Available via license: CC BY-NC-ND 4.0
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How to Cite:
Hashemi, S. F., & Khorsandi, H. (2022). Two-stage network structure with undesirable outputs
approach. International Journal of Health Sciences, 6(S7), 1499–1518.
https://doi.org/10.53730/ijhs.v6nS7.11557
International Journal of Health Sciences ISSN 2550-6978 E-ISSN 2550-696X © 2022.
Manuscript submitted: 9 April 2022, Manuscript revised: 18 June 2022, Accepted for publication: 27 July 2022
1499
Two-stage network structure with undesirable
outputs approach
Seyed Farid Hashemi
Department of Mathematics, Graduate Student of (M.Sc.)/Operations Research,
Central Tehran Branch, Islamic Azad University, Iran.
Hosna Khorsandi
Department of Marketing, Graduate Student of Business
management/transformation, Central Tehran Branch, Islamic Azad University,
Iran.
Abstract---Articles in which nonparametric data envelopment analysis
(DEA) is applied to estimate the efficiency based on a network
structure usually consider the desirable intermediary parts that are
the first stage’s outputs to be used as the second stage’s inputs. In
many real-life situations, the intermediary parameters include
desirable and undesirable outputs. This issue has drawn much
attention from DEA researchers in recent years. The main motivation
for this application is weak disposability for modeling the networks
with undesirable intermediary parts. Undesirable outputs have been
investigated in this study in two forms, i.e., the final outputs or
intermediary parameters. In both cases, the non-cooperative game
theory is offered for evaluating the relative efficiency of the executive
units. In 2008, a real case was explicated for 39 airports in Spain in
applied analysis of the proposed approaches.
Keywords---DEA, efficiency, desirable and undesirable output, leader-
follower game theory, concentrated model
.
Introduction
DEA is the science of mathematical planning based on the performance of the
decision-making units. In most DEA articles, a two-stage network structure is
considered with desirable scales for the first stage’s outputs to be used as the
second stage’s inputs [1]. In many real-life situations, the middle sections have
desirable and undesirable outputs. This issue has recently been taken into
account by DEA researchers. DEA was started by Charnes et al. [2] and expanded
by Bengar et al. [3]. Currently, DEA plays an essential role in the analysis of
undesirable outputs. The modeling’s undesirable factors have been considered
1500
not only for assessing efficiency and productivity but also for doing research
about, saying, pollution reduction. This approach has been challenged in the
writings by Hailu and Veeman [4], Grosskopf and Fare [5] and, also Hailu [6] and
Kuosmanen [7]. Based on the traditional approach, weak disposability modeling
(reduction of the undesirable outputs by lowering the production activities) has
been proposed by Shephard [8], who uses the unit abatement factor for all
activities observed in a sample. Kuosmanen [7] points out that applying an
integrated abatement factor with a concentration on reductive factors in the
companies to decrease emission costs is unreasonable. Podinovski and
Kuosmanen [9] created two other technologies for modeling weak disposability
based on allocated convexity theory. At present, inappropriate network structures
with undesirable outputs are deemed as the causes of weak disposability and, to
the best of the author’s knowledge, few works have been done based on DEA in
this regard with the consideration of the undesirable variables in the network
structure-based production systems. In real cases, the common generation of the
desirable and undesirable outputs causes the overall assessment and two-stage
network structure application to become problematic. In the studies by Cook et
al. [10], four topics have been offered for assessing the two-stage systems: the
standard DEA approach, efficiency analysis, network data envelopment analysis
and game theory approaches [11, 12]. As Hwang and Kao [13] pointed out, in a
topological study of standard DEA, every process is discussed as an independent
system. The implementation of these two processes is shown by the common
envelopment constraints in a calculation of unit system efficiency and overall
system efficiency; the product is the efficiencies of both of the two stages. These
two aspects have been considered in many of the researches such as those done
by Zhu about international companies [14], by Seiford and Zhu about banking
firms [15], by Lewis and Sexton on baseball [16] as well as by Wang et al. [17] on
IT articles. Liang et al. [18] performed more precise investigations on the two-
stage network structure for which a concept of the non-cooperative approach is
applied. This vista is specified through Stackelberg’s game theory or leader-
follower game theory.
