ma hema ics
A icle
Fuzzy Ranking Ne wo k DEA wi h Gene al S uc u e
Plácido Mo eno * and Sebas ián Lozano
School o Enginee ing, Uni e sidad de Se illa, E41092 Se ille, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed]
Recei ed: 18 July 2020; Accep ed: 8 Decembe 2020; Published: 14 Decembe 2020


Abs ac :
This pape ex ends wo uzzy anking da a en elopmen analysis (DEA) app oaches o
he case o gene al ne wo ks o p ocesses. The i s app oach p o ides an e iciency sco e o each
possibili y le el which equi es sol ing one linea p og am o each possibili y le el. The second
app oach is e en simple and p o ides an o e all e iciency sco e sol ing jus one linea p og am.
The p oposed app oaches a e es ed on wo da ase s om he li e a u e and compa ed wi h o he uzzy
ne wo k DEA app oaches. The esul s show ha he wo me hods p o ide e y highly co ela ed
e iciency es ima es which a e also consis en wi h hose o o he uzzy ne wo k DEA app oaches.
Keywo ds: e iciency assessmen ; ne wo k DEA; uzzy da a; uzzy anking; de uzzi ica ion
1. In oduc ion
Da a en elopmen analysis (DEA) is a well-known non-pa ame ic echnique, gene ally used o
assess he ela i e e iciency o a g oup decision making uni s (DMUs). Con en ional DEA conside s
ha each DMU consis s o jus a single p ocess. This p ocess consumes ce ain amoun s o inpu s and
p oduces ce ain amoun o ou pu s. The aim o DEA is o de ec ine iciencies, i.e., o check i he
same ou pu s can be ob ained wi h less inpu s he so-called inpu o ien a ion) o he same inpu s can
p oduce mo e ou pu s (ou pu o ien a ion). De ec ing all kinds o ine iciencies is impo an o bo h
compe i i eness and sus ainabili y.
The e a e di e en DEA models, depending on he o ien a ion, he me ic, and he e u ns o scale
(RTS) assump ions conside ed. Conside ing ce ain axioms (like con exi y o scalabili y, o example)
and applying he minimum ex apola ion p inciple, DEA can in e om he obse ed da a a so-called
p oduc ion possibili y se (PPS), which co esponds o he se o ope a ion poin s ha a e conside ed
easible. The PPS is o med using all linea combina ions o he obse ed DMUs. When a iable
e u ns o scale (VRS) a e assumed hen only con ex linea combina ions a e used. The non-domina ed
subse o he PPS de ines he e icien on ie . The DMUs ha all on he E icien F on ie (EF) a e
labelled e icien while hose ha do no a e e med ine icien and a e p ojec ed on o he EF. The
dis ance o a DMU o he EF is used o compu e an e iciency sco e so ha he u he om he EF
a DMU is (i.e., he la ge he inpu and ou pu imp o emen s ha can be achie ed) he lowe i s
e iciency sco e. Acco dingly, e icien DMUs ha e an e iciency sco e o one since o hem no inpu o
ou pu imp o emen s a e possible.
Di e en o con en ional DEA, ne wo k DEA looks a he in e nal s uc u e o he DMUs,
conside ing he DMU as a ne wo k o in e ela ed p ocesses which consume inpu s and p oduce
ou pu s bu also p oduce and consume in e media e p oduc s. As wi h con en ional DEA, many
di e en ne wo k DEA app oaches ha e also been p oposed (e.g., Kao [
1
]; Tone and Tsu sui [
2
,
3
]; Cook
e al. [
4
]; Lozano [
5
]). The ange o applica ions o ne wo k DEA has also g own acco dingly, spanning
banking (Lozano [
6
]), anspo a ion (e.g., Lozano e al. [
7
]), spo s (Mo eno and Lozano [
8
]), supply
chain (e.g., Lozano and Adenso-Díaz [9]), e c.
Ma hema ics 2020,8, 2222; doi:10.3390/ma h8122222 www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2020,8, 2222 2 o 18
All he abo e e e ences ha e deal wi h c isp da a, i.e., hey assume ha he e is no unce ain y
in he da a. This does no mean ha DEA canno wo k when he da a a e uzzy. On he con a y,
he e a e many di e en uzzy DEA app oaches (e.g., Kao and Liu [
10
], A ana-Jimenez e al. [
11
],
Saa i e al. [
