Review ArticleReview of Input Congestion Estimating Methods in DEA

Mohammad Khodabakhshi1 Farhad Hosseinzadeh Lotfi2 and Kourosh Aryavash3

1 Department of Mathematics Shahid Beheshti University Grand Campus Tehran 1983963113 Iran2 Science and Research Branch Islamic Azad University Tehran 1477893855 Iran3Department of Mathematics Faculty of Science Lorestan University Khorramabad 6815144316 Iran

Correspondence should be addressed to Mohammad Khodabakhshi mkhbakhshiyahoocom

Received 25 November 2013 Accepted 28 March 2014 Published 27 April 2014

Academic Editor S H Nasseri

Copyright copy 2014 Mohammad Khodabakhshi et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Congestion is an economic phenomenon in the production process where excessive amounts of the input cause a reduction of theoutput The motivation of this paper is to investigate and compare the methods provided for estimating congestion in the DEAliterature

1 Introduction

Data envelopment analysis (DEA) is one of the very popularoperations researchmethodswhich is designed for evaluatingthe efficiency of peer decision making units (DMUs) whenmultiple inputs and outputs are present CCR is the firstmodel of this tool which was proposed by Charnes et al [1]Then a variable returns to scale version of the CCR model(BCC) was introduced by Banker et al [2] To see the otherbasic DEA models refer to [3ndash6]

DEA can be used not only for estimating the performanceof units but also for solving other problems of decisionmakingmanagement and economicsOne of these problemsis to estimate the input congestion of DMUs If some inputsare used in amounts they cause output reduction and theninput congestion exists Such an occurrence may be found inmany economic activities For example a petroleum industryhas a difficulty in transporting oil and natural gas from itsproduction facilities to consumption areas because a pipelinenetwork among them has a capacity limit on transportationAnother example may be found in a transmission line ofelectricity that connects between generators and end usersThe transmission line has a line limit that determines notonly the amount of electricity but also the electricity price ina power trading market Also an excess of miners bumpinginto each other in an undergroundmine is an example wherea reduction in the number of miners can result in an increase

in the amount mined In what follows the exact definition ofcongestion in general case is provided

Definition 1 (input congestion) Input congestion occurswhenever the increasing of one or more inputs decreasessome outputs without improving other inputs or outputsConversely congestion occurs when decreasing of some ofthe inputs increases some outputs without worsening otherinputs or outputs

Definition 2 (weak congestion) A DMU evidences weakcongestion if some rather than all outputs increase as someinputs decrease without changing others

Definition 3 (strong congestion) A DMU evidences strongcongestion if all outputs increase as some inputs decreasewithout changing others

Notice that strong congestion implies weak congestionbut not vice versa In addition in a single input and asingle output case there is no distinction between strong andweak congestions Weak but not strong congestion occursonly for the case with more than one input or one outputThe necessary condition for the presence of congestion isinefficiency [7] So the efficiency is achieved if and only ifit is not possible to improve some inputs or outputs withoutworsening other inputs or outputs To clarify the relationship

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 963791 9 pageshttpdxdoiorg1011552014963791

2 Journal of Applied Mathematics

between input congestion and technical inefficiency weprovide the definition of technical inefficiency which is asfollows

Definition 4 (technical inefficiency) Technical inefficiency ispresent when it is possible to improve some inputs or outputswithout worsening other inputs or outputs

So far several methods formeasuring the effect of conges-tion have been developed under the framework of DEA Thecongestion estimating approaches were originated by Fareand Svensson [8] Fare and Grosskopf [9] and Fare et al[10] extended and developed a DEA model to calculate thecongestion amounts Hereafter this procedure is referred toas Farersquos approach This model is a radial approach in thatthe congestion effect is measured as the difference betweentechnologies under weak and strong disposability inputsBased on that model Byrnes et al [11 12] used examplesof coal mines to illustrate the decomposition of productionefficiency where the congestion effect is a component To seethe other references of this approach the reader is referred to[13ndash16] A big advantage of this approach is that it is possibleto decompose its measure of overall technical efficiency ina straightforward way into pure technical efficiency scaleefficiency and congestion efficiency It is the assumption thatcongestion is the difference between weak disposability andstrong disposability This approach fails to detect congestionin some cases especially when there is only one input Alsoit does not measure the amount of congestion

Cooper et al [17] published an alternative approach fordetermining input congestion that has some advantages toprevious method This method uses the slacks to determinethe congestion amount In this method the congestion effectis measured as the difference between the observed amountsand the expected amounts It was subsequently developedby Brockett et al [18] and Cooper et al [19] For simplicitythis procedure is referred to as Cooperrsquos approach Unlikethe radial approach which requires all inputs to reducein proportion to eliminate congestion this method allowseach input to reduce in different proportions This approachhas been applied to identify congesting inputs in Chineseindustries by Brockett et al [18] and Cooper et al [20]Later on Cooper et al [7] introduced a one-model versionof Cooperrsquos method To see the other references of thisapproach refer to [21ndash25]

The previous approaches express the congestion effect interms of excessive inputs According to the definition conges-tion occurswhen increases in some inputs results in decreasesin some outputs Hence congestion can also be measured asshortfalls in outputs Tone and Sahoo [26] and Wei and Yan[27] developed a method from the output point of view Thismodel hereafter is referred to as Tonersquos approach Sueyoshiand Sekitani [28] proposed amodified approachwhich is ableto measure the degree of congestion under the occurrence ofmultiple solutions This method is also a radial approach soa product form of efficiency decomposition is possible Fleggand Allen [29] applied these three approaches to examinewhether the rapid growth in the number of students in Britishuniversities has led to congestion One of the shortcomings of

Tonersquos method is its inability to identify the excessive amountof each input Also in the presence of alternative optimalsolutions this approach is incapable of detecting congestion(strong and weak) Furthermore in this approach all inputsand outputs of DMUs have been assumed positive while inreal applications data is often nonnegative Khoveyni et al[30] removed this drawback of Tonersquos approach

In DEA the inputs proportions changes are often basedon inputs reduction Apparently this subject is a reasonablejustification economically because of reducing the costs ofthe reduced inputs But in some cases inputs reduction suchas labor may be faced with social tensions as it happenedin China Jahanshahloo and Khodabakhshi [31 32] proposeda method to determine the sources and amounts of inputcongestion which is based on the relaxed combination ofinputs The framework of this method is similar to that ofCooper et al [7] Later on Khodabakhshi [33] provideda one-model approach of input congestion based on inputrelaxation model This approach reduces solving three prob-lems with the two-model approach introduced by Jahan-shahloo and Khodabakhshi [31] which is certainly importantfrom the computational point of view To see more referencesabout this approach the readers are referred to [34 35 37]Hereafter this procedure is referred to as Khodabakhshirsquosapproach Using this approach in textile industry of Chinashow that flexibility in using inputs causes output augmen-tation because of new inputs combination So it has morebenefits than the probable increasing cost of some inputsAlso Khodabakhshi [36] and Asgharian et al [38] proposedsome methods for determining the input congestion in thestochastic conditions (See also [37]) Jahanshahloo et al [39]and Khodabakhshi et al [40] proposed some methods forcomputing the congestion in DEA models with productionstrade-offs and weight restrictions

Noura et al [41] proposed another algorithm whichis equivalent to Cooperrsquos approach Their method requiresconsiderably less computation than Cooperrsquos approach Thismethod is called Nourarsquos approach

The above approaches only consider the desirable out-put But in the production process undesirable outputslike smoke pollution or waste are usually produced withdesirable outputs The desirable outputs are good outputsfor consumers and undesirable outputs are bad outputs forconsumers such as pollution Therefore congestion withundesirable outputs is different from that in the traditionalscenario Wu et al [42] combined the methods of Seifordand Zhu [43] and Wei and Yan [27] to propose a method fordetermining input congestion in the presence of undesirableoutputs which is named Wursquos approach

To see more references about input congestion measure-ment subject the readers can refer to [44ndash49]

The rest of the paper is structured as follows In Sections2ndash7 the approaches of Fare Cooper Tone KhodabakhshiNoura and Wu are briefly introduced The final sectionconcludes the paper

Journal of Applied Mathematics 3

2 Faumlrersquos Approach

Throughout this paper we deal with 119899 decision makingunits DMU119895 (119895 = 1 119899) each having 119898 inputs 119909119894119895 (119894 =

1 119898) for producing 119904 outputs 119910119903119895 (119903 = 1 119904) Theunderevaluation ofDMU is denoted byDMU119900 Also supposethat all inputs and outputs are nonnegative deterministicnumbers The efficiency score of DMU119900 under the strongdisposal technology is determined as follows

120579lowast= min 120579

st119899

sum

119895=1

119909119894119895120582119895 le 119909119894119900120579 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 119910119903119900 119903 = 1 119904

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0 119895 = 1 119899

(1)

Also the efficiency score of DMU119900 under the weakdisposal technology is determined as follows

120573lowast= min 120573

st119899

sum

119895=1

119909119894119895120582119895 = 119909119894119900120573 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 119910119903119900 119903 = 1 119904

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0 119895 = 1 119899

(2)

This measure identifies the presence of congestion pro-vided 120579

lowast120573lowastlt 1 and no congestion is present if 120579lowast120573lowast = 1

When there is only one input strong disposability and weakdisposability in input are the same therefore Farersquos approach-type congestion does not exist

3 Cooperrsquos Approach

The BCC model for measuring the efficiency has two varia-tions input and output orientation Since the input-orientedmodel may produce erroneous results in measuring conges-tion this method will concentrate on the output-orientationmodel [19] Using the output-oriented BCC model theefficiency of DMU119900 is evaluated as follows

max 120601119900 + 120576(

119898

sum

119894=1

119904minus119894 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119894 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119894 119904+119903 ge 0

(3)

Consider in model (3) that 120576 is a non-Archimedeanelement namely it is not a real number and defined to besmaller than any positive real number To avoid assigning avalue to 120576 this model can be solved via a two-stage procedure[32] In this procedure first 120601119900 is maximized while ignoringthe slacks in the objective Then 120601119900 is replaced with 120601

lowast119900

(optimal value of first stage) and the sum of the slacks ismaximized Using the optimal solution of the above modelwe have the following definitions

Definition 5 (efficiency) DMU119900 is efficient if 120601lowast119900 = 1 and119904minuslowast119894 = 119904

+lowast119903 = 0 for all 119894 119903

Inefficiency is a necessary condition for the presence ofcongestion Therefore if DMU119900 is found to be inefficient theoptimal solution of model (3) (120601lowast119900 120582

lowast119895 119904minuslowast119894 119904+lowast119903 ) is used to

determine the amount of technical inefficiency in inputs asfollows

max119898

sum

119894=1

120575minus119894

st 119909119894119900 minus 119904minuslowast119894 =

119899

sum

119895=1

119909119894119895120582119895 minus 120575minus119894 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120575minus119894 le 119904minuslowast119894 119894 = 1 119898

120575minus119894 120582119895 ge 0

(4)

Finally the optimum value of model (4) (120575+lowast119894 ) is used todetermine congestion amounts as follows

119904minus119888lowast119894 = 119904

minuslowast119894 minus 120575

minuslowast119894 119894 = 1 119898 (5)

Cooper et al [7] introduced the one-model versionof pervious method to deal with the congestion subjectSubstituting (5) into (4) we can rewrite the latter as follows

min119898

sum

119894=1

119904minus119888119894

st 119909119894119900 minus 119904minus119888119894 =

119899

sum

119895=1

119909119894119895120582119895 119894 = 1 119898

4 Journal of Applied Mathematics

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 ge 0 119894 = 1 119898

(6)

Using 120576 the models ((3) (6)) are combined into thefollowing single model

max 120601119900 + 120576(minus

119898

sum

119894=1

119904minus119888119894 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119888119894 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 119904+119903 ge 0

(7)

Theorem 6 Congestion is present if and only if in an optimalsolution 120601

lowast119900 120582lowast 119878+lowast 119878minus119888lowast of (7) at least one of the following

two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119878minus119888lowast gt 0 (1 le 119894 le 119898)

(ii) There exists at least one 119878+lowast119903 gt 0 (1 le 119903 le 119904) and atleast one 119878minus119888lowast gt 0 (1 le 119894 le 119898)

Proof See Cooper et al [7]

4 Tonersquos Approach

In the approach which was presented by Tone and Sahoo[26] it is assumed that all inputs and outputs are positiveThisapproach is based on the following production possibility set(PPS)

119875Convex

=

(119909 119910) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(8)

In this approach for measuring the congestion effectthe pure technical efficiency is firstly determined using thefollowing model

max 120583

st119899

sum

119895=1

119909119895120582119895 = 119909119900

119899

sum

119895=1

119910119895120582119895 minus 119902+= 119910119900120583

119899

sum

119895=1

120582119895 = 1

120582119895 119902+ge 0

(9)

Suppose that (1199090 119910119900) is the projection of DMU119900 which isobtained as

1199090 = 1199090

119910119900 = 120583lowast119910119900 + 119902

+lowast

(10)

In this case (1199090 119910119900) is on the strongly efficient frontier of119875Convex Here the definitions of congestion (strong and weak)from Tonersquos approach are assumed as follows

Definition 7 (strong congestion) A DMU119900 is strongly con-gested if there exists an activity (119909 119910) isin 119875Convex such that119909 = 120572119909119900 (with 0 lt 120572 lt 1) and 119910119900 ge 120573119910119900 (with 120573 gt 1)

Definition 8 (weak congestion) A DMU is (weakly) con-gested if it is strongly efficient and there exists an activityin 119875Convex that uses less resources in one or more inputs formaking more products in one or more outputs

First we consider DMU119900 which is strongly 119875Convex effi-cient Thus in each optimal solution of model (9) 120583lowast = 1

and 119902+lowast

= 0 Then the following mathematical program issolved for a specific DMU119900 Consider the following

max 1

119904

119904

sum

119903=1

119905+119903

119910119903119900

+ 120576(

1

119898

119898

sum

119894=1

119905minus119894

119909119894119900

)

st119899

sum

119895=1

119909119895120582119895 + 119905minus119894 = 119909119900

119899

sum

119895=1

119910119895120582119895 minus 119905+= 119910119900

119899

sum

119895=1

120582119895 = 1

120582119895 119905+ 119905minusge 0

(11)

where 120576 is a non-Archimedean small positive number

Journal of Applied Mathematics 5

Next let (120582lowast 119905minuslowast 119905+lowast) be an optimal solution of model(11) Then we will have the following cases

(i) If 119905+lowast = 0 DMU119900 has no congestion because thedecreasing in inputs cannot increase any outputs

(ii) If 119905+lowast = 0 119905minuslowast is also not zero since DMU119900 is strongly119875Convex efficient Consequently DMU119900 has (weak)congestion

5 Khodabakhshirsquos Approach

Jahanshahloo and Khodabakhshi [31 32] and Khodabakhshi[33] introduced the following model for improving output

max 120601119900 + 120576(

119898

sum

119894=1

119904minus1198941 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus1198941 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus1198941 119904+1198942 119904+119903 ge 0

(12)

This model is a version of output-oriented BCC modelwhich is not limited in using observed sources of evaluatingDMU While output-oriented BCC model is limited tothe using of observed sources of evaluating DMU inputrelaxation model by imposing a few changes on some inputscan produce outputs more than observed output or outputsuggested by BCC model Using the above model will bevery useful when attracting some inputs such as labor whichis necessary to solve employment problem in a society Theconditions of efficiency under the above model for DMU119900being evaluated become as follows

Definition 9 (efficiency) DMU119900 is efficient under model (12)if the following two conditions are satisfied

(i) 120601lowast119900 = 1

(ii) The optimal amounts of all slacks are zero

Using the optimal solution of model (12) (120601lowast119900 120582lowast119895

119904minuslowast1198941 119904+lowast1198942 119904+lowast119903 ) the following model is solved for determining

technical inefficiency of inputs

max119898

sum

119894=1

120575+119894

st 119909119894119900 minus 119904minuslowast1198941 + 119904

+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 minus 120575+119894 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120575+119894 le 119904minuslowast119894 119894 = 1 119898

120582119895120575+119894 ge 0

(13)Finally the amount of congestion for 119894th input is defined asfollows

119904minus119888119894 = 119904

minuslowast1198941 minus 120575

+lowast119894 119894 = 1 119898 (14)

where 120575+lowast119894 (119894 = 1 119898) is obtained by solving model (13)If the amount of total slacks (total inefficiencies) 119904minuslowast1198941 andnoncongesting input are the same then we do not have anyinput congestion In other words in this case the amount ofinput congestion is zero while if the amount of total slacksis more than noncongesting input then the input congestionwill exist for 119894th input It is obvious that if 119904+lowast1198942 is positive for119894th input there is no congestion for this input

Khodabakhshi [33] presented the one-model version ofthis approach The two models ((12) (13)) can be replaced bya single model which provides an alternative procedure todetermine input congestion Using relation (14) model (13)can be rewritten as follows

max119898

sum

119894=1

minus 119904minus119888119894

st 119909119894119900 minus 119904minus119888119894 + 119904+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 ge 0

(15)

If we consider the following model

max 120601119900 + 120576(

119898

sum

119894=1

minus 119904minus119888119894 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119888119894 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 120601119900119910119903119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904+1198942 119904minus119888119894 119904+119903 ge 0

(16)

6 Journal of Applied Mathematics

it is obvious that if (120601lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal

solution of (16) (120601lowast119900 119904+lowast2 119904+lowast) are part of an optimal solution

of (12) and (120582lowast 119904+lowast) is an optimal solution of (15) In other

words we can regard model (15) as part of a two-stageprocedure for solvingmodel (16) Using one-model approachwe reduced the solving of three problems with two-modelapproach to the solving of two problems with one-modelapproach This is certainly important from computationalpoint of view

Theorem 10 Congestion is present if and only if optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal solution of (16)

and at least one of the following conditions is satisfied

(i) 120601lowast119900 gt 1 and there exists at least one 119894 (1 le 119894 le 119898) suchthat 119904minus119888lowast119894 gt 0

(ii) There exists at least one 119903 (1 le 119903 le 119904) for which 119904+lowast119903 gt 0

and also one 119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

Theorem 11 Congestion is present if and only if for an optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) of (16) there is at least one

119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

6 Nourarsquos Approach

In this section the approach of Noura et al [41] is brieflyreviewed In this method first model (3) is solved and theoptimal solution 120601

lowast119900 120582lowast 119878+lowast 119878minuslowast is obtained Then the set 119864

is defined as follows

119864 = 119895 | 120601lowast119895 = 1 (17)

Among theDMUs in set119864 there exists at least one sayDMU119897that has the highest consumption in its first input componentcompared with the first input component of the remainingDMUs of set 119864 1199091119897 is denoted by 119909

lowast1 Then among the

DMUs in 119864 a DMU is found say DMU119905 that has the highestconsumption in its second input component compared to theremaining DMUs in 119864 1199092119905 is denoted by 119909

lowast2 In a similar

manner 119909lowast1 119909lowast2 119909

lowast119898 can be identified

Definition 12 Congestion is present if and only if in anoptimal solution 120601lowast119900 120582

lowast 119878+lowast 119878minuslowast of (3) for DMU119900 at least one

of the following two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119909119894119900 gt 119909lowast119894 (119894 =

1 119898)(ii) There exists at least one 119878+lowast119903 gt 0 (119903 = 1 119904) and at

least one 119909119894119900 gt 119909lowast119894 (119894 = 1 119898)

They denote the amount of congestion in the 119894th input ofDMUo by 119904

1198881015840

119894 where 119909119894119900 gt 119909lowast119894 and define it as

1199041198881015840

119894 = 119909119894119900 minus 119909lowast119894

(18)

