Int. J.IndustrialMathematisVol. 2,No. 3(2010)189-198
A Generalized Model in the Performane
Evaluation of Deision Making Sub-units
Z.Iravani
Department ofMathematis,Shahr-e-ReyBranh,IslamiAzadUniversity,Tehran,Iran.
Reeived28July2010;aepted11Otober2010.
|||||||||||||||||||||||||||||||-Abstrat
DataEnvelopmentAnalysis(DEA)evaluates theeÆienyof deisionmakingunitswith
multipleinputsand outputs. Sofar, anumberofDEAmodelshavebeendeveloped: The
CCRmodel,theBCCmodeland theFDH modelarewellknownasbasiDEA models.
Inmanyinstane,However,thedeisionmakingunitsan beseparatedintodierent
sub-units. In thispaper,we studya generalizedmodelforthisDMUsbydierent sub-units.
Keywords : Data envelopmentanalysis;Deisionmakingunits; Sub-units; EÆieny;Generalized
model
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1 Introdution
DataEnvelopmentAnalysis(DEA),originallyproposedbyCharnes,CooperandRodes
(1978 and 1979)[1℄, has beome one of the most widely used methods in management
siene. DEAmeasurestherelativeeÆienyofomparableentitiesalledDeision
mak-ing units(DMUs)essentially performing the same task using similar multiple inputs to
produe similar multiple outputs. The purpose of DEA is to empirially estimate the
so-alled eÆient frontier based on the set of available DMUs. A DMU is eÆient if
thereisnootherunit-existingorvirtualthataneitherproduemore outputsby
onsum-ing the same amount or less of inputs or produe the same amount or more of outputs
byonsuming lessor thesame amount of inputsas theDMU under onsideration. The
formerapproah isreferredto astheoutput orientedand thelatterastheinputoriented
DEA. DAE provides the user with informationaboutthe eÆient and ineÆient units,
as wellas the eÆieny sores and referene sets for ineÆient units. The result of the
DEA analysis, espeially theeÆieny sores, had pratial appliationsas performane
indiators ofDMUs.
In many instanes however, the deision making units an be separated into dierent
sub-units. f are and grosskpof [3℄, for example, look at a multi-stage proess where in
intermediate produt or output at one stage an be both nal produts and inputs to
the later stages of prodution. Those authors are not expliitly interested in obtaining
measuresof eÆienyat eahstage, butratherare onernedwithoverall eÆieny
mea-surement. Anotherexampleisduetoooketal[4℄andinvolvesmulti-omponenteÆieny
withsharedinputs.
Inthispaper,weproposeageneralizedmodelforDEAwhen DMUshassub-units,whih
an treat basi DEA models for this DMUs, speially, the CCR model, the BCC
model and theFDH modelina uniedway. In addition,weshowtheoretial properties
on relationshipsamong thismodel and those DEA models by sub-units,and this model
makes it possible to alulate the eÆieny of DMUs inorporating various preferene
strutureof deisionmakers.
The followingsetionsof thepaperprovidea sub-unitseÆienymeasurement.
2 Basi DEA models for sub-units
AssumethatwehavenDMUs,andaDMU
p
onsistsofbsub-units. alledDMSU.Eah
DMU
j
transformsresoures,orinputsintoproduts,oroutputsinpartiular,DMU
j ;2
jb 1,produesk
j
dierent typesof outputsand onsumesI
j
typesofexternalinputs
and I 0
j
types of internal inputs (i.e. a part of inputs oming from outside the whole
DMU and theother part omingfrom insidethe DMU ).The internalinputof DMS
j U
is output produed by the last DMSU
j
. The rst DMSU
1
onsumes the input vetor
X
1
and produes theoutput vetor Y
1
and thelastDMSU
b
onsumes theinternal input
vetorX
b
andtheexternalinputvetorX
b
produestheoutputvetorY
b
. AlltheDMSUs
onsidered have the same types of outputs and internal and external inputs. Espeially,
DMSU
j
;2 j b, onsumes I
j
types of external inputs X
j and I
0
j
types of internal
inputsX
j =Y
j 1
. Also DMSU
j
;2jb,produes k
j
typesof outputsY
j
. Seetheg.
1.