12
], Le
ó
n e al. [
13
], Le wo asi ikul e al. [
14
], Wang e al. [
15
], Soleimani-damaneh
e al. [
16
], A ana-Jimenez e al. [
17
], e c.). The eade is e e ed o Ha ami-Ma bini e al. [
18
] and
Em ouznejad e al. [19] o a axonomy and e iew o uzzy DEA app oaches.
O special ele ance a e hose uzzy DEA app oaches ha conside ne wo ks o p ocesses. Thus,
Kao and Liu [
20
] ex end Kao and Liu [
10
] app oach o wo-s age sys ems wi h uzzy da a. Liu [
21
]
p oposes a me hodology o ank uzzy wo-s age e iciencies, while Liu [
22
] adds weigh es ic ions
o Kao and Liu [
20
] uzzy wo-s age model. Ta ana and Khalili-Damghani [
23
] also wo k wi h Kao
and Liu [
20
] app oach bu implemen ing a leade - ollowe game heo y app oach o decompose he
wo-s age e iciency in o single-s age e iciencies. Ano he a ian is p esen ed in Hemma i e al. [
24
].
Wang e al. [
25
] adop a boo s apped unca ed- eg ession model o s udy he ela ionship be ween
uzzy wo-s age e iciencies and an exogenous a iable.
Conce ning non- adial ne wo k DEA app oaches, She meh e al. [
26
] adap he Ne wo k
Slack-Based Measu e (SBM) model by Tone and Tsu sui [
2
] o be able o deal wi h uzzy numbe s. Ol a
e al. [27] p opose a uzzy ex ension o he dynamic Ne wo k SBM app oach by Tone and Tsu sui [3].
Khalili-Damghani and Tagha i-Fa d [
28
] also s udy uzzy wo-s age DEA sys ems. Kao and
Lin [
29
] conside pa allel p ocesses and uzzy da a. Lozano [
30
,
31
] compu e p ocess e iciencies in
pa allel and wo-s age p ocess DEA sys ems, espec i ely. Kao [
32
] p oposes wo app oaches, namely
he membe ship g ade and he
α
-cu , o ne wo k DEA wi h uzzy da a. Lozano and Mo eno [
33
]
ex end he app oaches in Saa i e al. [
12
], Wang e al. [
15
], and Kao and Liu [
20
] o gene al ne wo ks o
p ocesses. Finally, Mi hedaya ian e al. [
34
] p esen a uzzy ne wo k DEA app oach wi h dual- ole
ac o s and undesi able ou pu s o e alua e g een supply chains.
Howe e , o he bes o ou knowledge, anking me hods needs u he de elopmen o ne wo k
DEA when dealing wi h uzzy da a and gene al ne wo k o p ocesses. In ac , ou wo k allows any
con igu a ion o he s ages ha o m he in e nal s uc u e o he DMU, which implies an imp o emen
o e o he models. I could also be a gued ha p e ious models seem o be complex enough o p e en
some esea che s om wo king wi h uzzy da a. The e o e, ou aim is o de elop a gene al uzzy
ne wo k DEA app oach wi hou adding unnecessa y complexi y.
The s uc u e o he pape is he ollowing. In Sec ion 2 he ne wo k DEA is in oduced, and a
c isp model is o mula ed. In Sec ion 3, he p oposed me hods a e p esen ed. In Sec ion 4 he esul s o
he applica ion o he p oposed me hods o wo da ase s om he li e a u e a e p esen ed. Finally,
Sec ion 5summa izes and concludes.
2. P elimina ies
In o de o acili a e a be e unde s anding o he p oposed uzzy anking ne wo k DEA
app oaches, his sec ion p esen s a b ie o e iew o ne wo k DEA models and uzzy numbe s.
2.1. C isp Ne wo k DEA Model
I is con enien o o mula e i s he c isp ne wo k DEA model, i.e., he ne wo k DEA model
wi hou uzzy da a. We will use he same no a ion ha Lozano [
5
] and Lozano and Mo eno [
33
].
This no a ion helps o o mula e he models in a e y compac o m.
Assume he DMUs o assess a e all s uc u ally homogeneous, i.e., all o hem ha e he same
numbe and ype o p ocesses. Each p ocess may consume a di e en subse o inpu s and may
p oduce a di e en subse o ou pu s. Le
I(p)
be he se o exogenous inpu s consumed by p ocess
p and, o each
i∈I(p)
, le
xp
ij
deno e he obse ed amoun o exogenous inpu i used by p ocess p
o DMU j. Simila ly, le
O(p)
he se o ou pu s p oduced by p ocess p and, o each
k∈O(p)
, le
yp
kj
deno e he amoun o ou pu k p oduced by p ocess p o DMU j. Le
PI(i)
be he se o p ocesses ha