Congestion is considered not present when 119909119894119900 le 119909lowast119894 and 119904

1198881015840

119894 =

0 The sum of all 1199041198881015840

119894 is the amount of congestion in DMU119900Noura et al [41] to establish general validity of their

method prove three theorems which show that Definition 12is equivalent to Cooper et alrsquos Theorem 6 where 119909119894119900 ge 119909

lowast119894

Also they prove another theorem which shows that thereexists no congestion in DMU119900 when 119909119894119900 le 119909

lowast119894 The computa-

tional effort of this method for calculating congestion is lessthan Cooperrsquos approach

7 Wursquos Approach

In this section the input congestion measurement subjectin presence of undesirable outputs is considered using theapproach of Wu et al [42] Assume that 119909 119910 119906 repre-sent the inputs desirable outputs and undesirable outputsrespectively Evidence of congestion in this method occurswhenever reducing some inputs 119909 can increase desirableoutputs119910 and decrease undesirable outputs 119906 simultaneouslyBefore measuring this kind of congestion we should developthe approach solving the problem of undesirable outputsSo far there have been several approaches for addressingthe undesirable output problem Wu et al [42] choose theapproach of Seiford and Zhu [43] to deal with undesirableoutputs The model is shown as follows

max 120575

st119899

sum

119895=1

119909119894119895120582119895 le 1199091198940 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(19)

where 120572119905 is a big enough positive number that canmake every119900119905119895 positive The fourth constraint is used to transform theundesirable output to a new variable The third constraint isused to constrain the new variable in the possible productionset Wu et al [42] proposed the following model for deter-mining the input congestion of DMU0 in the presence ofundesirable outputs

max 119911

st119899

sum

119895=1

119909119894119895120582119895 = 1199091198940 119894 = 1 119898

Journal of Applied Mathematics 7

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(20)

The production possibility set corresponding to thesemodels is as follows

119879 =

(119909 119910 119906) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

119906119895120582119895 le 119906

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(21)

Definition 13 If the optimal value of model (20) satisfies 119911lowast =1 then we call DMU0 weakly efficient

Definition 14 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 gt 1199100 gt 1199060 then the DMU evidences congestion

Definition 15 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 ge 1199100 ge 1199060 then the DMU evidences congestion

Theorem16 Aweakly efficientDMU0 ofmodel (20) evidencescongestion if and only if it is not weakly efficient in model (19)

Proof See Wu et al [42]

8 Conclusion

There are several approaches for dealing with input con-gestion measurement in DEA In this review six importantmethods are briefly considered and compared Whilst eachtechnique is useful in a specific area none of the method-ologies can be prescribed as the complete solution to thequestion of input congestionHowever there is need to designa complete method which can cover all of the followingproperties

(1) to determine both sources and amounts of inputcongestion

(2) to deal with congestion problem under the conditionthat all inputs and outputs are nonnegative values

(3) to measure the input congestion amounts in thepresence of the multiple optimal solutions

(4) to decompose the measure of overall technical effi-ciency into pure technical efficiency scale efficiencyand congestion efficiency

(5) to deal with the congestion problem in the presenceof the undesirable outputs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[2] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[3] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[4] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002

[5] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985

[6] K Tone ldquoA slacks-based measure of efficiency in data envelop-ment analysisrdquo European Journal of Operational Research vol130 no 3 pp 498ndash509 2001

[7] W W Cooper H Deng Z M Huang and S X Li ldquoA one-model approach to congestion in data envelopment analysisrdquoSocio-Economic Planning Sciences vol 36 no 4 pp 231ndash2382002

[8] R Fare and L Svensson ldquoCongestion of production factorsrdquoEconometrica vol 48 no 7 pp 1745ndash1752 1980

[9] R Fare and S Grosskopf ldquoMeasuring congestion in produc-tionrdquo Zeitschrift fur Nationalokonomie vol 43 no 3 pp 257ndash271 1983

[10] R Fare S Grosskopf and C A K Lovell The Measurementof Efficiency of Production Kluwer-Nijhoff Boston Mass USA1985

[11] P Byrnes R Fare and S Grosskopf ldquoMeasuring productiveefficiency an application to Illinois strip minesrdquo ManagementScience vol 30 no 6 pp 671ndash681 1984

[12] P Byrnes R Fare S Grosskopf and C A K Lovell ldquoTheeffect of unions on productivity US surface mining of coalrdquoManagement Science vol 34 no 9 pp 1037ndash1053 1988

[13] R Fare and S Grosskopf ldquoCongestion a noterdquo Socio-EconomicPlanning Sciences vol 32 no 1 pp 21ndash23 1998

[14] R Fare and S Grosskopf ldquoSlacks and congestion a commentrdquoSocio-Economic Planning Sciences vol 34 no 1 pp 27ndash33 2000

[15] R Fare and S Grosskopf ldquoDecomposing technical efficiencywith carerdquoManagement Science vol 46 no 1 pp 167ndash168 2000

8 Journal of Applied Mathematics

[16] R Fare and S Grosskopf ldquoWhen can slacks be used to identifycongestion An answer to W W Cooper L Seiford and J ZhurdquoSocio-Economic Planning Sciences vol 35 no 3 pp 217ndash2212001

[17] W W Cooper R G Thompson and R M Thrall ldquoIntroduc-tion extensions and new developments in DEArdquo Annals ofOperations Research vol 66 pp 3ndash45 1996

[18] P L Brockett W W Cooper H-C Shin and Y WangldquoInefficiency and congestion in Chinese production before andafter the 1978 economic reformsrdquo Socio-Economic PlanningSciences vol 32 no 1 pp 1ndash20 1998

[19] W W Cooper L M Seiford and J Zhu ldquoA unified additivemodel approach for evaluating inefficiency and congestionwith associated measures in DEArdquo Socio-Economic PlanningSciences vol 34 no 1 pp 1ndash25 2000

[20] W W Cooper H Deng B Gu S Li and R M Thrall ldquoUsingDEA to improve the management of congestion in Chineseindustries (1981ndash1997)rdquo Socio-Economic Planning Sciences vol35 no 4 pp 227ndash242 2001

[21] L Cherchye T Kuosmanen andT Post ldquoAlternative treatmentsof congestion in DEA a rejoinder to Cooper Gu and LirdquoEuropean Journal of Operational Research vol 132 no 1 pp 75ndash80 2001

[22] WWCooper BGu and S Li ldquoComparisons and evaluations ofalternative approaches to the treatment of congestion in DEArdquoEuropean Journal of Operational Research vol 132 no 1 pp 62ndash74 2001

[23] W W Cooper B Gu and S Li ldquoNote alternative treatments ofcongestion in DEAmdasha response to the Cherchye Kuosmanenand Post critiquerdquo European Journal of Operational Researchvol 132 no 1 pp 81ndash87 2001

[24] WW Cooper LM Seiford and J Zhu ldquoSlacks and congestionresponse to a comment by R Fare and S Grosskopfrdquo Socio-Economic Planning Sciences vol 35 no 3 pp 205ndash215 2001

[25] WW Cooper H Deng L M Seiford and J Zhu ldquoCongestionits identification and management with DEArdquo in Handbook ofData Envelopment Analysis W W Cooper L M Seiford and JZhu Eds pp 177ndash201 Kluwer Academic Boston Mass USA2004

[26] K Tone and B K Sahoo ldquoDegree of scale economies andcongestion a unified DEA approachrdquo European Journal ofOperational Research vol 158 no 3 pp 755ndash772 2004

[27] Q Wei and H Yan ldquoCongestion and returns to scale indata envelopment analysisrdquo European Journal of OperationalResearch vol 153 no 3 pp 641ndash660 2004

[28] T Sueyoshi and K Sekitani ldquoDEA congestion and returns toscale under an occurrence of multiple optimal projectionsrdquoEuropean Journal of Operational Research vol 194 no 2 pp592ndash607 2009

[29] A T Flegg and D O Allen ldquoDoes expansion cause congestionThe case of the older British universities 1994ndash2004rdquo EducationEconomics vol 15 no 1 pp 75ndash102 2007

[30] M Khoveyni R EslamiM Khodabakhshi G R Jahanshahlooand F H Hosseinzadeh ldquoRecognizing strong and weak con-gestion slack based in data envelopment analysisrdquo Computersamp Industrial Engineering vol 64 no 2 pp 731ndash738 2013

[31] G R Jahanshahloo and M Khodabakhshi ldquoSuitable combina-tion of inputs for improving outputs in DEA with determin-ing input congestion considering textile industry of Chinardquo

Applied Mathematics and Computation vol 151 no 1 pp 263ndash273 2004

[32] G R Jahanshahloo andMKhodabakhshi ldquoDetermining assur-ance interval for non-Archimedean element in the improvingoutputsmodel inDEArdquoAppliedMathematics and Computationvol 151 no 2 pp 501ndash506 2004

[33] M Khodabakhshi ldquoA one-model approach based on relaxedcombinations of inputs for evaluating input congestion inDEArdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 443ndash450 2009

[34] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007

[35] M Khodabakhshi and M Asgharian ldquoAn input relaxationmeasure of efficiency in stochastic data envelopment analysisrdquoApplied Mathematical Modelling vol 33 no 4 pp 2010ndash20232009

[36] 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

[37] M Khodabakhshi ldquoChance constrained additive input relax-ation model in stochastic data envelopment analysisrdquo Journal ofInformation and Systems Sciences vol 6 no 1 pp 99ndash112 2010

[38] M Asgharian M Khodabakhshi and L Neralic ldquoCongestionin stochastic data envelopment analysis an input relaxationapproachrdquo International Journal of Statistics and ManagementSystem vol 5 no 1 pp 84ndash106 2010

[39] G R Jahanshahloo M Khodabakhshi F H Lotfi and RM Goudarzi ldquoComputation of congestion in DEA modelswith productions trade-offs and weight restrictionsrdquo AppliedMathematical Sciences vol 5 no 14 pp 663ndash676 2011

[40] M Khodabakhshi R M Goudarzi M Y Maryaki and MHajiani ldquoA one-model approach for computation of congestionwith productions trade-offs and weight restrictionsrdquo Interna-tional Journal of Applied Operational Research vol 3 no 4 pp69ndash80 2013

[41] A A Noura F H Hosseinzadeh G R Jahanshahloo S FRashidi and B R Parker ldquoA new method for measuring con-gestion in data envelopment analysisrdquo Socio-Economic PlanningSciences vol 44 no 4 pp 240ndash246 2010

[42] J Wu Q An B Xiong and Y Chen ldquoCongestionmeasurementfor regional industries in China a data envelopment analysisapproach with undesirable outputsrdquo Energy Policy vol 57 pp7ndash13 2012

[43] L M Seiford and J Zhu ldquoModeling undesirable factors in effi-ciency evaluationrdquo European Journal of Operational Researchvol 142 no 1 pp 16ndash20 2002

[44] B Dervaux K Kerstens and P V Eeckaut ldquoRadial and non-radial static efficiency decompositions a focus on congestionmeasurementrdquo Transportation Research B Methodological vol32 no 5 pp 299ndash312 1998

[45] A T Flegg and D O Allen ldquoCongestion in the Chineseautomobile and textile industries revisitedrdquo Socio-EconomicPlanning Sciences vol 43 no 3 pp 177ndash191 2009

[46] C Kao ldquoCongestion measurement and elimination under theframework of data envelopment analysisrdquo International Journalof Production Economics vol 123 no 2 pp 257ndash265 2010

Journal of Applied Mathematics 9

[47] J Odeck ldquoCongestion ownership region of operation andscale their impact on bus operator performance in NorwayrdquoSocio-Economic Planning Sciences vol 40 no 1 pp 52ndash69 2006

[48] Q Wei and H Yan ldquoWeak congestion in output additive dataenvelopment analysisrdquo Socio-Economic Planning Sciences vol43 no 1 pp 40ndash54 2009

[49] T Sueyoshi and M Goto ldquoWeak and strong disposabilityvs natural and managerial disposability in DEA environmen-tal assessment comparison between Japanese electric powerindustry and manufacturing industriesrdquo Energy Economics vol34 no 3 pp 686ndash699 2012

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

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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

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

between input congestion and technical inefficiency weprovide the definition of technical inefficiency which is asfollows

Definition 4 (technical inefficiency) Technical inefficiency ispresent when it is possible to improve some inputs or outputswithout worsening other inputs or outputs

So far several methods formeasuring the effect of conges-tion have been developed under the framework of DEA Thecongestion estimating approaches were originated by Fareand Svensson [8] Fare and Grosskopf [9] and Fare et al[10] extended and developed a DEA model to calculate thecongestion amounts Hereafter this procedure is referred toas Farersquos approach This model is a radial approach in thatthe congestion effect is measured as the difference betweentechnologies under weak and strong disposability inputsBased on that model Byrnes et al [11 12] used examplesof coal mines to illustrate the decomposition of productionefficiency where the congestion effect is a component To seethe other references of this approach the reader is referred to[13ndash16] A big advantage of this approach is that it is possibleto decompose its measure of overall technical efficiency ina straightforward way into pure technical efficiency scaleefficiency and congestion efficiency It is the assumption thatcongestion is the difference between weak disposability andstrong disposability This approach fails to detect congestionin some cases especially when there is only one input Alsoit does not measure the amount of congestion

Cooper et al [17] published an alternative approach fordetermining input congestion that has some advantages toprevious method This method uses the slacks to determinethe congestion amount In this method the congestion effectis measured as the difference between the observed amountsand the expected amounts It was subsequently developedby Brockett et al [18] and Cooper et al [19] For simplicitythis procedure is referred to as Cooperrsquos approach Unlikethe radial approach which requires all inputs to reducein proportion to eliminate congestion this method allowseach input to reduce in different proportions This approachhas been applied to identify congesting inputs in Chineseindustries by Brockett et al [18] and Cooper et al [20]Later on Cooper et al [7] introduced a one-model versionof Cooperrsquos method To see the other references of thisapproach refer to [21ndash25]

The previous approaches express the congestion effect interms of excessive inputs According to the definition conges-tion occurswhen increases in some inputs results in decreasesin some outputs Hence congestion can also be measured asshortfalls in outputs Tone and Sahoo [26] and Wei and Yan[27] developed a method from the output point of view Thismodel hereafter is referred to as Tonersquos approach Sueyoshiand Sekitani [28] proposed amodified approachwhich is ableto measure the degree of congestion under the occurrence ofmultiple solutions This method is also a radial approach soa product form of efficiency decomposition is possible Fleggand Allen [29] applied these three approaches to examinewhether the rapid growth in the number of students in Britishuniversities has led to congestion One of the shortcomings of

Tonersquos method is its inability to identify the excessive amountof each input Also in the presence of alternative optimalsolutions this approach is incapable of detecting congestion(strong and weak) Furthermore in this approach all inputsand outputs of DMUs have been assumed positive while inreal applications data is often nonnegative Khoveyni et al[30] removed this drawback of Tonersquos approach

In DEA the inputs proportions changes are often basedon inputs reduction Apparently this subject is a reasonablejustification economically because of reducing the costs ofthe reduced inputs But in some cases inputs reduction suchas labor may be faced with social tensions as it happenedin China Jahanshahloo and Khodabakhshi [31 32] proposeda method to determine the sources and amounts of inputcongestion which is based on the relaxed combination ofinputs The framework of this method is similar to that ofCooper et al [7] Later on Khodabakhshi [33] provideda one-model approach of input congestion based on inputrelaxation model This approach reduces solving three prob-lems with the two-model approach introduced by Jahan-shahloo and Khodabakhshi [31] which is certainly importantfrom the computational point of view To see more referencesabout this approach the readers are referred to [34 35 37]Hereafter this procedure is referred to as Khodabakhshirsquosapproach Using this approach in textile industry of Chinashow that flexibility in using inputs causes output augmen-tation because of new inputs combination So it has morebenefits than the probable increasing cost of some inputsAlso Khodabakhshi [36] and Asgharian et al [38] proposedsome methods for determining the input congestion in thestochastic conditions (See also [37]) Jahanshahloo et al [39]and Khodabakhshi et al [40] proposed some methods forcomputing the congestion in DEA models with productionstrade-offs and weight restrictions

Noura et al [41] proposed another algorithm whichis equivalent to Cooperrsquos approach Their method requiresconsiderably less computation than Cooperrsquos approach Thismethod is called Nourarsquos approach

The above approaches only consider the desirable out-put But in the production process undesirable outputslike smoke pollution or waste are usually produced withdesirable outputs The desirable outputs are good outputsfor consumers and undesirable outputs are bad outputs forconsumers such as pollution Therefore congestion withundesirable outputs is different from that in the traditionalscenario Wu et al [42] combined the methods of Seifordand Zhu [43] and Wei and Yan [27] to propose a method fordetermining input congestion in the presence of undesirableoutputs which is named Wursquos approach

To see more references about input congestion measure-ment subject the readers can refer to [44ndash49]

The rest of the paper is structured as follows In Sections2ndash7 the approaches of Fare Cooper Tone KhodabakhshiNoura and Wu are briefly introduced The final sectionconcludes the paper

Journal of Applied Mathematics 3

2 Faumlrersquos Approach

Throughout this paper we deal with 119899 decision makingunits DMU119895 (119895 = 1 119899) each having 119898 inputs 119909119894119895 (119894 =

1 119898) for producing 119904 outputs 119910119903119895 (119903 = 1 119904) Theunderevaluation ofDMU is denoted byDMU119900 Also supposethat all inputs and outputs are nonnegative deterministicnumbers The efficiency score of DMU119900 under the strongdisposal technology is determined as follows

120579lowast= min 120579

st119899

sum

119895=1

119909119894119895120582119895 le 119909119894119900120579 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 119910119903119900 119903 = 1 119904

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0 119895 = 1 119899

(1)

Also the efficiency score of DMU119900 under the weakdisposal technology is determined as follows

120573lowast= min 120573

st119899

sum

119895=1

119909119894119895120582119895 = 119909119894119900120573 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 119910119903119900 119903 = 1 119904

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0 119895 = 1 119899

(2)

This measure identifies the presence of congestion pro-vided 120579

lowast120573lowastlt 1 and no congestion is present if 120579lowast120573lowast = 1

When there is only one input strong disposability and weakdisposability in input are the same therefore Farersquos approach-type congestion does not exist

3 Cooperrsquos Approach

The BCC model for measuring the efficiency has two varia-tions input and output orientation Since the input-orientedmodel may produce erroneous results in measuring conges-tion this method will concentrate on the output-orientationmodel [19] Using the output-oriented BCC model theefficiency of DMU119900 is evaluated as follows

max 120601119900 + 120576(

119898

sum

119894=1

119904minus119894 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119894 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119894 119904+119903 ge 0

(3)

Consider in model (3) that 120576 is a non-Archimedeanelement namely it is not a real number and defined to besmaller than any positive real number To avoid assigning avalue to 120576 this model can be solved via a two-stage procedure[32] In this procedure first 120601119900 is maximized while ignoringthe slacks in the objective Then 120601119900 is replaced with 120601

lowast119900

(optimal value of first stage) and the sum of the slacks ismaximized Using the optimal solution of the above modelwe have the following definitions

Definition 5 (efficiency) DMU119900 is efficient if 120601lowast119900 = 1 and119904minuslowast119894 = 119904

