Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic fuzzy input–output targets

Abstract

In this paper, we develop intuitionistic fuzzy data envelopment analysis (IFDEA) and dual IFDEA (DIFDEA) models based on \(\alpha \)- and \(\beta \)-cuts. We determine intuitionistic fuzzy (IF) efficiencies based on \(\alpha \)- and \(\beta \)-cuts. We develop an IF correlation coefficient (IFCC) between IF variables to validate the DIFDEA models. We propose an index ranking approach to rank the decision making units (DMUs). Also, we propose an approach to find the IF input–output targets which help to make inefficient DMUs as efficient DMUs in IF environment. Finally, an example and a health sector application are presented to illustrate and compare the proposed methods.

Full text

Soft Computing
https://doi.org/10.1007/s00500-018-3504-3
METHODOLOGIES AND APPLICATION
Development of intuitionistic fuzzy data envelopment analysis models
and intuitionistic fuzzy input–output targets
Alka Arya1·Shiv Prasad Yadav1
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract
In this paper, we develop intuitionistic fuzzy data envelopment analysis (IFDEA) and dual IFDEA (DIFDEA) models based
on α- and β-cuts. We determine intuitionistic fuzzy (IF) efficiencies based on α- and β-cuts. We develop an IF correlation
coefficient (IFCC) between IF variables to validate the DIFDEA models. We propose an index ranking approach to rank
the decision making units (DMUs). Also, we propose an approach to find the IF input–output targets which help to make
inefficient DMUs as efficient DMUs in IF environment. Finally, an example and a health sector application are presented to
illustrate and compare the proposed methods.
Keywords Data envelopment analysis ·Intuitionistic fuzzy efficiencies ·Ranking ·Intuitionistic fuzzy input–output targets
1 Introduction
Data envelopment analysis (DEA) is a linear programming
based nonparametric technique for measuring the relative
efficiencies of decision making units (DMUs) which utilize
multiple inputs to produce multiple outputs. Charnes et al.
(1978) proposed the CCR DEA model which determines
the performance efficiencies of DMUs. DMUs can be any
governmental agencies and nonprofitable organizations like
hospitals, educational institutions, banks, transportation etc.
The relative performance efficiency of a DMU is defined as
the ratio of its performance efficiency to the largest perfor-
mance efficiency. The relative performance efficiency of a
DMU lies in the range (0, 1]. There are some studies of crisp
DEA in different areas (Banker et al. 1984; Barnum et al.
2011; Charnes et al. 1978; Hollingsworth et al. 1999; Mogha
et al. 2014; Ramanathan and Ramanathan 2010).
The conventional DEA is limited to crisp input and crisp
output data. But real world applications have some input
and/or output data which possess some degree of fluctuation
Communicated by V. Loia.
BAlka Arya
alka1dma@gmail.com
Shiv Prasad Yadav
spyorfma@gmail.com
1Department of Mathematics, Indian Institute of Technology
Roorkee, Roorkee 247667, India
or imprecision or uncertainties. The fluctuation can take the
form of intervals, ordinal relations and fuzzy numbers. Fuzzy
set theory (Zadeh 1965) is an important tool to handle fluctu-
ations/uncertainties in real world problems. There are some
studies of fuzzy DEA (FDEA) in different areas (Moheb-
Alizadeh et al. 2011; Dotoli et al. 2015; Jahanshahloo et al.
2009; Kao and Liu 2000;Tsaietal.2010). In fuzzy set theory,
sum of the degree of membership (acceptance) and degree
of non-membership (rejection) of an element is equal to one,
i.e. the rejection value is equal to one minus the acceptance
value (Zou et al. 2016). But in real world problems, there is
possibility that the sum of the acceptance and rejection val-
ues of an element may come out to be less than one. Thus,
there remains some degree of hesitation. Fuzzy set theory
(Zimmermann 2011) is not appropriate to deal with such
problems; rather intuitionistic fuzzy set (IFS) theory is more
suitable.
IFS theory, proposed by Atanassov (1986), is an extension
of fuzzy set theory and has been found to be more useful
to deal with vagueness/uncertainty. The IFS considers both
the acceptance value and rejection value of an element such
that the sum of both values is less than one, i.e. it may have
hesitation. Since its invention/inception, the IFS theory has
received more and more attention and has been used in a wide
range of applications, such as reliability (Shu et al. 2006),
logic programming (Atanassov and Gargov 1998), decision
making (Li 2005), medical diagnosis (De et al. 2001), and
pattern recognition (Dengfeng and Chuntian 2002). Puri and
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A. Arya, S. P. Yadav
Yadav (2015) proposed an intuitionistic fuzzy optimistic and
pessimistic DEA models. Otay et al. (2017) proposed a new
multi-expert IFDEA and IF analytic hierarchy process (IF-
AHP) to determine the performance evaluation of healthcare
institutions. Hajiagha et al. (2013) proposed IFDEA model
with IF inputs and IF outputs using the aggregation operator.
Beauty of IFS theory over fuzzy set theory is that IFS
theory separates the degree of membership (acceptance) and
the degree of non-membership (rejection) of an element in
the set. With the help of IFS theory, we can decide about
the degree of acceptance, degree of rejection and degree of
hesitation for some quantity. For example, in health sector,
there exist two inputs: (i) number of beds and (ii) sum of
number of pathologists and laboratory technicians and two
outputs: (i) number of pathology operations and (ii) sum of
number of plaster and tubal ligation which possess some
degree of hesitation due to the difference in thought at the
management level and the hospital level. Moreover, under
the mentioned reasons, hospital management would be more
interested in running a hospital with less number of beds,
pathologist and doctors (employees) in order to reduce the
cost on beds, pathologist and doctors, whereas the hospital
manager may be interested in having more beds, patholo-
gist and doctors at the disposal of the hospital in order to
accommodate more patients, handle day-to-day increased
workload and overcome the profit reductions due to the inef-
ficiency of some existing beds, pathologist and doctors, i.e.
the number of beds, pathologist and doctors is likely to be
an undesirable attribute for the hospital management, how-
ever a desirable attribute for the hospital manager. So, the
difference of thought at management level and hospital level
may lead to the existence of hesitation in the patients, and
availability of beds, pathologist and doctors at hospital level.
This hesitation is responsible for both the membership and
non-membership degrees of the data for the number of beds
and doctors of a hospital. Hence, the number of beds, pathol-
ogist and doctors possesses IF behaviour at hospital level and
thus can be taken as IF input in DEA. The inputs and out-
puts possess some degree of hesitation due to the difference
in thought at the management level and the actual hospital
level. So, uncertainty in inputs and outputs can be well taken
as IFN.
In this paper, we develop DIFDEA models using α−,β-
cut to determine the IF efficiency and IF correlation coef-
ficient (IFCC) between IF variables and propose a ranking
approach to rank the DMUs. Also, this study determines the
IF input target and IF output target for inefficient DMUs.
The rest of the paper is organized as follows: Section 2
presents the preliminaries. Section 3presents the extension
of DEA to DIFDEA. Section 4presents the proposed IFCC to
validate the proposed DIFDEA models. Section 5presents
the proposed IF ranking approach. Section 6presents the
illustrative example and a health sector application. Last sec-
tion of the paper concludes the findings of this paper.
2 Preliminaries
This section includes some basic definitions and notions.
Definition 1 (Arya and Yadav (2017)) The performance effi-
ciency of a DMU is defined as the ratio of the weighted sum of
outputs (called virtual output) to the weighted sum of inputs
(called virtual input). Thus,
Performance efficiency =virtual output
virtual input .
The relative performance efficiency of a DMU is defined as
the ratio of its performance efficiency to the largest perfor-
mance efficiency. The relative performance efficiency of a
DMU lies in the range (0, 1]. DEA evaluates the relative per-
formance efficiency of a set of homogeneous DMUs (Arya
and Yadav 2017).
Definition 2 (Zimmermann 2011) A fuzzy number (FN) ˜
A
is defined as a convex normalized fuzzy set ˜
Aof the real line
Rwith membership function μ˜
Asuch that
•there exists exactly one x0∈Rwith μ˜
A(x0)=1. x0is
called the mean value of ˜
A,
•μ˜
Ais a piecewise continuous function on R.
Definition 3 (Arya and Yadav 2017) The triangular FN
(TFN) ˜
Ais a FN denoted by ˜
A=(al,am,au)and is defined
by the membership function μ˜
Agiven by
μ˜
A(x)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
x−al
am−al,al<x≤am,
au−x
au−am,am≤x<au,
0,elsewhere,
for all x∈R, where amis called the modal value and (al,au)
is called support of the TFN (al,am,au).
Definition 4 (Arithmetic operations on TFN) (Arya and
Yadav 2017)Let ˜
A1=(a1l,a1m,a1u)and ˜
A2=(a2l,
a2m,a2u)be two TFNs. Then, the arithmetic operations on
TFNs are given as follows:
•Adition: ˜
A1⊕˜
A2=(a1l+a2l,a1m+a2m,a1u+a2u).
•Subtraction: ˜
A1˜
A2=(a1l−a2u,a1m−a2m,a1u−a2l).
•Multiplication: ˜
A1⊗˜
A2≈(min(a1la2l,a1la2u,a1ua2l,
a1ua2u), a1ma2m,max(a1la2l,a1la2u,a1ua2l,a1ua2u)).
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Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic…
•Scalar multiplication:
λ˜
A1=(λa1l,λa1m,λa1u), for λ≥0,
(λa1u,λa1m,λa1l), for λ<0.
Definition 5 (Intuitionistic fuzzy set (IFS)) (Arya and Yadav
2018) Let us suppose that X is a universe of discourse. Then,
an IFS in Xis denoted by ˜
AIand is defined by ˜
AI=
{(x,μ˜
AI(x), ν ˜
AI(x)) ∀x∈X}, where μ˜
AI:X→[0,1]
and ν˜
AI:X→[0,1]represent the membership and non-
membership functions, respectively, of an element x in ˜
AI.
The values μ˜
AI(x)and ν˜
AI(x)represent the membership and
non-membership values of x being in ˜
AIwith the condition
that 0 ≤μ˜
AI(x)+ν˜
AI(x)≤1. The hesitation (indeter-
minacy) degree of an element x being in ˜
AIis defined as
π˜
AI(x)=1−μ˜
AI(x)−ν˜
AI(x)∀x∈X. Obviously 0 ≤
π˜
AI(x)≤1. If π˜
AI(x)=0, then ˜
AIis reduced to a fuzzy set.
Definition 6 (Normal IFS) (Arya and Yadav 2018)Let ˜
AI=
{(x,μ˜
AI(x), ν ˜
AI(x)) :x∈X}be an IFS. Then, ˜
AIis called
the normal IFS if ∃an x∈Xsuch that μ˜
AI(x)=1 and
ν˜
AI(x)=0.
Definition 7 (Convex IFS) (Arya and Yadav 2018)Let ˜
AI=
{(x,μ˜
AI(x), ν ˜
AI(x)) :x∈X}be an IFS. Then, ˜
AIis called
the Convex IFS if
(i) min (μ ˜
AI(x), μ ˜
AI(y)) ≤μ˜
AI(λx+(1−λ)y),∀x,y∈
Xand λ∈[0,1], i.e., μ˜
AIis quasi-concave function
over X.
(ii) max (ν ˜
AI(x), ν ˜
AI(y)) ≥ν˜
AI(λx+(1−λ)y),∀x,y∈X
and λ∈[0,1], i.e., ν˜
AIis quasi-convex function over X.
Definition 8 (α-cut)Theα-cut of an IFS ˜
AIis denoted by
AI
αand defined as AI
α={x:μ˜
AI(x)≥α}; α∈[0,1].
Definition 9 (β-cut)Theβ-cut of an IFS ˜
AIis denoted by
BI
βand defined as BI
β={x:ν˜
AI(x)≤β}; β∈[0,1].
Definition 10 (Intuitionistic fuzzy number (IFN)) (Arya and
Yadav 2018) The IFN is an extension of a FN (Zimmermann
2011) in IF environment. This is defined as follows:
Let ˜
AI={(x,μ ˜
AI(x), ν ˜
AI(x)) :x∈R}be an IFS, where
Ris the set of all real numbers. Then, ˜
AIis called the IFN if
the following conditions hold:
(i) There exists a unique x0∈Rsuch that μ˜
AI(x0)=1
and ν˜
AI(x0)=0,
(ii) ˜
AIis convex IFS,
(iii) μ˜
AIand ν˜
AIare piecewise continuous functions on R.
Mathematically, an IFS ˜
AI={(x,μ˜
AI(x), ν ˜
AI(x)) :x∈
R}is an IFN if μ˜
AIand ν˜
AIare piecewise continuous func-
tions from Rto [0,1]and 0 ≤μ˜
AI(x)+ν˜
AI(x)≤1,∀x∈R
given by
1
x()
I
A
x()
I
A
0
'l
a
l
a
m
a
u
a
'u
a
Fig. 1 Membership and non-membership functions of IFN ˜
AI
μ˜
AI(x)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
g1(x), al≤x<am,
1,x=am,
h1(x), am<x≤au,
0,elsewhere,
ν˜
AI(x)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
g2(x), al≤x<am,
0,x=am,
h2(x), am<x≤au,
1,elsewhere,
where amis called the mean value of ˜
AI;am−aland
au−amare called the left and right hand spreads of member-
ship function μ˜
AI, respectively; am−aland au−amare
called the left and right hand spreads of hesitation function
π˜
AI, respectively; g1and h1are called piecewise contin-
uous, increasing and decreasing functions in [al,am)and
(am,au], respectively; and g2and h2are called piecewise
continuous, strictly increasing and strictly decreasing func-
tions in [al,am)and (am,au], respectively. The IFN ˜
AI
is represented by ˜
AI=(al,am,au;al,am,au), where
al≤al≤am≤au≤au. Its graphical representation
is given in Figure 1.
Remark 1 For an IFN ˜
AI,theα- and β-cuts are closed inter-
vals as follows:
AI
α={x:μ˜
AI(x)≥α}=[a(α), b(α)]; α∈(0,1],
BI
β={x:ν˜
AI(x)≤β}=[a(β), b(β )]; β∈[0,1),
where a(α), b(β) are increasing functions of αand β,
respectively, and b(α), a(β) are decreasing functions of α
and β, respectively.
Definition 11 (Triangular intuitionistic fuzzy number
(TIFN)) (Arya and Yadav 2018) The TIFN ˜
AI=(al,am,au;
al,am,au)is an IFN with the membership function μ˜
AIand
non-membership function ν˜
AIgiven by
123