Ma hema ics 2020,8, 2222 3 o 18
consume he inpu i and
xij =P
p∈PI(i)
xp
ij
he o al amoun o inpu i consumed by DMU j. Le
PO(k)
be he se o p ocesses ha p oduce he ou pu k and
ykj =P
p∈PO(k)
yp
kj
he o al amoun o ou pu k
p oduced by DMU j.
A key ea u e o ne wo k DEA is ha , in addi ion o exogenous inpu s and ou pu s, he e gene ally
exis in e media e p oduc s ha a e p oduced by some p ocesses and consumed by o he s. Le
Pou ( )
be he se o p ocesses ha p oduce he in e media e p oduc and, o each
p∈Pou ( )
, le
zp
j
he
amoun o in e media e p oduc p oduced by p ocess p o DMU j. Analogously, le
Pin( )
be he se
o p ocesses ha consume he in e media e p oduc and, o each
p∈Pin( )
, le
zp
j
he amoun o
in e media e p oduc used by p ocess p o DMU j. Finally, le us de ine he se s
Rou (p)
and
Rin(p)
ha co espond o he in e media e p oduc s p oduced and consumed, espec i ely, by p ocess p.
Once he no a ion o equi ed da a has been in oduced le us conside he a iables. Fo he
adial, inpu -o ien ed model he a iables needed a e
θUni o m educ ion ac o o he inpu consump ion o DMU J
λp
jIn ensi y a iable o p ocess p o DMU j
EJ=Min θ(1)
subjec o
X
p∈PI(i)X
j
λp
jxp
ij ≤θX
p∈PI(i)
xp
iJ ∀i(2)
X
p∈PO(k)X
j
λp
jyp
kj ≥X
p∈PO(k)
yp
kJ ∀k(3)
X
p∈Pou ( )X
j
λp
jzp
j −X
p∈Pin( )X
j
λp
jzp
j ≥0∀ (4)
X
j
λp
j=1∀p(5)
λp
j≥0∀j∀pθ ee (6)
This c isp ne wo k DEA model (see Lozano [
5
], Lozano and Mo eno [
33
]) compu es he maximum
adial educ ion o he inpu s consumed by a gi en DMU J. No e ha his model has o be sol ed
as many imes as DMUs a e in he da ase . The idea is o compu e a easible ope a ing poin (i.e.,
wi hin he in e ed PPS) ha main ains he ou pu le el o DMU J bu educes all i s inpu s as much as
possible. The op imal alue o he
θ
a iable co esponds o he e iciency sco e o DMU J. The
λp
j
mul iplie s, no e ha he e is a speci ic se o hem o each p ocess p, de e mine he inpu s, ou pu s
and in e media e p oduc s o each p ocess. Thus, he linea combina ions o he obse ed da a de ine
a a ge easible ope a ing poin ha p oduces a leas he same amoun o ou pu , as equi ed by
Cons ain (3), while consuming a ac ion
θ
o he obse ed inpu s, as indica ed by Cons ain s (2).
Cons ain s (4) impose ha enough in e media e p oduc s a e gene a ed in e nally wi hin he sys em
o sa is y he in e nal demand o hose in e media e p oduc s. The e o e, he cons ain s gua an ee
ha he o al amoun o ou pu s p oduced by DMU J is no educed and ha he in e media e p oduc s
p oduced a e enough o supply hose p ocesses ha consume hem.
No e ha al hough, in he abo e model, i has been assumed ha all p ocesses exhibi a iable
e u ns o scale (VRS), o he RTS assump ions can be conside ed in which case Cons ain s (5) should
Ma hema ics 2020,8, 2222 4 o 18
be modi ied acco dingly (see Lozano [
5
]). Mo eo e , he abo e o mula ion co esponds o he inpu
o ien a ion. The co esponding model o he ou pu o ien a ion is shown in Appendix A.
2.2. Fuzzy Numbe s
Fo he sake o cla i y, an in oduc ion o uzzy numbe s is p o ided in his sec ion. Mo e de ailed
in o ma ion can be ound a Dubois and P ade [
35
]. A uzzy numbe
e
T
is a subse o he eal line
R
wi h membe ship unc ion µe
T:R→[0, 1]sa is ying he ollowing p ope ies:
(i)
The e exis s 0∈Rsuch ha µe
T( 0)=1;
(ii)
e
T
is uzzy con ex. In o he wo ds,
µe
T(γ 1+(1−γ) 2)≥minnµe
T( 1),µe
T( 2)o
, o any
1
,
2∈R
and γ∈[0, 1];
(iii)
e
Tis uppe semicon inuous on R, which means ha µ−1
e
T([∝, 1]) is closed o all ∝ ∈ [0, 1];
(i )
The suppo o is µe
Tbounded, i.e., he closu e o nx∈Rµe
T( )>0ois bounded.
In his pape , we deal wi h LR uzzy numbe s (LRFN) as in Le
ó
n e al. [
13
]. A uzzy numbe
e
T=h( )L,( )R,(β)L,(β)RiL,Ris a LRFN i i s membe ship unc ion has he ollowing s uc u e:
µe
T( )=





















