabili y and he e ogenei y a e also ex emely high. F om Appendix 1, i is appa en ha he alues o he ku osis indica o a e high, posi i e alues, which means ha he da a a e p edomina ing a ound he a e age alue. The alues o he skewness indica o a e also posi i e which indica es he densi y unc ions o he a iables ha e longe ails on he igh side, and he mass o he dis ibu ion is concen a ed on he le side. Figu e 1 p esen s he high a iabili y o he in es iga ed a iables which can be seen in Appendix 1. The desc ip i e s a is ics o he ou yea s’ a e age alues a e p esen ed in Table 1. The In e qua ile Range (IQR) indica o s, which show he ange o he middle 50% o he da a, we e calcula ed by using he a e age alues o he a iables. I is appa en om Table 1 ha he IQRs o gi en a iables ep esen only a ac ion o hei o al anges (0.72–7.43%). Ad- di ionally, mo e han 90% o he o al ange o he a iables is in he ou h qua ile. O e all, he s a is ical cha ac e is ics show simila endencies o he annual da a in Appendix 1. The a iables ha e huge a iabili y, and he mo e signi ican pa o hei o al ange is loca ed in Jou nal o Business Economics and Managemen , 2019, 20(6): 1238–1257 1245 Figu e 1. He e ogenei y ac oss yea s – di e ences among he companies (sou ce: au ho ’s calcula ion by R S udio) 1252 B. Baya aa e al. Measu ing pe o mance by in eg a ing k-medoids wi h DEA: Mongolian case Bi, G., Song, W., & Wu, J. (2014). A clus e ing me hod o e alua ing he en i onmen al pe o mance based on slacks-based measu e. Compu e s & Indus ial Enginee ing, 72, 169-177. h ps://doi.o g/10.1016/j.cie.2014.03.016 Boge o , P., & O o, L. (2011). In e na ional Se ies in Ope a ions Resea ch & Managemen Science: Vol. 157. Benchma king wi h DEA, SFA, and R. Sp inge , Bos on, MA. h ps://doi.o g/10.1007/978-1-4614-1900-6 Bousso iane, A., Dyson, R. G., & Thanassoulis, E. (1991). Applied da a en elopmen analysis. Eu opean Jou nal o Ope a ional Resea ch, 52(1), 1-15. h ps://doi.o g/10.1016/0377-2217(91)90331-O Cha nes, A., Coope , W. W., & Rhodes, E. (1978). Measu ing he e iciency o decision making uni s. Eu opean Jou nal o Ope a ional Resea ch, 2(6), 429-444. h ps://doi.o g/10.1016/0377-2217(78)90138-8 Coope , W., Sei o d, L., & Tone, K. (2006). In oduc ion o da a en elopmen analysis and i s uses. Re- ie ed om h p://link.sp inge .com/con en /pd /10.1007/0-387-29122-9.pd Dai, X., & Kuosmanen, T. (2014). Bes -p ac ice benchma king using clus e ing me hods: Applica ion o ene gy egula ion. Omega, 42(1), 179-188. h ps://doi.o g/10.1016/j.omega.2013.05.007 Fa ell, M. J. (1957). The measu emen o p oduc i e e iciency. Jou nal o he Royal S a is ical Socie y. Se ies A (Gene al), 120(3), 253. h ps://doi.o g/10.2307/2343100 Feny es, V., Ta nóczi, T., & Zsidó, K. (2015). Financial pe o mance e alua ion o ag icul u al en e - p ises wi h DEA me hod. P ocedia Economics and Finance, 32(15), 423-431. h ps://doi.o g/10.1016/S2212-5671(15)01413-6 Gandhi, G., & S i as