Determining the Degree of Preference Liking and Effectiveness of
DMUs over a Long Period of Time by Means of a New Approach based
on the Cross Efficiency Value Chain in Harmony with Fuzzy Arithmetic
BLERTA (KRISTO) NAZARKO
Department of Informatics and Technology,
European University of Tirana,
Kompleksi Xhura, Rruga Xhanfize Keko,
ALBANIA
Abstract: - In this paper, an alternative approach is presented for the evaluation of the likeability preference and
effectiveness of DMUs, based on the DEA and fuzzy DEA models. In the magnitudes of variable values
according to input-output levels, over the time period, some have not completely clear (fuzzy) values obtained
from perceptions and surveys. For a more realistic assessment of effectiveness and determination of the degree
of preference liking, to avoid accidental fluctuation values, and to get as close as possible to the trend of the
process's progress, dynamic analysis of smoothing of the time series is applied to the input-output value levels.
This is done according to a k-order moving average, determining the new levels of the input-output values. The
approach is applied in two phases. In the first phase, the efficiency value chain matrix is determined, applying
conventional DEA models with constant and variable returns to scale, evaluation of super efficiencies, fuzzy
efficiency, and cross-efficiency. The data and the comparison of the models are analyzed, focusing in particular
on the cross-efficiency value chain. In the second phase, fuzzy triangular numbers are composed of of the chain
of cross-efficiency values for each DMU. Then based on fuzzy arithmetic as well as the concept given by the
geometric probability model is determined and the transition matrix of the degree of preference liking, the
evaluation of the ranking is obtained according to the degree of preference liking of each DMU in relation to
other DMUs. In the paper, the contributions of the approach to the evaluation of the effectiveness and the
degree of preference liking with the relevant conclusions are highlighted.
Key-Words: - cross-efficiency, degree of preference liking, ranking, dynamic analysis, DEA models, fuzzy
triangular numbers, super efficiencies.
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1 Introduction
The approach presented is based on real-life data,
which aims to explore the distinctive features of
each DMU assessed as best practices as well as to
identify the impact of factors on the effectiveness
and the degree of preference likeability related to
the standard of living on the basis of the prefecture
and region. In Albania, according to regions and
prefectures, there are tangible differences in internal
migration as well as in the natural rate of population
growth. For the study analysis, the method of data
envelopment analysis (DEA) is used as a very
applied and powerful method in the study and
evaluation of the effectiveness, ranking, and
evaluation of the influence of the factors on the
efficiency value. In real life, the values of the
variable quantities are not all completely
determined, where DEA also shows "weakness" if
the inputs and outputs with which the DMUs
operate do not have completely clear quantitative
values or the data are vague as they can be those
given in the field of perceptions, surveys, etc.
Therefore, in addition to the basic DEA models with
constant returns to scale and variable returns to
scale, [1], [2], many models of its extension are also
applied, as well as Fuzzy DEA models. The study
analysis is over an extended period of time (2016-
2022), so the data can be presented formatted as a
time series, where seasonal variations, random
variations (or accidental variations influenced by the
conditions in which the observations are carried out)
are encountered, etc. In order to soften the variations
in the values of variable quantities and to get as
close as possible to the trend of the period,
decomposing time series and smoothing
transformations on time series is done, as is the
series of moving averages of order k, [3]. In the
multiplier model with linear programming in the
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DEA method, the DMUs choose their most
favorable weights, but the weights can have zero
values or very small values where it can be said that
specific inputs or outputs are ignored or
misinterpreted, and the performance of DMUs
decreasing and the power of distinguished between
them. To increase the distinguishing power in the
DEA rankings, many approaches have been applied,
such as the super-efficiency approach developed for
the first time by [4], as well as the connection of the
DEA performance with other approaches with
canonical correlation analysis such as [5]. The
approach called cross efficiency was presented for
the first time by [6] for evaluating the performance
of DMUs, but it can also be said as a likability
evaluation for management strategies. The
evaluation of efficiency of each DMU is evaluated
with its own weights, but also with the weights of
other units, which is called cross efficiency, where
each DMU is compared with every other unit in the
set of DMUs, then it is evaluated average efficiency
values for each DMUs. Applications of Cross-
efficiency can be found in many papers such as [7],
[8], [9], where in [9] the cross-efficiency evaluation
method is used for 102 DMUs (for the years 2012
and 2017), which use 4 inputs and 4 outputs, where
two of the outputs have a qualitative nature. Fuzzy
DEA is developed based on the theory of fuzzy sets.
