- Email: [email protected]
A non-radial directional distance method on classifying inputs and outputs in DEA: Application to banking industry
Accepted Manuscript
A non-radial directional distance method on classifying inputs and outputs in DEA: Application to banking industry Mehdi Toloo , Maryam Allahyar , Jana Hanˇclova PII: DOI: Reference:
S0957-4174(17)30638-3 10.1016/j.eswa.2017.09.034 ESWA 11555
To appear in:
Expert Systems With Applications
Received date: Revised date: Accepted date:
23 December 2016 14 May 2017 12 September 2017
Please cite this article as: Mehdi Toloo , Maryam Allahyar , Jana Hanˇclova , A non-radial directional distance method on classifying inputs and outputs in DEA: Application to banking industry, Expert Systems With Applications (2017), doi: 10.1016/j.eswa.2017.09.034
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Highlights A non-radial nor-oriented method is developed to deal with flexible measures.
Two optimistic and pessimistic approaches are proposed.
Each approach contains two individual and integrated models.
A case study of 61 banks in the Visegrad Four region validates the new models.
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A non-radial directional distance method on classifying inputs and outputs in DEA: Application to banking industry
Mehdi Toloo1
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Department of Systems Engineering, Technical University of Ostrava, Sokolska tř. 33, 702 00 Ostrava 1, Ostrava, Czech Republic E-mail: [email protected]
URL: http://homel.vsb.cz/~tol0013/
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Tel: (+420) 792 272 272
Maryam Allahyar
Young Researchers and Elite Club, Yadegar-e-Imam Khomeini (RAH)Shahre Rey Branch, Islamic Azad University, Tehran, Iran
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Email: [email protected]
Jana Hančlova
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Department of Systems Engineering, Technical University of Ostrava, Sokolska tř. 33, 702 00 Ostrava 1, Ostrava, Czech Republic
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Email: [email protected]
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Corresponding Author
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Abstract The original Data Envelopment Analysis (DEA) models have required an assumption that the
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status of all inputs and outputs be known exactly, whilst we may face a case with some flexible performance measures whose status is unknown. Some classifier approaches have been proposed in order to deal with flexible measures. This contribution develops a new classifier non-radial directional distance method with the aim of taking into account input contraction and output expansion, simultaneously, in the presence of flexible measures. To make the most appropriate decision for flexible measures, we suggest two pessimistic and optimistic
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approaches from both individual and summative points of view. Finally, a numerical real example in the banking system in the countries of the Visegrad Four (i.e. Czech Republic, Hungary, Poland, and Slovakia) is presented to elaborate applicability of the proposed method. Keywords: Data Envelopment Analysis; Directional distance function; Non-radial non-oriented models;
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Mixed integer linear programming; Flexible measure.
Introduction
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Data envelopment analysis (DEA) is a non-parametric approach based on mathematical programming for evaluating the relative performances of many different kinds of entities, commonly referred to as decision making units (DMUs), where the presence of multiple inputs
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and multiple outputs makes comparisons difficult. DEA was originally introduced by Charnes et al. (1978) (CCR model), built on the ideas of Farrell (1957), and extended by Banker et al. (1984) (BCC model). Nowadays, DEA is becoming a very important analysis tool and research
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method in various sciences such as management science, operational research, system engineering, decision analysis, etc. Input- and output-oriented models are two main approaches
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in DEA, which respectively measure the largest radial contraction of inputs and the largest radial expansion of outputs. Actually, the conventional DEA models developed with the assumption that the status of each measure is clearly stated as an input or an output variable. However, in the real world, certain variables, referred to as flexible measures, can play input role for some DMUs and output role for others. For example, “high-value customers” and “deposit” measures in a bank branch, “research income” measure in higher education application (Beasley, 1990), or “uptime” measure in evaluating robotics installations (Cook, Johnston and Mccutcheon, 1992) are considered as a flexible measure which could serve as either an input or an output. Bala and Cook (2003) presented a two-step procedure measurement tool for 1
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evaluating the performance of branches in the banking industry with flexible measures. In the first stage, all variables are assumed to be flexible and a discriminant model is used to designate the input/output status; the second stage performs the DEA analysis based on the variable designations chosen. Cook and Zhu (2007) introduced a method based on a fractional programming problem to determine whether a measure is an input or an output. However, Toloo (2009) claimed that their model may produce incorrect efficiency scores as result of
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introducing a large positive number to the model and introduced a revised mixed integer linear programming (MILP) model with the aim of excluding the large positive number from the model. Amirteimoori and Emrouznejad (2011), Toloo (2012, 2014) declared that one drawback in this method is the requirement to enter extra information to decide about the role of each variable. On the other hand, in the presence of alternative optimal solutions in classifier models, the results of selecting a flexible measure as an input or an output are the same for some DMUs and it is logical that not be taken into account for classifying inputs and outputs. Accordingly,
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Toloo (2012) referred to these cases as share cases and introduced a new classifier model that identifies share cases. In another study, Amirteimoori et al. (2013) developed a flexible slacksbased model to calculate the relative efficiency of DMUs in the presence of flexible measures. In order to allow the contraction in inputs and expansion in outputs, simultaneously, Chambers et al. (1996, 1998) introduced the directional distance function. In this approach, there is no need
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to distinguish between input-oriented and output-oriented approaches. Indeed, the traditional input- and output-oriented models are special cases of this concept. However, unlike the distance function is
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traditional DEA models, the relative efficiency score computed using the directional technology for efficient DMUs and ranges between
and
for inefficient DMUs.
Among the classic DEA models, the non-radial models have a higher discriminating power in
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evaluating the efficiencies of DMUs and seem to be more effective in performance assessment. Considering this feature, the current paper, inspired by the general non-radial directional distance model (DD model), proposes a new non-radial non-oriented classifier method in both
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envelopment and multiplier form in the presence of flexible measures. The suggested classifier directional distance model treats each flexible factor as input, output or both to evaluate individual DMUs from two pessimistic and optimistic viewpoints. Practically, in some cases,
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determining the status of a flexible measure may tie and in order to break the tie, the adopted models are developed from the summative perspective as well. Moreover, the proposed approach will be applied to evaluate banks in the countries of the Visegrad Four (V4) in 2014. V4 is an alliance of four central European states – Czech Republic (CZ), Hungary (HU), Poland (PL), and Slovakia (SK) – for the purposes of furthering their European integration, as well as for advancing military, economic and energy cooperation with one another. The efficiency of the financial system is playing a significant role from the 2
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viewpoints of macroeconomic and microeconomic. The performance of the banking sector from a macroeconomic perspective was influenced by the cost of financial intermediation and the stability of the entire financial system. From a microeconomic point of view, the evaluation of bank operations is important because foreign bank entry to a domestic financial market increases the competitiveness and is necessary to improve an institutional regulation. Evaluating the bank efficiency in the literature distinguishes between two basic approaches, the
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production and intermediation approaches that differ in defining the role of deposits as input or output in a production system of banks. This paper takes the deposits measure into consideration as a flexible measure.
The outline of the paper is organized as follows. In Section 2, we review two classifier methods based on classic DEA models. A new DEA-based classifier approach is introduced in both envelopment and multiplier forms in the presence of flexible measures in Section 3. The penultimate section illustrates the discrimination power of suggested models by a real case future research directions in Section 5.
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Preliminaries
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involving 61 bank branches in V4 countries. Finally, we have come to conclusion and some
This section contains two subsections. Firstly, we briefly review two classifier methods based on classic DEA models deal with flexible measures. Secondly, we provide some preliminary
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concept about DD model that inspires us in developing the suggested method in the rest of the
1.1
Flexible measures
Assume that there are (
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paper.
DMUs
) to produce
each using
semi-positive outputs
semi-positive inputs . The following pair of
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multiplier and envelopment models under constant returns to scale (CRS) assumption can be , respectively:
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applied to assess the radial input-efficiency of ∑
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∑ ∑
(1)
∑
∑ ∑
(2)
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where decision variables
and
are the weight of
is the intensity vector corresponds to
input and
output, respectively, and
. These two models are mutual dual (for more
details see Cooper et al. 2007). Moreover, assume that there are
semi-positive flexible measures
(
) whose
input/output statuses are unknown. To handle such flexible measures, Cook & Zhu (2007) and Toloo (2012) established the following MILP models by modifying models (1) and (2),
∑ ∑ ∑
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respectively: ∑ ∑
∑ ∑
∑
∑
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(3)
(4)
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∑ ∑ ∑ ∑
is a large positive number and the indicator variable
measure
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where
is an input (
) or an output (
designates that flexible
). The output-oriented version of above
models can easily be formulated (see Toloo (2012)). It should be noted that models (3) and (4)
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are not mutual dual. Toloo (2012) also showed that the existence of the alternative optimal solutions of classifier models for some DMUs could cause incorrect results and referred to these
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units as share cases, which must not be taken into account in overall classifying inputs and outputs.
1.2 Directional distance function Chambers et al. (1996, 1998) with the aim of generalizing of the existing distance functions and introducing a non-oriented approach suggested the following generic DD model, which assess under CRS technology:
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∑ ∑
(5)
where the nonzero directional vector (
)
enables the model to
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contract inputs and expand outputs simultaneously. Since data are semi-positive, a usual choice for the directional vector is the observed input and output levels. As a result, taking the direction vector (
)
into account, a well-known special case of model (5) can be
formulated as bellow:
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∑ ∑
(6)
From now on, “*” indicates the optimal values. It can be readily demonstrated that however unlike the traditional DEA models, any positive number between
. Furthermore,
can be
for inefficient DMUs. In other words, the value
is the
. Note that model (6) is an extension of traditional DEA models: The
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inefficiency score of
and
is efficient if
;
input-oriented CCR model (2) can be obtained if ( )
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output-oriented CCR model achieves when (
)
and analogously the .
Model (6) assumes that inputs are contracted and outputs are expanded at the same rate (proportionally) which it indeed signifies a radial measure of efficiency. However, in this study,
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we utilize the following non-radial version of model (6) which enables us to address nonproportionate improvement in inputs and outputs:
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∑
∑
∑ ∑
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(7)
In the formulation above the variables
and
are introduced to allow the non-proportionate
contraction of inputs and the non-proportionate expansion of outputs in the given directional vector
, respectively.
As inspection makes clear, we have score of
input and
and
. Indeed
and
are the inefficiency
output, respectively. Hence, the optimal solution of model (7) for 5
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tells us the amount of employed excess inputs shortfall outputs
and the amount of produced
. This model actually maximizes the sum of all input/output
inefficiencies. Here,
is efficient if
; otherwise it is inefficient in at least one of the components of
input or output vector.
∑ ∑
∑ ∑
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Consider the following dual model of model (7):
(8)
Since this model is non-oriented naturally, the weighted sum of both inputs and outputs are ∑
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included in the objective function which minimizes ∑
.
Models (7) and (8) are in fact envelopment and multiplier forms of non-radial DD models, respectively.
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Proposed non-radial classifier directional distance approach
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The main issue of this study is to facilitate the derivation of the input/output status of flexible measures. To pursue this objective, this section proposes a new classifier method based on non-
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radial DD model which combines both the input and output orientations. As a matter of fact, we adopt such model for reasons like the following: (i) Traditional classifier models are oriented and hence the role of a flexible measure in an
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input-oriented model may differ from an output-oriented model. However, DD models are non-oriented and hence extending classifier DD models accommodate the flexible measures by combining both input- and output-orientations.
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(ii) The special structure of formulations of envelopment and multiplier forms of DD models helps us to drive two classifier models and to decide the status of flexible measures from two different perspectives, i.e. pessimistic and optimistic.
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(iii) The main advantage of non-radial models is that they measure individual input reductions along with individual output expansions in order to provide information on the inefficiency score of each input and output.
In particular, this section develops a pair of non- radial classifier directional distance (CDD hereafter) models in both envelopment and multiplier forms, which introduce two pessimistic and optimistic approaches considering the individual and summative viewpoints. Individual models designate flexible measures in favor of each individual DMUs, meanwhile, summative 6
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models aim at accommodating flexible measures in favor of all DMUs, simultaneously. In other words, individual models are solved
times, one for each unit, while summative models are
applied just once to determine the overall role of flexible measures. We also provide two pessimistic and optimistic approaches for each individual and summative model. The former and latter approaches adjust the flexible measures status in line with maximizing and minimizing the possible distance from frontier in the given direction, respectively.