A. Arya, S. P. Yadav
I
A
1
I
A
0
'l
a
l
a
m
a
u
a
'u
a
Fig. 2 TIFN ˜
AI=(al,am,au;al,am,au)
μ˜
AI(x)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
x−al
am−al,al<x≤am,
au−x
au−am,am≤x<au,
0,elsewhere,
ν˜
AI(x)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
x−am
al−am,al<x≤am,
am−x
am−au,am≤x<au,
1,elsewhere,
where al,am,au,al,au∈Rsuch that al≤al≤am≤
au≤au. Its graphical representation is given in Figure 2.
Definition 12 (Positive IFN)Let ˜
AI=(al,am,au;al,
am,au)be an IFN. Then, ˜
AIis called a positive IFN if
al>0.
Definition 13 (Arithmetic operations on TIFN) (Arya and
Yadav 2018)Let ˜
A1
I=(a1l,a1m,a1u;a1
l,a1m,a1
u)and
˜
A2
I=(a2l,a2m,a2u;a2
l,a2m,a2
u)be two TIFNs. Then,
the arithmetic operations on TIFNs are given as follows:
(i) Addition: ˜
A1
I⊕˜
A2
I=(a1l+a2l,a1m+a2m,a1u+
a2u;a1
l+a2
l,a1m+a2m,a1
u+a2
u).
(ii) Multiplication: ˜
A1
I⊗˜
A2
I≈(a1la2l,a1ma2m,a1ua2u;
a1
la2
l,a1ma2m,a1
ua2
u), where ˜
A1
I,˜
A2
I>0.
(iii) Scalar multiplication: If λ∈R, then
λ˜
A1
I=(λa1l,λa1m,λa1u;λa1
l,λa1m,λa1
u), λ ≥0,
(λa1u,λa1m,λa1l;λa1
u,λa1m,λa1
l), λ ≤0.
3 Extension of DEA to DIFDEA models
DEA is a linear programming based methodology to measure
the relative efficiencies of DMUs when the production pro-
cess consists of multiple inputs and outputs. Assume that the
performance of a set of n homogeneous DMUs (DMUj; j=
1,2,3,...,n) is to be measured (Arya and Yadav 2017). Let
us suppose that each DMU utilizes m inputs to produce s out-
puts. Consider the jth DMU: DMU j,j=1,2,3, ..., n.Let
xij be the amount (value) of the ith input utilized and yrj be
the amount (value) of the rth output produced by the jth DMU,
j=1,2,3,...,n,i=1,2,3,...,m,r=1,2,3,...,s.
Then, the efficiency of DMU jis given by Charnes et al.
(1978)
Ej=
s

r=1
vrjyrj
m

i=1
uijxij
,j=1,2,...,n,
where uij and vrj are the weights corresponding to ith input
and rth output of DMU j, respectively.
In the CCR fractional program (FP) (Charnes et al. 1978),
the efficiency of the DMUjo is to be maximized subject to
the condition that the ratio of the virtual output to the virtual
input of every DMU should be less than or equal to unity. The
CCR (ratio) fractional DEA program and the corresponding
linear program (LP) for DMU joare given in Table 1. In these
models, the efficiency of DMU jo is denoted by Ejoand ε>0
is a non-Archimedean infinitesimal constant.
If DMUs have IF input and IF output data, then we
develop intuitionistic fuzzy DEA (IFDEA) models to deter-
mine the efficiencies of DMUs. Assume that the performance
of DMU j(j=1,2,...,n) is characterized by a production
process of m IF inputs ˜xI
ij;i=1,2,3,...,mto yield s IF
outputs ˜yI
rj;r=1,2,3,...,s. Let IF efficiency of DMUjo
be represented by ˜
EI
jo. Then, IFDEA model (Model 1) is
given as follows:
Model 1 (IFDEA)
max ˜
EI
jo=
s

r=1
vrj
o˜yI
rj
o
subject to
m

i=1
uij
o˜xI
ij
o=˜
1I,
s

r=1
vrj
o˜yI
rj −
m

i=1
uij
o˜xI
ij ≤˜
0I,j=1,2,3,...,n,
uij
o,v
rj
o≥ε, ∀i=1,2,3,...,m;∀r=1,2,3,...,s;ε
>0 is a non-Archimedean infinitesimal constant,
where uij
oand vrj
oare the weights corresponding to the ith
IF input and rth IF output, respectively.
123

Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic…
Table 1 Crisp DEA Models:
Fractional and LP forms Fractional DEA LP DEA
max Ejo=s
r=1vrj
oyrj
o
m
i=1uij
oxij
o
max Ejo=s
r=1vrj
oyrj
o
subject to subject to
s
r=1vrj
oyrj
m
i=1uij
oxij
≤1,j=1,2,3,...,nm
i=1uij
oxij
o=1
s
r=1vrj
oyrj −m
i=1uij
oxij ≤0,j=1,2,3,...,n
uij
o,v
rj
o≥ε, ∀i,ru
ij
o,v
rj
o≥ε, ∀i,r
3.1 Methodology for solving IFDEA model
Assume that IF input ˜xI
ij and IF output ˜yI
rj are TIFNs. Let
˜xI
ij =(xL
ij,xM
ij ,xU
ij;xL
ij ,xM
ij ,xU
ij )and ˜yI
rj =(yL
rj,yM
rj ,yU
rj;
yL
rj ,yM
rj ,yU
rj ). Then, Model 1 is transformed to the following
model (Model 2):
Model 2
max ˜
EI
jo=
s

r=1
vrj
oyL
rj
o,yM
rj
o,yU
rj
o;yL
rj
o,yM
rj
o,yU
rj
o
subject to
m

i=1
uij
oxL
ij
o,xM
ij
o,xU
ij
o;xL
ij
o,xM
ij
o,xU
ij
o=(1,1,1;1,1,1),
s

r=1
vrj
oyL
rj,yM
rj ,yU
rj;yL
rj ,yM
rj ,yU
rj 
−
m

i=1
uij
oxL
ij,xM
ij ,xU
ij;xL
ij ,xM
ij ,xU
ij 
≤(0,0,0;0,0,0), j=1,2,3,...,n,
uij
o,v
rj
o≥ε, ∀i,r.
3.2 Models based on ˛-cut
Replacing IF input ˜xI
ij and IF output ˜yI
rj by their α-cuts
[αxM
ij +(1−α)xL
ij,αxM
ij +(1−α)xU
ij]and [αyM
rj +(1−
α)yL
rj,αyM
rj +(1−α)yU
rj], respectively. Let the α-cut of ˜
EI
jo
be [EL
jo,α,EU
jo,α]. Then, Model 2 is reduced to the following
model (Model 3):
Model 3
max EL
jo,α,EU
jo,α=
s

r=1
vrj
oαyM
rj
o
+(1−α)yL
rj
o,αyM
rj
o+(1−α)yU
rj
o
subject to
m

i=1
uij
oαxM
ij
0+(1−α)xL
ij
o,αxM
ij
o+(1−α)xU
ij
o=[1,1],
s

r=1
vrj
oαyM
rj +(1−α)yL
rj,αyM
rj +(1−α)yU
rj
−
m

i=1
uij
oαxM
ij +(1−α)xL
ij,αxM
ij +(1−α)xU
ij
≤[0,0],j=1,2,3,...,n,
uij
o,v
rj
o≥ε, ∀i,r.
Model 3 is given in lower and upper bound model. The lower
bound model of Model 3 is given as below:
Model 4
max EL
jo,α =
s

r=1
vrj
oαyM
rj
o+(1−α)yL
rj
o
subject to
m

i=1
uij
oαxM
ij
0+(1−α)xU
ij
o=1,
s

r=1
vrj
oαyM
rj
o+(1−α)yL
rj
o
−
m

i=1
uij
oαxM
ij
0+(1−α)xU
ij
o≤0,
s

r=1
vrj
oαyM
rj +(1−α)yU
rj
−
m

i=1
uij
oαxM
ij +(1−α)xL
ij
≤0,j=1,2,3,...,n,j= jo,
uij
o,v
rj
o≥ε, ∀i,r.
Model 4 is a DEA model, and the levels of inputs and outputs
are now adjusted unfavourably to the evaluated DMUjoand
in favour of the other DMUs. For DMUjo, the outputs are
adjusted at their lower bounds and the inputs are adjusted at
123

A. Arya, S. P. Yadav
their upper bounds. For other DMUs, the outputs are adjusted
at their upper bounds and the inputs are favourably adjusted
at their lower bounds. Thus, the DMU jocomes to the worst
possible position compared with other DMUs based on α-cut.
The upper bound model of Model 3 is given as below:
Model 5
max EU
jo,α =
s

r=1
vrj
oαyM
rj
o+(1−α)yU
rj
o
subject to
m

i=1
uij
oαxM
ij
0+(1−α)xL
ij
o=1,
s

r=1
vrj
oαyM
rj
o+(1−α)yU
rj
o
−
m

i=1
uij
oαxM
ij
0+(1−α)xL
ij
o≤0,
s

r=1
vrj
oαyM
rj +(1−α)yL
rj
−
m

i=1
uij
oαxM
ij +(1−α)xU
ij
≤0,j=1,2,3,...,n,j= jo,
uij
o,v
rj
o≥ε, ∀i,r.
Model 5 is also a DEA model, where the levels of inputs
and outputs are adjusted in favour of the evaluated DMU jo
and in unfavour of other DMUs. For the evaluated DMU,
the outputs are adjusted at their upper bounds and the inputs
are adjusted at their lower bounds. Unfavourably for the other
DMUs, the outputs are adjusted at their lower bounds and the
inputs are adjusted at their upper bounds. Thus, the DMU jo
comes to the best possible position compared with the other
DMUs based on α-cut.
The dual of Model 4 is written as follows:
Model 6
min ξL
jo,α =θL
jo−⎛
⎝
s

r=1
S+
rj
o,α +
m

i=1
S−
ij
o,α⎞
⎠
subject to
θL
joαxM
ij
0+(1−α)xU
ij
o−λL
joαxM
ij
0+(1−α)xU
ij
o
−
n

j=1,= jo
λL
jαxM
ij +(1−α)xL
ij−S−
ij
o,α =0,∀i
λL
joαyM
rj
o+(1−α)yL
rj
o+
n

j=1,= jo
λL
jαyM
rj +(1−α)yU
rj
−S+
rj
o,α =αyM
rj
o+(1−α)yL
rj
o,∀r
λL
j,S−
ij
o,α,S+
rj
o,α ≥0,∀j=1,2,3,...,n,
j= jo,θ
L
jois unrestricted in sign.
In Model 6, S−
ij
o,α and S+
rj
o,α are the slack variables, λL
jis
the non-negative dual variable corresponding to the jth pri-
mal (Model 4) constraints, =1,2,3,…,n and θL
jois unrestricted
dual variable corresponding to the equation constraint in the
primal problem (Model 4).
The dual of Model 5 is written as follows:
Model 7
min ξU
jo,α =θU
jo−ε⎛
⎝
s

r=1
S+
rj
o,α +
m

i=1
S−
ij
o,α⎞
⎠
subject to
θU
joαxM
ij
o+(1−α)xL
ij
o−λU
joαxM
ij
0+(1−α)xL
ij
o
−
n

j=1,= jo
λU
jαxM
ij +(1−α)xU
ij−S−
ij
o,α =0,∀i
λU
joαyM
rj
o+(1−α)yU
rj
o+
n

j=1,= jo
λU
jαyM
rj +(1−α)yL
rj
−S+
rj
o,α =αyM
rj
o+(1−α)yU
rj
o,∀r
λU
j,S−
ij
o,α,S+
rj
o,α ≥0,∀j,θ
U
jois unrestricted in sign,
where λU
jis the non-negative dual variable corresponding to
the jth primal (Model 5) constraints, j=1,2,3,…,n and θU
jo
is unrestricted dual variable corresponding to the equation
constraint in the primal problem (Model 5). Models 6 and 7
are the proposed DIFDEA models based on α-cut and will
be denoted as PDIFDEAjo,α.
Definition 14 Efficient and inefficient DMUs based on α-cut,
DMU jois fully efficient if ξL∗
jo,α =1 for any α∈(0,1].
DMU jois efficient if ξU∗
jo,α =1 and ξL∗
jo,α <1 for any
α∈(0,1].
DMU jois inefficient if ξU∗
jo,α <1 for any α∈(0,1].
Axiom 3.1 The lower bound efficiency is less than or equal
to upper bound efficiency of DMU jo, i.e. ξL∗
jo,α ≤ξU∗
jo,α∀α∈
(0,1].
3.3 Models based on ˇ-cut
Replacing IF input ˜xI
ij and IF output ˜yI
rj by their β-
cuts ˜xI
ij =[βxL
ij +(1−β)xM
ij ,βxU
ij +(1−β)xM
ij ]and
˜yI
rj =[βyL
rj +(1−β)yM
rj ,βyU
rj +(1−β)yM
rj ], respec-
tively, Model 2 is reduced to the following model:
123

Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic…
Model 8
max EL
jo,β ,EU
jo,β =
s

r=1
v
rj
oβyL
rj
o
+(1−β)yM
rj
o,βyU
rj
o+(1−β)yM
rj
o
subject to
m

i=1
u
ij
oβxL
ij
o+(1−β)xM
ij
o,βxU
ij
o+(1−β)xM
ij
o=[1,1],
s

r=1
v
rj
oβyL
rj +(1−β)yM
rj ,βyU
rj +(1−β)yM
rj 
−
m

i=1
u
ij
oβxL
ij +(1−β)xM
ij ,βxU
ij +(1−β)xM
ij 
≤[0,0],j=1,2,3,...,n,j= jo,
u
ij
o,v

rj
o≥ε, ∀i,r.
Model 8 is given in lower and upper bound model. The lower
bound model of Model 8 is given as below:
Model 9
max EL
jo,β =
s

r=1
v
rj
oβyL
rj
o+(1−β)yM
rj
o
subject to
m

i=1
u
ij
oβxU
ij
o+(1−β)xM
ij
o=1,
s

r=1
v
rj
oβyL
rj
o+(1−β)yM
rj
o
−
m

i=1
u
ij
oβxU
ij
o+(1−β)xM
ij
o≤0,
s

r=1
v
rj
oβyU
rj +(1−β)yM
rj 
−
m

i=1
u
ij
oβxL
ij +(1−β)xM
ij ≤0,
j=1,2,3,...,n,j= jo,u
ij
o,v

rj
o≥ε, ∀i,r.
Model 9 is a DEA model, the levels of inputs and outputs
are now adjusted unfavourably to the evaluated DMU joand
in favour of the other DMUs. For DMUjo, the outputs are
adjusted at their lower bounds and the inputs are adjusted at
their upper bounds. For other DMUs, the outputs are adjusted
at their upper bounds and the inputs are favourably adjusted
at their lower bounds. Thus, the DMU jocomes to the worst
possible position compared to other DMUs based on β-cut.
The upper bound model of Model 8 is given as below:
Model 10
max EU
jo,β =
s