a a, R. (2014). Analysis and implemen a ion o modi ied K-medoids algo i hm o inc ease scalabili y and e iciency o la ge da ase . IJRET: In e na ional Jou nal o Resea ch in Enginee ing and Technology, 3(6), 150-153. h ps://doi.o g/10.15623/ij e .2014.0306027 Gand ud, C. (2015). Rep oducible esea ch wi h R and R s udio (2nd ed.). New Yo k: Chapman & Hall/ CRC. h ps://doi.o g/10.1201/b18546 G i in, J. E. J. P. A. (2011). Bayesian clus e ing o dis ibu ions in s ochas ic on ie analysis. Jou nal o P oduc i i y Analysis, 36(3), 275-283. h ps://doi.o g/10.1007/s11123-011-0213-7 Ha igan, J. A. (1989). Clus e ing algo i hms (pp. 331-335). John Wiley & Sons, Inc. h ps://doi.o g/10.1002/0471725382.sca d Ho-Kieu, D., Vo-Van, T., & Nguyen-T ang, T. (2018). Clus e ing o p obabili y densi y unc ions by new k-Medoids Me hod. Scien i ic P og amming, 2018, 7. h ps://doi.o g/10.1155/2018/2764016 Jahangoshai Rezaee, M., Jozmaleki, M., & Valipou , M. (2018). In eg a ing dynamic uzzy C-means, da a en elopmen analysis and a i icial neu al ne wo k o online p edic ion pe o mance o com- panies in s ock exchange. Physica A: S a is ical Mechanics and I s Applica ions, 489, 78-93. h ps://doi.o g/10.1016/j.physa.2017.07.017 Kassamba a, A. (2017). P ac ical guide o clus e analysis in R: unsupe ised machine lea ning. STHDA. Kau man, L., & Rousseeuw, P. J. (1987). Clus e ing by means o medoids. In Y. Dodge (Ed.), S a is ical Da a Analysis Based on he L 1-No m and Rela ed Me hods. Fi s In e na ional Con e ence. Else ie Science L d. Kian a , K., Ahadzadeh Namin, M., Alam Tab iz, A., Naja i, E., & Hosseinzadeh Lo i, F. (2017). Hyb id clus e analyzing and da a en elopmen analysis wi h in e al da a. Scien ia I anica, 25(5), 2904- 2911. h ps://doi.o g/10.24200/sci.2017.4482 Kim, B., Lee, H., & Kang, P. (2018). In eg a ing clus e alidi y indices based on da a en elopmen analysis. Applied So Compu ing, 64, 94-108. h ps://doi.o g/10.1016/j.asoc.2017.11.052 Lemos, C. A. A., Lins, M. P. E., & Ebecken, N. F. F. (2005). DEA implemen a ion and clus e ing analysis using he K-Means algo i hm. WIT T ansac ions on In o ma ion and Communica ion Technologies, 35, 321-329. Mas i, M. H. (2013). Pe o mance measu emen sys ems in se ice SME: A B unei case s udy. (PhDThe- sis). The Uni e si y o Manches e , Uni ed Kingdom. Jou nal o Business Economics and Managemen , 2019, 20(6): 1238–1257 1253 Mei, J. P., & Chen, L. (2011). Fuzzy ela ional clus e ing a ound medoids: A uni ied iew. Fuzzy Se s and Sys ems, 183(1), 44-56. h ps://doi.o g/10.1016/j. ss.2011.06.009 Mohammad, S., Zadegan, R., Mi zaie, M., & Sadoughi, F. (2013). Ranked k-medoids: A as and ac- cu a e ank-based pa i ioning algo i hm o clus e ing la ge da ase s. Knowledge-Based Sys ems, 39, 133-143. h ps://doi.o g/10.1016/j.knosys.2012.10.012 Munisamy-Do aisamy, S. (2004). Benchma king he pe o mance o UK elec ici y dis ibu ion ne wo k ope a o s: A s udy o quali y, e iciency and p oduc i