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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.
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Applied Economics Letters
ISSN: 1350-4851 (Print) 1466-4291 (Online) Journal homepage:
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:
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Published online: 13 May 2019.
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Combining multi-criteria and directional distances to decompose
non-compensatory measures of sustainable banking eciency
Thyago Celso Cavalcante Nepomuceno
, Cinzia Daraio
and Ana Paula Cabral Seixas Costa
Sapienza University of Rome, Roma, Italy;
Universidade Federal de Pernambuco, Pernambuco, Brazil
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 ineciency. 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
makers perspective. The methodology can be used to aid resource allocation and saving, identify
benchmarks for ecient practices and more generally for planning environmental policies in
many services and industrial organizations. We illustrate the method with an environmental
eciency evaluation of Brazilian Federal Saving Bank branches.
Directional distance
functions; data envelopment
analysis; multi-criteria
decision analysis; banks;
environmental eciency
C67; C44; D81; G21; Q29
I. Introduction
Since the introduction of the rst linear formula-
tions to measure the technical ineciency 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 eld (Daraio et al. 2019a,2019b). The assessment
of the eciency although is subject to a kind of
paradox. On the one hand, we have an objective
eciency measurement which is based on DEA, in
which the analyst does not choose a direction along
which to gauge the eciency: 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 justied.
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 benettooer a detailed representa-
tion of the multi-attribute production possibilities
to account for indierence, 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 exible nonpara-
metric way to gauge the performance of Decision
Making Units. They allow measuring the technical
eciency of DMUs by the choice of a direction
where outputs are expanded and inputs contracted
to reach the ecient frontier. The eciency scores
can be estimated by solving the following linear
programing model:
CONTACT Thyago Celso Cavalcante Nepomuceno Sapienza University of Rome, Roma, Italy
© 2019 Informa UK Limited, trading as Taylor & Francis Group
Dtx;y;gx;gyðÞ¼max β
zjyjr yor þβgyrr¼1;2;...;s
zjxji xoi βgxii¼1;2;...;m
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)dening the direction along which
the inputs must be contracted or the production
expanded to reach the ecient frontier. Equation
(1) represents the overall technical eciency (see
Färe, Grosskopf, and Zelenyuk 2008). The βcoe-
cient measures the absolute technical ineciency of
each decision unit. Ecient units have β= 0 i.e. they
are on the boundary of the production frontier. Note
that in the input-oriented case, the rst constraint in
the proposed model has gyr¼0, providing a cost
eciency direction solely as a function of the input
The absolute measure for the technical ine-
ciency (β) is strongly correlated with the direction
the decision maker chooses. A common direction
adopted by many practitioners in the productivity
and eciency analysis literature is the unit vector
(i.e. g x;y
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
denition 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
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
III. Outranking directions
Outranking methods bring outranking relations on
the set of available alternatives. They consider the
decision markers 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 diers to the other based on
the volume of information required, the decision
problem and according to the decision makers
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
exploring pairwise comparison with the denition
of concordance and discordance indices that can be
applied to practical real-world decision-making
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 ineciency (slacks) can be derived
as input contractions:
Dtx;y;gx;gyðÞ¼max β
zjyjr yor;r¼1;2;...;s
zjxji ð1βgxiÞxoi i¼1;2;...;m
Concordance index :Cai0
0if faiðÞþqafai0
1if faiðÞþpa<fai0
for any i
Dcordance index :Dai0
0if faiðÞþta<fai0
1if faiðÞþtafai0
for any i
If we consider the unit vector, the model
becomes a version of Shephards input distance
function (Färe and Grosskopf 2004). The function
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 dierent inputs so that the gain in contribu-
tion from one input must not be sucient to oset
a signicant 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 indierence threshold qis dened by the
decision maker or can be postulated as the weight-
ing standard deviation:
This is important to dene a complete elicitation
on the preference structure of the policy maker.
The manager freely denes 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-os
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 exibility of Directional Distance Functions
(DDF) in the eciency assessment provides
a tangible way to include the concerns of sustain-
ability into the evaluation of the nancial 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 denes the preference for sustainable
banking was dened 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, indierence and
veto (based on the prot contribution) were dened
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
dened 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 ine-
ciency. The benchmark units are those with zero
Table 1. Input criteria matrix and directions.
Thresholds/Inputs Cost Environment Impact Availability Accessibility Prot Contribution Directions gxi
Weights 10 8 6 6 Contribution Threshold = 5 -
Indierence 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
as reference for the linear combination from which
ecient 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 dened by the number of kilowatts, paper
reams and employees that can be reduced, respec-
tively, in order to achieve a sustainable-ecient pro-
duction of business transactions by each of the 26
decision units based on the predened preference
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 rst assessment. Some
units have become slightly more inecient than
before. This is due the dierent targets on the
frontier that dierent 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 ineciency.
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 signicant. Applying a sustainable
directionintheeciency targets for the branches
Table 2. Results of the assessments.
Traditional Analysis Sustainable Analysis
Ineciency slack.x1 slack.x2 slack.x3 Ineciency 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 eciency 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)
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 makers 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
exibility of Directional Distance Functions we pro-
pose a representation of the production process in the
eciency analysis considering complex trade-os.