1i ( )L≤ ≤( )R
L( )L−
(β)Li ( )L−(β)L≤ ≤( )L
R −( )R
(β)Ri ( )R≤ ≤( )R+(β)R
0o he wise
(7)
whe e
L,R:[0, 1]→[0, 1]
a e non-inc easing, con inuous shape unc ions wi h
L(0)=R(0)=
1 and
L(1)=R(1)=
0. Fu he mo e,
h( )L,( )Ri
consis s o he eal numbe s wi h he highes chance o
ealiza ion, while (β)Land (β)Ra e he le and igh sp ead, espec i ely.
No e ha LRFN includes as special cases he commonly used apezoidal uzzy numbe s
(T FN) and iangula uzzy numbe s (TFN). Thus, T FN use linea le and igh shape unc ions
L(α)=R(α)=
1
−α
. TFN use linea shape unc ions and, in addi ion, he le and igh
α=
1 alues
coincide, i.e., ( )L=( )R.
Finally, he α-cu s o a LRFN e
Ta e he in e als:
e
Tα=e
TL
α,e
TU
α
e
TL
α=( )L−L∗(α)·(β)L
e
TU
α=( )R+R∗(α)·(β)R
(8)
whe e he in e se shape unc ions a e de ined as
L∗(α)=suph:L(h)≥α
and
R∗(α)=
suph:R(h)≥α
. In he case o T FN and TFN hese in e se shape unc ions a e simply
L∗(α)=
R∗(α)=1−α.
3. P oposed Fuzzy Ranking Ne wo k DEA App oaches
In his sec ion, wo di e en uzzy anking me hods a e p esen ed. Bo h conside ha he inpu ,
ou pu and in e media e p oduc s a e LRFN:
e
Xp
ij =(xp
ijL
,xp
ijR
,βp
ijL
,βp
ijR)Li,Ri
e
Yp
kj =(yp
kjL
,yp
kjR
,ˆ
βp
kjL
,ˆ
βp
kjR)ˆ
Lk,ˆ
Rk
e
Zp
j =(zp
jL
,zp
jR
,ˆ
ˆ
βp
jL
,ˆ
ˆ
βp
jR)ˆ
ˆ
L ,ˆ
ˆ
R
(9)
Ma hema ics 2020,8, 2222 5 o 18
Acco ding o De ini ion (8), he α-cu s o hese LRFN e
Xp
ij,e
Yp
kj and e
Zp
j a e he in e als:
e
Xp
ijα="e
Xp
ijL
α,e
Xp
ijU
α#
e
Xp
ijL
α=xp
ijL
−L∗
i(α)·βp
ijL
e
Xp
ijU
α=xp
ijR
+R∗
i(α)·βp
ijR
(10)
e
Yp
kjα="e
Yp
kjL
α,e
Yp
kjU
α#
e
Yp
kjL
α=yp
kjL
−ˆ
L∗
k(α)·ˆ
βp
kjL
e
Yp
kjU
α=yp
kjR
+ˆ
R∗
k(α)·ˆ
βp
kjR
(11)
e
Zp
jα="e
Zp
jL
α,e
Zp
jU
α#
e
Zp
jL
α=zp
jL
−ˆ
ˆ
L
∗
(α)·ˆ
ˆ
βp
jL
e
Zp
jU
α=zp
jR
+ˆ
ˆ
R
∗
(α)·ˆ
ˆ
βp
jR
(12)
No e also ha i has been assumed ha o a gi en inpu , ou pu o in e media e p oduc he
le shape unc ion is he same o all DMUs and p ocesses and he same occu s wi h he igh shape
unc ion. This assump ion is commonly made (e.g., Le
ó
n e al. [
13
] o Soleimani-damaneh e al. [
16
])
and i is no oo es ic i e since i only assumes ha he da a o a ce ain ac o (inpu , ou pu o
in e media e p oduc ) a e desc ibed h ough LRFN o he same ype. This assump ion is impo an
because i means ha he linea combina ion o hese LRFN, using scala mul iplie s, is also a LRFN
wi h he same le and igh shape unc ions. Ma hema ically,
P
pP
j
λp
je
Xp
ij =