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2.1 Pessimistic CDD approach
In order to develop a non-radial envelopment DEA model for determining the input/output status of flexible measures, we consider model (7) and the variable measure . There would be two possible cases. If ∑
corresponding to flexible
treats as an input role, then the constraint
must be active; otherwise the constraint ∑
must
hold. A method to model the either-or constraints is introducing the indicator variable designates that
is an input, and
designates it as an output. As a result, to
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where
,
handle the either-or constraints, it is sufficient to impose the following constraints to model (7): ∑ ∑
If
is a large positive number. , then the constraint ∑
is active and the constraint ∑
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where
is weakened to ∑ binding. In this case,
treats as an input. We can analogously conclude that the situation is . Hence, we formulate the following non-radial non-oriented MILP model,
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reversed when
, which is always non-
which classifies flexible measures in the envelopment form: ∑
∑
)
(9)
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∑ ∑ ∑ ∑
(∑
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̅
It should be mentioned here that the direction vector in this model is where
and operator
element. In general, there are
multiples vectors
and
element by
various combinations for the direction vectors and the
following theorem proves that model (9) finds one direction with the maximum objective function value. Let
be the optimal objective value obtained by model (7) for
each combination. 7
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Theorem 1. ̅
{
}
Proof. By contradiction, suppose that ̅ Consider the
combination and let
{
where
}.
be the index of flexible measures, which treat as input.
Under these assumptions, model (7) can be rewritten as follows: ∑
)
∑ ∑ ∑ ∑
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∑
(∑
(10)
is at hand. Clearly,
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Suppose that model (10) is solved and the optimal solution ( (
is a feasible solution of model (9) where {
Corollary 1. If ̅
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However, the objective function value of this feasible solution, i.e. optimal objective function value ̅ , which is a contradiction. ■
, is larger than the
then an identical inefficiency score is achieved when the flexible measure is efficient by model (9) then it
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is considered either as input or output. In other words, if
is also efficient by models (7) and (8), no matter flexible measure is designated as an input or an output.
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In fact, Theorem 1 proves that model (9) obtains the optimal direction with the maximum sum of input/output inefficiencies among all
possible combinations.
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Now is time to decide on overall input/output status of flexible measures for each individual
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DMU. As the simplest method, one criterion to make a decision would be based on the majority choice among the DMUs. For , let ̅ and be the optimal objective value and the optimal values of the indicator variable corresponding to flexible measure l in model (9), respectively. The following criterion is taken into account, mathematically, to handle the flexible measure.
For flexible measure l, assign whole DMUs to the following two groups: { |̅ { |̅
} }
{ |̅
} 8
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Set
includes the index of DMUs that are efficient in model (9). Referencing to Corollary 1,
these DMUs are efficient either the flexible measure plays the role of an input or an output. So, logically, these DMUs must not be considered for classifying inputs and outputs. Flexible measure l should be designated as an output if | | Moreover, ties for determining the status of flexible measure
| | and as an input if | | happen when | |
| |.
| |. In this
case, an alternative criterion is needed to examine the issue from the point of view of the may be
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collection of all DMUs. Hence, ties for determining the status of flexible measure
broken by applying the following summative version of model (9) which carries out the simultaneous optimization the performance measure of all units and evaluates the pessimistic summative inefficiency corresponding each unit: ∑
∑
(∑
∑
)
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∑ ∑ (
)
∑
(
)
where the intensity variables
(
.
Note
),
(
that
and
the
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.
),
(
) for
will end up either as an input, if
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Theorem 2. Let corresponding to
solution (
),
, and in a single stage. As a , or an output, if
be the part of optimal solution ∑ ∑ ) ̅ .
(∑
in model (11) then
̅
Proof: By contradiction, we suppose that
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are corresponding to optimal
for summative model (11) accommodates flexible measure matter of fact, the flexible measure
(11)
where
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for
and variables
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∑
for
is a feasible solution to model (9) when
. It is easy to show that is evaluated. However, the
objective function value of this feasible solution is larger than the optimal objective function value ̅ , which is impossible. ■ 2.2 Optimistic CDD approach Referencing to Theorem 1, the formulated non-radial envelopment CDD model obtains the maximum possible inefficiency among all possible combinations for each DMU. Alternatively, 9
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we formulate non-radial multiplier CDD models, which give the minimum possible inefficiency for each DMU with the aim of increasing the discrimination power, which is needed for discriminating among efficient DMUs. To do this, we establish the following mixed integer nonlinear programming model extending model (8): ∑
(∑ ∑
)
∑
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∑
∑
(12)
Where the decision variable
, the following method can be applied to eliminate the product of a
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linear due to term
is the weight of flexible measure . Although model (12) is non‐
binary and a continuous variable: Let
be a indicator variable, and
be a continuous variable which
continuous variable is introduced to replace the product added to the model for forcing ̅ to take the value of
. Now a non-zero
. The following constraints must be
:
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̅ ̅ ̅ ̅ The validity of these constraints can be checked by examining all following two possible situations: If ̅
If
, then from the constraint ̅ and ̅
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we have ̅
are redundant.
, then the constraint ̅
or equivalently ̅
; subsequently the constraint ̅
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lead to ̅
We, therefore, utilize this method and let ̅
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, and also ̅
is redundant. for
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(12):
. In this case, other constraints
in order to linearize model
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∑
∑
(∑
∑
∑
∑
∑
̅
)
̅
(13)
In this model, if
, then
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̅ ̅ ̅
treats as an input and otherwise it treats as an output. The
following theorem proves that model (13) finds a direction with the minimum objective value possible combinations. Let
obtained by model (8) for the
} {
Proof. Suppose the
combination.
{
Theorem 3.
combination and let
be the optimal objective value
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among all
} and contrary to our claim
. Consider
be the index set of flexible measures which treat as input. Under
∑
(∑
∑
∑
Let (
)
∑
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∑
∑
(14)
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∑
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these assumptions, model (8) can be written as bellow:
be the optimal solution of model (14). Clearly, (
̅ is a feasible solution
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of model (13) where
̅
{
However, the objective function value of the feasible solution objective function value
is smaller than the optimal
, which is a contradiction. ■
The following theorem proves that the optimal objective value for the proposed envelopment CDD model (9) is greater than or equal to the optimal objective value for the proposed multiplier CDD model (13): Theorem 4.
̅ . 11
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Proof: According to Theorems 1 and 2, we have ̅ combination
and
for
. Since, for each combination, models (7) and (8) are mutually dual, it ̅ . which implies
then follows that
Theorem 4 lays emphasis on the fact that the proposed non-radial CDD models (9) and (13) are formulated with pessimistic and optimistic points of view, respectively, and hence if a unit is
Corollary 2. If ̅
, then
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efficient in the envelopment model (9), then it is also efficient in the multiplier model (13). .
It should be noted that the criterion based on the majority choice can be carried out here in order to designate the role (input or output) for each flexible measure by the optimal solution obtained in model (13). In order to break the possible ties, likewise to the aforementioned
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summative envelopment model (11), the following envelopment form is suggested with the aim of evaluating the optimistic summative inefficiency corresponding each unit: ∑ ∑
∑
̅
)
̅
(15)
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̅ ̅ ̅
,
and
represent the
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where
∑
M
∑
∑
(∑
input,
output, and
flexible measure weight for
. In fact, model (15) is an extended version of model (13) where the
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performance measure of all the DMUs is simultaneously considered. The optimal solution flexible measure (
for summative model (15) accommodates
in a single stage where
) for
, and
(
)
. The flexible measure
(
)
is an input, if
and otherwise it is an output. Theorem 4. Let ∑
be the optimal solution and ∑
̅
corresponding to
12
in model (15) then
∑ .
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Proof: By contradiction, we suppose that
for
is a feasible solution to model (10) when
. It is easy to show that is evaluated. However, the
objective function value of this feasible solution is smaller than the optimal objective function value
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, which is impossible. ■
Application
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To demonstrate the applications of proposed CDD models in both optimistic and pessimistic approaches (i.e. envelopment and multiplier forms as well as individual and summative models), we analyze the performance of banking industry in V4 countries. The economies of these countries have in common that have seen some changes after communism's collapse; the transition to a market economy, joining the European Union in May 2004 and especially the transformation of banking system. The transition from centrally planned economy to market economy had been accompanied with restructuring and liberalization of the banking system. It
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had been associated with the privatization of some banks, the entry of foreign-owned banks, deregulation of interest rates and changes in legislation.
Exploring Banking Efficiency is important from both macroeconomic and microeconomic standpoints. From the macroeconomic point of view, the efficiency of the banking sector affects the cost of financial intermediation and the stability of the entire financial system. From the
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microeconomic point of view, banking efficiency is particularly important for improving institutional regulation and supervision, and in particular for improving the competitiveness of banks. Increasing the efficiency of banks leads to a better distribution of financial resources, and
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thus better investment support and economic growth. Analysis of the banking performance considering a collection of inputs and outputs, which are
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clearly specified, discovers some problems in the system. However, the most controversial issue is the role of deposits, whether they should be classified as inputs or inputs. Boda and Zimkova (2015) investigate the bank efficiency in the Slovak banking industry using three the best-known
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approaches, the production, the intermediation, and the value-added ones. The production approach was founded by Benston (1965) and is based on assumption that the aim of banks is to
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produce deposits (liabilities) as well as loans (assets) and other services to customers. This service-oriented approach considers deposits as an output together with loans and the interest income. Hancock (1991) modified the production approach into the user-cost approach, where deposits are specified as both inputs and outputs of the cost/profit function of a bank. The second approach is the intermediation approach and was proposed by Sealey and Lindley (1977). This approach assumes that the main aim of a bank is to produce the intermediation services through the collection of deposits or other liabilities. The main banks are seen as production units, which transmute deposits into loans. They are also interested in the use of interest-earning assets and loans, securities and similar investments. The profit-oriented 13
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approach is a modification of the intermediation approach as the profit-maximizing tendency of the banks. The last value-added approach differs from previous ones in considering all liability and asset categories to have some output characteristics rather than distinguishing inputs from outputs in a mutually exclusive way. The major categories of produced deposits and loans are viewed as important outputs because they form a significant proportion of value added. The current study adopts the intermediation approach for evaluating bank efficiency in the
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commercial banks of V4 countries recorded in 2014. The economic environment has not been dramatically changed since 2014. The most V4 economies are slowly recovering from the global crisis and optimism among investors and consumers is growing. A positive effect on the banking sector was limited for a number of reasons. The effects of regulation are not only direct costs of implementation of regulation but also the changing business environment requesting of actual new information using data mining from disposal data at banks with help of
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technologies innovations.
There are 61 banks including 15 banks from CZ, 14 banks from HU, 23 banks from PL, and 9 banks from SK whose complete name and the country where they are located are provided in Table A.1 (see Appendix) in detail. We employed three groups of variables: three inputs (labor, physical capital, and credit risk ratio), three outputs (loans and advances, investments, and non-
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interest income), and one flexible measure (deposit), which are summarized in Table 1. =====[Please insert Table 1 around here]======
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The annual balance sheet and income statement data have been collected from the world banking information source Bankscope2. The data for bank branches are listed in Table 2. Mean values of inputs, outputs, and deposits along with the number of banks corresponding each
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country are summarized in Table A.2. in Appendix. =====[Please insert Table 2 around here]======
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In order to make a comparison between input- and output-oriented classifier models of Cook & Zhu (2007) and Toloo (2012), we apply these models to the data set. Table 3 reports the results.
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As can be seen, different assignations for the flexible measure are obtained through input- and output-oriented models of Cook & Zhu (2007) for banks
. To cope with this issue,
we now apply the proposed non-radial non-oriented CDD models by which the flexible measures are accommodated without the need for distinguishing between input- and outputorientation. 2
Bankscope combines comprehensive financial statements with a wide range of other banking intelligence including ratings, an analysis model, bank structures, news, Anti-Money-Laundering (AML) documentation and banking research. The website has information on 32000 banks and is the definitive tool for bank research and analysis. We refer the readers for more details visit https://bankscope.bvdinfo.com/.
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=====[Please insert Table 3 around here]====== Table 4 provides the obtained result from both suggested individual models (9) and (13). To be more specific, the calculated input/output inefficiencies through model (9) are also reported in Table 5. =====[Please insert Tables 4 and 5 around here]====== of Table 4 show the value of
as the sum of all
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The second and third columns
input/output inefficiencies for each DMU when deposit , i.e. , is considered as an input and an output, respectively (the columns labeled “
(input)” and “
(output)”). The optimal objective
function value of proposed individual models (9) and (13) along with the related optimal shown in the last four columns. We note that equalities ̅ =max and =min
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assured by Theorem 1 and Theorem 3, respectively.
are are
As can be extracted from the obtained results, models (9) and (13) identify 12 and 17 efficient banks, respectively. Moreover, 12 efficient DMUs are identified by both models which follows from Corollary 2. Essentially, more efficient units introduced by model (13) compared to model (9) which reveals the optimistic/pessimistic viewpoint of these models. For these 12 banks (six financial institutions from the Czech Republic and six banks from Poland), i.e.