Since a decade ago, increasing daily attention has been paid to productivity and
efficiency management methods. In this regard, the polluting undesirable outputs
should be taken into consideration. The primary objective of this research is the
application of the “weak disposability approach” for modeling a two-stage DEA
network through intermediaries’ undesirability measurements. The modeling’s
undesirable factors should be considered concerning efficiency and productivity
measurements and the abatement factor for air pollution. In real cases,
production accompanied by desirable and undesirable outputs causes problems
in assessing the efficiency and productivity in the two-stage network structures’
modeling. Shephard has used a traditional modeling approach (reducing
undesirable outputs by lowering the production activity rate); he points out that
the integrated reduction of production does not bring about cost reductions. This
research examines the DEA approach by applying weak disposability assumption
to the two-stage network structure with fundamental undesirability scales.
1501
Weak Disposability Technology:
Modeling the production activities’ undesirable outputs like emission of harmful
particles into the air and energy wastage in the power plants has drawn much
attention from researchers. Hailu and Veeman [4] expanded the non-parametrical
productivity analysis models that include undesirable outputs. They introduced
an informal uniformity condition to the technology and claimed that it could
better serve the weak disposability concept in DEA. Fare and Grosskopf showed
that the use of uniformity conditions for weak disposability modeling conflicts
with a law in physics.
According to Hailu and Veeman [4], the outputs are denoted by ,
good or desirable outputs by , or undesirable outputs by
. The technology includes all the possible entities (x, v, w) as shown
below:
Y={(x, v, w)} if x can generate (v, w).
Hailu and Veeman’s condition of uniformity can be applied as below:
If , then (1)
To showcase the assumption, the outputs are themselves defined as below:
According to Shephard, outputs are weakly accessible. In case the reduction in
the ratio of the outputs is found possible:
This presumption by Fare and Primont [17] for the weak disposability of the
outputs somewhat means that one of the outputs is possibly undesirable.
Assume that there are k numbers of DMUs; and, for DMUk, the data related to the
input vectors and the desirable and undesirable outputs include:
Also, assume that . The production technology can be
presented as shown in the following formula:
P(x)={(v, w) if x can produce (v, w) with
Definition 1: the outputs (desirable and undesirable) are weakly available if and
only if then
1502
Fare and Grosskopf [5] suggested the following technology assuming the variable
ratio of output to scale for weak disposability:
Blending the parameter θ into formula (2) depends on Shephard’s definition of
weak disposability. This parameter allows the simultaneous reduction of both the
good and bad outputs.
However, Kuosmanen claims that there is no special justification for applying
identical constriction factors to all the units. He also expresses that the correct
exertion of the principle of “weak disposability” entails using a distinct
constriction factor. These stated axioms are:
(A1): strong disposability between the inputs and good outputs; if
, then
(A2): weak disposability between the desired and undesired outputs; if
, then
(A3): T is convex.
As pointed out by Kuosmanen, this model uses an identical abatement coefficient
for all companies. To allow the use of a non-identical abatement coefficient for
private companies, he suggests the following construction technology:
It must be noted that formula (3) is a special case of (3) or . To calculate
weak disposability, the above formula introduces the abatement factor that
1503
leads to the gradual reduction of both the good and bad outputs by the same
section; hence it is in accordance with Shephard’s definition. It is noteworthy that
this is a certain section’s abatement factor, but it can also be applied for bringing
about non-uniform abatement in the whole company. Resultantly, formula (3) is
consistent with Shephard’s assumptions and definitions of weak disposability and
supports them. Free disposability of the goods’ inputs and outputs has been
modelled through unequal limitations according to x and v. Nonlinear Tk
technology can be recovered in a rectilinear form using a simpler and more
effective method.
Weak Disposability in the Process of Two-Stage Decision-Making:
A two-stage decision-making process, including mean desirable and undesirable
outputs’ measurements, has been introduced in this part. Imagine a two-stage
production process as exhibited in figure (1).
Figure (1): the overall structure of the two-stage network
Now, assume that there are k numbers of DMUs and, for the first stage, DMUs
are the data observed in the input vectors regarding the desirable and undesirable
outputs in the following order:
are the outputs used as the inputs to the next stage. The second stage is
fed with
and an external input vector .
