research article modified nonradial supper...
TRANSCRIPT
Research ArticleModified Nonradial Supper Efficiency Models
H Jahanshahloo1 F Hosseinzadeh Lotfi1 and M Khodabakhshi2
1 Department of Mathematics Science and Research Branch Islamic Azad University Tehran Iran2Department of Mathematics Faculty of Mathematical Sciences Shahid Beheshti University GC Tehran Iran
Correspondence should be addressed to H Jahanshahloo hodajahanshahloocom
Received 25 November 2013 Accepted 28 December 2013 Published 17 February 2014
Academic Editor S H Nasseri
Copyright copy 2014 H Jahanshahloo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Ranking Efficient Decision Making Units (DMUs) are an important issue in Data Envelopment Analysis (DEA) This is one ofthe main areas for the researcher Different methods for this purpose have been suggested Appearing nonzero slack in optimalsolution makes the method problematic In this paper we modify the nonradial supper efficiency model to remove this difficultySome numerical examples are solved by modified model
1 Introduction
Data Envelopment Analysis is a mathematical programmingtechnique which evaluates the relative efficiency of DMUsDEA classifies the DMUs into two different classes called setof efficient DMUs and the set of inefficient DMUs
Conventional DEA model cannot differentiate the effi-cient DMUs whose efficiency value is one Toward thisend different methods are suggested see [1ndash3] One ofthe important models is AP-Model which was proposed byAndersen and Petersen [4] and also see [5 6]
This model is widely used and the results are almostsatisfactory The main deficiencies of AP model are being asfollows
(1) infeasible for some kind of data(2) unstable in the sense that a small variation in data
causes big increase (degrease) in the result(3) not taking into account the nonzero slacks which
appear in optional solution (projection of omittedDMU is weak efficient in new PPS)
For removing these difficulties many researches have sug-gested different models for example see [7ndash9]
Tone [7] suggested the nonradial supper efficiencymodelThismodel fails to remove the 3rd deficiency In this paper wemodify the nonradial supper efficiency model that takes intoaccount nonzero slack that appears in optimal solution
The rest of the paper is organized as follows In Sections 2and 3 AP-model and Tonersquos model are discussed Section 4contains the modified model In Sections 5 and 6 inputand output oriented models are proposed Discussion andconclusion cover Section 7
2 Anderson Peterson (AP) Model
Consider 119899 decision making units DMU119895 (119895 = 1 119899)
which consumes 0 = 119909119895 ge 0 and 119909119895 isin 119877119898
(119895 = 1 119899)
vector as input to produce output vector 0 = 119910119895 ge 0 and119910119895 isin 119877
119904(119895 = 1 119899) The supper efficiency model may be
written as follows
120579lowast119901 = Min 120579
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 120579119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903119901 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
119904minus119894 ge 0 119894 = 1 119898
119904+119903 ge 0 119903 = 1 119904
(1)
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 919547 5 pageshttpdxdoiorg1011552014919547
2 Journal of Applied Mathematics
A
B
C
D
EF
Figure 1
The following example which has been taken from [7]shows the deficiency in case of having nonzero slacks inoptimal solution
Example 1 Consider the data given by Table 1
By using AP-Model
Sup 119860 = 120579lowast119860 = 1 119878
minuslowast2 = 4
Sup 119861 = 120579lowast119861 = 1260 All slacks are zero
Sup 119862 = 120579lowast119862 = 1133 All slacks are zero
Sup 119863 = 120579lowast119863 = 1250 119878
minuslowast1 = 75
Sup 119864 = 120579lowast119864 = 0750
Sup 119865 = 120579lowast119865 = 1
(2)
In other words the supper efficiencies are as follows
120579lowast119860 = 1 minus 4120576 (lowast)
120579lowast119861 = 1260
120579lowast119862 = 1133
120579lowast119863 = 125 minus 75120576 (lowast)
120579lowast119864 = 0750
120579lowast119865 = 1 (lowast)
(3)
Note These values are not correct in [7] because the resultshown in the paper has taken from original paper in which119865 is not included From Figure 1 it can be seen that the radialsupper Efficiency model is not able to rank 119865 which is non-extreme efficient and also is not able to rank 119863 and 119860 withnonzero slack in optimal solutions
Table 1
DMU 119860 119861 119862 119863 119864 119865
Input 1 2 2 5 10 10 35Input 2 12 8 5 4 6 65Output 1 1 1 1 1 1 1
Table 2
DMU 119860 119861 119862 119863 119864 119865 119866 119867
Input 1 4 7 8 4 2 10 12 10Input 2 3 3 1 2 4 1 1 15Output 1 1 1 1 1 1 1 1 1
3 Nonradial Supper EfficiencyModel (Tonersquos Model)
In 2002 Tone proposed the following nonradial supperefficiency model
120588lowast119901 = Min
(1119898)sum119898119894=1 (119909119894119909119894119901)
(1119904)sum119904119903=1 (119910119903119910119903119901)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 le 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 ge 119910119903 119903 = 1 119904
119909119894 ge 119909119894119901 119894 = 1 119898
119910119903 le 119910119903119901 119903 = 1 119904
119910119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(4)
Based on the SBMmodel (4) the nonradial supper efficiencymodel should be as follows
120574lowast119901 = Min
1 + (1119898)sum119898119894=1 (119904+119894 119909119894119901)
1 minus (1119904)sum119904119903=1 (119904minus119903 119910119903119901)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
119904+119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
(5)
We will show that (4) and (5) are not equivalent in the sensethat their optimal solutions are different