P
pP
j
λp
jxp
ijL
,P
pP
j
λp
jxp
ijR
,P
pP
j
λp
jβp
ijL
,P
pP
j
λp
jβp
ijR





Li,Ri
P
pP
j
λp
je
Yp
kj =





P
pP
j
λp
jyp
kjL
,P
pP
j
λp
jyp
kjR
,P
pP
j
λp
jˆ
βp
kjL
,P
pP
j
λp
jˆ
βp
kjR





ˆ
Lk,ˆ
Rk
P
pP
j
λp
je
Zp
j =





P
pP
j
λp
jzp
jL
,P
pP
j
λp
jzp
jR
,P
pP
j
λp
jˆ
ˆ
βp
jL
,P
pP
j
λp
jˆ
ˆ
βp
jR





ˆ
ˆ
L ,ˆ
ˆ
R
(13)
The α-cu s o hese LRFN P
pP
j
λp
je
Xp
ij,P
pP
j
λp
je
Yp
kj and P
pP
j
λp
je
Yp
kj a e hus he in e als:
P
pP
j
λp
je
Xp
ijα
=P
pP
j
λp
j(xp
ijL
−L∗
i(α)βp
ijL),P
pP
j
λp
j(xp
ijR
+R∗
i(α)βp
ijR)
P
pP
j
λp
je
Yp
kjα
=P
pP
j
λp
j(yp
kjL
−ˆ
L∗
k(α)ˆ
βp
kjL),P
pP
j
λp
j(yp
kjR
+ˆ
R∗
k(α)ˆ
βp
kjR)
P
pP
j
λp
je
Zp
jα
=P
pP
j
λp
j(zp
jL
−ˆ
ˆ
L
∗
(α)ˆ
ˆ
βp
jL),P
pP
j
λp
j(zp
jR
+ˆ
ˆ
R
∗
(α)ˆ
ˆ
βp
jR)
(14)
3.1. Fuzzy Ranking Me hod 1 (FRM1)
FRM1 uses he anking me hod in Tanaka e al. [
36
]. This me hod was p oposed in Le
ó
n e al. [
13
]
o con en ional (i.e., single-p ocess DEA). The s a ing poin is he o mula ion o he p oblem using
he uzzy da a. The objec i e is o educe all inpu s as much as possible wi h espec o he obse ed

