Investigation of Optimist and Pessimist Situations via DEA with Fuzzified Data: Banking Example

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1 Gaz nvesty Jounal of Scence G J Sc 28(4): (2015) Investgaton of Optmst and Pessmst Stuatons va DEA wth Fuzzfed Data: Bankng Example Ayhan GÖCÜKCÜ 1, 1 Süleyman Demel nvesty, Applcaton and Reseach Cente of Statstcal Consultancy,32220,Ispata, Tukey ABSTRACT Receved: 11/08/2015 Accepted: 03/11/2015 In ths study Data Envelopment Analyss (DEA) s appled to sample bankng data when the data s fuzzfed. In maket condtons, due to some easons, data s not always obvous and specfc. Heewth, t s focused on the fuzzness caused by the dvesfcaton of goods and sevces; examnes the applcablty of α-cut appoach and nvestgates the optmstc and pessmstc case of DEA fom customes and fms pespectve. Then, an applcaton scheme s poposed when the numbe of optons s excessve fo classcal methods. Key Wods: Fuzzy DEA, Bankng, Custome, Optmst and Pessmst Appoaches 1. INTRODCTION The pupose of ths study s to nvestgate the effcency of customes and fms n tems of optmst and pessmst Data Envelopment Analyss (DEA) appoach n fuzzfed envonment whch s known to be opposte to each othe. DEA s a classcal pefomance measuement tool ntoduced by Chanes et al. [1] and s a data-dven technque that uses obseved data to measue the elatve effcency of Decson Makng nts (DMs) wth vaous models and applcatons fo dffeent puposes and equements. The lack of any data pont s tedous and causes poblems n use and hence t s encounteed fequently n eal lfe. The data of one o moe DMs mght be lost, ncoect o fuzzfed. The soluton ncludes the use of fuzzy pogammng technques. Theefoe, though the use of the most optmstc, most pessmstc and most possble values, the pobablty dstbuton of the mssng data s obtaned and then fuzzy pogammng algothms ae used n the DEA. Fuzzy mathematcal pogammng poposed by Zadeh [2] s used fo a long tme n the lteatue and suveyed by Rommelfange [3], Heea and Vedegay [4] and Zmmemann [5] n dffeent dates. In the context of fuzzy DEA, the most compehensve study of the subject s Hatam-Mabn et al. [6] and Emouznejad and Tavana [7]. Coespondng autho, e-mal: ayhangolcukcu@sdu.edu.t

2 562 G J Sc, 28(4): (2015)/ Ayhan GÖCÜKCÜ In lteatue, dffeent aspects of bankng secto have been studed though DEA n abundance. In ths egad, Popovc [8], u et al. [9] and Paad and Zhu [10] ae the most ecent lteatue suveys. DEA has been used to study abundantly fo measung the sevce qualty o opeatng effcency of bank banches o fm level [11-15]. Chen et al. [16], takng nto account the maket sk, evaluated the banks n a fuzzy envonment. Bal and Gölcükcü [17] looked nto the secto fom the customes pespectve. Due to the effects of globalzaton, any county o a egon cannot be solated fom global events. Concuently, global socety s also nfluenced by countes o egons. The most stkng and ecent example s the wold css n 2008 caused by some fnancal actos. The actos of the maket manage the css by dffeent pcng polces o poduct dvesfcaton due to the expectatons of customes. Consequently, the poduct and/o pce optons that seve customes though dvesfcaton eaches such a lage quantty that t becomes mpossble to make dect compason. Ths s a knd of fuzzfcaton appled by fms fo nceasng poft. Bankng could also be consdeed n ths context. Although the sample data of ths study s about the bankng