Example 2 Consider the eight Decision Making Units(DMU) with two inputs and one output (see Table 2)
Journal of Applied Mathematics 3
A B
C
D
E
F G
H
K
RR998400
K998400E
998400
Figure 2
First the following are defined
119875119875119878 =
(119909 119910) | 119909 ge
119899
sum
119895=1
120582119895119909119895
119910 le
119899
sum
119895=1
120582119895119910119895 120582119895 ge 0 119895 = 1 119899
119875119875119878 =
(119909 119910) | 119909 ge
119899
sum
119895=1119895 = 119901
120582119895119909119895
119910 le
119899
sum
119895=1119895 = 119901
120582119895119910119895 120582119895 ge 0 119895 = 1 119899
119875119875119878 = 119875119875119878 cap (119909 119910) 119909 ge 119909119901 119910 le 119910119901
(6)
In above-mentioned example and shown in Figure 2
119875119875119878 = 119877119864119863119862119870 = 119878
119875119875119878 = 11987710158401198641015840119863119862119870 = 119878
119877101584011986410158401198701015840= 119878
(7)
We can see that 119878 is a proper subset of 119878 and 119878 is a propersubset of 119878 that is
119878 sub 119878 sub 119878 (8)
In this example the optimum value of objective function in(4) is
120588lowast119864 =
1
2
(
4
2
+
4
4
) = 15 (9)
and the optimal value of objective function in (5) is
120574lowast
119878=
1
2
(
4
2
+
2
4
) =
1
2
(2 +
1
2
) =
5
4
= 125 (10)
so (4) and (5) are not equivalent In other words theconstraints 119909 ge 119909119901 and 119910 le 119910119901 and 119910 ge 0 should be omittedfrom (4) Now we show that the nonradial supper efficiencymodel has the same difficulty as radial supper efficiencymodel in treating nonzero slacks in optimal solution Using
(5) the supper efficiency of 119864 119863 119862 may be evaluated asfollows
120574lowast119864 = Min (1 +
1
2
2
sum
119894=1
119904+119894
119909119894119864
)
st 4120582119860 + 7120582119861 + 8120582119862 + 4120582119863 + 10120582119865 + 12120582119866
+ 10120582119867 minus 119904+1 = 2
3120582119860 + 3120582119861 + 120582119862 + 2120582119863 + 120582119865 + 120582119866 + 15120582119867
minus 119904+2 = 4
120582119860 + 120582119861 + 120582119862 + 120582119863 + 120582119865 + 120582119866 + 120582119867 + 119904minus= 1
120582119860 120582119861 120582119862 120582119863 120582119865 120582119866 120582119867 119904+1 119904+2 119904minusge 0
120574lowast119864 = 1 +
1
2
(
2
2
+
0
4
) = 15
(11)
The projection of 119864 is on weak frontier
120574lowast119863 = Min (1 +
1
2
2
sum
119894=1
119904+119894
119909119894119863
)
st 4120582119860 + 7120582119861 + 8120582119862 + 2120582119864 + 10120582119865 + 12120582119866
+ 10120582119867 minus 119904+1 = 2
3120582119860 + 3120582119861 + 120582119862 + 4120582119864 + 120582119865 + 120582119866 + 15120582119867
minus 119904+2 = 4
120582119860 + 120582119861 + 120582119862 + 120582119864 + 120582119865 + 120582119866 + 120582119867 + 119904minus= 1
120582119860 120582119861 120582119862 120582119864 120582119865 120582119866 120582119867 119904+1 119904+2 119904minusge 0
120574lowast119863 = 1 +
1
2
(
2
4
+
0
3
) = 125
(12)
The projection is on strong frontier
120574lowast119862 = Min (1 +
1
2
2
sum
119894=1
119904+119894
119909119894119862
)
st 4120582119860 + 7120582119861 + 4120582119863 + 2120582119864 + 10120582119865
+ 12120582119866 + 10120582119867 minus 119904+1 = 8
3120582119860 + 3120582119861 + 2120582119863 + 4120582119864 + 120582119865
+ 120582119866 + 15120582119867 minus 119904+2 = 1
120582119860 + 120582119861 + 120582119863 + 120582119864 + 120582119865 + 120582119866 + 120582119867 + 119904minus= 1
120582119860 120582119861 120582119863 120582119864 120582119865 120582119866 120582119867 119904+1 119904+2 119904minusge 0
120574lowast119862 = 1 +
1
2
(
2
8
) = 1125
(13)
The projection is on strong frontierIn summery
120574lowast119862 = 1125
120574lowast119863 = 125
120574lowast119864 = 15
(14)
In the case the projection of omitted DMU119901 lied on weakfrontier nonzero slacks appear in optimal solution and thefollowing approach is suggested
4 Journal of Applied Mathematics
4 Modified Nonradial SupperEfficiency Model
First solve the following model
120574lowast119901 = Min
1 + (1119898)sum119898119894=1 (119904+119894 119909119894119901)
1 minus (1119904)sum119904119903=1 (119904minus119903 119910119903119901)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
119904+119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
(15)
The model (15) can be linearized by the suggested methodin [7] and solve by the simplex method
Suppose that
(119909 119910) = (119909119901 + 119904+lowast
119910119901 minus 119904minuslowast
) (16)
is the projection of DMU119901 on the frontier of 119875119875119878 and nowsolve the following model
119882lowast119901 = Max
(1119898)sum119898119894=1 (119904minus119894 119909119894)
(1119904)sum119904119903=1 (119904+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903 119903 = 1 119904
119904+119903 ge 0 119903 = 1 119904
119904minus119894 ge 0 119894 = 1 119898
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(17)
Set
120583lowast119901 = 120574lowast119901 minus 119882
lowast119901 (supper efficiency ofDMU119901) (18)
It is evident that if (119909 119910) is strongly efficient then 119882lowast119901 = 0
and 120583lowast119901 is supper efficiency score of DMU119901
5 Modified Input Oriented Nonradial SupperEfficiency Model
First the following model is solved
120585lowast119901 = Min (1 +
1
119898
119898
sum
119894=1
119904+119894
119909119894119901
)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+1015840119903 = 119910119903119901 119903 = 1 119904
119904+119894 ge 0 119894 = 1 119898
119904+1015840119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(19)