The final DMUk product is specified . There are two different
methods introduced for this decision-making process. A non-cooperative game
theory was introduced in the first method, and in the second method, attention
was directed at a concentrated model.
Leader-Follower Game (Play) Theory:
In this part, the leader-follower method has been developed for analyzing this
expanded two-stage structure. In the non-cooperative leader-follower game
theory, the leader is given a higher priority than the follower. The leader is a lot
better than the follower in this case.
1504
The algebraic model of the first stage’s production technology has been given
beneath:
Considering the uncertain variables μ and ρ, the above technology is linear.
Attention should be paid to radial size when using this well-described model by
which DMU0 output is intended to be measured under the conditions of
undesirable outputs’ abatement potentials. This optimum value is obtained based
on the following model:
The intended function minimizes the abatement factor to an equal ratio of all the
undesirable outputs along with preserving the current level of the desired inputs
and outputs. Model (5) is a linear programming problem and is always possible
and constrained.
Considering this point, a proposed network is efficient in one stage according to
the non-cooperative game theory if both stages are found separately efficient. In
case of the inefficiency of DMU0, we will have the following in the first stage:
1505
Where,
and
are the first and second constraints’ slack variables. Stage 1
can be improved by erasing undesirable inputs and outputs surplus and
compensating for the output shortages.
It is easy to show that the improved leader is now effective. Having the first stage’s
output enables the evaluation of the second stage while keeping the first stage’s
output. Following the lead of Kuosmanen, weak disposability can be formulated in
parts of numerical factor θ in all of the DMUs. Based on these presumptions, the
production set can be written in the following form:
Like the weak disposability behaviour, formula (7) uses the abatement factor ,
which reduces both the good and bad outputs in the same fraction based on Fare
and Graskopf’s definition.
Now, the very Kuosmanen’s method is applied for transforming the nonlinear
technology posited in (7) into a linear form.
1506
Then, are considered. Rearranging the conditions in (7), an
equivalent display of the construction technology in (7) is obtained as shown
underneath:
This technology is linear under the conditions that α and β variables are unclear.
Based on Stackelberg’s game theory for a two-stage process, the second stage
only considers the favorable solutions that can keep the first stage’s productive
configurations. The second stage uses a tripartite (made of three components) (x,
y and z) for a range within which the first stage’s outputs are scored desirable.
The following linear model is to be solved to evaluate the second stage.
1507
In this stage, the m-th desirable output, i.e.
, uses j-th undesirable
output, i.e. , as a constant. These constants are the first stage’s DMU0
outputs. Therefore, the right side of the two first formulas in the model (9) can
keep the first stage’s output. It must be noted that the system is effective if and
only if the two components' processes are effective.
Concentrated Model:
Under real-world circumstances, there are many cases in which sub-DMUs work
together to achieve a comprehensive system’s overall performance. For example,
the marketing and construction departments cooperate to maximize the
company’s profits. This section considers a concentrated method for evaluating
the relative performance of two-stage structures. In these methods, the two-stage
process is envisioned as a one-stage process that continuously estimates an
optimal plan for maximizing the overall output of the whole system. The mean
size is an undesirable output that should be abated in both stages. Although the
mean size is the first stage’s desirable output to be inputted into the second
stage, it might logically seem that it has to have been increased in the first stage
and reduced in the second stage. Considering the good outputs, v, there are two
logical behaviors:
i) One can remain unchanged because it has to be increased on the one
side and reduced simultaneously on the other side;
ii) The mean size is a system’s good product that its system can apply.
Thus, it appears that it can be used for the whole system to increase the desirable
output v. Herein, the author has used the second method for evaluating the two-
stage process. In case that it is intended to measure the performance of DMU0
regarding bringing about reductions in the undesirable outputs’ potentials and
decreases in outputs’ potentials, the following linear programming problem
should be solved:
S.T.
Stage One’s Constraints:
1508
S.T.
Stage Two’s Constraints:
Generic Constraints:
In the above formulas are constraints of the interventions
required for remaining dominant. The intended function can be decomposed into
two periods: the first period is
which is Russell’s input size,
the first stage’s bad output, and the second period is
, the
second stage’s size.
Final Undesirable Outputs:
This section considers the concentrated method for evaluating the temporal two-
stage process’s performance with the undesirable outputs as the final output. As
seen in figure (2), the first stage’s desirable size has been used for the second
stage. The mean undesirable size of the system is left unchanged in this stage.