Ma hema ics 2020,8, 2222 6 o 18
alues. This co esponds o Equa ions (15) and (16). No e ha he igh -hand side o (16) co esponds
o he obse ed inpu consump ion o DMU J (summed o all he p ocesses ha consume ha inpu )
while he le -hand side is he inpu consump ion o he a ge ope a ing poin (compu ed as a linea
combina ion o he inpu consump ion o all he DMUs). Simila ly, in Cons ain s (17), he igh -hand
side ep esen s he obse ed ou pu s o DMU J and he le -hand side is he co esponding amoun s
p oduced by he a ge ope a ing compu ed using a con ex linea combina ion o all he DMUs. Thus,
Cons ain s (17) impose ha he obse ed ou pu s a e no educed. Finally, Cons ain s (18) impose
ha he a ge ope a ing poin mus sa is y he in e media e p oduc s cons ain s ha gua an ee ha
he in e nal p oduc ion o in e media e p oduc s is enough o sa is y i s in e nal demand.
Min θ(15)
subjec o
X
p∈PI(i)X
j
λp
je
Xp
ij ≤θX
p∈PI(i)e
Xp
iJ ∀i(16)
X
p∈PO(k)X
j
λp
je
Yp
kj ≥X
p∈PO(k)e
Yp
kJ ∀k(17)
X
p∈Pou ( )X
j
λp
je
Zp
j −X
p∈Pin( )X
j
λp
je
Zp
j ≥0∀ (18)
cons ain s (5)and (6)
The Cons ain s (16)–(18) compa e wo uzzy quan i ies. The key idea in his me hod is how o
in e p e he inequali y, i.e., when o conside ha one uzzy quan i y is la ge han o equal o ano he .
Based on Tanaka e al. [
36
], Le
ó
n e al. [
13
] p oposed he use o he ollowing anking c i e ion o wo
uzzy numbe s e
Mand e
Na possibili y le el α:
e
M≥αe
N⇔ ∀h∈[α, 1]