Let
(119909 119910) = (119909119901 + 119904+lowast
119910119901 + 119904+lowast1015840
) (20)
Now solve the following problem
119882lowast119901 = Max
(1119898)sum119898119894=1 (119905minus119894 119909119894)
(1119904)sum119904119903=1 (119905+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119905minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119905+119903 = 119910119903 119903 = 1 119904
119905minus119894 ge 0 119894 = 1 119898
119905+119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(21)
Let120583lowast119901 = 120585lowast119901 minus 119882
lowast119901 (22)
6 Modified Output Oriented NonradialSupper Efficiency Model
Consider the following model
120578lowast119901 = Min (
1
1 minus (1119904)sum119904119894=1 (119904minus119903 119910119903119901)
)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus 1015840119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
119904minus1015840119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(23)
Journal of Applied Mathematics 5
Table 3
DMU 119862 119863 119864 119865 119866
AP 114 125 200 100 100SBM 1125 125 150 100 100Modified SBM 1125 125 125 090 073
Let
DMU119901 projection = (119909 119910) = (119909119901 minus 119904minus1015840lowast
119910119901 minus 119904minuslowast
) (24)
Now solve the following problem
119882lowast119901 = Max
(1119898)sum119898119894=1 (119904minus119894 119909119894)
(1119904)sum119904119903=1 (119904+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903 119903 = 1 119904
119904minus119894 ge 0 119894 = 1 119898
119904+119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(25)
Let
120588lowast119901 = 120578lowast119901 minus 119882
lowast119901 (26)
7 Conclusion
In this paper it has been shown that bothAP-Model and non-radial supper efficiency model are not able to rank DMUs ifthe projection of omitted DMU is weak efficient in 119875119875119878 Thenew method removes this difficulties and the example whichis solved by using the new method (modified method) con-firms the validity
The results for comparing the methods are shown inTable 3
Conflict of Interests
The authors declare that there is no conflict of interests regar-ding the publication of this paper
References
[1] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
[2] MKhodabakhshiMAsgharian andGNGregoriou ldquoAn inp-ut-oriented super-efficiency measure in stochastic data envel-opment analysis evaluating chief executive officers of US publicbanks and thriftsrdquo Expert Systems with Applications vol 37 no3 pp 2092ndash2097 2010
[3] G R Jahanshahloo andM Khodabakhshi ldquoUsing input-outputorientationmodel for determiningmost productive scale size inDEArdquo Applied Mathematics and Computation vol 146 no 2-3pp 849ndash855 2003
[4] P Andersen andN C Petersen ldquoA produce for ranking efficientunits in data envelopment analysisrdquoManagment Science vol 39pp 1261ndash1264 1993
[5] M Khodabakhshi ldquoA super-efficiency model based on impro-ved outputs in data envelopment analysisrdquoAppliedMathematicsand Computation vol 184 no 2 pp 695ndash703 2007
[6] M Khodabakhshi ldquoAn output oriented super-efficiency mea-sure in stochastic data envelopment analysis considering Ira-nian electricity distribution companiesrdquo Computers amp Indus-trial Engineering vol 58 no 4 pp 663ndash671 2010
[7] K Tone ldquoA slacks-based measure of super-efficiency in dataenvelopment analysisrdquo European Journal of Operational Rese-arch vol 143 no 1 pp 32ndash41 2002
[8] M Khodabakhshi ldquoSuper-efficiency in stochastic data envel-opment analysis an input relaxation approachrdquo Journal ofComputational and Applied Mathematics vol 235 no 16 pp4576ndash4588 2011
[9] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabian andF Nuri-Bahmani ldquoOn ranking decision making units with-common weights in DEArdquo Applied Mathematical Modelling Inpress
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
A
B
C
D
EF
Figure 1
The following example which has been taken from [7]shows the deficiency in case of having nonzero slacks inoptimal solution
Example 1 Consider the data given by Table 1
By using AP-Model
Sup 119860 = 120579lowast119860 = 1 119878
minuslowast2 = 4
Sup 119861 = 120579lowast119861 = 1260 All slacks are zero
Sup 119862 = 120579lowast119862 = 1133 All slacks are zero
Sup 119863 = 120579lowast119863 = 1250 119878
minuslowast1 = 75
Sup 119864 = 120579lowast119864 = 0750
Sup 119865 = 120579lowast119865 = 1
(2)
In other words the supper efficiencies are as follows
120579lowast119860 = 1 minus 4120576 (lowast)
120579lowast119861 = 1260
120579lowast119862 = 1133
120579lowast119863 = 125 minus 75120576 (lowast)
120579lowast119864 = 0750
120579lowast119865 = 1 (lowast)
(3)
Note These values are not correct in [7] because the resultshown in the paper has taken from original paper in which119865 is not included From Figure 1 it can be seen that the radialsupper Efficiency model is not able to rank 119865 which is non-extreme efficient and also is not able to rank 119863 and 119860 withnonzero slack in optimal solutions
Table 1
DMU 119860 119861 119862 119863 119864 119865
Input 1 2 2 5 10 10 35Input 2 12 8 5 4 6 65Output 1 1 1 1 1 1 1
Table 2
DMU 119860 119861 119862 119863 119864 119865 119866 119867
Input 1 4 7 8 4 2 10 12 10Input 2 3 3 1 2 4 1 1 15Output 1 1 1 1 1 1 1 1 1
3 Nonradial Supper EfficiencyModel (Tonersquos Model)
In 2002 Tone proposed the following nonradial supperefficiency model
120588lowast119901 = Min
(1119898)sum119898119894=1 (119909119894119909119894119901)
(1119904)sum119904119903=1 (119910119903119910119903119901)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 le 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 ge 119910119903 119903 = 1 119904