secto, othe sevce o poducton sectos could also face smla ccumstances. In ths context, companes ceate a fuzzy envonment by povdng multple optons though poduct o sevce dvesfcaton n ode to satsfy dffeent expectatons of the customes n some way changng the appeaance o use of the poduct. In the followng sectons, fstly, DEA and Fuzzy DEA methodology s gven wth a bef and α-cut appoach of Kao and u [18], whch s classcal fo the afoementoned type of poblems denoted wth an example. Secondly, the data vaables and an altenatve soluton s ntoduced wth an applcaton scheme and explaned why t ams to pesent a new poposal aganst Kao and u model [18]. The analyses and test esults ae gven wth tables n thd secton. Fnally, t s dscussed and concluded. 2. METHODS AND THE MATERIAS Data Envelopment Analyss s a classc method used to measue the elatve effcency of smlaly opeatng DMs whch have common nputs and outputs. It s a non-paametc technque and based on nea Pogammng (P). Due to ts pactcalty and vesatlty, DEA s accepted and spead a lot wth new models and wde applcaton aeas snce the emegng wok of Chanes et al. [1]. The ato model, whch depcts the dea of elatve effcency measuement of DMs, s tansfomed to P fom [1]; max s.t. m ; v vx 0 w m y m y v x j j 0 j 1,..., n; 1,..., s; 1,..., t (1) Geneal explanatons could be found n textbooks [19-21]. Recent developments and the extent of subject could be found n suveys and evews [6, 8-10, 22-26] Fuzzy DEA The decson-makes ae foced to choose between dffeent optons n many aspects of lfe. The dffeences between these optons could be mnuscule, o sometmes each opton cates to ou dffeent needs. In such cases, any decson make has fne-tune, eaange o wave cetan equements accodng to poty, thus, a fuzzy envonment appeas natually. At ths pont, Zadeh's [2] contbuton to the lteatue poduces a soluton. Fuzzy logc and fuzzy set concept s used n a vaety of aeas, the use of mathematcal pogammng s also qute common [3-5, 7, 27, 28]. In mathematcal pogammng fuzzness can be seen n; Technology matx Rght hand sde coeffcents Objectve functon Double o tple combnatons of above. In DEA, fuzzness can be obseved pmaly n data, namely technology matx. If all data s fuzzy, the objectve functon wll nheently be fuzzy, othewse only the DMs wth fuzzy data have fuzzy objectve functon. The effcency scoes may also be fuzzy f at least a fuzzy DM s on effcent fonte. Rght hand sde coeffcents ae taken as 0 due to model so t s awkwad to be fuzzy n DEA. Sengupta ntoduced the concept of fuzzness nto DEA [29, 30]. Besdes that, a few moe solutons exst fo vaous types of fuzzness. Hatam-Mabn et al. [6] classfed these poposals nto fou man types; Toleance appoach α-level based appoach Fuzzy ankng appoach The possblty appoach Some ecent woks othe than these appoaches ae classfed as an addton [31] unde the headng; Fuzzy athmetc

3 G J Sc, 28(4): (2015)/ Ayhan GÖCÜKCÜ 563 The Fuzzy Random/Type-2 Fuzzy Set Othe developments of Fuzzy DEA These appoaches nclude many dffeent applcatons fom each othe Classcal Method (α-cut Appoach) The best epesentng followng fuzzy DEA example was gven by Kao and u [18]. Thee ae fou hypothetcal decson makng unts named A, B, C, D and poducng 5, (5, 6, 8, 9), 9, 15 unts of outputs usng 1, 2, 3, 5 unts of nputs espectvely. Hee, (5, 6, 8, 9) s a tapezodal numbe as epesented n Fgue 1. the effcency of DMs whch ae efeencng ths unt ae also nexact. The