+lowast119903 = 0 for all 119894 119903

Inefficiency is a necessary condition for the presence ofcongestion Therefore if DMU119900 is found to be inefficient theoptimal solution of model (3) (120601lowast119900 120582

lowast119895 119904minuslowast119894 119904+lowast119903 ) is used to

determine the amount of technical inefficiency in inputs asfollows

max119898

sum

119894=1

120575minus119894

st 119909119894119900 minus 119904minuslowast119894 =

119899

sum

119895=1

119909119894119895120582119895 minus 120575minus119894 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120575minus119894 le 119904minuslowast119894 119894 = 1 119898

120575minus119894 120582119895 ge 0

(4)

Finally the optimum value of model (4) (120575+lowast119894 ) is used todetermine congestion amounts as follows

119904minus119888lowast119894 = 119904

minuslowast119894 minus 120575

minuslowast119894 119894 = 1 119898 (5)

Cooper et al [7] introduced the one-model versionof pervious method to deal with the congestion subjectSubstituting (5) into (4) we can rewrite the latter as follows

min119898

sum

119894=1

119904minus119888119894

st 119909119894119900 minus 119904minus119888119894 =

119899

sum

119895=1

119909119894119895120582119895 119894 = 1 119898

4 Journal of Applied Mathematics

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 ge 0 119894 = 1 119898

(6)

Using 120576 the models ((3) (6)) are combined into thefollowing single model

max 120601119900 + 120576(minus

119898

sum

119894=1

119904minus119888119894 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119888119894 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 119904+119903 ge 0

(7)

Theorem 6 Congestion is present if and only if in an optimalsolution 120601

lowast119900 120582lowast 119878+lowast 119878minus119888lowast of (7) at least one of the following

two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119878minus119888lowast gt 0 (1 le 119894 le 119898)

(ii) There exists at least one 119878+lowast119903 gt 0 (1 le 119903 le 119904) and atleast one 119878minus119888lowast gt 0 (1 le 119894 le 119898)

Proof See Cooper et al [7]

4 Tonersquos Approach

In the approach which was presented by Tone and Sahoo[26] it is assumed that all inputs and outputs are positiveThisapproach is based on the following production possibility set(PPS)

119875Convex

=

(119909 119910) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(8)

In this approach for measuring the congestion effectthe pure technical efficiency is firstly determined using thefollowing model

max 120583

st119899

sum

119895=1

119909119895120582119895 = 119909119900

119899

sum

119895=1

119910119895120582119895 minus 119902+= 119910119900120583

119899

sum

119895=1

120582119895 = 1

120582119895 119902+ge 0

(9)

Suppose that (1199090 119910119900) is the projection of DMU119900 which isobtained as

1199090 = 1199090

119910119900 = 120583lowast119910119900 + 119902

+lowast

(10)

In this case (1199090 119910119900) is on the strongly efficient frontier of119875Convex Here the definitions of congestion (strong and weak)from Tonersquos approach are assumed as follows

Definition 7 (strong congestion) A DMU119900 is strongly con-gested if there exists an activity (119909 119910) isin 119875Convex such that119909 = 120572119909119900 (with 0 lt 120572 lt 1) and 119910119900 ge 120573119910119900 (with 120573 gt 1)

Definition 8 (weak congestion) A DMU is (weakly) con-gested if it is strongly efficient and there exists an activityin 119875Convex that uses less resources in one or more inputs formaking more products in one or more outputs

First we consider DMU119900 which is strongly 119875Convex effi-cient Thus in each optimal solution of model (9) 120583lowast = 1

and 119902+lowast

= 0 Then the following mathematical program issolved for a specific DMU119900 Consider the following

max 1

119904

119904

sum

119903=1

119905+119903

119910119903119900

+ 120576(

1

119898

119898

sum

119894=1

119905minus119894

119909119894119900

)

st119899

sum

119895=1

119909119895120582119895 + 119905minus119894 = 119909119900

119899

sum

119895=1

119910119895120582119895 minus 119905+= 119910119900

119899

sum

119895=1

120582119895 = 1

120582119895 119905+ 119905minusge 0

(11)

where 120576 is a non-Archimedean small positive number

Journal of Applied Mathematics 5

Next let (120582lowast 119905minuslowast 119905+lowast) be an optimal solution of model(11) Then we will have the following cases

(i) If 119905+lowast = 0 DMU119900 has no congestion because thedecreasing in inputs cannot increase any outputs

(ii) If 119905+lowast = 0 119905minuslowast is also not zero since DMU119900 is strongly119875Convex efficient Consequently DMU119900 has (weak)congestion

5 Khodabakhshirsquos Approach

Jahanshahloo and Khodabakhshi [31 32] and Khodabakhshi[33] introduced the following model for improving output

max 120601119900 + 120576(

119898

sum

119894=1

119904minus1198941 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus1198941 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus1198941 119904+1198942 119904+119903 ge 0

(12)

This model is a version of output-oriented BCC modelwhich is not limited in using observed sources of evaluatingDMU While output-oriented BCC model is limited tothe using of observed sources of evaluating DMU inputrelaxation model by imposing a few changes on some inputscan produce outputs more than observed output or outputsuggested by BCC model Using the above model will bevery useful when attracting some inputs such as labor whichis necessary to solve employment problem in a society Theconditions of efficiency under the above model for DMU119900being evaluated become as follows

Definition 9 (efficiency) DMU119900 is efficient under model (12)if the following two conditions are satisfied

(i) 120601lowast119900 = 1

(ii) The optimal amounts of all slacks are zero

Using the optimal solution of model (12) (120601lowast119900 120582lowast119895

119904minuslowast1198941 119904+lowast1198942 119904+lowast119903 ) the following model is solved for determining

technical inefficiency of inputs

max119898

sum

119894=1

120575+119894

st 119909119894119900 minus 119904minuslowast1198941 + 119904

+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 minus 120575+119894 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120575+119894 le 119904minuslowast119894 119894 = 1 119898

120582119895120575+119894 ge 0

(13)Finally the amount of congestion for 119894th input is defined asfollows

119904minus119888119894 = 119904

minuslowast1198941 minus 120575

+lowast119894 119894 = 1 119898 (14)

where 120575+lowast119894 (119894 = 1 119898) is obtained by solving model (13)If the amount of total slacks (total inefficiencies) 119904minuslowast1198941 andnoncongesting input are the same then we do not have anyinput congestion In other words in this case the amount ofinput congestion is zero while if the amount of total slacksis more than noncongesting input then the input congestionwill exist for 119894th input It is obvious that if 119904+lowast1198942 is positive for119894th input there is no congestion for this input

Khodabakhshi [33] presented the one-model version ofthis approach The two models ((12) (13)) can be replaced bya single model which provides an alternative procedure todetermine input congestion Using relation (14) model (13)can be rewritten as follows

max119898

sum

119894=1

minus 119904minus119888119894

st 119909119894119900 minus 119904minus119888119894 + 119904+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 ge 0

(15)

If we consider the following model

max 120601119900 + 120576(

119898

sum

119894=1

minus 119904minus119888119894 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119888119894 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 120601119900119910119903119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904+1198942 119904minus119888119894 119904+119903 ge 0

(16)

6 Journal of Applied Mathematics

it is obvious that if (120601lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal

solution of (16) (120601lowast119900 119904+lowast2 119904+lowast) are part of an optimal solution

of (12) and (120582lowast 119904+lowast) is an optimal solution of (15) In other

words we can regard model (15) as part of a two-stageprocedure for solvingmodel (16) Using one-model approachwe reduced the solving of three problems with two-modelapproach to the solving of two problems with one-modelapproach This is certainly important from computationalpoint of view

Theorem 10 Congestion is present if and only if optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal solution of (16)

and at least one of the following conditions is satisfied

(i) 120601lowast119900 gt 1 and there exists at least one 119894 (1 le 119894 le 119898) suchthat 119904minus119888lowast119894 gt 0

(ii) There exists at least one 119903 (1 le 119903 le 119904) for which 119904+lowast119903 gt 0

and also one 119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

Theorem 11 Congestion is present if and only if for an optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) of (16) there is at least one

119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

6 Nourarsquos Approach

In this section the approach of Noura et al [41] is brieflyreviewed In this method first model (3) is solved and theoptimal solution 120601

lowast119900 120582lowast 119878+lowast 119878minuslowast is obtained Then the set 119864

is defined as follows

119864 = 119895 | 120601lowast119895 = 1 (17)

Among theDMUs in set119864 there exists at least one sayDMU119897that has the highest consumption in its first input componentcompared with the first input component of the remainingDMUs of set 119864 1199091119897 is denoted by 119909

lowast1 Then among the

DMUs in 119864 a DMU is found say DMU119905 that has the highestconsumption in its second input component compared to theremaining DMUs in 119864 1199092119905 is denoted by 119909

lowast2 In a similar

manner 119909lowast1 119909lowast2 119909

lowast119898 can be identified

Definition 12 Congestion is present if and only if in anoptimal solution 120601lowast119900 120582

lowast 119878+lowast 119878minuslowast of (3) for DMU119900 at least one

of the following two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119909119894119900 gt 119909lowast119894 (119894 =

1 119898)(ii) There exists at least one 119878+lowast119903 gt 0 (119903 = 1 119904) and at

least one 119909119894119900 gt 119909lowast119894 (119894 = 1 119898)

They denote the amount of congestion in the 119894th input ofDMUo by 119904

1198881015840

119894 where 119909119894119900 gt 119909lowast119894 and define it as

1199041198881015840

119894 = 119909119894119900 minus 119909lowast119894

(18)

Congestion is considered not present when 119909119894119900 le 119909lowast119894 and 119904

1198881015840

119894 =

0 The sum of all 1199041198881015840

119894 is the amount of congestion in DMU119900Noura et al [41] to establish general validity of their

method prove three theorems which show that Definition 12is equivalent to Cooper et alrsquos Theorem 6 where 119909119894119900 ge 119909

lowast119894

Also they prove another theorem which shows that thereexists no congestion in DMU119900 when 119909119894119900 le 119909

lowast119894 The computa-

tional effort of this method for calculating congestion is lessthan Cooperrsquos approach

7 Wursquos Approach

In this section the input congestion measurement subjectin presence of undesirable outputs is considered using theapproach of Wu et al [42] Assume that 119909 119910 119906 repre-sent the inputs desirable outputs and undesirable outputsrespectively Evidence of congestion in this method occurswhenever reducing some inputs 119909 can increase desirableoutputs119910 and decrease undesirable outputs 119906 simultaneouslyBefore measuring this kind of congestion we should developthe approach solving the problem of undesirable outputsSo far there have been several approaches for addressingthe undesirable output problem Wu et al [42] choose theapproach of Seiford and Zhu [43] to deal with undesirableoutputs The model is shown as follows

max 120575

st119899

sum

119895=1

119909119894119895120582119895 le 1199091198940 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(19)

where 120572119905 is a big enough positive number that canmake every119900119905119895 positive The fourth constraint is used to transform theundesirable output to a new variable The third constraint isused to constrain the new variable in the possible productionset Wu et al [42] proposed the following model for deter-mining the input congestion of DMU0 in the presence ofundesirable outputs

max 119911

st119899

sum

119895=1

119909119894119895120582119895 = 1199091198940 119894 = 1 119898

Journal of Applied Mathematics 7

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(20)

The production possibility set corresponding to thesemodels is as follows

119879 =

(119909 119910 119906) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

119906119895120582119895 le 119906

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(21)

Definition 13 If the optimal value of model (20) satisfies 119911lowast =1 then we call DMU0 weakly efficient

Definition 14 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 gt 1199100 gt 1199060 then the DMU evidences congestion

Definition 15 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 ge 1199100 ge 1199060 then the DMU evidences congestion

Theorem16 Aweakly efficientDMU0 ofmodel (20) evidencescongestion if and only if it is not weakly efficient in model (19)

Proof See Wu et al [42]

8 Conclusion

There are several approaches for dealing with input con-gestion measurement in DEA In this review six importantmethods are briefly considered and compared Whilst eachtechnique is useful in a specific area none of the method-ologies can be prescribed as the complete solution to thequestion of input congestionHowever there is need to designa complete method which can cover all of the followingproperties

(1) to determine both sources and amounts of inputcongestion

(2) to deal with congestion problem under the conditionthat all inputs and outputs are nonnegative values

(3) to measure the input congestion amounts in thepresence of the multiple optimal solutions

(4) to decompose the measure of overall technical effi-ciency into pure technical efficiency scale efficiencyand congestion efficiency

(5) to deal with the congestion problem in the presenceof the undesirable outputs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[2] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[3] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[4] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002

[5] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985

[6] K Tone ldquoA slacks-based measure of efficiency in data envelop-ment analysisrdquo European Journal of Operational Research vol130 no 3 pp 498ndash509 2001

[7] W W Cooper H Deng Z M Huang and S X Li ldquoA one-model approach to congestion in data envelopment analysisrdquoSocio-Economic Planning Sciences vol 36 no 4 pp 231ndash2382002

[8] R Fare and L Svensson ldquoCongestion of production factorsrdquoEconometrica vol 48 no 7 pp 1745ndash1752 1980

[9] R Fare and S Grosskopf ldquoMeasuring congestion in produc-tionrdquo Zeitschrift fur Nationalokonomie vol 43 no 3 pp 257ndash271 1983

[10] R Fare S Grosskopf and C A K Lovell The Measurementof Efficiency of Production Kluwer-Nijhoff Boston Mass USA1985

[11] P Byrnes R Fare and S Grosskopf ldquoMeasuring productiveefficiency an application to Illinois strip minesrdquo ManagementScience vol 30 no 6 pp 671ndash681 1984

[12] P Byrnes R Fare S Grosskopf and C A K Lovell ldquoTheeffect of unions on productivity US surface mining of coalrdquoManagement Science vol 34 no 9 pp 1037ndash1053 1988

[13] R Fare and S Grosskopf ldquoCongestion a noterdquo Socio-EconomicPlanning Sciences vol 32 no 1 pp 21ndash23 1998

[14] R Fare and S Grosskopf ldquoSlacks and congestion a commentrdquoSocio-Economic Planning Sciences vol 34 no 1 pp 27ndash33 2000

[15] R Fare and S Grosskopf ldquoDecomposing technical efficiencywith carerdquoManagement Science vol 46 no 1 pp 167ndash168 2000

8 Journal of Applied Mathematics

[16] R Fare and S Grosskopf ldquoWhen can slacks be used to identifycongestion An answer to W W Cooper L Seiford and J ZhurdquoSocio-Economic Planning Sciences vol 35 no 3 pp 217ndash2212001

[17] W W Cooper R G Thompson and R M Thrall ldquoIntroduc-tion extensions and new developments in DEArdquo Annals ofOperations Research vol 66 pp 3ndash45 1996

[18] P L Brockett W W Cooper H-C Shin and Y WangldquoInefficiency and congestion in Chinese production before andafter the 1978 economic reformsrdquo Socio-Economic PlanningSciences vol 32 no 1 pp 1ndash20 1998

[19] W W Cooper L M Seiford and J Zhu ldquoA unified additivemodel approach for evaluating inefficiency and congestionwith associated measures in DEArdquo Socio-Economic PlanningSciences vol 34 no 1 pp 1ndash25 2000

[20] W W Cooper H Deng B Gu S Li and R M Thrall ldquoUsingDEA to improve the management of congestion in Chineseindustries (1981ndash1997)rdquo Socio-Economic Planning Sciences vol35 no 4 pp 227ndash242 2001

[21] L Cherchye T Kuosmanen andT Post ldquoAlternative treatmentsof congestion in DEA a rejoinder to Cooper Gu and LirdquoEuropean Journal of Operational Research vol 132 no 1 pp 75ndash80 2001

[22] WWCooper BGu and S Li ldquoComparisons and evaluations ofalternative approaches to the treatment of congestion in DEArdquoEuropean Journal of Operational Research vol 132 no 1 pp 62ndash74 2001

[23] W W Cooper B Gu and S Li ldquoNote alternative treatments ofcongestion in DEAmdasha response to the Cherchye Kuosmanenand Post critiquerdquo European Journal of Operational Researchvol 132 no 1 pp 81ndash87 2001

[24] WW Cooper LM Seiford and J Zhu ldquoSlacks and congestionresponse to a comment by R Fare and S Grosskopfrdquo Socio-Economic Planning Sciences vol 35 no 3 pp 205ndash215 2001

[25] WW Cooper H Deng L M Seiford and J Zhu ldquoCongestionits identification and management with DEArdquo in Handbook ofData Envelopment Analysis W W Cooper L M Seiford and JZhu Eds pp 177ndash201 Kluwer Academic Boston Mass USA2004

[26] K Tone and B K Sahoo ldquoDegree of scale economies andcongestion a unified DEA approachrdquo European Journal ofOperational Research vol 158 no 3 pp 755ndash772 2004

[27] Q Wei and H Yan ldquoCongestion and returns to scale indata envelopment analysisrdquo European Journal of OperationalResearch vol 153 no 3 pp 641ndash660 2004

[28] T Sueyoshi and K Sekitani ldquoDEA congestion and returns toscale under an occurrence of multiple optimal projectionsrdquoEuropean Journal of Operational Research vol 194 no 2 pp592ndash607 2009

[29] A T Flegg and D O Allen ldquoDoes expansion cause congestionThe case of the older British universities 1994ndash2004rdquo EducationEconomics vol 15 no 1 pp 75ndash102 2007

[30] M Khoveyni R EslamiM Khodabakhshi G R Jahanshahlooand F H Hosseinzadeh ldquoRecognizing strong and weak con-gestion slack based in data envelopment analysisrdquo Computersamp Industrial Engineering vol 64 no 2 pp 731ndash738 2013

[31] G R Jahanshahloo and M Khodabakhshi ldquoSuitable combina-tion of inputs for improving outputs in DEA with determin-ing input congestion considering textile industry of Chinardquo

Applied Mathematics and Computation vol 151 no 1 pp 263ndash273 2004

[32] G R Jahanshahloo andMKhodabakhshi ldquoDetermining assur-ance interval for non-Archimedean element in the improvingoutputsmodel inDEArdquoAppliedMathematics and Computationvol 151 no 2 pp 501ndash506 2004

[33] M Khodabakhshi ldquoA one-model approach based on relaxedcombinations of inputs for evaluating input congestion inDEArdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 443ndash450 2009

[34] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007

[35] M Khodabakhshi and M Asgharian ldquoAn input relaxationmeasure of efficiency in stochastic data envelopment analysisrdquoApplied Mathematical Modelling vol 33 no 4 pp 2010ndash20232009

[36] 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

[37] M Khodabakhshi ldquoChance constrained additive input relax-ation model in stochastic data envelopment analysisrdquo Journal ofInformation and Systems Sciences vol 6 no 1 pp 99ndash112 2010

[38] M Asgharian M Khodabakhshi and L Neralic ldquoCongestionin stochastic data envelopment analysis an input relaxationapproachrdquo International Journal of Statistics and ManagementSystem vol 5 no 1 pp 84ndash106 2010

[39] G R Jahanshahloo M Khodabakhshi F H Lotfi and RM Goudarzi ldquoComputation of congestion in DEA modelswith productions trade-offs and weight restrictionsrdquo AppliedMathematical Sciences vol 5 no 14 pp 663ndash676 2011

[40] M Khodabakhshi R M Goudarzi M Y Maryaki and MHajiani ldquoA one-model approach for computation of congestionwith productions trade-offs and weight restrictionsrdquo Interna-tional Journal of Applied Operational Research vol 3 no 4 pp69ndash80 2013

[41] A A Noura F H Hosseinzadeh G R Jahanshahloo S FRashidi and B R Parker ldquoA new method for measuring con-gestion in data envelopment analysisrdquo Socio-Economic PlanningSciences vol 44 no 4 pp 240ndash246 2010