Therefore, in this method, improving the first stage’s output through increasing
the number of the outputs influences the second stage’s output. In other words,
the size quantities from the mean to the optimal are considered the final output
and are not viewed as the second stage’s inputs.
1509
Figure (2): two-stage network structure with final undesirable outputs
The process can be seen in situations like evaluating the relationships between
micro-and macro-level productions in the supply chain management methods or
airport modelling operations. The use of DEA for modelling such situations
decreases the inputs, increases the desirable outputs, and decreases the
undesirable outputs. Therefore, the upcoming sections present a concentrated
DEA network model with variable output ratio to scale and weak disposability
attributes for optimal final outputs. The first stage of this model takes the
following form:
(11)
S.T.
Stage One’s Constraints:
S.T.
Stage Two’s Constraints:
1510
Generic Constraints:
In the above formulas, the constraints are the requirements for
gaining domination. The objective function can be attached to the middle of Amin
in the Thatcher Section: the first part is Russell’s input size and the first stage’s
undesirable output as the input to the second stage; the second part is Russell’s
input size for the second stage.
Problem Solving:
To specify the practical concept of the proposed methods, we will apply methods
to real cases like 39 Spanish airports in 2008, as cited in Lozano et al.
After formulating the method’s framework, it has to be showcased through
empirical methods. It has been found in an analysis of the productivity based on
parametric and nonparametric techniques that the airport’s performance is
suffering now for a long time, with the initial research interests having been
incited by the high socioeconomic significance of the issue.
The existing DEA studies about the airport’s framework have considered an
airport a single process. The slack-based DEA network method has been proposed
by Yu et al., but it does not consider the undesirable outputs. As stated by Lozano
et al., consideration of undesirable outputs increases the analysis ratio and ends
in a fairer performance evaluation. This research has been motivated by testing
the application of disposability in DEA network modelling based on mean
undesirable sizes. To reach a second suggestion, the methods posited in a case
study of 39 Spanish airports have been taken into account, as cited in Lozano et
al.
Similar to prior research on airport performance evaluation, airport processes are
divided into two stages:
Airplane movement process
Airplane loading process
The three inputs of the first stage are seminally specified: the whole of a region’s
airport (x1), the precinct’s capacity (x2) and the number of exit gates (x3). As for
1511
the second stage’s output, annual passenger motions (y1) and the number of
landing times (y2) have been considered. Amongst the inputs added for the second
stage, the number of luggage belts (z1) and the number of check-in-out counters
(z2) can be pointed out. The cross-mean of the airplanes’ takeoff and landing
numbers has been considered as the mean desirable size, the mean number of
the delayed flights (w1), and the density of passengers until the announcement of
the second flight (w2) have been considered as the mean undesirable sizes. As
discussed by Lozano et al., these mean w1 and w2 sizes are the first stage’s final
outputs. The important point of this research and the primary difference between
the method proposed herein and the other methods is that this research’s
proposed method considers that the number of delayed flights and the density of
delayed flights’ passengers in the airport directly influence the second stage
(airport loading process). These undesirable outputs have been used herein as the
inputs of the second stage. To see how the weak disposability assumption
influences the two-stage structures, both the cooperative and non-cooperative
game methods have been considered. At first, the leader-follower game method
proposed in the related articles was applied, and it is assumed that the leader
should be the airplane movement process.
The following Table presents the first stage’s desirable scores as drawn on the
non-cooperative method posited herein in leader-follower game theory with
desirable amounts for the objective points.
