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Trang 1OMEGA Int J of Mgmt Sci., Vol 19, No 6, pp 549-557, 1991 0305-0483/91 $3.00+0.00 Printed in Great Britain All rights reserved Copyright © 1991 Pergamon Press plc
Bank Branch Operating Efficiency:
A Comparative Application of DEA and
the Loglinear Model
D I G I O K A S University of Athens and Commercial Bank of Greece
(Received September 1990; in revised form 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 g a l e 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 program-
ming, retums to scale
1 INTRODUCTION ming This function differs from DEA in that it
assumes only one output resulting from multiple DATA ENVELOPMENT ANALYSTS (DEA) (see inputs The use of the Cobb-Douglas function Charnes et al [7]) is a mathematical program- in banking is supported by other research ming technique used to estimate the efficiency of [10,13] In Section 3, the Loglinear model decision making units (DMUs) Since the first
(Cobb-Douglas), which is applied only to single publications of DEA in 1978, the literature has
output-multiple input situations, is presented expanded rapidly, reporting further theoretical
The data is presented in Section 4 As the developments as well as numerous applications estimates of different characteristics of the pro- [5, 12] Later developments [1, 3,4] have ex-
duction correspondence provided by these two tended DEA to the estimation of cost and methods are commonly employed for policy production correspondences This method is inferences, their comparison is useful and inter- presented and discussed in Section 2 of the
esting In Section 5, the results of DEA are paper Until now, the results from the appli- compared to those from the Loglinear model cation of DEA to the banking sector have not This section contains also our findings on econ- been compared to those coming from the appli- omies of scale Finally, a few concluding re- cation of other classical estimation methods of marks are offered in Section 6
a production function In a study of North
Carolina hospitals, results of DEA have been
compared to those from traditional econometric 2 DATA ENVELOPMENT ANALYSIS techniques, e.g econometric modelling using the
translog cost function by Banker et al [2] In the In general, DEA measures efficiency by esti- present paper, the Cobb-Douglas function has mating an empirical production function which been estimated using mathematical program- represents the highest values of outputs/benefits
549
Trang 2550 Giokaa Bank Branch Operating Efficiency
that could be generated by inputs/resources as Finally, DEA supplies information about given by the observed input-output vectors, relatively inefficient branches, regarding specific DEA is composed of several mathematical inputs that they overutilise or outputs that they models sharing the principle of envelopment, underproduce, depending on whether the objec-
As Golany [11] points out, "DEA is quickly tive of the decision makers is input minimisation emerging as the leading method for efficient or output maximisation (see Appendix C) evaluation, in terms of both the number of Assuming that the reference point on the research papers published and the number of production function for an inefficient D M U applications to real world problems" The orig- will be a convex combination of the observed
inal model proposed by Charnes et al [7], efficient DMUs, Banker et al [1] provide a
known as CCR, was applied to the banking linear programming model (known as BCC) context by several authors [6, 14-16] The which permits examination of productive detailed formulation of the CCR model is given efficiencies and returns to scale (see
The application of the CCR model to a set of
branches allows the comparison of each one of 3 L O G L I N E A R FUNCTION E S T I M A T I O N them to the rest of the data set and the elici-
tation of a number of conclusions: In many of the studies of economies of scale First, the technique assigns to all the branches in banking, output is assumed to be produced being evaluated efficiency ratings as follows: according to a Cobb Douglas production func-
tion [10, 13] This function has the desirable
ho = 1, signifying relative efficiency; property of being transformable into a logarith-
ho < 1, signifying relative inefficiency, mic 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 formu-
study That is, ho = 1 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 < 1 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; there- efficiency 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 rela- branch if the latter was making use of
available resources in the best (rela- tively 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 do 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:
Second, DEA identifies the efficiency refer- Xo.A,t A~ A,3 • A,,
ence set for each inefficient branch This is the
set of relatively efficient branches to which it has