119909119894 ge 119909119894119901 119894 = 1 119898
119910119903 le 119910119903119901 119903 = 1 119904
119910119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(4)
Based on the SBMmodel (4) the nonradial supper efficiencymodel should be as follows
120574lowast119901 = Min
1 + (1119898)sum119898119894=1 (119904+119894 119909119894119901)
1 minus (1119904)sum119904119903=1 (119904minus119903 119910119903119901)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
119904+119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
(5)
We will show that (4) and (5) are not equivalent in the sensethat their optimal solutions are different
Example 2 Consider the eight Decision Making Units(DMU) with two inputs and one output (see Table 2)
Journal of Applied Mathematics 3
A B
C
D
E
F G
H
K
RR998400
K998400E
998400
Figure 2
First the following are defined
119875119875119878 =
(119909 119910) | 119909 ge
119899
sum
119895=1
120582119895119909119895
119910 le
119899
sum
119895=1
120582119895119910119895 120582119895 ge 0 119895 = 1 119899
119875119875119878 =
(119909 119910) | 119909 ge
119899
sum
119895=1119895 = 119901
120582119895119909119895
119910 le
119899
sum
119895=1119895 = 119901
120582119895119910119895 120582119895 ge 0 119895 = 1 119899
119875119875119878 = 119875119875119878 cap (119909 119910) 119909 ge 119909119901 119910 le 119910119901
(6)
In above-mentioned example and shown in Figure 2
119875119875119878 = 119877119864119863119862119870 = 119878
119875119875119878 = 11987710158401198641015840119863119862119870 = 119878
119877101584011986410158401198701015840= 119878
(7)
We can see that 119878 is a proper subset of 119878 and 119878 is a propersubset of 119878 that is
119878 sub 119878 sub 119878 (8)
In this example the optimum value of objective function in(4) is
120588lowast119864 =
1
2
(
4
2
+
4
4
) = 15 (9)
and the optimal value of objective function in (5) is
120574lowast
119878=
1
2
(
4
2
+
2
4
) =
1
2
(2 +
1
2
) =
5
4
= 125 (10)
so (4) and (5) are not equivalent In other words theconstraints 119909 ge 119909119901 and 119910 le 119910119901 and 119910 ge 0 should be omittedfrom (4) Now we show that the nonradial supper efficiencymodel has the same difficulty as radial supper efficiencymodel in treating nonzero slacks in optimal solution Using
(5) the supper efficiency of 119864 119863 119862 may be evaluated asfollows
120574lowast119864 = Min (1 +
1
2
2
sum
119894=1
119904+119894
119909119894119864
)
st 4120582119860 + 7120582119861 + 8120582119862 + 4120582119863 + 10120582119865 + 12120582119866
+ 10120582119867 minus 119904+1 = 2
3120582119860 + 3120582119861 + 120582119862 + 2120582119863 + 120582119865 + 120582119866 + 15120582119867
minus 119904+2 = 4
120582119860 + 120582119861 + 120582119862 + 120582119863 + 120582119865 + 120582119866 + 120582119867 + 119904minus= 1
120582119860 120582119861 120582119862 120582119863 120582119865 120582119866 120582119867 119904+1 119904+2 119904minusge 0
120574lowast119864 = 1 +
1
2
(
2
2
+
0
4
) = 15
(11)
The projection of 119864 is on weak frontier
120574lowast119863 = Min (1 +
1
2
2
sum
119894=1
119904+119894
119909119894119863
)
st 4120582119860 + 7120582119861 + 8120582119862 + 2120582119864 + 10120582119865 + 12120582119866
+ 10120582119867 minus 119904+1 = 2
3120582119860 + 3120582119861 + 120582119862 + 4120582119864 + 120582119865 + 120582119866 + 15120582119867
minus 119904+2 = 4
120582119860 + 120582119861 + 120582119862 + 120582119864 + 120582119865 + 120582119866 + 120582119867 + 119904minus= 1
120582119860 120582119861 120582119862 120582119864 120582119865 120582119866 120582119867 119904+1 119904+2 119904minusge 0
120574lowast119863 = 1 +
1
2
(
2
4
+
0
3
) = 125
(12)
The projection is on strong frontier
120574lowast119862 = Min (1 +
1
2
2
sum
119894=1
119904+119894
119909119894119862
)
st 4120582119860 + 7120582119861 + 4120582119863 + 2120582119864 + 10120582119865
+ 12120582119866 + 10120582119867 minus 119904+1 = 8
3120582119860 + 3120582119861 + 2120582119863 + 4120582119864 + 120582119865
+ 120582119866 + 15120582119867 minus 119904+2 = 1
120582119860 + 120582119861 + 120582119863 + 120582119864 + 120582119865 + 120582119866 + 120582119867 + 119904minus= 1
120582119860 120582119861 120582119863 120582119864 120582119865 120582119866 120582119867 119904+1 119904+2 119904minusge 0
120574lowast119862 = 1 +
1
2
(
2
8
) = 1125
(13)
The projection is on strong frontierIn summery
120574lowast119862 = 1125
120574lowast119863 = 125
120574lowast119864 = 15
(14)
In the case the projection of omitted DMU119901 lied on weakfrontier nonzero slacks appear in optimal solution and thefollowing approach is suggested
4 Journal of Applied Mathematics
4 Modified Nonradial SupperEfficiency Model
First solve the following model
120574lowast119901 = Min
1 + (1119898)sum119898119894=1 (119904+119894 119909119894119901)
1 minus (1119904)sum119904119903=1 (119904minus119903 119910119903119901)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
119904+119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
(15)
The model (15) can be linearized by the suggested methodin [7] and solve by the simplex method
Suppose that
(119909 119910) = (119909119901 + 119904+lowast
119910119901 minus 119904minuslowast
) (16)
is the projection of DMU119901 on the frontier of 119875119875119878 and nowsolve the following model
119882lowast119901 = Max
(1119898)sum119898119894=1 (119904minus119894 119909119894)
(1119904)sum119904119903=1 (119904+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903 119903 = 1 119904
119904+119903 ge 0 119903 = 1 119904
119904minus119894 ge 0 119894 = 1 119898
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(17)
Set
120583lowast119901 = 120574lowast119901 minus 119882