For notational purpose,let y (p)
j
;j = 1;:::;b, denote the output vetors produed by jth
sub-DMUof DMU
p
inwhih
Y (p)
j
=(y (p)
j1 ;:::;y
(p)
j;k
j );
Also,letX (p)
j
and X
j (p)
;j =2;:::;b,denote I
j and I
0
j
-dimensionalvetorsofexternaland
internalinputs to jth sub-DMUof DMU
p
, respetively, inwhih
X (p)
j
=(x (p)
j1
;:::;x (p)
j;I
j );
x (p)
j =(x
(p)
j1 ;:::;x
(p)
j;I 0
j )=(y
(p)
j 1;1 ;:::;y
(p)
j 1;I 0
j );
Hene,a measure ofaggregate performanee (a)
p
an berepresentedby
e (a)
p =
(1)T
y (p)
1 +
(2)T
y (p)
2
++ (b)T
y (p)
b
v (1)T
x (p)
1 +v
(2)T
x (p)
2
++v (b)T
x (p)
+v (1)T
y (p)
1
++v (b 1)T
and performanefor eahsub-unitsof DMUpan be representedby
e (1)
p =
(1)T
y (p)
1
v (1)T
x (p)
1
e (i)
p =
(i)T
y (p)
i
v (i)T
x (p)
i +v
(i 1)T
y (p)
i 1
; i=2; ;b:
Theorem 2.1. The aggregate eÆieny e (a)
p
is a onvex ombination of DMSU 0
s
eÆ-ieny.
Proof: Theproof isin[2℄.
Theorem 2.2. DMUpis eÆienyi all of DMUSp areeÆieny.
Proof: Theproof isstraightforward.
Then we have thefollowingmathematial programmingproblem:
max e
(a)
p
s:t: e
(a)
j
1; j=1;:::;n
e (i)
j
1; j=1;:::;b; j=1;:::;n
(i)
2
1
; i=1;:::;b
(v (i)
;v (i)
)2
2
; i=1;:::;b
(2.1)
The sets
1
and
2
are assurane regions denedby any restritionsimposed on
multi-pliers[4℄. The model(2.1) an beexpressed inthefollowingform
Max
b
X
i=1
(i)T
y (p)
i
S:t:
b
X
j=1 v
(i)T
x (p)
i +
b 1
X
j=1 v
(i)T
y (p)
i =1;
b
X
j=1
(i)T
y (j)
i b
X
j=1 v
(i)T
x (j)
i
b 1
X
i=1 v
(i)T
y (j)
i
0;j=1;:::;n
(i)T
y (j)
i v
(i)T
x (j)
i v
(i 1)T
y (j)
i 1
0; i=2; ;b;j=1; ;n
(i)
2
1
i=1; ;b
(v (i)
;v (i)
)2 i=1; ;b
The formof
1 and
2
dependson how
1
and
2
arestrutured.
Nowthe CCRmodelinpresent DMU'sfollowsas:
Min
S:t:
n
X
j=1
j x
(j)
i +
n
X
j=1
ij x
(j)
i
x
(j)
i
; i=1; ;b
n
X
j=1
j y
(j)
i +
n
X
j=1
ij y
(j)
i
y
(p)
i
; i=1; ;b 1
n
X
j=1
j y
(j)
i n
X
j=1
ij y
(j)
i y
(p)
i
; i=1; ;b
j
0; j=1; ;n
ij
0; i=1; ;b j =1; ;n:
(2.3)
AndtheBCC modelinpresent DMU'sfollowsas:
Min
S:t: n
X
j=1
j x
(j)
i +
n
X
j=1
ij x
(j)
i
x
(j)
i
; i=1; ;b
n
X
j=1
j y
(j)
i +
n
X
j=1
ij y
(j)
i
y
(p)
i
; i=1; ;b 1
n
X
j=1
j y
(j)
i n
X
j=1
ij y
(j)
i y
(p)
i
; i=1; ;b
n
X
j=1
j +
b
X
i=1 n
X
j=1
ij =1;
j
0; j=1; ;n
ij
0; i=1; ;b; j =1; ;n:
The multiplierform ofthe BCCmodelinpresent DMSUfollows as:
Max
b
X
i=1
(i)T
y (p)
i +u
0
S:t:
b
X
i=1 v
(i)T
x (p)
i +
b 1
X
i=1 v
(i)T
y (p)
i =1;
b
X
i=1
(i)T
y (j)
i b
X
i=1 v
(i)T
x (j)
i
b 1
X
i=1 v
(i)T
y (j)
i +u
0
0;j =1;:::;n
(i)T
y (j)
i v
(i)T
x (j)
i v
(i 1)T
y (j)
i 1 +u
0
0; i=2; ;b;j =1; ;n
(i)
2
1
i=1; ;b
(v (i)
;v (i)
)2
2
i=1; ;b
(2.5)
'
&
$
% #
" !