r=1
v
rj
oβyU
rj
o+(1−β)yM
rj
o
subject to
m

i=1
u
ij
oβxL
ij
o+(1−β)xM
ij
o=1,
s

r=1
v
rj
oβyU
rj
o+(1−β)yM
rj
o
−
m

i=1
u
ij
oβxL
ij
o+(1−β)xM
ij
o≤0,
s

r=1
v
rj
oβyL
rj +(1−β)yM
rj 
−
m

i=1
u
ij
oβxU
ij +(1−β)xM
ij ≤0,
j=1,2,...,n,j= jo,u
ij
o,v

rj
o≥ε, ∀i,r.
Model 10 is also a DEA model, where the levels of inputs
and outputs are adjusted in favour of the evaluated DMU jo
and in unfavour of other DMUs. For the evaluated DMU jo,
the outputs are adjusted at their upper bounds and the inputs
are adjusted at their lower bounds. Unfavourably for the other
DMUs, the outputs are adjusted at their lower bounds and the
inputs are adjusted at their upper bounds. Thus, the DMU jo
comes to the best possible position compared to other DMUs
based on β-cut.
The dual of Model 9 is written as follows:
Model 11
min ξL
jo,β =θL
jo−s

r=1
S+
rj
o+
M

i=1
S−
ij
o
subject to
θL
joβxU
ij
o+(1−β)xM
ij
o−λL
joβxU
ij
o−(1−β)xM
ij
o
−
n

j=1,= jo
λL
jβxL
ij +(1−β)xM
ij −S−
ij
o,β =0,∀i
λL
joβyL
rj
o+(1−β)yM
rj
o+
n

j=1,= jo
λL
jβyU
rj +(1−β)yM
rj 
−S+
rj
o,β =βyL
rj
o+(1−β)yM
rj
o,∀r
λL
j≥0∀j;S−
ij
o,S+
rj
o≥0,∀i,r;θL
jois unrestricted in sign.
The dual of Model 10 is written as follows:
123

A. Arya, S. P. Yadav
Model 12
min ξU
jo,β =θU
jo−s

r=1
S+
rj
o+
m

i=1
S−
ij
o
subject to
θU
joβxL
ij
o+(1−β)xM
ij
o−λU
joβxL
ij
o+(1−β)xM
ij
o
−
n

j=1,= jo
λU
jβxU
ij +(1−β)xM
ij −S−
ij
o=0,∀i,
λU
jβyU
rj
o+(1−β)yM
rj
o+
n

j=1,= jo
λU
jβyL
rj +(1−β)yM
rj 
−S+
rj
o=βyU
rj
o+(1−β)yM
rj
o,∀r,
λU
j≥0∀j;S−
ij
o,S+
rj
o,≥0∀i,r;θU
jois unrestricted in sign.
Models 11 and 12 are the proposed dual IFDEA (DIFDEA)
models based on β-cut and will be denoted as
PDIFDEAjo,β.
Definition 15 Efficient and inefficient DMUs based on β-cut,
DMU jois fully efficient if ξL∗
jo,β =1 for any β∈[0,1).
DMU jois efficient if ξU∗
jo,β =1 and ξL∗
jo,β <1 for any
β∈[0,1).
DMU jois inefficient if ξU∗
jo,β <1 for any β∈[0,1).
Axiom 3.2 The lower bound efficiency is less than or equal
to upper bound efficiency of DMU jo, i.e. ξL∗
jo,β ≤ξU∗
jo,β ∀β∈
[0,1).
3.4 IF input targets and IF output targets
The main objective of DEA models is to identify the efficient
and inefficient DMUs and to suggest to make the inefficient
DMUs as efficient DMUs. An inefficient DMU can become
efficient using adjusting inputs (called input targets) and/or
adjusting outputs (called output targets). The ith input target
and rth output target for DMU j0are denoted by ¯xij
0and ¯yrj
0,
respectively, and are defined by Agarwal (2014)
¯xij
0=θ∗
ij
0xij
0−S−
ij
0,¯yrj
0=yrj
0+S+
rj
0
If inputs and outputs are IFNs, then the adjusting IF inputs
(called IF input target) and/or adjusting IF outputs (called
IF output target) for DMU j0are denoted by ¯
˜xI
ij
0and ¯
˜yrj
0,
respectively, and are defined by
¯
˜xI
ij
0=˜
θI∗
j0˜xI
ij
0−S−
ij
0,¯
˜yI
rj
0=¯
˜yI
rj
0+S+
rj
0.
4 Proposed IF correlation coefficients (IFCCs)
to validate the proposed DIFDEA models
Definition 16 (Isotonicity test) If positive correlation coef-
ficients between input–output data are found (Avkiran et al.
2008), the selection of inputs and outputs is justified.
Avkiran et al. (2008) and Tsai et al. (2006) proposed the iso-
tonicity test to validate the conventional DEA models. Puri
and Yadav (2013) proposed the fuzzy correlation coefficients
between fuzzy input–output data. In this paper, we are con-
cerned with the evaluation of ξL
jo,α,ξU
jo,α,ξL
jo,β and ξU
jo,β on
the basis of the IF input–output data. To ensure the validity of
the proposed DIFDEA models, we find the IFCCs between
IF variables. To the best of our knowledge, in the literature,
nobody has proposed the IFCCs. Therefore, in this paper,
we propose the IFCCs between IF variables using expected
values (Hung and Wu 2001).
4.1 Expected interval and expected value of an IFN
Let ˜
AI=(am;am−al,au−am;am−bl,bu−am)be
an IFN with membership and non-membership functions
μ˜
AI(x)and ν˜
AI(x), respectively, given by
μ˜
AI(x)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
g1(x), al≤x<am,
1,x=am,
h1(x), am<x≤au,
0,elsewhere.
ν˜
AI(x)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
g2(x), bl≤x<am,
0,x=am,
h2(x), am<x≤bu,
1,elsewhere.
The expected interval (Grzegorzewski 2003)of ˜
AIis the
crisp interval EI(˜
AI)given by EI(˜
AI)=[E∗L(˜
AI),
E∗U(˜
AI)], where
E∗L(˜
AI)=bl+am
2+1
2am
bl
g2(x)dx−1
2am
al
g1(x)dx,
(4.1)
E∗U(˜
AI)=am+bu
2+1
2au
am
h1(x)dx−1
2bu
am
h2(x)dx.
(4.2)
The expected value of an IFN is given by
EV(˜
AI)=E∗L(˜
AI)+E∗U(˜
AI)
2.(4.3)
Theorem 1 Let ˜
AI=(al,am,au;bl,am,bu)be a TIFN.
Then, EV (˜
AI)=al+bl+4am+au+bu
8.
123

Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic…
Proof Using (4.1), we have E∗L(˜
AI)=al+2am+bl
4.
Using (4.2), we have E∗U(˜
AI)=au+2am+bu
4.
Therefore, EV(˜
AI)=al+bl+4am+au+bu
8.
4.2 Proposed IFCC between IF variables
Definition 17 (correlation coefficients between variables)
Let xand ybe two crisp variables and (x,y)assume the
values (xi,yi), i=1,2,3,...,n. Then, the correlation
coefficient between xand yis denoted by C(x,y)and is
defined by
C(x,y)=
n
n

i=1
xiyi−
n

i=1
xi
n

i=1
yi
n
n

i=1
xi2−n

i=1
xi2
.n
n

i=1
yi2−n

i=1
yi2.
(4.4)
Let ˜xIand ˜yIbe two IF variables and (˜xI,˜yI)assume the
values (˜xI
i,˜yI
i), i=1,2,3,...,n. Then, the IFCC between
˜xIand ˜yIis denoted by ˜
CI(˜xI,˜yI)and is defined by
˜
CI(˜xI,˜yI)
=
n
n

i=1
˜xiI˜yiI−
n

i=1
˜xiIn

i=1
˜yiI
n
n

i=1˜xiI2−n

i=1
˜xiI2
.n
n

i=1˜yiI2−n

i=1
˜yiI2.
(4.5)
The square of a positive TIFN ˜
AIis given by
(˜
AI)2=˜
AI.˜
AI
=(al,am,au;al,am,au).(al,am,au;al,am,au)
=((al)2,(am)2,(au)2;(al)2,(am)2,(au)2). (4.6)
It is difficult to apply (4.5) if data are large. To obtain the
IFCC between the IF variables, we propose a new method
using the expected value approach as described below:
Let ˜xI
i=(xl
i,xm
i,xu
i;xl
i,xm
i,xu
i)and ˜yI
i=(yl
i,ym
i,yu
i;
yl
i,ym
i,yu
i)be TIFNs. Then, the α-cut of ˜xI
iis the interval
[xL
i,α,xU
i,α], where xL
i,α =αxm
i+(1−α)xl
i, and xU
i,α =
αxm
i+(1−α)xu
i,α∈(0,1].
Similarly, the α-cut of ˜yI
iis the interval [yL
i,α,yU
i,α], where
yL
i,α =αym
i+(1−α)yl
i, and yU
i,α =αym
i+(1−α)yu
i,
α∈(0,1].
The expected interval (EI) of the IFCC ˜
CI(˜xI,˜yI)
based on α-cuts of ˜xI
iand ˜yI
iis defined as the interval
CEI
α(˜xI,˜yI)=[CL
α(˜xI,˜yI), CU
α(˜xI,˜yI)], where
CL
α˜xI,˜yI
=
n
n

i=1
xL
i,α yL
i,α −
n

i=1
xL
i,α
n

i=1
yL
i,α



n
n

i=1xL
i,α 2−n

i=1
xL
i,α 2
.


n
n

i=1yL
i,α 2−n

i=1
yL
i,α 2.
(4.7)
CU
α˜xI,˜yI
=
n
n

i=1
xU
i,α yU
i,α −
n

i=1
xU
i,α
n

i=1
yU
i,α



n
n

i=1xU
i,α 2−n

i=1
xU
i,α 2
.


