The Importance of Discriminant Metrics in Efficiency Analysis
KELLY PATRICIA MURILLO
Department of Mathematics,
University of Aveiro,
Campus Universitário de Santiago, 3810-193 Aveiro,
PORTUGAL
Abstract: - In this article, a non-parametric deterministic method that combines efficiency models with
mathematical techniques to examine decision units is applied. In order to better understand the calculated
efficiencies, characterize and identify possible improvements in less efficient units, four discrimination metrics
are proposed. The metrics are determined by how the efficiency index is calculated. The metric that best
represents data and allows for more detailed analysis of results is taken as a reference to build a new metric
with a more complete structure. The latter allows a general characterization of the decision unit in the context
studied. The methodology presented in this study is discussed through an empirical application, which allows
examining the efficiency of European countries in production sectors.
Key-Words: - Metrics analysis, efficiency models, Data envelope analysis, Production sectors.
Received: April 26, 2023. Revised: October 6, 2023. Accepted: October 18, 2023. Published: October 27, 2023.
1 Introduction
In studies involving data analysis, additional
mathematical tools are often applied to analysis
models with the aim of differentiating the database
more strictly. In efficiency analysis, there are many
aspects to consider for establishing a strong
structure that allows a complete and reliable study.
Part of the data analysis involves identifying units
with highly differentiated characteristics, which may
interfere in some way with the results obtained.
This is because units, whose input/output values
correspond precisely to the minimum/maximum in
which the entire data set varies, may be worse/better
positioned relative to the majority. Regarding
variables, their classification beyond being inputs or
outputs should be considered. The analysis should
establish whether the variables are stationary,
desirable, undesirable, intermediate, etc. On the
other hand, the efficiency models are determined
according to the context, the available database, and
the objectives. This determines whether a linear or
nonlinear, parametric, static, or dynamic model will
be used, etc.
However, it is very common in this type of
analysis to emphasize the most appropriate selection
of the data, the correct variables, and more adjusted
models moving to the background, crucial aspects
such as the metrics used in the study, forgetting that
these can decisively influence the results obtained as
well as the other aspects mentioned.
In the literature, efficiency measurement has
been based on parametric and nonparametric border
analyses. The two most commonly used methods are
the regression analysis approach, which leads to the
use of econometric methods, and the Data
Envelopment Analysis DEA which uses linear
programming, [1], [2]. DEA allows solving
problems of simultaneous maximization of products
or simultaneous minimization of inputs, building an
optimal production frontier, and comparing each
observation unit against the expected optimum. The
DEA model is based on radial contractions at all
undesirable inputs and outputs and radial expansions
at all desirable outputs.
Recently, in some studies to evaluate efficiency
where it is required to establish inefficiency indices
in each variable individually, the Multidirectional
Efficiency Analysis MEA, initially proposed in, [3],
has been used. This model allows the reduction of
inputs and expansion of outputs, looking for a
separate potential improvement in each input
variable and each output variable.
The use of the most suitable non-parametric
model for the intended analysis depends on the
objectives of each particular study. DEA and MEA
are the most important models currently used to
measure efficiency. [4] compares a set of public and
private schools using DEA. In, [5], DEA is used to
evaluate the financial performance of the textile
industry in Haryana, located in the northern part of
India. The authors in, [6], build on the existing
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studies in Dynamic Network Data Envelopment
Analysis (DNDEA) and proposing a sequential
structure incorporating dual-role characteristics of
the production factors.
In, [7], DEA techniques were applied to examine
the comparative efficiency of higher education
institutions. Other interesting studies with DEA are,
[8], [9], [10], [11].
Regarding MEA, in, [12], the use of MEA allows
investigating how railway reforms affect the
inefficiencies of specific cost factors. In, [13], MEA
is used to assess the level of energy and
environmental efficiency and the trend of China’s
transport sector. In, [14], both integrated MEA
efficiency levels and efficiency standards were
detected, which are represented by the specific
MEA efficiency of the variable according to each
type of emission or discharge of industrial pollutants
from major cities in China. In, [15], the efficiency
of European countries in the context of Circular
Economy, is examined, considering the sector
of plastic. Other interesting studies with
applications in MEA are, [16], [17].
The metrics we present correspond to the need to
establish a well-structured and complete
methodology that allows evaluating efficiency from
various angles and clearly establishing which factors
must be improved. Metrics allow us to examine the
factors that influence the behavior of decision units
in any context, such as public health, business
efficiency, educational quality, energy efficiency,
and the circular economy, among others.