lowast119901 (supper efficiency ofDMU119901) (18)
It is evident that if (119909 119910) is strongly efficient then 119882lowast119901 = 0
and 120583lowast119901 is supper efficiency score of DMU119901
5 Modified Input Oriented Nonradial SupperEfficiency Model
First the following model is solved
120585lowast119901 = Min (1 +
1
119898
119898
sum
119894=1
119904+119894
119909119894119901
)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+1015840119903 = 119910119903119901 119903 = 1 119904
119904+119894 ge 0 119894 = 1 119898
119904+1015840119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(19)
Let
(119909 119910) = (119909119901 + 119904+lowast
119910119901 + 119904+lowast1015840
) (20)
Now solve the following problem
119882lowast119901 = Max
(1119898)sum119898119894=1 (119905minus119894 119909119894)
(1119904)sum119904119903=1 (119905+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119905minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119905+119903 = 119910119903 119903 = 1 119904
119905minus119894 ge 0 119894 = 1 119898
119905+119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(21)
Let120583lowast119901 = 120585lowast119901 minus 119882
lowast119901 (22)
6 Modified Output Oriented NonradialSupper Efficiency Model
Consider the following model
120578lowast119901 = Min (
1
1 minus (1119904)sum119904119894=1 (119904minus119903 119910119903119901)
)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus 1015840119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
119904minus1015840119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(23)
Journal of Applied Mathematics 5
Table 3
DMU 119862 119863 119864 119865 119866
AP 114 125 200 100 100SBM 1125 125 150 100 100Modified SBM 1125 125 125 090 073
Let
DMU119901 projection = (119909 119910) = (119909119901 minus 119904minus1015840lowast
119910119901 minus 119904minuslowast
) (24)
Now solve the following problem
119882lowast119901 = Max
(1119898)sum119898119894=1 (119904minus119894 119909119894)
(1119904)sum119904119903=1 (119904+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903 119903 = 1 119904
119904minus119894 ge 0 119894 = 1 119898
119904+119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(25)
Let
120588lowast119901 = 120578lowast119901 minus 119882
lowast119901 (26)
7 Conclusion
In this paper it has been shown that bothAP-Model and non-radial supper efficiency model are not able to rank DMUs ifthe projection of omitted DMU is weak efficient in 119875119875119878 Thenew method removes this difficulties and the example whichis solved by using the new method (modified method) con-firms the validity
The results for comparing the methods are shown inTable 3
Conflict of Interests
The authors declare that there is no conflict of interests regar-ding the publication of this paper
References
[1] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
[2] MKhodabakhshiMAsgharian andGNGregoriou ldquoAn inp-ut-oriented super-efficiency measure in stochastic data envel-opment analysis evaluating chief executive officers of US publicbanks and thriftsrdquo Expert Systems with Applications vol 37 no3 pp 2092ndash2097 2010
[3] G R Jahanshahloo andM Khodabakhshi ldquoUsing input-outputorientationmodel for determiningmost productive scale size inDEArdquo Applied Mathematics and Computation vol 146 no 2-3pp 849ndash855 2003
[4] P Andersen andN C Petersen ldquoA produce for ranking efficientunits in data envelopment analysisrdquoManagment Science vol 39pp 1261ndash1264 1993
[5] M Khodabakhshi ldquoA super-efficiency model based on impro-ved outputs in data envelopment analysisrdquoAppliedMathematicsand Computation vol 184 no 2 pp 695ndash703 2007
[6] M Khodabakhshi ldquoAn output oriented super-efficiency mea-sure in stochastic data envelopment analysis considering Ira-nian electricity distribution companiesrdquo Computers amp Indus-trial Engineering vol 58 no 4 pp 663ndash671 2010
[7] K Tone ldquoA slacks-based measure of super-efficiency in dataenvelopment analysisrdquo European Journal of Operational Rese-arch vol 143 no 1 pp 32ndash41 2002
[8] M Khodabakhshi ldquoSuper-efficiency in stochastic data envel-opment analysis an input relaxation approachrdquo Journal ofComputational and Applied Mathematics vol 235 no 16 pp4576ndash4588 2011
[9] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabian andF Nuri-Bahmani ldquoOn ranking decision making units with-common weights in DEArdquo Applied Mathematical Modelling Inpress
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
A B
C
D
E
F G
H
K
RR998400
K998400E
998400
Figure 2
First the following are defined
119875119875119878 =
(119909 119910) | 119909 ge
119899
sum
119895=1
120582119895119909119895
119910 le
119899
sum
119895=1
120582119895119910119895 120582119895 ge 0 119895 = 1 119899
119875119875119878 =
(119909 119910) | 119909 ge
119899
sum
119895=1119895 = 119901
120582119895119909119895
119910 le
119899
sum
119895=1119895 = 119901
120582119895119910119895 120582119895 ge 0 119895 = 1 119899
119875119875119878 = 119875119875119878 cap (119909 119910) 119909 ge 119909119901 119910 le 119910119901
(6)
In above-mentioned example and shown in Figure 2
119875119875119878 = 119877119864119863119862119870 = 119878
119875119875119878 = 11987710158401198641015840119863119862119870 = 119878
119877101584011986410158401198701015840= 119878
(7)
We can see that 119878 is a proper subset of 119878 and 119878 is a propersubset of 119878 that is
119878 sub 119878 sub 119878 (8)
In this example the optimum value of objective function in(4) is
120588lowast119864 =
1
2
(
4
2
+
4
4
) = 15 (9)
and the optimal value of objective function in (5) is
120574lowast
119878=
1
2
(
4
2
+
2
4
) =
1
2
(2 +
1
2
) =
5
4
= 125 (10)