The eciency 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 identication of processes, policies and
actions that benchmark units adopt to minimize the
environmental impact.
Disclosure statement
No potential conict of interest was reported by the authors.
Thyago Celso Cavalcante Nepomuceno
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This paper presents the initial results of a creative-innovative potential assessment in a group of Brazilian cities. The proposal is based on two pillars: open data platforms and multicriteria analysis. Through the open data platforms it was possible to get data and dimension the criteria and related indicators used in the study, while the application of the multicriteria methods PROMETHEE I and II created a ranking of the cities from the perspective of their creativity and innovativeness.
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Evaluation by ranking/rating of data based on a multitude of indicators typically calls for multi-criteria decision analyses (MCDA) methods. MCDA methods often, in addition to indicator values, require further information, typically subjective. This paper presents a partial-order methodology as an alternative to analyze multi-indicator systems (MIS) based on indicator values that are simultaneously included in the analyses. A non-technical introduction of main concepts of partial order is given, along with a discussion of the location of partial order between statistics and MCDA. The paper visualizes examples of a ‘simple’ partial ordering of a series of chemicals to explain, in this case, unexpected behavior. Further, a generalized method to deal with qualitative inputs of stakeholders/decision makers is suggested, as well as how to disclose peculiar elements/outliers. The paper finishes by introducing formal concept analysis (FCA), which is a variety of partial ordering that allows exploration and thus the generation of implications between the indicators. In the conclusion and outlook section, take-home comments as well as pros and cons in relation to partial ordering are discussed.
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For strategic reasons, benchmarking best practices from efficient competitors is not usual in many data envelopment analysis (DEA) applications. Even for industries composed of multiple branches, providing information about efficient practices for their peers can jeopardize results for those branches if they compete for market, resources or recognition by the central administration. In this work, a time‐series adaptation for the DEA directional model is proposed as an alternative for coping with this problem. The methodological approach has three stages for this benchmarking to occur: Data, Information and Knowledge Extraction. In the first stage, we compare the same unit in different moments to identify efficient periods instead of efficient competitors. As a result, successful performance strategies are investigated using the bibliometric coupling of employees' relevant statements in the second and third stages. The application in a branch of the Brazilian Federal Savings Bank allowed an internal benchmarking of efficient periods when specific performance incentives, innovative processes, competitive strategies, and human resource changes were adopted for improving the unit's performance.
The Brazilian Federal Savings Bank (Caixa Econômica Federal—CEF) is the biggest government-owned bank in Latin America and the fourth globally in terms of assets. This work reports an empirical assessment of best practices conducted in one of the bank's units situated in Pernambuco. This empirical investigation is developed with a combination of Time-Series Data Envelopment Analysis for efficient (time) peers, human interactions, and the PROMETHEE methodology for outranking the best practices and strategies in accordance with the support of employees, sustainability impact and response to minimizing COVID-19 propagation during open hours.
Rapidly changing environments place different players at the vortex of the innovation process. Therefore, in the digital age, strong businesses are sometimes built on perceptions and on the approval of the community. The shift from linear value chains to ecosystems is likely to occur in 4.0 organizations adopting service or customer orientations, according to their participation in networked ecosystems. Moving from organization-centered innovation to ecosystem co-creation will approach individuals and institutions thus enhancing sustainable and smart product development along with trust. Embedded innovation is a self-sustained process in which firms and stakeholders interact in a common environment creating a common identity. Empirical results reinforce the role of open innovation strategies and the user community as pillars of sustainable innovation ecosystems. Policy actions need to reinforce these ecosystems as they will generate employment encompassing innovative and inclusive growth, fostering the resilience of societies and environmental sustainability.
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This contribution is the first attempt to systematically review all empirical surveys that so far have been made available in the broad field of efficiency and productivity analysis using frontier estimation methodologies. We provide a systematic bibliometric review on the many empirical surveys in the field of efficiency and productivity analysis, the most relevant concepts, areas, overlaps, and potentials to explore from its introduction to the most recent surveys. We combine the United Nations’ International Standard Industrial Classification (ISIC) taxonomy for the economic activity with the Journal of Economic Literature (JEL) classification system to classify the empirical surveys and to identify the current gaps in the literature. In addition to the most relevant/generic potential areas for applications (according to the United Nation's ISIC), this methodology provides a cluster analysis with the most relevant concepts that have been considered so far (according to the JEL codes). This overview brings an interesting guide for future work to develop the whole field.
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The software available to implement and carry out efficiency analysis is crucial for the diffusion of efficiency frontier techniques among applied researchers and policy makers. The implementation of up‐to‐date productivity and efficiency analysis is indeed important to advance our knowledge in many fields, ranging from the public and regulated sectors to the private ones. This contribution fills a gap in the existing literature and surveys the currently available options to estimate a variety of frontier methodologies using either general or dedicated programs. We outline directions for future research.
