Int. J.IndustrialMathematisVol. 2,No. 4(2010)263-270
Ranking Stohasti EÆient DMUs based on
Reliability
G.R.Jahanshahloo a
,M.H. Behzadi b
,M.Mirbolouki a
(a)Departmentof Mathematis,SieneandResearhBranh,IslamiAzadUniversity,Tehran,Iran.
(b)Department ofStatistis,SieneandResearh Branh, IslamiAzadUniversity,Tehran, Iran.
Reeived6June2010;revised25Otober 2010;aepted30Otober2010.
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Abstrat
Data Envelopment Analysis (DEA) is a nonparametri mathematial programming
ap-proah for evaluating eÆieny of a set of deision making units (DMUs). Sine the
number of eÆient DMUs is more than one, there is a neessity of having methods to
disriminate between eÆient DMUs. In lassi DEA it is assumed that data have
de-terministivalues. Throughreal world appliationsthisassumptionmaynotbe satised.
Ontheotherhand,datamayhavestohastiessene. Inthispaperamethodforranking
stohastiDMUs issuggested whihisbased onthereliabilityofeÆienyof DMUs.
Us-ingnumerialexample, we demonstrate howto usetheresults.
Keywords : Data envelopment analysis; Quadrati programming; Ranking; Stohasti
program-ming.
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1 Introdution
Data Envelopment Analysis(DEA)oneptswereoriginatedbyCharneset al. [3℄ after
thatthisideawasextended toan approahforevaluatingtherelativeeÆienyofDMUs
[6,10℄. TheDMUsusuallyuseasetofresoures,referredtoasinputindies,andtransform
themintoasetofoutomes,referredtoasoutputindies. DEAmodelsdivideDMUsinto
twoategories: eÆientDMUsandineÆientDMUs. Inmanyappliations,weknowthat
usuallyseveral DMUsareeÆient. Todisriminatebetweenthese eÆient DMUsranking
methodologies had been initiated [9℄. Sexton et al. [11℄ were pioneers in ranking eld.
They introdued a ranking method based on a ross-eÆieny. Andersen and Petersen
[1℄ evaluated a DMUs eÆieny by exludingit from produtionpossibilityset and they
started suppereÆieny eld. They triedto disriminatebetween these eÆient DMUs,
byusingdierent eÆienysoreslargerthan1.0. Cooketal. [4℄developedprioritization
models to rank the eÆient units in DEA. Torgersen et al. [12℄ obtained a omplete
ranking of eÆient DMUs by measuring their importane as a benhmark for ineÆient
DMUs.
Inmanyappliationsthroughrealworld,dataofproblemareimpreise. Oneof
meth-odsin onfronting with these kinds of data is onsidering them as stohasti orrandom
variables. Cooperetal. [5℄appliedstohastivariablesinDEAmodels. Theyalsodened
stohasti eÆient DMUs. Huangand Li[8℄ introduedstohasti dominaneonditions.
behzadiet al. [2℄and HosseinzadehLotet al. [7℄ proposedmethodsforranking
stohas-ti eÆient DMUs. In thispaper we propose a ranking approah based on Huang an Li
[8℄. With this method an stohasti eÆient DMU has a higher rank if it is dominated
withlessererror.
The paper is organized as follows: First the preliminaries on stohasti models and
stohastieÆienyareprovidedandthenamodelforrankingDMUsbasedonstohasti
reliabilityisintrodued. Usingnumerialexample,wedemonstrate howtousetheresult.