Table 1
The result of the stage one’s efficiency and planning evaluation
Airport
e1α
v1
w1
w2
x1
x2
x3
Acoruna
0.32
51
17.719
0
395.939
1
7731.322
5
78375.82
86
5.000
4.000
Albacete
0.20
14
2.1130
11.6833
277.2762
112666.8
227
2.000
1.993
9
Alicante
0.55
64
81.097
0
4242.34
24
79263.36
03
135000.0
00
25.78
24
16.00
0
Almeria
0.26
88
18.280
0
399.492
8
5416.975
6
80251.93
78
7.000
3.608
9
Asturias
0.27
28
18.371
0
357.426
9
6519.220
2
76282.52
83
7.000
3.608
9
Badajoz
1
4.0330
137.000
0
2365.400
0
171000.0
00
1.000
2.000
Barcelona
1
321.69
30
33036.0
00
645924.6
000
475020.0
000
121.0
00
65.00
00
Bilbao
0.37
30
61.682
0
1713.02
03
30159.97
49
133926.9
648
13.85
23
11.92
93
Cordoba
1
9.6040
14.0000
254.4000
62100.00
0
23.00
0
1.000
El Hierro
1
4.0775
0
27.0000
641.6000
37500.00
0
3.000
2.000
1512
Fuertevent
ura
0.27
82
44.552
0
1090.72
15
20083.66
16
116641.8
814
18.81
45
10.00
0
Girona-
Costa Brav
0.34
81
49.927
0
1737.71
46
34.916.36
75
108000.0
00
12.68
57
6.638
7
Canaria
Gran.
1
116.25
20
7463.00
000
136380.7
000
139500.0
00
55.00
0
38.00
0
Granada-
Jaen
0.43
68
19.279
0
415.381
1
7804.796
3
711154.5
099
11.00
0
3.000
Ibiza
1
57233
0
6193.00
0
152840.1
000
126000.0
00
25.00
0
12.00
0
Jerez
1
50.551
0
1174.00
00
19292.20
00
103500.0
00
9.000
5.000
La Gomera
1
3.3930
17.0000
420.7000
45000.00
0
3.000
2.000
La Palma
1
20.109
0
423.000
0
8286.000
0
99000.00
0
5.000
5.000
Lazarote
0.37
36
53.375
0
1906.95
21
37991.68
74
108000.0
00
11.95
47
6.397
8
Leon
0.13
44
5.7050
59.4138
966.6842
66542.60
79
5.000
1.903
9
Madrid
Barajas
1
469.74
60
52526.0
000
903860.0
000
927000.0
00
263.0
00
230.0
00
Malaga
1
119.82
10
15548.0
000
277663.8
000
144000.0
00
43.00
0
30.00
0
Melilla
1
10959
0
218.000
2979.600
0
64260.00
0
5.000
2.000
Murcia
0.29
25
19.339
0
395.772
26
7097.728
8
73638.71
12
5.000
3.710
5
Palmade
Mallorca
0.65
52
193.37
90
17060.9
591
328590.2
194
267737.2
523
78.91
53
47.04
21
Pamplona
1
12.971
0
666.000
0
11691.80
00
99315.00
0
7.000
2.000
Reus
1
26.676
0
943.000
0
18240.80
00
110475.0
00
5.000
5.000
Salamanca
1
12.450
0
427.000
0
6626.100
0
150000.0
00
6.000
2.000
San
Sebastian
0.30
76
12.282
0
219.302
0
3439.934
2
57029.67
77
6.000
2.222
0
Santander
0.36
91
19.198
0
370.625
5
6586.355
1
73730.15
53
8.00
3.271
8
Santiago
0.19
18
21.925
0
382.976
0
6583.588
5
79197.02
76
16.00
0
2.756
6
Saragossa
0.20
23
14.584
0
221.525
2
3954.956
8
65860.75
82
12
2.282
6
Seville
0.99
56
65.067
0
2555.65
39
50859.10
52
120646.3
647
17.61
25
10
Tenerife
North
1
67.800
0
1783.00
00
32637.00
00
153000.0
00
16.00
0
16.00
0
Tenerife
0.44
60.779
2357.03
49715.24
140286.7
18.13
13.97
1513
South
86
0
39
72
337
09
79
Valencia
1
96.795
0
4998.00
00
102719.2
00
144000.0
00
35.00
0
18.00
0
Valladolid
0.25
36
13.002
0
213.812
3
3743.770
2
54999.44
63
7.000
2.491
3
Vigo
0.21
89
17.934
0
236.052
8
5603.127
0
60209.18
49
8.000
2.652
5
Vitoria
0.17
51
12.225
0
117.117
9
2028.257
5
64601.99
37
18.00
0
1.615
5
Similar results have been reported for the second stage, as exhibited in Table (2).
