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OMEGA Int J of Mgmt Sci., Vol 19, No 6, pp 549-557, 1991 Printed in Great Britain All rights reserved 0305-0483/91 $3.00+0.00 Copyright © 1991 Pergamon Press plc Bank Branch Operating Efficiency: A Comparative Application of DEA and the Loglinear Model DI GIOKAS University of Athens and Commercial Bank of Greece (Received September 1990; in revisedform February 1990 In this paper, a comparison regarding the operational efficiency of individual branches of a bank is made, through the application to the same body of data of two different estimation methods: (i) Data Envelopment Analysis (DEA) and (ii) Loglinear Model Analysis In addition to that, the study examines whether operations in the bank branches were conducted in regions of increasing, constant or decreasing returns to gale The DEA results suggest that increasing, constant or decreasing returns to scale may be observed in different regions of the production function, whereas the Loglincar model suggests that increasing returns to scale are in operation Key worda banking, data envelopment analysis, loglinear estimation, mathematical programming, retums to scale INTRODUCTION DATA ENVELOPMENT ANALYSTS (DEA) (see Charnes et al [7]) is a mathematical programming technique used to estimate the efficiency of decision making units (DMUs) Since the first publications of DEA in 1978, the literature has expanded rapidly, reporting further theoretical developments as well as numerous applications [5, 12] Later developments [1, 3,4] have extended DEA to the estimation of cost and production correspondences This method is presented and discussed in Section of the paper Until now, the results from the application of DEA to the banking sector have not been compared to those coming from the application of other classical estimation methods of a production function In a study of North Carolina hospitals, results of DEA have been compared to those from traditional econometric techniques, e.g econometric modelling using the translog cost function by Banker et al [2] In the present paper, the Cobb-Douglas function has been estimated using mathematical program549 ming This function differs from DEA in that it assumes only one output resulting from multiple inputs The use of the Cobb-Douglas function in banking is supported by other research [10,13] In Section 3, the Loglinear model (Cobb-Douglas), which is applied only to single output-multiple input situations, is presented The data is presented in Section As the estimates of different characteristics of the production correspondence provided by these two methods are commonly employed for policy inferences, their comparison is useful and interesting In Section 5, the results of DEA are compared to those from the Loglinear model This section contains also our findings on economies of scale Finally, a few concluding remarks are offered in Section DATA ENVELOPMENT ANALYSIS In general, DEA measures efficiency by estimating an empirical production function which represents the highest values of outputs/benefits Giokaa Bank Branch Operating Efficiency 550 that could be generated by inputs/resources as given by the observed input-output vectors, DEA is composed of several mathematical models sharing the principle of envelopment, As Golany [11] points out, "DEA is quickly emerging as the leading method for efficient evaluation, in terms of both the number of research papers published and the number of applications to real world problems" The original model proposed by Charnes et al [7], known as CCR, was applied to the banking context by several authors [6, 14-16] The detailed formulation of the CCR model is given in Appendix A The application of the CCR model to a set of branches allows the comparison of each one of them to the rest of the data set and the elicitation of a number of conclusions: First, the technique assigns to all the branches being evaluated efficiency ratings as follows: ho = 1, signifying relative efficiency; ho < 1, signifying relative inefficiency, Finally, DEA supplies information about relatively inefficient branches, regarding specific inputs that they overutilise or outputs that they underproduce, depending on whether the objective of the decision makers is input minimisation or output maximisation (see Appendix C) Assuming that the reference point on the production function for an inefficient D M U will be a convex combination of the observed efficient DMUs, Banker et al [1] provide a linear programming model (known as BCC) which permits examination of productive efficiencies and