[42] J Wu Q An B Xiong and Y Chen ldquoCongestionmeasurementfor regional industries in China a data envelopment analysisapproach with undesirable outputsrdquo Energy Policy vol 57 pp7ndash13 2012

[43] L M Seiford and J Zhu ldquoModeling undesirable factors in effi-ciency evaluationrdquo European Journal of Operational Researchvol 142 no 1 pp 16ndash20 2002

[44] B Dervaux K Kerstens and P V Eeckaut ldquoRadial and non-radial static efficiency decompositions a focus on congestionmeasurementrdquo Transportation Research B Methodological vol32 no 5 pp 299ndash312 1998

[45] A T Flegg and D O Allen ldquoCongestion in the Chineseautomobile and textile industries revisitedrdquo Socio-EconomicPlanning Sciences vol 43 no 3 pp 177ndash191 2009

[46] C Kao ldquoCongestion measurement and elimination under theframework of data envelopment analysisrdquo International Journalof Production Economics vol 123 no 2 pp 257ndash265 2010

Journal of Applied Mathematics 9

[47] J Odeck ldquoCongestion ownership region of operation andscale their impact on bus operator performance in NorwayrdquoSocio-Economic Planning Sciences vol 40 no 1 pp 52ndash69 2006

[48] Q Wei and H Yan ldquoWeak congestion in output additive dataenvelopment analysisrdquo Socio-Economic Planning Sciences vol43 no 1 pp 40ndash54 2009

[49] T Sueyoshi and M Goto ldquoWeak and strong disposabilityvs natural and managerial disposability in DEA environmen-tal assessment comparison between Japanese electric powerindustry and manufacturing industriesrdquo Energy Economics vol34 no 3 pp 686ndash699 2012

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

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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

2 Faumlrersquos Approach

Throughout this paper we deal with 119899 decision makingunits DMU119895 (119895 = 1 119899) each having 119898 inputs 119909119894119895 (119894 =

1 119898) for producing 119904 outputs 119910119903119895 (119903 = 1 119904) Theunderevaluation ofDMU is denoted byDMU119900 Also supposethat all inputs and outputs are nonnegative deterministicnumbers The efficiency score of DMU119900 under the strongdisposal technology is determined as follows

120579lowast= min 120579

st119899

sum

119895=1

119909119894119895120582119895 le 119909119894119900120579 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 119910119903119900 119903 = 1 119904

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0 119895 = 1 119899

(1)

Also the efficiency score of DMU119900 under the weakdisposal technology is determined as follows

120573lowast= min 120573

st119899

sum

119895=1

119909119894119895120582119895 = 119909119894119900120573 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 119910119903119900 119903 = 1 119904

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0 119895 = 1 119899

(2)

This measure identifies the presence of congestion pro-vided 120579

lowast120573lowastlt 1 and no congestion is present if 120579lowast120573lowast = 1

When there is only one input strong disposability and weakdisposability in input are the same therefore Farersquos approach-type congestion does not exist

3 Cooperrsquos Approach

The BCC model for measuring the efficiency has two varia-tions input and output orientation Since the input-orientedmodel may produce erroneous results in measuring conges-tion this method will concentrate on the output-orientationmodel [19] Using the output-oriented BCC model theefficiency of DMU119900 is evaluated as follows

max 120601119900 + 120576(

119898

sum

119894=1

119904minus119894 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119894 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119894 119904+119903 ge 0

(3)

Consider in model (3) that 120576 is a non-Archimedeanelement namely it is not a real number and defined to besmaller than any positive real number To avoid assigning avalue to 120576 this model can be solved via a two-stage procedure[32] In this procedure first 120601119900 is maximized while ignoringthe slacks in the objective Then 120601119900 is replaced with 120601

lowast119900

(optimal value of first stage) and the sum of the slacks ismaximized Using the optimal solution of the above modelwe have the following definitions

Definition 5 (efficiency) DMU119900 is efficient if 120601lowast119900 = 1 and119904minuslowast119894 = 119904

+lowast119903 = 0 for all 119894 119903

Inefficiency is a necessary condition for the presence ofcongestion Therefore if DMU119900 is found to be inefficient theoptimal solution of model (3) (120601lowast119900 120582

lowast119895 119904minuslowast119894 119904+lowast119903 ) is used to

determine the amount of technical inefficiency in inputs asfollows

max119898

sum

119894=1

120575minus119894

st 119909119894119900 minus 119904minuslowast119894 =

119899

sum

119895=1

119909119894119895120582119895 minus 120575minus119894 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120575minus119894 le 119904minuslowast119894 119894 = 1 119898

120575minus119894 120582119895 ge 0

(4)

Finally the optimum value of model (4) (120575+lowast119894 ) is used todetermine congestion amounts as follows

119904minus119888lowast119894 = 119904

minuslowast119894 minus 120575

minuslowast119894 119894 = 1 119898 (5)

Cooper et al [7] introduced the one-model versionof pervious method to deal with the congestion subjectSubstituting (5) into (4) we can rewrite the latter as follows

min119898

sum

119894=1

119904minus119888119894

st 119909119894119900 minus 119904minus119888119894 =

119899

sum

119895=1

119909119894119895120582119895 119894 = 1 119898

4 Journal of Applied Mathematics

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 ge 0 119894 = 1 119898

(6)

Using 120576 the models ((3) (6)) are combined into thefollowing single model

max 120601119900 + 120576(minus

119898

sum

119894=1

119904minus119888119894 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119888119894 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 119904+119903 ge 0

(7)

Theorem 6 Congestion is present if and only if in an optimalsolution 120601

lowast119900 120582lowast 119878+lowast 119878minus119888lowast of (7) at least one of the following

two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119878minus119888lowast gt 0 (1 le 119894 le 119898)

(ii) There exists at least one 119878+lowast119903 gt 0 (1 le 119903 le 119904) and atleast one 119878minus119888lowast gt 0 (1 le 119894 le 119898)

Proof See Cooper et al [7]

4 Tonersquos Approach

In the approach which was presented by Tone and Sahoo[26] it is assumed that all inputs and outputs are positiveThisapproach is based on the following production possibility set(PPS)

119875Convex

=

(119909 119910) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(8)

In this approach for measuring the congestion effectthe pure technical efficiency is firstly determined using thefollowing model

max 120583

st119899

sum

119895=1

119909119895120582119895 = 119909119900

119899

sum

119895=1

119910119895120582119895 minus 119902+= 119910119900120583

119899

sum

119895=1

120582119895 = 1

120582119895 119902+ge 0

(9)

Suppose that (1199090 119910119900) is the projection of DMU119900 which isobtained as

1199090 = 1199090

119910119900 = 120583lowast119910119900 + 119902

+lowast

(10)

In this case (1199090 119910119900) is on the strongly efficient frontier of119875Convex Here the definitions of congestion (strong and weak)from Tonersquos approach are assumed as follows

Definition 7 (strong congestion) A DMU119900 is strongly con-gested if there exists an activity (119909 119910) isin 119875Convex such that119909 = 120572119909119900 (with 0 lt 120572 lt 1) and 119910119900 ge 120573119910119900 (with 120573 gt 1)

Definition 8 (weak congestion) A DMU is (weakly) con-gested if it is strongly efficient and there exists an activityin 119875Convex that uses less resources in one or more inputs formaking more products in one or more outputs

First we consider DMU119900 which is strongly 119875Convex effi-cient Thus in each optimal solution of model (9) 120583lowast = 1

and 119902+lowast

= 0 Then the following mathematical program issolved for a specific DMU119900 Consider the following

max 1

119904

119904

sum

119903=1

119905+119903

119910119903119900

+ 120576(

1

119898

119898

sum

119894=1

119905minus119894

119909119894119900

)

st119899

sum

119895=1

119909119895120582119895 + 119905minus119894 = 119909119900

119899

sum

119895=1

119910119895120582119895 minus 119905+= 119910119900

119899

sum

119895=1

120582119895 = 1

120582119895 119905+ 119905minusge 0

(11)

where 120576 is a non-Archimedean small positive number

Journal of Applied Mathematics 5

Next let (120582lowast 119905minuslowast 119905+lowast) be an optimal solution of model(11) Then we will have the following cases

(i) If 119905+lowast = 0 DMU119900 has no congestion because thedecreasing in inputs cannot increase any outputs

(ii) If 119905+lowast = 0 119905minuslowast is also not zero since DMU119900 is strongly119875Convex efficient Consequently DMU119900 has (weak)congestion

5 Khodabakhshirsquos Approach

Jahanshahloo and Khodabakhshi [31 32] and Khodabakhshi[33] introduced the following model for improving output

max 120601119900 + 120576(

119898

sum

119894=1

119904minus1198941 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus1198941 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus1198941 119904+1198942 119904+119903 ge 0

(12)

This model is a version of output-oriented BCC modelwhich is not limited in using observed sources of evaluatingDMU While output-oriented BCC model is limited tothe using of observed sources of evaluating DMU inputrelaxation model by imposing a few changes on some inputscan produce outputs more than observed output or outputsuggested by BCC model Using the above model will bevery useful when attracting some inputs such as labor whichis necessary to solve employment problem in a society Theconditions of efficiency under the above model for DMU119900being evaluated become as follows

Definition 9 (efficiency) DMU119900 is efficient under model (12)if the following two conditions are satisfied

(i) 120601lowast119900 = 1

(ii) The optimal amounts of all slacks are zero

Using the optimal solution of model (12) (120601lowast119900 120582lowast119895

119904minuslowast1198941 119904+lowast1198942 119904+lowast119903 ) the following model is solved for determining

technical inefficiency of inputs

max119898

sum

119894=1

120575+119894

st 119909119894119900 minus 119904minuslowast1198941 + 119904

+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 minus 120575+119894 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120575+119894 le 119904minuslowast119894 119894 = 1 119898

120582119895120575+119894 ge 0

(13)Finally the amount of congestion for 119894th input is defined asfollows

119904minus119888119894 = 119904

minuslowast1198941 minus 120575

+lowast119894 119894 = 1 119898 (14)

where 120575+lowast119894 (119894 = 1 119898) is obtained by solving model (13)If the amount of total slacks (total inefficiencies) 119904minuslowast1198941 andnoncongesting input are the same then we do not have anyinput congestion In other words in this case the amount ofinput congestion is zero while if the amount of total slacksis more than noncongesting input then the input congestionwill exist for 119894th input It is obvious that if 119904+lowast1198942 is positive for119894th input there is no congestion for this input

Khodabakhshi [33] presented the one-model version ofthis approach The two models ((12) (13)) can be replaced bya single model which provides an alternative procedure todetermine input congestion Using relation (14) model (13)can be rewritten as follows

max119898

sum

119894=1

minus 119904minus119888119894

st 119909119894119900 minus 119904minus119888119894 + 119904+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 ge 0

(15)

If we consider the following model

max 120601119900 + 120576(

119898

sum

119894=1

minus 119904minus119888119894 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119888119894 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 120601119900119910119903119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904+1198942 119904minus119888119894 119904+119903 ge 0

(16)

6 Journal of Applied Mathematics

it is obvious that if (120601lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal

solution of (16) (120601lowast119900 119904+lowast2 119904+lowast) are part of an optimal solution

of (12) and (120582lowast 119904+lowast) is an optimal solution of (15) In other

words we can regard model (15) as part of a two-stageprocedure for solvingmodel (16) Using one-model approachwe reduced the solving of three problems with two-modelapproach to the solving of two problems with one-modelapproach This is certainly important from computationalpoint of view

Theorem 10 Congestion is present if and only if optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal solution of (16)

and at least one of the following conditions is satisfied

(i) 120601lowast119900 gt 1 and there exists at least one 119894 (1 le 119894 le 119898) suchthat 119904minus119888lowast119894 gt 0

(ii) There exists at least one 119903 (1 le 119903 le 119904) for which 119904+lowast119903 gt 0

and also one 119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

Theorem 11 Congestion is present if and only if for an optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) of (16) there is at least one

119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

6 Nourarsquos Approach

In this section the approach of Noura et al [41] is brieflyreviewed In this method first model (3) is solved and theoptimal solution 120601

lowast119900 120582lowast 119878+lowast 119878minuslowast is obtained Then the set 119864

is defined as follows

119864 = 119895 | 120601lowast119895 = 1 (17)

Among theDMUs in set119864 there exists at least one sayDMU119897that has the highest consumption in its first input componentcompared with the first input component of the remainingDMUs of set 119864 1199091119897 is denoted by 119909

lowast1 Then among the

DMUs in 119864 a DMU is found say DMU119905 that has the highestconsumption in its second input component compared to theremaining DMUs in 119864 1199092119905 is denoted by 119909

lowast2 In a similar

manner 119909lowast1 119909lowast2 119909

lowast119898 can be identified

Definition 12 Congestion is present if and only if in anoptimal solution 120601lowast119900 120582

lowast 119878+lowast 119878minuslowast of (3) for DMU119900 at least one

of the following two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119909119894119900 gt 119909lowast119894 (119894 =

1 119898)(ii) There exists at least one 119878+lowast119903 gt 0 (119903 = 1 119904) and at

least one 119909119894119900 gt 119909lowast119894 (119894 = 1 119898)

They denote the amount of congestion in the 119894th input ofDMUo by 119904

1198881015840

119894 where 119909119894119900 gt 119909lowast119894 and define it as

1199041198881015840

119894 = 119909119894119900 minus 119909lowast119894

(18)

Congestion is considered not present when 119909119894119900 le 119909lowast119894 and 119904

1198881015840

119894 =

0 The sum of all 1199041198881015840

119894 is the amount of congestion in DMU119900Noura et al [41] to establish general validity of their

method prove three theorems which show that Definition 12is equivalent to Cooper et alrsquos Theorem 6 where 119909119894119900 ge 119909

lowast119894

Also they prove another theorem which shows that thereexists no congestion in DMU119900 when 119909119894119900 le 119909

lowast119894 The computa-

tional effort of this method for calculating congestion is lessthan Cooperrsquos approach

7 Wursquos Approach

In this section the input congestion measurement subjectin presence of undesirable outputs is considered using theapproach of Wu et al [42] Assume that 119909 119910 119906 repre-sent the inputs desirable outputs and undesirable outputsrespectively Evidence of congestion in this method occurswhenever reducing some inputs 119909 can increase desirableoutputs119910 and decrease undesirable outputs 119906 simultaneouslyBefore measuring this kind of congestion we should developthe approach solving the problem of undesirable outputsSo far there have been several approaches for addressingthe undesirable output problem Wu et al [42] choose theapproach of Seiford and Zhu [43] to deal with undesirableoutputs The model is shown as follows

max 120575

st119899

sum

119895=1

119909119894119895120582119895 le 1199091198940 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(19)

where 120572119905 is a big enough positive number that canmake every119900119905119895 positive The fourth constraint is used to transform theundesirable output to a new variable The third constraint isused to constrain the new variable in the possible productionset Wu et al [42] proposed the following model for deter-mining the input congestion of DMU0 in the presence ofundesirable outputs

max 119911

st119899

sum

119895=1

119909119894119895120582119895 = 1199091198940 119894 = 1 119898

Journal of Applied Mathematics 7

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(20)

The production possibility set corresponding to thesemodels is as follows

119879 =

(119909 119910 119906) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

119906119895120582119895 le 119906

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(21)

Definition 13 If the optimal value of model (20) satisfies 119911lowast =1 then we call DMU0 weakly efficient

Definition 14 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 gt 1199100 gt 1199060 then the DMU evidences congestion

Definition 15 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 ge 1199100 ge 1199060 then the DMU evidences congestion

Theorem16 Aweakly efficientDMU0 ofmodel (20) evidencescongestion if and only if it is not weakly efficient in model (19)

Proof See Wu et al [42]

8 Conclusion

There are several approaches for dealing with input con-gestion measurement in DEA In this review six importantmethods are briefly considered and compared Whilst eachtechnique is useful in a specific area none of the method-ologies can be prescribed as the complete solution to thequestion of input congestionHowever there is need to designa complete method which can cover all of the followingproperties

(1) to determine both sources and amounts of inputcongestion

(2) to deal with congestion problem under the conditionthat all inputs and outputs are nonnegative values

(3) to measure the input congestion amounts in thepresence of the multiple optimal solutions

(4) to decompose the measure of overall technical effi-ciency into pure technical efficiency scale efficiencyand congestion efficiency

(5) to deal with the congestion problem in the presenceof the undesirable outputs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[2] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[3] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[4] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002

[5] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985

[6] K Tone ldquoA slacks-based measure of efficiency in data envelop-ment analysisrdquo European Journal of Operational Research vol130 no 3 pp 498ndash509 2001

[7] W W Cooper H Deng Z M Huang and S X Li ldquoA one-model approach to congestion in data envelopment analysisrdquoSocio-Economic Planning Sciences vol 36 no 4 pp 231ndash2382002

[8] R Fare and L Svensson ldquoCongestion of production factorsrdquoEconometrica vol 48 no 7 pp 1745ndash1752 1980

[9] R Fare and S Grosskopf ldquoMeasuring congestion in produc-tionrdquo Zeitschrift fur Nationalokonomie vol 43 no 3 pp 257ndash271 1983

[10] R Fare S Grosskopf and C A K Lovell The Measurementof Efficiency of Production Kluwer-Nijhoff Boston Mass USA1985

[11] P Byrnes R Fare and S Grosskopf ldquoMeasuring productiveefficiency an application to Illinois strip minesrdquo ManagementScience vol 30 no 6 pp 671ndash681 1984

[12] P Byrnes R Fare S Grosskopf and C A K Lovell ldquoTheeffect of unions on productivity US surface mining of coalrdquoManagement Science vol 34 no 9 pp 1037ndash1053 1988

[13] R Fare and S Grosskopf ldquoCongestion a noterdquo Socio-EconomicPlanning Sciences vol 32 no 1 pp 21ndash23 1998

[14] R Fare and S Grosskopf ldquoSlacks and congestion a commentrdquoSocio-Economic Planning Sciences vol 34 no 1 pp 27ndash33 2000

[15] R Fare and S Grosskopf ldquoDecomposing technical efficiencywith carerdquoManagement Science vol 46 no 1 pp 167ndash168 2000

8 Journal of Applied Mathematics

[16] R Fare and S Grosskopf ldquoWhen can slacks be used to identifycongestion An answer to W W Cooper L Seiford and J ZhurdquoSocio-Economic Planning Sciences vol 35 no 3 pp 217ndash2212001

[17] W W Cooper R G Thompson and R M Thrall ldquoIntroduc-tion extensions and new developments in DEArdquo Annals ofOperations Research vol 66 pp 3ndash45 1996

[18] P L Brockett W W Cooper H-C Shin and Y WangldquoInefficiency and congestion in Chinese production before andafter the 1978 economic reformsrdquo Socio-Economic PlanningSciences vol 32 no 1 pp 1ndash20 1998

[19] W W Cooper L M Seiford and J Zhu ldquoA unified additivemodel approach for evaluating inefficiency and congestionwith associated measures in DEArdquo Socio-Economic PlanningSciences vol 34 no 1 pp 1ndash25 2000

[20] W W Cooper H Deng B Gu S Li and R M Thrall ldquoUsingDEA to improve the management of congestion in Chineseindustries (1981ndash1997)rdquo Socio-Economic Planning Sciences vol35 no 4 pp 227ndash242 2001

[21] L Cherchye T Kuosmanen andT Post ldquoAlternative treatmentsof congestion in DEA a rejoinder to Cooper Gu and LirdquoEuropean Journal of Operational Research vol 132 no 1 pp 75ndash80 2001