[10] is the first to present the Fuzzy set, also [11], in
addition to the generalization of the conventional
Fuzzy set, connected it with the so-called
membership function, giving the concept of
linguistic variable. The authors [12] and [13] give
the classification of approaches applied in Fuzzy
DEA, classifying them into 6 types. [14] provides an
approach to Fuzzy DEA in a form characterized by
numbers reflected through perception, also
proposing an extension of the Fuzzy DEA model in
the relationship between DEA and linear regression.
The ranking is related to the comparison of Fuzzy
numbers. [15] makes a comparison of Fuzzy
numbers based on the concept of probability, giving
examples compared with other approaches. [16]
develops the approach of programming possibilities
with a certain level of possibility based on three
components, so that Fuzzy numbers are realistic to
represent approximations and use the concept of
possibility by comparing fuzzy numbers. [17]
developed the approach in the case of Fuzzy linear
programming and linear programming with multiple
objectives, giving a modified model for each case.
Based on different applications for the ranking of
fuzzy numbers, in the coefficient of variation of the
distance of the central point and the initial point [18]
proposes a modification of the approach based on
the distance called sign distance. [19] proposes a
new ranking function for the ranking of the real
number and the fuzzy number with an acceptance
rate and then extends it to the ranking of two fuzzy
numbers. The ranking of fuzzy numbers is
interpreted as an instrument in many application
models. To evaluate the measurement of efficiency
using the concept of the set of fuzzy numbers in the
context of DEA, [20] brings fuzzy mathematical
programming, to contribute to an optimal solution in
the evaluation of efficiency, fuzzy regression to
illustrate and types of different options that are
available. Efficiency evaluation and ranking of
DMUs with Fuzzy data, where the CCR fuzzy
model is transformed into a crisp linear
programming problem applying α-cut approach
illustrated and with numerical examples is given in
[21]. [22] presents fuzzy DEA models based on
fuzzy arithmetic formulated as a linear
programming where the fuzzy efficiency of
decision-making units can be evaluated and an
analytical approach of fuzzy ranking developed
according to fuzzy rank efficiencies for performance
evaluation. [23] proposed finding a common set of
weights in fuzzy DEA by evaluating the upper
bounds of the weights in the solution of the problem
presented in linear programming., demonstrate the
flexibility of the procedure illustrated and with
examples. In [24] a fuzzy expected value approach
is proposed for DEA analysis, in which we first
obtain the weights of the values for the inputs and
outputs. These weights are used to measure the
optimistic and pessimistic efficiency of DMUs.
Then the geometric mean is evaluated. Fuzzy
models are built based on fuzzy arithmetic and α-
level sets, determining the ranking approach for
fuzzy efficiencies. [25] provides a model of fuzzy
DEA dynamics in a study to compare discriminating
power and perceived improvement with the aim of
improving the performance of DMUs operating with
56 railways in computational time and
discriminating power. [26] presents the model in the
fuzzy context to evaluate efficiency and productivity
in an uncertain environment with different α levels,
where decision-makers can evaluate economic and
environmental factors in the selection of sustainable
suppliers with a probability distribution. [27]
presents a new approach for priorities in the process
of fuzzy analytical hierarchy, where the fuzzy nature
of the data is maintained in all the steps of the
approach, further determines the level of
consistency, gives the pairwise comparison matrix
with appropriate index with the aim of selecting a
better ventilation system. Considering the input and
output data that may be inaccurate in [28] a possible
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approach to solving DEA models with fuzzy data is
proposed, where the fuzzy data are constructed as
trapezoidal fuzzy numbers to evaluate the technical
efficiency. To study the impact of undesirable
factors in a banking system where 12 DMUs operate
[29] analyses their effectiveness by proposing the
integration of the cross-efficiency model from DEA
and the α-cut model of fuzzy DEA. In the approach
presented in [30], the approach for an extension of
DEA is presented, where more components are
operated with variables that take the form of fuzzy
numbers for the evaluation of the fuzzy technical
efficiency and the ranking of DMUs, giving linear
programming in numerical applications. The
application of Fuzzy DEA in a two-phase process
with multiple objectives investigating the effects on
the efficiency value with the application of multiple
linear regression is seen in [31]. [32] presents an
approach based on the lexicographic language in a
linear program with many objectives for efficiency
evaluation in fuzzy efficiency models. This paper
presents an alternative approach for evaluating the
performance of DMUs along a time course
according to a given period with the evaluation of
the degree of preference based on Fuzzy arithmetic.
