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Applied Economics Letters
ISSN: 1350-4851 (Print) 1466-4291 (Online) Journal homepage: https://www.tandfonline.com/loi/rael20
Combining multi-criteria and directional distances
to decompose non-compensatory measures of
sustainable banking efficiency
Thyago Celso Cavalcante Nepomuceno, Cinzia Daraio & Ana Paula Cabral
Seixas Costa
To cite this article: Thyago Celso Cavalcante Nepomuceno, Cinzia Daraio & Ana Paula Cabral
Seixas Costa (2019): Combining multi-criteria and directional distances to decompose non-
compensatory measures of sustainable banking efficiency, Applied Economics Letters, DOI:
10.1080/13504851.2019.1616051
To link to this article: https://doi.org/10.1080/13504851.2019.1616051
Published online: 13 May 2019.
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ARTICLE
Combining multi-criteria and directional distances to decompose
non-compensatory measures of sustainable banking efficiency
Thyago Celso Cavalcante Nepomuceno
a,b
, Cinzia Daraio
a
and Ana Paula Cabral Seixas Costa
b
a
Sapienza University of Rome, Roma, Italy;
b
Universidade Federal de Pernambuco, Pernambuco, Brazil
ABSTRACT
Non compensatory choices are widespread in the economics, strategic management and decision
making. Nevertheless, many assessments of productivity still fail to consider non-compensatory
preference structures in the measure of the technical inefficiency. This paper proposes
a preference elicitation schema, typical of Multi-criteria decision analysis, for the selection of
the directional vector in the assessment of a sustainable productivity. The direction choice is
based on the weighted aggregation of concordance indexes for each decision criteria on each
individual input, such that it represents an index of relative importance according to the decision
maker’s perspective. The methodology can be used to aid resource allocation and saving, identify
benchmarks for efficient practices and more generally for planning environmental policies in
many services and industrial organizations. We illustrate the method with an environmental
efficiency evaluation of Brazilian Federal Saving Bank branches.
KEYWORDS
Directional distance
functions; data envelopment
analysis; multi-criteria
decision analysis; banks;
environmental efficiency
JEL CLASSIFICATION
C67; C44; D81; G21; Q29
I. Introduction
Since the introduction of the first linear formula-
tions to measure the technical inefficiency and pro-
ductivity of Decision Making Units (DMUs) from
Charnes, Cooper, and Rhodes (1978), the so called
Data Envelopment Analysis (DEA), countless mod-
els, applications and software tools have popularized
the field (Daraio et al. 2019a,2019b). The assessment
of the efficiency although is subject to a kind of
paradox. On the one hand, we have an objective
efficiency measurement which is based on DEA, in
which the analyst does not choose a direction along
which to gauge the efficiency: the direction is
imposed by the DEA linear programme according
to which DMUs have to contract inputs (or expand
outputs) in a radial proportionate way. On the other
hand, we have a Directional Distance framework
(Färe and Grosskopf 2004) in which the analyst is
free to choose the preferred direction to expand the
outputs or to reduce the inputs. The path towards
the frontier and the related benchmarking are
imposed by this direction. To avoid arbitrary
choices, the selected direction must be justified.
This manuscript explores a combination of
Directional Distance Functions (DDF) techniques
with the outranking Multi-criteria (MCDA) elici-
tation methods to handle non-compensatory
choices. The mathematical approach proposed
here has the benefittooffer a detailed representa-
tion of the multi-attribute production possibilities
to account for indifference, preference and veto
thresholds, which may support policy makers to
obtain insights on their own preferences and
values. The application on the Brazilian Federal
Saving Bank highlights the contribution of the
proposed approach to classify sustainable units,
identify sustainable practices, processes and
determine the optimal input-output relationship.