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, we have
that means these
banks are efficient no matter the deposit measure is designated as an input or an output. Therefore, these banks are not taken into account in the overall decision on the role of deposit as
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input or output. Large financial institutions such as ČSOB, Česká spořitelna, private PPF, Komerční banka, Raiffeisenbank and the Czech-Moravian Guarantee and Development Bank are in a group of efficient banks in the Czech Republic. Moreover, in Poland, the efficient group
PT
consists large banks (i.e. Bank Zachodni WBK, ING Bank Slaski and Bank Handlowy w Warszawie) and also some small special banks (SGB Bank, RBS Bank, and Bank of Tokyo).
CE
Boxplot is a convenient way to graphically depict groups of data through their quartiles. Figure 1 shows boxplot for ̅ =max and =min . In terms of inefficiency comparison by model (9), the best banks and financial institutions for year 2014 are in the Czech Republic
AC
(median 3.57). They are also followed by Hungarian banks (27.39), Polish banks (41.91) and last are banks in Slovakia (60.02). A similar analysis for model (13) testifies that the best situation is in the Czech Banks (median is 0.000), followed by Hungarian banks (5.91), Polish banks (13.78) and least effective banks are in Slovakia (24.77). The obtained results of the comparison of banking inefficiency in the EU countries using the proposed pessimistic and optimistic models are consistent with empirical studies and developments in financial systems in these countries, which have been affected by the global financial crisis since 2008. The Czech banking system is one of the most resilient systems in the 15
ACCEPTED MANUSCRIPT
EU due to the high liquidity of the sector, which is dominated by deposits from clients and is less dependent on interbank market financing. The Hungarian banking system was one of the smallest banking system in the EU with a large bank concentration, but with the development of this system, new types of credit and financial institutions were entering the market. The macroeconomic function of the transfer of deposits into loans and the microeconomic function of providing a complex of banking services were the most successful in the Czech and
CR IP T
Hungarian banks. The second group consists of Polish and Slovak banks. Poland has the largest banking industry out of the Visegrad countries and this banking system is focused on domestic business and plays an important role in financing private households, SMEs, big infrastructure projects. Polish banking sector is also owned by foreign-owned institution. The main reason for the low-efficiency of Polish banks is the loan quality and not a well-developed payment system. The high level of current earnings means that companies had no need to take out new loans, being able to fund investment and going operations internally. The size of Slovak banking
AN US
system is rather small and influenced by largest banks. The most of the banks have the
AC
CE
PT
ED
M
universal banking license.
Figure 1. Boxplot for (Ineff_M9) = ̅ =max
and (Ineff_M13) =
=min
As can be seen from the results in Table 5, the variations of inefficiency are significant. Accordingly, for the sake of illustration, we only consider boxplot with the inefficiency axis in the range 0-250 in Figure 1 to be more readable. On the one hand, through model (9), ignoring 12 share cases and based on the majority choice among 49 banks, all these banks treat the flexible measure as an output and hence deposit 16
ACCEPTED MANUSCRIPT
should be considered as an output with a pessimistic standpoint. On the other hand, applying model (13), in all 49 banks the flexible measure plays an input role and hence deposit should be considered as an input with an optimistic point of view. All in all, a profit-oriented banking system is desirable from the pessimistic standpoint; meanwhile, the optimistic viewpoint follows a service-oriented banking system. Returning to Table 4, we observe that there are some banks possess very high inefficiency score
CR IP T
from the pessimistic standpoint whereas these banks have better performance from the optimistic view. This significant variance is logical since, following from Theorems 1 and 3, the pessimistic model (9) finds a specific combination (among
various combinations) which
maximizes the sum of inefficiency scores while the optimistic model (13) introduces the combination with minimum sum of inefficiency scores. Moreover, some banks have recived high inefficiency score from both optimistic and pessimistic views. In order to find the main
AN US
sources of inefficiency, one can refer to Table 5 which provides useful information as a management guide to improve efficiency in performance of each inefficient bank. For example, Bank29 (Post Bank, SK) is a bank with high inefficiency score from both optimistic and pessimistic perspectives and referencing to Table 5 reveals that its first output (loans and advances) is the main source of inefficiency. In fact, the value of the first output of this bank is
M
400 which is significantly low compared to the values of other its input/ouput measures. Although there is no tie for determining the status of deposit in this example, we apply the summative models as well in order to see the results from the summative inefficiency
ED
perspective. After solving models (11) and (15), ( , the optimal objective function value and
)
and (
)
, are obtained, respectively. By comparing
the obtained results through both individual and summative envelopment (or multiplier) forms
PT
for this case study, deposit ends up the same role. However, for 61 banks, the final results are obtained by running 122 individual models (envelopment ant multiplier form), whereas,
CE
summative inefficiency needs to solve only two models for any number of DMU. It should be noted that there is no guarantee that the same results are always achieved through individual and summative version, as mentioned in Cook & Zhu (2007), because the summative
AC
approach might be overly sensitive to extreme DMUs, or possibly to the larger DMUs.
4
Conclusion
One of the controversial issues in many applications in DEA literature is the existence of flexible measure which can serve as either an input or output. Some studies have been done to classify inputs and outputs in DEA. This paper deployed a new classifier method based on directional distance function. As the main feature of this study, we developed a pair of non-radial CDD 17
ACCEPTED MANUSCRIPT
models in both envelopment and multiplier forms which introduce two pessimistic and optimistic approaches from both individual and summative viewpoints. Individual approach accommodates the flexible measure in favor of each DMU; nevertheless, in order to break the possible ties for classifying inputs and outputs, we have extended summative approach which designates the flexible measures in favor of all DMUs, simultaneously. Finally, to validate the discrimination power of the new procedure, an empirical study on the banking industry in V4
CR IP T
countries has been presented in which the deposit factor is taken into consideration as the flexible measure. It is found that deposit is treated as an output and as an input from the pessimistic and optimistic viewpoints, respectively, by both individual and summative models. Finally, extending the classifier models in the presence of imprecise, negative, or stochastic data can be considered as some interesting future research directions. Moreover, dealing with Malmquist index in the presence of selective measures is an alternative research direction.
AN US
Estimating the state of returns to scale in the company of flexible measure under variable returns to scale technology also could be a motivating topic for further study. Acknowledgements The
research
was
supported
by
the
European
Social
Fund
within
the
project
M
CZ.1.07/2.3.00/20.0296, the Czech Science Foundation through project No. 16-17810S as well as the SGS Project No. SP2017/141 VŠB-Technical University of Ostrava. All support is greatly
ED
acknowledged. References
PT
Amirteimoori, A., & Emrouznejad, A. (2011). Flexible measures in production process: A DEAbased approach. RAIRO - Operations Research, 45(1), 63–74.
CE
Amirteimoori, A., Emrouznejad, A., & Khoshandam, L. (2013). Classifying flexible measures in data envelopment analysis: A slack-based measure. Measurement, 46(10), 4100–4107.
AC
Bala, K., & Cook, W. D. (2003). Performance measurement with classification information: An enhanced additive DEA model. Omega, 31, 439–450. Banker, R. D., Charnes, A., & and Cooper, W. W. (1984). Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis. Management Science, 30(9), 1078– 1092. Beasley, J. E. (1990). Comparing university departments, 18(2), 171–183. Benston, G. J. (1965). Branch banking and economies of scale. The Journal of Finance, 20(2), 312– 331.
18
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Boda, M., & Zimková, E. (2015). Efficiency in the Slovak Banking Industry: A Comparison of Three Approaches. Prague Economic Papers, 2015(4), 434–451. Chambers, R. G., Chung, Y., & Färe, R. (1996). Benefit and Distance Functions. Journal of Economic Theory, 70(2), 407–419.
CR IP T
Chambers, R. G., Chung, Y., & Färe, R. (1998). Profit, Directional Distance Functions, and Nerlovian Efficiency. Journal of Optimization Theory and Applications, 98(2), 351–364. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444. Cook, W. D., Johnston, D. A., & Mccutcheon, D. (1992). Implementations of Robotics Identifying Efficient Implementers. Omega-International Journal of Management Science, 20(2), 227–239.
AN US
Cook, W. D., & Zhu, J. (2007). Classifying inputs and outputs in data envelopment analysis. European Journal of Operational Research, 180(2), 692–699. Cooper, W. W., Seiford, L. M., & Tone, K. (2007). Data envelopment analysis: A comprehensive text with models, applications, references and DEA-solver software (2nd edi.). Springer US.
M
Farrell, M. J. (1957). The Measurement of Productive Efficiency. Journal of the Royal Statistical Society. Series A (General), 120(3), 253.
ED
Hancock, D. (1991). A Theory of Production for the Financial Firm. Dordrecht: Kluwer Academic Publishers. Sealey, C. W., & Lindley, J. T. (1977). Inputs, outputs, and A theory of prouduction and cost at depository financial institutions. The Journal of Finance, 32(4), 1251–1266.
PT
Toloo, M. (2009). On classifying inputs and outputs in DEA: A revised model. European Journal of Operational Research (Vol. 198).
CE
Toloo, M. (2012). Alternative solutions for classifying inputs and outputs in data envelopment analysis. Computers & Mathematics with Applications, 63(6), 1104–1110.
AC
Toloo, M. (2014). Notes on classifying inputs and outputs in data envelopment analysis : a comment. European Journal of Operational Research, 198(2009), 1625–1628.