2 Preliminaries
Consider n DMUs with ~
X
j = (~x
1j ;:::;x~
mj ) and
~
Y
j = (~y
1j ;:::;y~
sj
) as random input
and output vetors of DMU
j
, j = 1:::;n. Assume that X
j = (x
1j ;:::;x
mj
) and Y
j =
(y
1j ;:::;y
sj
) stand for orresponding vetors of expeted values of input and output for
every DMU
j
. All input and output omponents have been onsidered to be normally
distributed. Thehaneonstrainedversion ofinputorientedstohastiBCC modelisas
follows:
min
s:t: pf n
X
j=1
j ~ y
rj y~
ro
g1 ; r=1;:::;s;
pf n
X
j=1
j ~ x
ij ~x
io
g1 ; i=1;:::;m;
n
X
j=1
j =1;
j
0; j=1;:::;n:
(2.1)
Model(2.1)anbeonvertedintothefollowingtwo-stagemodelwithequalityonstraints:
min "( s
X
r=1 s
+
r +
m
X
i=1 s
i )
s:t: pf n
X
j=1
j ~ y
rj s
+
r y~
ro
g=1 ; r=1;:::;s;
pf n
X
j=1
j ~ x
ij +s
i ~x
io
g=1 ; i=1;:::;m;
n
X
j=1
j =1;
s
i
0; s +
r
0; i=1;:::;m; r =1;:::;s:
0; j=1;:::;n:
where in the above models, p means \probability" and is a predetermined number
between0and 1. Onbasisofnormaldistributionharateristis, thedeterministimodel
for(2.2) an beattained asfollows:
min "( s X r=1 s + r + m X i=1 s i ) s:t: n X j=1 j y rj s + r + 1 ()u o r =y ro
; r =1;:::;s;
n X j=1 j x ij +s i 1 ()v I i =x io
; i=1;:::;m;
n X j=1 j =1; s i 0; s + r
0; i=1;:::;m; r=1;:::;s;
j
0; j=1;:::;n:
(2.3) where (u o r ) 2 = n X j=1 j6=o n X k =1 k 6=o j k ov(~y rj ;y~
rk )+2(
o 1) n X j=1 j6=o j ov(~y rj ;y~
ro )+( o 1) 2 var(~y ro ); and (v I i ) 2 = n X j=1 j6=o n X k =1 k 6=o j k ov(~x ij ;x~
ik )+2(
o ) n X j=1 j6=o j ov(~x ij ;x~
io )+( o ) 2 var(~x io ):
Here, is the umulative distribution funtion of the standard normal distribution and
1
(),isitsinverseinlevelof. Theabovemodelisaquadratinonlinearprogramming
model.
Denition 2.1. DMU
o
is stohasti eÆient if and only if in the optimal solution of
model (2.3), the following onditions areboth satised:
(i)
=1
(ii) Slak values are all zeros.
3 Reliability ranking method
Let ~
X = (~x
1 ;:::;x~
n ) and
~
Y = (~y
1 ;:::;y~
n
) be the matrixes of input vetors and output
vetors respetively, and X =(x
1 ;:::;x
n
) and Y =( y
1 ;:::;y
n
) betheirmean matrixes.
Huangand Li[5℄ dened-stohasti eÆient DMUasfollows,
Denition 3.1. DMU
o
is -stohasti eÆientif it is eÆientwithreliability of atleast
(1 ).
The above denition indiates that an stohasti eÆient DMU may be dominated
witha probabilityof at most. Ontheother handDMU
o
is -stohastieÆient ifthe
followingexpressionishold and stritlyforat leastone.
By thespeiations ofsets and probabilityoneptsitis resultedthat, 8 < : n X j=1 j ~ x j x~ o ; n X j=1 j ~ y j y~ o 9 = ; 8 < : 1 T ( n X j=1 j ~ x j ~ x o )+1
T
(y~
o n X j=1 j ~ y j )<0
9 = ; : where1 T
=(1;:::;1). From the above expressionwe have,
P 0 n X j=1 j ~ x j x~ o ; n X j=1 j ~ y j y~ o 1 A P 0 1 T ( n X j=1 j ~ x j ~ x o )+1
T
(y~
o n X j=1 j ~ y j )<0
1
A
:
Thereforethefollowing expressionis theneessaryonditionfor-stohastieÆieny.