[22] WWCooper BGu and S Li ldquoComparisons and evaluations ofalternative approaches to the treatment of congestion in DEArdquoEuropean Journal of Operational Research vol 132 no 1 pp 62ndash74 2001

[23] W W Cooper B Gu and S Li ldquoNote alternative treatments ofcongestion in DEAmdasha response to the Cherchye Kuosmanenand Post critiquerdquo European Journal of Operational Researchvol 132 no 1 pp 81ndash87 2001

[24] WW Cooper LM Seiford and J Zhu ldquoSlacks and congestionresponse to a comment by R Fare and S Grosskopfrdquo Socio-Economic Planning Sciences vol 35 no 3 pp 205ndash215 2001

[25] WW Cooper H Deng L M Seiford and J Zhu ldquoCongestionits identification and management with DEArdquo in Handbook ofData Envelopment Analysis W W Cooper L M Seiford and JZhu Eds pp 177ndash201 Kluwer Academic Boston Mass USA2004

[26] K Tone and B K Sahoo ldquoDegree of scale economies andcongestion a unified DEA approachrdquo European Journal ofOperational Research vol 158 no 3 pp 755ndash772 2004

[27] Q Wei and H Yan ldquoCongestion and returns to scale indata envelopment analysisrdquo European Journal of OperationalResearch vol 153 no 3 pp 641ndash660 2004

[28] T Sueyoshi and K Sekitani ldquoDEA congestion and returns toscale under an occurrence of multiple optimal projectionsrdquoEuropean Journal of Operational Research vol 194 no 2 pp592ndash607 2009

[29] A T Flegg and D O Allen ldquoDoes expansion cause congestionThe case of the older British universities 1994ndash2004rdquo EducationEconomics vol 15 no 1 pp 75ndash102 2007

[30] M Khoveyni R EslamiM Khodabakhshi G R Jahanshahlooand F H Hosseinzadeh ldquoRecognizing strong and weak con-gestion slack based in data envelopment analysisrdquo Computersamp Industrial Engineering vol 64 no 2 pp 731ndash738 2013

[31] G R Jahanshahloo and M Khodabakhshi ldquoSuitable combina-tion of inputs for improving outputs in DEA with determin-ing input congestion considering textile industry of Chinardquo

Applied Mathematics and Computation vol 151 no 1 pp 263ndash273 2004

[32] G R Jahanshahloo andMKhodabakhshi ldquoDetermining assur-ance interval for non-Archimedean element in the improvingoutputsmodel inDEArdquoAppliedMathematics and Computationvol 151 no 2 pp 501ndash506 2004

[33] M Khodabakhshi ldquoA one-model approach based on relaxedcombinations of inputs for evaluating input congestion inDEArdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 443ndash450 2009

[34] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007

[35] M Khodabakhshi and M Asgharian ldquoAn input relaxationmeasure of efficiency in stochastic data envelopment analysisrdquoApplied Mathematical Modelling vol 33 no 4 pp 2010ndash20232009

[36] 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

[37] M Khodabakhshi ldquoChance constrained additive input relax-ation model in stochastic data envelopment analysisrdquo Journal ofInformation and Systems Sciences vol 6 no 1 pp 99ndash112 2010

[38] M Asgharian M Khodabakhshi and L Neralic ldquoCongestionin stochastic data envelopment analysis an input relaxationapproachrdquo International Journal of Statistics and ManagementSystem vol 5 no 1 pp 84ndash106 2010

[39] G R Jahanshahloo M Khodabakhshi F H Lotfi and RM Goudarzi ldquoComputation of congestion in DEA modelswith productions trade-offs and weight restrictionsrdquo AppliedMathematical Sciences vol 5 no 14 pp 663ndash676 2011

[40] M Khodabakhshi R M Goudarzi M Y Maryaki and MHajiani ldquoA one-model approach for computation of congestionwith productions trade-offs and weight restrictionsrdquo Interna-tional Journal of Applied Operational Research vol 3 no 4 pp69ndash80 2013

[41] A A Noura F H Hosseinzadeh G R Jahanshahloo S FRashidi and B R Parker ldquoA new method for measuring con-gestion in data envelopment analysisrdquo Socio-Economic PlanningSciences vol 44 no 4 pp 240ndash246 2010

[42] J Wu Q An B Xiong and Y Chen ldquoCongestionmeasurementfor regional industries in China a data envelopment analysisapproach with undesirable outputsrdquo Energy Policy vol 57 pp7ndash13 2012

[43] L M Seiford and J Zhu ldquoModeling undesirable factors in effi-ciency evaluationrdquo European Journal of Operational Researchvol 142 no 1 pp 16ndash20 2002

[44] B Dervaux K Kerstens and P V Eeckaut ldquoRadial and non-radial static efficiency decompositions a focus on congestionmeasurementrdquo Transportation Research B Methodological vol32 no 5 pp 299ndash312 1998

[45] A T Flegg and D O Allen ldquoCongestion in the Chineseautomobile and textile industries revisitedrdquo Socio-EconomicPlanning Sciences vol 43 no 3 pp 177ndash191 2009

[46] C Kao ldquoCongestion measurement and elimination under theframework of data envelopment analysisrdquo International Journalof Production Economics vol 123 no 2 pp 257ndash265 2010

Journal of Applied Mathematics 9

[47] J Odeck ldquoCongestion ownership region of operation andscale their impact on bus operator performance in NorwayrdquoSocio-Economic Planning Sciences vol 40 no 1 pp 52ndash69 2006

[48] Q Wei and H Yan ldquoWeak congestion in output additive dataenvelopment analysisrdquo Socio-Economic Planning Sciences vol43 no 1 pp 40ndash54 2009

[49] T Sueyoshi and M Goto ldquoWeak and strong disposabilityvs natural and managerial disposability in DEA environmen-tal assessment comparison between Japanese electric powerindustry and manufacturing industriesrdquo Energy Economics vol34 no 3 pp 686ndash699 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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4 Journal of Applied Mathematics

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 ge 0 119894 = 1 119898

(6)

Using 120576 the models ((3) (6)) are combined into thefollowing single model

max 120601119900 + 120576(minus

119898

sum

119894=1

119904minus119888119894 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119888119894 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 119904+119903 ge 0

(7)

Theorem 6 Congestion is present if and only if in an optimalsolution 120601

lowast119900 120582lowast 119878+lowast 119878minus119888lowast of (7) at least one of the following

two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119878minus119888lowast gt 0 (1 le 119894 le 119898)

(ii) There exists at least one 119878+lowast119903 gt 0 (1 le 119903 le 119904) and atleast one 119878minus119888lowast gt 0 (1 le 119894 le 119898)

Proof See Cooper et al [7]

4 Tonersquos Approach

In the approach which was presented by Tone and Sahoo[26] it is assumed that all inputs and outputs are positiveThisapproach is based on the following production possibility set(PPS)

119875Convex

=

(119909 119910) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(8)

In this approach for measuring the congestion effectthe pure technical efficiency is firstly determined using thefollowing model

max 120583

st119899

sum

119895=1

119909119895120582119895 = 119909119900

119899

sum

119895=1

119910119895120582119895 minus 119902+= 119910119900120583

119899

sum

119895=1

120582119895 = 1

120582119895 119902+ge 0

(9)

Suppose that (1199090 119910119900) is the projection of DMU119900 which isobtained as

1199090 = 1199090

119910119900 = 120583lowast119910119900 + 119902

+lowast

(10)

In this case (1199090 119910119900) is on the strongly efficient frontier of119875Convex Here the definitions of congestion (strong and weak)from Tonersquos approach are assumed as follows

Definition 7 (strong congestion) A DMU119900 is strongly con-gested if there exists an activity (119909 119910) isin 119875Convex such that119909 = 120572119909119900 (with 0 lt 120572 lt 1) and 119910119900 ge 120573119910119900 (with 120573 gt 1)

Definition 8 (weak congestion) A DMU is (weakly) con-gested if it is strongly efficient and there exists an activityin 119875Convex that uses less resources in one or more inputs formaking more products in one or more outputs

First we consider DMU119900 which is strongly 119875Convex effi-cient Thus in each optimal solution of model (9) 120583lowast = 1

and 119902+lowast

= 0 Then the following mathematical program issolved for a specific DMU119900 Consider the following

max 1

119904

119904

sum

119903=1

119905+119903

119910119903119900

+ 120576(

1

119898

119898

sum

119894=1

119905minus119894

119909119894119900

)

st119899

sum

119895=1

119909119895120582119895 + 119905minus119894 = 119909119900

119899

sum

119895=1

119910119895120582119895 minus 119905+= 119910119900

119899

sum

119895=1

120582119895 = 1

120582119895 119905+ 119905minusge 0

(11)

where 120576 is a non-Archimedean small positive number

Journal of Applied Mathematics 5

Next let (120582lowast 119905minuslowast 119905+lowast) be an optimal solution of model(11) Then we will have the following cases

(i) If 119905+lowast = 0 DMU119900 has no congestion because thedecreasing in inputs cannot increase any outputs

(ii) If 119905+lowast = 0 119905minuslowast is also not zero since DMU119900 is strongly119875Convex efficient Consequently DMU119900 has (weak)congestion

5 Khodabakhshirsquos Approach

Jahanshahloo and Khodabakhshi [31 32] and Khodabakhshi[33] introduced the following model for improving output

max 120601119900 + 120576(

119898

sum

119894=1

119904minus1198941 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus1198941 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus1198941 119904+1198942 119904+119903 ge 0

(12)

This model is a version of output-oriented BCC modelwhich is not limited in using observed sources of evaluatingDMU While output-oriented BCC model is limited tothe using of observed sources of evaluating DMU inputrelaxation model by imposing a few changes on some inputscan produce outputs more than observed output or outputsuggested by BCC model Using the above model will bevery useful when attracting some inputs such as labor whichis necessary to solve employment problem in a society Theconditions of efficiency under the above model for DMU119900being evaluated become as follows

Definition 9 (efficiency) DMU119900 is efficient under model (12)if the following two conditions are satisfied

(i) 120601lowast119900 = 1

(ii) The optimal amounts of all slacks are zero

Using the optimal solution of model (12) (120601lowast119900 120582lowast119895

119904minuslowast1198941 119904+lowast1198942 119904+lowast119903 ) the following model is solved for determining

technical inefficiency of inputs

max119898

sum

119894=1

120575+119894

st 119909119894119900 minus 119904minuslowast1198941 + 119904

+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 minus 120575+119894 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120575+119894 le 119904minuslowast119894 119894 = 1 119898

120582119895120575+119894 ge 0

(13)Finally the amount of congestion for 119894th input is defined asfollows

119904minus119888119894 = 119904

minuslowast1198941 minus 120575

+lowast119894 119894 = 1 119898 (14)

where 120575+lowast119894 (119894 = 1 119898) is obtained by solving model (13)If the amount of total slacks (total inefficiencies) 119904minuslowast1198941 andnoncongesting input are the same then we do not have anyinput congestion In other words in this case the amount ofinput congestion is zero while if the amount of total slacksis more than noncongesting input then the input congestionwill exist for 119894th input It is obvious that if 119904+lowast1198942 is positive for119894th input there is no congestion for this input

Khodabakhshi [33] presented the one-model version ofthis approach The two models ((12) (13)) can be replaced bya single model which provides an alternative procedure todetermine input congestion Using relation (14) model (13)can be rewritten as follows

max119898

sum

119894=1

minus 119904minus119888119894

st 119909119894119900 minus 119904minus119888119894 + 119904+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 ge 0

(15)

If we consider the following model

max 120601119900 + 120576(

119898

sum

119894=1

minus 119904minus119888119894 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119888119894 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 120601119900119910119903119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904+1198942 119904minus119888119894 119904+119903 ge 0

(16)

6 Journal of Applied Mathematics

it is obvious that if (120601lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal

solution of (16) (120601lowast119900 119904+lowast2 119904+lowast) are part of an optimal solution

of (12) and (120582lowast 119904+lowast) is an optimal solution of (15) In other

words we can regard model (15) as part of a two-stageprocedure for solvingmodel (16) Using one-model approachwe reduced the solving of three problems with two-modelapproach to the solving of two problems with one-modelapproach This is certainly important from computationalpoint of view

Theorem 10 Congestion is present if and only if optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal solution of (16)

and at least one of the following conditions is satisfied

(i) 120601lowast119900 gt 1 and there exists at least one 119894 (1 le 119894 le 119898) suchthat 119904minus119888lowast119894 gt 0

(ii) There exists at least one 119903 (1 le 119903 le 119904) for which 119904+lowast119903 gt 0

and also one 119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

Theorem 11 Congestion is present if and only if for an optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) of (16) there is at least one

119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

6 Nourarsquos Approach

In this section the approach of Noura et al [41] is brieflyreviewed In this method first model (3) is solved and theoptimal solution 120601

lowast119900 120582lowast 119878+lowast 119878minuslowast is obtained Then the set 119864

is defined as follows

119864 = 119895 | 120601lowast119895 = 1 (17)

Among theDMUs in set119864 there exists at least one sayDMU119897that has the highest consumption in its first input componentcompared with the first input component of the remainingDMUs of set 119864 1199091119897 is denoted by 119909

lowast1 Then among the

DMUs in 119864 a DMU is found say DMU119905 that has the highestconsumption in its second input component compared to theremaining DMUs in 119864 1199092119905 is denoted by 119909

lowast2 In a similar

manner 119909lowast1 119909lowast2 119909

lowast119898 can be identified

Definition 12 Congestion is present if and only if in anoptimal solution 120601lowast119900 120582

lowast 119878+lowast 119878minuslowast of (3) for DMU119900 at least one

of the following two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119909119894119900 gt 119909lowast119894 (119894 =

1 119898)(ii) There exists at least one 119878+lowast119903 gt 0 (119903 = 1 119904) and at

least one 119909119894119900 gt 119909lowast119894 (119894 = 1 119898)

They denote the amount of congestion in the 119894th input ofDMUo by 119904

1198881015840

119894 where 119909119894119900 gt 119909lowast119894 and define it as

1199041198881015840

119894 = 119909119894119900 minus 119909lowast119894

(18)

Congestion is considered not present when 119909119894119900 le 119909lowast119894 and 119904

1198881015840

119894 =

0 The sum of all 1199041198881015840

119894 is the amount of congestion in DMU119900Noura et al [41] to establish general validity of their

method prove three theorems which show that Definition 12is equivalent to Cooper et alrsquos Theorem 6 where 119909119894119900 ge 119909

lowast119894

Also they prove another theorem which shows that thereexists no congestion in DMU119900 when 119909119894119900 le 119909

lowast119894 The computa-

tional effort of this method for calculating congestion is lessthan Cooperrsquos approach

7 Wursquos Approach

In this section the input congestion measurement subjectin presence of undesirable outputs is considered using theapproach of Wu et al [42] Assume that 119909 119910 119906 repre-sent the inputs desirable outputs and undesirable outputsrespectively Evidence of congestion in this method occurswhenever reducing some inputs 119909 can increase desirableoutputs119910 and decrease undesirable outputs 119906 simultaneouslyBefore measuring this kind of congestion we should developthe approach solving the problem of undesirable outputsSo far there have been several approaches for addressingthe undesirable output problem Wu et al [42] choose theapproach of Seiford and Zhu [43] to deal with undesirableoutputs The model is shown as follows

max 120575

st119899

sum

119895=1

119909119894119895120582119895 le 1199091198940 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(19)

where 120572119905 is a big enough positive number that canmake every119900119905119895 positive The fourth constraint is used to transform theundesirable output to a new variable The third constraint isused to constrain the new variable in the possible productionset Wu et al [42] proposed the following model for deter-mining the input congestion of DMU0 in the presence ofundesirable outputs

max 119911

st119899

sum

119895=1

119909119894119895120582119895 = 1199091198940 119894 = 1 119898

Journal of Applied Mathematics 7

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(20)

The production possibility set corresponding to thesemodels is as follows

119879 =

(119909 119910 119906) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

119906119895120582119895 le 119906

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(21)

Definition 13 If the optimal value of model (20) satisfies 119911lowast =1 then we call DMU0 weakly efficient

Definition 14 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 gt 1199100 gt 1199060 then the DMU evidences congestion

Definition 15 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 ge 1199100 ge 1199060 then the DMU evidences congestion

Theorem16 Aweakly efficientDMU0 ofmodel (20) evidencescongestion if and only if it is not weakly efficient in model (19)

Proof See Wu et al [42]

8 Conclusion

There are several approaches for dealing with input con-gestion measurement in DEA In this review six importantmethods are briefly considered and compared Whilst eachtechnique is useful in a specific area none of the method-ologies can be prescribed as the complete solution to thequestion of input congestionHowever there is need to designa complete method which can cover all of the followingproperties

(1) to determine both sources and amounts of inputcongestion

(2) to deal with congestion problem under the conditionthat all inputs and outputs are nonnegative values

(3) to measure the input congestion amounts in thepresence of the multiple optimal solutions

(4) to decompose the measure of overall technical effi-ciency into pure technical efficiency scale efficiencyand congestion efficiency

(5) to deal with the congestion problem in the presenceof the undesirable outputs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[2] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[3] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[4] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002

[5] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985

[6] K Tone ldquoA slacks-based measure of efficiency in data envelop-ment analysisrdquo European Journal of Operational Research vol130 no 3 pp 498ndash509 2001

[7] W W Cooper H Deng Z M Huang and S X Li ldquoA one-model approach to congestion in data envelopment analysisrdquoSocio-Economic Planning Sciences vol 36 no 4 pp 231ndash2382002

[8] R Fare and L Svensson ldquoCongestion of production factorsrdquoEconometrica vol 48 no 7 pp 1745ndash1752 1980

[9] R Fare and S Grosskopf ldquoMeasuring congestion in produc-tionrdquo Zeitschrift fur Nationalokonomie vol 43 no 3 pp 257ndash271 1983

[10] R Fare S Grosskopf and C A K Lovell The Measurementof Efficiency of Production Kluwer-Nijhoff Boston Mass USA1985

[11] P Byrnes R Fare and S Grosskopf ldquoMeasuring productiveefficiency an application to Illinois strip minesrdquo ManagementScience vol 30 no 6 pp 671ndash681 1984

[12] P Byrnes R Fare S Grosskopf and C A K Lovell ldquoTheeffect of unions on productivity US surface mining of coalrdquoManagement Science vol 34 no 9 pp 1037ndash1053 1988

[13] R Fare and S Grosskopf ldquoCongestion a noterdquo Socio-EconomicPlanning Sciences vol 32 no 1 pp 21ndash23 1998

[14] R Fare and S Grosskopf ldquoSlacks and congestion a commentrdquoSocio-Economic Planning Sciences vol 34 no 1 pp 27ndash33 2000

[15] R Fare and S Grosskopf ldquoDecomposing technical efficiencywith carerdquoManagement Science vol 46 no 1 pp 167ndash168 2000

8 Journal of Applied Mathematics

[16] R Fare and S Grosskopf ldquoWhen can slacks be used to identifycongestion An answer to W W Cooper L Seiford and J ZhurdquoSocio-Economic Planning Sciences vol 35 no 3 pp 217ndash2212001

[17] W W Cooper R G Thompson and R M Thrall ldquoIntroduc-tion extensions and new developments in DEArdquo Annals ofOperations Research vol 66 pp 3ndash45 1996

[18] P L Brockett W W Cooper H-C Shin and Y WangldquoInefficiency and congestion in Chinese production before andafter the 1978 economic reformsrdquo Socio-Economic PlanningSciences vol 32 no 1 pp 1ndash20 1998