e
ML
h≥e
NL
h
e
MR
h≥e
NR
h
(19)
Following his c i e ion, o each possibili y le el
α∈[0, 1]
an inpu -o ien ed e iciency sco e
EJ(α)
can be compu ed o each DMU J using he ollowing model:
EJ(α)=Min θ(20)
subjec o
X
p∈PI(i)X
j
λp
j·xp
ijL
≤θ·X
p∈PI(i)xp
iJL∀i(21)
X
p∈PI(i)X
j
λp
j·xp
ijR
≤θ·X
p∈PI(i)xp
iJR∀i(22)
X
p∈PI(i)X
j
λp
j·e
Xp
ijL
α
≤θ·X
p∈PI(i)e
Xp
iJL
α
∀i(23)
X
p∈PI(i)X
j
λp
j·e
Xp
ijU
α
≤θ·X
p∈PI(i)e
Xp
iJU
α
∀i(24)
X
p∈PO(k)X
j
λp
j·yp
kjL
≥X
p∈PO(k)yp
kJL
∀k(25)
Ma hema ics 2020,8, 2222 7 o 18
X
p∈PO(k)X
j
λp
j·yp
kjR
≥X
p∈PO(k)yp
kJR
∀k(26)
X
p∈PO(k)X
j
λp
j·e
Yp
kjL
α
≥X
p∈PO(k)e
YkJL
α∀k(27)
X
p∈PO(k)X
j
λp
j·e
Yp
kjU
α
≥X
p∈PO(k)e
YkJU
α∀k(28)
X
p∈Pou ( )X
j
λp
j·zp
jL
−X
p∈Pin( )X
j
λp
j·zp
jL
≥0∀ (29)
X
p∈Pou ( )X
j
λp
j·zp
jR
−X
p∈Pin( )X
j
λp
j·zp
jR
≥0∀ (30)
X
p∈Pou ( )X
j
λp
j·e
Zp
jL
α
−X
p∈Pin( )X
j
λp
j·e
Zp
jL
α
≥0∀ (31)
X
p∈Pou ( )X
j
λp
j·e
Zp
jU
α
−X
p∈Pin( )X
j
λp
j·e
Zp
jU
α
≥0∀ (32)
cons ain s (5)and (6)
This uzzy anking app oach has he d awback, as Soleimani-damaneh e al. [
16
] indica es o
con en ional uzzy DEA, o ha ing many cons ain s, mo e so in he case o uzzy ne wo k DEA (due
o he exis ence o in e media e p oduc s). Howe e , he numbe o cons ain s can be educed o
almos hal (because o many cons ain s being edundan ) in he case o symme ic TFN, in which
case he app op ia e model o use is he ollowing:
EJ(α)=Min θ(33)
subjec ocons ain s (23),(24),(27),(28),(31),(32),(5)and (6)
Same as in León e al. [13], he ollowing holds o he e iciency sco es compu ed by FRM1:
EJ(α1)≥EJ(α2)∀α1< α2
EJ(α2)=1⇒EJ(α1)=1∀α1< α2
(34)
Thus, he e iciency sco es
EJ(α)
ep esen a mono onous non-inc easing unc ion o he possibili y
le el
α
. In o de o ank he e iciency o he di e en DMUs he ollowing a ea c i e ion is p oposed.
A DMU J anks abo e ano he DMU J’ in e ms o e iciency i he a ea below
EJ(α)
is g ea e han ha
he a ea below
EJ0(α)
. The a ea below
EJ(α)
is
A eaEJ=R1
0EJ(α)dα
which in he common case ha
EJ(α)is compu ed o se e al disc e e possibili y le els αq educes o
A eaEJ=PαqEJαq
Pαq1.
Finally, le us men ion ha , same as in Le
ó
n e al. [
13
], he e iciency o DMU J can also be exp essed
as a uzzy se whose membe ship unc ion is µEJ(θ)=supnα:EJ(α)=θo.
Ma hema ics 2020,8, 2222 8 o 18
3.2. Fuzzy Ranking Me hod 2 (FRM2)
FRM2 uses he anking me hod in Yao and Wu [
37
]. This me hod was p oposed in
Soleimani-damaneh e al. [
16
] o con en ional (i.e., single-p ocess DEA). As be o e, he key is
how o in e p e he Inequali ies (16)–(18), i.e., when o conside ha one uzzy quan i y is la ge han
o equal o ano he . Yao and Wu [
37
] de ined he signed dis ance be ween wo uzzy numbe s
e
M
and
e
Nas
de
M,e
N=1
2Z1
0e
ML
α+e
MU
α−e
NL
α−e
NR
αdα(35)
and p oposed he ollowing anking c i e ion:
e
M≥e
N⇔de
M,e
N≥0
e
M≤e
N⇔de
M,e
N≤0(36)
Mo eo e , Soleimani-damaneh e al. [
16
] showed he signed dis ance be ween wo LRFN
e
M=
n(m)L,(m)R,(β)L,(β)RoL,Rand e
N=n(n)L,(n)R,(γ)L,(γ)RoL,Rcan be exp essed as
de
M,e
N=(m)L+(m)R−(n)L−(n)R+h(γ)L−(β)LiZ1
0
L∗(α)dα+h(β)R−(γ)RiZ1
0
R∗(α)dα(37)
and ha he esul ing model is equi alen o he co esponding c isp DEA model using app op ia ely
de uzzi ied ac o alues.
Thus, om he LRFN o each inpu , ou pu and in e media e p oduc a de uzzi ied alue is
compu ed as
xp
ij =1
2·(xp
ijL
+xp
ijR
+βp
ijRR1
0R∗
i(α)dα−βp
ijLR1
0L∗
i(α)dα)
yp
kj =1
2·(yp
kjL
+yp
kjR
+ˆ
βp
kjRR1
0R∗
k(α)dα−ˆ
βp
kjLR1
0L∗
k(α)dα)
zp
j =1
2·(zp
jL
+zp
jR
+ˆ
ˆ
βp
jRR1
0R∗
(α)dα−ˆ
βp
jLR1
0L∗
(α)dα)
(38)
whe e
L∗
i(α)
,
L∗
k(α)
,
L∗
(α)
and
R∗
i(α)
,
R∗
k(α)
,
R∗
(α)
a e all in e se shape unc ions. In he case o T FN
and TFN L∗(α)=R∗(α)=1−αwhich means ha :
R1
0L∗
i(α)dα=R1
0L∗
k(α)dα=R1
0L∗
(α)dα=1
2∀i∀k∀
R1
0R∗
i(α)dα=R1
0R∗
k(α)dα=R1
0R∗
(α)dα=1
2∀i∀k∀ (39)
and he e o e
xp
ij =1
2·