returns to scale (see Appendix A) LOGLINEAR FUNCTION ESTIMATION In many of the studies of economies of scale in banking, output is assumed to be produced according to a Cobb Douglas production function [10, 13] This function has the desirable property of being transformable into a logarithmic linear function that will allow the co- It must be emphasised that the assessment of efficients to be estimated by solving a linear the branches is conditional on the data set under programming (LP) model The detailed formustudy That is, ho = signifies a branch which is lation of this LP model is given in Appendix B efficient compared to the performance of the The solution of this linear programming other branches in the data set, while ho < problem allows for: implies relative inefficiency For a branch to be assessed as relatively inefficient, it means that (i) The estimation of the coefficients of the the data set contains branches (or combinations function showing the best relative of branches) displaying greater efficiency; thereefficiency From these coefficients we fore, on the basis of available information, the can derive the number of transactions performance of the branch can be improved, which could be undertaken by each When, however, a branch is assessed as relabranch if the latter was making use of available resources in the best (relatively efficient, this only implies that there are no branches (or combinations of branches) in the tively) productive way data set performing more efficiently; it is still (ii) The measure of efficiency for each possible that such branches exist, and have branch (a percent of utilisation of its simply not been selected for examination in the resources) which is given by the index: data set B, Second, DEA identifies the efficiency referXo.A,t .A~ A,3 • A,, ence set for each inefficient branch This is the set of relatively efficient branches to which it has been most directly compared in calculating its DATA efficiency rating This facilitates the exploration The sample which was used consists of 17 of the nature of inefficiencies of the branch in branches I of a regional division of the Commerquestion, by comparing it to a selected subset of cial Bank of Greece and the data refer to the more efficient branches in the study year 1988 Only those inputs which concern bank branches directly were used, ignoring bank 1In order to safeguard Bank confidentiality, we not mention the names of individual bank branches, which are recorded under code number, overheads, since the objective o f this analysis was to evaluate of the use of inputs consumed Omega, Vol 19, No directly at the branch Inputs employed are the following: (i) Labour (personhours) This variable summarises the actual work (in personhours) employed during the operation of the branches and includes administrative/processing personnel, marketing officers and branch management All branches employ approximately the same proportion of administrative/processing personnel in their staff The number of personhours captures all hours worked by the personnel of the branch including overtime and detached labour, (ii) Operating expenses (drachmas) This variable expresses the consumption of a range of inputs by the bank branch and covers all the operating expenses of the branches, such as those for telephone, electricity, stationery and other supplies This variable is measured in monetary terms, due to difficulty of grouping together the widely dissimilar entities that comprise it Note that all purchases are performed centrally by the bank and hence all branches are charged the same prices Expenses for salaries, rent and depreciationofbuildingshavenot been included, since personnel expenses and office space have already been covered by the other two inputs, (iii) Utilised branch space (square metres) This variable shows the contribution of available space to the production of the bank branch It should be noted that only space which is in productive use has been included, It was decided not to include any inputs reflecting market conditions as all the branches in the data set are located in Athens, they belong to the same regional division of the Commercial Bank and operate in reasonably similar markets, As outputs we defined bank products offered to customers and more specifically the complete number of transactions (total 72)performed in each branch, In order to apply DEA, transactions were grouped according to the section of the branch which performs them and the weighted number O M I~ 19/6.