[19] W W Cooper L M Seiford and J Zhu ldquoA unified additivemodel approach for evaluating inefficiency and congestionwith associated measures in DEArdquo Socio-Economic PlanningSciences vol 34 no 1 pp 1ndash25 2000

[20] W W Cooper H Deng B Gu S Li and R M Thrall ldquoUsingDEA to improve the management of congestion in Chineseindustries (1981ndash1997)rdquo Socio-Economic Planning Sciences vol35 no 4 pp 227ndash242 2001

[21] L Cherchye T Kuosmanen andT Post ldquoAlternative treatmentsof congestion in DEA a rejoinder to Cooper Gu and LirdquoEuropean Journal of Operational Research vol 132 no 1 pp 75ndash80 2001

[22] WWCooper BGu and S Li ldquoComparisons and evaluations ofalternative approaches to the treatment of congestion in DEArdquoEuropean Journal of Operational Research vol 132 no 1 pp 62ndash74 2001

[23] W W Cooper B Gu and S Li ldquoNote alternative treatments ofcongestion in DEAmdasha response to the Cherchye Kuosmanenand Post critiquerdquo European Journal of Operational Researchvol 132 no 1 pp 81ndash87 2001

[24] WW Cooper LM Seiford and J Zhu ldquoSlacks and congestionresponse to a comment by R Fare and S Grosskopfrdquo Socio-Economic Planning Sciences vol 35 no 3 pp 205ndash215 2001

[25] WW Cooper H Deng L M Seiford and J Zhu ldquoCongestionits identification and management with DEArdquo in Handbook ofData Envelopment Analysis W W Cooper L M Seiford and JZhu Eds pp 177ndash201 Kluwer Academic Boston Mass USA2004

[26] K Tone and B K Sahoo ldquoDegree of scale economies andcongestion a unified DEA approachrdquo European Journal ofOperational Research vol 158 no 3 pp 755ndash772 2004

[27] Q Wei and H Yan ldquoCongestion and returns to scale indata envelopment analysisrdquo European Journal of OperationalResearch vol 153 no 3 pp 641ndash660 2004

[28] T Sueyoshi and K Sekitani ldquoDEA congestion and returns toscale under an occurrence of multiple optimal projectionsrdquoEuropean Journal of Operational Research vol 194 no 2 pp592ndash607 2009

[29] A T Flegg and D O Allen ldquoDoes expansion cause congestionThe case of the older British universities 1994ndash2004rdquo EducationEconomics vol 15 no 1 pp 75ndash102 2007

[30] M Khoveyni R EslamiM Khodabakhshi G R Jahanshahlooand F H Hosseinzadeh ldquoRecognizing strong and weak con-gestion slack based in data envelopment analysisrdquo Computersamp Industrial Engineering vol 64 no 2 pp 731ndash738 2013

[31] G R Jahanshahloo and M Khodabakhshi ldquoSuitable combina-tion of inputs for improving outputs in DEA with determin-ing input congestion considering textile industry of Chinardquo

Applied Mathematics and Computation vol 151 no 1 pp 263ndash273 2004

[32] G R Jahanshahloo andMKhodabakhshi ldquoDetermining assur-ance interval for non-Archimedean element in the improvingoutputsmodel inDEArdquoAppliedMathematics and Computationvol 151 no 2 pp 501ndash506 2004

[33] M Khodabakhshi ldquoA one-model approach based on relaxedcombinations of inputs for evaluating input congestion inDEArdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 443ndash450 2009

[34] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007

[35] M Khodabakhshi and M Asgharian ldquoAn input relaxationmeasure of efficiency in stochastic data envelopment analysisrdquoApplied Mathematical Modelling vol 33 no 4 pp 2010ndash20232009

[36] 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

[37] M Khodabakhshi ldquoChance constrained additive input relax-ation model in stochastic data envelopment analysisrdquo Journal ofInformation and Systems Sciences vol 6 no 1 pp 99ndash112 2010

[38] M Asgharian M Khodabakhshi and L Neralic ldquoCongestionin stochastic data envelopment analysis an input relaxationapproachrdquo International Journal of Statistics and ManagementSystem vol 5 no 1 pp 84ndash106 2010

[39] G R Jahanshahloo M Khodabakhshi F H Lotfi and RM Goudarzi ldquoComputation of congestion in DEA modelswith productions trade-offs and weight restrictionsrdquo AppliedMathematical Sciences vol 5 no 14 pp 663ndash676 2011

[40] M Khodabakhshi R M Goudarzi M Y Maryaki and MHajiani ldquoA one-model approach for computation of congestionwith productions trade-offs and weight restrictionsrdquo Interna-tional Journal of Applied Operational Research vol 3 no 4 pp69ndash80 2013

[41] A A Noura F H Hosseinzadeh G R Jahanshahloo S FRashidi and B R Parker ldquoA new method for measuring con-gestion in data envelopment analysisrdquo Socio-Economic PlanningSciences vol 44 no 4 pp 240ndash246 2010

[42] J Wu Q An B Xiong and Y Chen ldquoCongestionmeasurementfor regional industries in China a data envelopment analysisapproach with undesirable outputsrdquo Energy Policy vol 57 pp7ndash13 2012

[43] L M Seiford and J Zhu ldquoModeling undesirable factors in effi-ciency evaluationrdquo European Journal of Operational Researchvol 142 no 1 pp 16ndash20 2002

[44] B Dervaux K Kerstens and P V Eeckaut ldquoRadial and non-radial static efficiency decompositions a focus on congestionmeasurementrdquo Transportation Research B Methodological vol32 no 5 pp 299ndash312 1998

[45] A T Flegg and D O Allen ldquoCongestion in the Chineseautomobile and textile industries revisitedrdquo Socio-EconomicPlanning Sciences vol 43 no 3 pp 177ndash191 2009

[46] C Kao ldquoCongestion measurement and elimination under theframework of data envelopment analysisrdquo International Journalof Production Economics vol 123 no 2 pp 257ndash265 2010

Journal of Applied Mathematics 9

[47] J Odeck ldquoCongestion ownership region of operation andscale their impact on bus operator performance in NorwayrdquoSocio-Economic Planning Sciences vol 40 no 1 pp 52ndash69 2006

[48] Q Wei and H Yan ldquoWeak congestion in output additive dataenvelopment analysisrdquo Socio-Economic Planning Sciences vol43 no 1 pp 40ndash54 2009

[49] T Sueyoshi and M Goto ldquoWeak and strong disposabilityvs natural and managerial disposability in DEA environmen-tal assessment comparison between Japanese electric powerindustry and manufacturing industriesrdquo Energy Economics vol34 no 3 pp 686ndash699 2012

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

Next let (120582lowast 119905minuslowast 119905+lowast) be an optimal solution of model(11) Then we will have the following cases

(i) If 119905+lowast = 0 DMU119900 has no congestion because thedecreasing in inputs cannot increase any outputs

(ii) If 119905+lowast = 0 119905minuslowast is also not zero since DMU119900 is strongly119875Convex efficient Consequently DMU119900 has (weak)congestion

5 Khodabakhshirsquos Approach

Jahanshahloo and Khodabakhshi [31 32] and Khodabakhshi[33] introduced the following model for improving output

max 120601119900 + 120576(

119898

sum

119894=1

119904minus1198941 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus1198941 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 119910119903119900120601119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus1198941 119904+1198942 119904+119903 ge 0

(12)

This model is a version of output-oriented BCC modelwhich is not limited in using observed sources of evaluatingDMU While output-oriented BCC model is limited tothe using of observed sources of evaluating DMU inputrelaxation model by imposing a few changes on some inputscan produce outputs more than observed output or outputsuggested by BCC model Using the above model will bevery useful when attracting some inputs such as labor whichis necessary to solve employment problem in a society Theconditions of efficiency under the above model for DMU119900being evaluated become as follows

Definition 9 (efficiency) DMU119900 is efficient under model (12)if the following two conditions are satisfied

(i) 120601lowast119900 = 1

(ii) The optimal amounts of all slacks are zero

Using the optimal solution of model (12) (120601lowast119900 120582lowast119895

119904minuslowast1198941 119904+lowast1198942 119904+lowast119903 ) the following model is solved for determining

technical inefficiency of inputs

max119898

sum

119894=1

120575+119894

st 119909119894119900 minus 119904minuslowast1198941 + 119904

+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 minus 120575+119894 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120575+119894 le 119904minuslowast119894 119894 = 1 119898

120582119895120575+119894 ge 0

(13)Finally the amount of congestion for 119894th input is defined asfollows

119904minus119888119894 = 119904

minuslowast1198941 minus 120575

+lowast119894 119894 = 1 119898 (14)

where 120575+lowast119894 (119894 = 1 119898) is obtained by solving model (13)If the amount of total slacks (total inefficiencies) 119904minuslowast1198941 andnoncongesting input are the same then we do not have anyinput congestion In other words in this case the amount ofinput congestion is zero while if the amount of total slacksis more than noncongesting input then the input congestionwill exist for 119894th input It is obvious that if 119904+lowast1198942 is positive for119894th input there is no congestion for this input

Khodabakhshi [33] presented the one-model version ofthis approach The two models ((12) (13)) can be replaced bya single model which provides an alternative procedure todetermine input congestion Using relation (14) model (13)can be rewritten as follows

max119898

sum

119894=1

minus 119904minus119888119894

st 119909119894119900 minus 119904minus119888119894 + 119904+lowast1198942 =

119899

sum

119895=1

119909119894119895120582119895 119894 = 1 119898

120601lowast119900119910119903119900 + 119904

+lowast119903 =

119899

sum

119895=1

119910119903119895120582119895 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904minus119888119894 ge 0

(15)

If we consider the following model

max 120601119900 + 120576(

119898

sum

119894=1

minus 119904minus119888119894 minus

119898

sum

119894=1

119904+1198942 +

119904

sum

119903=1

119904+119903)

st 119909119894119900 =

119899

sum

119895=1

119909119894119895120582119895 + 119904minus119888119894 minus 119904+1198942 119894 = 1 119898

0 =

119899

sum

119895=1

119910119903119895120582119895 minus 120601119900119910119903119900 minus 119904+119903 119903 = 1 119904

1 =

119899

sum

119895=1

120582119895

120582119895 119904+1198942 119904minus119888119894 119904+119903 ge 0

(16)

6 Journal of Applied Mathematics

it is obvious that if (120601lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal

solution of (16) (120601lowast119900 119904+lowast2 119904+lowast) are part of an optimal solution

of (12) and (120582lowast 119904+lowast) is an optimal solution of (15) In other

words we can regard model (15) as part of a two-stageprocedure for solvingmodel (16) Using one-model approachwe reduced the solving of three problems with two-modelapproach to the solving of two problems with one-modelapproach This is certainly important from computationalpoint of view

Theorem 10 Congestion is present if and only if optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal solution of (16)

and at least one of the following conditions is satisfied

(i) 120601lowast119900 gt 1 and there exists at least one 119894 (1 le 119894 le 119898) suchthat 119904minus119888lowast119894 gt 0

(ii) There exists at least one 119903 (1 le 119903 le 119904) for which 119904+lowast119903 gt 0

and also one 119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

Theorem 11 Congestion is present if and only if for an optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) of (16) there is at least one

119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

6 Nourarsquos Approach

In this section the approach of Noura et al [41] is brieflyreviewed In this method first model (3) is solved and theoptimal solution 120601

lowast119900 120582lowast 119878+lowast 119878minuslowast is obtained Then the set 119864

is defined as follows

119864 = 119895 | 120601lowast119895 = 1 (17)

Among theDMUs in set119864 there exists at least one sayDMU119897that has the highest consumption in its first input componentcompared with the first input component of the remainingDMUs of set 119864 1199091119897 is denoted by 119909

lowast1 Then among the

DMUs in 119864 a DMU is found say DMU119905 that has the highestconsumption in its second input component compared to theremaining DMUs in 119864 1199092119905 is denoted by 119909

lowast2 In a similar

manner 119909lowast1 119909lowast2 119909

lowast119898 can be identified

Definition 12 Congestion is present if and only if in anoptimal solution 120601lowast119900 120582

lowast 119878+lowast 119878minuslowast of (3) for DMU119900 at least one

of the following two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119909119894119900 gt 119909lowast119894 (119894 =

1 119898)(ii) There exists at least one 119878+lowast119903 gt 0 (119903 = 1 119904) and at

least one 119909119894119900 gt 119909lowast119894 (119894 = 1 119898)

They denote the amount of congestion in the 119894th input ofDMUo by 119904

1198881015840

119894 where 119909119894119900 gt 119909lowast119894 and define it as

1199041198881015840

119894 = 119909119894119900 minus 119909lowast119894

(18)

Congestion is considered not present when 119909119894119900 le 119909lowast119894 and 119904

1198881015840

119894 =

0 The sum of all 1199041198881015840

119894 is the amount of congestion in DMU119900Noura et al [41] to establish general validity of their

method prove three theorems which show that Definition 12is equivalent to Cooper et alrsquos Theorem 6 where 119909119894119900 ge 119909

lowast119894

Also they prove another theorem which shows that thereexists no congestion in DMU119900 when 119909119894119900 le 119909

lowast119894 The computa-

tional effort of this method for calculating congestion is lessthan Cooperrsquos approach

7 Wursquos Approach

In this section the input congestion measurement subjectin presence of undesirable outputs is considered using theapproach of Wu et al [42] Assume that 119909 119910 119906 repre-sent the inputs desirable outputs and undesirable outputsrespectively Evidence of congestion in this method occurswhenever reducing some inputs 119909 can increase desirableoutputs119910 and decrease undesirable outputs 119906 simultaneouslyBefore measuring this kind of congestion we should developthe approach solving the problem of undesirable outputsSo far there have been several approaches for addressingthe undesirable output problem Wu et al [42] choose theapproach of Seiford and Zhu [43] to deal with undesirableoutputs The model is shown as follows

max 120575

st119899

sum

119895=1

119909119894119895120582119895 le 1199091198940 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(19)

where 120572119905 is a big enough positive number that canmake every119900119905119895 positive The fourth constraint is used to transform theundesirable output to a new variable The third constraint isused to constrain the new variable in the possible productionset Wu et al [42] proposed the following model for deter-mining the input congestion of DMU0 in the presence ofundesirable outputs

max 119911

st119899

sum

119895=1

119909119894119895120582119895 = 1199091198940 119894 = 1 119898

Journal of Applied Mathematics 7

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(20)

The production possibility set corresponding to thesemodels is as follows

119879 =

(119909 119910 119906) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

119906119895120582119895 le 119906

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(21)

Definition 13 If the optimal value of model (20) satisfies 119911lowast =1 then we call DMU0 weakly efficient

Definition 14 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 gt 1199100 gt 1199060 then the DMU evidences congestion

Definition 15 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 ge 1199100 ge 1199060 then the DMU evidences congestion

Theorem16 Aweakly efficientDMU0 ofmodel (20) evidencescongestion if and only if it is not weakly efficient in model (19)

Proof See Wu et al [42]

8 Conclusion

There are several approaches for dealing with input con-gestion measurement in DEA In this review six importantmethods are briefly considered and compared Whilst eachtechnique is useful in a specific area none of the method-ologies can be prescribed as the complete solution to thequestion of input congestionHowever there is need to designa complete method which can cover all of the followingproperties

(1) to determine both sources and amounts of inputcongestion

(2) to deal with congestion problem under the conditionthat all inputs and outputs are nonnegative values

(3) to measure the input congestion amounts in thepresence of the multiple optimal solutions

(4) to decompose the measure of overall technical effi-ciency into pure technical efficiency scale efficiencyand congestion efficiency

(5) to deal with the congestion problem in the presenceof the undesirable outputs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[2] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[3] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[4] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002

[5] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985

[6] K Tone ldquoA slacks-based measure of efficiency in data envelop-ment analysisrdquo European Journal of Operational Research vol130 no 3 pp 498ndash509 2001

[7] W W Cooper H Deng Z M Huang and S X Li ldquoA one-model approach to congestion in data envelopment analysisrdquoSocio-Economic Planning Sciences vol 36 no 4 pp 231ndash2382002

[8] R Fare and L Svensson ldquoCongestion of production factorsrdquoEconometrica vol 48 no 7 pp 1745ndash1752 1980

[9] R Fare and S Grosskopf ldquoMeasuring congestion in produc-tionrdquo Zeitschrift fur Nationalokonomie vol 43 no 3 pp 257ndash271 1983

[10] R Fare S Grosskopf and C A K Lovell The Measurementof Efficiency of Production Kluwer-Nijhoff Boston Mass USA1985

[11] P Byrnes R Fare and S Grosskopf ldquoMeasuring productiveefficiency an application to Illinois strip minesrdquo ManagementScience vol 30 no 6 pp 671ndash681 1984

[12] P Byrnes R Fare S Grosskopf and C A K Lovell ldquoTheeffect of unions on productivity US surface mining of coalrdquoManagement Science vol 34 no 9 pp 1037ndash1053 1988

[13] R Fare and S Grosskopf ldquoCongestion a noterdquo Socio-EconomicPlanning Sciences vol 32 no 1 pp 21ndash23 1998

[14] R Fare and S Grosskopf ldquoSlacks and congestion a commentrdquoSocio-Economic Planning Sciences vol 34 no 1 pp 27ndash33 2000

[15] R Fare and S Grosskopf ldquoDecomposing technical efficiencywith carerdquoManagement Science vol 46 no 1 pp 167ndash168 2000

8 Journal of Applied Mathematics

[16] R Fare and S Grosskopf ldquoWhen can slacks be used to identifycongestion An answer to W W Cooper L Seiford and J ZhurdquoSocio-Economic Planning Sciences vol 35 no 3 pp 217ndash2212001

[17] W W Cooper R G Thompson and R M Thrall ldquoIntroduc-tion extensions and new developments in DEArdquo Annals ofOperations Research vol 66 pp 3ndash45 1996

[18] P L Brockett W W Cooper H-C Shin and Y WangldquoInefficiency and congestion in Chinese production before andafter the 1978 economic reformsrdquo Socio-Economic PlanningSciences vol 32 no 1 pp 1ndash20 1998

[19] W W Cooper L M Seiford and J Zhu ldquoA unified additivemodel approach for evaluating inefficiency and congestionwith associated measures in DEArdquo Socio-Economic PlanningSciences vol 34 no 1 pp 1ndash25 2000

[20] W W Cooper H Deng B Gu S Li and R M Thrall ldquoUsingDEA to improve the management of congestion in Chineseindustries (1981ndash1997)rdquo Socio-Economic Planning Sciences vol35 no 4 pp 227ndash242 2001

[21] L Cherchye T Kuosmanen andT Post ldquoAlternative treatmentsof congestion in DEA a rejoinder to Cooper Gu and LirdquoEuropean Journal of Operational Research vol 132 no 1 pp 75ndash80 2001

[22] WWCooper BGu and S Li ldquoComparisons and evaluations ofalternative approaches to the treatment of congestion in DEArdquoEuropean Journal of Operational Research vol 132 no 1 pp 62ndash74 2001

[23] W W Cooper B Gu and S Li ldquoNote alternative treatments ofcongestion in DEAmdasha response to the Cherchye Kuosmanenand Post critiquerdquo European Journal of Operational Researchvol 132 no 1 pp 81ndash87 2001

[24] WW Cooper LM Seiford and J Zhu ldquoSlacks and congestionresponse to a comment by R Fare and S Grosskopfrdquo Socio-Economic Planning Sciences vol 35 no 3 pp 205ndash215 2001

[25] WW Cooper H Deng L M Seiford and J Zhu ldquoCongestionits identification and management with DEArdquo in Handbook ofData Envelopment Analysis W W Cooper L M Seiford and JZhu Eds pp 177ndash201 Kluwer Academic Boston Mass USA2004

