O.R. Theory and it s Application Multi-Component Commercial Bank Branch Progress and Regress: An Application of Data Envelopment Analysis

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1 Inernaional Mahemaical Forum, 1, 2006, no. 33, O.R. Theory and i s Alicaion Muli-Comonen Commercial Bank Branch Progress and Regress: An Alicaion of Daa Envelomen Analysis A. Divandari Dearmen of Managemen, Tehran Universiy, Tehran, Iran G. R. Jahanshahloo and F. Hosseinzadeh Lofi 1 Dearmen of Mahemaics, Science and Research Branch Islamic Azad Universiy, Tehran, Iran 2 H. Nikoomaram Dearmen of Managemen, Science and Research Branch Islamic Azad Universiy, Tehran, Iran F. Rezai Balf Dearmen of Mahemaics, Islamic Azad Universiy, Qaemshahr, Iran Absrac The urose of he resen sudy is o develo a mehod used o evaluae he rogress and regress of cammercial bank differen branches during differen eriods. I has o be menioned ha daa envelomen analysis is used o develo such a mehod. The resuls seem o be quie saisfacory from managerial oin of view. Mahemaics Subjec Classificaion: Oeraions Research, 90. Keywords: Daa Envelomen Analysis, Progress, Regress 1 Corresonding auhor, addresses: hosseinzadeh lofi@yahoo.com 2 Tel.: , Fax: , P.O. Box 14155/775 and 14155/4933, Pos code:

2 1636 A. Divandari e al 1. Inroducion In evaluaing he relaive efficiency of a se of decision making unis, you may refer o comare he resuls in wo searae eriods o find ou heir rogress and regress. Daa envelomen analysis is used for his urose. Considering a secial funcion, i seems ha he resuls are very realisic and saisfacory from managerial oin of view. The regress and rogress of differen comonens for differen branches enables he manager o ake relevan decisions for hese comonens. This aer conains following secions: in secion 2 basic conces are exlained, he roosed model for calculaing rogress and regress is u forward in secion 3, and he alicaion of his mehod in Bank Mella branches, are discussed in he las secion is conclusion. 2. Background Suose ha, {DMU j : j =1,..., n} is a se of decision-making unis, and DMU j consumes m inu X =(x 1j,..., x mj ) o roduce s ouu Y = (y 1j,..., y sj ) as non-negaive amouns, and X 0,Y 0,X 0,Y 0. The roducion ossibiliy se (PPS) is creaed in associaion wih osulaes, non-emy, consan reurns o scale, convexiy and lausibiliy as follow: T c = {(X, Y ):X n j λ jx j,y n j λ jy j } The fronier creaed wih each roducion ossibiliy se is called he resonden efficiency of PPS. To assess efficien unis, heir disance from boarder is measured. If i is locaed on he fronier i is called an efficien uni, oherwise i would be inefficien. To find ou if DMU is locaed on he efficiency fronier or no, he following linear rogramming roblem is solved. Max z = U T Y s.. U T Y j V T X j 0, j =1,..., n (1) V T X =1 U, V 1ɛ If in oimal soluion (1), z = 1, hen DMU is efficien oherwise i would be inefficien. ( In his aer radial efficiency is considered). Now, consider ha each decision making uni consiss of wo comonens. Suose ha,x 1 is he secific inu vecor for he firs comonen,x 2 is secific inu vecor for he second comonen, X is he shared inu vecor for boh comonens,y 1 is he secific ouu vecor for he firs comonen, Y 2 is he secific ouu vecor for he second comonen, and Ȳ is he shared ouu vecor for boh comonens. Le α X,(1 α) X are he vecors of shared inu and βȳ,(1 β)ȳ are share ouu of comonen 1 and 2 resecively. The aggregaed efficiency and comonen efficiency is obained by solving he following roblem.

