research article modified nonradial supper...

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 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


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 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)


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


) (24)


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


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

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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 119863 = 120579lowast119863 = 1250 119878

minuslowast1 = 75

Sup 119864 = 120579lowast119864 = 0750

Sup 119865 = 120579lowast119865 = 1

(2)

In other words the supper efficiencies are as follows


120579lowast119861 = 1260

120579lowast119862 = 1133


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)


A B

C

D

E

F G

H

K

RR998400

K998400E

998400

Figure 2


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)


119875119875119878 = 119877119864119863119862119870 = 119878

119875119875119878 = 11987710158401198641015840119863119862119870 = 119878

119877101584011986410158401198701015840= 119878

(7)


119878 sub 119878 sub 119878 (8)


120588lowast119864 =

1

2

(

4

2

+

4

4

) = 15 (9)


120574lowast

119878=

1

2

(

4

2

+

2

4

) =

1

2

(2 +

1

2

) =

5

4

= 125 (10)



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)


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)


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)


120574lowast119862 = 1125

120574lowast119863 = 125

120574lowast119864 = 15

(14)





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)


Suppose that

(119909 119910) = (119909119901 + 119904+lowast


) (16)


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







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)


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)


lowast119901 (22)



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)


Table 3

DMU 119862 119863 119864 119865 119866


Let



) (24)


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


lowast119901 (26)

7 Conclusion





References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A B

C

D

E

F G

H

K

RR998400

K998400E

998400

Figure 2


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)


119875119875119878 = 119877119864119863119862119870 = 119878

119875119875119878 = 11987710158401198641015840119863119862119870 = 119878

119877101584011986410158401198701015840= 119878

(7)


119878 sub 119878 sub 119878 (8)


120588lowast119864 =

1

2

(

4

2

+

4

4

) = 15 (9)


120574lowast

119878=

1

2

(

4

2

+

2

4

) =

1

2

(2 +

1

2

) =

5

4

= 125 (10)



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)


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)


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)


120574lowast119862 = 1125

120574lowast119863 = 125

120574lowast119864 = 15

(14)





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)


Suppose that

(119909 119910) = (119909119901 + 119904+lowast


) (16)


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







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)


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)


lowast119901 (22)



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)


Table 3

DMU 119862 119863 119864 119865 119866


Let



) (24)


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


lowast119901 (26)

7 Conclusion





References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












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)


Suppose that

(119909 119910) = (119909119901 + 119904+lowast


) (16)


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







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)


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)


lowast119901 (22)



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)


Table 3

DMU 119862 119863 119864 119865 119866


Let



) (24)


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


lowast119901 (26)

7 Conclusion





References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 3

DMU 119862 119863 119864 119865 119866


Let



) (24)


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


lowast119901 (26)

7 Conclusion





References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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