[26] K Tone and B K Sahoo ldquoDegree of scale economies andcongestion a unified DEA approachrdquo European Journal ofOperational Research vol 158 no 3 pp 755ndash772 2004

[27] Q Wei and H Yan ldquoCongestion and returns to scale indata envelopment analysisrdquo European Journal of OperationalResearch vol 153 no 3 pp 641ndash660 2004

[28] T Sueyoshi and K Sekitani ldquoDEA congestion and returns toscale under an occurrence of multiple optimal projectionsrdquoEuropean Journal of Operational Research vol 194 no 2 pp592ndash607 2009

[29] A T Flegg and D O Allen ldquoDoes expansion cause congestionThe case of the older British universities 1994ndash2004rdquo EducationEconomics vol 15 no 1 pp 75ndash102 2007

[30] M Khoveyni R EslamiM Khodabakhshi G R Jahanshahlooand F H Hosseinzadeh ldquoRecognizing strong and weak con-gestion slack based in data envelopment analysisrdquo Computersamp Industrial Engineering vol 64 no 2 pp 731ndash738 2013

[31] G R Jahanshahloo and M Khodabakhshi ldquoSuitable combina-tion of inputs for improving outputs in DEA with determin-ing input congestion considering textile industry of Chinardquo

Applied Mathematics and Computation vol 151 no 1 pp 263ndash273 2004

[32] G R Jahanshahloo andMKhodabakhshi ldquoDetermining assur-ance interval for non-Archimedean element in the improvingoutputsmodel inDEArdquoAppliedMathematics and Computationvol 151 no 2 pp 501ndash506 2004

[33] M Khodabakhshi ldquoA one-model approach based on relaxedcombinations of inputs for evaluating input congestion inDEArdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 443ndash450 2009

[34] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007

[35] M Khodabakhshi and M Asgharian ldquoAn input relaxationmeasure of efficiency in stochastic data envelopment analysisrdquoApplied Mathematical Modelling vol 33 no 4 pp 2010ndash20232009

[36] 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

[37] M Khodabakhshi ldquoChance constrained additive input relax-ation model in stochastic data envelopment analysisrdquo Journal ofInformation and Systems Sciences vol 6 no 1 pp 99ndash112 2010

[38] M Asgharian M Khodabakhshi and L Neralic ldquoCongestionin stochastic data envelopment analysis an input relaxationapproachrdquo International Journal of Statistics and ManagementSystem vol 5 no 1 pp 84ndash106 2010

[39] G R Jahanshahloo M Khodabakhshi F H Lotfi and RM Goudarzi ldquoComputation of congestion in DEA modelswith productions trade-offs and weight restrictionsrdquo AppliedMathematical Sciences vol 5 no 14 pp 663ndash676 2011

[40] M Khodabakhshi R M Goudarzi M Y Maryaki and MHajiani ldquoA one-model approach for computation of congestionwith productions trade-offs and weight restrictionsrdquo Interna-tional Journal of Applied Operational Research vol 3 no 4 pp69ndash80 2013

[41] A A Noura F H Hosseinzadeh G R Jahanshahloo S FRashidi and B R Parker ldquoA new method for measuring con-gestion in data envelopment analysisrdquo Socio-Economic PlanningSciences vol 44 no 4 pp 240ndash246 2010

[42] J Wu Q An B Xiong and Y Chen ldquoCongestionmeasurementfor regional industries in China a data envelopment analysisapproach with undesirable outputsrdquo Energy Policy vol 57 pp7ndash13 2012

[43] L M Seiford and J Zhu ldquoModeling undesirable factors in effi-ciency evaluationrdquo European Journal of Operational Researchvol 142 no 1 pp 16ndash20 2002

[44] B Dervaux K Kerstens and P V Eeckaut ldquoRadial and non-radial static efficiency decompositions a focus on congestionmeasurementrdquo Transportation Research B Methodological vol32 no 5 pp 299ndash312 1998

[45] A T Flegg and D O Allen ldquoCongestion in the Chineseautomobile and textile industries revisitedrdquo Socio-EconomicPlanning Sciences vol 43 no 3 pp 177ndash191 2009

[46] C Kao ldquoCongestion measurement and elimination under theframework of data envelopment analysisrdquo International Journalof Production Economics vol 123 no 2 pp 257ndash265 2010

Journal of Applied Mathematics 9

[47] J Odeck ldquoCongestion ownership region of operation andscale their impact on bus operator performance in NorwayrdquoSocio-Economic Planning Sciences vol 40 no 1 pp 52ndash69 2006

[48] Q Wei and H Yan ldquoWeak congestion in output additive dataenvelopment analysisrdquo Socio-Economic Planning Sciences vol43 no 1 pp 40ndash54 2009

[49] T Sueyoshi and M Goto ldquoWeak and strong disposabilityvs natural and managerial disposability in DEA environmen-tal assessment comparison between Japanese electric powerindustry and manufacturing industriesrdquo Energy Economics vol34 no 3 pp 686ndash699 2012

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

6 Journal of Applied Mathematics

it is obvious that if (120601lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal

solution of (16) (120601lowast119900 119904+lowast2 119904+lowast) are part of an optimal solution

of (12) and (120582lowast 119904+lowast) is an optimal solution of (15) In other

words we can regard model (15) as part of a two-stageprocedure for solvingmodel (16) Using one-model approachwe reduced the solving of three problems with two-modelapproach to the solving of two problems with one-modelapproach This is certainly important from computationalpoint of view

Theorem 10 Congestion is present if and only if optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) is an optimal solution of (16)

and at least one of the following conditions is satisfied

(i) 120601lowast119900 gt 1 and there exists at least one 119894 (1 le 119894 le 119898) suchthat 119904minus119888lowast119894 gt 0

(ii) There exists at least one 119903 (1 le 119903 le 119904) for which 119904+lowast119903 gt 0

and also one 119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

Theorem 11 Congestion is present if and only if for an optimalsolution (120601

lowast119900 120582lowast 119904minus119888lowast1 119904+lowast2 119904+lowast) of (16) there is at least one

119894 (1 le 119894 le 119898) such that 119904minus119888lowast119894 gt 0

Proof See Khodabakhshi [33]

6 Nourarsquos Approach

In this section the approach of Noura et al [41] is brieflyreviewed In this method first model (3) is solved and theoptimal solution 120601

lowast119900 120582lowast 119878+lowast 119878minuslowast is obtained Then the set 119864

is defined as follows

119864 = 119895 | 120601lowast119895 = 1 (17)

Among theDMUs in set119864 there exists at least one sayDMU119897that has the highest consumption in its first input componentcompared with the first input component of the remainingDMUs of set 119864 1199091119897 is denoted by 119909

lowast1 Then among the

DMUs in 119864 a DMU is found say DMU119905 that has the highestconsumption in its second input component compared to theremaining DMUs in 119864 1199092119905 is denoted by 119909

lowast2 In a similar

manner 119909lowast1 119909lowast2 119909

lowast119898 can be identified

Definition 12 Congestion is present if and only if in anoptimal solution 120601lowast119900 120582

lowast 119878+lowast 119878minuslowast of (3) for DMU119900 at least one

of the following two conditions is satisfied

(i) 120601lowast119900 gt 1 and there is at least one 119909119894119900 gt 119909lowast119894 (119894 =

1 119898)(ii) There exists at least one 119878+lowast119903 gt 0 (119903 = 1 119904) and at

least one 119909119894119900 gt 119909lowast119894 (119894 = 1 119898)

They denote the amount of congestion in the 119894th input ofDMUo by 119904

1198881015840

119894 where 119909119894119900 gt 119909lowast119894 and define it as

1199041198881015840

119894 = 119909119894119900 minus 119909lowast119894

(18)

Congestion is considered not present when 119909119894119900 le 119909lowast119894 and 119904

1198881015840

119894 =

0 The sum of all 1199041198881015840

119894 is the amount of congestion in DMU119900Noura et al [41] to establish general validity of their

method prove three theorems which show that Definition 12is equivalent to Cooper et alrsquos Theorem 6 where 119909119894119900 ge 119909

lowast119894

Also they prove another theorem which shows that thereexists no congestion in DMU119900 when 119909119894119900 le 119909

lowast119894 The computa-

tional effort of this method for calculating congestion is lessthan Cooperrsquos approach

7 Wursquos Approach

In this section the input congestion measurement subjectin presence of undesirable outputs is considered using theapproach of Wu et al [42] Assume that 119909 119910 119906 repre-sent the inputs desirable outputs and undesirable outputsrespectively Evidence of congestion in this method occurswhenever reducing some inputs 119909 can increase desirableoutputs119910 and decrease undesirable outputs 119906 simultaneouslyBefore measuring this kind of congestion we should developthe approach solving the problem of undesirable outputsSo far there have been several approaches for addressingthe undesirable output problem Wu et al [42] choose theapproach of Seiford and Zhu [43] to deal with undesirableoutputs The model is shown as follows

max 120575

st119899

sum

119895=1

119909119894119895120582119895 le 1199091198940 119894 = 1 119898

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(19)

where 120572119905 is a big enough positive number that canmake every119900119905119895 positive The fourth constraint is used to transform theundesirable output to a new variable The third constraint isused to constrain the new variable in the possible productionset Wu et al [42] proposed the following model for deter-mining the input congestion of DMU0 in the presence ofundesirable outputs

max 119911

st119899

sum

119895=1

119909119894119895120582119895 = 1199091198940 119894 = 1 119898

Journal of Applied Mathematics 7

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(20)

The production possibility set corresponding to thesemodels is as follows

119879 =

(119909 119910 119906) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

119906119895120582119895 le 119906

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(21)

Definition 13 If the optimal value of model (20) satisfies 119911lowast =1 then we call DMU0 weakly efficient

Definition 14 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 gt 1199100 gt 1199060 then the DMU evidences congestion

Definition 15 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 ge 1199100 ge 1199060 then the DMU evidences congestion

Theorem16 Aweakly efficientDMU0 ofmodel (20) evidencescongestion if and only if it is not weakly efficient in model (19)

Proof See Wu et al [42]

8 Conclusion

There are several approaches for dealing with input con-gestion measurement in DEA In this review six importantmethods are briefly considered and compared Whilst eachtechnique is useful in a specific area none of the method-ologies can be prescribed as the complete solution to thequestion of input congestionHowever there is need to designa complete method which can cover all of the followingproperties

(1) to determine both sources and amounts of inputcongestion

(2) to deal with congestion problem under the conditionthat all inputs and outputs are nonnegative values

(3) to measure the input congestion amounts in thepresence of the multiple optimal solutions

(4) to decompose the measure of overall technical effi-ciency into pure technical efficiency scale efficiencyand congestion efficiency

(5) to deal with the congestion problem in the presenceof the undesirable outputs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[2] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[3] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[4] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002

[5] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985

[6] K Tone ldquoA slacks-based measure of efficiency in data envelop-ment analysisrdquo European Journal of Operational Research vol130 no 3 pp 498ndash509 2001

[7] W W Cooper H Deng Z M Huang and S X Li ldquoA one-model approach to congestion in data envelopment analysisrdquoSocio-Economic Planning Sciences vol 36 no 4 pp 231ndash2382002

[8] R Fare and L Svensson ldquoCongestion of production factorsrdquoEconometrica vol 48 no 7 pp 1745ndash1752 1980

[9] R Fare and S Grosskopf ldquoMeasuring congestion in produc-tionrdquo Zeitschrift fur Nationalokonomie vol 43 no 3 pp 257ndash271 1983

[10] R Fare S Grosskopf and C A K Lovell The Measurementof Efficiency of Production Kluwer-Nijhoff Boston Mass USA1985

[11] P Byrnes R Fare and S Grosskopf ldquoMeasuring productiveefficiency an application to Illinois strip minesrdquo ManagementScience vol 30 no 6 pp 671ndash681 1984

[12] P Byrnes R Fare S Grosskopf and C A K Lovell ldquoTheeffect of unions on productivity US surface mining of coalrdquoManagement Science vol 34 no 9 pp 1037ndash1053 1988

[13] R Fare and S Grosskopf ldquoCongestion a noterdquo Socio-EconomicPlanning Sciences vol 32 no 1 pp 21ndash23 1998

[14] R Fare and S Grosskopf ldquoSlacks and congestion a commentrdquoSocio-Economic Planning Sciences vol 34 no 1 pp 27ndash33 2000

[15] R Fare and S Grosskopf ldquoDecomposing technical efficiencywith carerdquoManagement Science vol 46 no 1 pp 167ndash168 2000

8 Journal of Applied Mathematics

[16] R Fare and S Grosskopf ldquoWhen can slacks be used to identifycongestion An answer to W W Cooper L Seiford and J ZhurdquoSocio-Economic Planning Sciences vol 35 no 3 pp 217ndash2212001

[17] W W Cooper R G Thompson and R M Thrall ldquoIntroduc-tion extensions and new developments in DEArdquo Annals ofOperations Research vol 66 pp 3ndash45 1996

[18] P L Brockett W W Cooper H-C Shin and Y WangldquoInefficiency and congestion in Chinese production before andafter the 1978 economic reformsrdquo Socio-Economic PlanningSciences vol 32 no 1 pp 1ndash20 1998

[19] W W Cooper L M Seiford and J Zhu ldquoA unified additivemodel approach for evaluating inefficiency and congestionwith associated measures in DEArdquo Socio-Economic PlanningSciences vol 34 no 1 pp 1ndash25 2000

[20] W W Cooper H Deng B Gu S Li and R M Thrall ldquoUsingDEA to improve the management of congestion in Chineseindustries (1981ndash1997)rdquo Socio-Economic Planning Sciences vol35 no 4 pp 227ndash242 2001

[21] L Cherchye T Kuosmanen andT Post ldquoAlternative treatmentsof congestion in DEA a rejoinder to Cooper Gu and LirdquoEuropean Journal of Operational Research vol 132 no 1 pp 75ndash80 2001

[22] WWCooper BGu and S Li ldquoComparisons and evaluations ofalternative approaches to the treatment of congestion in DEArdquoEuropean Journal of Operational Research vol 132 no 1 pp 62ndash74 2001

[23] W W Cooper B Gu and S Li ldquoNote alternative treatments ofcongestion in DEAmdasha response to the Cherchye Kuosmanenand Post critiquerdquo European Journal of Operational Researchvol 132 no 1 pp 81ndash87 2001

[24] WW Cooper LM Seiford and J Zhu ldquoSlacks and congestionresponse to a comment by R Fare and S Grosskopfrdquo Socio-Economic Planning Sciences vol 35 no 3 pp 205ndash215 2001

[25] WW Cooper H Deng L M Seiford and J Zhu ldquoCongestionits identification and management with DEArdquo in Handbook ofData Envelopment Analysis W W Cooper L M Seiford and JZhu Eds pp 177ndash201 Kluwer Academic Boston Mass USA2004

[26] K Tone and B K Sahoo ldquoDegree of scale economies andcongestion a unified DEA approachrdquo European Journal ofOperational Research vol 158 no 3 pp 755ndash772 2004

[27] Q Wei and H Yan ldquoCongestion and returns to scale indata envelopment analysisrdquo European Journal of OperationalResearch vol 153 no 3 pp 641ndash660 2004

[28] T Sueyoshi and K Sekitani ldquoDEA congestion and returns toscale under an occurrence of multiple optimal projectionsrdquoEuropean Journal of Operational Research vol 194 no 2 pp592ndash607 2009

[29] A T Flegg and D O Allen ldquoDoes expansion cause congestionThe case of the older British universities 1994ndash2004rdquo EducationEconomics vol 15 no 1 pp 75ndash102 2007

[30] M Khoveyni R EslamiM Khodabakhshi G R Jahanshahlooand F H Hosseinzadeh ldquoRecognizing strong and weak con-gestion slack based in data envelopment analysisrdquo Computersamp Industrial Engineering vol 64 no 2 pp 731ndash738 2013

[31] G R Jahanshahloo and M Khodabakhshi ldquoSuitable combina-tion of inputs for improving outputs in DEA with determin-ing input congestion considering textile industry of Chinardquo

Applied Mathematics and Computation vol 151 no 1 pp 263ndash273 2004

[32] G R Jahanshahloo andMKhodabakhshi ldquoDetermining assur-ance interval for non-Archimedean element in the improvingoutputsmodel inDEArdquoAppliedMathematics and Computationvol 151 no 2 pp 501ndash506 2004

[33] M Khodabakhshi ldquoA one-model approach based on relaxedcombinations of inputs for evaluating input congestion inDEArdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 443ndash450 2009

[34] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007

[35] M Khodabakhshi and M Asgharian ldquoAn input relaxationmeasure of efficiency in stochastic data envelopment analysisrdquoApplied Mathematical Modelling vol 33 no 4 pp 2010ndash20232009

[36] 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

[37] M Khodabakhshi ldquoChance constrained additive input relax-ation model in stochastic data envelopment analysisrdquo Journal ofInformation and Systems Sciences vol 6 no 1 pp 99ndash112 2010

[38] M Asgharian M Khodabakhshi and L Neralic ldquoCongestionin stochastic data envelopment analysis an input relaxationapproachrdquo International Journal of Statistics and ManagementSystem vol 5 no 1 pp 84ndash106 2010

[39] G R Jahanshahloo M Khodabakhshi F H Lotfi and RM Goudarzi ldquoComputation of congestion in DEA modelswith productions trade-offs and weight restrictionsrdquo AppliedMathematical Sciences vol 5 no 14 pp 663ndash676 2011

[40] M Khodabakhshi R M Goudarzi M Y Maryaki and MHajiani ldquoA one-model approach for computation of congestionwith productions trade-offs and weight restrictionsrdquo Interna-tional Journal of Applied Operational Research vol 3 no 4 pp69ndash80 2013

[41] A A Noura F H Hosseinzadeh G R Jahanshahloo S FRashidi and B R Parker ldquoA new method for measuring con-gestion in data envelopment analysisrdquo Socio-Economic PlanningSciences vol 44 no 4 pp 240ndash246 2010

[42] J Wu Q An B Xiong and Y Chen ldquoCongestionmeasurementfor regional industries in China a data envelopment analysisapproach with undesirable outputsrdquo Energy Policy vol 57 pp7ndash13 2012

[43] L M Seiford and J Zhu ldquoModeling undesirable factors in effi-ciency evaluationrdquo European Journal of Operational Researchvol 142 no 1 pp 16ndash20 2002

[44] B Dervaux K Kerstens and P V Eeckaut ldquoRadial and non-radial static efficiency decompositions a focus on congestionmeasurementrdquo Transportation Research B Methodological vol32 no 5 pp 299ndash312 1998

[45] A T Flegg and D O Allen ldquoCongestion in the Chineseautomobile and textile industries revisitedrdquo Socio-EconomicPlanning Sciences vol 43 no 3 pp 177ndash191 2009

[46] C Kao ldquoCongestion measurement and elimination under theframework of data envelopment analysisrdquo International Journalof Production Economics vol 123 no 2 pp 257ndash265 2010

Journal of Applied Mathematics 9

[47] J Odeck ldquoCongestion ownership region of operation andscale their impact on bus operator performance in NorwayrdquoSocio-Economic Planning Sciences vol 40 no 1 pp 52ndash69 2006

[48] Q Wei and H Yan ldquoWeak congestion in output additive dataenvelopment analysisrdquo Socio-Economic Planning Sciences vol43 no 1 pp 40ndash54 2009