3 O.R. Theory and Alicaion 1637 Max e a = U 1 Y 1 + U 1 (βȳ)+u 2 Y 2 e 1 j = U 1 Y 1 e 2 j = U 2 Y 2 + U 2 (1 β)ȳ V 1 X+ 1 V 1 (α X )+V 2 X+ 2 V 2 (1 α) X j + U 1 (βȳ j ) V 1 Xj 1+ V 1 (α X j ) j + U 2 (1 β)ȳj V 2 Xj 2+ 1, j =1,..., n (2) V 2 (1 α) X j 1, j =1,..., n U 1 1ɛ, U 2 1ɛ, Ū 1 1ɛ, Ū 2 1ɛ V 1 1ɛ, V 2 1ɛ, V 1 1ɛ, V 2 1ɛ Consider he subsiuions Ū 1 β = U 3, Ū 2 (1 β) =U 4, V 1 α = V 3, V ( 1 α) =V 4, We can ransform roblem (2) as linear form follows: Max z = U 1 Y 1 + U 2 Y 2 + U 3 Ȳ + U 4 Ȳ s.. V 1 X 1 + V 2 X 2 + V 3 X + V 4 X =1 U 1 Yj 1 + U 3 Ȳ j V 1 Xj 1 V 3 Xj 0, j =1,..., n (3) U 2 Yj 2 + U 4 Ȳ j V 2 Xj 2 V 4 Xj 0, j =1,..., n U 1 1ɛ, U 2 1ɛ, U 3 1βɛ,U 4 1(1 β)ɛ V 1 1ɛ, V 2 1ɛ, V 3 1αɛ, V 4 1(1 α)ɛ The oimal soluion model (3) he efficiency of he firs, second, and aggregae comonens by using he following formulas: Efficiency of he firs comonen= e 1 = U 1 Y 1 +U 3Ȳ V 1 X+V 1 3 X (4) Efficiency of he second comonen= e 2 = U 2 Y 2 +U 4Ȳ V 2 X 2 +V 4 X Efficiency of he aggregae comonen = e a = U 1 Y 1 +U 3Ȳ+U 2 Y 2 +U 4Ȳ V 1 X 1 +V 3 X +V 2 X 2 +V 4 X 3. Progress and Regress of Comonens Suose ha we are going o assess he siuaion of a DMU in wo differen eriods of ime or we wan o comare is imrovemen in hese wo eriods. Since he relaive efficiency of each eriod is calculaed according o ha secific eriod, assessing he relaive efficiency of each uni in wo indeenden eriods of ime doesn give us he accurae informaion abou heir rogress and regress. Suose ha a decision making uni, such as, was relaively efficien in firs eriod and also roved efficien in second eriod, comared wih all unis. Whereas he value of is efficiency in boh eriods is 1, no rogress and no regress occur, bu we don know if he efficiency fronier, relaive o he revious eriod, is in beer siuaion or worse.

4 1638 A. Divandari e al Here, a mehod is roosed o solve his roblem. To measure he amoun of rogress and regress in wo eriods of ime, jus comare he efficiency of he second eriod wih he efficiency fronier of he firs eriod, or we can say, comare is relaive efficiency wih he revious efficiency fronier, in his way he efficiency of he firs and second eriods are comared wih each oher and we obain a crieria o evaluae he rogress and regress of he unis. Consider he following figure: Y 2 O Figure 1. Progress and Regress 1 X In Figure 1, you see five decisions making unies roduce an ouu using single inu. Circles show he siuaion of hese unis in he firs eriod, and squares show heir siuaion in he second eriod. The fronier is he efficiency fronier in he firs eriod. The efficiency of each uni can be deermined by measuring is disance from he fronier, nearer unis is more efficien. For examle, uni 5 in he second eriod is nearer o he fronier han he firs eriod, so his uni has made rogress, and in his way, uni 4 shows regress. Generally, in Figure 1, unies 1 and 5 made rogress and unis 2 and 4 made regress, and uni 3 remains in he same siuaions wihou rogress and regress. To assess he rogress and regress of a uni, aly he above menioned rocedure o i. Now, consider ha we are going o measure he rogress and regress of hese unies in wo eriods of ime and comare he changes of a eriod wih he changes of anoher eriod, for examle we comare his summer wih las summer and suose ha he daa is gahered monhly. The firs se is calculaing he efficiency of each decision making uni in he firs eriod. The following model calculaes he efficiency of each comonen of DMU in he eriod = 1,..., T.