19
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APPENDIX Table A.1. Commercial banks of V4 banking sector subject to the analysis Country
Bank name
DMUs
Ceskoslovenska Obchodni Banka-
32
CZ
Ceska Sporitelna
33
PL
OTP Bank Plc
34
1
CZ
2
CZ
3
HU
4
PL
5
CZ
Komercni Banka
36
6
PL
Bank Zachodni WBK
37
7
PL
mBank
38
8
PL
ING Bank Slaski-Capital Group
9
CZ
10
PL
Bank Polska Kasa Opieki SA-Bank Pekao
35
PL
HU CZ
Unicredit Bank Czech Republic and Slovakia Getin Noble Bank
Czech Moravian Guarantee and Develpoment Bank Santander Consumer Bank SGB Bank
Budapest Bank Nyrt-Budapest Hitel-és Fejleszési Bank Nyrt Modra pyramida stavebni sporitelna as Raiffeisen
PL
Euro Bank
39
SK
Sberbank Slovensko
40
SK
Prima banka Slovensko
41
CZ
Air Bank
42
Bank Millennium
Bank name
CZ
AN US
PL
CSOB
M
11
Country
CR IP T
DMUs
HU
Bank of Hungarian Savings Cooperatives LimitedTAKAREKBANK
SK
Vseobecna Uverova Banka
43
PL
Idea Bank
13
PL
Bank Handlowy w Warszawie
44
HU
FHB Kereskedelmi Bank Zrt
14
SK
Tatra Banka
45
PL
Bank Pocztowy
ED
12
HU
K&H Bank Zrt
46
SK
OTP Banka Slovensko
16
PL
Bank BPH
47
PL
MBank Hipoteczny
17
PL
Nordea Bank Polska
48
CZ
Expobank CZ
18
HU
Erste Bank Hungary Nyrt
49
PL
HSBC Bank Polska
19
HU
MKB Bank Zrt
50
PL
RBS Bank (Polska)
51
HU
KDB Bank Europe
21
Ceskoslovenska obchodna banka-
CE
20
PT
15
SK
CSOB
HU
Raiffeisen Bank Zrt
52
PL
FM Bank PBP
HU
CIB Bank Ltd-CIB Bank Zrt
53
CZ
Equa Bank
23 24
PL
Alior Bank Spólka Akcyjna
54
SK
Privatbanka
HU
UniCredit Bank Hungary Zrt
55
PL
Pekao Bank Hipoteczny
25
PL
BNP Paribas Bank Polska
56
HU
Bank of China (Hungary)
26
CZ
GE Money Bank
57
HU
27
PL
Bank Ochrony Srodowiska
58
PL
28
CZ
J&T Banka
59
HU
AC
22
20
MagNet Hungarian Community Bank Bank of Tokyo - Mitsubishi UFJ (Polska) Magyar Cetelem Bank Rt
ACCEPTED MANUSCRIPT
29
SK
Post Bank JSC-Postova Banka
60
SK
CSOB Stavebna Sporitelna
30
CZ
PPF banka
61
CZ
Evropsko-ruska banka
31
CZ
Stavební Sporitelna Ceské Sporitelny
Table A.2. Mean value of data for each country in 2014 Country
banks
Inputs PE
FA
Flexible CRR
DEP
15
75519
84461
49.43
8340805
HU
14
92061
112030
53.64
4773693
PL
23
94523
58398
65.12
6806943
SK
9
43178
41233
61.91
3311167
V4 group
61
81709
74583
58.16
6201704
LA
M ED PT CE AC
21
SEC
NEA
1542066
2430671
597703
318278
1542982
496632
308651
1928066
573831
149533
1039011
216800
590683
1832105
509307
AN US
CZ
Output
CR IP T
No. of
ACCEPTED MANUSCRIPT
TABLES Table 1. Description of three groups of variables Measures
Name
Description
Input variables =PE
Total personnel expenses
Physical capital
=FA
Fixed assets
Credit risk
=CR
Net loans/Total assets
=DEP
Deposit and short-term funding
Loans and Advances
=LA
Loans and advances to banks
Investments
=SEC
Other securities
Non-interest income
=NEA
Non-earning assets
CR IP T
Labor
Flexible Variable Deposit
AN US
Output variables
Table 2. Data for the 61 commercial banks in V4 countries in 2014 256231
283822
49.21
2
328507
516325
50.5
3
686796
879261
60.16
4
447771
382676
5
262864
286920
6
331134
7
194588
8
223067
1038336
9760384
3328338
26480895
919459
8683189
3083983
63.94
30924048
2438522
8512511
1878279
54.74
25475452
4587331
6096395
2016568
152290
64.18
20419727
542477
5651880
2473068
170827
65.41
19595973
835646
6284417
827197
147307
55.97
17622988
444088
5920254
2047469
94073
69871
62.4
12979049
2405106
3159656
448750
91023
77806
75.38
13336015
332169
2222990
1043667
131683
39288
73.25
11518691
424106
2088996
966800
CE
11
10033454
2746297
PT
10
6977311
29304187
M
1
9
31888107
ED
DMU
12
107800
122100
65.54
8831600
771700
2806200
252900
13
152122
92587
33.55
7931612
852185
4780990
683453
14
93100
67.41
7579000
151600
1937000
785500
97719
141144
45.8
6598928
263473
3423997
682015
16
140494
77325
68.77
3292296
118130
1617585
614698
17
55418
24363
82.45
7040428
192445
562362
584028
18
88419
36216
60.03
6622321
444410
2152468
377061
19
70382
205181
64.23
4884371
330247
785638
977851
20
66600
71800
67.58
4870600
42200
1682800
173900
21
86637
37672
66.12
5075290
93241
1238500
608247
22
78387
75933
67.73
4962492
429687
944721
510011
AC
110300
15
22
110811
51783
76.94
5140868
61195
816315
428223
24
54822
89401
51.45
5004387
572462
1949398
87976
25
67093
30261
78.53
3340130
19066
627818
391222
26
80332
26972
72.4
3418511
40603
832440
454181
27
42803
28094
65.45
2947753
36688
1131536
297671
28
25441
7691
45.92
3313358
471603
1297045
360764
29
43300
21700
48.8
3331200
400
1536900
406200
30
8456
1531
30.12
3174855
925601
936682
755753
31
6816
14069
36.8
3450439
1204393
975134
92214
32
7946
5030
14.5
3258977
2328455
698347
20338
33
45475
16226
80.45
2500825
131370
358193
155612
34
15070
2504
38.49
2806127
145116
1502488
231758
35
74221
54960
62.58
2408961
197956
733082
152612
36
10657
13865
51.95
2663425
1030343
324648
69838
37
8310
2078
47.8
2760803
55037
1407155
94182
38
41792
17261
76.26
2373643
224945
264787
132887
39
19900
19400
70.28
1589000
225100
167200
185600
40
15900
20200
62.1
1639000
104600
516500
78400
41
12720
8675
44.45
1661198
70345
782980
146121
42
16346
21817
24.23
1694606
385220
791736
175860
43
18513
16463
65.62
7051
475974
103701
44
11898
14269
36.23
1576177
476847
590476
56752
45
22244
10761
68.48
1590237
8739
443438
96921
46
16500
21400
74.33
1220300
34100
254100
55100
14681
49
10087
50
AN US
M
2191
84.58
394712
5368
145839
20992
23228
75.36
973367
185857
55460
12965
1180
46.51
758177
34883
340402
88471
PT
5489
48
1472071
5754
361
39.2
614265
66203
317845
74219
6697
1634
52.71
659852
139002
36815
186582
16502
CE
47
CR IP T
23
ED
ACCEPTED MANUSCRIPT
51
1305
57.2
576290
7697
226439
47018
53
12746
3470
69.92
596810
57705
70709
67101
54
5400
1300
33.13
547500
11100
387300
10900
55
2961
120
92.93
114783
10761
16346
1926
56
3221
8846
35.32
311966
170943
63159
13390
57
3729
978
36.65
354212
13818
206997
28652
58
2143
169
44.34
248031
160114
36111
8835
59
9573
1115
87.75
197242
19134
1567
11860
60
2900
100
67.98
192300
5000
63100
2700
61
3007
3364
35.37
193540
45006
29585
60096
AC
52
23
ACCEPTED MANUSCRIPT
Table 3. Input-oriented v.s. output-oriented classifier models (Input-oriented)
Cook & Zhu
Toloo
0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0
0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0
0 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
ED
20 21 22
PT
23
AC
CE
25 26 27 28 29 30 31 32 33 34 35 36 37
AN US
2
CR IP T
Toloo
1
24
(Output-oriented)
Cook & Zhu
M
DMU
24
ACCEPTED MANUSCRIPT
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
AC
CE
PT
ED
61
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 1
CR IP T
40
1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 1
AN US
39
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0
M
38
25
ACCEPTED MANUSCRIPT
Table 4. Results and input/output status of deposit for each bank Model (9)
Model (7) (input)
(output)
Model (13)
̅ =max
=min
1
0
0
0
0
2
0
0
0
0
3
3.79
6.03
6.03
1
4
2.78
2.78
2.78
0
5
0.59
0
0.59
0
6
0
0
0
0
7
8.85
10.83
10.83
1
0
0
0
0
9
3.57
3.57
10
22.79
25.67
11
16.35
18.36
12
9.17
15.34
0
0
14
27.33
60.02
15
4.53
25.03
16
0
17 18 19
7.23
20 21
3.79
0
2.78
0
0
1
0
1
8.85
0
0
1
0
3.52
0
25.67
1
22.79
0
18.36
1
16.35
0
15.34
1
9.17
0
0
0
0
0
60.02
1
27.33
0
25.03
1
4.53
0
86.24
86.24
1
0
0
23.85
61.74
61.74
1
23.85
0
7.69
20
20
1
7.69
0
26.91
26.91
1
7.23
0
59.16
258.68
258.68
1
59.16
0
28.83
91.58
91.58
1
28.83
0
ED
PT
22
0
3.57
M
13
1
0
AN US
8
0
CR IP T
DMU
7.81
27.87
27.87
1
7.81
0
50.09
190.01
190.01
1
50.09
0
12.25
20.01
20.01
1
12.25
0
25
91.74
596.76
596.76
1
91.74
0
26
42.19
248.19
248.19
1
42.19
0
27
30.08
265.43
265.43
1
30.08
0
28
2.41
9.73
9.73
1
2.41
0
29
1975.06
15484.94
15484.94
1
1975.06
0
30
0
0
0
0
0
0
31
0
0
0
0
0
1
32
0
0
0
0
0
0
33
13.78
76.62
76.62
1
13.78
0
34
0
0
0
0
0
0
35
7.8
60.03
60.03
1
7.8
0
23
AC
CE
24
26
ACCEPTED MANUSCRIPT
9.14
17.04
17.04
1
9.14
0
37
0
0
0
0
0
0
38
9.46
56.58
56.58
1
9.46
0
39
6.58
38.25
38.25
1
6.58
0
40
9.44
48.51
48.51
1
9.44
0
41
5.54
51.73
51.73
1
5.54
0
42
0.83
10.47
10.47
1
0.83
0
43
102.59
758.9
758.9
1
102.59
0
44
4.58
19.73
19.73
1
4.58
0
45
102.42
605.33
605.33
1
102.42
0
46
24.77
150.37
150.37
1
24.77
0
47
33.55
220.96
220.96
1
33.55
0
48
24.26
142.03
142.03
1
24.26
0
49
5.51
29.24
29.24
1
5.51
0
50
0
0
0
0
0
0
51
0
30.74
30.74
1
0
0
52
28.51
125.13
125.13
1
28.51
0
53
8.15
69.73
69.73
1
8.15
0
54
0
105.35
105.35
1
0
0
55
18.79
41.91
41.91
1
18.79
0
56
4.15
30.7
30.7
1
4.15
0
57
2.9
50.65
50.65
1
2.9
0
58
0
0
0
0
0
0
59
78.81
565.52
565.52
1
78.81
0
60
25.62
28.82
28.82
1
25.62
0
61
0
31.13
31.13
1
0
0
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36
27
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Table 5. The calculated input/output inefficiencies DMU 0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
3
0.47
0.4
0
0.37
4.43
4
0.47
0.15
0
0.09
2.07
5
0.31
0.05
0
0.21
0
6
0
0
0
0
0
7
0.25
0
0
0.19
10.36
8
0
0
0
0
0
0.44
0.38
0
0.23
0
0
11
0.65
0
0
12
0
0.13
0
13
0
0
0
14
0.29
0
0
15
0
0.12
0
16
0.51
0
0
17
0.26
0
18
0.53
0
19
0
0.59
20
0.16
21
0.5
22
0.13
0.2
0.12
0.36
0
0
0
0.02
0
0
0
0.03
0
0
0
0
2.43
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9 10
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1
0.22
24.38
0.84
0
0.19
16.91
0.6
0
1.41
12.26
0.98
0.57
0
0
0
57.39
1.14
0
1.34
23.34
0.23
0
4.06
80.19
1.48
0
1.31
54.65
5.51
0
0
1
18.07
0.4
0
0
1.98
20.56
3.77
0
0.37
0
2.47
254.37
1.31
0
0
0
1.68
87.83
1.58
0
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0
1.2
ED
0
0
2.39
22.12
3.23
0
0
0
2.41
183.22
3.9
0
0
0.23
0
0.7
2.03
0.36
16.68
0.34
0
0
3.98
587.54
4.89
0
26
0.51
0
0
3.36
241.3
3.02
0
27
0.1
0
0
3.81
259.74
1.78
0
28
0.3
0
0
1.11
6.97
0.34
1.01
29
0.34
0
0
1.95
15482.17
0.48
0
30
0
0
0
0
0
0
0
31
0
0
0
0
0
0
0
32
0
0
0
0
0
0
0
33
0.26
0
0
4.35
58.34
7.94
5.73
34
0
0
0
0
0
0
0
35
0.15
0
0
5.92
49.18
4.41
0.37
36
0
0.86
0.27
0.5
0.13
2.64
12.64
24
AC
CE
25
PT
0
0.49
23
28
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0
0
0
0
0
0
0
38
0.2
0
0
4.66
35.52
10.87
5.34
39
0
0.39
0.44
4.11
23.64
9.67
0
40
0
0.51
0.52
2.97
42.78
1.72
0
41
0
0.14
0.43
2.12
48.57
0.47
0
42
0
0.35
0
2.07
7.53
0.51
0
43
0
0.31
0.47
4.18
751.5
2.45
0
44
0
0.67
0
1.51
1.35
0.98
15.22
45
0
0
0.22
4.58
590.58
3.79
6.17
46
0
0.52
0.59
4.55
139.99
4.72
0
47
0
0
0.82
4.45
201.51
2.75
11.42
48
0
0.89
0.31
4.66
7.65
28.32
100.2
49
0.35
0
0.5
2.23
19.45
1.12
5.58
50
0
0
0
0
0
0
0
51
0
0.04
0.33
2.46
0.37
27.55
0
52
0.51
0
0
3.56
107.29
2.3
11.48
53
0
0
0.41
7.17
31.36
18.14
12.65
54
0
0
0.46
2.75
64.04
0.5
37.6
55
0.78
0
0.97
1.17
5.74
3.49
29.76
56
0
0.93
0.68
2.88
1.06
4.65
20.5
57
0
0
0.67
3.02
58
0
0
59
0
60
0.78
61
0
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37
0.92
8.6
0
0
0
0
0
0.42
10.1
27.01
490.54
37.45
0
0.96
0.06
10.49
0
16.52
5.28
16.12
8.4
0
ED
M
37.44
0
0.82
AC
CE
PT
0.52
29
A non-radial directional distance method on classifying inputs and outputs in DEA: Application to banking industry Mehdi Toloo , Maryam Allahyar , Jana Hanˇclova PII: DOI: Reference:
S0957-4174(17)30638-3 10.1016/j.eswa.2017.09.034 ESWA 11555
To appear in:
Expert Systems With Applications
Received date: Revised date: Accepted date:
23 December 2016 14 May 2017 12 September 2017
Please cite this article as: Mehdi Toloo , Maryam Allahyar , Jana Hanˇclova , A non-radial directional distance method on classifying inputs and outputs in DEA: Application to banking industry, Expert Systems With Applications (2017), doi: 10.1016/j.eswa.2017.09.034
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Highlights A non-radial nor-oriented method is developed to deal with flexible measures.