#
" !
#
" !
Fig.1. TheDMUpbyDMSUs
-?
? ? ? DMSU
1
DMSU
2
DMSU
b x
(p)
1
x (p)
2
x (p)
b
y (p)
1 =x
(p)
1
y (p)
2 =x
(p)
2
y (p)
b =x
(p)
b
Y q
q
q
3 A generalized model in present sub-units
In this setion, we formulate the generalized model in present sub-units, based on a
domination struture and dene a new eÆieny in thismodel. Next we establish
rela-tionshipsbetweenthisgeneralized model andbasiDEA modelsmentionedin setion2.
Now, we formulate a generalized DEAmodelin present sub-unitsbyemploying the
aug-mentedThebyshevseularizingfuntion[5℄. Thismodel,whihanevaluatetheeÆieny
inseveral basimodelswhih arespeialases forallDMUs, followsas:
Max
S:t: e
d
j +(
b
X
i=1
(i)T
(y (p)
i y
(j)
i )+
b
X
i=1 v
(i)T
( x (p)
i +x
(j)
i )+
b 1
X
i=1 v
(i)
( y (p)
i +y
(j)
i ))
(i)T
y (j)
i v
(i)T
x (j)
i v
(i 1)T
y (j)
i 1
0; i=2; ;b;j=1; ;n
b
X
i=1
(i)
+ b
X
i=1 v
(i)
+ b 1
X
i=1 v
(i)
=1
(i)
0; i=1; ;b
v (i)
0; i=1; ;b
v (i)
0; i=1; ;b 1
(3.6)
where >0 is appropriately given aording to given problems, and e
d
j
(j =1; ;n) is
denedbyfollowing:
e
d
j
= max
i=1;;bt=1;;b 1 f
(i)
(y (p)
i y
(j)
i );v
(i)
( x (p)
i +x
(j)
i )+v
(t)
( y (p)
t +y
(j)
t
)g (3.7)
Note thatwhen j=p then0.
Denition 3.1. ( eÆieny) For a given positive number , DMU
o
is dened to
be eÆieny if and only if the optimal value to the problem (3.6) is equal to zero.
Otherwise, DMUpis said to be ineÆieny.
Theorem 3.1. If 6=0 the existene DMU where dominated DMUp.
Proof: Let6=0,byontraditionsupposethatthere isnotDMUwheredominated
DMUp.
Ontheother hand, forall j we have
2
6
4 Y
(j)
X (j)
(j) 3
7
5 2
6
4 Y
(p)
X (p)
(p) 3
7
We denote Z
j =
2
6
4 Y
(j)
X (j)
X (j)
3
7
5. Therefore
Z (j)
Z (p)
(8j): (3.8)
Andfrom inequalitiesofthemodel(3.6) inpresent sub-unitsforall DMUs we have
e
d
j +(
b
X
i=1
(i)T
(y (p)
i y
(j)
i )+
b
X
i=1 v
(i)T
( x (p)
i +x
(j)
i )+
b
X
i=1 v
(i)T
(y (p)
i y
(j)
i ))
But,<0,andiffreevariable,thenneessaryandsuÆientonditionforexisteneabove
inequality(for somej6=p)is
e
d
j +(
b
X
i=1
(i)T
(y (p)
i y
(j)
i )+
b
X
i=1 v
(i)T
( x (p)
i +x
(j)
i )+
b
X
i=1 v
(i)T
(y (p)
i y
(j)
i
))<0
We have
e
d
j
+(;v;v) 2
6
4 Y
(p)
i Y
(j)
i
X (p)
i +X
(j)
i
X (p)
i +X
(j)
i 3
7
5
<0 (for somej 6=p):
That iswe have thefollowing
e
d
j
+(;v;v)(Z (i)
p Z
(i)
j
)<0 (for somej 6=p):
Nowby(3.8) and >0 and (;v;v)0wemusthave
e
d
j
<0 (for somej 6=):
Andby denition e
d
j
(for somej6=p)we have:
e
d
j
= max
i=1;;bt=1;;b 1 f
(i)
(y (p)
i y
(j)
i );v
(i)
( x (p)
i +x
(j)
i )+v
(t)
(y (p)
t y
(j)
t
)g<0:
Hene by(;v;v)0, Z (i)
p Z
(i)
j
<0 (forsome j6=p). Whereontradition by (3.8).
