# Transmission properties of microwaves
at an optical Weyl point
in a three-dimensional chiral photonic crystal

Shun Takahashi  1,* Souma Tamaki  1 Kenichi Yamashita  1 Takuya Yamaguchi  1
Tetsuya Ueda  1 and Satoshi Iwamoto2,3  1Kyoto Institute of Technology, Sakyo,
Kyoto 606-8585, Japan
2Research Center for Advanced Science and Technology, The University of Tokyo,
Meguro, Tokyo 153-8505, Japan
3Institute of Industrial Science, The University of Tokyo, Meguro, Tokyo
153-8505, Japan<EMAIL_ADDRESS>
###### Abstract

Microwave transmission measurements were performed for a three-dimensional
(3D) layer-by-layer chiral photonic crystal (PhC), whose photonic band
structure contains 3D singular points, Weyl points. For the frequency and
wavevector in the vicinity of a Weyl point, the transmitted intensity was
found to be inversely proportional to the square of the propagation length. In
addition, the transmitted wave was well-collimated in the plane parallel to
the PhC layers, even for point-source incidence. When a plane wave was
incident on the PhC containing metal scatters, the planar wavefront was
reconstructed after the transmission, indicating a cloaking effect.

††journal: oe††articletype: Research Article

##  1 Introduction

Singularity in a band structure indicates various exotic phenomena in
electrical transport as well as in classical wave propagation. One of the
famous singularities is the Dirac point in two-dimensional (2D) layered
materials such as graphene. At a Dirac point, two linearly dispersive bands
intersect in the 2D momentum space, and ideally induce massless transport,
which has been studied for high electron mobility [1]. In addition, because a
topological charge appears by lifting a Dirac point degeneracy, topological
edge states have been intensively studied not only in electronics [1, 2] but
also in photonics [3] and phononics [4].

As a three-dimensional (3D) counterpart of a Dirac point, a Weyl point has
also been studied intensively after its experimental discoveries in condensed
matter physics [5, 6] and microwave photonics [7]. To form Weyl points, the
necessary condition in the studied systems is to break either spatial
inversion ($P$) symmetry or time-reversal ($T$) symmetry [2]. Particularly for
classical waves that are usually insensitive to external magnetic fields for
breaking $T$, most of the experimental reports have adopted to break $P$ in
various 3D structures for elastic waves [8, 9, 10, 11], microwave [7, 12, 13,
14, 15], and infrared light [16, 17, 18]. The use of synthetic dimensions for
Weyl points has also been theoretically studied in layered metamaterials [19,
20]. As is the case with Dirac points, topological surface states appear by
selecting a particular 2D momentum plane and lifting the Weyl degeneracy
apparently [12, 16, 13, 14, 10, 8, 9, 11]. In four-dimensional band
structures, these surface-state modes form open arcs, known as Fermi arcs or
surface-state arcs, connecting two Weyl points with opposite signs of
topological charges [2, 13, 10, 8, 11]. Recent reports show one-dimensional
(1D) Hinge states derived from Weyl points [21, 22]. In addition, by using
dispersion near Weyl points, negative refraction has been reported for elastic
waves [8, 11, 23].

Another exotic phenomenon at such singular points is the characteristic power
law of electric current or photonic transmission intensity. At a Dirac point,
electrical conductivity has been analytically revealed to be constant in both
diffusive [24] and ballistic [25, 26] systems, which is also known as pseudo-
diffusive transport. Thus, electric current is inversely proportional to the
length of conductors. In photonic systems, the transmission intensity (photon
current) is inversely proportional to the propagation length $L$ [27, 28, 29].
In contrast, at the Weyl point, the electrical conductivity has been
theoretically found to be $\propto$ $1/L$ in both diffusive [30] and ballistic
[31] systems. Analogous to a Dirac point, the transmission intensity at an
optical Weyl point may exhibit a power law $\propto$ $1/L^{2}$, although such
propagation properties inside a 3D structure at a Weyl point have not been
studied. In addition, based on the pseudo-diffusive propagation at Dirac
points, phase reconstruction was demonstrated by measuring a planar wavefront
of the transmitted field through 2D phononic crystals for a plane-wave
injection in the direction parallel to a Dirac-point wavevector [32, 33]. The
zero-refractive-index effects due to accidental Dirac points at the
$\it{\Gamma}$ point can also provide wavefront control [34, 35]. However, such
characteristic wavefronts in transmitted waves for Weyl points have not yet
been investigated.

In this study, we fabricated all-dielectric 3D chiral photonic crystal (PhC)
possessing Weyl points in the photonic band structure, and measured microwave
transmission at a frequency of $\sim$ 5 GHz in the vicinity of a Weyl point.
We found unique transmission properties at the Weyl point, such as a
transmission dip at the frequency, and a power law $\propto$ $1/L^{2}$ of the
microwave intensity inside the PhC. In addition, the transmitted wave was
found to be well-collimated in the plane parallel to the PhC layers, even for
point-source incidence. We also demonstrated a reconstructed planar wavefront
for plane-wave incidence even when the chiral PhC contained metal inclusions,
indicating a cloaking effect. These results may be significant steps for a
deeper understanding of Weyl physics in the future.

##  2 Sample structure

Figure 1:  (a) Schematic of the studied 3D chiral PhC. Parts of the structure
are removed for clarity. (b) Top view of the chiral PhC. To distinguish the
stacked layers, the rods for each layer were of different colours. (c)
Photonic band structure of the chiral PhC for the lowest four bands. The
studied Weyl point at $K$ point is pointed by a red arrow. The inset shows
half of the first Brillouin zone ($k_{z}$ $>$ 0). The magnified band structure
near the Weyl point is shown in (d).

So far, various 3D periodic structures with broken $P$ symmetry have been
studied to measure Weyl points. Among them, we adopted a chiral woodpile
structure in this study (Fig. 1(a)). In our previous report for this PhC [36],
we calculated the photonic band structure and found 3D point degeneracies. In
the vicinity of these point degeneracies, we further calculated the section
Chern numbers defined for a 2D section of a 3D Brillouin zone. These numbers
showed non-zero values by changing wavevectors, confirming that the point
degeneracies were Weyl points. In addition, we proposed the optimum structural
parameters of the PhC for practical measurements of the Weyl points. Here in
this study, to measure one of these Weyl points at microwave operation
frequency $\sim$ 5 GHz, the following structural parameters were set by using
the scaling law of PhCs [37].

The studied layer-by-layer chiral PhC in air is schematically shown in Figs.
1(a) and (b). Each layer was composed of periodically arranged dielectric rods
whose refractive indices are $n$ = 3.5 and widths are 4.8 mm. The rods were
arranged in a period of 10$\sqrt{3}$ mm, and the thickness of each layer was
6.4 mm. These patterned layers were stacked one by one after an in-plane
rotation of $60^{\circ}$ for each stack. Thus, three layers construct a single
helical unit with pitch $p$ = 19.2 mm. The top view of PhC in Fig. 1(b) shows
a triangular lattice, and we set an in-plane period $a$ = 20 mm.