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 Commer- question, 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 do not
mention the names of individual bank branches, which o v e r h e a d s , since the objective o f this a n a l y s i s
a r e recorded under code number, was to evaluate of the use of inputs consumed
Trang 3Omega, Vol 19, No 6 551 directly at the branch Inputs employed are the of transactions produced at each section was
(i) Labour (personhours) This variable and Capital Transfers, Category B = Section
of credit, Category C = Section of Foreign summarises the actual work (in per- Receipts) Transactions were weighted with co- sonhours) employed during the oper-
ation of the branches and includes efficients of equivalence defined by the Domestic administrative/processing personnel, Operations and Branches Division on the basis
of their experience from the workings of bank marketing officers and branch man- branches Estimation of the Cobb-Douglas type agement All branches employ ap-
proximately the same proportion of function of maximum relative efficiency requires administrative/processing personnel in all bank branch products to be aggregated into
a single one This aggregation was performed their staff The number of personhours with the help of the above mentioned captures all hours worked by the per- coefficients of equivalence In this way, the sonnel of the branch including over-
time and detached labour, weighted number of transactions performed by
a bank branch was defined as a single output
(ii) Operating expenses (drachmas) This
The data are reproduced in Appendix D variable expresses the consumption of
a range of inputs by the bank branch
of the branches, such as those for
The DEA results are summarised in Table 1 telephone, electricity, stationery and
The results from the CCR model give the overall other supplies This variable is technical and scale efficiency which is less than measured in monetary terms, due to or equal to the pure (input) technical efficiency difficulty of grouping together the measured by the BCC model [1] We will use widely dissimilar entities that comprise results of the CCR model to examine operating
it Note that all purchases are per- efficiency of individual branches, whereas re- formed centrally by the bank and suits of the BCC model will be used to examine hence all branches are charged the
economies of scale Thus, the results of the CCR same prices Expenses for salaries, rent model [column (2)] indicate that 12 branches are and depreciationofbuildingshavenot relatively inefficient, that is they have an been included, since personnel ex-
efficiency rating of less than 1 A branch is penses and office space have already found to be inefficient if it is possible to con- been covered by the other two inputs, struct a reference branch as a linear combi-
(iii) Utilised branch space (square nation of other branches, such that the reference
metres) This variable shows the con-
tribution of available space to the performs at least as many transactions while production of the bank branch It using less inputs than the real branch As Sher- should be noted that only space which man and Gold [14] point out " the efficiency
rating does not rank order the branches, but
is in productive use has been included, rather suggests the degree of inefficiency of a
It was decided not to include any inputs branch compared with its efficiency reference reflecting market conditions as all the branches set" Hence, K3 is about 86% efficient com-
in the data set are located in Athens, they pared to K2, K4 and KS K15 is also about 86% belong to the same regional division of the efficient but compared to K2 and K4 Generally, Commercial Bank and operate in reasonably this means that both K3 and K15 could reduce
As outputs we defined bank products offered without reducing their outputs The perform-
to customers and more specifically the complete ance of inefficient branches can be improved number of transactions (total 72)performed in either by increasing outputs or by cutting
In order to apply DEA, transactions were inputs that inefficient branches overutilise or grouped according to the section of the branch outputs that they underproduce are summarised which performs them and the weighted number in Table 2 More specifically, the degree to
Trang 4552 Giokas Bank Branch Operating E:ciency
Table 1
Uf
K I0 0.878 K2, K4, K 17 0.885 K2, K4, K6, K 17 - 0.06 0.796
which inputs can be reduced is indicated by the situation and do not take into account possible numbers in colums (3), (4) and (5) Numbers in particular differences in branch operations columns (6), (7) and (8) show how much extra The degrees of relative efficiency estimated by output an inefficient branch could generate with DEA [column (2) of Table 1] in general, are the same level of inputs if it moved to the not significantly different to the measures of efficient frontier, e.g branch KI0 can increase efficiency which were estimated by the loglinear its outputs by 13.9, 13.9 and 46.9%, respect- model [column (7)] It should be emphasised ively Application of the loglinear model gave that the above results have been produced by the efficiency measures which are presented in two completely different methods, and thus column (7) of Table 12 According to the esti- would not be expected to be in total agreement mated measures of efficiency two branches (e.g Still, a comparison may prove to be useful in K2 and K6) achieved the maximum possible establishing certain useful conclusions