In this paper, four metrics are distributed in four
approaches, which allow for different visualizations
of the efficiency index against the given database.
The index is calculated from a non-parametric and
deterministic model to measure the technical
efficiency of decision units. After selecting the
metric that best represents the data and allows a
more detailed analysis of the results, a fifth metric is
defined from that. This new metric has a more
complete structure, which allows a general
characterization of the decision unit in the studied
context.
To visualize the effects of the metrics studied,
particularly in this work, we examined the
efficiency of European countries in production
sectors compared to their waste management
capacity.
The remainder of this document is presented as
follows: In the next section, the model for
calculating the efficiency index is presented, and,
based on this, the four proposed approaches are
discussed. In Section 3, the efficiency of 26
European countries in three productive sectors is
examined. In Section 4, the general comments and
the final observations are established.
2 Methodology and Characterization
of Metrics
In the literature, we find different models to
examine the efficiency of decision units, which
basically depend on the characteristics of the
database and the objective of the study. This can be
oriented to the input, oriented to the output, with a
constant scale, or with a variant scale, among other
features.
It should be noted that, for the purpose of this
study, the efficiency indicator of each decision unit
can be obtained using either of the two non-
parametric and deterministic methods mentioned
above: the traditional DEA, [18], or the last-model
MEA, [19]. Specifically, about the DEA. The most
commonly used DEA models are the DEA-CCR
model, introduced in, [1], and the DEA-BCC model,
introduced in, [2]. The DEA-CCR model assumes
constant scale returns, and the DEA-BCC model, on
the other hand, allows variable scale returns. The
MEA model, on the other hand, can be adjusted to
use VRS (a model with variable scale returns) or
CRS (a model with constant scale returns),
according to the objectives of the problem in
context.
In order to facilitate reading this article, unify the
concepts, and structure the metrics to be studied, we
introduce an efficiency index, the one determined by
the DEA-CCR model. Note, however, that another
model (such as DEA-BCC or MEA) could be
selected. This is because the model to use to
calculate the index depends on the problem to be
studied, the orientation, the scale, and the type of
study to be done. The selected model has been the
one that best corresponds to the characteristics of
the problem to be studied, allowing us to concretely
visualize the differences in the glass, paper, and
plastic production sectors discussed in the numerical
simulation.
Consider a decision unit. Suppose that any unit
produces outputs, using ,
inputs.
Definition 2.1 Let be a given
database with set of values .
The technical efficiency index for a specific
observation,
, is the
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optimal solution
of the linear problem
P(
):
such that
where u are the weights of the outputs; v are the
weights of the inputs and,.
The efficiency index obtained in Definition 2.1 is
a value
. The linear program (1)-(4) is
executed for the decision units of the study and an
index is obtained for each. This index allows us to
compare the performance of each unit, according to
the variables considered in the study.
2.1 Effective Metrics Characterization
In this section, different metrics will be distributed
in approaches that will allow different visualizations
of the efficiency index compared to the given
database. The index is calculated from a non-
parametric and deterministic model to measure the
technical efficiency of the decision units.
Based on the efficiency model described in the
previous section, five metrics are considered in this
study: M1, M2, M3, M4, and M5. The first four
(M1–M4) are related to the four approaches E1, E2,
E3, and E4, which are determined by how efficiency
indices
are calculated:
(E1) Efficiency calculation, where metric M1
considers all study variables;
(E2) Efficiency calculation with the M2 metric,
in which only a representative subset of the database
is considered, obtained through Principal
Component Analysis (PCA);
(E3) Calculation of efficiency, considering a
composition metric. The M3 metric is defined in
combination with the results of the E1 and E2
approaches;
(E4) Calculation of the efficiency index, with the
metric M4, defined as a metric of composition by
ranges.
A fifth metric, M5, is defined based on the best
results obtained in previous approaches. The M5
metric presents a composition structure and is
divided by ranges (Section 2.2). This new metric has
a more complete structure, which allows a general
characterization of the decision unit in the studied
context.
To continue, we will describe the metrics and
approaches specifically.
Let be a database given with
, where
represents the inputs and represents the
products. Consider a subset of the
database with , where
y represent the inputs and
outputs selected by the PCA.
(E1) Efficiency calculation, considering all study
variables.