5 O.R. Theory and Alicaion 1639 Max Z = T U 1 Y 1 s.. V 1 X 1 U 1 Y 1 j + U 2 Y 2 + U 3 Ȳ + U 4 Ȳ + V 3 X + V 4 X =1 + V 2 X 2 + U 3 Ȳj V 1 X 1 j V 3 X j 0, j =1,..., n (5) U 2 Yj 2 + U 4 Ȳj Xj 2 V 4 X j 0, j =1,..., n U 1 1ɛ, U 2 1ɛ, U 3 1βɛ,U 4 1(1 β)ɛ, =1,..., T V 1 1ɛ, V 2 1ɛ, V 3 1αɛ, V 4 1(1 α)ɛ, =1,..., T. By solving roblem (5), accumulaed efficiency of DMU in he oal eriod and he efficiency of he firs and second comonens in each eriod would be calculaed as follows: The efficiency of he firs comonen in sage = e 1 The efficiency of he second comonen in sage = e 2 The efficiency of he aggregae comonen in sage = e a = U 1 Y 1 V 1 X 1 +U 3Ȳ +U 2 Y 2 +U 4Ȳ +V 3 X +V 2 X 2 +V 4 X = U 1 Y 1+U 3Ȳ V 1 X 1+V 3 X = U 2 Y 2 +U 4Ȳ V 2 X 2 +V 4 X Now, according o he roosed model, in he new eriod, each decision making uni would be comared wih all unis of he firs eriod which were efficien in a leas one comonen and a leas in one single. Suose e j is he new efficiency ( in he second duraion) and e j is he old one (in he firs duraion), hus: Where e j >e j, DMU j shows rogress. Where e j = e j, here isn any rogress and regress. Where e j <e j, DMU j shows regress. Now, we are going o roose a crierion for measuring rogress and regress. Consider ha he roorion e j e j shows he amoun of growh (no rogress or regress). In lieraure discussing evaluaion of rogress and regress, he following formula has been used: ρ = e e e In which e and e, are relaive efficiency of DMU in eriod +1 and resecively. We used his formula, he resul was no acceable from managerial oin of view. The following examle was u forward o convince us ha he resul is no realisic. This examle considers wo branches called A and B. suoses he relaive efficiency of an in eriod + 1 and is 0.3 and 0.1, so he rogress is: ρ = =2

6 1640 A. Divandari e al Tha means 200% rogress. Now consider branch B wih relaive efficiency of 0.95, 0.85 in eriods +1 and. The rogress is: ρ = = From managerial oin of view, erformance of branch B is much beer han erformance of branch A, since when a DMU is close o fronier i is very hard o imrove is erformance, comared wih a branch ha is far from he efficiency fronier, so we were asked o modify he formula. The grou had a long discussion; finally he funcion which is used below was suggesed for his urose. We hink ha his is no he only funcion ha can be used for calculaing Progress or Regress. Y O e j e j Figure 2. Progress is defined as he area under he curve from e j o e j X Suose ha f is a funcion, so he amoun of rogress is defined as he area under he curve from e j o e j. Therefore, we have: The rogress of DMU j = ρ = e j e j f(x)dx (6) The following is one of he funcions ha we can roose for his urose: f(x) = 2x x+1 By relacing i in (6), rogress can be calculaed from he following equaion: ρ j = 2[(e j e j )+ln( e j+1 e j +1)] (7)

7 O.R. Theory and Alicaion 1641 Where e j e j, inegral would be negaive and i imlies he regress of DMU j, and if e j = e j, according o (5) we would have ρ j = 0 and here is no any rogress and regress. And in case e j e j, inegral would be osiive and i imlies he rogress of DMU j. So we have he following Table: Table 1. The crieria for deermining rogress or regress ρ j Siuaion ρ j > 0 I shows rogress ρ j =0 Neiher rogress nor regress ρ j < 0 I shows regress 4. The daa and analyzing he resuls This aer will assess all he branches of Bank Mella, all over he counry. Bank Mella has 1946 branches classified ino 6 differen levels called, disincive branches, 1 s degree, 2 nd degree, 3 rd degree, 4 h degree, 5 h degree, and 6 h degree, resecively. As you see in Table 2, we allocaed secific se of branches o each level. Table 2. Classificaion of branches Se of branches Branches under assessmen Insincive degree All he branches Insincive degree All he branches 1 s, 2 nd, 3 rd, 4 h, 5 h, and 6 h degrees 1 s degree 2 nd, 3 rd, 4 h, 5 h, and 6 h degrees 2 nd degree 3 rd, 4 h, 5 h, and 6 h degrees 3 rd degree 4 h, 5 h, and 6 h degrees 4 h degree 5 h, and 6 h degrees 5 h degree 6 h degree 6 h degree Table 3.a and Table 3.b show you he inended indicaors for calculaing rogress and regress, resecively. The daa of all he branches, during 6 monhs in 2002 is gahered and his duraion is divided ino wo 3-monhs eriods, so we comared he erformance of he branches in he sring wih heir erformance in he summer. Acually, July is comared wih Aril, Augus wih May, and Seember wih June, o calculae he rogress or regress of each branch. You can see he summarized resuls in Table 4. Table 3.a. The resul of rogress Degree Firs comonen Second comonen accumulaion

