Ad-a154 244 Foundations Of Data Envelopment Analysis For 1/1 Pareto .
AD-A154 244UNCLASSIFIEDFOUNDATIONS OF DATA ENVELOPMENT ANALYSIS FORPARETO-KOOPARNS EFFICIENT EM.(U) TEXAS UNIV AT AUSTINCENTER FOR CYBERNETIC STUDIES A CHARNES ET AL. FEB 85F/B 5/1CCS- 54 MDA9B3-3-K-O32Eu.llliEEEEEEEEEE1/1NL
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-Research Report CCS 504FOUNDATIONS OF DATA ENVELOPMENT ANALYSISFOR PARETO-KOOPMANS EFFICIENT EMPIRICALPRODUCTION FUNCTIONSbyA. CharnesW.W. CooperB. GolanyInL. SeifordJ. Stutz*CENTER FORCYBERNETICSTUDIESThe University of Texas Austin,Texas 78712SELECDTICMAY 221985lIBJ'TON TAEM ENA pproved km public rleagAD)jstiutuoa UnliraitediSbjjd)(J IN JNU34O) iV (J3Jf1U0HdJH
.7SResearch Report CCS 504FOUNDATIONS OF DATA ENVELOPMENT ANALYSISFOR PARETO-KOOPMANS EFFICIENT EMPIRICALPRODUCTION FUNCTIONS'.byA* CharnesW.W. CooperB. GolanyL. SeifordJ. Stutz**.November 1984Revised February 1985DTICMA*University of Miami**I0This research was partly supported by NSF Grant SES-8408134 and USARIContract M0A903-83-K-0312 with the Center for Cybernetic Studies, TheUniversity of Texas at Austin. Reproduction in whole or in part ispermitted for any purpose of the United States Government.CENTER FOR CYBERNETIC STUDIESA. Charnes, DirectorCollege of Business Administration 5.202The Uni-/ersity of Texas at Austind-stin, Texas 78712(512) 471-1821kpptvoed.ONSTATEMLN1rp-ublic zloD4.stributLOfl UnILMitedL.---.A
T7 .o.777777FOUNDATIONS OF DATA ENVELOPMENT ANALYSIS FORPARETO-KOOPMANS EFFICIENT EMPIRICAL PRODUCTION FUNCTIONSbyA. Charnes, W.W. Cooper, B. Golany, L. Seiford, J. StutzAbstractThe construction and analysis of Pareto-efficient frontierproduction functions by a new Data Envelopment Analysis method ispresented in the context of new theoretical characterizations of theinherent structure and capabilities of such empirical production functions.Contrasts and connections with other developments, including solutions ofsome remaining problems, are made re aspects such as inforatics, economiesof scale, isotonicity and non-concavity, discretionary and nondiscretionary inputs, piecewise linearity, partial derivatives andCobb-Douglas properties of the functions.Non-Archimedean constructs arenot required.A.- Keyv,Words\Pareto-Koopmans EfficiencyY-' ."[,Efficiency Analysis).,Frontier Production Functions5"-Data Envelopment Analysis.".9.Lanld/arSpouialOriginal:"An Empirical DEA Production Function" by A. Charnes,W.W. Cooper & L. Seiford, April 1981. : ," ' '.r ,",,'.* ,.'', - ,', '/",,.,a"
.",.*.o.,'INTRODUCTIONClassically, the economic theory of production is heavily based onthe conceptual use of the Pareto-efficiency (or Pareto-optimal) frontier ofproduction possibility sets to define "the" production function.The workof R. Shephard [27], [28] under restrictions on the mathematicalstructure of production possibility sets and cost relations, developed anelegant "transform" theory between production aspects and cost aspects [17].This was applied to various classes of explicitly given parametric functionalforms and problems of statistical estimation of parameters from data wereconsidered in classical statistical contexts especially by successors likeS. Afriat, D. Aigner, F. Forsund [1, 2, 24].1These efforts were almostexclusively for single output functions.M. J. Farrell in [22] partly responding to the inadequacies ofseparate indices of labor productivity, capital productivity, etc., undertookwhat he referred to as an "activity analysis" approach 2 that could deal moreadequately with the problem.Possibly because of the limitations of theelaborate matrix inversion routines he was employing, Farrell confined hisnumerical examples and discussion to single output situations, although hedid formulate the multiple output case.If anything, the need for beingable to deal directly with multiple outputs has grown ever-more urgent withthe continuing growth of the not-for-profit and service sectors in theUnited States and other "western type" economies.As we have elsewhere1Theneed for some-method of estimating frontier functions had beenpointed out as early as 1935 by Frisch [25] in his statistical study ofchocolate production in France.2 In thesense of the original work by T. C. Koopmans that is describedin Chapter IX of [12] where it was also accorded an explicit linear programmingformulation and interpretation.'.]-.-i,.'"., ''' '-.'''''.I.