P 0 1 T ( n X j=1 j ~ x j ~ x o )+1
T
(y~
o n X j=1 j ~ y j )<0
1
A
: (3.4)
Let ~
h=1 T
( ~
X x~
o )+1
T
(y~
o ~
Y). Thereforeexpression(3.4)isequaltoP
~
h0
.
ItanbeonvertedtoanequalformbyaddingnonnegativevariablesasP
~
h0
= ".
Applyingslakvariablesresultsthat,
P
~
hs 0
=: (3.5)
~
h hasnormal distributionwithparameters h and 2
h
where,
h=E( ~
h )=1 T
(X x
o )+1
T (y o Y); 2 h
=Var( ~
h )=1 T
1+1 T
0
1+2(1 T 00 1): (3.6) and =[ ik ℄ mm ; ik =Cov( n P j=1 j x ij x io ; n P j=1 j x kj x ko ); 0 =[ 0 rk ℄ ss ; 0 rk =Cov(y ro n P j=1 j y rj ;y ko n P j=1 j y kj ); 00 =[ 00 ir ℄ ms ; 00 ir =Cov( n P j=1 j x ij x io ;y ro n P j=1 j y rj ):
From expressions(3.5) and (3.6) wehave,
Thusdeterministi equivalentof (3.4) is h s 0
+
h
1
()=0:
SupposeDMU
o
isanstohastieÆientDMU.Thereforefrom denition2.1andoptimal
solutionofmodel(2.3),
n
P
j=1
j x
ij
1
()v
i =x
io
;i=1;:::;m;
n
P
j=1
j y
rj +
1
()u
r =y
ro
;r=1;:::;s:
Let be the optimality solutions spae of model (2.3). In our ranking method we seek
the minimumlevel of errorin whih a DMUis dominated probabilistiallyon set . i.e.
we seektheoptimalsolutionof thefollowingmodel:
=min
s:t:
n
P
j=1
j x
ij
1
()v
i =x
io
; i=1;:::;m;
n
P
j=1
j y
rj +
1
()u
r =y
ro
; r =1;:::;s;
h s 0
+ 1
()w=0;
v 2
i =
n
P
j=1
j6=o n
P
k=1
k6=o
j
k Cov(~x
ij ;x~
ik )+(
o 1)
2
Var(~x
io )
+2(
o 1)
n
P
j=1
j6=o
j Cov(~x
ij ;x~
io
); i=1;:::;m;
u 2
r =
n
P
j=1
j6=o n
P
k=1
k6=o
j
k Cov(~y
rj ;y~
rk )+(
o 1 )
2
Var(y~
ro )
+2(
o 1)
n
P
j=1
j6=o
j Cov(~y
rj ;y~
ro
);r=1;:::;s;
w 2
=1 T
1+1 T
0
1+2 1 T
00
1
;
w0;
j
0;j=1;:::;n;
v
i 0;u
r
0;i=1;:::;m;r =1;:::;s:
(3.7)
Model(3.7) is anonlinear programmingwhihis notlear beause of theexistene of ill
onverted to thefollowingmodel:
=min
s:t:
n
P
j=1
j x
ij
1
()v
i =x
io
; i=1;:::;m;
n
P
j=1
j y
rj +
1
()u
r =y
ro
; r=1;:::;s;
h s 0
+w=0;
v 2
i =
n
P
j=1
j6=o n
P
k=1
k6=o
j
k Cov(~x
ij ;x~
ik )+(
o 1)
2
Var(x~
io )
+2(
o 1)
n
P
j=1
j6=o
j Cov(x~
ij ;x~
io
); i=1;:::;m;
u 2
r =
n
P
j=1
j6=o n
P
k=1
k6=o
j
k Cov(y~
rj ;y~
rk )+(
o 1 )
2
Var(y~
ro )
+2(
o 1)
n
P
j=1
j6=o
j Cov(y~
rj ;y~
ro
);r =1;:::;s;
w 2
=1 T
1+1 T
0
1+2 1 T
00
1
;
3:8 3:8;
w0;
j
0;j =1;:::;n;
v
i 0;u
r
0;i=1;:::;m;r =1;:::;s:
(3.8)
Model(3.8)isanonlinearquadratiprogrammingmodel. Theonstraint 3:8 +3:8
isaddedtopreventunboundednessofthismodel. Theoptimalobjetivefuntionofmodel
(3.8) is ourreliabilityranking indiator. Less
indiates better rank.