-D 551 of transactions produced at each section was calculated (Category A = Section of Deposits and Capital Transfers, Category B = Section of credit, Category C = Section of Foreign Receipts) Transactions were weighted with coefficients of equivalence defined by the Domestic Operations and Branches Division on the basis of their experience from the workings of bank branches Estimation of the Cobb-Douglas type function of maximum relative efficiency requires all bank branch products to be aggregated into a single one This aggregation was performed with the help of the above mentioned coefficients of equivalence In this way, the weighted number of transactions performed by a bank branch was defined as a single output The data are reproduced in Appendix D RESULTS The DEA results are summarised in Table The results from the CCR model give the overall technical and scale efficiency which is less than or equal to the pure (input) technical efficiency measured by the BCC model [1] We will use results of the CCR model to examine operating efficiency of individual branches, whereas resuits of the BCC model will be used to examine economies of scale Thus, the results of the CCR model [column (2)] indicate that 12 branches are relatively inefficient, that is they have an efficiency rating of less than A branch is found to be inefficient if it is possible to construct a reference branch as a linear combination of other branches, such that the reference performs at least as many transactions while using less inputs than the real branch As Sherman and Gold [14] point out " the efficiency rating does not rank order the branches, but rather suggests the degree of inefficiency of a branch compared with its efficiency reference set" Hence, K3 is about 86% efficient compared to K2, K4 and KS K15 is also about 86% efficient but compared to K2 and K4 Generally, this means that both K3 and K15 could reduce the resources they utilise by approximately 14% without reducing their outputs The performance of inefficient branches can be improved either by increasing outputs or by cutting inputs The DEA results regarding specific inputs that inefficient branches overutilise or outputs that they underproduce are summarised in Table More specifically, the degree to Giokas Bank Branch Operating E:ciency 552 Table CRR model DEA Branch code (I) BCC model Efficiency efficiency rating reference set (2) (3) Uf efficiency rating reference set Indicator of returns to scale DEA (4) Efficiency (5) (6) Loglinear efficiency rating (7) KI K2 K3 0.951 1.000 0.859 K2, K4, K8 K2 K2, K4, K8 0.989 !.000 1.000 K2, K3, K6, K8, KI7 K2 K3 -0.14 0.00 -0.32 0.988 1.000 0.982 K4 K5 K6 K7 K8 K9 K I0 KI I 1.000 0.683 0.984 1.000 1.000 0.764 0.878 0.903 K4 K2, K K2, K4 K7 K8 K2, K 17 K2, K4, K 17 K2, K4 1.000 0.805 1.000 1.000 1.000 0.942 0.885 0.932 K4 K2, K3 K6 K7 K8 K2, K3, K 17 K2, K4, K6, K 17 K2, K 0.00 -0.25 - 0.09 0.00 0.00 - 0.35 - 0.06 0.42 0.910 0.734 1.000 0.749 0.632 0.550 0.796 0.822 K12 KI3 KI4 " KI5 KI6 KI7 0.984 0.662 0.726 0.869 0.574 1.000 K2, K4, KS, KI7 K2 K4, KS, KI7 K2, K4 K2, KS, K17 KI7 0.995 0.720 0.917 0.876 0.658 1.000 K2, K4, K6, K8, KI7 K2, K4, K7 K3, K8, KI7 K2, K4 K3, K8, KIT KI7 -0.09 -0.21 -0.72 0.56 -0.42 0.00 0.903 0.498 0.631 0.745 0.387 0.889 which inputs can be reduced is indicated by the numbers in colums (3), (4) and (5) Numbers in columns (6), (7) and (8) show how much extra output an inefficient branch could generate with the same level of inputs if it moved to the efficient frontier, e.g branch KI0 can increase its outputs by 13.9, 13.9 and 46.9%, respectively Application of the loglinear model gave the efficiency measures which are presented in column (7) of Table 12 According to the estimated measures of efficiency two branches (e.g K2 and K6) achieved the maximum possible efficiency, whereas the rest of the branches fall within a broad range of relative efficiency (measures of efficiency are within the range 0.988-0.387) It should be noted that the coefficients have been estimated under the minimisation of total deviation between the observed and maximum production and the obtained measures of relative efficiency classify the branches in one single scale In other words, it is possible to compare two bank branches on the basis of the estimated measures of efficiency and to derive certain conclusions regarding their relative efficiency It should be mentioned, however, that these results reflect an average ZThe Loglinear model that was eventually selected gave the following estimated equation: Q, = A°i~ ' A~~'~ A~3 where: A~, A~2,A~3 are the personhours, operating ex- peases (in thousands o f drachmas) and square metres