After debugging the database on which the study
will be conducted, it is important to take a first look
at the results on all the selected data, applying the
M1 metric (Definition 2.1.1). This involves
calculating the efficiency index with all variables,
analyzing the results, and examining whether they
make sense within the context being studied.
Definition 2.1.1: Let be a database
with . The efficiency index of
each DMU , is defined as the value
: optimal solution of (5)
(E2) Efficiency calculation, considering a subset
of the initial database.
For this approach, the M2 metric (Definition
2.1.2) is used. Thus, the relevance of each variable
in the model must be established first. An efficiency
study consistent with the results depends largely on
the relevance of the variables considered in the
study.
The most representative variables
can be determined by statistical techniques such as
Principal Component Analysis (PCA). We selected
the most relevant variables for this study, using the
PCA, accompanied by a dimensionality test called
test-dim. The PCA analysis, proposed in, [20],
transforms a series of correlated variables into a
series of uncorrelated variables, [21]. Once the PCA
is complete, an attenuation test is performed. This
allows testing the number of axes in a multivariate
analysis. The procedure is based on the calculation
of the RV coefficient, [22].
Definition 2.1.2: Let be a subset
of with . The
efficiency index of each DMU k N, is defined as
the value,
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: optimal solution of (6)
(E3) Efficiency calculation, considering a
composition metric.
The M3 metric (Definition 2.1.3) used in this
approach is defined in combination with the results
of the E1 and E2 approaches.
Definition 2.1.3: Let be a database
with , and a
subset of the database with .
The efficiency index of each DMU , is
defined as the value
(7)
where
is the optimal solution of
with , and
is the optimal solution of
with . Then
and
are the values in (5) and (6), respectively.
(E4) Calculation of the efficiency index, with a
metric of composition by ranges.
The M4 metric (Definition 2.1.4) in this
approach is defined in combination with the results
of the E1 and E2 approaches, differentiating
between units that are fully efficient (
) and
the other remaining units (
). The latter
correspond to efficient but not fully efficient units
and inefficient units.
Definition 2.1.4: Let be a database
given with , and
a subset of the database with .
The efficiency index of each DMU , is
defined as the value
where
is the optimal solution of
with ; and
is the optimal solution
of with . Therefore
and
are the values in (5) and (6),
respectively.
Note that when all variables are relevant to the
study, we have to
. Then the metrics M1
and M2 are equal, and therefore the focus E1 and E2
become only one. On the other hand, if all units are
fully efficient (
), we have
; the
metrics M3 and M4 are the same, and then the
approaches E3 and E4 are the same. Clearly, the
latter case would lose interest, not analysis, and it
would be necessary to establish other metrics that
would allow us to differentiate and characterize the
database more strictly.
2.2 Composite Metric Characterization
Once it has been decided which is the metric that
best represents the data and allows a better analysis
of results, a more complete metric is proposed that
allows a characterization of each decision unit in the
context studied (considering all areas of study).
Consider that all units k, from the previous
section, present the following structure:
, a decision unit,
identifying a sector , a country and a year
. Suppose that any unit produces
outputs, using inputs.
Next, we will define the M5 metric (Definition
2.2.1) with weights for each sector from the M4
metric.
Definition 2.2.1: Let be a database
and a subset with
, the efficiency index of each DMU
, is defined as the value
with
where the values correspond to the weights
that are attributed to each sector
is the
optimal solution of with
in ;
is the optimal solution of
with in . Therefore,
the indices
correspond to the value
in (8).
Considering the sector ;
and
are the
values in (5) and (6), respectively.
3 Numerical Application
To visualize the effects of the metrics studied,
particularly in this work, we will examine the
efficiency of European countries in production
sectors compared to their waste management
capacity. The metrics defined in the previous
section are applied in three production sectors: (S1)
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glass production, (S2) paper production, and (S3)
plastic production.
Twenty-six European countries are considered as
decision-making units for the analysis: Austria
(AT), Belgium (BE), Bulgaria (BG), Cyprus (CY),
Czech Republic (CZ), Denmark (DK), Estonia
(USA), Finland (FI), France (FR), Germany (DE),
Greece (EL), Hungary (HU), Ireland (IE), Italy (IT),
Latvia (LV), Luxembourg (LU), Malta (MT),
Norway (NO), the Netherlands (NL), Poland (PL),
Portugal (PT), Romania (RO), Slovakia (SK),
Slovenia (SI), Spain (ES) and the United Kingdom
(UK). Acronyms in brackets represent the
nomenclature of each country, used throughout the
article.