[49] T Sueyoshi and M Goto ldquoWeak and strong disposabilityvs natural and managerial disposability in DEA environmen-tal assessment comparison between Japanese electric powerindustry and manufacturing industriesrdquo Energy Economics vol34 no 3 pp 686ndash699 2012

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 7

119899

sum

119895=1

119910119903119895120582119895 ge 1199101199030120575 119903 = 1 119904

119899

sum

119895=1

119900119905119895120582119895 ge 1199001199050120575 119905 = 1 119896

119900119905119895 = minus119906119905119895 + 120572119905 119895 = 1 119899

119899

sum

119895=1

120582119895 = 1

120582119895 ge 0

(20)

The production possibility set corresponding to thesemodels is as follows

119879 =

(119909 119910 119906) |

119899

sum

119895=1

119909119895120582119895 = 119909

119899

sum

119895=1

119910119895120582119895 ge 119910

119899

sum

119895=1

119906119895120582119895 le 119906

119899

sum

119895=1

120582119895 = 1 120582119895 ge 0

(21)

Definition 13 If the optimal value of model (20) satisfies 119911lowast =1 then we call DMU0 weakly efficient

Definition 14 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 gt 1199100 gt 1199060 then the DMU evidences congestion

Definition 15 Assume that DMU0 is defined as weaklyefficient by model (20) If it has (119909 119910 ) isin 119879 119909 le 1199090 and119909 = 1199090 119910 ge 1199100 ge 1199060 then the DMU evidences congestion

Theorem16 Aweakly efficientDMU0 ofmodel (20) evidencescongestion if and only if it is not weakly efficient in model (19)

Proof See Wu et al [42]

8 Conclusion

There are several approaches for dealing with input con-gestion measurement in DEA In this review six importantmethods are briefly considered and compared Whilst eachtechnique is useful in a specific area none of the method-ologies can be prescribed as the complete solution to thequestion of input congestionHowever there is need to designa complete method which can cover all of the followingproperties

(1) to determine both sources and amounts of inputcongestion

(2) to deal with congestion problem under the conditionthat all inputs and outputs are nonnegative values

(3) to measure the input congestion amounts in thepresence of the multiple optimal solutions

(4) to decompose the measure of overall technical effi-ciency into pure technical efficiency scale efficiencyand congestion efficiency

(5) to deal with the congestion problem in the presenceof the undesirable outputs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[2] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[3] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[4] N Adler L Friedman and Z Sinuany-Stern ldquoReview ofranking methods in the data envelopment analysis contextrdquoEuropean Journal of Operational Research vol 140 no 2 pp249ndash265 2002

[5] A Charnes W W Cooper B Golany L Seiford and JStutz ldquoFoundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functionsrdquo Journal ofEconometrics vol 30 no 1-2 pp 91ndash107 1985

[6] K Tone ldquoA slacks-based measure of efficiency in data envelop-ment analysisrdquo European Journal of Operational Research vol130 no 3 pp 498ndash509 2001

[7] W W Cooper H Deng Z M Huang and S X Li ldquoA one-model approach to congestion in data envelopment analysisrdquoSocio-Economic Planning Sciences vol 36 no 4 pp 231ndash2382002

[8] R Fare and L Svensson ldquoCongestion of production factorsrdquoEconometrica vol 48 no 7 pp 1745ndash1752 1980

[9] R Fare and S Grosskopf ldquoMeasuring congestion in produc-tionrdquo Zeitschrift fur Nationalokonomie vol 43 no 3 pp 257ndash271 1983

[10] R Fare S Grosskopf and C A K Lovell The Measurementof Efficiency of Production Kluwer-Nijhoff Boston Mass USA1985

[11] P Byrnes R Fare and S Grosskopf ldquoMeasuring productiveefficiency an application to Illinois strip minesrdquo ManagementScience vol 30 no 6 pp 671ndash681 1984

[12] P Byrnes R Fare S Grosskopf and C A K Lovell ldquoTheeffect of unions on productivity US surface mining of coalrdquoManagement Science vol 34 no 9 pp 1037ndash1053 1988

[13] R Fare and S Grosskopf ldquoCongestion a noterdquo Socio-EconomicPlanning Sciences vol 32 no 1 pp 21ndash23 1998

[14] R Fare and S Grosskopf ldquoSlacks and congestion a commentrdquoSocio-Economic Planning Sciences vol 34 no 1 pp 27ndash33 2000

[15] R Fare and S Grosskopf ldquoDecomposing technical efficiencywith carerdquoManagement Science vol 46 no 1 pp 167ndash168 2000

8 Journal of Applied Mathematics

[16] R Fare and S Grosskopf ldquoWhen can slacks be used to identifycongestion An answer to W W Cooper L Seiford and J ZhurdquoSocio-Economic Planning Sciences vol 35 no 3 pp 217ndash2212001

[17] W W Cooper R G Thompson and R M Thrall ldquoIntroduc-tion extensions and new developments in DEArdquo Annals ofOperations Research vol 66 pp 3ndash45 1996

[18] P L Brockett W W Cooper H-C Shin and Y WangldquoInefficiency and congestion in Chinese production before andafter the 1978 economic reformsrdquo Socio-Economic PlanningSciences vol 32 no 1 pp 1ndash20 1998

[19] W W Cooper L M Seiford and J Zhu ldquoA unified additivemodel approach for evaluating inefficiency and congestionwith associated measures in DEArdquo Socio-Economic PlanningSciences vol 34 no 1 pp 1ndash25 2000

[20] W W Cooper H Deng B Gu S Li and R M Thrall ldquoUsingDEA to improve the management of congestion in Chineseindustries (1981ndash1997)rdquo Socio-Economic Planning Sciences vol35 no 4 pp 227ndash242 2001

[21] L Cherchye T Kuosmanen andT Post ldquoAlternative treatmentsof congestion in DEA a rejoinder to Cooper Gu and LirdquoEuropean Journal of Operational Research vol 132 no 1 pp 75ndash80 2001

[22] WWCooper BGu and S Li ldquoComparisons and evaluations ofalternative approaches to the treatment of congestion in DEArdquoEuropean Journal of Operational Research vol 132 no 1 pp 62ndash74 2001

[23] W W Cooper B Gu and S Li ldquoNote alternative treatments ofcongestion in DEAmdasha response to the Cherchye Kuosmanenand Post critiquerdquo European Journal of Operational Researchvol 132 no 1 pp 81ndash87 2001

[24] WW Cooper LM Seiford and J Zhu ldquoSlacks and congestionresponse to a comment by R Fare and S Grosskopfrdquo Socio-Economic Planning Sciences vol 35 no 3 pp 205ndash215 2001

[25] WW Cooper H Deng L M Seiford and J Zhu ldquoCongestionits identification and management with DEArdquo in Handbook ofData Envelopment Analysis W W Cooper L M Seiford and JZhu Eds pp 177ndash201 Kluwer Academic Boston Mass USA2004

[26] K Tone and B K Sahoo ldquoDegree of scale economies andcongestion a unified DEA approachrdquo European Journal ofOperational Research vol 158 no 3 pp 755ndash772 2004

[27] Q Wei and H Yan ldquoCongestion and returns to scale indata envelopment analysisrdquo European Journal of OperationalResearch vol 153 no 3 pp 641ndash660 2004

[28] T Sueyoshi and K Sekitani ldquoDEA congestion and returns toscale under an occurrence of multiple optimal projectionsrdquoEuropean Journal of Operational Research vol 194 no 2 pp592ndash607 2009

[29] A T Flegg and D O Allen ldquoDoes expansion cause congestionThe case of the older British universities 1994ndash2004rdquo EducationEconomics vol 15 no 1 pp 75ndash102 2007

[30] M Khoveyni R EslamiM Khodabakhshi G R Jahanshahlooand F H Hosseinzadeh ldquoRecognizing strong and weak con-gestion slack based in data envelopment analysisrdquo Computersamp Industrial Engineering vol 64 no 2 pp 731ndash738 2013

[31] G R Jahanshahloo and M Khodabakhshi ldquoSuitable combina-tion of inputs for improving outputs in DEA with determin-ing input congestion considering textile industry of Chinardquo

Applied Mathematics and Computation vol 151 no 1 pp 263ndash273 2004

[32] G R Jahanshahloo andMKhodabakhshi ldquoDetermining assur-ance interval for non-Archimedean element in the improvingoutputsmodel inDEArdquoAppliedMathematics and Computationvol 151 no 2 pp 501ndash506 2004

[33] M Khodabakhshi ldquoA one-model approach based on relaxedcombinations of inputs for evaluating input congestion inDEArdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 443ndash450 2009

[34] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007

[35] M Khodabakhshi and M Asgharian ldquoAn input relaxationmeasure of efficiency in stochastic data envelopment analysisrdquoApplied Mathematical Modelling vol 33 no 4 pp 2010ndash20232009

[36] 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

[37] M Khodabakhshi ldquoChance constrained additive input relax-ation model in stochastic data envelopment analysisrdquo Journal ofInformation and Systems Sciences vol 6 no 1 pp 99ndash112 2010

[38] M Asgharian M Khodabakhshi and L Neralic ldquoCongestionin stochastic data envelopment analysis an input relaxationapproachrdquo International Journal of Statistics and ManagementSystem vol 5 no 1 pp 84ndash106 2010

[39] G R Jahanshahloo M Khodabakhshi F H Lotfi and RM Goudarzi ldquoComputation of congestion in DEA modelswith productions trade-offs and weight restrictionsrdquo AppliedMathematical Sciences vol 5 no 14 pp 663ndash676 2011

[40] M Khodabakhshi R M Goudarzi M Y Maryaki and MHajiani ldquoA one-model approach for computation of congestionwith productions trade-offs and weight restrictionsrdquo Interna-tional Journal of Applied Operational Research vol 3 no 4 pp69ndash80 2013

[41] A A Noura F H Hosseinzadeh G R Jahanshahloo S FRashidi and B R Parker ldquoA new method for measuring con-gestion in data envelopment analysisrdquo Socio-Economic PlanningSciences vol 44 no 4 pp 240ndash246 2010

[42] J Wu Q An B Xiong and Y Chen ldquoCongestionmeasurementfor regional industries in China a data envelopment analysisapproach with undesirable outputsrdquo Energy Policy vol 57 pp7ndash13 2012

[43] L M Seiford and J Zhu ldquoModeling undesirable factors in effi-ciency evaluationrdquo European Journal of Operational Researchvol 142 no 1 pp 16ndash20 2002

[44] B Dervaux K Kerstens and P V Eeckaut ldquoRadial and non-radial static efficiency decompositions a focus on congestionmeasurementrdquo Transportation Research B Methodological vol32 no 5 pp 299ndash312 1998

[45] A T Flegg and D O Allen ldquoCongestion in the Chineseautomobile and textile industries revisitedrdquo Socio-EconomicPlanning Sciences vol 43 no 3 pp 177ndash191 2009

[46] C Kao ldquoCongestion measurement and elimination under theframework of data envelopment analysisrdquo International Journalof Production Economics vol 123 no 2 pp 257ndash265 2010

Journal of Applied Mathematics 9

[47] J Odeck ldquoCongestion ownership region of operation andscale their impact on bus operator performance in NorwayrdquoSocio-Economic Planning Sciences vol 40 no 1 pp 52ndash69 2006

[48] Q Wei and H Yan ldquoWeak congestion in output additive dataenvelopment analysisrdquo Socio-Economic Planning Sciences vol43 no 1 pp 40ndash54 2009

[49] T Sueyoshi and M Goto ldquoWeak and strong disposabilityvs natural and managerial disposability in DEA environmen-tal assessment comparison between Japanese electric powerindustry and manufacturing industriesrdquo Energy Economics vol34 no 3 pp 686ndash699 2012

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

8 Journal of Applied Mathematics

[16] R Fare and S Grosskopf ldquoWhen can slacks be used to identifycongestion An answer to W W Cooper L Seiford and J ZhurdquoSocio-Economic Planning Sciences vol 35 no 3 pp 217ndash2212001

[17] W W Cooper R G Thompson and R M Thrall ldquoIntroduc-tion extensions and new developments in DEArdquo Annals ofOperations Research vol 66 pp 3ndash45 1996

[18] P L Brockett W W Cooper H-C Shin and Y WangldquoInefficiency and congestion in Chinese production before andafter the 1978 economic reformsrdquo Socio-Economic PlanningSciences vol 32 no 1 pp 1ndash20 1998

[19] W W Cooper L M Seiford and J Zhu ldquoA unified additivemodel approach for evaluating inefficiency and congestionwith associated measures in DEArdquo Socio-Economic PlanningSciences vol 34 no 1 pp 1ndash25 2000

[20] W W Cooper H Deng B Gu S Li and R M Thrall ldquoUsingDEA to improve the management of congestion in Chineseindustries (1981ndash1997)rdquo Socio-Economic Planning Sciences vol35 no 4 pp 227ndash242 2001

[21] L Cherchye T Kuosmanen andT Post ldquoAlternative treatmentsof congestion in DEA a rejoinder to Cooper Gu and LirdquoEuropean Journal of Operational Research vol 132 no 1 pp 75ndash80 2001

[22] WWCooper BGu and S Li ldquoComparisons and evaluations ofalternative approaches to the treatment of congestion in DEArdquoEuropean Journal of Operational Research vol 132 no 1 pp 62ndash74 2001

[23] W W Cooper B Gu and S Li ldquoNote alternative treatments ofcongestion in DEAmdasha response to the Cherchye Kuosmanenand Post critiquerdquo European Journal of Operational Researchvol 132 no 1 pp 81ndash87 2001

[24] WW Cooper LM Seiford and J Zhu ldquoSlacks and congestionresponse to a comment by R Fare and S Grosskopfrdquo Socio-Economic Planning Sciences vol 35 no 3 pp 205ndash215 2001

[25] WW Cooper H Deng L M Seiford and J Zhu ldquoCongestionits identification and management with DEArdquo in Handbook ofData Envelopment Analysis W W Cooper L M Seiford and JZhu Eds pp 177ndash201 Kluwer Academic Boston Mass USA2004

[26] K Tone and B K Sahoo ldquoDegree of scale economies andcongestion a unified DEA approachrdquo European Journal ofOperational Research vol 158 no 3 pp 755ndash772 2004

[27] Q Wei and H Yan ldquoCongestion and returns to scale indata envelopment analysisrdquo European Journal of OperationalResearch vol 153 no 3 pp 641ndash660 2004

[28] T Sueyoshi and K Sekitani ldquoDEA congestion and returns toscale under an occurrence of multiple optimal projectionsrdquoEuropean Journal of Operational Research vol 194 no 2 pp592ndash607 2009

[29] A T Flegg and D O Allen ldquoDoes expansion cause congestionThe case of the older British universities 1994ndash2004rdquo EducationEconomics vol 15 no 1 pp 75ndash102 2007

[30] M Khoveyni R EslamiM Khodabakhshi G R Jahanshahlooand F H Hosseinzadeh ldquoRecognizing strong and weak con-gestion slack based in data envelopment analysisrdquo Computersamp Industrial Engineering vol 64 no 2 pp 731ndash738 2013

[31] G R Jahanshahloo and M Khodabakhshi ldquoSuitable combina-tion of inputs for improving outputs in DEA with determin-ing input congestion considering textile industry of Chinardquo

Applied Mathematics and Computation vol 151 no 1 pp 263ndash273 2004

[32] G R Jahanshahloo andMKhodabakhshi ldquoDetermining assur-ance interval for non-Archimedean element in the improvingoutputsmodel inDEArdquoAppliedMathematics and Computationvol 151 no 2 pp 501ndash506 2004

[33] M Khodabakhshi ldquoA one-model approach based on relaxedcombinations of inputs for evaluating input congestion inDEArdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 443ndash450 2009

[34] M Khodabakhshi ldquoA super-efficiency model based onimproved outputs in data envelopment analysisrdquo AppliedMathematics and Computation vol 184 no 2 pp 695ndash7032007

[35] M Khodabakhshi and M Asgharian ldquoAn input relaxationmeasure of efficiency in stochastic data envelopment analysisrdquoApplied Mathematical Modelling vol 33 no 4 pp 2010ndash20232009

[36] 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

[37] M Khodabakhshi ldquoChance constrained additive input relax-ation model in stochastic data envelopment analysisrdquo Journal ofInformation and Systems Sciences vol 6 no 1 pp 99ndash112 2010

[38] M Asgharian M Khodabakhshi and L Neralic ldquoCongestionin stochastic data envelopment analysis an input relaxationapproachrdquo International Journal of Statistics and ManagementSystem vol 5 no 1 pp 84ndash106 2010

[39] G R Jahanshahloo M Khodabakhshi F H Lotfi and RM Goudarzi ldquoComputation of congestion in DEA modelswith productions trade-offs and weight restrictionsrdquo AppliedMathematical Sciences vol 5 no 14 pp 663ndash676 2011

[40] M Khodabakhshi R M Goudarzi M Y Maryaki and MHajiani ldquoA one-model approach for computation of congestionwith productions trade-offs and weight restrictionsrdquo Interna-tional Journal of Applied Operational Research vol 3 no 4 pp69ndash80 2013

[41] A A Noura F H Hosseinzadeh G R Jahanshahloo S FRashidi and B R Parker ldquoA new method for measuring con-gestion in data envelopment analysisrdquo Socio-Economic PlanningSciences vol 44 no 4 pp 240ndash246 2010

[42] J Wu Q An B Xiong and Y Chen ldquoCongestionmeasurementfor regional industries in China a data envelopment analysisapproach with undesirable outputsrdquo Energy Policy vol 57 pp7ndash13 2012

[43] L M Seiford and J Zhu ldquoModeling undesirable factors in effi-ciency evaluationrdquo European Journal of Operational Researchvol 142 no 1 pp 16ndash20 2002

[44] B Dervaux K Kerstens and P V Eeckaut ldquoRadial and non-radial static efficiency decompositions a focus on congestionmeasurementrdquo Transportation Research B Methodological vol32 no 5 pp 299ndash312 1998

[45] A T Flegg and D O Allen ldquoCongestion in the Chineseautomobile and textile industries revisitedrdquo Socio-EconomicPlanning Sciences vol 43 no 3 pp 177ndash191 2009

[46] C Kao ldquoCongestion measurement and elimination under theframework of data envelopment analysisrdquo International Journalof Production Economics vol 123 no 2 pp 257ndash265 2010

Journal of Applied Mathematics 9

[47] J Odeck ldquoCongestion ownership region of operation andscale their impact on bus operator performance in NorwayrdquoSocio-Economic Planning Sciences vol 40 no 1 pp 52ndash69 2006

[48] Q Wei and H Yan ldquoWeak congestion in output additive dataenvelopment analysisrdquo Socio-Economic Planning Sciences vol43 no 1 pp 40ndash54 2009

[49] T Sueyoshi and M Goto ldquoWeak and strong disposabilityvs natural and managerial disposability in DEA environmen-tal assessment comparison between Japanese electric powerindustry and manufacturing industriesrdquo Energy Economics vol34 no 3 pp 686ndash699 2012

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 9

[47] J Odeck ldquoCongestion ownership region of operation andscale their impact on bus operator performance in NorwayrdquoSocio-Economic Planning Sciences vol 40 no 1 pp 52ndash69 2006

[48] Q Wei and H Yan ldquoWeak congestion in output additive dataenvelopment analysisrdquo Socio-Economic Planning Sciences vol43 no 1 pp 40ndash54 2009

[49] T Sueyoshi and M Goto ldquoWeak and strong disposabilityvs natural and managerial disposability in DEA environmen-tal assessment comparison between Japanese electric powerindustry and manufacturing industriesrdquo Energy Economics vol34 no 3 pp 686ndash699 2012

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