efficiency, whereas the rest of the branches fall The biggest difference between the two within a broad range of relative efficiency efficiency indexes is observed in some bank (measures of efficiency are within the range branches which have been characterised as 0.988-0.387) It should be noted that the co- productive by DEA, whereas the estimated efficients have been estimated under the mini- efficiency measures of the loglinear model are misation of total deviation between the significantly below unity This is mainly ob- observed and maximum production and the served in branches K7, K8 and K17 In order to obtained measures of relative efficiency classify explain this phenomenon, it should be stated the branches in one single scale In other words, that the concept of efficiency is treated some-
it is possible to compare two bank branches on what differently under DEA and the loglinear the basis of the estimated measures of efficiency model Both models measure the operating and to derive certain conclusions regarding efficiency of branches in relation to that of other their relative efficiency It should be mentioned, branches The loglinear model, however, treats however, that these results reflect an average bank branches as if they produce a single
product and classifies them on a single efficiency scale, whereas DEA examines combinations of
following estimated equation: under DEA, some branches with "unusual"
Q, = 1 A°i ~ ' A~ ~'~ A~3 combinations cannot be compared directly to a
peases (in t h o u s a n d s o f drachmas) a n d square metres o f ised as branches of maximum efficiency
Trang 5Omega, Vol 19, No 6 553
Table 2
PH: Personhours
OE: Operating expenses
sq m: Square metres of branch space
A: Weighted -number of transactions done by section of deposits and capital transfers
B: Weighted number of transactions done by section of credit
C: Weighted number of transactions done by section of foreign receipts
Note: Numbers in parentheses indicate the % of excess inputs or deficient outputs of a branch compared with its efficient reference set
certain inputs of production Thus, the loglinear as Counter-Branch As for branch K16, accord- model practically ignores branch space in the ing to internal information, it is being kept in evaluation of maximum efficiency In an operation despite operational problems, due to analogous manner, DEA's solution most of the its geographical position
time shows that the space of bank branches is
not a key variable affecting production This is Economies o f scale
clearly proved by the fact that the most import- The returns to scale for a particular observed ant variables influencing bank branch efficiency input-output mix may be examined using DEA are personhours and/or the operating bank by estimating the sign of the variable Uo (see
In conclusion, from a comparison of the to scale are indicated for uo<O, constant results of the two methods (as shown in Table 1) returns for Uo = 0, and decreasing returns for
it becomes clear that branches K5, K9, K10, Uo > 0 The results in Table 1 [column (6)]
K l l , K13, KI4, K15 and KI6 exhibit special indicate that only 2 branches experience de- problems of operating efficiency Therefore, it is creasing returns to scale, 5 have constant returns necessary to focus the attention of Bank de- to scale and the majority have increasing re- cision makers to those branches in order to turns Berg et al [6] point out that the convexity locate the reasons of such low efficiency and of the frontier ensures that increasing returns then take corrective measures For their assist- will be more frequent at smaller branches ance, apart from the information of Table 1, Because of the fact that the sample of 17 bank there is additional information provided by branches consisted of satellite (small) and not DEA regarding the specific inputs which are centre (large) branches, the above results give overutilised at the low-efficiency branches support to those of Berg and others that small