classcal method elated to ou study s α-cut appoach poposed by Kao and u [18]. If the values of nputs {x j } and outputs {y j }could appoxmately be known, they could be epesented as fuzzy sets X j and Ỹ k whch have a membeshp functon of µ X j and µ Ỹk espectvely as follows; X x, X x S X îj j X îj j îj îj Y y, Y y S Y (2) îk k Y îk k îk îk Fgue 1: Gaphcal epesentaton of the example of Kao and u As ndcated n Fgue 1, as long as the output of DM B s less than 7.5 the effcent fonte s the connectng lne between A and D. Snce the effcency of a decson makng unt s measued as the dstance between the unt and the effcent fonte the effcency scoe of A, C, and D could be attaned as 1.0, 0.9, and 1.0 espectvely. On the othe hand the effcency of B dffes fom 5/7.5=0.666 to 7.5/7.5=1 whle ts output dffes fom 5 to 7.5. If the output of B exceeds 7.5, the effcent fonte s boken nto two peces as shown n fgue 1. Moeove, the effcency of DM C s evaluated accodng to ths fonte. As ndcated above example f any obsevaton s not exact, the effcency of coespondng DM s not exact. Futhemoe, f ths DM s on effcent fonte Wheen S(X j) and S(Ỹ k ) ae the suppot set of X j and Ỹ k as a unvesal set of k th output and j th nput of th DM. Consequently, the effcency of DM 0 could be evaluated by the followng model; max s.t; m ; v vx w m Y m Y v X j j 0 j 1,..., n; 1,..., s; 1,..., t (3) In the model above, all the values ae taken as fuzzy. Manly, f a value s fuzzy, based on the est of the data, a tangula membeshp functon s evaluated wth mnmum, aveage and maxmum values. Ths values coesponds to most pessmst, most possble and most optmstc values espectvely. Once membeshp functons ae evaluated fo all fuzzy values, the esults could be obtaned [18]. Bascally, α-cut appoach depends on obtanng a set of possble values. Suppose that;, Xîj mn xj S X îj, max X xj S X îj x îj X j xj S X îj X j X j Xîj j x j, Yk mn yk S Yk, max Y yk S Yk k Yk yk S Yîk Yk Y k y Yk k yk (4a) (4b)

4 564 G J Sc, 28(4): (2015)/ Ayhan GÖCÜKCÜ epesents the α-cut set of X j and Ỹ k and ndcates the α-possblty of coespondng nputs and outputs. As a esult Kao and u [18] gave the followng sngle step DEA fomulaton. max s.t , fo KVB 0 m Y v X 0, j 1,..., n; j 0 m ; v v w m Y X m Y v X j j j j 1,..., s; 1,..., t max s.t , fo KVB 0 m Y v X 0, j 1,..., n; j 0 m ; v v w m Y X m Y v X j j j j 1,..., s; 1,..., t Theeafte, the membeshp functons w, w w (6) (5a) (5b) fo dffeent α-levels than can be acheved by the soluton of above (5a) and (5b) model fo evey DM and evey fuzzy values. Data and Poposed Model Data elated to ou study wee collected sepaately fo each bank ove the ntenet. All banks that ae opeatng and povdng sevces to ndvdual customes n Tukey ae ncluded n the data set but they ae coded wth numeals n ode not to cause a conflct of nteest. About 20 banks gants loans to ndvdual customes unde the headngs of consume cedt (TKT), vehcle puchase loans (TST) and housng loans (KON). Futhemoe, they accept deposts of Tuksh a (T), S Dolla (SD) and Euo (ER) fom ndvdual customes. Hee, loan ates ae detemned as nput fo custome and output fo banks. But concenng the customes, they ae outputs fo customes and nputs fo banks. Thee s a vast numbe of loan/nteest ates optons offeed to customes dependng on tems o the amount of money. In such case, as could be seen n Table 1-2, the abundance of optons may cause a fuzzy envonment due to the numbe of compasons that have to be made among choces. Table 1: The numbe of optons va date and amounts that ae gven by banks to custome accodng to vaables Bank No Vaables T SD ERO TKT KON TST Total Numbe of Compason