.-.o-o.-.-.-.--. -.-.2observed,one can otherwise do almost nothing with important parts ofnot-for-profit and governmental activities without becoming involved inarbitrary weighting schemes.Even attempts at measuring and evaluating"total factor productivity" will fall short of what is required since thisis inherently a single output concept.Farrell was right, therefore, inre-orienting the direction of development toward the various types ofefficiencies that he referred to as "technical", "scale", and "allocativeefficiencies".2Building on the individual firm or country ( Decision Making Unit)evaluations of Farrell and the engineering ratio idea of efficiency measurefor a single input and output, efficiency analysis in its managerial aspectsand its constructible extensions to multi-input, multi-output situationswas initiated by Charnes, Cooper, and Rhodes in [15] and [16].Subsequentextensions and elaborations by the former pair with other students andcolleagues were made in [14], [18], [19], and [20] with more detailedattention to classical economic aspects and deeper analysis of the production function side of the mathematical duality structure and Data EnvelopmentAnalysis first presented in the original CCR work.The CCR ratio measuresand the variants of Farrell, Shephard, Fare, et al, on the dual linearprogramming side require, however, non-Archimedean constructs for rigoroustheory and usage.Their solution methods also do not easily provide impor-tant needed properties of their associated empirical production functions.Thus, in this paper we introduce as basic the idea of Paretooptimality with respect to an empirically defined production possibility set.ISee Banker, Bowlin., Charnes, and Cooper [5].2 Thisterm was introduced in [15] and contracted to DMU because itwas recognized that something like it was needed to refer to publicsector organizations.--- . . %%,. - %- % .'. .:-."-'' . -'.,.-. '".".*.'o"%'. .--."""""-"".',.m-.'.-. .
7.73We characterize the mathematical structures permitted under our minimalK-Properties such asassumptions and contrast these with work by others.isotonicity, non-concavity, economies of scale, piece-wise linearity, CobbDouglas forms, discretionary and non-discretionary inputs are treated througha new Data Envelopment Analysis method and informatics which permits aconstructive development of an empirical production function and its partialderivatives without loss of efficiency analysis or use of non-Archimedeanfield extensions.'-.A- -.*.*"7.1-
4EMPIRICAL FUNCTION SETTING AND GENERATIONBy an "empirical" function we shall mean a vector function whosevalues are known at a finite number of points and whose values at otherpoints in its domain are given by linear (usually convex)of values at known points.combinationsThe points in the domain are "inputs," thecomponent values of the vector function "outputs."We shall assume thatinputs are so chosen that convex combinations of input values for eachinput are meaningful input values.We assume this for output values as well.In efficiency analysis, observations are generated by a finitenumber of "DMU"s, or "productive," or "response" units, all of which havethe same inputs and outputs.obtained for each unit.A relative efficiency rating is to beTypically, observations over time will be madeof each unit and the results of efficiency analyses will be employed toassist in managing each of the units.m inputs.We assume n units, s outputs and.-The observed values are to be non-negative (sometimes positive)numbers.We shall employ the notation X., Y. for respectively the observedvectors of the inputs and outputs of the jth DMU.By X and Y we shallmean the matrices whose n column vectors are respectively those of then DMU's.We use x, y, A, s to denote column vectors also with .ethe columnvector of ones.We use "T" as a superscript to denote the transpose, e.g.T.e is the row vector of ones.letters.Point sets are denoted by (capital) scriptThe inequality symbol between vectors means that inequalityholds for each component.-. .,.-. -.--. t.-.q'- .S,. . . . . . . '- , . .-- . .,iiIiI