Two optimistic and pessimistic approaches are proposed.
Each approach contains two individual and integrated models.
A case study of 61 banks in the Visegrad Four region validates the new models.
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A non-radial directional distance method on classifying inputs and outputs in DEA: Application to banking industry
Mehdi Toloo1
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Department of Systems Engineering, Technical University of Ostrava, Sokolska tř. 33, 702 00 Ostrava 1, Ostrava, Czech Republic E-mail: [email protected]
URL: http://homel.vsb.cz/~tol0013/
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Tel: (+420) 792 272 272
Maryam Allahyar
Young Researchers and Elite Club, Yadegar-e-Imam Khomeini (RAH)Shahre Rey Branch, Islamic Azad University, Tehran, Iran
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Email: [email protected]
Jana Hančlova
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Department of Systems Engineering, Technical University of Ostrava, Sokolska tř. 33, 702 00 Ostrava 1, Ostrava, Czech Republic
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Email: [email protected]
1
Corresponding Author
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Abstract The original Data Envelopment Analysis (DEA) models have required an assumption that the
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status of all inputs and outputs be known exactly, whilst we may face a case with some flexible performance measures whose status is unknown. Some classifier approaches have been proposed in order to deal with flexible measures. This contribution develops a new classifier non-radial directional distance method with the aim of taking into account input contraction and output expansion, simultaneously, in the presence of flexible measures. To make the most appropriate decision for flexible measures, we suggest two pessimistic and optimistic
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approaches from both individual and summative points of view. Finally, a numerical real example in the banking system in the countries of the Visegrad Four (i.e. Czech Republic, Hungary, Poland, and Slovakia) is presented to elaborate applicability of the proposed method. Keywords: Data Envelopment Analysis; Directional distance function; Non-radial non-oriented models;
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Mixed integer linear programming; Flexible measure.
Introduction
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Data envelopment analysis (DEA) is a non-parametric approach based on mathematical programming for evaluating the relative performances of many different kinds of entities, commonly referred to as decision making units (DMUs), where the presence of multiple inputs
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and multiple outputs makes comparisons difficult. DEA was originally introduced by Charnes et al. (1978) (CCR model), built on the ideas of Farrell (1957), and extended by Banker et al. (1984) (BCC model). Nowadays, DEA is becoming a very important analysis tool and research
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method in various sciences such as management science, operational research, system engineering, decision analysis, etc. Input- and output-oriented models are two main approaches
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in DEA, which respectively measure the largest radial contraction of inputs and the largest radial expansion of outputs. Actually, the conventional DEA models developed with the assumption that the status of each measure is clearly stated as an input or an output variable. However, in the real world, certain variables, referred to as flexible measures, can play input role for some DMUs and output role for others. For example, “high-value customers” and “deposit” measures in a bank branch, “research income” measure in higher education application (Beasley, 1990), or “uptime” measure in evaluating robotics installations (Cook, Johnston and Mccutcheon, 1992) are considered as a flexible measure which could serve as either an input or an output. Bala and Cook (2003) presented a two-step procedure measurement tool for 1
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evaluating the performance of branches in the banking industry with flexible measures. In the first stage, all variables are assumed to be flexible and a discriminant model is used to designate the input/output status; the second stage performs the DEA analysis based on the variable designations chosen. Cook and Zhu (2007) introduced a method based on a fractional programming problem to determine whether a measure is an input or an output. However, Toloo (2009) claimed that their model may produce incorrect efficiency scores as result of
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introducing a large positive number to the model and introduced a revised mixed integer linear programming (MILP) model with the aim of excluding the large positive number from the model. Amirteimoori and Emrouznejad (2011), Toloo (2012, 2014) declared that one drawback in this method is the requirement to enter extra information to decide about the role of each variable. On the other hand, in the presence of alternative optimal solutions in classifier models, the results of selecting a flexible measure as an input or an output are the same for some DMUs and it is logical that not be taken into account for classifying inputs and outputs. Accordingly,
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Toloo (2012) referred to these cases as share cases and introduced a new classifier model that identifies share cases. In another study, Amirteimoori et al. (2013) developed a flexible slacksbased model to calculate the relative efficiency of DMUs in the presence of flexible measures. In order to allow the contraction in inputs and expansion in outputs, simultaneously, Chambers et al. (1996, 1998) introduced the directional distance function. In this approach, there is no need
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to distinguish between input-oriented and output-oriented approaches. Indeed, the traditional input- and output-oriented models are special cases of this concept. However, unlike the distance function is
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traditional DEA models, the relative efficiency score computed using the directional technology for efficient DMUs and ranges between
and
for inefficient DMUs.
Among the classic DEA models, the non-radial models have a higher discriminating power in
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evaluating the efficiencies of DMUs and seem to be more effective in performance assessment. Considering this feature, the current paper, inspired by the general non-radial directional distance model (DD model), proposes a new non-radial non-oriented classifier method in both
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envelopment and multiplier form in the presence of flexible measures. The suggested classifier directional distance model treats each flexible factor as input, output or both to evaluate individual DMUs from two pessimistic and optimistic viewpoints. Practically, in some cases,
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determining the status of a flexible measure may tie and in order to break the tie, the adopted models are developed from the summative perspective as well. Moreover, the proposed approach will be applied to evaluate banks in the countries of the Visegrad Four (V4) in 2014. V4 is an alliance of four central European states – Czech Republic (CZ), Hungary (HU), Poland (PL), and Slovakia (SK) – for the purposes of furthering their European integration, as well as for advancing military, economic and energy cooperation with one another. The efficiency of the financial system is playing a significant role from the 2
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viewpoints of macroeconomic and microeconomic. The performance of the banking sector from a macroeconomic perspective was influenced by the cost of financial intermediation and the stability of the entire financial system. From a microeconomic point of view, the evaluation of bank operations is important because foreign bank entry to a domestic financial market increases the competitiveness and is necessary to improve an institutional regulation. Evaluating the bank efficiency in the literature distinguishes between two basic approaches, the
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production and intermediation approaches that differ in defining the role of deposits as input or output in a production system of banks. This paper takes the deposits measure into consideration as a flexible measure.
The outline of the paper is organized as follows. In Section 2, we review two classifier methods based on classic DEA models. A new DEA-based classifier approach is introduced in both envelopment and multiplier forms in the presence of flexible measures in Section 3. The penultimate section illustrates the discrimination power of suggested models by a real case future research directions in Section 5.
1
Preliminaries
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involving 61 bank branches in V4 countries. Finally, we have come to conclusion and some
This section contains two subsections. Firstly, we briefly review two classifier methods based on classic DEA models deal with flexible measures. Secondly, we provide some preliminary
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concept about DD model that inspires us in developing the suggested method in the rest of the
1.1
Flexible measures
Assume that there are (
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paper.
DMUs
) to produce
each using
semi-positive outputs
semi-positive inputs . The following pair of
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multiplier and envelopment models under constant returns to scale (CRS) assumption can be , respectively:
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applied to assess the radial input-efficiency of ∑
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∑ ∑
(1)
∑
∑ ∑
(2)
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where decision variables
and
are the weight of
is the intensity vector corresponds to
input and
output, respectively, and
. These two models are mutual dual (for more
details see Cooper et al. 2007). Moreover, assume that there are
semi-positive flexible measures
(
) whose
input/output statuses are unknown. To handle such flexible measures, Cook & Zhu (2007) and Toloo (2012) established the following MILP models by modifying models (1) and (2),
∑ ∑ ∑
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respectively: ∑ ∑
∑ ∑
∑
∑
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(3)
(4)
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∑ ∑ ∑ ∑
is a large positive number and the indicator variable
measure
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where
is an input (
) or an output (
designates that flexible
). The output-oriented version of above
models can easily be formulated (see Toloo (2012)). It should be noted that models (3) and (4)
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are not mutual dual. Toloo (2012) also showed that the existence of the alternative optimal solutions of classifier models for some DMUs could cause incorrect results and referred to these
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units as share cases, which must not be taken into account in overall classifying inputs and outputs.
1.2 Directional distance function Chambers et al. (1996, 1998) with the aim of generalizing of the existing distance functions and introducing a non-oriented approach suggested the following generic DD model, which assess under CRS technology:
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∑ ∑
(5)
where the nonzero directional vector (
)
enables the model to
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contract inputs and expand outputs simultaneously. Since data are semi-positive, a usual choice for the directional vector is the observed input and output levels. As a result, taking the direction vector (
)
into account, a well-known special case of model (5) can be
formulated as bellow:
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∑ ∑
(6)
From now on, “*” indicates the optimal values. It can be readily demonstrated that however unlike the traditional DEA models, any positive number between
. Furthermore,
can be
for inefficient DMUs. In other words, the value
is the
. Note that model (6) is an extension of traditional DEA models: The
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inefficiency score of
and
is efficient if
;
input-oriented CCR model (2) can be obtained if ( )
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output-oriented CCR model achieves when (
)
and analogously the .
Model (6) assumes that inputs are contracted and outputs are expanded at the same rate (proportionally) which it indeed signifies a radial measure of efficiency. However, in this study,
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we utilize the following non-radial version of model (6) which enables us to address nonproportionate improvement in inputs and outputs:
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∑
∑
∑ ∑
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(7)
In the formulation above the variables
and
are introduced to allow the non-proportionate
contraction of inputs and the non-proportionate expansion of outputs in the given directional vector
, respectively.
As inspection makes clear, we have score of
input and
and
. Indeed
and
are the inefficiency
output, respectively. Hence, the optimal solution of model (7) for 5
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tells us the amount of employed excess inputs shortfall outputs
and the amount of produced
. This model actually maximizes the sum of all input/output
inefficiencies. Here,
is efficient if
; otherwise it is inefficient in at least one of the components of
input or output vector.
∑ ∑
∑ ∑
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Consider the following dual model of model (7):
(8)
Since this model is non-oriented naturally, the weighted sum of both inputs and outputs are ∑
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included in the objective function which minimizes ∑
.
Models (7) and (8) are in fact envelopment and multiplier forms of non-radial DD models, respectively.
2
Proposed non-radial classifier directional distance approach
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The main issue of this study is to facilitate the derivation of the input/output status of flexible measures. To pursue this objective, this section proposes a new classifier method based on non-
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radial DD model which combines both the input and output orientations. As a matter of fact, we adopt such model for reasons like the following: (i) Traditional classifier models are oriented and hence the role of a flexible measure in an
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input-oriented model may differ from an output-oriented model. However, DD models are non-oriented and hence extending classifier DD models accommodate the flexible measures by combining both input- and output-orientations.
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(ii) The special structure of formulations of envelopment and multiplier forms of DD models helps us to drive two classifier models and to decide the status of flexible measures from two different perspectives, i.e. pessimistic and optimistic.
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(iii) The main advantage of non-radial models is that they measure individual input reductions along with individual output expansions in order to provide information on the inefficiency score of each input and output.
In particular, this section develops a pair of non- radial classifier directional distance (CDD hereafter) models in both envelopment and multiplier forms, which introduce two pessimistic and optimistic approaches considering the individual and summative viewpoints. Individual models designate flexible measures in favor of each individual DMUs, meanwhile, summative 6
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models aim at accommodating flexible measures in favor of all DMUs, simultaneously. In other words, individual models are solved
times, one for each unit, while summative models are
applied just once to determine the overall role of flexible measures. We also provide two pessimistic and optimistic approaches for each individual and summative model. The former and latter approaches adjust the flexible measures status in line with maximizing and minimizing the possible distance from frontier in the given direction, respectively.