4 An appliation
In this setion, we onsider 10 branhes of an Iranian bank with two stohasti
in-puts and two stohasti outputs and runthe mentioned modelin order to fully rank the
stohasti eÆient units. Inthismodel,\payable benet"and \delayed requisitions" are
inputsand\amountofdeposits"and\reeivedbenet"areoutputs. Thesedatabasedon
onsidering ten suessive months have normal distribution and their saled parameters
arepresentedinTable1. We want to assess thetotal performaneof these units. Inthis
example these DMUs have been assessedby usingmodel (2.3). Then stohasti eÆient
DMUshavebeenrankedbytheirineÆienysoresbyapplyingmodel(3.8). Weonsider
Table 1
Preditedinputsandoutputs.
inputs outputs
xij N(;) xij N(;) yrj N(;) yrj N(;)
X1,1 N(18.79,9.41) X2,1 N(7.28,0.76) Y1,1 N(49.6,6.93) Y2,1 N(4.7,0.64)
X1,2 N(44.3,25.3) X2,2 N(1.11,0.15) Y1,2 N(73.13,3.62) Y2,2 N(1.85,0.15)
X1,3 N(19.73,16.63) X2,3 N(19.2,0.69) Y1,3 N(108.04,15.02) Y2,3 N(6.06,0.12)
X1,4 N(17.43,11.06) X2,4 N(59.47,0.92) Y1,4 N(44.97,3.71) Y2,4 N(4.9,1.29)
X1,5 N(10.38,4.59) X2,5 N(12.23,7.74) Y1,5 N(31.63,6.24) Y2,5 N(2.78,0.66)
X1,6 N(16.67,10.42) X2,6 N(568.63,37.42) Y1,6 N(71.98,8.37) Y2,6 N(13.19,3.03)
X1,7 N(25.46,13.67) X2,7 N(552.85,20.78) Y1,7 N(78.05,13.99) Y2,7 N(7.79,1.89)
X1,8 N(123.06,65.3) X2,8 N(14.78,0.25) Y1,8 N(219.69,19.38) Y2,8 N(35.3,3.92)
X1,9 N(36.16,19.59) X2,9 N(361.88,34.11) Y1,9 N(86.25,6.95) Y2,9 N(17.64,1.92)
X1,10 N(46.41,23.06) X2,10 N(12.81,0.62) Y1,10 N(194.58,42.15) Y2,10 N(25.9,3.52)
Table 2
Results
eÆientDMU b
RANK
DMU3 -1.02 3
DMU5 -0.98 4
DMU6 -2.08 1
DMU10 -1.19 2
5 Conlusion
ThereareseveralmodelsinDEAeldwhihhavebeenformulatedforevaluatingeÆieny
and ranking DMUs in various elds with dierent data suh as: deterministi, interval,
fuzzy, e.t.. In real world appliation managers may enounter the data whih are not
deterministi. Nowadays theextentofprobabilityhasasigniantimportane. Therefore
DEA models have been extended to stohasti data by researhers. Thus the neessity
of having modelsthat are able to rank DMUs has been under onsideration. They have
been assessed and in suh way they have dened the stohasti eÆient DMUs. In this
paperonbasisoftheeÆienyreliabilityofeÆientDMUs,amodelforrankingstohasti
eÆientDMUshasbeenpresented. Thismodelisaquadratiprogrammingmodelandthe
objetivefuntionisafuntionof,whihistheleveloferrorthatshouldbedeterminedby
themanagers. Inthispaperwehave appliednormaldistributions. Dierent distributions
aswellasnormal distributionan beonsideredfrom thispoint ofview.
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