II. Directional distance function
Directional Distances are a very flexible nonpara-
metric way to gauge the performance of Decision
Making Units. They allow measuring the technical
efficiency of DMUs by the choice of a direction
where outputs are expanded and inputs contracted
to reach the efficient frontier. The efficiency scores
can be estimated by solving the following linear
programing model:
CONTACT Thyago Celso Cavalcante Nepomuceno nepomuceno@dis.uniroma1.it Sapienza University of Rome, Roma, Italy
APPLIED ECONOMICS LETTERS
https://doi.org/10.1080/13504851.2019.1616051
© 2019 Informa UK Limited, trading as Taylor & Francis Group
Dtx;y;gx;gyðÞ¼max β
s:t:X
k
j¼1
zjyjr yor þβgyrr¼1;2;...;s
X
k
j¼1
zjxji xoi βgxii¼1;2;...;m
X
k
j¼1
zj¼1;zj0;j¼1;2;...;k
(1)
Where xji and yjr are the inputs and outputs
for each junit, and gxi;gyr
are the input
vector g(x1;x2;...;xm) and the output vector
g(y1;y2;...;ys)defining the direction along which
the inputs must be contracted or the production
expanded to reach the efficient frontier. Equation
(1) represents the overall technical efficiency (see
Färe, Grosskopf, and Zelenyuk 2008). The βcoeffi-
cient measures the absolute technical inefficiency of
each decision unit. Efficient units have β= 0 i.e. they
are on the boundary of the production frontier. Note
that in the input-oriented case, the first constraint in
the proposed model has gyr¼0, providing a cost
efficiency direction solely as a function of the input
reductions.
The absolute measure for the technical ineffi-
ciency (β) is strongly correlated with the direction
the decision maker chooses. A common direction
adopted by many practitioners in the productivity
and efficiency analysis literature is the unit vector
(i.e. g x;y
ðÞ
¼(1,1)). This vector evaluates the con-
traction in the inputs or the expansion in the
outputs equally in the same direction (Färe and
Grosskopf 2004). Other feasible choices for the
direction proposed in the literature include: the
definition of the direction for the expansion/con-
traction as proportions of the mean (Fukuyama
and Weber 2017); infer the direction through
a data-driven way (Daraio and Simar 2014,
2016); choosing the distance function according
to a reference direction g x;y
ðÞ
=g
ðxR;yRÞ;where the
direction is determined by the proportion of the
outputs and inputs from the reference decision
unit ‘R’(Briec 1997); or to determine the direc-
tional vector endogenously, i.e. as a part of the
linear problem (Färe, Grosskopf, and Whittaker
2013).
III. Outranking directions
Outranking methods bring outranking relations on
the set of available alternatives. They consider the
decision marker’s preferences for each alternative
instead of a single value. The most prominent out-
ranking methods are the ELECTRE (Elimination
and Choice Translating Algorithm), designed by
Benayoun, Roy, and Sussman (1966), and the
PROMETHEE (Preference Ranking Organization
Method for Enrichment of Evaluations) developed
by Brans, Vincke, and Mareschal (1986) and co-
workers. Each method differs to the other based on
the volume of information required, the decision
problem and according to the decision maker’s
preference structure.
Recent improved developments on non-
compensatory outranking methods have been pro-
posed to evaluate job-satisfaction (Peng and Wang
2018), hotel location selection (Ji, Zhang, and Wang
2018A), Resource Allocation in Public Universities
(Martins, de Almeida, and Morais 2019), problems
with hesitant interval-valued fuzzy sets (Wang et al.
2017) and outsourcing relations (Ji, Zhang, and
Wang 2018B; de Carvalho, Poleto, and Seixas
2018).Thefeasibilityfromthesemethodspermits
exploring pairwise comparison with the definition
of concordance and discordance indices that can be
applied to practical real-world decision-making
problems.
Based on the outranking procedures (see Belton
and Stewart 2002) we propose the following
adapted model for the preference decomposition
into a directional input vector of which the abso-
lute technical inefficiency (slacks) can be derived
as input contractions:
Dtx;y;gx;gyðÞ¼max β
s:t:X
k
j¼1
zjyjr yor;r¼1;2;...;s
X
k
j¼1
zjxji ð1βgxiÞxoi i¼1;2;...;m
X
k
j¼1
zj¼1;zj0;j¼1;2;...;k
(2)
where:
2T. C. C. NEPOMUCENO ET AL.
gxi
ðÞ
¼i1ðÞ
1X
m
i¼1PA
a¼1waCai0
;iðÞ
PA
a¼1wi
!