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2.1 Pessimistic CDD approach
In order to develop a non-radial envelopment DEA model for determining the input/output status of flexible measures, we consider model (7) and the variable measure . There would be two possible cases. If ∑
corresponding to flexible
treats as an input role, then the constraint
must be active; otherwise the constraint ∑
must
hold. A method to model the either-or constraints is introducing the indicator variable designates that
is an input, and
designates it as an output. As a result, to
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where
,
handle the either-or constraints, it is sufficient to impose the following constraints to model (7): ∑ ∑
If
is a large positive number. , then the constraint ∑
is active and the constraint ∑
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where
is weakened to ∑ binding. In this case,
treats as an input. We can analogously conclude that the situation is . Hence, we formulate the following non-radial non-oriented MILP model,
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reversed when
, which is always non-
which classifies flexible measures in the envelopment form: ∑
∑
)
(9)
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∑ ∑ ∑ ∑
(∑
PT
̅
It should be mentioned here that the direction vector in this model is where
and operator
element. In general, there are
multiples vectors
and
element by
various combinations for the direction vectors and the
following theorem proves that model (9) finds one direction with the maximum objective function value. Let
be the optimal objective value obtained by model (7) for
each combination. 7
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Theorem 1. ̅
{
}
Proof. By contradiction, suppose that ̅ Consider the
combination and let
{
where
}.
be the index of flexible measures, which treat as input.
Under these assumptions, model (7) can be rewritten as follows: ∑
)
∑ ∑ ∑ ∑
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∑
(∑
(10)
is at hand. Clearly,
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Suppose that model (10) is solved and the optimal solution ( (
is a feasible solution of model (9) where {
Corollary 1. If ̅
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However, the objective function value of this feasible solution, i.e. optimal objective function value ̅ , which is a contradiction. ■
, is larger than the
then an identical inefficiency score is achieved when the flexible measure is efficient by model (9) then it
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is considered either as input or output. In other words, if
is also efficient by models (7) and (8), no matter flexible measure is designated as an input or an output.
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In fact, Theorem 1 proves that model (9) obtains the optimal direction with the maximum sum of input/output inefficiencies among all
possible combinations.
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Now is time to decide on overall input/output status of flexible measures for each individual
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DMU. As the simplest method, one criterion to make a decision would be based on the majority choice among the DMUs. For , let ̅ and be the optimal objective value and the optimal values of the indicator variable corresponding to flexible measure l in model (9), respectively. The following criterion is taken into account, mathematically, to handle the flexible measure.
For flexible measure l, assign whole DMUs to the following two groups: { |̅ { |̅
} }
{ |̅
} 8
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Set
includes the index of DMUs that are efficient in model (9). Referencing to Corollary 1,
these DMUs are efficient either the flexible measure plays the role of an input or an output. So, logically, these DMUs must not be considered for classifying inputs and outputs. Flexible measure l should be designated as an output if | | Moreover, ties for determining the status of flexible measure
| | and as an input if | | happen when | |
| |.
| |. In this
case, an alternative criterion is needed to examine the issue from the point of view of the may be
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collection of all DMUs. Hence, ties for determining the status of flexible measure
broken by applying the following summative version of model (9) which carries out the simultaneous optimization the performance measure of all units and evaluates the pessimistic summative inefficiency corresponding each unit: ∑
∑
(∑
∑
)
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∑ ∑ (
)
∑
(
)
where the intensity variables
(
.
Note
),
(
that
and
the
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.
),
(
) for
will end up either as an input, if
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Theorem 2. Let corresponding to
solution (
),
, and in a single stage. As a , or an output, if
be the part of optimal solution ∑ ∑ ) ̅ .
(∑
in model (11) then
̅
Proof: By contradiction, we suppose that
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are corresponding to optimal
for summative model (11) accommodates flexible measure matter of fact, the flexible measure
(11)
where
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for
and variables
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∑
for
is a feasible solution to model (9) when
. It is easy to show that is evaluated. However, the
objective function value of this feasible solution is larger than the optimal objective function value ̅ , which is impossible. ■ 2.2 Optimistic CDD approach Referencing to Theorem 1, the formulated non-radial envelopment CDD model obtains the maximum possible inefficiency among all possible combinations for each DMU. Alternatively, 9
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we formulate non-radial multiplier CDD models, which give the minimum possible inefficiency for each DMU with the aim of increasing the discrimination power, which is needed for discriminating among efficient DMUs. To do this, we establish the following mixed integer nonlinear programming model extending model (8): ∑
(∑ ∑
)
∑
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∑
∑
(12)
Where the decision variable
, the following method can be applied to eliminate the product of a
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linear due to term
is the weight of flexible measure . Although model (12) is non‐
binary and a continuous variable: Let
be a indicator variable, and
be a continuous variable which
continuous variable is introduced to replace the product added to the model for forcing ̅ to take the value of
. Now a non-zero
. The following constraints must be
:
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̅ ̅ ̅ ̅ The validity of these constraints can be checked by examining all following two possible situations: If ̅
If
, then from the constraint ̅ and ̅
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we have ̅
are redundant.
, then the constraint ̅
or equivalently ̅
; subsequently the constraint ̅
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lead to ̅
We, therefore, utilize this method and let ̅
10
, and also ̅
is redundant. for
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(12):
. In this case, other constraints
in order to linearize model
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∑
∑
(∑
∑
∑
∑
∑
̅
)
̅
(13)
In this model, if
, then
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̅ ̅ ̅
treats as an input and otherwise it treats as an output. The
following theorem proves that model (13) finds a direction with the minimum objective value possible combinations. Let
obtained by model (8) for the
} {
Proof. Suppose the
combination.
{
Theorem 3.
combination and let
be the optimal objective value
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among all
} and contrary to our claim
. Consider
be the index set of flexible measures which treat as input. Under
∑
(∑
∑
∑
Let (
)
∑
ED
∑
∑
(14)
PT
∑
M
these assumptions, model (8) can be written as bellow:
be the optimal solution of model (14). Clearly, (
̅ is a feasible solution
AC
CE
of model (13) where
̅
{
However, the objective function value of the feasible solution objective function value
is smaller than the optimal
, which is a contradiction. ■
The following theorem proves that the optimal objective value for the proposed envelopment CDD model (9) is greater than or equal to the optimal objective value for the proposed multiplier CDD model (13): Theorem 4.
̅ . 11
ACCEPTED MANUSCRIPT
Proof: According to Theorems 1 and 2, we have ̅ combination
and
for
. Since, for each combination, models (7) and (8) are mutually dual, it ̅ . which implies
then follows that
Theorem 4 lays emphasis on the fact that the proposed non-radial CDD models (9) and (13) are formulated with pessimistic and optimistic points of view, respectively, and hence if a unit is
Corollary 2. If ̅
, then
CR IP T
efficient in the envelopment model (9), then it is also efficient in the multiplier model (13). .
It should be noted that the criterion based on the majority choice can be carried out here in order to designate the role (input or output) for each flexible measure by the optimal solution obtained in model (13). In order to break the possible ties, likewise to the aforementioned
AN US
summative envelopment model (11), the following envelopment form is suggested with the aim of evaluating the optimistic summative inefficiency corresponding each unit: ∑ ∑
∑
̅
)
̅
(15)
PT
ED
̅ ̅ ̅
,
and
represent the
CE
where
∑
M
∑
∑
(∑
input,
output, and
flexible measure weight for
. In fact, model (15) is an extended version of model (13) where the
AC
performance measure of all the DMUs is simultaneously considered. The optimal solution flexible measure (
for summative model (15) accommodates
in a single stage where
) for
, and
(
)
. The flexible measure
(
)
is an input, if
and otherwise it is an output. Theorem 4. Let ∑
be the optimal solution and ∑
̅
corresponding to
12
in model (15) then
∑ .
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Proof: By contradiction, we suppose that
for
is a feasible solution to model (10) when
. It is easy to show that is evaluated. However, the
objective function value of this feasible solution is smaller than the optimal objective function value
3
, which is impossible. ■
Application
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To demonstrate the applications of proposed CDD models in both optimistic and pessimistic approaches (i.e. envelopment and multiplier forms as well as individual and summative models), we analyze the performance of banking industry in V4 countries. The economies of these countries have in common that have seen some changes after communism's collapse; the transition to a market economy, joining the European Union in May 2004 and especially the transformation of banking system. The transition from centrally planned economy to market economy had been accompanied with restructuring and liberalization of the banking system. It
AN US
had been associated with the privatization of some banks, the entry of foreign-owned banks, deregulation of interest rates and changes in legislation.
Exploring Banking Efficiency is important from both macroeconomic and microeconomic standpoints. From the macroeconomic point of view, the efficiency of the banking sector affects the cost of financial intermediation and the stability of the entire financial system. From the
M
microeconomic point of view, banking efficiency is particularly important for improving institutional regulation and supervision, and in particular for improving the competitiveness of banks. Increasing the efficiency of banks leads to a better distribution of financial resources, and
ED
thus better investment support and economic growth. Analysis of the banking performance considering a collection of inputs and outputs, which are
PT
clearly specified, discovers some problems in the system. However, the most controversial issue is the role of deposits, whether they should be classified as inputs or inputs. Boda and Zimkova (2015) investigate the bank efficiency in the Slovak banking industry using three the best-known
CE
approaches, the production, the intermediation, and the value-added ones. The production approach was founded by Benston (1965) and is based on assumption that the aim of banks is to
AC
produce deposits (liabilities) as well as loans (assets) and other services to customers. This service-oriented approach considers deposits as an output together with loans and the interest income. Hancock (1991) modified the production approach into the user-cost approach, where deposits are specified as both inputs and outputs of the cost/profit function of a bank. The second approach is the intermediation approach and was proposed by Sealey and Lindley (1977). This approach assumes that the main aim of a bank is to produce the intermediation services through the collection of deposits or other liabilities. The main banks are seen as production units, which transmute deposits into loans. They are also interested in the use of interest-earning assets and loans, securities and similar investments. The profit-oriented 13
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approach is a modification of the intermediation approach as the profit-maximizing tendency of the banks. The last value-added approach differs from previous ones in considering all liability and asset categories to have some output characteristics rather than distinguishing inputs from outputs in a mutually exclusive way. The major categories of produced deposits and loans are viewed as important outputs because they form a significant proportion of value added. The current study adopts the intermediation approach for evaluating bank efficiency in the
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commercial banks of V4 countries recorded in 2014. The economic environment has not been dramatically changed since 2014. The most V4 economies are slowly recovering from the global crisis and optimism among investors and consumers is growing. A positive effect on the banking sector was limited for a number of reasons. The effects of regulation are not only direct costs of implementation of regulation but also the changing business environment requesting of actual new information using data mining from disposal data at banks with help of
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technologies innovations.
There are 61 banks including 15 banks from CZ, 14 banks from HU, 23 banks from PL, and 9 banks from SK whose complete name and the country where they are located are provided in Table A.1 (see Appendix) in detail. We employed three groups of variables: three inputs (labor, physical capital, and credit risk ratio), three outputs (loans and advances, investments, and non-
M
interest income), and one flexible measure (deposit), which are summarized in Table 1. =====[Please insert Table 1 around here]======
ED
The annual balance sheet and income statement data have been collected from the world banking information source Bankscope2. The data for bank branches are listed in Table 2. Mean values of inputs, outputs, and deposits along with the number of banks corresponding each
PT
country are summarized in Table A.2. in Appendix. =====[Please insert Table 2 around here]======
CE
In order to make a comparison between input- and output-oriented classifier models of Cook & Zhu (2007) and Toloo (2012), we apply these models to the data set. Table 3 reports the results.
AC
As can be seen, different assignations for the flexible measure are obtained through input- and output-oriented models of Cook & Zhu (2007) for banks
. To cope with this issue,
we now apply the proposed non-radial non-oriented CDD models by which the flexible measures are accommodated without the need for distinguishing between input- and outputorientation. 2
Bankscope combines comprehensive financial statements with a wide range of other banking intelligence including ratings, an analysis model, bank structures, news, Anti-Money-Laundering (AML) documentation and banking research. The website has information on 32000 banks and is the definitive tool for bank research and analysis. We refer the readers for more details visit https://bankscope.bvdinfo.com/.
14
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=====[Please insert Table 3 around here]====== Table 4 provides the obtained result from both suggested individual models (9) and (13). To be more specific, the calculated input/output inefficiencies through model (9) are also reported in Table 5. =====[Please insert Tables 4 and 5 around here]====== of Table 4 show the value of
as the sum of all
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The second and third columns
input/output inefficiencies for each DMU when deposit , i.e. , is considered as an input and an output, respectively (the columns labeled “
(input)” and “
(output)”). The optimal objective
function value of proposed individual models (9) and (13) along with the related optimal shown in the last four columns. We note that equalities ̅ =max and =min
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assured by Theorem 1 and Theorem 3, respectively.
are are
As can be extracted from the obtained results, models (9) and (13) identify 12 and 17 efficient banks, respectively. Moreover, 12 efficient DMUs are identified by both models which follows from Corollary 2. Essentially, more efficient units introduced by model (13) compared to model (9) which reveals the optimistic/pessimistic viewpoint of these models. For these 12 banks (six financial institutions from the Czech Republic and six banks from Poland), i.e.