Table 2
Results of the second stage’s efficiency and planning evaluation
Airport
e2α
z1
z2
y1
y2
Acoruna
0.4607
4.6072
0.6493
1174.97
283.571
Albacete
0.2504
1.0015
0.0005
22.2416
8.924
Alicante
0.9174
38.5329
5.6282
9578.304
17236.65
Almeria
0.2617
4.4384
0.6436
1024.303
35.593
Asturias
0.5167
5.6841
0.8312
1530.245
139.465
Badajoz
0.2955
1.1821
0.0321
81.01
1.9719
Barcelona
1
143
19
30272.08
103996.5
Bilbao
0.5672
20.41197
3.6713
4172.903
4486.831
Cordoba
1
1
0
22.23
0
El Hierro
0.3123
1.5614
0.1077
195.425
171.717
Fuerteventura
0.4631
15.745
2.7926
4492.003
2722.661
Girona-Costa
Brava
1
18
3
5510.97
184.127
Gran Canaria
0.5619
48.324
7.1008
10212.12
33695.25
Granada-Jaen
0.4449
5.3385
0.7667
1422.014
66.889
Ibiza
0.4165
19.9924
3.3321
4647.36
4992.297
Jerez
1
12
2
1203.817
90.428
La Gomera
0.2124
1.062
0.0113
41.89
7.863
La Palma
0.406
5.2782
0.812
1151.357
1277.264
Lanzarote
0.4055
19.8717
3.2444
5438.178
5429.589
Leon
0.4382
1.3146
0.0562
123.183
15.979
As shown in Table (2), 17 airports are found effective in the first stage when the
airplane movement process is considered the leader. Preserving these effective
airports for the second stage, seven other airports are found effective in the
second stage. Having a glance at the second column in Tables (1) and (2) makes it
clear that only four airports are efficient in the general sense of the term
(Barcelona, Cordoba, Jerez and Madrid Airports). It is seen in the observation of
columns 4 and 5 in Table (1) that the first and second undesirable outputs have
undergone reductions, respectively, for 973.05 and 18441.61.
A concentrated model [10] has also been used for the airport data according to the
results reported in tables (3) and (4). The first three columns in Table (3) report
1514
the outputs’ scores in the course of the stages. The seven last columns of this
Table show the influence factors. As seen therein, out of 39 airports, 24 are
completely efficient. It is discerned in a test of columns 6 and 7 in Table (1),
columns 4 and 5 in Table (2) and columns 6 and 7 in Table (4) that the
intermediaries’ reductions in the first and second undesirable outputs’ abatement
stages are considerably larger in non-cooperative method than the intermediaries’
reductions through the use of the cooperative method meaning that, in this
example, the non-cooperative method, significantly reduces the undesirable
outputs.
We use the concentrated model for the airport data based on the results reported
in tables (3) and (4). The first three columns in these tables give the outputs’
scores during the stages. The seven last columns of these tables display the
influence factors.
Table 3
Total output along with the total results
Airport
eα(tota
l)
eα(1)
eα(2)
β1
β2
θ1β
3
θ2
φ2