For this chiral PhC, the photonic band structure calculated using the plane-
wave expansion method is shown in Fig. 1(c) and (d) for the lowest four bands.
Deduced from the triangular lattice from the top view of PhC, a Dirac-point-
like degeneracy occurs at the $K$ and $K^{\prime}$ points for the second and
third bands. In contrast to Dirac points in 2D systems, this point degeneracy
is lifted from $K$ to $H$ by varying the wavevector $k_{z}$ parallel to the
helical axis, indicating a 3D point degeneracy. This 3D point degeneracy was
confirmed to be a Weyl point by calculating the section Chern number [36]. The
frequency of this Weyl point is 4.78 GHz for the designed PhC.

This 3D structure was fabricated by drilling a dielectric board based on
polytetrafluoroethylene with $n$ = 3.5 ($\pm$0.1) at $\sim$ 5 GHz. The
dissipation factor of this material is as small as $\sim$0.002 at
approximately 5 GHz. We carefully selected this dielectric material with $n$ =
3.5, comparable to that of semiconductor materials such as GaAs or Si, for our
future applications to telecommunication bands by scaling the 3D PhC to a sub-
micron period [38, 39, 40, 41]. The dimensions of the fabricated PhC were the
same as those of the designed structure, as shown in Fig. 2. The fabrication
error was $\pm$0.2 mm. We prepared three types of in-plane patterns, where
rods were rotated by $60^{\circ}$ for each pattern. The number of in-plane
periods is eight in the top pattern in Fig. 2(a), and the patterns are
surrounded by a self-suspended frame. By stacking these layers in the
appropriate order, we fabricated a 3D chiral PhC with left-handed chirality,
as shown in Fig. 2(b). The entire size of the chiral PhC is 152 mm in-plane
square and 96 mm height corresponding to five helical periods. Owing to the
finite diameter of the drill, the rod width near the frame has a relatively
large fabrication error of $\sim$3 mm, which is much smaller than the target
wavelength of $\sim$60 mm (5 GHz). In addition, the frame has three small
holes with a diameter of 3.3 mm to align the stacked layers.

Figure 2:  (a) Photo of the fabricated three layers with periodic rod
patterns. The rods in each layer are rotated by $60^{\circ}$. A 100-yen coin
is also placed for comparison. (b), (c) Photo of the stacked 3D PhC from top
and bird’s eye view, respectively.

##  3 Measurement setups

For this fabricated PhC, we performed microwave transmission measurements in
the $\it{\Gamma}$-$K$ direction using a network analyser (N5224A, Agilent
Technologies). As an incident source, we first used a loop antenna with a
diameter of 3 mm. The loop antenna works as a magnetic dipole generating
transverse-electric (TE)-polarised waves, where the electric field oscillates
in the $x$-$y$ plane parallel to the PhC layers. In the phase-reconstruction
measurement, we used a horn antenna to generate a plane wave with TE
polarisation. For the dipole (plane-wave) incidence, the centre of the loop
(horn) antenna was positioned 3 (60) mm away from the 2D centre of the PhC
side surface of incidence.

For spatially resolved detection of the TE-polarised microwave, another loop
antenna was mechanically scanned in a 3D spatial resolution of 2.5 mm. For the
intensity measurement inside the PhC, the detection antenna was inserted into
the triangular holes. From the network analyser, we obtained one of the
scattering parameters, $S_{21}$, between the source loop/horn antenna (port 1)
and the detection loop antenna (port 2) in the frequency range of 4 - 6 GHz
with the resolution of 0.04 GHz. We then extracted amplitude and phase of the
transmitted microwave at each frequency. During the measurements inside
(outside) the PhC, we attached absorber sheets, 15 dB absorption at
approximately 5 GHz, on the surfaces of the PhC, except for the incidence and
top (transmission) surfaces.

##  4 Transmission intensity at the Weyl point

Figure 3:  (a) Schematic of the measurement for the transmission spectrum. (b)
Measured transmission spectrum (left panel) and corresponding band structure
in the $k_{x}$ direction (right panel).

First, we measured a transmission spectrum for the dipole incidence. The
transmitted intensity was measured from 60 mm (single wavelength) away from
the backside of the PhC, as schematically shown in Fig. 3(a). Figure 3(b)
shows the transmission spectrum in the left panel, accompanied with the
calculated band structure in the $k_{x}$ direction in the right panel. The
details of the band calculation are shown in Supplement 1. In the transmission
spectrum, the intensity was decreased by 15 dB at approximately 4.75 GHz,
corresponding to the Weyl point frequency in the right panel. Because the
density of states at an ideal Weyl point is zero [30, 42], this intensity
decrease experimentally indicates the existence of the Weyl point. In the
high-frequency region at approximately 5.50 GHz, only a small transmission
intensity was detected because of the symmetry mismatch in the spatial
distribution of the electromagnetic field between the TE incidence and the
third lowest band for $k_{x}a$ < $4\pi/3$ (shown in Supplement 1). Similar
spectra were obtained at different detection positions for 30 mm < $x$ < 120
mm away from the PhC surface. From this agreement between the experimental
spectrum and the numerical band structure, we confirmed that the Weyl point
exists at approximately 4.75 GHz.

Thereafter, we measured the microwave intensity inside the PhC by inserting
the detection antenna into the six triangular holes, as schematically shown in
Fig. 4(a). The height of the detection antenna was scanned from -5 to +5 mm
relative to the height of the source. Figure 4(b) shows the detected intensity
as a function of the distance from the source, $L$, for three particular
frequencies; 4.38 GHz, 4.74 GHz, and 5.50 GHz, corresponding to the symmetry-
matched band, Weyl point, and symmetry-mismatched (anti-symmetric) band,
respectively. The symbols (error bars) indicate the average (distributed)
intensity for each height of the detection antenna. The measured intensity was
normalised to the value at $L$ = 25 mm for each frequency. At 4.38 GHz, the
intensity was large and almost independent of $L$ owing to the symmetry-
matched band. At 5.50 GHz, the intensity decreased exponentially because the
symmetric input field could not efficiently excite the anti-symmetric Bloch
mode. In contrast, at 4.74 GHz, the intensity was inversely proportional to
the square of $L$, which coincides with the theoretical results for ballistic
electronic systems [31]. This power law is clearly different from that for
Dirac points in 2D photonic systems where the transmission intensity $\propto$
$1/L$ [27, 28], implying that the transport at the Weyl point was not pseudo-
diffusive. Such difference in the power law stems from the dispersion of the
density of states in the vicinity of a Dirac/Weyl point [30]. The density of
states shows a linear dispersion for a Dirac point, whereas that shows a
quadratic dispersion for a Weyl point. Numerical calculations also showed the
power law $\propto$ $1/L^{2}$ (shown in Supplement 1). These experimental and
numerical results demonstrate a unique aspect of optical Weyl points, and a
rigorous theoretical analysis of a photonic system is expected.

Figure 4:  (a) Schematic of the intensity measurement for the detection
position dependence inside the PhC. (b) Microwave intensity inside the PhC as
a function of distance from the source, $L$. The detected intensity was
normalised to the value at $L$ = 25 mm for each frequency. The two dotted
lines indicate the power-law of $L^{-1}$ and $L^{-2}$.