M
, we have
that means these
banks are efficient no matter the deposit measure is designated as an input or an output. Therefore, these banks are not taken into account in the overall decision on the role of deposit as
ED
input or output. Large financial institutions such as ČSOB, Česká spořitelna, private PPF, Komerční banka, Raiffeisenbank and the Czech-Moravian Guarantee and Development Bank are in a group of efficient banks in the Czech Republic. Moreover, in Poland, the efficient group
PT
consists large banks (i.e. Bank Zachodni WBK, ING Bank Slaski and Bank Handlowy w Warszawie) and also some small special banks (SGB Bank, RBS Bank, and Bank of Tokyo).
CE
Boxplot is a convenient way to graphically depict groups of data through their quartiles. Figure 1 shows boxplot for ̅ =max and =min . In terms of inefficiency comparison by model (9), the best banks and financial institutions for year 2014 are in the Czech Republic
AC
(median 3.57). They are also followed by Hungarian banks (27.39), Polish banks (41.91) and last are banks in Slovakia (60.02). A similar analysis for model (13) testifies that the best situation is in the Czech Banks (median is 0.000), followed by Hungarian banks (5.91), Polish banks (13.78) and least effective banks are in Slovakia (24.77). The obtained results of the comparison of banking inefficiency in the EU countries using the proposed pessimistic and optimistic models are consistent with empirical studies and developments in financial systems in these countries, which have been affected by the global financial crisis since 2008. The Czech banking system is one of the most resilient systems in the 15
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EU due to the high liquidity of the sector, which is dominated by deposits from clients and is less dependent on interbank market financing. The Hungarian banking system was one of the smallest banking system in the EU with a large bank concentration, but with the development of this system, new types of credit and financial institutions were entering the market. The macroeconomic function of the transfer of deposits into loans and the microeconomic function of providing a complex of banking services were the most successful in the Czech and
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Hungarian banks. The second group consists of Polish and Slovak banks. Poland has the largest banking industry out of the Visegrad countries and this banking system is focused on domestic business and plays an important role in financing private households, SMEs, big infrastructure projects. Polish banking sector is also owned by foreign-owned institution. The main reason for the low-efficiency of Polish banks is the loan quality and not a well-developed payment system. The high level of current earnings means that companies had no need to take out new loans, being able to fund investment and going operations internally. The size of Slovak banking
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system is rather small and influenced by largest banks. The most of the banks have the
AC
CE
PT
ED
M
universal banking license.
Figure 1. Boxplot for (Ineff_M9) = ̅ =max
and (Ineff_M13) =
=min
As can be seen from the results in Table 5, the variations of inefficiency are significant. Accordingly, for the sake of illustration, we only consider boxplot with the inefficiency axis in the range 0-250 in Figure 1 to be more readable. On the one hand, through model (9), ignoring 12 share cases and based on the majority choice among 49 banks, all these banks treat the flexible measure as an output and hence deposit 16
ACCEPTED MANUSCRIPT
should be considered as an output with a pessimistic standpoint. On the other hand, applying model (13), in all 49 banks the flexible measure plays an input role and hence deposit should be considered as an input with an optimistic point of view. All in all, a profit-oriented banking system is desirable from the pessimistic standpoint; meanwhile, the optimistic viewpoint follows a service-oriented banking system. Returning to Table 4, we observe that there are some banks possess very high inefficiency score
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from the pessimistic standpoint whereas these banks have better performance from the optimistic view. This significant variance is logical since, following from Theorems 1 and 3, the pessimistic model (9) finds a specific combination (among
various combinations) which
maximizes the sum of inefficiency scores while the optimistic model (13) introduces the combination with minimum sum of inefficiency scores. Moreover, some banks have recived high inefficiency score from both optimistic and pessimistic views. In order to find the main
AN US
sources of inefficiency, one can refer to Table 5 which provides useful information as a management guide to improve efficiency in performance of each inefficient bank. For example, Bank29 (Post Bank, SK) is a bank with high inefficiency score from both optimistic and pessimistic perspectives and referencing to Table 5 reveals that its first output (loans and advances) is the main source of inefficiency. In fact, the value of the first output of this bank is
M
400 which is significantly low compared to the values of other its input/ouput measures. Although there is no tie for determining the status of deposit in this example, we apply the summative models as well in order to see the results from the summative inefficiency
ED
perspective. After solving models (11) and (15), ( , the optimal objective function value and
)
and (
)
, are obtained, respectively. By comparing
the obtained results through both individual and summative envelopment (or multiplier) forms
PT
for this case study, deposit ends up the same role. However, for 61 banks, the final results are obtained by running 122 individual models (envelopment ant multiplier form), whereas,
CE
summative inefficiency needs to solve only two models for any number of DMU. It should be noted that there is no guarantee that the same results are always achieved through individual and summative version, as mentioned in Cook & Zhu (2007), because the summative
AC
approach might be overly sensitive to extreme DMUs, or possibly to the larger DMUs.
4
Conclusion
One of the controversial issues in many applications in DEA literature is the existence of flexible measure which can serve as either an input or output. Some studies have been done to classify inputs and outputs in DEA. This paper deployed a new classifier method based on directional distance function. As the main feature of this study, we developed a pair of non-radial CDD 17
ACCEPTED MANUSCRIPT
models in both envelopment and multiplier forms which introduce two pessimistic and optimistic approaches from both individual and summative viewpoints. Individual approach accommodates the flexible measure in favor of each DMU; nevertheless, in order to break the possible ties for classifying inputs and outputs, we have extended summative approach which designates the flexible measures in favor of all DMUs, simultaneously. Finally, to validate the discrimination power of the new procedure, an empirical study on the banking industry in V4
CR IP T
countries has been presented in which the deposit factor is taken into consideration as the flexible measure. It is found that deposit is treated as an output and as an input from the pessimistic and optimistic viewpoints, respectively, by both individual and summative models. Finally, extending the classifier models in the presence of imprecise, negative, or stochastic data can be considered as some interesting future research directions. Moreover, dealing with Malmquist index in the presence of selective measures is an alternative research direction.
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Estimating the state of returns to scale in the company of flexible measure under variable returns to scale technology also could be a motivating topic for further study. Acknowledgements The
research
was
supported
by
the
European
Social
Fund
within
the
project
M
CZ.1.07/2.3.00/20.0296, the Czech Science Foundation through project No. 16-17810S as well as the SGS Project No. SP2017/141 VŠB-Technical University of Ostrava. All support is greatly
ED
acknowledged. References
PT
Amirteimoori, A., & Emrouznejad, A. (2011). Flexible measures in production process: A DEAbased approach. RAIRO - Operations Research, 45(1), 63–74.
CE
Amirteimoori, A., Emrouznejad, A., & Khoshandam, L. (2013). Classifying flexible measures in data envelopment analysis: A slack-based measure. Measurement, 46(10), 4100–4107.
AC
Bala, K., & Cook, W. D. (2003). Performance measurement with classification information: An enhanced additive DEA model. Omega, 31, 439–450. Banker, R. D., Charnes, A., & and Cooper, W. W. (1984). Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis. Management Science, 30(9), 1078– 1092. Beasley, J. E. (1990). Comparing university departments, 18(2), 171–183. Benston, G. J. (1965). Branch banking and economies of scale. The Journal of Finance, 20(2), 312– 331.
18
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Boda, M., & Zimková, E. (2015). Efficiency in the Slovak Banking Industry: A Comparison of Three Approaches. Prague Economic Papers, 2015(4), 434–451. Chambers, R. G., Chung, Y., & Färe, R. (1996). Benefit and Distance Functions. Journal of Economic Theory, 70(2), 407–419.
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Chambers, R. G., Chung, Y., & Färe, R. (1998). Profit, Directional Distance Functions, and Nerlovian Efficiency. Journal of Optimization Theory and Applications, 98(2), 351–364. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444. Cook, W. D., Johnston, D. A., & Mccutcheon, D. (1992). Implementations of Robotics Identifying Efficient Implementers. Omega-International Journal of Management Science, 20(2), 227–239.
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Cook, W. D., & Zhu, J. (2007). Classifying inputs and outputs in data envelopment analysis. European Journal of Operational Research, 180(2), 692–699. Cooper, W. W., Seiford, L. M., & Tone, K. (2007). Data envelopment analysis: A comprehensive text with models, applications, references and DEA-solver software (2nd edi.). Springer US.
M
Farrell, M. J. (1957). The Measurement of Productive Efficiency. Journal of the Royal Statistical Society. Series A (General), 120(3), 253.
ED
Hancock, D. (1991). A Theory of Production for the Financial Firm. Dordrecht: Kluwer Academic Publishers. Sealey, C. W., & Lindley, J. T. (1977). Inputs, outputs, and A theory of prouduction and cost at depository financial institutions. The Journal of Finance, 32(4), 1251–1266.
PT
Toloo, M. (2009). On classifying inputs and outputs in DEA: A revised model. European Journal of Operational Research (Vol. 198).
CE
Toloo, M. (2012). Alternative solutions for classifying inputs and outputs in data envelopment analysis. Computers & Mathematics with Applications, 63(6), 1104–1110.
AC
Toloo, M. (2014). Notes on classifying inputs and outputs in data envelopment analysis : a comment. European Journal of Operational Research, 198(2009), 1625–1628.