xp
ijL
+xp
ijR
+βp
ijR
−βp
ijL
2








yp
kj =1
2·







yp
kjL
+yp
kjR
+ˆ
βp
kjR
−ˆ
βp
kjL
2








zp
j =1
2·







zp
jL
+zp
jR
+ˆ
ˆ
βp
jR
−ˆ
ˆ
βp
jL
2








(40)
Ma hema ics 2020,8, 2222 9 o 18
Mo eo e , in he case o symme ic T FN and TFN, he abo e exp essions educe o
xp
ij =xp
ijL+xp
ijR
2
yp
kj =yp
kjL+yp
kjR
2
zp
j =zp
jL+zp
jR
2
(41)
In any case, once he de uzzi ied alues o he inpu , ou pu and in e media e p oduc s ha e been
compu ed, he ollowing LP is sol ed:
EJ=Min θ(42)
subjec o
X
p∈PI(i)X
j
λp
jxp
ij ≤θ·X
p∈PI(i)
xp
iJ ∀i(43)
X
p∈PO(k)X
j
λp
jyp
kj ≥X
p∈PO(k)
yp
kJ ∀k(44)
X
p∈Pou ( )X
j
λp
jzp
j −X
p∈Pin( )X
j
λp
jzp
j ≥0∀ (45)
cons ain s (5)and (6)
yielding, o each DMU J, a single e iciency sco e EJ.
No e ha he, in he end, abo e model is jus Models (1)–(6) applied o he de uzzi ied alues
o he inpu s, ou pu s and in e media e p oduc s. I s in e p e a ion is simila , i.e., he model aims a
minimizing he amoun o inpu s consumed by a a ge easible ope a ing poin ha is compu ed as
linea con ex combina ion o all he DMUs. The a ge ope a ing poin mus main ain he ou pu le el
o DMU J and i is also equi ed ha he in e media e p oduc s p oduced mus be g ea e han he
amoun consumed. As men ioned abo e, he di e ence be ween his model and Models (1)–(6) is
ha in he la e he obse ed da a we e c isp while in he abo e model he c isp inpu , ou pu , and
in e media e p oduc alues used o all DMUs (including DMU J) a e ob ained om he obse ed
uzzy da a h ough a de uzzi ica ion p ocess. As wi h he p e ious app oach he co esponding
ou pu -o ien ed model is o mula ed in Appendix A.
4. Illus a ion o P oposed App oaches
In o de o illus a e he p oposed app oach wo da ase s om he li e a u e will be used. The
i s one co esponds o a simple wo-s age sys em and was used in Kao and Liu [
20
]. This da ase
in ol es 24 DMUs (which ep esen Taiwanese non-li e insu ance companies) and conside s wo inpu s
(Ope a ing expenses and Insu ance expenses), wo in e media e p oduc s (Di ec w i en p emiums
and Reinsu ance p emiums) and wo ou pu s (Unde w i ing p o i and In es men p o i ). The uppe
and lowe e iciency limi s compu ed by Kao and Liu [
20
], which co espond o he uppe and lowe
e iciency es ima es o
α
=0, espec i ely, a e shown in Table 1. In addi ion, Table 1shows he
es ima es o
α
=1. Because in his da ase we a e dealing wi h TFN, he uppe and lowe e iciencies
o α=1 coincide.
The e iciency es ima es and he co esponding anking index compu ed by he p oposed FRM1
app oach a e also shown in Table 1. To compa e wi h he esul s in Kao and Liu [
20
], cons an e u ns
o scale (CRS) in all p ocesses and inpu o ien a ion a e conside ed. Al hough o FRM1 only h ee
possibili y le els a e shown in Table 1, calcula ions we e made o ele en le els, i.e.,
α
anges om 0 o
1 wi h 0.1 inc emen s. No e ha o FRM1 he es ima ed e iciency sco e o each possibili y le el is
Ma hema ics 2020,8, 2222 16 o 18
X
p∈PO(k)X
j
λp
j·e
Yp
kjL
α
≥γ·X
p∈PO(k)e
YkJL
α∀k(A12)
X
p∈PO(k)X
j
λp
j·e
Yp
kjU
α
≥γ·X
p∈PO(k)e
YkJU
α∀k(A13)
cons ain s (5),(29),(30),(31),(32)and (A4)
•FRM2 (ou pu o ien a ion) EJ−1=Max γ(A14)
subjec o
X
p∈PI(i)X
j
λp
jxp
ij ≤X
p∈PI(i)
xp
iJ ∀i(A15)
X
p∈PO(k)X
j
λp
jyp
kj ≥γ·X
p∈PO(k)
yp
kJ ∀k(A16)
cons ain s (5),(45)and (A4)
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