##  5 Spatial distribution of the transmitted wave

Figure 5:  (a) Schematic of the $x$-$y$ detection plane at the backside of the
PhC. (b), (c) Spatial distribution of the transmitted wave at 4.76 GHz and
4.16 GHz, respectively.  Figure 6:  (a) Schematic of the $x$-$z$ detection
plane. (b), (c) Spatial distribution of the transmitted wave at 4.76 GHz and
4.16 GHz, respectively.

We investigated spatial distribution of the transmitted wave at the backside
of the PhC. We first measured both the amplitude and phase of the transmitted
microwave in the $x$-$y$ plane at the same height as the dipole source, as
schematically shown in Fig. 5(a). Figures 5(b) and (c) show the spatial
distribution of the transmitted wave at the frequency of the Weyl point (4.76
GHz) and in the continuous band (4.16 GHz), respectively. Whereas a spherical
wavefront was detected in (c), a planar wavefront was observed in (b), even
for the point-source incidence in the 2D plane. Note that such an output with
a planar wavefront was never observed in the $\it{\Gamma}$-$M$ direction at
4.76 GHz (shown in Supplement 1), nor at different frequencies from the Weyl
point.

We also measured the spatial distribution of the transmitted wave in the
$x$-$z$ plane at the same $y$ position as the dipole source, as schematically
shown in Fig. 6(a). Figures 6(b) and (c) shows the mapped transmitted waves
for the $y$ slice at 4.76 GHz and 4.16 GHz, respectively. Fig. 6(c) at 4.16
GHz again shows a clear spherical wavefront, whereas Fig. 6(b) at the Weyl
point frequency does not show a clear planar wavefront, but shows a spherical
wavefront. From these results, the output from the chiral PhC at the Weyl
point frequency was found to have a cylindrical wavefront.

To explain these output wavefronts for the dipole input, we first consider a
$k$-space distribution of the Bloch modes in the PhC. As shown schematically
in Fig. 7(a), we numerically projected the Bloch modes at the Weyl point
frequency in the first Brillouin zone onto the $k_{x}$ = $K$ plane from
$\it{\Gamma}$ point. Figure 7(b) shows the projection result with discrete
plots because the first Brillouin zone was divided into $16^{3}$ meshes in our
calculation of the equi-frequency dispersion curves. Thereafter, we calculated
the angles between $\overrightarrow{\it{\Gamma}K}$ and the vectors from the
dipole source to the sides of the output surface. For the $k$ vectors
corresponding to these angles, the collection of their components ($k_{y}$,
$k_{z}$) at $k_{x}$ = $K$ represents the red dotted square in Fig. 7(b). Here,
we refer to this square as the output aperture, and the projected modes
outside this square are all absorbed by the absorber sheets attached to the
PhC.

Based on the numerical results shown in Fig. 7(b), we first discuss the
propagation in the in-plane directions at $k_{z}$ = 0. In the output aperture,
there is only a single mode of the Weyl point. Because waves from the dipole
source can be represented by a sum of Fourier series of plane waves with
various wavevectors, one of the Fourier components whose wavevector is same as
that of the Weyl point can transmit through the PhC. Therefore, the PhC
filtered the single plane wave and the planar wavefront was obtained, as shown
in Fig. 5(b). Further numerical calculations also yielded a well-collimated
output (shown in Supplement 1). In contrast, in the out-of-plane propagation,
there are continuous modes along the $k_{z}$ axis in the output aperture in
Fig. 7(b). Thus, the transmitted wave showed a spherical wavefront. In the
case of the ideal Weyl point [14], the transmitted wave could exhibit a planar
wavefront in both the in-plane and out-of-plane directions.

Figure 7:  (a) Schematic of the projection on the $k_{x}$ = $K$ plane from
$\it{\Gamma}$ for the Bloch modes at the Weyl point frequency. (b) Projection
result of (a). In the red dotted square corresponding to the output aperture
of the PhC, the projected Bloch modes exist only on the $k_{z}$ axis
continuously.

By utilising the well-collimated output for the point-source input in the
$x$-$y$ plane at the Weyl point frequency, we demonstrated a cloaking effect
in the 2D plane, where phases of plane-wave incidence are reconstructed after
transmission through a scattering material. We intentionally inserted four
regular triangular pillars made of aluminium into the holes of the PhC, as
shown in Fig. 8. Each side of the triangle and the length of the pillars are
11.7 mm and 96 mm, respectively, which are the same dimensions as the
triangular holes of the PhC. For this PhC containing these four scatterers, we
applied a plane wave in the $\it{\Gamma}$-$K$ direction by using the horn
antenna, and measured the spatial distribution of the transmitted wave at the
backside of the PhC. Figure 8 shows a planar wavefront output despite the
existence of the scatterers, indicating a phase-reconstruction effect. This
result could be interpreted as a cloaking effect in the direction parallel to
that of the Weyl-point wavevector.

Figure 8:  Spatial distribution of the transmitted wave for the PhC containing
four Al pillars. The input plane wave at 4.76 GHz was generated by the horn
antenna, and the detection antenna was scanned in the $x$-$y$ plane at the
central height of the PhC.

##  6 Summary

We investigated the microwave transmission near the frequency and wavevector
of the Weyl point in the all-dielectric 3D chiral PhC. The intensity
measurement inside the PhC exhibited the spatial power law $\propto 1/L^{2}$,
which coincides with the previous theoretical report on electronic systems
[31]. After transmission through the PhC, we observed an output with a planar
wavefront even for the point-source input in the plane parallel to the PhC
layers, because the PhC filtered the single plane-wave mode of the Weyl point.
This 2D well-collimated output was not disturbed by metal inclusions in the
PhC, which could be applied to the cloaking effect. These results may provide
important insights for the future Weyl physics and applications to control
microwave propagation such as the cloaking effect, a Veselago lens [23] based
on the negative refraction [8], and topologically protected propagation
channels including 1D Hinge states [21, 22].

## Funding

This work was supported by Grants-in-Aid for Scientific Research, Nos.
17H06138, 15H05700, 18K18857, and JST-CREST JPMJCR19T1.

## Acknowledgments

The authors would like to thank M. Koshino for his fruitful discussions.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

##

See Supplement 1 for supporting content.

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∎

11institutetext:  Awadh Pratap Singh22institutetext: Department of
Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India.
22email<EMAIL_ADDRESS>33institutetext: Shiv Prasad Yadav
44institutetext: Department of Mathematics, Indian Institute of Technology
Roorkee, Roorkee-247667, India.
44email<EMAIL_ADDRESS>
#  A Two-stage Network Data Envelopment Analysis: An Education Sector
Application

Awadh Pratap Singh  Shiv Prasad Yadav

(Received: date / Accepted: date)

###### Abstract

In general, data envelopment analysis (DEA) works like a black box that does
not provide any adequate detail to identify the specific reason for
inefficiency in decision-making units (DMUs). The motivation of this study is
to analyze the cause of the inefficiency of DMUs in a decision process with
the help of a two-stage relational network DEA model. In the current study, a
two-stage relational network DEA model is applied to measure the performance
of DMUs for the whole process and each stage independently. In general, past
studies used conventional DEA models in education sectors to analyze the
performances of educational institutions. In the current study, by considering
quantitative attributes to measure the performance of Indian institutes of
management (IIMs) by using network DEA, we develop a procedure that captures
both quality and quantity.