5A'HYPOGRAPH EMPIRICAL PRODUCTION POSSIBILITY SETGiven the (empirical) points (X.,Y.). j l,.,n with (mxl) "input"vectors X. ) 0 and (sxl) "output" vectors Y. ) 0, we define the "empiricalproduction set" PE to be the convex hull of these points i.e.nn(2.1)PE{(xy)x" Xj, yj 1VUJYji,3" We extend it to our "empirical production possibility set" QE by addir- toP all points with inputs in P, and outputs not greater than some outputin PE i.e.(2.2)QE{(xy)x -x,y 4y for some E}eNote that QE is contained in (e.g. is smaller than) every productionpossibility set heretofore employed, i.e. those studied by Farrell [22],Shephard [28], Banker, Charnes and Cooper [4], Fare, et al. [21], etc.'Wealso use fewer axioms than the others, including even Banker [3] who, upto this point in journal publication, had used the most parsimonious axiomsystem to characterize the production possibility sets of DEA1.TheFarrell, Shephard, Fare sets are (truncated) cones; the BCC set (whennot also a cone) adds to QE the set((x,y))x ,yy for some () a QE}.These relations may be visualized in the schematic plot ofFigure 1:1ourusage antedates his since we used this in the Ph.D. and otherwork with Rhodes (the "R" of CCR).
6y/"///C/"/Figure 1Iwhere QE PE U A,the BCC set is QEU B, and the Farrell, Shephard, F reset is QE U B U C.',Q denote the sets corresponding to PEand QEwhen only theLetoutput yr is the ordinate.Evidently a frontier function fr (X) is determined1by(2.3)fr(X) max Y4 for (X,Yr)EiThen,Theorem 0: QEis the hypograph of fr(X) over {x :(x,y)E} se QE U B U C.'.- ., . . . . . . -. , . .- . - . - . ,Proof:TheH4th of setfr(X)co is:.-.-.-e. :'Ihypographdenot, -Qre thep ndsetn to P. andwhe., only'-'" -" th-e,." "- - -Let VE denote(x : (x,y) c QE}.It is the domai.i (the input set)of our empirical frontier functions.--Theorem 1:(x)A is a concave, piecewise linear function onQEPE,
-7-Proof:A necessary and sufficient condition for f (x) to be concave isthat its hypograph is a convex set (cf.Rockefellar [26], or FenchelThe piecewise linearity also follows from the constructioncombinations of the empirical points (X.,Y.), j 1,. ,n.33Z3]).of QE by all convexUWe further observe explicitly that no use whatever has been made ofnon-negativity of input and output values in the sets, functions or proofof Theorems 0 and 1. Therefore, they hold without this restriction--afact we shall employ elsewhere.Also, no assumptions have been made about the properties of anyunderlying function, or function hypograph, from which the (X.,Yj) of ourempirical construct may be considered samples.Theorem 1 shows, therefore,that any empirical (maximum) frontier function is the "concave cap" functionof its graph. . .-,.---,.,,-'-.---'"