o f branch space, in branch i, respectively O~ is the maxim u m volume o f transactions which can be produced by branch i situation and not take into account possible particular differences in branch operations The degrees of relative efficiency estimated by DEA [column (2) of Table 1] in general, are not significantly different to the measures of efficiency which were estimated by the loglinear model [column (7)] It should be emphasised that the above results have been produced by two completely different methods, and thus would not be expected to be in total agreement Still, a comparison may prove to be useful in establishing certain useful conclusions The biggest difference between the two efficiency indexes is observed in some bank branches which have been characterised as productive by DEA, whereas the estimated efficiency measures of the loglinear model are significantly below unity This is mainly observed in branches K7, K8 and K17 In order to explain this phenomenon, it should be stated that the concept of efficiency is treated somewhat differently under DEA and the loglinear model Both models measure the operating efficiency of branches in relation to that of other branches The loglinear model, however, treats bank branches as if they produce a single product and classifies them on a single efficiency scale, whereas DEA examines combinations of multiple inputs and multiple outputs Thus, under DEA, some branches with "unusual" combinations cannot be compared directly to a reference branch and, as a result, are characterised as branches of maximum efficiency Both methods of estimation of efficiency gave similar results regarding the significance of Omega, Vol 19, No 553 Table Inefficient branch ( 1) Efficiency rating (2) KI K3 K5 K6 K9 KI0 KII KI2 KI3 KI4 KI5 KI6 0.951 0.859 0.683 0.984 0.764 0.878 0.903 0.984 0.662 0.726 0.869 0.574 PH: OE: sq m: A: B: C: Excess inputs of a branch compared with its reference set PH OE sq m (3) (4) (5) 1685 2902 11703 447 6350 5766 5603 687 25466 10145 7399 18674 (4.9) (14.1) (31.7) (I.6) (23.6) (12.2) (9.7) (1.6) (51.2) (33.8) (13.1) (42.6) Personhours Operating expenses Square metres of branch space Weighted -number of transactions done Weighted number of transactions done Weighted number of transactions done Note: Numbers in parentheses indicate the % 320 (4.9) 487 (14.1) 2764 (31.7) 71 (I.6) 10764 (73.4) 922 (12.2) 1164 (9.7) 116 (1.6) 5147 (47.2) 1166 (27.4) 1163 (13.1) 8550 (67.4) by by by of 252 248 554 195 220 345 326 651 1371 239 361 476 (42.6) (58.1) (66.7) (41.1) (49.2) (45.2) (42.7) (58.7) (33.8) (49.9) (43.0) (59.4) Deficient outputs of a branch compared with its efficient reference set A B C (6) (7) (8) 13764 21780 111721 3317 48372 41275 49597 4924 118822 41824 54652 96581 (5.1) (16.4) (46.4) (I.6) (30.8) (13.9) (10.7) (1.6) (51.0) (37.7) (15.0) (74.2) 785 1910 6888 4556 4665 2328 14716 199 5381 3266 8745 15154 (8.7) (51.7) (144.7) (142.9) (30.9) (13.9) (545.4) (1.6) (86.1) (37.6) (137.3) (74.2) 578 1143 3779 4209 10704 5640 5948 392 7129 7454 8502 20877 (5.1) (16.4) (46.7) (74.9) (801.2) (46.9) (45.0) (I.6) (151.6) (37.7) (78.8) (74.2) section of deposits and capital transfers section of credit section of foreign receipts excess inputs or deficient outputs of a branch compared with its efficient reference set certain inputs of production Thus, the loglinear model practically ignores branch space in the evaluation of maximum efficiency In an analogous manner, DEA's solution most of the time shows that the space of bank branches is not a key variable affecting production This is clearly proved by the fact that the most important variables influencing bank branch efficiency are personhours and/or the operating bank expenses In conclusion, from a comparison of the results of the two methods (as shown in Table 1) it becomes clear that branches K5, K9, K10, K l l , K13, KI4, K15 and KI6 exhibit special problems of operating efficiency Therefore, it is necessary to focus the attention of Bank decision makers to those branches in order to locate the reasons of such low efficiency and then take corrective measures For their assistance, apart from the information of Table 1, there is additional information provided by DEA regarding the specific inputs which are overutilised at the low-efficiency branches (Table 2) Finally, evaluating the above results a year later, it is of interest to observe that for at least half of the branches which were characterised as inefficient (i.e KS, K9, K10, K l l , K13, K14, K15, K16) an analogous opinion had been formed independently bybankexecutives More