n
n

i=1yU
i,α 2−n

i=1
yU
i,α 2.
(4.8)
Observe that CL
α(˜xI,˜yI)and CU
α(˜xI,˜yI)are the correlation
coefficients based on the data (xL
i,α,yL
i,α)and (xU
i,α,yU
i,α),
respectively, i=1,2,3,...,n. These correlation coeffi-
cients satisfy the following properties:
1. CL
α(˜xI,˜yI)∈[−1,1]and CU
α(˜xI,˜yI)∈[−1,1]
∀α∈(0,1].
2. CL
α(˜xI,˜yI)=1 and CU
α(˜xI,˜yI)=1if ˜xI=˜yI
∀α∈(0,1].
3. CL
α(˜xI,˜yI)=CL
α(˜yI,˜xI)and CU
α(˜xI,˜yI)=CU
α(˜yI,˜xI)
∀α∈(0,1].
The expected value (EV) of IFCC ˜
CI(˜xI,˜yI)based on α-cut
is denoted by CEV
α(˜xI,˜yI)and is defined by
CEV
α(˜xI,˜yI)=1
2CL
α(˜xI,˜yI)+CU
α(˜xI,˜yI),α∈(0,1].
(4.9)
The β-cut of ˜xI
iis the interval [xL
i,β ,xU
i,β ], where xL
i,β =
βxl
i+(1−β)xm
i, and xU
i,β =βxu
i+(1−β)xm
i∀β∈[0,1).
Similarly, the β-cut of ˜yI
iis the interval [yL
i,β ,yU
i,β ], where
yL
i,β =βyl
i+(1−β)ym
i, and yU
i,β =βyu
i+(1−β)ym
i∀β∈
[0,1).
The EI of IFCC ˜
CI(˜xI,˜yI)based on β-cuts of ˜xI
iand
˜yI
iis defined as the interval CEI
β(˜xI,˜yI)=[CL
β(˜xI,˜yI),
CU
β(˜xI,˜yI)], where
CL
β(˜xI,˜yI)
=
n
n

i=1
xL
i,β yL
i,β −
n

i=1
xL
i,β
n

i=1
yL
i,β



n
n

i=1xL
i,β 2−n

i=1
xL
i,β 2
.


n
n

i=1yL
i,β 2−n

i=1
yL
i,β 2.
(4.10)
123

A. Arya, S. P. Yadav
CU
β(˜xI,˜yI)
=
n
n

i=1
xU
i,β yU
i,β −
n

i=1
xU
i,β
n

i=1
yU
i,β



n
n

i=1xU
i,β 2−n

i=1
xU
i,β 2
.


n
n

i=1yU
i,β 2−n

i=1
yU
i,β 2.
(4.11)
Observe that CL
β(˜xI,˜yI)and CU
β(˜xI,˜yI)are the correlation
coefficients based on the data (xL
i,β ,yL
i,β )and (xU
i,β ,yU
i,β ),
respectively, i=1,2,3,...,n. These correlation coeffi-
cients satisfy the following properties:
1. CL
β(˜xI,˜yI)∈[−1,1]and CU
β(˜xI,˜yI)∈[−1,1]
∀β∈[0,1).
2. CL
β(˜xI,˜yI)=1 and CU
β(˜xI,˜yI)=1if ˜xI=
˜yI∀β∈[0,1).
3. CL
β(˜xI,˜yI)=CL
β(˜yI,˜xI)and CU
β(˜xI,˜yI)=
CU
β(˜yI,˜xI)∀β∈[0,1).
The EV of IFCC ˜
CI(˜xI,˜yI)based on β-cut is denoted by
CEV
β(˜xI,˜yI)and is defined by
CEV
β(˜xI,˜yI)=1
2CL
β(˜xI,˜yI)
+CU
β(˜xI,˜yI),β∈[0,1). (4.12)
The EV of IFCC ˜
CI(˜xI,˜yI)based on α- and β-cuts is
denoted by CEV
α,β (˜xI,˜yI)and is defined by
CEV
α,β (˜xI,˜yI)=CEV
α(˜xI,˜yI)+CEV
β(˜xI,˜yI)
2(4.13)
Theorem 2 Let [CL
α(˜xI,˜yI), CU
α(˜xI,˜yI)]and [CL
β(˜xI,˜yI),
CU
β(˜xI,˜yI)]be the EIs of the IFCC ˜
C(˜xI,˜yI)based
on α- and β-cuts, respectively. Then, C EV
α,β (˜xI,˜yI)=
CL
α(˜xI,˜yI)+CU
α(˜xI,˜yI)+CL
β(˜xI,˜yI)+CU
β(˜xI,˜yI)
4.
Proof The EV of IFCC ˜
CI(˜xI,˜yI)based on α- and β-cuts
is given by
CEV
α,β (˜xI,˜yI)=CEV
α(˜xI,˜yI)+CEV
β(˜xI,˜yI)
2
Using (5.9) and (5.12), we get
CEV
α,β (˜xI,˜yI)=1
2CL
α(˜xI,˜yI)+CU
α(˜xI,˜yI)
2
+
CL
β(˜xI,˜yI)+CU
β(˜xI,˜yI)
2⎤
⎦
=
CL
α(˜xI,˜yI)+CU
α(˜xI,˜yI)+CL
β(˜xI,˜yI)+CU
β(˜xI,˜yI)
4

CEV
α,β (˜xI,˜yI)satisfies the following properties:
1. CEV
α,β (˜xI,˜yI)∈[−1,1].
2. CEV
α,β (˜xI,˜yI)=1if ˜xI=˜yI.
3. CEV
α,β (˜xI,˜yI)=CEV
α,β (˜yI,˜xI).
Remark If the value of CEV
α,β (˜xI,˜yI)for each ˜xIand ˜yI
is positive, then the proposed DIFDEA models are consistent
and inclusion of IF variables is justified.
5 Proposed intuitionistic fuzzy ranking
approach
Ranking has an important role in DEA. Some definitions are
as follows:
Definition 18 In DEA, the ranking index (Chen and Klein
1997)forthe jth DMU is:
Rj=n
i=0(Ej)U
αi−c
n
i=0(Ej)U
αi−c−n
i=0(Ej)L
αi−d;n→∞,
where (Ej)L
αiand (Ej)U
αiare the lower bound and upper
bound efficiencies of the jth DMU, respectively, for αi∈
(0,1];c=min
αi
(Ej)L
αiand d=max
αi
(Ej)U
αi.
Definition 19 Let A=[a,b]and B=[c,d]be two inter-
vals. Then, the difference A−Bof Aand Bis defined as
A−B=[a−d,b−c].
5.1 Methodology
The proposed index ranking (PIR) method evaluates the effi-
ciencies of DMUs.
To the best of our knowledge, in the DEA literature, there
is no ranking approach for IFNs.
The proposed method uses α- and β-cuts to rank the
DMUs.
(i) Index based on α-cut
Suppose aαi=min
jξL
j,αiand bαi=max
jξU
j,αifor α∈(0,1].
Obviously, ξU
j,αi−aαiis positive for all j for any α∈(0,1]
and (ξ L
j,αi−bαi)is negative for all j for any αi∈(0,1]. Thus,
the index value is defined by
Ij=n
i=0(ξU
j,αi−aαi)
n
i=0(ξU
j,αi−aαi)−n
i=0(ξ L
j,αi−bαi).(5.1)
123

Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic…
(ii) Index based on β-cut
Suppose cβi=min
jξL
j,βiand dβi=max
jξU
j,βi.
Obviously, ξU
j,βi−dβiis negative and ξL
j,βi−cβiis positive
for all j for any βi∈[0,1). Thus, the index value is defined
by
I
j=n
i=0(ξ U
j,βi−dβi)
n
i=0(ξ U
j,βi−dβi)−n
i=0(ξ L
j,βi−cβi).(5.2)
(iii) Now, we construct the composite index of IF efficiencies
based on Ijand I
jfor the jth DMU is given by
ICj=ηIj+(1−η) I
j(5.3)
where η∈(0,1)is a parameter depending on the decision-
maker’s intention. Generally, ηis taken as 0.5.
5.1.1 Algorithm for the PIR approach
Based on the above analysis, we suggest the following algo-
rithm for ranking:
Step 1 Determine both ξL
jo,α and ξU
jo,α given by Model 6
and Model 7, respectively, for each DMU jo,jo=
1,2,3,...,n.
Step 2 Determine both ξL
jo,β and ξU
jo,β given by Model 11
and Model 12, respectively, for each DMU jo,jo=
1,2,3,...,n.
Step 3 Determine both Ijoand I
jogiven by (5.1) and (5.2),
respectively for each DMU jo,jo=1,2,3,...,n.
Step 4 Determine ICjogiven by (5.3) for each DMU jo,
jo=1,2,3,...,n.
Step 5 Rank the DMUs according to the decreasing values
of ICjo.
This PIR method is suitable for the IFN efficiencies because
it is based on α- and β-cut and it can handle the large quantity
of IFNs.
The flowchart showing the overview of the proposed methods
is given in Figure 3.
5.2 Merits of the proposed methods over existing
methods
The proposed method determines the efficiencies of DMUs
in interval form based on α-cut and β-cut. The proposed
IFCC is used to validate the proposed models based on IF
variables. But, the fuzzy DEA (FDEA) models determine
the efficiencies of DMUs based on α-cut only. The proposed
method uses α-cut and β-cut to rank the DMUs. The proposed
ranking method gives the aggregate rank for all α-cuts and
β-cuts. But, the existing ranking methods in FDEA (Arya
and Yadav (2017,2018)) of DMUs based on α-cut only and
existing ranking methods in IFDEA (Puri and Yadav (2015),
Daneshvar Rouyendegh (2011)) do not use α-cut and β-cut.
6 Numerical examples
In this section, to ensure the validity of the proposed models,
we consider an illustrative example and a health sector appli-
cation. The efficiencies obtained by the proposed models will
be termed as proposed efficiencies (PEs).
6.1 An illustrative example:
Let there be 5 DMUs having two IF inputs and two IF outputs
which are represented as TIFNs. The IF input and IF output
data are listed in Table 2.
6.1.1 Determining efficiencies of DMUs
The IFCC between IF variables (IF inputs–outputs) are deter-
mined using (4.13) and are shown in Table 3. Table 3shows
the lower and upper bounds of each expected interval based
on α- and β-cuts. Also, the corresponding expected values
come out to be positive. Therefore, the inclusion of the IF
input and IF output data are justified, and the DIFDEA mod-
els are consistent.
The ξL
j,α,ξU
j,α,ξ
L
j,β and ξU
j,β for each DMU jare calcu-
lated using Models 6, 7, 11 and 12 for different αand β∈
[0,1], respectively. The results are shown in Table 4.Byusing
the software Lingo, the values of the ξL
j,α ,ξU
j,α,ξ
L
j,β and ξU
j,β
for α, β =0(0.25)1.0 are calculated (see Table 4) for each
DMU.
The composite index values for 5 DMUs are IC(DMU1)
=0.3569;IC(DMU2)=0.38;IC(DMU3)=0.3587;
IC(DMU4)=0.455;IC(DMU5)=0.3435. The DMUs
are ranked by using PIR discussed in Sect. 5as DMU4 >
DMU2 >DMU3 >DMU1 >DMU5.
6.1.2 Comparison of proposed efficiencies and crisp
efficiencies
To validate the proposed efficiencies through PDIFDEA, the
proposed efficiencies are compared with the efficiencies of
crisp DEA (crisp efficiencies) and are given in Table 5.In
Table 5, the efficiencies of DMUs are found to be smaller by
PDIFDEA compared to crisp DEA. In Table 5,DMUs2,4
and 5 are efficient in crisp DEA, but these are inefficient with
efficiency scores 0.38, 0.455 and 0.3435 using PDIFDEA,
respectively. Therefore, PDIFDEA is more realistic rather
than crisp DEA. Crisp DEA and PDIFDEA may give the
same efficiencies for certain data. Crisp DEA does not deal
with the uncertainty/vagueness, but PDIFDEA deals with the
123