Finally, evaluating the above results a year noted that when centre branches were added to later, it is of interest to observe that for at least the sample, the results showed that 80% of them half of the branches which were characterised as operate under constant returns to scale inefficient (i.e KS, K9, K10, K l l , K13, K14, It is interesting to recall at this point that K15, K16) an analogous opinion had been when aloglinear function (Cobb-Douglas)was formed independently bybankexecutives More used to estimate operating efficiency in the specifically, during 1989, branches K9 and K14 aggregate data, the existence of increasing re- were merged due to recognised problematic turns to scale was confirmed, since the satisfac- behaviour (as well as due to the small distance tion of xl + xz + x3 > 1 (where xi are the between them), whereas K13 was characterised coefficients of the loglinear model), implies this
Trang 6554 Giokas Bank Branch Operating Efficiency
Table 3 Operating ratios of bank branches used per transaction Based on these ratios, it
Operating expenses appears that some of the inefficiencies identified per 100 with DEA may be due to scale economies DEA Transactions transactions related to personnel and supply usage
Branch etiiciency per in thousands of
K4 (3) 1.000 7.86 (4) 1.97 (16) In this study, DEA was compared to another
K6 (11) 0.984 7.60 (6) 2,09 (15) method of estimation of relative branch K7 (13) 1.000 6.79 (10) 3,S4 (4) efficiency, the loglinear model, and a critical
KI0 (6) 0.878 6.90 (8) 2.32 (12) between the two methods was carried out The
KI2(5) 0.984 7.75(5) 2.15 (14) results of the two methods did not exhibit
KI5 (4) 0.869 6.72 (11) 2.34 (10) basic advantage of the loglinear model is the KI6 (14) 0.574 4.08 (17) 7.10 (2) fact that it can rank branches on a single scale
The numbers in brackets are the rank orders (in decreasing order) o f relative efficiency from which the priority
the weighted number of transactions they process [in efficiency can be deduced On the other hand,
column (I)];
the number of transactions per personhour [in column (3)1; DEA facilitates the evaluation of efficiency
Operating expenses per 100 transactions [in column (4) 1 of bank branches which cannot be compared
directly, and has the added advantage that it takes into account the structure of inputs in result In the particular estimated model, branches and gives more detailed indications
xt + x2 + x3 = 1, 26 It was e.g, observed that regarding the inputs which exhibit special the existence of increasing returns to scale in problems in non-productive branches
some areas is not compensated by decreasing Application of these two mathematical pro- returns to scale observed in some other gramming methods to the same body of data
We further investigated the scale economies cial Bank), reveals some differences and simi- issue by considering two types of ratios, similar larities regarding economies of scale More
to those used by Sherman and Gold [14]: num- specifically, the results of the loglinear model ber of transactions per personhour and operat- suggest that increasing returns to scale exist, ing expenses per 100 transactions (see Table 3) whereas the results of DEA show that increas- The calculation of the rank order correlation ing, constant and decreasing returns to scale between the numbers in column (1) and each of exist, with a larger percentage of bank branches columns (3) and (4), confirmed that there is exhibiting increasing returns to scale Therefore, significant correlation between any of the ratios in the sample of bank branches which was and the size of the branch The rank correlation examined (small to medium size branches) econ- between the size of the branch (in terms of total omies of scale seem to be in operation, indi- weighted transactions) and transactions per per- eating that operating efficiency can possibly be sonhour is 0.66 (significance level = 0.0081) improved if the bank size increases This is also Thus, labour economies due to size appear to supported by the fact that 80% of large explain inefficiencies, as the correlation co- branches (centre) exhibit constant economies of efficient suggests that laiger branches process scale
more transactions per personhour than smaller In conclusion, both methods which were pre- branches The rank order correlation between sented in this work, offer direct information the size of the branch and operating expenses towards a central bank problem, that of bank per 100 transactions is - 0 4 4 (significance efficiency It appears that DEA is quite useful, level = 0.0793) This means that smaller since it can detect more details in the structure branches are more costly, suggesting that bank of the offered bank services, as well as provide
branch size may have influence on the supplies essential guidance in bank efficiency control
Trang 7Omega, Vol 19, No 6 555 Therefore, the information which is provided subject to
i - !
A P P E N D I X A u,, v~ ;~ * and u o u n c o n s t r a i n e d in sign (6)
D E A Models The equivalent linear programming model with
C o m m o n notation used in D E A models, which the above fractional programming model
is replaced is presented in Appendix C The term
ho is the relative efficiency o f branch o;
o is the branch being assessed from Uo was interpreted by the authors as an indicator the set of j = 1 n bank o f returns to scale More specifically, the