ThisontraditionassertsthatthereisnotexisteneDMUwheredominatedDMU,and
theproofisomplete.
4 Relationships between generalized modelandBCC (CCR)
model in present sub- units
In this setion, we establish theoretial properties on relationships among eÆienies in
thebasiDEA modeland generalized modelinpresent sub-units.
Proof: SupposethatDMUp is-eÆient forsome suÆientlylargepositive . That
isfor alloptimalsolutionwe have:
0= e d j
+(;v;v)(Z
p Z
j ):
The neessary and suÆient ondition for some suÆiently large positive number for
thisinequalityfollowsas:
( Z p Z j 0 e d j 0 (4.9)
Then we have
( ;v ;v )(Z p Z j )0:
Therefore ( ;v ;v )Z p ( ;v ;v )Z j
)0: (4.10)
Supposethat (v;v) X p X p
= then( v ; v ) X p X p =1.
We denote u 0 = ( ;v ;v )Z p
. Heneby(4.10) wehave
( ;v ;v )Z j +u 0
0)( ; v ; v )Z j + u 0 0 Therefore( ; v ; v
)is a feasiblesolutionforBCC model,inpresent sub-unitswherethe
valueof objetive funtionisone. Then DMUpis eÆientinpresentDMU's.
Now by additional restrition P b i=1 (i)T (y (p) i ) = P b i=1 v (p)T (x (j) i ) + P b 1 i=1 v (i) (y (p) i ) we
studythegeneralized modelin present sub-unitsforallDMUs
Max S:t: e d j +( b X i=1 (i)T (y (p) i y (j) i )+ b X i=1 v (i)T ( x (p) i +x (j) i )+ b 1 X i=1 v (i) ( y (p) i +y (j) i )) ( b X i=1 (i) (y (p) i )= b X i=1 v (i)T (x (p) i + b 1 X i=1 v (i) (y (p) i ) (i)T y (j) i v (i)T x (j) i v (i 1)T y (j) i 1 0; b X i=1 (i) + b X i=1 v (i) + b 1 X i=1 v (i) =1 (i)
0; i=1; ;b
v (i)
0; i=1; ;b
v (i)
0; i=1; ;b 1;
(4.11)
Proof: Supposethat DMUpis-eÆientforsome suÆient largepositive. That is
forall solution( ^
; ;^ v
; ^
v)wehave b
=0
0= ^
e
d
j
+(^;v;^ ^
v)(Z (p)
Z (j)
)
The neessaryand suÆient onditionforthis formulais
(
Z (p)
Z (j)
0
e
d
j 08j
(4.12)
But, we supposethatvx^ (p)
+bvx (p)
= then
^ v
x
(p)
+ b
v
x
(p)
=1: (4.13)
Now y^ (p)
=vx^ (p)
+ ^
v x (p)
therefore ^
y
(p)
= ^ v
x
(p)
+ ^
v
x
(p)
and by (4.13) we have
^
y
(p)
=1 (4.14)
and by(4.12) we have
( ;^ ^v; ^
v)(Z (p)
Z (j)
)0
(^;v;^ ^
v)Z (p)
( ;^ ^v; ^
v)Z (j)
0
Hene
(^;v;^ ^
v)Z (j)
0:
Then
^
y
(j) ^ v
x
(j) ^
v
x
(j)
0
and
b
X
i=1 ^
(i)T
y
(j) b
X
i=1 ^ v
(i)T
x
(j)
i
b 1
X
i=1 ^
v (i)T
y
(j)
i
0; j=1; ;n:
Therefore( ^
;
^ v
;
^
v
)isafeasiblesolutionforCCRmodelinpresentsub-unitsandthevalue
of objetive funtionis ^
y
(p)
=1. Then DMUpis eÆient inpresent sub-units.
5 Conlusion
In this paper, we have suggested the GDEA model for performane evaluation based
on parametri domination struture and dened eÆieny in the GDEAmodel. The
methodpresentedherean beusedfortheanalysis ofanyreal situationwherea DMUis
general-Referenes
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