A. Arya, S. P. Yadav
Sources:
Literature review
Experts’ opinion
Selection of the relevant input
and output data variables for
performance evaluation of the
DMUs Selection Approaches:
Data Collection
Identify crisp/ fuzzy/intuitionistic fuzzy Data
Representation of crisp data in
intuitionistic fuzzy form
Final input and output data
set in the form of TIFNs
Fuzzification of data using
Experts’ opinions
Performance evaluation using dual intuitionistic fuzzy DEA (DIFDEA) approach
Selection of DIFDEA models
(-cut, -cut method)
Lower and upper bound DIFDEA
models based on -cut
Lower and upper bound DIFDEA
models based on -cut
Intuitionistic fuzzy efficiencies based on - cut
Intuitionistic fuzzy efficiencies based on -cut
Rank the DMUs
ation
d
Proposed ranking approach on the basis of IF efficiencies
Proposed intuitionistic fuzzy correlation coefficients (IFCCs)
between IF variables
Proposed IF input target and
IF output target
Fig. 3 Overview of the proposed method
uncertainty/vagueness. Therefore, PDIFDEA is more effi-
cient rather than crisp DEA.
6.1.3 Determining IF input–output targets of DMUs
Finally, we obtain the IF input targets and IF output targets
discussed in Sect. 3.4 whichareshowninTable6.
From input targets, we conclude that
(i) for DMU 1, the IF inputs have to be decreased
from (3.5,4,4.5;3.2,4.0,4.7)and (1.9,2.1,2.3;1.7,
2.1,2.5)to (2.18,3.4,4.5;1.57,3.4,4.7)and (1.18,
1.8,2.3;0.93,1.8,2.5), respectively, to become
efficient,
(ii) for DMU 2, the IF inputs have to be decreased from
(2.9,2.9,2.9;2.9,2.9,2.9)and (1.4,1.5,1.6;1.3,
123

Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic…
Table 2 IF inputs and IF outputs for 5 DMUs
DMUs IF inputs IF outputs
˜x1I˜x2I˜y1I˜y2I
1 (3.5, 4.0, 4.5; 3.2, 4.0, 4.7 ) (1.9, 2.1, 2.3; 1.7, 2.1, 2.5) (2.4, 2.6, 2.8; 2.2, 2.6, 3) (3.8, 4.1, 4.4; 3.6, 4.1, 4.6)
2 (2.9, 2.9, 2.9; 2.9, 2.9, 2.9) (1.4, 1.5, 1.6; 1.3, 1.5, 1.8) (2.2, 2.2, 2.2; 2.2, 2.2, 2.2) (3.3, 3.5, 3.7; 3.1, 3.5, 3.9)
3 (4.4, 4.9, 5.4; 4.2, 4.9, 5.6) (2.2, 2.6, 3.0; 2.1, 2.6, 3.2) (2.7, 3.2, 3.7; 2.5, 3.2, 3.9) (4.3, 5.1, 5.9; 4.1, 5.1, 6.2)
4 (3.4, 4.1, 4.8; 3.1, 4.1, 4.9) (2.2, 2.3, 2.4; 2.1, 2.3, 2.6) (2.5, 2.9, 3.3; 2.4, 2.9, 3.6) (5.5, 5.7, 5.9; 5.3, 5.7, 6.1)
5 (5.9, 6.5, 7.1; 5.6, 6.5, 7.2) (3.6, 4.1, 4.6; 3.5, 4.1, 4.7) (4.4, 5.1, 5.8; 4.2, 5.1, 6.6) (6.5, 7.4, 8.3; 5.6, 7.4, 9.2)
Table 3 IFCCs between IF inputs–outputs
CL
0.1CR
0.1CL
0.1CR
0.1CEV
0.1,0.1
˜xI
1˜xI
2˜yI
1˜yI
2˜xI
1˜xI
2˜yI
1˜yI
2˜xI
1˜xI
2˜yI
1˜yI
2˜xI
1˜xI
2˜yI
1˜yI
2˜xI
1˜xI
2˜yI
1˜yI
2
˜xI
11 0.95 0.96 0.77 1 0.97 0.96 0.96 1 0.93 0.94 0.63 1 0.97 0.95 0.94 1 0.96 0.95 0.83
˜xI
20.95 1 0.97 0.92 0.97 1 0.90 0.96 0.93 1 0.96 0.85 0.97 1 0.99 0.97 0.96 1 0.95 0.92
˜yI
10.96 0.97 1 0.83 0.96 0.90 1 0.92 0.94 0.96 1 0.72 0.95 0.99 1 0.99 0.95 0.96 1 0.87
˜yI
20.77 0.92 .83 1 0.96 0.96 0.93 1 0.63 0.85 0.72 1 0.94 0.97 0.99 1 0.83 0.92 0.87 1
Table 4 The IF efficiencies for α, β =0(0.25)1.0
α, β DMU 1 DMU 2 DMU 3 DMU 4 DMU 5
[ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ][ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ][ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ][ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ][ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ]
0 [0.62,1] [0.85,0.85] [0.83,1] [1,1] [0.57,1] [0.86,0.86] [0.85,1] [1,1] [0.64,1] [1,1]
0.25 [0.7,1] [0.77,0.97] [0.91,1] [1,1] [0.64,1] [0.75,0.98] [0.94,1] [0.91,1] [0.73,1] [0.91,1]
0.5 [0.76,0.96] [0.68,1] [0.99,1] [0.87,1] [0.72,1] [0.64,1] [1,1] [0.84,1] [0.84,1] [0.75,1]
0.75 [0.81,0.9] [0.58,1] [1,1] [0.75,1] [0.79,0.93] [0.53,1] [1,1] [0.76,1] [0.97,1] [0.62,1]
1.0 [0.85,0.85] [0.48,1] [1,1] [0.65,1] [0.86,.86] [0.45,1] [1,1] [0.68,1] [1,1] [0.52,1]
1.5,1.8)to (2.4,2.9,2.9;1.89,2.9,2.9)and (1.17,1.5,
1.6;0.84,1.5,1.8), respectively, to become efficient,
(iii) for DMU 3, the IF inputs have to be decreased
from (4.4,4.9,5.4;4.2,4.9,5.6)and (2.2,2.6,3;2.1,
2.6,3.2)to (2.5,4.21,5.4;1.84,4.21,5.6)and (1.25,
2.24,3.0,0.94,2.24,3.2), respectively, to become effi-
cient,
(iv) for DMU 4, the IF inputs have to be decreased from
(3.4,4.1,4.8;3.1,4.1,4.9)and (2.2,2.3,2.4;2.1,
2.3,2.6)to (2.89,4.1,4.8;2.11,4.1,4.9)and (1.87,
2.3,2.4;1.43,2.3,2.6), respectively, to become effi-
cient,
(v) for DMU 5, the IF inputs have to be decreased from
(5.9,6.5,7.1;5.6,6.5,7.2)and (3.6,4.1,4.6;3.5,
4.1,4.7)to (3.77,6.5,7.1;3.38,6.5,7.2)and (2.3,4.1,
4.6;1.82,4.1,4.7), respectively, to become efficient.
6.2 Health sector application
This is a real life application in health sector. The most impor-
tant role in the economy of any country is health care of
Table 5 Comparison of proposed efficiencies and crisp efficiencies
DMUs Proposed efficiency Crisp efficiency Difference
10.3569 0.8548 0.4979
20.38 1 0.62
30.3587 0.8607 0.502
40.455 1 0.545
50.3435 1 0.6565
rural and urban areas. Health care is of three types: primary
(in which individuals and families are directly connected to
health system), secondary (in which patients from primary
health care are referred to specialists in higher hospitals for
treatment) and tertiary health care (in which specialized con-
sultative care is provided usually on referral from primary
and secondary medical care). The performance of hospitals
has become a major concern of planners and policy-makers in
India. The Uttar Pradesh (U.P) state is one of the largest states
of India. It has 18 divisions. Meerut is one of them which has
123