so (4) and (5) are not equivalent In other words theconstraints 119909 ge 119909119901 and 119910 le 119910119901 and 119910 ge 0 should be omittedfrom (4) Now we show that the nonradial supper efficiencymodel has the same difficulty as radial supper efficiencymodel in treating nonzero slacks in optimal solution Using
(5) the supper efficiency of 119864 119863 119862 may be evaluated asfollows
120574lowast119864 = Min (1 +
1
2
2
sum
119894=1
119904+119894
119909119894119864
)
st 4120582119860 + 7120582119861 + 8120582119862 + 4120582119863 + 10120582119865 + 12120582119866
+ 10120582119867 minus 119904+1 = 2
3120582119860 + 3120582119861 + 120582119862 + 2120582119863 + 120582119865 + 120582119866 + 15120582119867
minus 119904+2 = 4
120582119860 + 120582119861 + 120582119862 + 120582119863 + 120582119865 + 120582119866 + 120582119867 + 119904minus= 1
120582119860 120582119861 120582119862 120582119863 120582119865 120582119866 120582119867 119904+1 119904+2 119904minusge 0
120574lowast119864 = 1 +
1
2
(
2
2
+
0
4
) = 15
(11)
The projection of 119864 is on weak frontier
120574lowast119863 = Min (1 +
1
2
2
sum
119894=1
119904+119894
119909119894119863
)
st 4120582119860 + 7120582119861 + 8120582119862 + 2120582119864 + 10120582119865 + 12120582119866
+ 10120582119867 minus 119904+1 = 2
3120582119860 + 3120582119861 + 120582119862 + 4120582119864 + 120582119865 + 120582119866 + 15120582119867
minus 119904+2 = 4
120582119860 + 120582119861 + 120582119862 + 120582119864 + 120582119865 + 120582119866 + 120582119867 + 119904minus= 1
120582119860 120582119861 120582119862 120582119864 120582119865 120582119866 120582119867 119904+1 119904+2 119904minusge 0
120574lowast119863 = 1 +
1
2
(
2
4
+
0
3
) = 125
(12)
The projection is on strong frontier
120574lowast119862 = Min (1 +
1
2
2
sum
119894=1
119904+119894
119909119894119862
)
st 4120582119860 + 7120582119861 + 4120582119863 + 2120582119864 + 10120582119865
+ 12120582119866 + 10120582119867 minus 119904+1 = 8
3120582119860 + 3120582119861 + 2120582119863 + 4120582119864 + 120582119865
+ 120582119866 + 15120582119867 minus 119904+2 = 1
120582119860 + 120582119861 + 120582119863 + 120582119864 + 120582119865 + 120582119866 + 120582119867 + 119904minus= 1
120582119860 120582119861 120582119863 120582119864 120582119865 120582119866 120582119867 119904+1 119904+2 119904minusge 0
120574lowast119862 = 1 +
1
2
(
2
8
) = 1125
(13)
The projection is on strong frontierIn summery
120574lowast119862 = 1125
120574lowast119863 = 125
120574lowast119864 = 15
(14)
In the case the projection of omitted DMU119901 lied on weakfrontier nonzero slacks appear in optimal solution and thefollowing approach is suggested
4 Journal of Applied Mathematics
4 Modified Nonradial SupperEfficiency Model
First solve the following model
120574lowast119901 = Min
1 + (1119898)sum119898119894=1 (119904+119894 119909119894119901)
1 minus (1119904)sum119904119903=1 (119904minus119903 119910119903119901)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
119904+119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
(15)
The model (15) can be linearized by the suggested methodin [7] and solve by the simplex method
Suppose that
(119909 119910) = (119909119901 + 119904+lowast
119910119901 minus 119904minuslowast
) (16)
is the projection of DMU119901 on the frontier of 119875119875119878 and nowsolve the following model
119882lowast119901 = Max
(1119898)sum119898119894=1 (119904minus119894 119909119894)
(1119904)sum119904119903=1 (119904+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903 119903 = 1 119904
119904+119903 ge 0 119903 = 1 119904
119904minus119894 ge 0 119894 = 1 119898
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(17)
Set
120583lowast119901 = 120574lowast119901 minus 119882
lowast119901 (supper efficiency ofDMU119901) (18)
It is evident that if (119909 119910) is strongly efficient then 119882lowast119901 = 0
and 120583lowast119901 is supper efficiency score of DMU119901
5 Modified Input Oriented Nonradial SupperEfficiency Model
First the following model is solved
120585lowast119901 = Min (1 +
1
119898
119898
sum
119894=1
119904+119894
119909119894119901
)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+1015840119903 = 119910119903119901 119903 = 1 119904
119904+119894 ge 0 119894 = 1 119898
119904+1015840119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(19)
Let
(119909 119910) = (119909119901 + 119904+lowast
119910119901 + 119904+lowast1015840
) (20)
Now solve the following problem
119882lowast119901 = Max
(1119898)sum119898119894=1 (119905minus119894 119909119894)
(1119904)sum119904119903=1 (119905+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119905minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119905+119903 = 119910119903 119903 = 1 119904
119905minus119894 ge 0 119894 = 1 119898
119905+119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(21)
Let120583lowast119901 = 120585lowast119901 minus 119882
lowast119901 (22)
6 Modified Output Oriented NonradialSupper Efficiency Model
Consider the following model
120578lowast119901 = Min (
1
1 minus (1119904)sum119904119894=1 (119904minus119903 119910119903119901)
)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus 1015840119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
119904minus1015840119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(23)
Journal of Applied Mathematics 5
Table 3
DMU 119862 119863 119864 119865 119866
AP 114 125 200 100 100SBM 1125 125 150 100 100Modified SBM 1125 125 125 090 073
Let
DMU119901 projection = (119909 119910) = (119909119901 minus 119904minus1015840lowast
119910119901 minus 119904minuslowast
) (24)
Now solve the following problem
119882lowast119901 = Max
(1119898)sum119898119894=1 (119904minus119894 119909119894)