###### Keywords:

Network DEA Efficiency IIMs Education sector efficiencies Research

##  1 Introduction

Data envelopment analysis (DEA) is a data-supported nonparametric approach
that is used to measure the relative performance of decision-making units
(DMUs) cook2009data . Charnes, Cooper, and Rhodes charnes1978measuring
developed a model called the CCR DEA Model to measure the performance of DMUs.
The efficiency of DMU is defined as the ratio of output to input (efficiency =
output/input). The relative efficiency of a DMU is defined as, the ratio of
its efficiency to the largest efficiency under consideration. It lies in the
interval (0,1]. A DMU with an efficiency score of 1 is called efficient DMU
otherwise inefficient DMU.

Education and research play a vital role in the development of any nation. In
the current study, we will try to measure the performance efficiencies of IIMs
with the help of the network DEA model, particularly the two-stage network
DEA. The main reason for using the two-stage network DEA model over the
conventional CCR DEA model is to understand the causes behind the
inefficiencies of DMUs. In general, DEA works like a black box that does not
provide any adequate detail to identify the specific reason for inefficiency
in DMUs. The motivation of this study is to analyze the cause of the
inefficiency of DMUs in a decision process with the help of a two-stage
relational network DEA model. There are many types of research available in
literature where the network DEA model is developed and used for the
performance evaluation of DMUs. Kao kao2014network  reported a detailed review
on network DEA which consists of the DMUs with series, parallel and mixed
structure. Application of network DEA approach can be found in several fields
like education sector, banking sector, healthcare sector, hotel industries,
etc. Tan and Despotis tan2021investigation  investigated the efficiency of the
hotel industry by using a network DEA model. Kao and Hung kao2008efficiency ,
Arya and Singh arya2021development , and Ma and Chen ma2018evaluating
proposed a two-stage parallel series network and provided the application of
proposed models in the healthcare sector. Puri et al. puri2017new , Lofti and
Ghasemi lotfi2013multi  also used multi-component DEA in banking sector.

In past studies, many researchers singh2021performance ; arcelus1997efficiency
; sinuany1994academic ; bessent1982application ; tomkins1988experiment ;
tyagi2009relative  used conventional DEA in education sector. As Lee and
Worthington lee2016network  explained in their work that higher educational
institutions are the more complex structure that uses multiple inputs to
produce multiple outputs. The research process is also complicated. As, in
some stage, some output of research process become the inputs for the next
stages to produce final outputs. Many researchers applied network DEA models
for performance evaluation in the education sector. Lee and Worthington
lee2016network  applied a network DEA model for evaluating the research
performance of Australian universities. In this study, they developed a
university research production model. The 1st stage uses input (full-time
equivalent (FTE) academic staff and capital stock) to produce output as the
number of publications. In the 2nd stage, the number of publications is used
to produce the final output (grant secured). Our study is also motivated by
Lee and Worthington’s study. Tavares et al. tavares2021proposed  proposed a
multistage network DEA model for performance evaluation of higher educational
institutions in Brazil. In this study, they proposed an approach that is used
to measure the efficiencies of 45 Brazilian federal universities. Yang et al.
yang2018measuring  also applied a two-stage network DEA model for analyzing
the reasons behind the inefficiencies of Chinese research universities.
Moreno-Gómez et al. moreno2019measuring  measured efficiencies of 78 Colombian
universities across the period 2015–2017 by using a two-stage network DEA
model. Monfared and Safi monfared2013network , proposed a set of performance
indicators to enable efficient analysis of academic activities and apply
network DEA structure to analyze the performance of the academic colleges at
Alzahra University in Iran.

There are many kinds of research available in the field of network DEA, but
the application of network DEA in the field of the education sector is still
very limited. In the current study, by considering quantitative attributes to
measure the performance of Indian institutes of management (IIMs) by using
network DEA, we develop a procedure that captures both quality and quantity.
In the current study, we developed a research production model to elaborate
the research process in IIMs. With the help of a two-stage relational network
DEA model performance analysis is done. The results are compared with the
conventional DEA model. In this study, we aim to know the reasons for the
inefficiencies of DMUs with the help of the network DEA model.

The rest of the paper is organized into 4 sections. Introduction is given in
Section 1. A two-stage relational network DEA model and supported statements
are given in Section 2. The conceptual framework of the proposed two-stage
research production model and education sector application with results is
discussed in Section 3. Finally, some concluding remarks and future scope are
given in Section 4.

##  2  Two-stage relational network DEA model

Suppose there are $n$ homogeneous DMUs ($DMU_{j};j=1,2,3,...,n$), which
consumes $m$ inputs, $x_{ij},\,i=1,2,3,...,m$ to produce $s$ outputs
$y_{rj},\,r=1,2,3,...,s$. Then the efficiency score of the $k^{th}$ DMU can be
defined with the help of the following CCR DEA modelcooper2007data :

Model 1

: For $k=1,2,3,...,n,$

$\displaystyle\max
E_{k}=\frac{\sum_{r=1}^{s}y_{rk}v_{rk}}{\sum_{i=1}^{m}x_{ik}u_{ik}}$ (1)
$\displaystyle\mbox{subject
to}~{}~{}\frac{\sum_{r=1}^{s}y_{rj}v_{rk}}{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq 1;$
(2) $\displaystyle u_{ik}\geq\varepsilon\,\,\forall
i,v_{rk}\geq\varepsilon\,\,\forall r,\,\,\,\varepsilon>0;$

where $\varepsilon$ is the non-Archimedean infinitesimal; $y_{rj}$ is the
amount of the $r^{th}$ output produced by the $j^{th}$ DMU; $x_{ij}$ is the
amount of the $i^{th}$ input used by the $j^{th}$ DMU; $u_{ik}$ and $v_{rk}$
are the weights corresponding to the $i^{th}$ input and $r^{th}$ output
respectively.

Definition 1:charnes1978measuring

$E_{k}$ is called the CCR efficiency of $DMU_{k}$. $E_{k}$ lies in the
interval (0,1]. Let $E_{k}^{*}$ be the optimal value of $E_{k}$. Then
$DMU_{k}$ is said to be CCR-efficient if $E_{k}^{*}=1$, an, $DMU_{k}$ is said
to be CCR-inefficient if $E_{k}^{*}<1$.

Now, we consider a production process composed of two sub-processess as
depicted in Fig 1. The whole process consists of two stages. The 1st stage
uses m inputs $x_{ik},~{}i=1,2,...,m$ to produce p intermediate products
$z_{dk},~{}d=1,2,3,...,p$. The 2nd stage uses p intermediate products
$z_{dk},~{}d=1,2,3,...,p$ to produce s outputs $y_{rk},~{}r=1,2,3,...,s$. The
whole process uses m inputs $x_{ik},~{}i=1,2,...,m$ to produce s outputs
$y_{rk},~{}r=1,2,3,...,s$. Moreover, the outputs of 1st stage (i.e. $z_{dk}$)
are the inputs of the 2nd stage. The efficiency of 1st and 2nd stage can be
calculated by the conventional CCR DEA model explained in Two-stage relational
network DEA model. Suppose the efficiency of $k^{th}$ DMU in 1st stage is
$E_{k}^{1}$ and in the 2nd stage is $E_{k}^{2}$. Then the 1st and 2nd stage
efficiencies of $DMU_{k}$ can be calculated with the help of Two-stage
relational network DEA model and Two-stage relational network DEA model as
follows kao2008efficiency :

Figure 1: A two-stage series production process system with input x, output y,
and intermediate products z.