A. Arya, S. P. Yadav
Table 6 IF input target and IF output target for 5 DMUs
DMUs IF input targets IF output targets
¯
˜xI
1¯
˜xI
2¯
˜yI
1¯
˜yI
2
1 (2.18, 3.4, 4.5; 1.57, 3.4, 4.7 ) (1.18, 1.8, 2.3; 0.93, 1.8, 2.5) (2.4, 2.6, 2.8; 2.2, 2.6, 3) (3.8, 4.1, 4.4; 3.6, 4.1, 4.6)
2 (2.4, 2.9, 2.9; 1.89, 2.9, 2.9) (1.17, 1.5, 1.6; 0.84, 1.5, 1.8) (2.2, 2.2, 2.2; 2.2, 2.2, 2.2) (3.3, 3.5, 3.7; 3.1, 3.5, 3.9)
3 (2.5, 4.21, 5.4; 1.84, 4.21, 5.6) (1.25, 2.24, 3.0; 0.94, 2.24, 3.2) (2.7, 3.2, 3.7; 2.5, 3.2, 3.9) (4.3, 5.1, 5.9; 4.1, 5.1, 6.2)
4 (2.89, 4.1, 4.8; 2.11, 4.1, 4.9) (1.87, 2.3, 2.4; 1.43, 2.3, 2.6) (2.5, 2.9, 3.3; 2.4, 2.9, 3.6) (5.5, 5.7, 5.9; 5.3, 5.7, 6.1)
5 (3.77, 6.5, 7.1; 3.38, 6.5, 7.2) (2.3, 4.1, 4.6; 1.82, 4.1, 4.7) (4.4, 5.1, 5.8; 4.2, 5.1, 6.6) (6.5, 7.4, 8.3; 5.6, 7.4, 9.2)
Table 7 Hospital notations S.No. Hospital Code Hospital names
1 BC Baghpat under CMO
2 BD Baghpat District Hospital
3 BLC Bulandshahr under CMO
4 BLD Bulandshahr District Hospital
5 BLFD Bulandshahr Female District Hospital
6 BLK Bulandshahr Ch- Khurja
7 GC Ghaziabad under CMO
8 GD Ghaziabad District Hospital
9 GFD Ghaziabad Female District Hospital
10 GS Ch- Sanjay Nagar
11 GBC Gautam Budh Nagar under CMO
12 GBD Gautam Budh Nagar District Hospital
13 MC Meerut under CMO
14 MD Meerut District Hospital
15 MFD Meerut Female District Hospital
16 HCM Hapur under CMO
Table 8 IF input and IF output data for 16 hospitals. Source: Administrative Office, Meerut district, Uttar Pradesh, India.
DMUs IF inputs IF outputs
˜x1I˜x2I˜y1I˜y2I
BC (140,144,150;135,144,155 ) (2,2,8;1,2,15) (83085,83089,83092;83082,83089,83096) (6,7,9;5,7,12)
BD (18,20,25;15,20,28) (2,3,5;1,3,10) (14105,14107,14110;14103,14107,14115) (897,900,904;895,900,908)
BLC (442,446,449;440,446,451) (2,3,6;1,3,10) (115855,115858,115862;115852,115858,115865) (1910,1913,1915;1908,1913,1919)
BLD (145,149,152;140,149,155) (1,1,4;1,1,8) (165120,165121,165125;165118,165121,165129) (1560,1563,1567;1558,1563,1570)
BLFD (58,60,63;55,60,65) (1,1,6;1,1,10) (34095,34099,340105;34092,34099,340110) (1620,1622,1626;1617,1622,1630)
BLK (65,68,70;62,68,75) (1,1,6;1,1,10) (12185,12189,12195;12180,12189,12199) (1280,1283,1287;1277,1283,1290)
GC (122,124,126;120,124,130) (6,7,10;4,7,15) (65290,65293,65298;65286,65293,652102) (215,218,222;210,218,227)
GD (162,166,168;160,166,172) (3,5,8;2,5,12) (258750,258754,258758;258745,258754,258765) (423,426,428;420,426,432)
GFD (65,68,75;62,68,80) (1,1,5;1,1,10) (77852,77856,77859;77850,77856,77865) (1160,1164,1168;1155,1164,1175)
GS (98,100,105;95,100,115) (2,2,6;1,2,12) (79720,79725,79729;79718,79725,79735) (910,913,916;905,913,920)
GBC (130,132,135;125,132,145) (2,2,7;1,2,13) (25385,25387,25392;25380,25387,25397) (1756,1761,1765;1750,1761,1770)
GBD (97,100,105;95,100,110) (2,3,8;1,3,14) (297445,297449,297453;297440,297449,297458) (4000,4004,4008;3995,4004,4012)
MC (248,250,255;245,250,260) (6,8,12;5,8,15) (61190,61192,61196;61185,61192,611102) (635,638,642;630,638,648)
MD (248,250,255;245,250,258) (6,8,13;4,8,15) (129432,129435,129438;129430,129435,129445) (1050,1052,1057;1046,1052,1062)
MFD (90,93,95;85,93,100) (2,2,8;1,2,13) (78275,78278,78282;78272,78278,78288) (1600,1606,1610;1595,1606,1610)
HCM (100,104,110;95,104,120) (1,1,5;1,1,10) (44902,44906,44909;44900,44906,44916) (2110,2113,2116;2106,2113,2120)
123

Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic…
Table 9 IFCCs between IF variables
CL
0.1CR
0.1CL
0.1CR
0.1CEV
0.1,0.1
˜x1I˜x2I˜y1I˜y2I˜x1I˜x2I˜y1I˜y2I˜x1I˜x2I˜y1I˜y2I˜x1I˜x2I˜y1I˜y2I˜x1I˜x2I˜y1I˜y2I
˜xI
11 0.39 0.23 0.002 1 0.36 0.023 0.001 1 0.41 0.23 0.001 1 0.33 0.23 0.001 1 0.37 0.18 0.0012
˜xI
20.39 1 0.08 0.003 0.36 1 0.026 0.002 0.41 1 0.19 0.003 0.33 1 0.57 0.003 0.37 1 0.22 0.0002
˜yI
10.23 0.08 1 0.38 0.023 0.026 1 0.37 0.23 0.19 1 0.38 0.23 0.57 1 0.001 0.18 0.22 1 0.28
˜yI
20.002 0.003 0.38 1 0.001 0.002 0.37 1 0.001 0.003 0.38 1 0.001 0.003 0.001 1 0.0012 0.0002 0.28 1
Table 10 The IF efficiencies
based on α, β =0(0.25)0.5of
16 hospitals
α, β →0 0.25 0.5
DMUs [ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ][ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ][ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ]
BC [0.094,0.99] [0.34,0.34] [0.12,0.82] [0.19,0.79] [0.18,0.63] [0.17,0.99]
BD [0.87,1] [1,1] [0.92,1] [1,0.99] [0.98,1] [0.91,0.99]
BLC [0.16,1] [0.35,0.35] [0.2,1] [0.23,0.99] [0.24,1] [0.14,0.99]
BLD [0.27,1] [1,1] [0.36,1] [0.5,1] [0.55,1] [0.33,1]
BLFD [0.53,1] [0.99,0.99] [0.56,1] [0.62,1] [0.59,1] [0.53,1]
BLK [0.38,1] [0.75,0.75] [0.41,1] [0.44,1] [0.44,1] [0.38,1]
GC [0.08,0.28] [0.18,0.18] [0.11,0.23] [0.17,0.62] [0.16,0.18] [0.16,1]
GD [0.26,1] [0.52,0.52] [0.34,1] [0.51,1] [0.49,1] [0.47,1]
GFD [0.37,1] [0.76,0.76] [0.38,1] [0.39,1] [0.392,1] [0.37,1]
GS [0.22,1] [0.37,0.37] [0.23,0.84] [0.25,0.8] [0.25,0.67] [0.24,1]
GBC [0.29,1] [0.52,0.52] [0.31,1] [0.31,1] [0.32,1] [0.28,1]
GBD [0.94,1] [1,1] [1,1] [1,1] [1,1] [1,1]
MC [0.064,0.25] [0.082,0.082] [0.068,0.19] [0.08,0.48] [0.077,0.15] [0.075,1]
MD [0.12,0.52] [0.17,0.17] [0.13,0.41] [0.17,0.32] [0.16,0.31] [0.16,0.58]
MFD [0.37,1] [0.54,0.54] [0.38,1] [0.4,1] [0.4,1] [0.37,1]
HCM [0.44,1] [1,1] [0.46,1] [0.48,1] [0.48,1] [0.42,1]
Table 11 The IF efficiencies
based on α, β =0.75(0.25)1.0
of 16 hospitals
α, β →0.75 1.0 Composite index Rank
DMUs [ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ][ξL
jo,α,ξU
jo,α][ξL
jo,β ,ξU
jo,β ]ICj0
BC [0.21,0.47] [0.12,0.99] [0.34,0.34] [0.086,0.99] 0.5253 6
BD [1,1] [0.83,0.99] [1,1] [0.76,0.99] 0.3963 12
BLC [0.28,0.71] [0.1,0.99] [0.35,0.35] [0.09,1] 0.5301 5
BLD [0.87,1] [0.28,1] [1,1] [0.25,1] 0.3259 14
BLFD [0.62,0.68] [0.46,1] [0.65,0.65] [0.43,1] 0.2308 16
BLK [0.46,1] [0.31,1] [0.75,0.75] [0.28,1] 0.3872 13
GC [0.17,0.18] [0.11,1] [0.18,0.18] [0.081,1] 0.6765 1
GD [0.52,0.77] [0.32,1] [0.52,0.52] [0.24,1] 0.4342 9
GFD [0.41,1] [0.35,1] [0.76,0.76] [0.3,1] 0.4017 11
GS [0.26,0.49] [0.21,1] [0.37,0.37] [0.18,1] 0.5428 3
GBC [0.36,0.83] [0.24,1] [0.52,0.52] [0.2,1] 0.4707 8
GBD [1,1] [0.91,1] [1,1] [0.79,1] 0.4167 10
MC [0.081,0.11] [0.065,1] [0.082,0.082] [0.057,1] 0.6517 2
MD [0.17,0.23] [0.12,0.91] [0.17,0.17] [0.11,1] 0.5155 7
MFD [0.42,0.81] [0.35,1] [0.54,0.54] [0.34,1] 0.5345 4
HCM [0.72,1] [0.35,1] [1,1] [0.29,1] 0.2979 15
123