specifically, during 1989, branches K9 and K14 were merged due to recognised problematic behaviour (as well as due to the small distance between them), whereas K13 was characterised as Counter-Branch As for branch K16, according to internal information, it is being kept in operation despite operational problems, due to its geographical position Economies o f scale The returns to scale for a particular observed input-output mix may be examined using DEA by estimating the sign of the variable Uo (see BCC model in Appendix A) Increasing returns to scale are indicated for uo The results in Table [column (6)] indicate that only branches experience decreasing returns to scale, have constant returns to scale and the majority have increasing returns Berg et al [6] point out that the convexity of the frontier ensures that increasing returns will be more frequent at smaller branches Because of the fact that the sample of 17 bank branches consisted of satellite (small) and not centre (large) branches, the above results give support to those of Berg and others that small branches show increasing returns to scale It is noted that when centre branches were added to the sample, the results showed that 80% of them operate under constant returns to scale It is interesting to recall at this point that when aloglinear function (Cobb-Douglas)was used to estimate operating efficiency in the aggregate data, the existence of increasing returns to scale was confirmed, since the satisfaction of xl + xz + x3 > (where xi are the coefficients of the loglinear model), implies this 554 Branch code (1) KI (7) K2 (I) K3 (16) K4 (3) K6 (11) K7 (13) KS (8) Giokas Bank Branch Operating Efficiency Table Operating ratiosof bank branches Operating expenses per 100 DEA Transactions transactions etiiciency rating (2) per pcrsonhour (3) in thousands of drachmas (4) 0.951 1.000 0.859 8.38 (2) 10.14 (I) 6.96 (7) 2.26 (13) 2.33 (11) 2.41 (9) 1.000 0.984 1.000 1.000 7.86 (4) 7.60 (6) 6.79 (10) 5.60 (14) 6.87 (9) 1.97(16) 2,09 (15) 3,S4 (4) 3.11 (6) K5 (9) 0.683 3.44 (5) K9 (15) 0.764 6.44 (12) 8.46 (t) KI0 (6) KI I (2) KI2(5) 0.878 0.903 0.984 6.90 (8) 8.26 (3) 7.75(5) KI3 (10) KI4 (17) 0.662 0.726 4.90 (15) 4.64 (16) KI5 (4) KI6 (14) 0.869 0.574 6.72 (11) 4.08 (17) 2.32 (12) 2.52 (8) 2.15 (14) 4.47 (3) 3.06 (7) 2.34 (10) 7.10 (2) used per transaction Based on these ratios, it appears that some of the inefficiencies identified with DEA may be due to scale economies related to personnel and supply usage C O N C L U S I O N S In this study, DEA was compared to another method of estimation of relative branch efficiency, the loglinear model, and a critical comparison of advantages and disadvantages between the two methods was carried out The results of the two methods did not exhibit significant differences Briefly speaking, the basic advantage of the loglinear model is the fact that it can rank branches on a single scale KIT (12) 1.000 6.39 (13) 1.92 (17) The numbers in brackets are the rank orders (in decreasing order) of the branches with respect to: the weighted number of transactions they process [in column (I)]; the number of transactions per personhour [in column (3)1; Operating expenses per 100 transactions [in column (4) which must be given in addressing problems of efficiency can be deduced On the other hand, DEA facilitates the evaluation of efficiency of bank branches which cannot be compared result In the particular estimated model, xt + x2 + x3 = 1, 26 It was e.g, observed that the existence of increasing returns to scale in some areas is not compensated by decreasing returns to scale observed in some other areas We further investigated the scale economies issue by considering two types of ratios, similar to those used by Sherman and Gold [14]: number of transactions per personhour and operating expenses per 100 transactions (see Table 3) The calculation of the rank order correlation between the numbers in column (1) and each of columns (3) and (4), confirmed that there is significant correlation between any of the ratios and the size of the branch The rank correlation between the size of the branch (in terms of total weighted transactions) and transactions per personhour is 0.66 (significance level = 0.0081) Thus, labour economies due to size appear to explain inefficiencies, as the correlation coefficient suggests that laiger branches process more transactions per personhour than smaller branches The rank order correlation between the size of the branch and operating expenses per 100 transactions is - 4 (significance level = 0.0793) This means that smaller branches are