5 G J Sc, 28(4): (2015)/ Ayhan GÖCÜKCÜ Total ,90E+39 The exponental numbe on the lowe ght hand sde cone of the table s the total numbes of compasons that have to be made among choces. It eveals that the stuaton s extemely vague fo custome. Ths can be consdeed as fuzzfcaton n some way. Moeove, the empty cells coloed dak epesent the DMs whch ae not gvng coespondng sevce. E.g. Bank 5, 7, 13, 14 and 16 dd not offe housng loans. Bank 4 also dd not gve vehcle puchase loans etc. Thee ae eght banks wth mssng sevces lkewse, namely banks 5, 7, 8, 10, 12, 13, 15 and 16. Ths s anothe souce of fuzzness fo customes who want to evaluate the best sevcng bank. nde the assumpton that dsegadng the amount and tem constants wll not pose a poblem fo customes, the numbe compasons ae shown n Table 2 whch ncludes only the maxmum and mnmum value as a choce. The numbe on the lowe ght hand sde s stll exponental and keeps ts vagueness. α-cut appoach of Kao and u (17) cannot gve a pactcal soluton fo custome so t s used by banks to mpove the poftablty aganst custome. Thus, nsstng on ths appoach s useless. Table 2: The numbe of optons va date and amounts that ae gven by banks to custome accodng to vaables when only max. and mn. taken Bank No Vaables T SD ERO TKT KON TST Total Numbe of Compason

6 566 G J Sc, 28(4): (2015)/ Ayhan GÖCÜKCÜ Total ,5112E+23 As mentoned above and as s seen n Table 1 and 2, data of some banks ae not avalable dependng upon the absence of coespondng sevce. Consdeng that the DEA depend on all the data obseved, absence of data as mentoned ceates an addtonal fuzzness whch seems to affect the esults of the analyss of all DMs. Customes who want to choose a bank o a bank manage who wants to see the stuaton n the sevces secto wll not each the goal unless the mssng data s estoed. Howeve, t s not possble to examne all the cases mentoned above, such as usng the classcal appoach. In ths nstance, dffeent fom the appoaches suggested n the lteatue [32-41] a method can be poposed s to nvestgate optmstc and pessmstc cases. It should be evaluated sepaately fo customes and companes as ndcated n Table 3. Table 3: Method applcaton scheme If data s avalable; Custome Oented Fm Oented If data s not avalable; Custome Oented Fm Oented Aveagng Explanaton Optmst Pessmst Maxmzaton Poblem (poft / etun) Output: Depost ate Maxmum Mnmum Input: oan Rate Mnmum Maxmum Maxmzaton Poblem (poft / etun) Output: oan Rate Maxmum Mnmum Input: Depost ate Mnmum Maxmum Maxmzaton Poblem (poft / etun) Output: Depost ate Secto Maxmum Secto Mnmum Input: oan Rate Secto Mnmum Secto Maxmum Maxmzaton Poblem (poft / etun) Output: oan Rate Secto Mnmum Secto Maxmum Input: Depost ate Secto Maxmum * Secto Mnmum To pevent the effcent fonte fom non-sevcng DMs Null cell eplaced wth elevant aveage of secto. * The DM s attaned to effcent fonte as elevant sevce s gven by the bank Null cell eplaced wth elevant aveage of secto. Even the mssng values fuzzfes the effcency of banks, the nteest ates of loans cannot be lowe than the secto mnmum o moe than the secto maxmum. Theefoe, the most pessmstc value could be taken as obseved mnmum; the most optmstc values could be taken as obseved maxmum and most possble values could be taken as secto aveage as mentoned n the method applcaton scheme gven n Table RESTS In ou study, DEA (1) model s solved sepaately fo each case accodng to the above applcaton schema lsted n Table 3. Results based on dffeent expectatons of custome and banks ae gven n Table 4. The dak cells of the tables epesent the DMs wth mssng values whch ae the cause of fuzzness. The notceable ponts of the tables ae the effect of fuzzy