The analysis was done over a period of eleven
years (2006–2016). Input and output variables are
selected for waste generation, recovery, and
recycling in order to examine the performance of
countries in relation to the circular economy of the
three production sectors (Table 1).
Table 1. Inputs/outputs variables
If we consider the study in terms of a circular
economy, CO2 emissions and waste, would be
undesirable variables. Therefore, it becomes
necessary to use complement variables instead.
These variables are defined as the maximum value
of the variable in a complete database minus the
value of the variable for the unit under
consideration, [23]. In order to obtain the efficiency
indices, the other variables are used in their normal
form and complement the CO2 and waste emissions.
The data is organized according to each
approach. A software package developed by Python
called pyDEA, [24], is used for data processing and
the calculation of efficiency indices. Specifically,
for the E2 approach, the database was reduced to
labor and energy consumed inputs and to waste,
recycling, and recovery outputs. Calculations are
made for all countries and sectors throughout the
study period. In Table 2, the results obtained in each
metric for the years 2006 and 2016 in sector S1 are
reflected.
We rank efficiency indices in four ranges:
The differences in the metrics applied are
noticeable. In fact, in 2006, on metric M1, 61,5% of
the countries studied had an efficiency index in the
range ; 30,7% in and 7,6% in . None of the
countries obtained with this metric an index below
0.63. According to M2, the result was 42,3% in ,
15,3% in , 3,8% in and 38,4% in. The
indices with this metric were the lowest of the four
approaches, with a score of 0,18.
About the metric M3, 46,1% of countries have an
index in ; 15,3% in; 26,9% in and 11,5%
in. According to M4, 46,1% have in ; 7,6%
in; 3,8% in and 42,3% in. The second-
highest percentage occurred in the last rank.
Considering decision units with indexes in the
ranges or , efficient, and index units in the
ranges or as inefficient. With M1, 92,2% of
the units in 2006 are efficient; with M2, 57,6%; with
M3, 61,4% and with M4 only 53,7%. This leads to
the conclusion that M4 metric is stricter in unit
discrimination.
Table 2. Comparison of metrics in S1, 2006 and
2016
3.1 Relative Position
Undoubtedly, another interesting aspect is to
analyze how both change the position in an
efficiency ranking according to the metric applied.
These differences can be examined in Table 3 for
2006, sector S1.
BG is located in the 3rd position when the M1
metric is used, in the 21º position when the M2
metric is used, in the 17º position when the M3
metric is used, and in the 15º position when the M4
metric is used. LV is located in the 10th position
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when metric M1 is used, in position 15º when
metric M2 is used, in position 15º when metric M3
is used, and in position 13º when metric M4 is used.
The case of PL is relevant to this analysis and
clearly shows the objective of this study. PL is
placed in 13th position when the M1 metric is used,
18º when the M2 metric is used, 16º when the M3
metric is used, and 14º when the M4 metric is used.
It is therefore fully efficient in the M1 metric,
inefficient in the M2 metric, and efficient in the M3
and M4 metrics.
Table 3. Countries' relative positions M1-M4,
2006
On the other hand, it should be noted that the
relative position of decision units does not
necessarily provide all the information on the index.
For example, the countries NL, FI, ES, CY, RO, EL,
CZ, EE, HU, SK, and SI have the same positions
(16 to 26, respectively) in M1 and M4 metrics.
However, the scores are very different
for M1, and
for M4.
Countries BE, DE, DK, FR, IT, LU, MT, NO, PT,
IE, AT and UK, have the same positions (1-12) in
the metrics M2, M3, and M4. However, scores are
only different for M2 (
). The
indices for these 12 countries calculated with M3
(
) and M4 (
)
are equal.
3.2 Evolution over Time
Figure 1, Figure 2 and Figure 3 show the
efficiency results per year (2006-2016),
according to each approach (E1-E4) and sectors
S1-S3, respectively.
Each year is represented by a color, and each
country corresponds to a column within the graph.
The lines represent the variability of the results
obtained for each country in each year. When the
lines are very close in a certain country, it means
that throughout the study period, their efficiency
values were very similar, such as the cases of LV
(Figure 1), PT (Figure 2), and IE (Figure 3). On the
contrary, if the lines are very separated in a certain
country, it means that there are great differences in
efficiency results throughout the study period, such
as BG, HU, and RO (Figure 1, Figure 2 and Figure
3). Looking at the results simultaneously, we can
see that E4 is the approach with the greatest
variability in the three sectors over the years.
