262MEHRABIAN,ALIREZAEE AND JAHANSHAHLOO The paper unfolds as follows.Section2represents Andersen/Petersen’s model and presents the proposed model,both in the production possibility set(PPS).Section3presents an illustrative example.Section4presents an empirical investigation.A summary is given in Section5.2.Efficiency analysis by an alternative measureThere are n DMUs to be evaluated,each consumes varying amounts of m different inputs to produce s different outputs.In the model formulation,X p and Y p denote,respectively, the nonnegative vectors of input and output values for DMU p.Definition.The production possibility set(PPS)is the set{(X t,Y t)|the outputs Y t can be produced with the inputs X t}.The set of n DMUs of actual production possibility(X j,Y j),j=1,...,n is considered. Our focus is on the empirically defined production possibility set T with constant returns assumption that is specified by the following postulates:Postulate1(Ray Unboundedness).If(X t,Y t)∈T,then(λX t,λY t)∈T for allλ≥0. Postulate2(Convexity).If(X t,Y t)∈T and(X u,Y u)∈T,then(λX t+(1−λ)X u,λY t+(1−λ)Y u)∈T for allλ∈[0,1].Postulate3(Monotonicity).If(X t,Y t)∈T and X u≥X t,Y u≤Y t,then(X u,Y u)∈T. Postulate4(Inclusion of Observations).The observed(X j,Y j)∈T for all j=1,...,n. Postulate5(Minimum extrapolation).If a production possibility set T satisfies Postulates 1,2,3,and4,then T⊂T .The unique production possibility set with constant returns assumption determined by the above postulates is given by:T=(X t,Y t)|X t≥nj=1λj X j,Y t≤nj=1λj Y j,λj≥0for j=1,...,nFigure1represents a production possibility set,T,for the simplest case of single input and single output.For efficiency evaluation relative to the set T,we consider the following two mathe-matical programs:θ∗p=min a pη∗p=min w p+1subject to:subject to:(a p X p,Y p)∈T(X p+w p1,Y p)∈TThefirst program yields the measure of efficiency introduced by Charnes et al.[6].Hence, we call it the CCR-model.The second program is a new measure of efficiency that weA COMPLETE EFFICIENCY RANKING 263Figure 1.Production possibility set.introduce.For the set T above,these programs can be rewritten:θ∗p =min a p η∗p =min w p +1subject to:subject to:n j =1λj X j ≤a p X p ,nj =1λj X j ≤X p +w p 1,nj =1λj Y j ≥Y p ,n j =1λj Y j ≥Y p ,λj ≥0,j =1,...,n λj ≥0,j =1,...,n where 1is a vector of units.Note that the feasible solution of a p =1,λp =1,and λj =0,j =1,...,n ,j =p for the CCR-model implies that θ∗p ≤1and similarly the feasible solution of w p =0,λp =1,and λj =0,j =1,...,n ,j =p for the alternative formulation implies that η∗p ≤1.Also θ∗p ,η∗p ≥0,since for θ∗p or η∗p less than zero the corresponding input constraints will be inconsistent.Therefore θ∗p and η∗p lie between 0and 1.To rank the relative efficiency of DMUs with unit efficiency,Andersen and Petersen propose that evaluated unit be excluded from the mathematical program,leading to the264MEHRABIAN,ALIREZAEE AND JAHANSHAHLOO following programs depending on the unit p be evaluated:the AP-model the MAJ-modela∗p=min a p j∗p=min w p+1subject to:subject to:nj=1 j=p λj X j≤a p X p,nj=1j=pλj X j≤X p+w p1,nj=1 j=p λj Y j≥Y p,nj=1j=pλj Y j≥Y p,λj≥0,j=1,...,nλj≥0,j=1,...,nAlthough the optimal objective function values for the MAJ-model depend upon the units of measurement of input data,X j,j=1,...,n,unit independence is obtained by normalization1,that is,dividing input data by the maximum input(for each input).Note that for the case of full inefficiency,that is,when a DMU uses maximum inputs,for no production,both models provide zero scores,and for the case of full efficiency,both models provide not less than one scores.Hence,a∗p and j∗p lie between0and+∞.The following feasibility condition for the MAJ-model is readily verified. Proposition.(Necessary and sufficient condition for feasibility in the MAJ-model).The MAJ-model is feasible for evaluation of DMU p with output vector Y p≥0if and only if for each r,r=1,...,s,either y r p=0or there exists a DMU j,j=p,such that y r j=0. The hypothesis of the proposition is not sufficient for feasibility of the AP-model.Dula and Hichmen[10]presents a list of cases that leads to the infeasible problems.3.Illustrative exampleConsider the example given in Table1.There are5DMUs(A,B,C,D,and E)each consume two inputs to produce two outputs with constant returns assumption.We considerthree different instances of unit A,denoted A1,A2,and A3in Table1.DMU A1,DMU A2,parison test data.DMUs→A1A2A3B C D Einput1201510102input288854612output11111221output22221112A COMPLETE EFFICIENCY RANKING265Table2.Results.The CCR-model The AP-model The MAJ-modelDMU A1100%147%127.6%DMU A2100%infeasible131.0%DMU A3100%2000%130.9%and DMU A3are compared one at a time,with all other DMUs(B,C,D,and E)by theCCR-model,the AP-model,and the MAJ-model.Table2presents the results.The AP-model for evaluation of DMU A2leads to an infeasible problem because itsfirstinput is zero and for evaluation of DMU A3leads to a large score because of a relativelysmall value for itsfirst input.These difficulties can be removed by the MAJ-model.putational experienceIn Jahanshahloo and Alirezaee[11],the evaluation of teaching in the University for Teacher Education was considered.Teaching inputs were expressed in teacher hours and classified in terms of two inputs,professorial staff and instructors.Teaching outputs were expressed in student hours and classified in terms of two outputs,course enrollments in undergraduate and graduate studies.There are six efficient departments whose rankings appear in Table3.The AP-model is infeasible for Department of Women’s Physical,the9th DMU,and Institute of Mathematics,the19th DMU,which are indicated by asterisks.Note that these DMUs have zero inputs.Also,the Department of Theology and Islamic Culture,the2nd DMU,has an overestimation score in the AP-model because of relatively small input.The MAJ-model evaluates the9th and19th DMUs so that they have explicit ranking,and also the2nd DMU now has a more acceptable rank.For 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