The evaluation of the efficiency value chain is given
by the application of DEA models, CCR efficiency,
super efficiency, Cross-efficiency value chain, and
Fuzzy efficiency where the respective rankings are
determined. The data of variable quantities are
processed in what is called statistics smoothing of
the time series. This is done as the data may not be
clearly defined (obtained from perceptions or from
surveys). The evaluation of the efficiency according
to the different models is done to provide the most
detailed performance evaluation, comparing the
models and the advantage of the application of
Cross efficiency in the composition of fuzzy
numbers and the application in the evaluation of the
degree of preference in relation to the others
models. For the evaluation of the degree of
preference based on Fuzzy arithmetic, from the
chain of cross-efficiency values evaluated in the
relevant time course, the composition of fuzzy
numbers is done first. In harmony with fuzzy
arithmetic, for two Fuzzy triangular numbers (),
where is again a triangular fuzzy number,
which can be projected geometrically as a location,
and then the geometric model of the probability of
the event is applied, enabling presentation of the
matrix of degree of preference. This alternative
approach can be applied in certain fields in the
evaluation of the degree of preference of DMUs that
operate along a given time course.
2 Methodology
The study of this work in the basic conception is
based on first evaluating the data in their dynamic
analysis, displayed during a period of time (which
can be obtained from perceptions and surveys). By
evaluating the efficiency of DMUs according to the
values obtained from data processing according to a
module k (or order k), , determined according
to the moving average (dynamic time series
analysis). The data is applied to DEA and Fuzzy
DEA models for evaluating effectiveness, ranking
DMUs, and relevant analysis by comparing the
models applied, and determining the advantages of
each model. Over an extended period of time, data
can be viewed as a time series for any variable. If
the time series is given followed by consecutive
values X1, X2, …, XN, we determine the “series” of
values according to the module of the moving
averages. The series of values according to the
model is given by the countable sums, [33],
[34]:
:
,
, …
,
.
These values, defined as input-output levels, are
applied for a more realistic assessment and
examination of the periodic trend of instantaneous
changes and the evaluation of the effectiveness of
each DMU according to the relevant DEA models.
By successively following the chain of efficiency
values with the data according to the module,
the matrix of the chain with efficiency values is also
built.
Efficiency values are calculated by applying the
DEA model input oriented according to constant
and variable returns to scale (CRS and VRS), [35]:
= min (1)
s.t: -
0 =1,2,…,m,
= min (2)
s.t
=1,2,…,m,
,
0
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The matrix is constructed with the chain of
efficiency values as below, also giving the indicator
of the change in the efficiency value (Ind. EC).
=
,
where, Ind. EC =
To increase the distinguishing power in the
ranking of DMUs (where several DMUs with the
efficiency 1), the super-efficiency
evaluation model is applied as follows [4], [35]:
min
st:
i=1,2,…,m;
(3)
, r =1,2..,s;
, j
Based on the conclusions of the efficiency
values (CRS and VRS), the scale efficiency is also
determined, which also enables the classification of
inefficiencies, [36]. For each DMU, the impact of
variable factors on efficiency values is also
determined using the formula:
W() =
100% ( is the
efficiency value according to the I-th input.)
Following the study, to determine the degree of
preference likeability based on efficiency values,
triangular fuzzy numbers are composed (from
matrices of efficiency values of one model).
Fuzzy background:
The definition of fuzzy set theory was first proposed
by [10]. Based on this concept, the field of
applications has been expanded in the presentation
of solutions to problems that use it in research
operations and other areas from the practical life of
the real world, where the data are not completely
clearly defined. Fuzzy sets are related to a
characteristic function called the membership
function. In accordance with the definitions given
by [37], [38] are given:
Definition 1: A Fuzzy set given over a collection
of objects X is defined by the set of ordered pairs
= {(x, : xX}, where is the
membership function, of the membership degree
value of each element x and that : X [0,1].