Therefore, the M4 metric allows for greater
discrimination among decision units.
In Figure 4, Figure 5 and Figure 6, the country
indices calculated in the M4 metric for sectors S1–
S3 are represented, respectively. Each year is
represented by a colored line, and each country
corresponds to a section within the graph. Each of
the 10 circles in the graph represents a value on the
scale from 0 to 1. The closer each colored line is to
the center, the lower the efficiency value obtained in
that year. For example, if we analyze the LV and
RO countries, we can see that the score changes
from year to year in greater proportion in some
sectors. However, we also find countries like LU
where efficiency is always 1, independent of the
approach.
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Fig. 1: Decision units in S1, E1-E4 approaches
Fig. 2: Decision units in S2, E1-E4 approaches
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Fig. 3: Decision units in S3, E1-E4 approaches
Fig. 4: Efficiency in S1, M4
Fig. 5: Efficiency in S2, M4
Fig. 6: Efficiency in S3, M4
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Below, we introduce the calculations with the
M5 metric. For this, it is necessary to define the
values , corresponding to the weights, that are
attributed to each sector . In this study, we will
give the same importance in the calculation of
efficiency to the three sectors (glass, plastic, and
paper); however, the values can be assigned
differently.
Consider
. Then the index
to be calculated is
(12)
Table 4 shows the results obtained in (12) with
the M5 metric. For this study, all units (26
countries) and all sectors (S1–S3) are considered.
Table 4. Global performance (%), M5 metric
Performance percentages by rank in each metric
are shown in Table 5.
Table 5. Metrics by rank (%), year 2006
The differences between the metrics are
noticeable, and the importance of the M5 metric in
relation to each of the previous ones is reflected. For
example, with the M1 metric, 92,2% of the sample
is efficient; with the M2 metric, 57,6%; with the M3
metric, 61,4%; with the M4 metric, only 53,7%; and
with the M5 metric, 57,7%.
4 Conclusions
The study is based on the DEA-CCR model and
shows the importance of selecting the most
appropriate metric in each context for measuring
efficiency.
In this work, different metrics are considered,
determined by how efficiency indices
are calculated.
Specifically, five metrics are considered: M1, M2,
M3, M4, and M5. The first four (M1–M4) are
related to four approaches: E1, E2, E3, and E4,
respectively. Next, it is proposed to build the fifth
metric, M5, which it considers the strictest of the
four previous metrics.
In numerical simulation, three sectors are
considered: (S1) glass production, (S2) paper
production, and (S3) plastic production. The study
involved 26 European countries for eleven years
(2006–2016). The variables were selected for waste
generation, recovery, and recycling in the three
sectors.
The differences between the five metrics and the
three sectors are very noticeable. The lowest
efficiency levels are in the M2 metric, followed by
the M4 metric, then M3, and the highest levels in
the M1 metric. The last metric defined in the M4
approach is the one with the most data variability,
allowing for more detailed analysis. With this
metric, it is possible to more strictly differentiate
efficient units, establishing a ranking even when the
units are all on the border.
The results show that a metric not analyzed in
depth can give the wrong idea that the decision units
studied are mostly (or entirely) efficient. It is
important to consider that in addition to calculating
the efficiency index, in this type of study, it is
possible to make additional estimates and
calculations to obtain the maximum amount of
information from the database.
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WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2023.20.208
Kelly Patricia Murillo
E-ISSN: 2224-2899
2434
Volume 20, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Kelly Patricia Murillo has written, reviewed, and
actively participated in all the publication stages of
this manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This work is supported by the Center for Research
and Development in Mathematics and Applications
(CIDMA) through the Portuguese Foundation for
Science and Technology (FCT - Fundação para a
Ciência e a Tecnologia), UIDB/04106/2020,
UIDP/04106/2020, and by national funds (OE),
through FCT, I.P., in the scope of the framework
contract foreseen in the numbers 4, 5 and 6 of the
article 23, of the Decree-Law 57/2016, of August
29, changed by Law 57/2017, of July 19.
Conflict of Interest
The author declares no conflict of interest.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2023.20.208
Kelly Patricia Murillo
E-ISSN: 2224-2899
2435
Volume 20, 2023