(1119904)sum119904119903=1 (119904+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903 119903 = 1 119904
119904minus119894 ge 0 119894 = 1 119898
119904+119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(25)
Let
120588lowast119901 = 120578lowast119901 minus 119882
lowast119901 (26)
7 Conclusion
In this paper it has been shown that bothAP-Model and non-radial supper efficiency model are not able to rank DMUs ifthe projection of omitted DMU is weak efficient in 119875119875119878 Thenew method removes this difficulties and the example whichis solved by using the new method (modified method) con-firms the validity
The results for comparing the methods are shown inTable 3
Conflict of Interests
The authors declare that there is no conflict of interests regar-ding the publication of this paper
References
[1] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
[2] MKhodabakhshiMAsgharian andGNGregoriou ldquoAn inp-ut-oriented super-efficiency measure in stochastic data envel-opment analysis evaluating chief executive officers of US publicbanks and thriftsrdquo Expert Systems with Applications vol 37 no3 pp 2092ndash2097 2010
[3] G R Jahanshahloo andM Khodabakhshi ldquoUsing input-outputorientationmodel for determiningmost productive scale size inDEArdquo Applied Mathematics and Computation vol 146 no 2-3pp 849ndash855 2003
[4] P Andersen andN C Petersen ldquoA produce for ranking efficientunits in data envelopment analysisrdquoManagment Science vol 39pp 1261ndash1264 1993
[5] M Khodabakhshi ldquoA super-efficiency model based on impro-ved outputs in data envelopment analysisrdquoAppliedMathematicsand Computation vol 184 no 2 pp 695ndash703 2007
[6] M Khodabakhshi ldquoAn output oriented super-efficiency mea-sure in stochastic data envelopment analysis considering Ira-nian electricity distribution companiesrdquo Computers amp Indus-trial Engineering vol 58 no 4 pp 663ndash671 2010
[7] K Tone ldquoA slacks-based measure of super-efficiency in dataenvelopment analysisrdquo European Journal of Operational Rese-arch vol 143 no 1 pp 32ndash41 2002
[8] M Khodabakhshi ldquoSuper-efficiency in stochastic data envel-opment analysis an input relaxation approachrdquo Journal ofComputational and Applied Mathematics vol 235 no 16 pp4576ndash4588 2011
[9] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabian andF Nuri-Bahmani ldquoOn ranking decision making units with-common weights in DEArdquo Applied Mathematical Modelling Inpress
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
4 Modified Nonradial SupperEfficiency Model
First solve the following model
120574lowast119901 = Min
1 + (1119898)sum119898119894=1 (119904+119894 119909119894119901)
1 minus (1119904)sum119904119903=1 (119904minus119903 119910119903119901)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
119904+119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
(15)
The model (15) can be linearized by the suggested methodin [7] and solve by the simplex method
Suppose that
(119909 119910) = (119909119901 + 119904+lowast
119910119901 minus 119904minuslowast
) (16)
is the projection of DMU119901 on the frontier of 119875119875119878 and nowsolve the following model
119882lowast119901 = Max
(1119898)sum119898119894=1 (119904minus119894 119909119894)
(1119904)sum119904119903=1 (119904+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903 119903 = 1 119904
119904+119903 ge 0 119903 = 1 119904
119904minus119894 ge 0 119894 = 1 119898
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(17)
Set
120583lowast119901 = 120574lowast119901 minus 119882
lowast119901 (supper efficiency ofDMU119901) (18)
It is evident that if (119909 119910) is strongly efficient then 119882lowast119901 = 0
and 120583lowast119901 is supper efficiency score of DMU119901
5 Modified Input Oriented Nonradial SupperEfficiency Model
First the following model is solved
120585lowast119901 = Min (1 +
1
119898
119898
sum
119894=1
119904+119894
119909119894119901
)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 minus 119904+119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+1015840119903 = 119910119903119901 119903 = 1 119904
119904+119894 ge 0 119894 = 1 119898
119904+1015840119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(19)
Let
(119909 119910) = (119909119901 + 119904+lowast
119910119901 + 119904+lowast1015840
) (20)
Now solve the following problem
119882lowast119901 = Max
(1119898)sum119898119894=1 (119905minus119894 119909119894)
(1119904)sum119904119903=1 (119905+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119905minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119905+119903 = 119910119903 119903 = 1 119904
119905minus119894 ge 0 119894 = 1 119898
119905+119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(21)
Let120583lowast119901 = 120585lowast119901 minus 119882
lowast119901 (22)
6 Modified Output Oriented NonradialSupper Efficiency Model
Consider the following model
120578lowast119901 = Min (
1
1 minus (1119904)sum119904119894=1 (119904minus119903 119910119903119901)
)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus 1015840119894 = 119909119894119901 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 + 119904minus119903 = 119910119903119901 119903 = 1 119904
119904minus1015840119894 ge 0 119894 = 1 119898
119904minus119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(23)
Journal of Applied Mathematics 5
Table 3
DMU 119862 119863 119864 119865 119866
AP 114 125 200 100 100SBM 1125 125 150 100 100Modified SBM 1125 125 125 090 073
Let
DMU119901 projection = (119909 119910) = (119909119901 minus 119904minus1015840lowast
119910119901 minus 119904minuslowast
) (24)
Now solve the following problem
119882lowast119901 = Max
(1119898)sum119898119894=1 (119904minus119894 119909119894)