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The Z-number provides an adequate and reliable description of cognitive information. The nature of Z-numbers is complex, however, and important issues in Z-number computation remain to be addressed. This study focuses on developing a computationally simple method with Z-numbers to address multicriteria decision-making (MCDM) problems. Processing Z-numbers requires the direct computation of fuzzy and probabilistic uncertainties. We used an effective method to analyze the Z-number construct. Next, we proposed some outranking relations of Z-numbers and defined the dominance degree of discrete Z-numbers. Also, after analyzing the characteristics of elimination and choice translating reality III (ELECTRE III) and qualitative flexible multiple criteria method (QUALIFLEX), we developed an improved outranking method. To demonstrate this method, we provided an illustrative example concerning job-satisfaction evaluation. We further verified the validity of the method by a criteria test and comparative analysis. The results demonstrate that the method can be successfully applied to real-world decision-making problems, and it can identify more reasonable outcomes than previous methods. This study overcomes the high computational complexity in existing Z-number computation frameworks by exploring the pairwise comparison of Z-numbers. The method inherits the merits of the classical outranking method and considers the non-compensability of criteria. Therefore, it has remarkable potential to address practical decision-making problems involving Z-information.
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Keen competition drives hotel companies to enhance their position. One way to do this is to select a proper hotel location. However, hotel location selection is a complex problem. This study establishes a multi-criteria hotel location selection method. In this method, cognitive information is depicted by multi-hesitant fuzzy linguistic term sets (MHFLTSs). Moreover, the method considers the non-compensation of criteria. It introduces the elimination and choice translating reality (ELECTRE) method. Notably, the method utilizes projection to define concordance and discordance indices. A case study and comparative study are performed in this study. They exhibit the feasibility of the method. Results of the studies show that the method can solve such problems, and they reveal the method’s advantages. One theoretical contribution lies in the characterization of cognitive information. MHFLTSs can handle vacillation of decision-makers caused by their complex cognition, and they express both conformity and divergence of opinions during cognitive processes. Our method has the advantages of the ELECTRE method. In addition, the ELECTRE method is improved by introducing the projection. The proposed method is promising in hotel location selection. Moreover, it is a potential option to address cognitive computation.
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For decision makers, expressing their opinions through subintervals of [0, 1] is sometimes easier than using crisp numbers. This study defines some outranking relations derived by ELECTRE III for hesitant interval-valued fuzzy sets (HIVFSs). The properties of these outranking relations are discussed in detail. The concordance and discordance indexes of HIVFSs are then proposed. Ranking approach is developed based on the outranking relations of HIVFSs to handle multi-criteria decision-making problems. Finally, we provide practical examples, as well as sensitivity analysis and comparison with other techniques.
This study aims to demonstrate how the design of a decision support system (DSS) can improve the process of internal resource allocation in Brazil public universities. Currently, there are not any kind of general DSS for such a problem. To do so, the analysis is carried out by identifying the general model from the Brazilian Ministry of Education and the models from every federal university, finding similarities between each model, and dividing the models into categories, according to their similarities. Thus, a DSS resource allocation model prototype was proposed. The perspectives are to contribute to the decision problem of how to allocate resources properly faced by Brazilians public universities, take safer and reliable decisions, seeking to reduce uncertainties and to maximize their results.
Outsourcing has become a common and important link for enterprises. When selecting an outsourcing provider, single-valued neutrosophic linguistic sets can successfully express qualitative and fuzzy information. Furthermore, the selection of an outsourcing provider is a multi-criteria decision-making problem that can be tackled by the multi-attribute border approximation area comparison (MABAC) method. In MABAC, criteria are assumed to be compensatory. However, criteria may be non-compensatory in outsourcing provider selection. Thus, this paper introduces the main idea of the elimination and choice translating reality (ELECTRE) method. An MABAC–ELECTRE method is established under single-valued neutrosophic linguistic environments. This method uses the mean-squared deviation weight method to obtain the weights of criteria. Moreover, an illustrative example is conducted to explain the procedure of the MABAC–ELECTRE method. A comparative analysis verifies its feasibility in solving problems with non-compensatory criteria.
In this chapter we provide a directional distance DEA framework that extends standard DEA.
In efficiency analysis the assessment of the performance of Decision-Making Units (DMUs) relays on the selection of the direction along which the distance from the efficient frontier is measured. Directional Distance Functions (DDFs) represent a flexible way to gauge the inefficiency of DMUs. Permitting the selection of a direction towards the efficient frontier is often useful in empirical applications. As a matter of fact, many papers in the literature have proposed specific DDFs suitable for different contexts of application. Nevertheless, the selection of a direction implies the choice of an efficiency target which is imposed to all the analysed DMUs. Moreover, there exist many situations in which there is no a priori economic or managerial rationale to impose a subjective efficiency target. In this paper we propose a data-driven approach to find out an 'objective' direction along which to gauge the inefficiency of each DMU. Our approach permits to take into account for the heterogeneity of DMUs and their diverse contexts that may influence their input and/or output mixes. Our method is also a data-driven technique for benchmarking each DMU. We describe how to implement our framework and illustrate its usefulness with simulated and real data sets.