Model 2

: For $k=1,2,3,...,n,$

$\displaystyle\max
E_{k}^{1}=\frac{\sum_{d=1}^{p}z_{dk}w_{dk}}{\sum_{i=1}^{m}x_{ik}u_{ik}}$ (3)
$\displaystyle\mbox{subject
to}~{}~{}\frac{\sum_{d=1}^{p}z_{dj}w_{dk}}{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq 1;$
(4) $\displaystyle u_{ik}\geq\varepsilon\,\,\forall
i,w_{dk}\geq\varepsilon\,\,\forall p,\,\,\,\varepsilon>0.$

Model 3

: For $k=1,2,3,...,n,$

$\displaystyle\max
E_{k}^{2}=\frac{\sum_{r=1}^{s}y_{rk}v_{rk}}{\sum_{d=1}^{p}z_{dk}w^{\prime}_{dk}}$
(5) $\displaystyle\mbox{subject
to}~{}~{}\frac{\sum_{r=1}^{s}y_{rj}v_{rk}}{\sum_{d=1}^{p}z_{dj}w^{\prime}_{dk}}\leq
1;$ (6) $\displaystyle v_{rk}\geq\varepsilon\,\,\forall
r,w^{\prime}_{dk}\geq\varepsilon\,\,\forall p,\,\,\,\varepsilon>0.$

Note that the weights $w_{dk}$ and $w^{\prime}_{dk}$ used in Two-stage
relational network DEA model and Two-stage relational network DEA model are
same lee2016network . The reason is, input of sub-process 2 to be the expected
output of sub-process 1 while calculating the expected output of the whole
process. When this concept is generalized to the case of multiple intermediate
products, it requires the aggrigated value of intermediate products, which is
$\sum_{d=1}^{p}w_{dk}z_{dk}$ to be the same kao2008efficiency . Hence Two-
stage relational network DEA model can be re-written as follows:

Model 4

: For $k=1,2,3,...,n,$

$\displaystyle\max
E_{k}^{2}=\frac{\sum_{r=1}^{s}y_{rk}v_{rk}}{\sum_{d=1}^{p}z_{dk}w_{dk}}$ (7)
$\displaystyle\mbox{subject
to}~{}~{}\frac{\sum_{r=1}^{s}y_{rj}v_{rk}}{\sum_{d=1}^{p}z_{dj}w_{dk}}\leq 1;$
(8) $\displaystyle v_{rk}\geq\varepsilon\,\,\forall
r,w_{dk}\geq\varepsilon\,\,\forall p,\,\,\,\varepsilon>0.$

With the help of Two-stage relational network DEA model, efficiency of the
whole process can be calculated independently as the 1st stage efficiency
model (Two-stage relational network DEA model) and 2nd stage efficiency model
(Two-stage relational network DEA model) are essentially same. (Two-stage
relational network DEA model) and (Two-stage relational network DEA model)
give the 1st and second stage efficiencies of independent network DEA model.
To link these two sub-processes with the whole process, a model must be
constructed that describe the series relationship of the whole process with
the two sub-processes.

Suppose, for $DMU_{k}$; $u_{ik}^{*}$, $v_{rk}^{*}$, $w_{dk}^{*}$ are the
optimal weights (multipliers) that are obtained while calculating overall
efficiency $E_{k}$, sub-process efficiencies $E_{k}^{1}$, and $E_{k}^{2}$.
Then we have,

$\displaystyle
E_{k}=\frac{\sum_{r=1}^{s}y_{rk}v_{rk}^{*}}{\sum_{i=1}^{m}x_{ik}u_{ik}^{*}}$
(9) $\displaystyle
E_{k}^{1}=\frac{\sum_{d=1}^{p}z_{dk}w_{dk}^{*}}{\sum_{i=1}^{m}x_{ik}u_{ik}^{*}}$
(10) $\displaystyle
E_{k}^{2}=\frac{\sum_{r=1}^{s}y_{rk}v_{rk}^{*}}{\sum_{d=1}^{p}z_{dk}w_{dk}^{*}}$
(11)

From Eqs. 9,10 & 11, we have a relation among $E_{k}$, $E_{k}^{1}$, and
$E_{k}^{2}$ as;

$\textbf{$E_{k}=E_{k}^{1}\times E_{k}^{2}$}.\\\ $ (12)

Based on this concept, let us incorporate the ratio constraints of the two
sub-processes into conventional DEA model i.e., Two-stage relational network
DEA model, to get a two-stage relational network DEA model. This will give the
overall efficiency $E_{k}$, to a two-stage relational network DEA model,
taking into account the series relationship of two sub-processes as follows:

Model 5

: For $k=1,2,3,...,n,$

$\displaystyle\max
E_{k}=\frac{\sum_{r=1}^{s}y_{rk}v_{rk}}{\sum_{i=1}^{m}x_{ik}u_{ik}}$ (13)
$\displaystyle\mbox{subject
to}~{}~{}\frac{\sum_{r=1}^{s}y_{rj}v_{rk}}{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq 1;$
(14)
$\displaystyle\frac{\sum_{d=1}^{p}z_{dj}w_{dk}}{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq
1;$ (15)
$\displaystyle\frac{\sum_{r=1}^{s}y_{rj}v_{rk}}{\sum_{d=1}^{p}z_{dj}w_{dk}}\leq
1;$ (16) $\displaystyle u_{ik}\geq\varepsilon\,\,\forall
i,v_{rk}\geq\varepsilon\,\,\forall r,\,\,\,w_{dk}\geq\varepsilon\,\,\forall
p,\varepsilon>0.$

Now, Two-stage relational network DEA model can be written in linear form as
follows:

Model 6

: For $k=1,2,3,...,n,$

$\displaystyle\max E_{k}={\sum_{r=1}^{s}y_{rk}v_{rk}}$ (17)
$\displaystyle\mbox{subject to}~{}~{}{\sum_{i=1}^{m}x_{ik}u_{ik}}=1;$ (18)
$\displaystyle{\sum_{r=1}^{s}y_{rj}v_{rk}}-{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq
0;$ (19)
$\displaystyle{\sum_{d=1}^{p}z_{dj}w_{dk}}-{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq
0;$ (20)
$\displaystyle{\sum_{r=1}^{s}y_{rj}v_{rk}}-{\sum_{d=1}^{p}z_{dj}w_{dk}}\leq
0;$ (21) $\displaystyle u_{ik}\geq\varepsilon\,\,\forall
i,v_{rk}\geq\varepsilon\,\,\forall r,\,\,\,w_{dk}\geq\varepsilon\,\,\forall
p,\varepsilon>0.$

It is quite possible that the optimal solution (optimal multipliers) obtained
after solving Two-stage relational network DEA model are not unique.
Therefore, the relation $E_{k}=E_{k}^{1}\times E_{k}^{2}$ would not be unique.
This makes comparison of $E_{k}^{1}$ and $E_{k}^{2}$ between all DMUs not the
same basis. One solution to this problem is to find a set of multipliers that
produce the largest $E_{k}^{1}$ while maintaning the $E_{k}$ obtained from
Two-stage relational network DEA model.