φ1
Acoruna
0.7654
0.838
3
0.692
5
1
1
1
0.6
0.5
9
0.8
2
0.7
6
Albacete
0.7698
0.785
8
0.753
7
0.9
1
1
0.9
9
0.6
5
0.3
8
1
0.9
9
Alicante
1
1
1
1
1
1
1
1
1
1
Almeria
0.4154
0.457
3
0.373
5
0.5
3
0.4
7
0.6
2
0.3
6
0.3
0.4
3
0.2
Asturias
0.5718
0.576
4
0.567
1
0.8
2
1
0.4
8
0.3
0.4
9
1
0.6
8
Badajoz
1
1
1
1
1
1
1
1
1
Barcelona
1
1
1
1
1
1
1
1
1
1
Bilbao
0.6734
0.701
7
0.645
1
0.6
5
0.8
1
0.5
2
0.5
2
0.8
1
0.7
1
Cordoba
0.875
1
0.75
1
1
1
1
1
1
0
El Hierro
1
1
1
1
1
1
1
1
1
1
Fuerteventu
ra
0.5945
0.651
3
0.537
6
0.8
4
0.5
6
1
0.2
2
0.4
4
0.7
2
0.5
8
Girona-
Costa Brava
1
1
1
1
1
1
1
1
1
1
Gran
Canaria
1
1
1
1
1
1
1
1
1
1
Granada-
Jaen
1
1
1
1
1
1
1
1
1
1
Ibiza
0.5885
0.68
0.497
1
1
0.6
9
0.9
2
0.4
5
0.3
3
0.5
7
0.6
3
Jarez
1
1
1
1
1
1
1
1
1
1
La Gomera
1
1
1
1
1
1
1
1
1
1
La Palma
1
1
1
1
1
1
1
1
1
1
Lanzarote
0.7394
0.790
0.688
1
0.8
0.6
0.7
0.7
0.5
0.7
1515
3
4
8
6
1
7
1
Leon
1
1
1
1
1
1
1
1
1
1
Madrid
Barajas
1
1
1
1
1
1
1
1
1
1
Malaga
1
1
1
1
1
1
1
1
1
1
Melilla
1
1
1
1
1
1
1
1
1
1
Murcia
1
1
1
1
1
1
1
1
1
1
Palma De
Mallorca
1
1
1
1
1
1
1
1
1
1
Pamplona
1
1
1
1
1
1
1
1
1
1
Reus
1
1
1
1
1
1
1
1
1
1
Salamanca
1
1
1
1
1
1
1
1
1
1
San
Sebastian
0.5552
0.620
6
0.489
7
0.8
4
0.8
8
0.7
2
0.3
5
0.3
2
0.7
4
0.5
5
Santander
0.5968
0.586
6
0.607
0.7
3
0.8
3
0.6
0.4
2
0.3
6
0.8
6
0.7
9
Santiago
0.4022
0.427
9
0.376
4
0.6
6
0.6
4
0.3
6
0.2
5
0.2
3
0.5
7
0.4
5
Saragossa
1
1
1
1
1
1
1
1
1
1
Seville
1
1
1
1
1
1
1
1
1
1
Tenerife
North
1
1
1
1
1
1
1
1
1
1
Tenerife
South
0.7181
0.794
6
0.641
5
1
0.7
5
1
0.6
5
0.5
7
0.6
0.7
4
Valencia
1
1
1
1
1
1
1
1
1
1
Valladolid
0.439
0.238
4
0.439
6
0.3
7
0.7
8
0.4
6
0.3
2
0.2
6
0.6
0.5
8
Vigo
0.5056
0.565
2
0.446
0.7
5
1
0.5
8
0.2
6
0.2
4
0.6
9
0.6
Vitoria
1
1
1
1
1
1
1
1
1
1
Table (4) presents the results of the concentrated points.
Table 4
Results of the concentrated points
Airport
x1*
x2*
x3*
v1*
w1*
w2*
z1*
z2*
y1*
y2*
Acoruna
87.30
0
5
4
17.7
2
727.
78
14132.
09
8.2
3
2.2
7
1174.
97
283.5
7
Albacete
1471
38.4
2
1.9
9
2.11
37.5
1
523.12
4
0.9
9
123.3
8
70.59
Alicante
135.0
00
31
16
81.1
7624
14244
5.8
42
9
9578.
3
5982.
31
Almeria
7677
3.82
7
3.1
2
18.2
8
399.
26
6143.8
6
7.2
5
1.6
2
1024.
3
304.2
6
Asturias
8082
4.01
7
4.3
18.3
7
390.
88
6914.4
8
11
2.0
4
1530.
24
472.3
3
Badajoz
171.0
00
1
2
4.03
137
2365.4
4
1
81.01
0
1516
Barcelon
a
475.0
20
12
1
65
321.
69
32.0
36
64592
4.6
14
3
19
3027
2.08
1039
96.5
Bilbao
3.213
354
16.
77
12
61.6
8
3410
.8
43649.
86
29.
04
4.9
6
4173.
9
1083
6.53
Cordoba
62.10
0
23
1
9.6
14
254.4
1
0
22.23
0
El Hierro
37.50
0
3
2
4.78
27
641.6
5
1
195.4
3
171.7
2
Fuerteve
ntura
1291
37.3
19.
07
10
42.5
5
1706
.24
30046.
04
22.
53
4.6
2
4492
6765.
66
Girona-
Costa
Brava
108.0
00
17
7
49.9
3
4992
10030
5.6
18
3
5510.
97
184.1
3
Gran
Canaria
139.5
00
55
38
116.