j is the number of branches, scale are indicated for Uo < 0, constant returns
for Uo = 0, and decreasing returns to scale for
j = l n;
r is the number of outputs, u° > 0
r = l , , s ;
i = 1 m;
yrj is observed output r at branch j Loglinear M o d e l
(r = 1, 2 s); If Q represents the maximum volume of
x o is observed input i at branch j transactions which can be produced when (i = 1 , 2 , , m ) ; inputs Ai, A:,A 3 Am are used, the
E is a small positive number The
Cobb Douglas production function will take value o f ¢ used in our study was
the following form:
E = 10-6;
_ X l x 2 x 3 x m
v~, u, are virtual multipliers for input i Q - X o A I A 2 A 3 .- A, ( I ) and output r, respectively A reasonable target set by a bank is to approxi-
1 C C R model mate by the entire set of branches the maximum
F o r each bank branch the following model is possible productivity which is emanating from
can be stated as follows:
~ u,.y,o
Max ho "~ (I) Define x0, xt, x2, x3 x,~ which minimise the
v,.x, total deviation between the observed and maxi-
subject to
a
~'+ u,'y,j Z = X (x° + a,, x t + ai2.x 2 + a,3.x , + + a,=x,,, 13+) (2)
u,,v~>~(, i = 1 m, r = l s (3) xo+a~t.xl+a~2.x2+aifx3+ +a x,,>~[3, '
through a series of transformations (see Appen-
n the number of branches;
F o r each bank branch the following model is B~ the number of actual transactions in
~ u , y , , - u A~ the quantity of input j in branch i;
Trang 8E~iciency
The C C R and B C C models are t r a n s f o r m e d into ~ u,.y ~ I (5)
a L P m o d e l as follows [I, 7, 12] ,-,
r = l i - - |
In the case where o u t p u t e n h a n c e m e n t is
emphasised for each branch, solve: : = l n and u,, v, > ~ Vr, i (6)
r - I
u,'y,, - ~ v,'x~ ~ O, j -~ ! n and u, v i >~, (3) i-,
r - I i ~ l
In the case where input reduction is emphasised, ~ u,.y,i - ~ v, x¢ - Uo <~ o,
the f o r m u l a t i o n is written as: ,-s ,-,
j = 1 n and u,, v , ~ c
Input-Output Data for 17 Bank Branches
PH: " Personhours
OE: Operating expenses (in thousands of drachmas)
sq rn: Square metres of branch space
A: Weighted No of transactions performed by section of deposits and capital transfers
B: Weighted No of transactions performed by section of credit
C: Weighted No of transactions performed by section of foreign receipts
D: Total weighted No of transactions performed by each branch
I would like to t h a n k D r M Vassiloglou for her helpful
suggestions which have greatly improved the presentation 1 Banker RD, Charnes A a n d Cooper W W (1984) Some and theoretical scope o f the paper A n y errors or omissions models for estimating technical and scale inefficiencies remain the a u t h o r ' s responsibility The c o m m e n t s o f a in D a t a Envelopment Analysis Mgmt Sci 30(9),
Trang 9Omega, Vol 19, No 6 557
2 Banker RD, Conrad RF and Strauss RP (1986) A 10 Clark JA (1984) Estimation of economies of scale in comparative application of Data Envelopment Analysis banking using a generalized functional form J Man
and Translog Methods: An illustrative study of Hospi- Cr Bank 16(1), 53-68
tal production Mgmt Sci 32(1), 30-44 11 Golany B (1988) An interactive MOLP procedure for
3 Banker RD and Maindiratta A (1986) Piecewise log- the extension of DEA to effectiveness analysis J Opl linear estimation of efficient production surfaces Mgmt Res Soc 39(8), 725-734
4 Banker RE) and Morey RC (1986) The use ofcategori- for DEA Omega 17(3), 237-250
cal vadables in Data Envelopment Analysis Mgmt Sci 13 Prastacos GP (1986) (Ed.) Management science and 32(12), 1613-1627 information technology in the banking sector: A review
5 Banker RD, Charnes A, Cooper WW, Swarts J and In Proceedings of the 7th conference of the Greek
Thomas DA (1989) An introduction to Data Envelop- Operational Research Association, Greek Operational
merit Analysis with some of its models and their uses Research Society, Athens (Publication in greek),
6 Berg AS, Forsund FR and Jansen ES (1989) Bank 14 Sherman H D a n d Gold F (1985)Bank branch operating output measurement and the construction of best prac- efficiency: Evaluation with Data Envelopment Analysis
tice frontiers Paper presented at the 16th Meeting of J Banking Fin 9, 297-315
the European Finance Association, Stockholm, August 15 Smullen J (1989) The application of Data Envelopment
31-September 2, 1989 and at the Econometric Society's Analysis to branch planning and management ap-
European Meeting, Munich, September 4-8, 1989 praisal Paper presented at the European Working Group
7 Charnes A, Cooper WW and Rhodes E (1978) on Operational Research, Eurobanking 1989
Measuring the efficiency of decision making units Eur 16 Vassiloglou M and Giokas D (1990) A study of the
8 Charnes A, Cooper WW and Rhodes E (1979) Short Data Envelopment Analysis J Opl Res Soc 41(7),
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9 Charnes A and Cooper WW (1985) Preface to topics ADI~RV.SS ~OR CORnESPONDENCE: Dr Dimitris Giokas, Univer-
in Data Envelopment Analysis, Ann Ops Res 2, sity of Athens, Department of Economics, 8 Pesmazoglou