Model 7

: For $k=1,2,3,...,n,$

$\displaystyle\max E_{k}^{1}={\sum_{d=1}^{p}z_{dk}w_{dk}}$ (22)
$\displaystyle\mbox{subject to}~{}~{}{\sum_{i=1}^{m}x_{ik}u_{ik}}=1;$ (23)
$\displaystyle{\sum_{r=1}^{s}y_{rk}v_{rk}}-E_{k}{\sum_{i=1}^{m}x_{ik}u_{ik}}=0;$
(24)
$\displaystyle{\sum_{r=1}^{s}y_{rj}v_{rk}}-{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq
0;$ (25)
$\displaystyle{\sum_{d=1}^{p}z_{dj}w_{dk}}-{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq
0;$ (26)
$\displaystyle{\sum_{r=1}^{s}y_{rj}v_{rk}}-{\sum_{d=1}^{p}z_{dj}w_{dk}}\leq
0;$ (27) $\displaystyle u_{ik}\geq\varepsilon\,\,\forall
i,v_{rk}\geq\varepsilon\,\,\forall r,\,\,\,w_{dk}\geq\varepsilon\,\,\forall
p,\varepsilon>0.$

After calculating $E_{k}^{1}$ with the help of Two-stage relational network
DEA model, second stage efficiency efficiency $E_{k}^{2}$ can be calculated
easily by using the relation ( Eq. 12) as $E_{k}^{2}=E_{k}/E_{k}^{1}$.
Silmilarly, if decision-maker is more concerned for second stage efficiency
then, one can formulated model for calculating $E_{k}^{2}$ by the following
model as follows:

Model 8

: For $k=1,2,3,...,n,$

$\displaystyle\max E_{k}^{2}={\sum_{r=1}^{s}y_{rk}v_{rk}}$ (28)
$\displaystyle\mbox{subject to}~{}~{}{\sum_{d=1}^{p}z_{dk}w_{dk}}=1;$ (29)
$\displaystyle{\sum_{r=1}^{s}y_{rk}v_{rk}}-E_{k}{\sum_{i=1}^{m}x_{ik}u_{ik}}=0;$
(30)
$\displaystyle{\sum_{r=1}^{s}y_{rj}v_{rk}}-{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq
0;$ (31)
$\displaystyle{\sum_{d=1}^{p}z_{dj}w_{dk}}-{\sum_{i=1}^{m}x_{ij}u_{ik}}\leq
0;$ (32)
$\displaystyle{\sum_{r=1}^{s}y_{rj}v_{rk}}-{\sum_{d=1}^{p}z_{dj}w_{dk}}\leq
0;$ (33) $\displaystyle u_{ik}\geq\varepsilon\,\,\forall
i,v_{rk}\geq\varepsilon\,\,\forall r,\,\,\,w_{dk}\geq\varepsilon\,\,\forall
p,\varepsilon>0.$

The efficiency of 1st stage $E_{k}^{1}$ can be calculated easily by using the
relation ( Eq. 12) as $E_{k}^{1}=E_{k}/E_{k}^{2}$.

###### Theorem 2.1

For $DMU_{k}$, if $E_{k}=1~{}\implies~{}E_{k}^{1}=E_{k}^{2}=1$. Where
$E_{k}^{1}$ and $E_{k}^{2}$ are the optimal values of 1st and 2nd stage
efficiencies obtained from Two-stage relational network DEA model and Two-
stage relational network DEA model.

Proof:

Suppose $(u_{ik}^{*},v_{rk}^{*},w_{dk}^{*}~{}\forall~{}i,r,d)$ be an optimal
solution of Two-stage relational network DEA model then we have

$E_{k}={\sum_{r=1}^{s}y_{rk}v_{rk}^{*}}$,

and

$\displaystyle{\sum_{i=1}^{m}x_{ik}u_{ik}^{*}}=1;$ (34)
$\displaystyle{\sum_{r=1}^{s}y_{rj}v_{rk}^{*}}-{\sum_{i=1}^{m}x_{ij}u_{ik}^{*}}\leq
0;$ (35)
$\displaystyle{\sum_{d=1}^{p}z_{dj}w_{dk}^{*}}-{\sum_{i=1}^{m}x_{ij}u_{ik}^{*}}\leq
0;$ (36)
$\displaystyle{\sum_{r=1}^{s}y_{rj}v_{rk}^{*}}-{\sum_{d=1}^{p}z_{dj}w_{dk}^{*}}\leq
0;$ (37) $\displaystyle u_{ik}^{*}\geq\varepsilon\,\,\forall
i,v_{rk}^{*}\geq\varepsilon\,\,\forall
r,\,\,\,w_{dk}^{*}\geq\varepsilon\,\,\forall p,\varepsilon>0.$

With $E_{k}=1$, we have, for specific $DMU_{k}:$

$1={\sum_{r=1}^{s}y_{rk}v_{rk}^{*}}\leq{\sum_{d=1}^{p}z_{dk}w_{dk}^{*}}\leq{\sum_{i=1}^{m}x_{ik}u_{ik}^{*}}=1$
(38)

Therefore, from Eq. (38) it is possible that
${\sum_{r=1}^{s}y_{rk}v_{rk}^{*}}={\sum_{d=1}^{p}z_{dk}w_{dk}^{*}}=1$. In
other words, the optimal solution
$(u_{ik}^{*},v_{rk}^{*},w_{dk}^{*}~{}\forall~{}i,r,d)$ can make the objective
function values of Two-stage relational network DEA model and Two-stage
relational network DEA model are equal to 1. It is easy to varify that
$(u_{ik}^{*},v_{rk}^{*},w_{dk}^{*}~{}\forall~{}i,r,d)$ can satisfy all the
constraints of Two-stage relational network DEA model and Two-stage relational
network DEA model simultaneously.
Thus, we have $E_{k}^{1}=E_{k}^{2}=1$ for $DMU_{k}.$

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\square$

##  3 Education sector application

For the application of the network DEA, we consider the problem of checking
the performance efficiency of the Indian Institutes of Management (IIMs) in
India. IIMs are administrative and research educational institutions in India.
Primarily they offer undergraduate, postgraduate, doctoral, and other
additional courses. IIMs have been declared the most important institutions in
the country by The Ministry of Education Govt. of India.

###  3.1 Conceptual framework

Education and research play a vital role in the development of any nation. In
the current study, we will try to measure the performance efficiencies of IIMs
with the help of the network DEA model, particularly the two-stage network
DEA. The main reason for using the two-stage network DEA model over the
conventional CCR DEA model is to understand the causes behind the
inefficiencies of DMUs. The network DEA model used for the current study is
depicted in Fig 2. Here, in this study, we constructed a two-stage research
production model of IIMs. The research production model in IIMs is complex.
Fig 2 depicts our network DEA model for IIMs. This model consists of two
stages. In the 1st stage, the model consumes $x_{ij}$ inputs to produce
$z_{dj}$ intermediate products. Outputs of 1st stage start working as inputs
in 2nd stage.

Figure 2: A two-stage series production process system with inputs
$x_{1},x_{2},x_{3}$, output y, and intermediate products z.