THE EMPIRICAL PARETO-EFFICIENT PRODUCTION FUNCTIONA Pareto-efficient (ninimum) point for a finite set of functionsg 1 (x),.,g,(x) is a point xsuch that there is no other Doint x in th-domain of these functions such that(3.1)gk(x) gk(x*)k l,.,K,with at least one strict inequality.Charnes and Cooper in [5J, ChapterI)showed that x* is Pareto-efficient iff x* is an optimal solution to themathematical (goal) program1K(3.2)min F gk(x)x k 1subject to gk(x) gk(x*) , k l,.This was employed by Ben-Israel, Ben-Tal and Charnes in [7 ] to developthe currently strongest necessary and sufficient conditiqns for a Paretominimum in convex programming.Utilizing (3.2) we can now define and construct, im(or ex-)plicitlythe Pareto-efficient empirical (frontier) productionfunction.Because of Koopmans' work in this area (see [111), we shall useinterchangeably the designations "Pareto-efficient" and "Pareto-Koopmansefficienteetc. in this paper.Other usages of (3.2) to generalizations suchas the "functional efficiency" of Charnes and Cooper Ill]will not bedeveloped here.First, by (3.2), the Pareto- efficient points among our nempirical points can be determined.The empirical Pareto-optimal functionis then defined on the convex hull of thefr inputs by convex combinationsof the "output" values.Note that the convex hull of the Pareto-efficientpoints might not include all of PE since only the doubled line portionof the frontier corresponds to Pareto-efficient points.IFor a formal definition of goal programming and some of its history,A.and Cooper (10).See charnes,. .r.%-? . .r.T2.:. .-.-.,.- ".? ,.-.-.- . -. , . . . . ".- '. .w.,,-V"
9Since for efficient production we wish to maximize on outputs whileminimizing on inputs, our relevant gk(x) include both outputs and inputs, e.g.-gk(x )(33)-xi ,k s i, i 1,.,mfor(x,y).e QEFor the optimization in (3.2) we clearly need only consider (x,y) E PErather than QE*Thus the constraint inequalities in (3.2) are for a testpoint (x*,y*):x xX*y y*(3.4)and we have, since these are the envelopment constraints of Data EnvelopmentAnalysis for an observed input vector x and corresponding output vector yThe envelopment constraints of Data Envelopment Analysis inTheorem 2:production analysis are the Charnes-Cooper constraints of (3.2) for-testingPareto-Koopmans efficiency of an empirical production point.In no way is what we call "Data Envelopment Analysis" restricted tolinear constant returns to scale functions or to truncated cone domains.(3.2), Data Envelopment Analysis applies to much moregeneralEvconvex functions, function domains and other situations than the currentempirical production function one.To test an empirical "input-output" point (X0,Y ) for Pareto-Koopmans0efficiency, the C2 (Charnes and Cooper) test of (3.2) becomesminX,S ,S-- subject to YXeXC Y-s" XXX, s ,s-.'s--XX(3.5)where -eT yx eTx0.-Y X.[ ,. ,yn].nn.
23yAI"I "xFigure 3Informatically, we can do this by applying transformations of1form g. (y ) the g( (y,-j). (Where,with a20 to obtain possible new facets in min yProblems do arise, of course, .on whether one gets spuriousempirical frontier portions in this manner for empirical points whichshould "really" be inefficient.Evidently such non-concave portions areportions of increasing returns to scale if they are truly on the frontier.
22we choose the form of the inputs so that an increase in an input shouldnot decrease the outputs.But even here we need still more to determinethe non-concave portions of an isotonic function.For example, in Figure 2an isotonic function is plotted together with the resulting concave cap(largedashed lines) obtained as the empirical function-yl QE,II'x.Figure 2As suggested in our original (1981) paper I , non-concavity can beexploredby applying (output) component by component -strictly concavetransformations g,.,to obtain g,,(y)instead of y. so that g(yI(x)) wouldbe concave and our plot might look like1 "AnEmpirical DEA Production Function" by A. Charnes, W.W. Cooper,and L. Seiford, April 1981, CCS 396, Center for Cybernetic Studies, University of Texas, Austin, Texas.'.'.-,.". ". .".-.lI. q .