8 1642 A. Divandari e al 20 Disincive Grade Grade Grade Grade Grade Grade Table 3.b. The resul of regress Degree Firs comonen Second comonen accumulaion Disincive Grade Grade Grade Grade Grade Grade These Tables consis of wo columns, he firs column is he degree of he branch along wih heir number, and he second column we have he number of branches wih rogress in Table 3.a and he number of branches wih regress conains branches wih regress in Table 3.b. The second column in Table 3.a and Table 3.b were each divided ino hree columns which are corresonden o he firs comonen, second comonen, and accumulaion. Each of hese columns also conains hree smaller columns. Column 1 shows he number of branches which made rogress or regress in July comared wih Aril, column 2 shows he number of branches which made rogress or regress in Augus comared wih May, and column 3 shows he same informaion abou Seember and June. The difference beween he oal number of branches in a given degree, wih he number of branches wih rogress and regress, is he number of branches which made neiher rogress nor regress. For examle, in 6 h degree, firs comonen, hird column, 3 branches showed rogress and 12 branches showed regress, and as he oal number of 6 h degree branches is 17, we have 2 branches wihou any rogress or regress in his monh. According o hese Tables, mos of he disincive branches showed rogress in he firs comonen, while in oher degrees boh comonens were affeced equally. I shows ha disincive branches focus more on resource absorion which is he firs

9 O.R. Theory and Alicaion 1643 comonen, and ry o be successful in his area. While, in oher degrees, he rogress facor in boh comonens (resource absorion and giving services) acs equally. Consider ha disincive branches have aricular cusomers such as large comanies and facories and ac, so hey should be more acive in absorbing resources and avoid giving unnecessary services. As can be seen in Table 3.a, in all degrees, he number of branches which show rogress in July, comared wih Aril, is much more han oher monhs because usually a he beginning of each fiscal year (Iranian fiscal year begins in Aril), commercial banks exerience a deflaion rooed in he economic deflaion of he sociey and i would be modified laer. Anoher imoran oin in his Table is he remarkable rogress of branches in 4 h degree. In Table 4, we have he ercenage of branches wih rogress in accumulaive efficiency in differen degrees. 4 h degree wih 74% rogress is in highes level and he lowes level belongs o 6 h degree. Table 4. The highes level and he lowes level of rogress Degree Disincive Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Percenage %60 %51 %52 %63 %74 %53 %41 By sudying imoran inus and ouus such as resources, ineress, faciliies and ac, he reason of high rogress ercenage in 3 rd and 4 h degrees were deeced. We found ou ha here is a narrow difference beween hese daa in moderae degrees wih a higher degree, so hey are moivaed o imrove heir ouu and be romoed o a higher degree. While for a branch in 2 nd degree rising o a higher degree is no so easy and here is a considerable difference beween is daa and hose of 1 s degree and ha s why heir moivaion o imrove heir erformance is less han moderae degrees. 5 h and 6 h degrees are in such a bad condiion, maybe because of undesirable geograhical and environmenal siuaion, ha are no able o comee wih oher degrees. 5. Conclusion In his aer, a mehod is roosed for measuring he rogress and regress in differen eriods of ime and i has been alied o he branches of Bank Mella. This mehod ried o remove he sudden changes of daa in some branches o reduce he remarkable changes in rogress and regress of hose branches. However, he sudden changes ha each branch exerience a he end of he fiscal year, roduces a dro in he erformance of Aril, ha s why he number of rogressed branches in July is much more han Aril. To reduce he effec of such evens on he measuring rocess he erformance of he

10 1644 A. Divandari e al branches is assessed in differen eriods of ime. The resuls show ha he rogress of moderae branches is more han he ohers, and ha s because of he comeiion exiss beween hem for rising o a higher degree. Bu in lower levels such as 5 h and 6 h degrees, here is more regress han rogress, and we wonder ha if hey should be allowed o coninue heir aciviies. References Banker R. D., A. Charnes and W. W. Cooer (1984) Some model for esimaing echnical and scale inefficiencies in DEA, Managemen Science, Vol. 30 (9), Charnes A., W. W. Cooer and E. Rhodes (1978) Measuring he efficiency of DMUs, Euroean Journal of Oeraional Research, Vol. 2, Cook W. D., M. Hababou and H. J. H. Tuener (2000) Mulicomonen Efficiency Measurmen and Shared Inus in Daa Envelomen Analysis: An Alicaion o Sales and Service Performance in Bank Branches, Journal of Produciviy Analysis, Vol. 14, Cooer W. W., L. M. Seiford and K. Tone (2000a), Daa Envelomen Analysis Kluwer Academic Publicaion, Boson, Dordrech, London. Jahanshahloo G. R., F. Hosseinzadeh Lofi, N. Shoja, G. Tohidi and S. Razavian,(2004), The ouu esimaion of a DMU according o imrovemen of is efficiency Alied Mahemaics and Comuaion, Vol. 147, Wei Q. L., J. Zhang and X. Zhang (2000), An inverse DEA model for inu/ouu esimae, Euroean Journal of Oeraional Research, Vol. 121, Received: Seember 25, 2005