(1119904)sum119904119903=1 (119904+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903 119903 = 1 119904
119904minus119894 ge 0 119894 = 1 119898
119904+119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(25)
Let
120588lowast119901 = 120578lowast119901 minus 119882
lowast119901 (26)
7 Conclusion
In this paper it has been shown that bothAP-Model and non-radial supper efficiency model are not able to rank DMUs ifthe projection of omitted DMU is weak efficient in 119875119875119878 Thenew method removes this difficulties and the example whichis solved by using the new method (modified method) con-firms the validity
The results for comparing the methods are shown inTable 3
Conflict of Interests
The authors declare that there is no conflict of interests regar-ding the publication of this paper
References
[1] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
[2] MKhodabakhshiMAsgharian andGNGregoriou ldquoAn inp-ut-oriented super-efficiency measure in stochastic data envel-opment analysis evaluating chief executive officers of US publicbanks and thriftsrdquo Expert Systems with Applications vol 37 no3 pp 2092ndash2097 2010
[3] G R Jahanshahloo andM Khodabakhshi ldquoUsing input-outputorientationmodel for determiningmost productive scale size inDEArdquo Applied Mathematics and Computation vol 146 no 2-3pp 849ndash855 2003
[4] P Andersen andN C Petersen ldquoA produce for ranking efficientunits in data envelopment analysisrdquoManagment Science vol 39pp 1261ndash1264 1993
[5] M Khodabakhshi ldquoA super-efficiency model based on impro-ved outputs in data envelopment analysisrdquoAppliedMathematicsand Computation vol 184 no 2 pp 695ndash703 2007
[6] M Khodabakhshi ldquoAn output oriented super-efficiency mea-sure in stochastic data envelopment analysis considering Ira-nian electricity distribution companiesrdquo Computers amp Indus-trial Engineering vol 58 no 4 pp 663ndash671 2010
[7] K Tone ldquoA slacks-based measure of super-efficiency in dataenvelopment analysisrdquo European Journal of Operational Rese-arch vol 143 no 1 pp 32ndash41 2002
[8] M Khodabakhshi ldquoSuper-efficiency in stochastic data envel-opment analysis an input relaxation approachrdquo Journal ofComputational and Applied Mathematics vol 235 no 16 pp4576ndash4588 2011
[9] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabian andF Nuri-Bahmani ldquoOn ranking decision making units with-common weights in DEArdquo Applied Mathematical Modelling Inpress
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Table 3
DMU 119862 119863 119864 119865 119866
AP 114 125 200 100 100SBM 1125 125 150 100 100Modified SBM 1125 125 125 090 073
Let
DMU119901 projection = (119909 119910) = (119909119901 minus 119904minus1015840lowast
119910119901 minus 119904minuslowast
) (24)
Now solve the following problem
119882lowast119901 = Max
(1119898)sum119898119894=1 (119904minus119894 119909119894)
(1119904)sum119904119903=1 (119904+119903 119910119903)
st119899
sum
119895=1119895 = 119901
120582119895119909119894119895 + 119904minus119894 = 119909119894 119894 = 1 119898
119899
sum
119895=1119895 = 119901
120582119895119910119903119895 minus 119904+119903 = 119910119903 119903 = 1 119904
119904minus119894 ge 0 119894 = 1 119898
119904+119903 ge 0 119903 = 1 119904
120582119895 ge 0 119895 = 1 119899 119895 = 119901
(25)
Let
120588lowast119901 = 120578lowast119901 minus 119882
lowast119901 (26)
7 Conclusion
In this paper it has been shown that bothAP-Model and non-radial supper efficiency model are not able to rank DMUs ifthe projection of omitted DMU is weak efficient in 119875119875119878 Thenew method removes this difficulties and the example whichis solved by using the new method (modified method) con-firms the validity
The results for comparing the methods are shown inTable 3
Conflict of Interests
The authors declare that there is no conflict of interests regar-ding the publication of this paper
References
[1] M Khodabakhshi and K Aryavash ldquoRanking all units in dataenvelopment analysisrdquo Applied Mathematics Letters vol 25 no12 pp 2066ndash2070 2012
[2] MKhodabakhshiMAsgharian andGNGregoriou ldquoAn inp-ut-oriented super-efficiency measure in stochastic data envel-opment analysis evaluating chief executive officers of US publicbanks and thriftsrdquo Expert Systems with Applications vol 37 no3 pp 2092ndash2097 2010
[3] G R Jahanshahloo andM Khodabakhshi ldquoUsing input-outputorientationmodel for determiningmost productive scale size inDEArdquo Applied Mathematics and Computation vol 146 no 2-3pp 849ndash855 2003
[4] P Andersen andN C Petersen ldquoA produce for ranking efficientunits in data envelopment analysisrdquoManagment Science vol 39pp 1261ndash1264 1993
[5] M Khodabakhshi ldquoA super-efficiency model based on impro-ved outputs in data envelopment analysisrdquoAppliedMathematicsand Computation vol 184 no 2 pp 695ndash703 2007
[6] M Khodabakhshi ldquoAn output oriented super-efficiency mea-sure in stochastic data envelopment analysis considering Ira-nian electricity distribution companiesrdquo Computers amp Indus-trial Engineering vol 58 no 4 pp 663ndash671 2010
[7] K Tone ldquoA slacks-based measure of super-efficiency in dataenvelopment analysisrdquo European Journal of Operational Rese-arch vol 143 no 1 pp 32ndash41 2002
[8] M Khodabakhshi ldquoSuper-efficiency in stochastic data envel-opment analysis an input relaxation approachrdquo Journal ofComputational and Applied Mathematics vol 235 no 16 pp4576ndash4588 2011
[9] S Ramezani-Tarkhorani M Khodabakhshi S Mehrabian andF Nuri-Bahmani ldquoOn ranking decision making units with-common weights in DEArdquo Applied Mathematical Modelling Inpress
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of