Definition 2: The set of elements of x (collection)
such that the value of the membership function is
equal to 1,=1, represents Core ( ), core of
Fuzzy set.
Mathematically, this can be given by the
equation core(= {x:=1}.
Definition 3: (α-cut) (Figure 1)
Let it be the geometric projection of a
trapezoidal number Fuzzy = (, where
are all real numbers (
).
Fig. 1: core, –cut, and support of fuzzy set, [38]
The α-cut set of a Fuzzy number (trapezoidal)
is the set given and denoted ={x:
}. Let us express the membership function,
=
where
and
are membership functions
of the left side and the right side. If according to (α-
cut) we have = then we say that:
x =
for x [, = 1.
In Figure 1, where the geometric projection of
the Fuzzy number is given, core (), cut, and
support of are presented respectively. Support of
Fuzzy set is presented when α-cut we have α=0, 0-
cut. For a Fuzzy triangular number = (
we have the following :
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=
Let two Fuzzy numbers be given: =
( and = ( and k a
scalar [38], [37], [39]:
a) Addition of two fuzzy numbers: = =
().
b) Subtraction of two fuzzy numbers: = =
().
c) Multiplication of a fuzzy number by a scalar
number:= =
• Positive Fuzzy Triangular Numbers and Negative
Fuzzy Triangular Numbers
A triangular fuzzy number given =
( is classified as positive if
(i=1,2,3) and classified as negative if
(i=1,2,3).
They are classified as partial negative if
.
If we have n DMUs, where the levels of inputs
and outputs can also be characterized by fuzzy
triangular numbers (= (
,
,
) and =
(
,
,
).The values of fuzzy efficiencies
() [22] are calculated from the models
(4), (5) and (6).
max =
st:
= 1 (4)
-
0
, r ,, i.
max =
st:
= 1 (5)
-
0
, r ,, i.
max =
st:
= 1 (6)
-
0
, r ,, i.
To evaluate the ranking of DMUs, the
geometric mean is evaluated =
[40].
• Cross-efficiency evaluation
In addition to the evaluation of efficiencies
according to the above models, the cross efficiency
evaluation model is also applied, which has a better
distinguishing power for ranking DMUs, [35].
In conventional DEA models, DMUs have the
nature of self-evaluation in the selection of weights
for each input and output, where we can have
several efficient evaluated DMUs, where Ef=1.
The Cross efficiency model is the approach
presented by [6] and [41]. The efficiency values of
each DMUs with the cross efficiency model are
evaluated not only with their own weights, but also
with the input -output weights of other units. The
average representing the Cross efficiency result for
a DMUj (j=1, 2,...,n) is given:
=
,
where this average is the average of the values
according to the column presented in the Cross
efficiency matrix, [41]. From the cross efficiency
value chain, the corresponding efficiency value
chain matrix is formatted, where the harmonic cross
efficiency for the given period and the
corresponding rankings are also evaluated. From the
cross efficiency value chain matrix for each DMU,
the triangular fuzzy numbers , are composed,
where (, where this matrix fully
enables this composition, which is advantageous
over other models.
After determining the triangular fuzzy numbers
from the cross efficiency values, based on fuzzy
arithmetic and the geometric model of the concept
of event probability, the approach for determining
the degree of preference likeability is then applied.
In the following figures, the cases of geometric
projections of triangular fuzzy numbers are given
when their comparison is required. If two triangular
fuzzy numbers = () and = (
according to fuzzy arithmetic and fuzzy triangular
number =( = ()
where for each of them their geometric projections
can be given in a coordinate plane. For each case, it
can be judged according to the concept of the
geometric model of the probability of event A, the
probability of this event is estimated. Case 1: In the
Figure 2, the geometric projection of the two fuzzy
numbers = () = (5, 7, 9.5) and =
( = (1, 2, 3.5) is given. The fuzzy
triangular number =( = (1.5, 5, 8.5), where
it seems that the fuzzy triangular number is
classified as positive, since each of its components
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