Dai0
;iðÞ;
"iÞi0
ðÞ2I0j9faiðÞ:Dai0
;iðÞg
(3)
And:
Concordance index :Cai0
;iðÞ¼
0if faiðÞþqafai0
ðÞ
1if faiðÞþpa<fai0
ðÞ
for any i
(4)
Dcordance index :Dai0
;iðÞ¼
0if faiðÞþta<fai0
ðÞ
1if faiðÞþtafai0
ðÞ
for any i
If we consider the unit vector, the model
becomes a version of Shephard’s input distance
function (Färe and Grosskopf 2004). The function
fai0
ðÞ
represents the score of the input ibeing eval-
uated for the decision criteria a, compared to all
the other inputs faiðÞ. The veto threshold tworks
to constraint the degree of compensation among
the different inputs so that the gain in contribu-
tion from one input must not be sufficient to offset
a significant lack of contribution from the other.
After the aggregation procedure, the compensa-
tion among the inputs is reduced (given a small
veto) or abolished (given a high veto threshold).
The indifference threshold qis defined by the
decision maker or can be postulated as the weight-
ing standard deviation:
qi¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
m1X
m
i¼1
ðwi
wiÞ2
s(5)
This is important to define a complete elicitation
on the preference structure of the policy maker.
The manager freely defines weights according to
the degree of the global importance he/she attri-
butes to the reduction of each iinputs according
to some decision criteria in a regular production
process. This procedure produces concordance and
discordance matrices by the pairwise comparisons
in (4), which implies the inexistence of trade-offs
between scaling factors, leaving the decision
maker free to choose any quantitative measure of
any scale. Lastly, the (aggregate) concordance and
discordance matrices are multiplied (see Equation
(3)) to produce a value between 0 and 1 that
represents the non-compensatory contraction on
each individual input.
IV. Application on the sustainable banking
The flexibility of Directional Distance Functions
(DDF) in the efficiency assessment provides
a tangible way to include the concerns of sustain-
ability into the evaluation of the financial institu-
tions technical performance. As an application in
the banking industry, we have collected data on 26
units of the Brazilian Federal Saving Bank (Caixa
Econômica Federal), which is the largest state-
owned bank in the Latin America. A weighting
scheme that defines the preference for sustainable
banking was defined by the manager from one of
the decision units with the elicitation of 4 decision
criteria (Cost, Environmental Impact, Availability
and Accessibility to the inputs) to compare the
inputs utilized to produce business transactions.
The thresholds for preference, indifference and
veto (based on the profit contribution) were defined
by the analyst. The inputs considered are the elec-
tricity consumption, printing services and employ-
ees. Table 1 presents the scores for each decision
criteria, the information on the thresholds and the
defined direction from the pairwise comparison (4)
and aggregation (3).
Table 2 brings the optimal level of contraction
from the slacks in the input distance function, which
represents the absolute measure of technical ineffi-
ciency. The benchmark units are those with zero
slacks(A,G,H,O,R,S,X,Y,andZ).Theyserve
Table 1. Input criteria matrix and directions.
Thresholds/Inputs Cost Environment Impact Availability Accessibility Profit Contribution Directions gxi
ðÞ
Weights 10 8 6 6 Contribution Threshold = 5 -
Indifference 20,000 0.000000001 0.3 0.3 -
Preference 35,000 0.000000002 0.5 0.5 -
Electricity (x1) 194,676.96 0.000000009 5 5 2 0.7
Printing (x2) 39,597.11 0.000000724 4 4 2 0.466,667
Employees (x3) 2,239,581.84 0.000000000 3 1 10 0
APPLIED ECONOMICS LETTERS 3
as reference for the linear combination from which
efficient practices might be inferred. The saving
potential from the sustainable analysis is compared
to the traditional DDF analysis. The slacks from x1
to x3 are defined by the number of kilowatts, paper
reams and employees that can be reduced, respec-
tively, in order to achieve a sustainable-efficient pro-
duction of business transactions by each of the 26
decision units based on the predefined preference
structure.
The gains to the ecosystem are translated in the
last row. Especially concerning the potential water
saving by the hydroelectric provision of electricity,
more than one billion liters of water can be saved
with the reduction on electricity consumption by
the 26 bank branches during one year, instead of
reducing employees in the first assessment. Some
units have become slightly more inefficient than
before. This is due the different targets on the
frontier that different directions aim to achieve.
Lastly, in the Table 3 we compare our results with
those obtained by other classical and most used
methods to measure the technical inefficiency.