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APPENDIX Table A.1. Commercial banks of V4 banking sector subject to the analysis Country
Bank name
DMUs
Ceskoslovenska Obchodni Banka-
32
CZ
Ceska Sporitelna
33
PL
OTP Bank Plc
34
1
CZ
2
CZ
3
HU
4
PL
5
CZ
Komercni Banka
36
6
PL
Bank Zachodni WBK
37
7
PL
mBank
38
8
PL
ING Bank Slaski-Capital Group
9
CZ
10
PL
Bank Polska Kasa Opieki SA-Bank Pekao
35
PL
HU CZ
Unicredit Bank Czech Republic and Slovakia Getin Noble Bank
Czech Moravian Guarantee and Develpoment Bank Santander Consumer Bank SGB Bank
Budapest Bank Nyrt-Budapest Hitel-és Fejleszési Bank Nyrt Modra pyramida stavebni sporitelna as Raiffeisen
PL
Euro Bank
39
SK
Sberbank Slovensko
40
SK
Prima banka Slovensko
41
CZ
Air Bank
42
Bank Millennium
Bank name
CZ
AN US
PL
CSOB
M
11
Country
CR IP T
DMUs
HU
Bank of Hungarian Savings Cooperatives LimitedTAKAREKBANK
SK
Vseobecna Uverova Banka
43
PL
Idea Bank
13
PL
Bank Handlowy w Warszawie
44
HU
FHB Kereskedelmi Bank Zrt
14
SK
Tatra Banka
45
PL
Bank Pocztowy
ED
12
HU
K&H Bank Zrt
46
SK
OTP Banka Slovensko
16
PL
Bank BPH
47
PL
MBank Hipoteczny
17
PL
Nordea Bank Polska
48
CZ
Expobank CZ
18
HU
Erste Bank Hungary Nyrt
49
PL
HSBC Bank Polska
19
HU
MKB Bank Zrt
50
PL
RBS Bank (Polska)
51
HU
KDB Bank Europe
21
Ceskoslovenska obchodna banka-
CE
20
PT
15
SK
CSOB
HU
Raiffeisen Bank Zrt
52
PL
FM Bank PBP
HU
CIB Bank Ltd-CIB Bank Zrt
53
CZ
Equa Bank
23 24
PL
Alior Bank Spólka Akcyjna
54
SK
Privatbanka
HU
UniCredit Bank Hungary Zrt
55
PL
Pekao Bank Hipoteczny
25
PL
BNP Paribas Bank Polska
56
HU
Bank of China (Hungary)
26
CZ
GE Money Bank
57
HU
27
PL
Bank Ochrony Srodowiska
58
PL
28
CZ
J&T Banka
59
HU
AC
22
20
MagNet Hungarian Community Bank Bank of Tokyo - Mitsubishi UFJ (Polska) Magyar Cetelem Bank Rt
ACCEPTED MANUSCRIPT
29
SK
Post Bank JSC-Postova Banka
60
SK
CSOB Stavebna Sporitelna
30
CZ
PPF banka
61
CZ
Evropsko-ruska banka
31
CZ
Stavební Sporitelna Ceské Sporitelny
Table A.2. Mean value of data for each country in 2014 Country
banks
Inputs PE
FA
Flexible CRR
DEP
15
75519
84461
49.43
8340805
HU
14
92061
112030
53.64
4773693
PL
23
94523
58398
65.12
6806943
SK
9
43178
41233
61.91
3311167
V4 group
61
81709
74583
58.16
6201704
LA
M ED PT CE AC
21
SEC
NEA
1542066
2430671
597703
318278
1542982
496632
308651
1928066
573831
149533
1039011
216800
590683
1832105
509307
AN US
CZ
Output
CR IP T
No. of
ACCEPTED MANUSCRIPT
TABLES Table 1. Description of three groups of variables Measures
Name
Description
Input variables =PE
Total personnel expenses
Physical capital
=FA
Fixed assets
Credit risk
=CR
Net loans/Total assets
=DEP
Deposit and short-term funding
Loans and Advances
=LA
Loans and advances to banks
Investments
=SEC
Other securities
Non-interest income
=NEA
Non-earning assets
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Labor
Flexible Variable Deposit
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Output variables
Table 2. Data for the 61 commercial banks in V4 countries in 2014 256231
283822
49.21
2
328507
516325
50.5
3
686796
879261
60.16
4
447771
382676
5
262864
286920
6
331134
7
194588
8
223067
1038336
9760384
3328338
26480895
919459
8683189
3083983
63.94
30924048
2438522
8512511
1878279
54.74
25475452
4587331
6096395
2016568
152290
64.18
20419727
542477
5651880
2473068
170827
65.41
19595973
835646
6284417
827197
147307
55.97
17622988
444088
5920254
2047469
94073
69871
62.4
12979049
2405106
3159656
448750
91023
77806
75.38
13336015
332169
2222990
1043667
131683
39288
73.25
11518691
424106
2088996
966800
CE
11
10033454
2746297
PT
10
6977311
29304187
M
1
9
31888107
ED
DMU
12
107800
122100
65.54
8831600
771700
2806200
252900
13
152122
92587
33.55
7931612
852185
4780990
683453
14
93100
67.41
7579000
151600
1937000
785500
97719
141144
45.8
6598928
263473
3423997
682015
16
140494
77325
68.77
3292296
118130
1617585
614698
17
55418
24363
82.45
7040428
192445
562362
584028
18
88419
36216
60.03
6622321
444410
2152468
377061
19
70382
205181
64.23
4884371
330247
785638
977851
20
66600
71800
67.58
4870600
42200
1682800
173900
21
86637
37672
66.12
5075290
93241
1238500
608247
22
78387
75933
67.73
4962492
429687
944721
510011
AC
110300
15
22
110811
51783
76.94
5140868
61195
816315
428223
24
54822
89401
51.45
5004387
572462
1949398
87976
25
67093
30261
78.53
3340130
19066
627818
391222
26
80332
26972
72.4
3418511
40603
832440
454181
27
42803
28094
65.45
2947753
36688
1131536
297671
28
25441
7691
45.92
3313358
471603
1297045
360764
29
43300
21700
48.8
3331200
400
1536900
406200
30
8456
1531
30.12
3174855
925601
936682
755753
31
6816
14069
36.8
3450439
1204393
975134
92214
32
7946
5030
14.5
3258977
2328455
698347
20338
33
45475
16226
80.45
2500825
131370
358193
155612
34
15070
2504
38.49
2806127
145116
1502488
231758
35
74221
54960
62.58
2408961
197956
733082
152612
36
10657
13865
51.95
2663425
1030343
324648
69838
37
8310
2078
47.8
2760803
55037
1407155
94182
38
41792
17261
76.26
2373643
224945
264787
132887
39
19900
19400
70.28
1589000
225100
167200
185600
40
15900
20200
62.1
1639000
104600
516500
78400
41
12720
8675
44.45
1661198
70345
782980
146121
42
16346
21817
24.23
1694606
385220
791736
175860
43
18513
16463
65.62
7051
475974
103701
44
11898
14269
36.23
1576177
476847
590476
56752
45
22244
10761
68.48
1590237
8739
443438
96921
46
16500
21400
74.33
1220300
34100
254100
55100
14681
49
10087
50
AN US
M
2191
84.58
394712
5368
145839
20992
23228
75.36
973367
185857
55460
12965
1180
46.51
758177
34883
340402
88471
PT
5489
48
1472071
5754
361
39.2
614265
66203
317845
74219
6697
1634
52.71
659852
139002
36815
186582
16502
CE
47
CR IP T
23
ED
ACCEPTED MANUSCRIPT
51
1305
57.2
576290
7697
226439
47018
53
12746
3470
69.92
596810
57705
70709
67101
54
5400
1300
33.13
547500
11100
387300
10900
55
2961
120
92.93
114783
10761
16346
1926
56
3221
8846
35.32
311966
170943
63159
13390
57
3729
978
36.65
354212
13818
206997
28652
58
2143
169
44.34
248031
160114
36111
8835
59
9573
1115
87.75
197242
19134
1567
11860
60
2900
100
67.98
192300
5000
63100
2700
61
3007
3364
35.37
193540
45006
29585
60096
AC
52
23
ACCEPTED MANUSCRIPT
Table 3. Input-oriented v.s. output-oriented classifier models (Input-oriented)
Cook & Zhu
Toloo
0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0
0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0
0 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
ED
20 21 22
PT
23
AC
CE
25 26 27 28 29 30 31 32 33 34 35 36 37
AN US
2
CR IP T
Toloo
1
24
(Output-oriented)
Cook & Zhu
M
DMU
24
ACCEPTED MANUSCRIPT
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
AC
CE
PT
ED
61
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 1
CR IP T
40
1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 1
AN US
39
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0
M
38
25
ACCEPTED MANUSCRIPT
Table 4. Results and input/output status of deposit for each bank Model (9)
Model (7) (input)
(output)
Model (13)
̅ =max
=min
1
0
0
0
0
2
0
0
0
0
3
3.79
6.03
6.03
1
4
2.78
2.78
2.78
0
5
0.59
0
0.59
0
6
0
0
0
0
7
8.85
10.83
10.83
1
0
0
0
0
9
3.57
3.57
10
22.79
25.67
11
16.35
18.36
12
9.17
15.34
0
0
14
27.33
60.02
15
4.53
25.03
16
0
17 18 19
7.23
20 21
3.79
0
2.78
0
0
1
0
1
8.85
0
0
1
0
3.52
0
25.67
1
22.79
0
18.36
1
16.35
0
15.34
1
9.17
0
0
0
0
0
60.02
1
27.33
0
25.03
1
4.53
0
86.24
86.24
1
0
0
23.85
61.74
61.74
1
23.85
0
7.69
20
20
1
7.69
0
26.91
26.91
1
7.23
0
59.16
258.68
258.68
1
59.16
0
28.83
91.58
91.58
1
28.83
0
ED
PT
22
0
3.57
M
13
1
0
AN US
8
0
CR IP T
DMU
7.81
27.87
27.87
1
7.81
0
50.09
190.01
190.01
1
50.09
0
12.25
20.01
20.01
1
12.25
0
25
91.74
596.76
596.76
1
91.74
0
26
42.19
248.19
248.19
1
42.19
0
27
30.08
265.43
265.43
1
30.08
0
28
2.41
9.73
9.73
1
2.41
0
29
1975.06
15484.94
15484.94
1
1975.06
0
30
0
0
0
0
0
0
31
0
0
0
0
0
1
32
0
0
0
0
0
0
33
13.78
76.62
76.62
1
13.78
0
34
0
0
0
0
0
0
35
7.8
60.03
60.03
1
7.8
0
23
AC
CE
24
26
ACCEPTED MANUSCRIPT
9.14
17.04
17.04
1
9.14
0
37
0
0
0
0
0
0
38
9.46
56.58
56.58
1
9.46
0
39
6.58
38.25
38.25
1
6.58
0
40
9.44
48.51
48.51
1
9.44
0
41
5.54
51.73
51.73
1
5.54
0
42
0.83
10.47
10.47
1
0.83
0
43
102.59
758.9
758.9
1
102.59
0
44
4.58
19.73
19.73
1
4.58
0
45
102.42
605.33
605.33
1
102.42
0
46
24.77
150.37
150.37
1
24.77
0
47
33.55
220.96
220.96
1
33.55
0
48
24.26
142.03
142.03
1
24.26
0
49
5.51
29.24
29.24
1
5.51
0
50
0
0
0
0
0
0
51
0
30.74
30.74
1
0
0
52
28.51
125.13
125.13
1
28.51
0
53
8.15
69.73
69.73
1
8.15
0
54
0
105.35
105.35
1
0
0
55
18.79
41.91
41.91
1
18.79
0
56
4.15
30.7
30.7
1
4.15
0
57
2.9
50.65
50.65
1
2.9
0
58
0
0
0
0
0
0
59
78.81
565.52
565.52
1
78.81
0
60
25.62
28.82
28.82
1
25.62
0
61
0
31.13
31.13
1
0
0
AN US
M
AC
CE
PT
ED
CR IP T
36
27
ACCEPTED MANUSCRIPT
Table 5. The calculated input/output inefficiencies DMU 0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
3
0.47
0.4
0
0.37
4.43
4
0.47
0.15
0
0.09
2.07
5
0.31
0.05
0
0.21
0
6
0
0
0
0
0
7
0.25
0
0
0.19
10.36
8
0
0
0
0
0
0.44
0.38
0
0.23
0
0
11
0.65
0
0
12
0
0.13
0
13
0
0
0
14
0.29
0
0
15
0
0.12
0
16
0.51
0
0
17
0.26
0
18
0.53
0
19
0
0.59
20
0.16
21
0.5
22
0.13
0.2
0.12
0.36
0
0
0
0.02
0
0
0
0.03
0
0
0
0
2.43
AN US
9 10
CR IP T
1
0.22
24.38
0.84
0
0.19
16.91
0.6
0
1.41
12.26
0.98
0.57
0
0
0
57.39
1.14
0
1.34
23.34
0.23
0
4.06
80.19
1.48
0
1.31
54.65
5.51
0
0
1
18.07
0.4
0
0
1.98
20.56
3.77
0
0.37
0
2.47
254.37
1.31
0
0
0
1.68
87.83
1.58
0
M
0
1.2
ED
0
0
2.39
22.12
3.23
0
0
0
2.41
183.22
3.9
0
0
0.23
0
0.7
2.03
0.36
16.68
0.34
0
0
3.98
587.54
4.89
0
26
0.51
0
0
3.36
241.3
3.02
0
27
0.1
0
0
3.81
259.74
1.78
0
28
0.3
0
0
1.11
6.97
0.34
1.01
29
0.34
0
0
1.95
15482.17
0.48
0
30
0
0
0
0
0
0
0
31
0
0
0
0
0
0
0
32
0
0
0
0
0
0
0
33
0.26
0
0
4.35
58.34
7.94
5.73
34
0
0
0
0
0
0
0
35
0.15
0
0
5.92
49.18
4.41
0.37
36
0
0.86
0.27
0.5
0.13
2.64
12.64
24
AC
CE
25
PT
0
0.49
23
28
ACCEPTED MANUSCRIPT
0
0
0
0
0
0
0
38
0.2
0
0
4.66
35.52
10.87
5.34
39
0
0.39
0.44
4.11
23.64
9.67
0
40
0
0.51
0.52
2.97
42.78
1.72
0
41
0
0.14
0.43
2.12
48.57
0.47
0
42
0
0.35
0
2.07
7.53
0.51
0
43
0
0.31
0.47
4.18
751.5
2.45
0
44
0
0.67
0
1.51
1.35
0.98
15.22
45
0
0
0.22
4.58
590.58
3.79
6.17
46
0
0.52
0.59
4.55
139.99
4.72
0
47
0
0
0.82
4.45
201.51
2.75
11.42
48
0
0.89
0.31
4.66
7.65
28.32
100.2
49
0.35
0
0.5
2.23
19.45
1.12
5.58
50
0
0
0
0
0
0
0
51
0
0.04
0.33
2.46
0.37
27.55
0
52
0.51
0
0
3.56
107.29
2.3
11.48
53
0
0
0.41
7.17
31.36
18.14
12.65
54
0
0
0.46
2.75
64.04
0.5
37.6
55
0.78
0
0.97
1.17
5.74
3.49
29.76
56
0
0.93
0.68
2.88
1.06
4.65
20.5
57
0
0
0.67
3.02
58
0
0
59
0
60
0.78
61
0
AN US
CR IP T
37
0.92
8.6
0
0
0
0
0
0.42
10.1
27.01
490.54
37.45
0
0.96
0.06
10.49
0
16.52
5.28
16.12
8.4
0
ED
M
37.44
0
0.82
AC
CE
PT
0.52
29