7 G J Sc, 28(4): (2015)/ Ayhan GÖCÜKCÜ 567 values. These values affect not only the effcency of DMs wth mssng/fuzzy values, but also the effcency of othe DMs. The Kuskal-Walls test esults gven n Table 5 also confm these esults. Table 4: DEA esults, accodng to the ncluson fom of mssng data nto analyss Bank Case 1 Case 2 Case 3 No CP CO FP FO CP CO FP FO CP CO FP FO 1 0,139 0,473 0, ,309 0,782 0, ,288 0,922 0, ,199 0, ,756 0,423 0, ,756 0,387 0,815 0,97 0, ,593 0,459 0,814 0,385 0,97 0,688 0,811 0,385 0,966 0,928 0,801 0, ,412 0, ,578 0,768 0, ,578 0,768 0, , ,254 0,887 0, ,514 0,887 0, , ,303 0,48 0,861 0,535 0,645 0,887 0,806 0,535 0, ,737 0, ,767 0, ,971 0,767 0,385 0,982 0,92 0,763 0, ,545 0, ,582 0, ,762 0, ,686 0,532 0,822 0, ,731 0,819 0, ,819 0, ,382 0, ,39 0,71 0, ,544 0,745 0, , ,219 0,397 0,996 0,761 0,474 0,69 0,972 0,761 0,435 0,796 0,911 0, ,597 0,936 0,586 0, ,586 0, ,683 0, ,383 0,801 0, ,487 0,972 0, , ,492 0,31 1 0, , ,326 0,996 0, , ,314 0,85 0,873 0,81 0, ,858 0,962 0,359 0,99 0,826 0, ,365 0,779 0,983 0,814 0,398 0,951 0,979 0,814 0,382 0,788 0,934 0, ,549 0, ,327 0,999 0,8 1 0,327 0,992 0,922 0,969 0, ,641 0, ,53 0,964 0, ,53 0,964 0,71 1 0, ,572 0,38 1 0, , , ,76 0,95 0, ,708 0,524 0,846 0, ,698 0,846 0, ,858 0,846 0,478 Case 1: The DEA esults, when the secto maxmum s used as mssng values. Case 2: The DEA esults, when the secto mnmum s used as mssng values. Table 5: Kuskal-Walls test esults Case 3: The DEA esults, when the secto aveagng used as mssng values. CO: Custome Optmst; CP: Custome Pessmst; FO: Fm Optmst; FP: Fm Pessmst H 0 p Decson 1 The dstbuton of custome pessmst DEA esults ove cases ae the same 0,034 H 0 ejected 2 The dstbuton of custome optmst DEA esults ove cases ae the same 0,004 H 0 ejected 3 The dstbuton of fm pessmst DEA esults ove cases ae the same 0,587 H 0 not ejected 4 The dstbuton of fm optmst DEA esults ove cases ae the same 0,406 H0 not ejected Sgnfcance value taken as %5.

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9 568 G J Sc, 28(4): (2015)/ Ayhan GÖCÜKCÜ 4. CONCSION AND DISCSSION Effcent opeatons of the economc actos n an economy ae of vtal mpotance fo a county. Patculaly, n the case of a fuzzy envonment facng some dffcultes n decson makng s nevtable. Essentally, fuzzy DEA s a useful tool and gude that any decson make can use n mentoned cases. Accodng to the esults of Kuskal-Walls test (Table 5), the DEA esults obtaned by the applcaton of scheme gven n Table 3 s statstcally the same fo banks, namely n almost evey stuaton, the banks ae effcent. Snce the data used n ths study ae detemned fom banks, t would not be wse fo the banks to offe sevces that would ende them neffcent. Nevetheless, decsons mplemented by the banks obvously affect the customes effcency. These esults also ndcate that, even f the vaables ae the same, evaluaton of effcency n tems of fms and customes s dffeent. The pecepton that, n