Three models are considered in the comparison
analysis: The traditional Constant Return to Scale
CCR model from Charnes, Cooper, and Rhodes
(1978); the Non-Discretionary DEA model from
Charnes et al. (1985), considering the employees as
the variable beyond the managerial control of units;
and the traditional Färe and Grosskopf (2004) DDF
model. Both the inputs for electricity and printing
have the greater environmental impact reduction in
the sustainable non-compensatory analysis. The
impact in the environment in terms of water saving
and trees are significant. Applying a sustainable
directionintheefficiency targets for the branches
Table 2. Results of the assessments.
Unit
Traditional Analysis Sustainable Analysis
Inefficiency slack.x1 slack.x2 slack.x3 Inefficiency slack.x1 slack.x2 slack.x3
A00000 0 00
B 6.32 3.29 0 3.97 6.32 9.66 31.43 0
C 7.86 9.23 77.93 0.65 7.86 81.32 240.5 0
D 11.36 0 59.3 5.84 11.36 11.53 119.98 0
E 21.9 249.37 600.24 16.93 21.9 277.41 681.07 0
F 6.72 0 129.89 2.84 6.92 0 159.59 0
G00000 0 00
H00000 0 00
I 13.48 0 50.88 10.28 13.48 1.99 69.72 0
J 13.11 6.78 0 8.51 13.11 34.66 67.83 0
K 10.55 0 162.53 5.22 10.55 30.56 246.08 0
L 14.92 36.28 126.01 10.71 14.92 57.71 184.85 0
M 11.47 0 79.62 8.31 11.48 0.18 94.64 0
N 20.17 0 252.19 17.01 20.18 0 285.86 0
O00000 0 00
P 5.37 0 1.09 2.81 5.37 2.58 8.53 0
Q 17.47 9.51 248.69 13.46 17.47 26.04 299.5 0
R00000 0 00
S00000 0 00
T 5.68 0 101.03 3.32 5.68 0 139.99 0
U 1.35 0 3.73 0 1.35 0 23.32 0
V 7.38 11.06 12.92 4.82 7.38 12.49 19.74 0
W 5.25 0 6.87 2.54 5.25 0 28.87 0
X00000 0 00
Y00000 0 00
Z00000 0 00
Sum 180.36 325.52 1912.93 117.21 180.58 546.13 2701.5 0
Environmental Impact Reduction (Kilowatt | Reams | Employee): 220,610.00 788.57 -
Environmental Impact Reduction (Litters of Water | Trees): 1,270,713,600.00 39.43 -
Table 3. Comparison among efficiency models.
Models Electricity (x1)* Printing (x2)** Employees (x3)
CCR-DEA 12,916.64 1038.39 11.89
Non-Discretionary 509,111.00 2636.37 0.00
Traditional DDF 325,520.00 1912.93 117.21
Non-Compensatory DDF 546,130.00 2701.50 0.00
Saving Potential *** (213,229,440 to 3,071,308,963.4) (3.26 to 83.16) -
*Kilowatt; ** Reams; *** Litters of Water (Electricity) and Trees (Printing)
4T. C. C. NEPOMUCENO ET AL.
can save from 213 million (compared to the Non-
Discretionary method) to 3 billion (compared to the
traditional CCR model) litters of water. Likewise, it
has a saving potential from 3 (compared to the Non-
Discretionary method) to 83 (compared to the tradi-
tional CCR model) trees.
V. Conclusion
In this paper we propose a representation of the
multi-attribute production possibility which includes
the decision maker’s preferences and value judgments
over acceptable targets. By combining Multi-criteria
methods (that allows us to elicitate scores for the
global importance of the resources, limiting or abol-
ishing the compensation among the inputs) with the
flexibility of Directional Distance Functions we pro-
pose a representation of the production process in the
efficiency analysis considering complex trade-offs.
The efficiency projection from the preference schema
in this evaluation compels managers to impose nar-
rower constraints in the usage of some (environmen-
tal-related) resources than traditional frontier
assessments. In return, it allows a sustainable gain
with the identification of processes, policies and
actions that benchmark units adopt to minimize the
environmental impact.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Thyago Celso Cavalcante Nepomuceno http://orcid.org/
0000-0001-8327-6472
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