i y. Quali y. Wa wick Business School. Na ayana, G. S., & Vasuma hi, D. (2018). An A ibu es simila i y-based K-Medoids clus e ing ech- nique in da a mining. A abian Jou nal o Science and Enginee ing, 43(8), 3979-3992. h ps://doi.o g/10.1007/s13369-017-2761-2 Neely, A., G ego y, M., & Pla s, K. (1995). Pe o mance measu emen sys em design. In e na ional Jou nal o Ope a ions & P oduc ion Managemen , 15(4), 80-116. h ps://doi.o g/10.1108/01443579510083622 Om ani, H., Sha aa , K., & Em ouznejad, A. (2018). An in eg a ed uzzy clus e ing coope a i e game da a en elopmen analysis model wi h applica ion in hospi al e iciency. Expe Sys ems wi h Ap- plica ions, 114, 615-628. h ps://doi.o g/10.1016/j.eswa.2018.07.074 Pa k, H. S., & Jun, C. H. (2009). A simple and as algo i hm o K-medoids clus e ing. Expe Sys ems wi h Applica ions, 36(2), 3336-3341. h ps://doi.o g/10.1016/j.eswa.2008.01.039 Pa el, A., & Singh, P. (2013). New App oach o K-mean and K-medoids Algo i hm. In e na ional Jou nal o Compu e Applica ions Technology and Resea ch, 2(1), 1-5. h ps://doi.o g/10.7753/IJ- CATR0201.1001 Po, R.-W., Guh, Y.-Y., & Yang, M.-S. (2009). A new clus e ing app oach using da a en elopmen analy- sis. Eu opean Jou nal o Ope a ional Resea ch. 1. h ps://doi.o g/10.1016/j.ejo .2008.10.022 Rokach, L., & Maimon, O. (2010). Chap e 15 – Clus e ing me hods. In The da a mining and knowledge disco e y handbook (pp. 321-352). Sp inge . h ps://doi.o g/10.1007/0-387-25465-X_15 Sood, M., & Bansal, S. (2013). K-Medoids clus e ing echnique using ba algo i hm. In e na ional Jou - nal o Applied In o ma ion Sys ems, 5(8), 20-22. h ps://doi.o g/10.5120/ijais13-450965 Sudi , E. F. (1996). E ec i eness, quali y and e iciency: a managemen o ien ed app oach. Sp inge . h ps://doi.o g/10.1007/978-94-009-1828-3 Thaka e, S. Y., & Bagal, S. B. (2015). Pe o mance e alua ion o K-means clus e ing algo i hm wi h a ious dis ance me ics. In e na ional Jou nal o Compu e Applica ions, 110(11), 975-8887. h ps://doi.o g/10.5120/19360-0929 Ueasin, N., Liao, S. Y., & Wongchai, A. (2015). The echnical e iciency o ice husk powe gene a ion in Thailand: compa ing da a en elopmen analysis and s ochas ic on ie analysis. Ene gy P ocedia, 75, 2757-2763. h ps://doi.o g/10.1016/j.egyp o.2015.07.518 Vinco á, I. K. (2005). Using dea models o measu e e iciency. Bia ec, (1), 24-28. Wu, J. (2012). Sp inge Theses: ecognizing ou s anding (Phd Resea ch). Ad ances in K-means Clus e ing: a da a mining hinking. Sp inge . h ps://doi.o g/10.1007/978-3-642-29807-3 Xu, R., & Wunsch, D. C. (2008). Clus e ing. Wiley. h ps://doi.o g/10.1002/9780470382776 Zhang, Q., & Couloigne , I. (2005). A new and e icien K-Medoid algo i hm o spa ial clus e ing. In O. Ge asi, e al. (Eds.), Compu a ional Science and I s Applica ions – ICCSA 2005. ICCSA 2005. Lec u e No es in Compu e Science ( ol. 3482, pp. 181-189). Sp inge . h ps://doi.o g/10.1007/11424857_20 1254 B. Baya aa e al. Measu ing