more costly, suggesting that bank branch size may have influence on the supplies directly, and has the added advantage that it takes into account the structure of inputs in branches and gives more detailed indications regarding the inputs which exhibit special problems in non-productive branches Application of these two mathematical programming methods to the same body of data (branches of a regional division of the Commercial Bank), reveals some differences and similarities regarding economies of scale More specifically, the results of the loglinear model suggest that increasing returns to scale exist, whereas the results of DEA show that increasing, constant and decreasing returns to scale exist, with a larger percentage of bank branches exhibiting increasing returns to scale Therefore, in the sample of bank branches which was examined (small to medium size branches) economies of scale seem to be in operation, indieating that operating efficiency can possibly be improved if the bank size increases This is also supported by the fact that 80% of large branches (centre) exhibit constant economies of scale In conclusion, both methods which were presented in this work, offer direct information towards a central bank problem, that of bank efficiency It appears that DEA is quite useful, since it can detect more details in the structure of the offered bank services, as well as provide essential guidance in bank efficiency control of relative efficiency from which the priority 555 Omega, Vol 19, No Therefore, the information which is provided from these two methodologies can play an important role in the formulation of administrative decisions for bank branches subject to ~ u,.y~-Uo ,-, i-! u,, v~ ;~ * and u o u n c o n s t r a i n e d in sign AP PENDIX A D E A Models Common notation used in DEA models, is the relative efficiency of branch o; is the branch being assessed from the set of j = n bank branches; is the number of branches, ho o j j=l r i yrj xo E v~, u, (5) l V~'X¢ n; (6) The equivalent linear programming model with which the above fractional programming model is replaced is presented in Appendix C The term Uo was interpreted by the authors as an indicator of returns to scale More specifically, the authors have shown that increasing returns to scale are indicated for Uo < 0, constant returns for Uo = 0, and decreasing returns to scale for is the number of outputs, r=l, ,s; is the number of inputs, i = m; is observed output r at branch j u° > (r = 1, s); is observed input i at branch j (i = , , , m ) ; is a small positive number The value of ¢ used in our study was E = 10-6; are virtual multipliers for input i If Q represents the maximum volume of transactions which can be produced when inputs Ai, A:,A Am are used, the Cobb Douglas production function will take the following form: and output r, respectively A reasonable target set by a bank is to approxi- C C R model For each bank branch the following model is solved: APPENDIX B Loglinear M o d e l _ Q-Xo Xl AI x2 x3 A2 A3 - xm A, (I) mate by the entire set of branches the maximum possible productivity which is emanating from the production function (1) Thus the problem can be stated as follows: ~ u,.y,o Max ho "~ (I) v,.x, "~ subject to Define x0, xt, x2, x3 x,~ which minimise the total deviation between the observed and maximum possible production a Z = X (x° + a,, x t + ai2.x2 + a,3.x , + + a,=x,,, 13+) (2) ~'+ u,'y,j ,=l ~< (2) ~v,'x,~ u,,v~>~(, i=1 m, r = l s (3) The above fractional programming model is replaced with a linear programming equivalent, through a series of transformations (see Appendix C) i-i under the restrictions: xo+a~t.xl+a~2.x2+aifx3+ +a x, ~>0 where: n fli = ag = xo = In Xo m For each bank branch the following model is solved: B~ Max h° -'= i ~ v: x ~- I A~ (4) i = 1, 2, n (3) (4) the number of branches; the number of inputs; the number of actual transactions in branch i; the quantity of input j in branch i; In Bi; lnAg; B C C model ~ u , y , , - u .x,,>~[3,' Giokas Bank Branch Operating E~iciency 556 APPENDIX C subject to The C C R and B C C models are transformed into a L P m o d e l as follows [I, 7, 12] ~ u,.y ~ I ,-, CCR M o d e l (5) - r =~l u,.y,j+i~- - | v,'x¢>>.O, In the case where output enhancement is emphasised for each branch, solve: Max ho = ~ u,.y,~ : = l n and u,, v, > ~ Vr, i (I) (6) B C C M o d e l Max h = ~ u , ' y , , - u, subject to (7) r-I ~ v,.x,, = I (2) subject to ~.v,'x = u,'y,,- ~ v,'x~~ O, j -~ ! n and u, vi>~, r-I (3) (8) I i-, i~l In the case where input reduction is emphasised, the formulation is written as: ~ u,.y,i - ~ v, x¢ - Uo