case fm s optmst the custome wll be pessmst o n case the custome s optmst the fm wll be pessmst s naccuate. As a consequence, fms and customes evaluate the self-effcent fontes ndependent fom each othe and f the two pats both am to be effcent at the same tme, they must come to an ageement. Ths study attempts and poposes a way of compomse that would make customes and fms effcent and satsfed. Evaluatng a sngle model s left to the futue eseach. CONFICT OF INTEREST No conflct of nteest was declaed by the authos. REFERENCES [1] Chanes, A., Coope, W.W., and Rhodes, E., "Measung the effcency of decson makng unts", Eu. J. Ope. Res., 2(6): , (1978). [2] Zadeh,.A., "Fuzzy sets", Infomaton and contol, 8(3): , (1965). [3] Rommelfange, H., "Fuzzy lnea pogammng and applcatons", Eu. J. Ope. Res., 92(3): , (1996). [4] Heea, F. and Vedegay, J., "Fuzzy sets and opeatons eseach: pespectves", Fuzzy Sets and Systems, 90(2): , (1997). [5] Zmmemann, H.J., "Fuzzy set theoy", Wley Intedscplnay Revews: Computatonal Statstcs, 2(3): , (2010). [6] Hatam-Mabn, A., Emouznejad, A., and Tavana, M., "A taxonomy and evew of the fuzzy data envelopment analyss lteatue: Two decades n the makng", Eu. J. Ope. Res., 214(3): , (2011). [7] Emouznejad, A. and Tavana, M., Pefomance Measuement wth Fuzzy Data Envelopment Analyss, Spnge, (2014). [8] Popovc, M.-C., "A Suvey on Bank Effcency. Reseach Wth Data Envelopment Analyss and Stochastc Fonte Analyss", SEA Pactcal Applcaton of Scence, (01): , (2013). [9] u, J.S., et al., "A suvey of DEA applcatons", Omega, 41(5): , (2013). [10] Paad, J.C. and Zhu, H., "A suvey on bank banch effcency and pefomance eseach wth data envelopment analyss", Omega, 41(1): 61-79, (2013). [11] Pu, J. and Yadav, S.P., "A fuzzy DEA model wth undesable fuzzy outputs and ts applcaton to the bankng secto n Inda", Expet Systems wth Applcatons, 41(14): , (2014). [12] Pu, J. and Yadav, S.P., Fuzzy mx-effcency n fuzzy data envelopment analyss and ts applcaton, n Pefomance Measuement wth Fuzzy Data Envelopment Analyss, Spnge, , (2014). [13] Tavana, M. and Khall-Damghan, K., "A new two-stage Stackelbeg fuzzy data envelopment analyss model", Measuement, 53: , (2014). [14] Wang, Y.-M. and Chn, K.-S., "Fuzzy data envelopment analyss: A fuzzy expected value appoach", Expet Systems wth Applcatons, 38(9): , (2011). [15] Xa, Q., ang,., and Yang, F., Integatng Fuzzy Intemedate Factos n Supply Chan Effcency Evaluaton, n Pefomance Measuement wth Fuzzy Data Envelopment Analyss, Spnge, , (2014). [16] Chen, Y.-C., et al., "The analyss of bank busness pefomance and maket sk Applyng Fuzzy DEA", Economc Modellng, 32: , (2013). [17] Bal, H. and Gölcükcü, A., "Data Envelopment Analyss: an applcaton to Tuksh bankng ndusty", Mathematcal and Computatonal Applcatons, 7(1): 65-72, (2002). [18] Kao, C. and u, S.-T., "Fuzzy effcency measues n data envelopment analyss", Fuzzy sets and systems, 113(3): , (2000). [19] Chanes, A., Data Envelopment Analyss: Theoy, Methodology, and Applcatons: Theoy, Methodology and Applcatons, Spnge Scence & Busness Meda, (1994). [20] Coope, W.W., Sefod,.M., and Tone, K., Data envelopment analyss: a compehensve text wth models, applcatons, efeences and DEA-solve softwae, Spnge Scence & Busness Meda, (2007). [21] Zhu, J., Data Envelopment Analyss: A Handbook of Models and Methods, Spnge, (2015).

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