pe o mance by in eg a ing k-medoids wi h DEA: Mongolian case APPENDIX 1 Desc ip i e s a is ics o a iables (Million ug ik) Yea s The name o s a is ical cha ac e is ics Cu en asse s Non- cu en asse s Asse s Re enue Cos o goods sold Ope a- ional cos s A e - ax p o i 2012 Minimum 0 0 9 0 0 0 –8,840 1. Qua ile 165 420 782 200 19 94 0 Median 854 1,286 2,154 924 376 422 11 3. Qua ile 5,281 8,306 13,588 8,658 4,353 1,224 453 Maximum 105,495 267,542 291,633 229,514 142,790 39,669 80,021 Mean 6,689 13,945 20,633 13,492 8,562 2,478 2,022 S anda d de ia ion 15,083 38,744 47,659 36,149 21,906 6,306 9,455 Coe icien o a ia ion 225.50% 277.84% 230.98% 267.92% 255.86% 254.45% 467.67% Skewness 4.09 4.58 3.67 4.25 3.92 3.96 6.66 Ku osis 20.53 23.50 14.82 19.70 17.33 16.58 49.95 2013 Minimum 3 5 21 2 0 0 –5,259 1. Qua ile 171 492 892 258 96 104 –44 Median 1,095 1,296 3,390 1,291 800 430 21 3. Qua ile 6,161 10,409 14,928 7,411 6,201 2,029 298 Maximum 105,217 360,649 402,889 233,290 155,390 48,554 62,096 Mean 8,185 18,160 26,345 16,966 11,295 3,301 1,792 S anda d de ia ion 18,880 50,712 63,663 42,024 27,915 8,509 8,849 Coe icien o a ia ion 230.68% 279.25% 241.65% 247.69% 247.15% 257.79% 493.81% Skewness 3.56 4.54 3.87 3.42 3.54 3.81 5.52 Ku osis 13.34 23.73 16.38 11.78 12.95 14.58 31.46 2014 Minimum 2 5 23 4 0 0 –14,174 1. Qua ile 168 510 1,000 372 107 117 –41 Median 1,233 1,692 5,063 1,535 750 441 11 3. Qua ile 7,490 8,858 16,276 9,145 7,355 2,334 463 Maximum 101,021 233,295 334,316 255,895 154,619 66,141 43,794 Mean 7,745 17,161 24,907 15,618 10,505 3,849 1,101 S anda d de ia ion 16,728 41,785 55,186 39,714 25,103 10,424 6,425 Coe icien o a ia ion 215.98% 243.48% 221.57% 254.29% 238.96% 270.82% 583.78% Skewness 3.48 3.43 3.61 4.10 3.77 4.07 4.63 Ku osis 13.15 11.88 14.36 18.66 15.82 17.54 26.48 Jou nal o Business Economics and Managemen , 2019, 20(6): 1238–1257 1255 Yea s The name o s a is ical cha ac e is ics Cu en asse s Non- cu en asse s Asse s Re enue Cos o goods sold Ope a- ional cos s A e - ax p o i 2015 Minimum 1 5 24 0 0 0 –3,505 1. Qua ile 205 587 1,153 234 82 143 –61 Median 1,386 1,783 4,707 1,053 618 432 22 3. Qua ile 8,326 10,634 19,092 8,444 6,378 2,036 348 Maximum 88,985 420,903 448,809 240,637 191,124 72,455 30,072 Mean 8,586 23,549 32,135 17,208 12,154 3,912 835 S anda d de ia ion 18,465 62,275 74,707 44,021 31,411 10,846 4,128 Coe icien o a ia ion 215.06% 264.44% 232.48% 255.83% 258.44% 277.27% 494.11% Skewness 3.24 4.04 3.50 3.58 3.77 4.32 5.51 Ku osis 10.26 18.83 13.02 12.86 15.18 20.19 32.52 End o Appendix 1 1256 B. Baya aa e al. Measu ing pe o mance by in eg a ing k-medoids wi h DEA: Mongolian case APPENDIX 2 De e mina ion o he op imal clus e numbe s by silhoue e me hod Op imal clus e numbe in 2012 Op imal clus e numbe in 2013 Op imal clus e numbe in 2014 Op imal clus e numbe in 2015 Jou nal o Business Economics and Managemen , 2019, 20(6): 1238–1257 1257 APPENDIX 3 The clus e ing plo