25
7463
13638
0.7
86
19
1021
2.12
3369
5.25
Granada
-Jaen
134.5
50
11
3
19.2
8
951
17868.
8
12
3
1422.
01
66.89
Ibiza
126.0
00
17.
3
11.
09
57.2
3
2788
.87
80977.
44
27.
58
8.0
2
4647.
36
7518.
65
Jarez
103.5
00
9
5
50.5
5
1174
19292.
2
13
3
1303.
82
90.43
La
Gomera
45.00
0
3
2
3.39
17
420.7
5
1
41.89
7.86
La Palma
99.00
0
5
5
20.1
1
423
8286
13
2
1151.
36
1277.
26
Lanzarot
e
108.0
00
19.
12
10.
94
53.3
8
3885
.92
72199.
4
28.
05
5.6
8
5438.
18
5429.
59
Leon
94.50
0
5
2
5.7
442
7191.5
3
1
123.1
8
15.98
Madrid
Barajas
927.0
00
26
3
23
0
269.
75
52.5
26
90836
0
48
4
53
5082
6.49
3291
86.6
Malaga
144.0
00
43
30
119.
82
15.5
48
27776
63.8
85
16
1281
3.47
4800.
27
Melilla
64.26
0
5
3
10.9
6
218
2979.6
4
1
314.6
2
386.3
4
Murcia
138.0
00
5
5
19.3
4
1344
24103.
1
18
4
1876.
26
2.73
Palma
De
Mallorca
295.6
50
86
68
193.
38
26.0
38
50148
6
20
4
16
2283
2.86
2139
5.79
Pamplon
a
99.31
5
7
2
12.9
7
666
11691.
8
4
1
434.4
8
52.94
Reus
110.4
75
5
5
26.6
8
943
18240.
8
8
3
1278.
07
119.8
5
Salaman
ca
150.0
00
6
2
12.4
5
427
6626.1
4
2
60.1
0
San
Sebastia
n
6612
5.17
5.2
7
2.1
6
12.2
8
250.
42
3540.2
8
4.4
7
1.0
9
403.1
9
373.5
7
1517
Santand
er
7578
0.8
6.6
4
2.9
8
19.2
419.
81
6468.5
8
6.8
9
1.5
7
856.6
1
307.3
Santiago
9497
4.94
10.
18
4.3
6
21.9
4
497.
53
7975.6
10.
84
2.2
7
1917.
47
2418.
8
Saragoss
a
302.3
10
12
3
14.5
8
1095
19547.
6
6
2
594.9
5
2143
8.89
Seville
151.2
00
23
10
65.0
7
2567
51084.
9
42
6
4392.
62
6102.
26
Tenerife
North
153.0
00
16
16
67.8
1783
22.637
37
5
4236.
62
2078
1.67
Tenerife
South
144.0
00
23.
21
22
60.7
8
3419
.17
62889.
89
52.
29
10.
42
8251.
99
1731
1.92
Valencia
144.0
00
35
18
96.8
4998
10271
9.2
42
8
5779.
34
1322
5.8
Valladoli
d
6742
2.79
5.4
8
2.2
8
13
268.
3
3853.3
6
4.8
1
1.1
6
479.6
9
365.1
5
Vigo
8142
6.74
8
3.4
6
17.9
3
392.
26
6140.4
1
8.2
9
1.7
9
1278.
86
1481.
94
Vitoria
1575
00
18
3
12.2
3
669
11585.
8
7
2
67.82
3498
9.73
Conclusion
The analysis of the two-stage network structure efficiency has drawn many DEA
researchers’ attention in recent years. The extant studies on the network DEA
demonstrate the existence of undesirable and desirable products in two-stage
processes. The previous approaches in network DEA could not evaluate the
efficiency properly. The present study used a two-stage DEA approach for
analyzing the implementation of these processes along with the undesirable
intermediary parameters. Two different two-stage structures were considered and
used in every structure for different cases (one with harmful outputs as the final
outputs and the other with harmful outputs as the intermediary parameters). The
present article’s quotient of the research body in this field is its application of
weak disposability assumption in two-stage processes in case of desirable and
undesirable outputs. The approach above has been explicated through a set of
real data procured about 39 Spanish airports for 2008.
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