In our research production model, we use three inputs ($x_{1}$: financial
resources, $x_{2}$: number of Ph.D. students, $x_{1}$: number of faculty
members) in 1st stage. In research, these three parameters play a very crucial
role. For any Ph.D. student, it is always challenging to research without a
guide and financial support. This is the reason why we considered these three
parameters as inputs in 1st stage. The output of 1st stage or intermediate
product is considered as the ’number of publications’ ($z_{1}$). This is the
most obvious output for any research production model. In India, Ph.D.
students have to publish a few research papers to submit their thesis. These
publications validate the choice of inputs as, Ph.D. students, faculties, and
financial resources. The output of the 1st stage $z_{1}$ is used as an
intermediate input in the 2nd stage to produce the final output $y_{1}$
(research grant). Since in the present study, we are focusing only on research
activities so we are not very much bothered about other parameters. For any
IIM, it is very important to win ample research grants so that they can
continue their research process. IIMs get their research grant from two types
of research projects; first as the sponsored research projects, and second as
consultancy research projects. In our study, we consider the research grant
index ($y_{1}$) as the sum of research grants obtained from sponsored research
projects and consultancy research projects. It is a widespread perception that
can be treated as the fact that if any IIMs publishes more research papers
then that IIM can win more research grants. Hence the choice of intermediate
input $z_{1}$ is the best choice for the final output $y_{1}$ for our two-
stage research production model.

###  3.2 Results and discussion

In the current study, IIMs are considered as our DMUs. The data is collected
from the NIRF website launched by The Ministry of Education on 29th September
2015. Data is collected for the year 2020-21. Input, intermediate input, and
output data for 13 IIMs for the application two-stage network DEA model is
given in Table 1.

Table 1: Inputs (x), intermediate product (z), and output(y) for 13 IIMs

DMU | IIM Name | State |  Inputs | Intermediate product | Output
---|---|---|---|---|---
|  |  | $\qquad{x_{1}}$ | ${x_{2}}$ | ${x_{3}}$ | ${\qquad\qquad z_{1}}$ | $\qquad{y_{1}}$
$D_{1}$ | IIM Bangalore | Karnataka | 2783060307 | 100 | 116 |  81 | 1302218468
$D_{2}$ | IIM Ahmedabad | Gujrat | 2282349127 | 128 | 110 |  43 | 34953142
$D_{3}$ | IIM Calcutta | West Bengal | 1372017597 | 97 | 91 |  94 | 261107000
$D_{4}$ | IIM Lucknow | Uttar Pradesh | 1405523203 | 178 | 104 |  89 | 10643365
$D_{5}$ | IIM Indore | Madhya Pradesh | 1323781000 | 120 | 139 |  124 | 165125234
$D_{6}$ | IIM Kozhikode | Kerala | 1009040766 | 91 | 109 |  47 | 112206310
$D_{7}$ | IIM Udaipur | Rajasthan | 675751470 | 13 | 42 |  27 | 1091766
$D_{8}$ | IIM Tiruchirapalli | Tamilnadu | 366034016 | 22 | 55 |  25 | 6119078
$D_{9}$ | IIM Raipur | Chhatisgarh | 724040010 | 21 | 36 |  40 | 9499507
$D_{10}$ | IIM Rohtak | Haryana | 343393901 | 53 | 92 |  47 | 26337325
$D_{11}$ | IIM Shillong | Meghalaya | 373246687 | 23 | 32 |  44 | 5971808
$D_{12}$ | IIM Kashipur | Uttarakhand | 337690753 | 54 | 45 |  31 | 8121608
$D_{13}$ | IIM Ranchi | Jharkhand | 287052918 | 66 | 42 |  65 | 696200

Table 2: Efficiency scores and ranks (in parantheses) of two-stage network DEA
model

DMU | IIM Name | Two-stage relational network DEA model
---|---|---
|  | $~{}~{}~{}E_{k}$ | $~{}~{}~{}E_{k}^{1}$ | $~{}~{}~{}E_{k}^{2}$
$D_{1}$ | IIM Bangalore | 0.4973(1) | 0.4973(9) |  1(1)
$D_{2}$ | IIM Ahmedabad | 0.0135(7) | 0.2668(11) | 0.0506(5)
$D_{3}$ | IIM Calcutta | 0.1235(2) | 0.7147(4) | 0.1728(2)
$D_{4}$ | IIM Lucknow | 0.0041(11) | 0.5529(7) | 0.0074(11)
$D_{5}$ | IIM Indore | 0.0568(3) | 0.6857(5) | 0.0823(4)
$D_{6}$ | IIM Kozhikode | 0.0507(4) | 0.3417(10) | 0.1485(3)
$D_{7}$ | IIM Udaipur | 0.0025(12) | 1(1) | 0.0025(12)
$D_{8}$ | IIM Tiruchirapalli | 0.0090(8) | 0.5934(6) | 0.0152(8)
$D_{9}$ | IIM Raipur | 0.0145(6) | 0.9809(2) | 0.0148(9)
$D_{10}$ | IIM Rohtak | 0.0268(5) | 0.7703(3) | 0.0348(6)
$D_{11}$ | IIM Shillong | 0.0084(9) | 1(1) | 0.0084(10)
$D_{12}$ | IIM Kashipur | 0.0082(10) | 0.5064(8) | 0.0163(7)
$D_{13}$ | IIM Ranchi | 0.0007(13) | 1(1) | 0.0007(13)

In our research production model, the efficiency of 1st stage measures the
performance of IIMs based on publications, while the 2nd- stage measures the
performance on the basis of the research grant won by any IIM. By applying
Two-stage relational network DEA model, the overall efficiency of two-stage
relational network DEA model for 13 IIMs is calculated. The results are shown
in Table 2 under the subcolumn $E_{k}$ of the column two-stage relational
network DEA model. Since, to win an attractive research grant is of more
concern for any IIM so that the research process can be performed without any
obstacle, we choose to measure $E_{k}^{2}$ first with the help of $E_{k}$ and
after this calculate $E_{k}^{1}$ via $E_{k}^{1}={E_{k}}/{E_{k}^{2}}$. The
choice of selecting performance measures as a research grant or a number of
publications is completely dependent upon the decision-maker. Here, the
research grants are our primary concern, while publications can be the topic
of paramount concern for some other decision-maker. This choice is not going
to affect the 1st and 2nd stage efficiency scores of a two-stage relational
network DEA model.