A. Arya, S. P. Yadav
Table 12 Comparison of
proposed efficiencies and crisp
efficiencies
DMUs Proposed efficiency Crisp efficiency Difference
BC 0.5253 0.3427 −0.1826
BD 0.3963 1 0.6037
BLC 0.5301 0.3512 −0.1789
BLD 0.3259 1 0.6741
BLFD 0.2308 0.6505 0.4197
BLK 0.3872 0.7474 0.3602
GC 0.6765 0.177 −0.4995
GD 0.4342 0.524 0.0898
GFD 0.4017 1 0.5983
GS 0.5428 0.3672 −0.1756
GBC 0.4707 0.519 0.0483
GBD 0.4167 1 0.5833
MC 0.6517 0.0825 −0.5692
MD 0.5155 0.174 −0.3415
MFD 0.5345 0.5426 0.0081
HCM 0.2979 1 0.7021
6 districts named as Baghpat, Bulandshahr, Ghaziabad, GB
Nagar, Meerut and Hapur. Each district has some public hos-
pitals. Total number of public hospitals in Meerut division is
16. In this paper, we discuss the performance efficiency of
public hospitals which are in Meerut division. Table 7gives
the public hospitals in Meerut division.
6.2.1 Variables and data selection
In this study, we have taken two inputs: (i) total number of
beds (say ˜x1I) and (ii) sum of number of pathologists and
number of laboratory technicians (say ˜x2I) and two outputs:
(i) number of pathology operations (say ˜y1I) and (ii) sum
of number of plaster and number of tubal ligation (say ˜y2I)
of 16 hospitals which possess some degree of hesitation due
to the difference in thought at the management level and the
actual hospital level. So, uncertainty in input data and output
data at hospital level can be well taken as TIFN. The IF input
and IF output data are provided by the administrative office,
Meerut district, Uttar Pradesh, India, for the calender year
2013–2014, and it is shown in Table 8.
6.2.2 Determining efficiencies of hospitals
The IFCCs between IF variables (IF inputs–outputs) are
determined using (4.13) and are shown in Table 9. Table 9
shows the lower and upper bounds of each expected interval
based on α- and β-cuts. Also, the corresponding expected
values come out to be positive. Therefore, the inclusion of
the IF input and IF output data is justified and the DIFDEA
models are consistent.
The IF efficiencies of all hospitals are evaluated using
Models 6, 7, 11 and 12 for different αand β−values, which
are shown in Tables 10 and 11. The composite index ICj
of IF efficiencies ξL
j,α,ξU
j,α,ξ
L
j,β and ξU
j,β for each DMU j
is calculated and shown in Tables 10 and 11. The ranks of
the hospitals using the PIR approach are obtained as GC >
MC >GS >MFD >BLC >BC >MD >GBC >GD >
GBD >GFD >BD >BLK >BLD >HCM >BLFD.
Thus, GC is the best performer hospital and BLFD is the
worst performer hospital.
6.2.3 Comparison of proposed efficiencies and crisp
efficiencies
To validate the proposed efficiencies of hospitals through
PDIFDEA, the proposed efficiencies are compared with the
efficiencies of crisp DEA (crisp efficiencies) and are given
in Table 12. Table 12 shows that the efficiencies of hospi-
tals are smaller by PDIFDEA compared to crisp DEA. In
Table 12, hospitals BD, BLD, GFD, GBD and HCM are effi-
cient in crisp DEA, but these hospitals are inefficient with
efficiency scores 0.3963, 0.3259, 0.4017, 0.4167 and 0.2979
using PDIFDEA, respectively. Therefore, PDIFDEA is more
realistic rather than crisp DEA. Crisp DEA and PDIFDEA
may give same efficiencies for certain data. Crisp DEA
does not deal with the uncertainty/vagueness, but PDIFDEA
deals with the uncertainty/vagueness. Therefore, PDIFDEA
is more efficient rather than crisp DEA. Hence, we preferred
PDIFDEA rather than crisp DEA.
123

Development of intuitionistic fuzzy data envelopment analysis models and intuitionistic…
Table 13 IF input target and IF output target data for 16 hospitals
DMUs IF input target IF output target
¯
˜xI
1¯
˜xI
2¯
˜yI
1¯
˜yI
2
BC (13.16,48.96,148.5;11.61,48.96,153.45) (0.18,0.68,7.92;0.07,0.68,14.85) (83085,83089,83092;83082,83089,83096) (6,7,9;5,7,12)
BD (15.7,20,25;15.2,20,27.72) (1.74,2,5;0.76,2,9.9) (14105,14107,14110;14103,14107,14115) (897,900,904;895,900,908)
BLC (70.72,156.1,449;40.92,156.1,451) (0.32,1.05,6;0.09,1.05,9.9) (115855,115858,115862;115852,115858,115865) (1910,1913,1915;1908,1913,1919)
BLD (39.15,149,152;35,149,155) (0.27,1,4;0.25,1,8) (165120,165121,165125;165118,165121,165129) (1560,1563,1567;1558,1563,1570)
BLFD (30.74,39,63;23.65,39,65) (0.53,0.65,6;0.43,0.65,10) (34095,34099,340105;34092,34099,340110) (1620,1622,1626;1617,1622,1630)
BLK (24.7,51,65;17.36,51,75) (0.38,0.75,6;0.28,0.75,10) (12185,12189,12195;12180,12189,12199) (1280,1283,1287;1277,1283,1290)
GC (9.76,22.32,35.28;9.72,22.32,130) (0.48,1.26,2.8;0.32,1.26,15) (65290,65293,65298;65286,65293,652102) (215,218,222;210,218,227)
GD (142.12,86.32,168;38.4,86.32,172) (0.78,2.6,8;0.48,2.6,12) (258750,258754,258758;258745,258754,258765) (423,426,428;420,426,432)
GFD (24,35.36,75;18.6,35.36,80) (0.37,0.76,5;0.3,0.76,10) (77852,77856,77859;77850,77856,77865) (1160,1164,1168;1155,1164,1175)
GS (21.56,37,105;17.1,37,115) (0.44,0.74,6;0.18,0.74,12) (79720,79725,79729;79718,79725,79735) (910,913,916;905,913,920)
GBC (37.7,68.64,135;24.87,68.64,145) (2,2,7;1,2,13) (25385,25387,25392;25380,25387,25397) (1756,1761,1765;1750,1761,1770)
GBD (98.7,100,105;75.05,100,110) (1.88,3,8;0.79,3,14) (297445,297449,297453;297440,297449,297458) (4000,4004,4008;3995,4004,4012)
MC (15.87,20.5,63.75;13.96,20.5,260) (0.36,0.65,3;0.28,0.65,15) (61190,61192,61196;61185,61192,611102) (635,638,642;630,638,648)
MD (29.76,42.5,132.6;26.95,42.5,258) (0.72,1.36,6.76;0.44,1.36,15) (129432,129435,129438;129430,129435,129445) (1050,1052,1057;1046,1052,1062)
MFD (33.3,50.22,95;28.9,50.22,100) (0.74,1.18,8;0.34,1.18,13) (78275,78278,78282;78272,78278,78288) (1600,1606,1610;1595,1606,1610)
HCM (44,100,104;28.31,100,120) (0.44,1,5;0.29,1,10) (44902,44906,44909;44900,44906,44916) (2110,2113,2116;2106,2113,2120)
123

A. Arya, S. P. Yadav
6.2.4 Determining IF input–output targets of hospitals
Finally, we obtain the IF input target and IF output target data
whichareshowninTable13. Using these IF input target and
IF output target, we make the inefficient hospitals as effi-
cient hospitals. For the inefficient hospital BC, the IF inputs
have to be decreased from (140,144,150;135,144,155)
and (2,2,8;1,2,15)to (13.16,48.96,148.5;11.61,48.96,
153.45)and (0.18,0.68,7.92;0.07,0.68,14.85), respec-
tively, to become efficient. Similarly, we find the IF inputs
decreased values for other inefficient hospitals and are shown
in Table 13.
Conclusion
In this paper, we have determined the performance efficien-
cies of DMUs. The real world applications data have some
degree of uncertainties. To deal with such data, we have con-
sidered them as TIFNs. We have developed IFDEA models
based on α−,β-cuts. Four DIFDEA models (Models 6, 7,
11 and 12) have been developed to determine the perfor-
mance efficiencies of the DMUs. Next, we have developed
IF index ranking approach for DIFDEA models. This rank-
ing approach is efficient and effective for the large number of
IF input–output data. We have also proposed the targets for
the DMUs with IF inputs–outputs. Finally, an example and
a health sector application are presented to illustrate the pro-
posed models. To ensure the validity of the proposed models,
we have considered the performance of 16 hospitals in the
Meerut zone of India with two IF inputs: total number of beds,
sum of number of pathologists and number of laboratory
technicians, and two IF outputs: number of pathology opera-
tions, sum of number of plaster and number of tubal ligation.
GC is determined as the best performer hospital with high
level of efficiencies, and BLFD is the worst performer hospi-
tal with low level of efficiencies. We also determined the IF
input–output targets data for inefficient DMUs by which it is
found that how an inefficient hospital is made efficient hospi-
tal. PDIFDEA has realistic point of view better representing
inefficient performance efficiencies, but crisp DEA has an
optimistic point of view to the same problem. By extending
to IF environment, the DEA method is more effective for
real world applications in the sense that it covers hesitation
also.
Limitations and Future Research Plan
This paper has some limitations. The proposed models are
studied under the constant returns to scale (CRS). We plan to
extend these models to the variable returns to scale (VRS).
The uncertainty in this paper is limited to TIFNs. We plan
to use the trapezoidal IFNs and interval valued intuitionistic
fuzzy sets to determine the efficiencies of real world appli-
cations.
Acknowledgements The authors are thankful to the Ministry of Human
Resource Development (MHRD), the Govt. of India, India, with grant
number MHR-02-23-200-44, for financial support in pursuing this
research. The authors are also thankful to Mr. Tajender, ARO, Admin-
istrative Office, Meerut, India, for providing the valuable data of the
hospitals.
Compliance with ethical standards
Conflict of interest Alka Arya has received research grants from Min-
istry of Human Resource Development (MHRD), Govt. of India, India.
Shiv Prasad Yadav declares that he has no conflict of interest.
Ethical approval This article does not contain any studies with human
participants performed by any of the authors.
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