Now we will analyze the results obtained from applying Two-stage relational
network DEA model, Two-stage relational network DEA model, and Two-stage
relational network DEA model on the data given in Table 1. The results are
given under the column named as ’two-stage relational network DEA model’. Note
that in Table 2, none of the 13 IIMs, performed efficiently in both stages.
IIM Bangalore achieved the highest overall efficiency score of 0.4973 while,
IIM Ranchi achieved the lowest overall efficiency score of 0.0007. For the 1st
stage, there are 3 IIMs named, IIM Udaipur, IIM Shillong, and IIM Ranchi,
which perform efficiently; as they have an efficiency score $E_{k}^{1}=1.$ In
the 2nd stage, IIM Bangalore is the only IIM that is efficient. As we have
already derived a relation $E_{k}=E_{k}^{1}\times E_{k}^{2}$ in Eq. (12),
indicates that the overall efficiency is the product of 1st and 2nd stage
efficiencies. So there is no $E_{k}$ greater than its corresponding
$E_{k}^{1}~{}or~{}E_{k}^{2}$. It is to be noted that IIM Udaipur ($D_{7}$),
IIM Shillong ($D_{11}$), and IIM Ranchi ($D_{13}$) perform efficiently in 1st
stage but not efficiently in the 2nd stage. It means that these IIMs are good
at publishing research but still not able to win an attractive research grant.
Whereas IIM Bangalore ($D_{1}$) does not perform well in the 1st stage but in
the 2nd stage it is performing efficiently. This indicates that it is a bit
easy for IIM Bangalore to win ample research grants. It is observable from
Table 2 that except for IIM Bangalore, the efficiency of the 1st stage is
higher than the efficiency of the 2nd stage for all IIMs. This indicates that
the low $E_{k}$ of the whole process is mainly due to the low $E_{k}^{2}$ of
the 2nd stage (i.e., research grant). Since $E_{k}$ is always less than or
equal to $E_{k}^{1}$, and $E_{k}^{2}$ so it will be more informative to look
at the ranks, which are the numbers in brackets in Table 2, over the
efficiency scores. If any IIM has similar ranks in
$E_{k},~{}E_{k}^{1},~{}and~{}E_{k}^{2}$ then it can be said that the
particular IIM performs evenly in the whole process and in both the sub-
processes. Hence, with the help of a two-stage relational network DEA, we are
in the position to understand the reason behind the inefficiency of any IIM.

Kao and Hung kao2008efficiency  explained logically that $E_{k}$ for any IIM
should lie either between $E_{k}^{1},~{}and~{}E_{k}^{2}$ or in the
neighborhood of $E_{k}^{1},~{}and~{}E_{k}^{2}$. The main reason for this is
that the performance of the whole process is the aggregation of two sub-
processes. Here in Table 2, it can be seen that out of 13 IIMs the ranks of 10
IIMs lies between $E_{k}^{1},~{}and~{}E_{k}^{2}$, and the ranks of remaining 3
IIMs lies in the neighborhood of $E_{k}^{1},~{}or~{}E_{k}^{2}$.

Now, with the help of two-stage relational network DEA model we are some what
able to understand the actual reasons of inefficiencies in DMUs. As we know
that the DEA works like a black box that does not provide any adequate detail
to identify the specific reason for inefficiency in decision-making units
(DMUs). After knowing the reasons of inefficiecies, now we are interested in
the comparison of the efficiency of the whole research production process with
the two-stage relational network DEA model ($E_{k}$) and conventional CCR DEA
model ($E_{k}^{CCR}$). Actually, we wanted to understand what will happen if
the overall efficiency of DMUs is calculated by taking sub-stages of the model
in mind and without bothering about sub-stages. The comparison is done and the
results are shown in Table 3. The results depicted in Table 3 shows that the
conventional CCR efficiency ($E_{k}^{CCR}$) is higher than two-stage
relational network DEA efficiency ($E_{k}$). But it will be more appropriate
to compare the ranks of IIMs rather than their efficiency scores because the
relational model has two more sets of constraints than the conventional CCR
DEA model. These additional sets of constraints represent the 1st and 2nd
stage efficiency scores. The 1st and 2nd stage efficiency score can be
affecting factors in the overall efficiency scores of IIMs in the two-stage
relational network DEA model as explained earlier. Kao and Hung
kao2008efficiency , mathematically, explained in detail why this deviation
occurs in the $E_{k}$ and $E_{k}^{CCR}$ by constructing the dual model of the
relational two-stage network DEA model. A Spearman’s rank zar2005spearman
correlation coefficient is calculated for $E_{k}$ and $E_{k}^{CCR}$. The
correlation coefficient $(\rho=0.91758)$ indicates that efficiencies obtained
from both two-stage relational and conventional CCR DEA models are highly
correlated. This means that the efficiency scores from both models produce
almost the same rankings. This can be easily observed in Table 3.

Table 3: Overall efficiency scores and ranks (in parantheses) of two-stage
network DEA model and conventional CCR DEA model

DMU | IIM Name | Two-stage relational network DEA model | Conventional CCR DEA model
---|---|---|---
|  | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E_{k}$ | $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E_{k}^{CCR}$
$D_{1}$ | IIM Bangalore |  0.4973(1) |  1(1)
$D_{2}$ | IIM Ahmedabad |  0.01349(7) |  0.0327(10)
$D_{3}$ | IIM Calcutta |  0.1235(2) |  0.4067(2)
$D_{4}$ | IIM Lucknow |  0.0041(11) |  0.0162(11)
$D_{5}$ | IIM Indore |  0.0568(3) |  0.2666(3)
$D_{6}$ | IIM Kozhikode |  0.0507(4) |  0.2377(4)
$D_{7}$ | IIM Udaipur |  0.0025(12) |  0.0064(12)
$D_{8}$ | IIM Tiruchirapalli |  0.0090(8) |  0.0357(7)
$D_{9}$ | IIM Raipur |  0.0145(6) |  0.0347(8)
$D_{10}$ | IIM Rohtak |  0.0268(5) |  0.1639(5)
$D_{11}$ | IIM Shillong |  0.0084(9) |  0.0342(9)
$D_{12}$ | IIM Kashipur |  0.0082(10) |  0.0514(6)
$D_{13}$ | IIM Ranchi |  0.0007(13) |  0052(13)

##  4 Conclusions

The main objective of the current study is to illustrate the process of
research of IIMs with the help of the proposed research production model. This
study applies a two-stage relational network DEA model for the performance
analysis of IIMs. In our proposed research production model there are two sub-
processes. As we know that higher educational institutions are complex
structures that use multiple inputs to produce multiple outputs. The research
process is also complicated. As, in some stage, some output of research
process become the inputs for the next stages to produce final outputs. Our
proposed research production model explains the use of inputs and outputs in
sub-stages very interestingly. Hence, it is pretty reasonable to consider the
research production model for IIM in sub-stages, not as simply one that
transforms input to produce outputs. As we know that in general, data
envelopment analysis (DEA) works like a black box that does not provide any
adequate detail to identify the specific reason for inefficiency in decision-
making units (DMUs).

Our study shows that the efficiency level of IIMs is higher in the
conventional DEA model than the two-stage relational network DEA model. The
reason for the deviation in both types of efficiency levels is explained in
our study. So it will be more appropriate to compare the ranks of IIMs rather
than their efficiency scores. A Spearman’s rank correlation coefficient is
calculated for $E_{k}$ and $E_{k}^{CCR}$. The correlation coefficient
$(\rho=0.91758)$ indicates that efficiencies obtained from both two-stage
relational and conventional CCR DEA models are highly correlated. This means
that the efficiency scores from both models produce almost the same rankings.
But due to the use of the two-stage relational network DEA model, we are now
able to know the reasons behind the inefficiencies of IIMs. We found that many
IIMs performed well on the 1st stage but not on the 2nd stage. Based on this,
we can conclude that for IIMs it is easier to produce research publications
than to win attractive research grants. In a real-world situation, input-
output data can be imprecise. The future research direction can lead us to the
development and application of a two-stage network DEA model in a fuzzy
environment.

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