21c dfor r R , wherec xdIn generalwhenanxifYr YIn gnrlweanxdccdr fo.eR whrRUR (1,2,.,s}, then the cone of isotonic directions (wl,.,Ws)iisspecified by(s I[rYre R ()y.y-.rr)re R- (YHomogeneous production functions play an important role in theeconomics literature.Thereby, whether or not a function for whichf(px) Paf(x), with p 0, had economies of scale would be decided by thevalue of the exponent a. More generally, increasing or decreasing "returnsto scale" would be present respectively, at i if f(px) pf(x) orf(p ) pf(x) for p 1 at points pR in a small neighborhood of R. TheBCC paper [4] gives a criterion for deciding this (with production possibilityset QEU B UC or QEU B) but does not give us the rates of change.Because of our preceding theorems, however, we know that empiricalPareto-efficient functions are c-d-isotonic on facets and concave in eachcomponent function regardless of the nature of the underlying productionpossibility set.Thereby, we automatically anticipatelower and lowerreturns to scale in going from facet to facet with increasing eTx. Andour partial derivatives can give us explicitly the rates of change in eachobserved facet.Practically, our choices of inputs are generally made with theexpectation that the underlying Pareto-optimal function is isotonic, i.e.,. . .
20Now we proveTheorem 4:If a Pareto-efficient empirical production function hasonly a single output,then it is an isotone function.Proof:Suppose xa 4 Xb for Pareto-efficient (xya) and (xb yb)bBy definition of Pareto-efficiency for (xb yb),some component outputbaayj can exceed yb only if some other output yk ySince there isonly one output, then ya A f(Xa) 4 yb A f(xb).fix b ) for Pareto-efficient points, i.e. theThus xa Xb implies f(xa)isotone property.QED.To show that a multiple-output Pareto-efficient production function.need not. be isotone, consider the following one-input, two-output examplewith 3 sample points a, b, c:ax 1bc23Y,521y2574Input-output points a and b are*Pareto-efficient, c is not.We do not haveisotonicity since xa xb, but fl(xa) .ya fI(xb) 0 y.If now we "project" the outputs of a and b along the direction givenby wI and w2 for Yl and y2 we obtain the single output 5w, 5w2 for aand 2w1 7w2 for b. Requiring7w2 2w ;0 5w2 5w.implies w2 ; 3/2 w1 as the cone of directions (w1 , w2 ) which yields anisotonic relation.U-
19Theorem 3:Every facet of the empirical Pareto-efficient function relationconsists of Pareto-efficient input-output points.Proof:Each facet corresponds to a basic optimal solution to thefollowing linear program (C2-test) for some sample inputoutput point (XO , Yo):minYSeT-eT s -eT s"T-, 4 eTs-s -s-XX-s-yY0-X00. 0eXA, swhere Y(YI "".'.Yn),Xs0(XV,.,Xn), with optimal s* O, s*' O.Every(Xj, Y.) in the optimal basis is Pareto-efficient since (1) the optimal dualevaluators determined by the basic solution do not depend on the right handside vector (-Xo, Yo), and (2), replacing (-X0, Yo) by any (-X., Y) inthe basis preserves the feasibility of the basis for solution with thenew right hand side.Next, if we C2 -test any convex combination of the basic (-X., Y.)T-TTe.g. (-XB, YB)T0Bwith GB ) 0, eT GB - 1 for Pareto-efficiency ad thisreference set, i.e. insert it in place of (-X0 , Yo)T, an optimal X* isthen simply GB (plus s ,s0) with this same basis.0Thus the wholefacet is Pareto-efficient.Q.E.D.-.-.
18(B) Isotonicity and Economies of ScaleTo date the structure, or "geometry", of empirical Pareto-efficientproduction functions has received little attention.The structuredepends on and varies with the "production possibility" or "reference"set chosen.Here we make a beginning for our new set QE and leave tolater research more in-depth and broader explorations including those forother reference sets.In many practical situations we try to choose inputs and outputswith the thought that the underlying empirical Pareto-efficient functionshould be "isotone" (which means "order-preserving").(vector) function f(x) is isotone if xWhat in fact can be the case?By definition axb implies f(xa)f(xb). IWe show here that in the single outputsituation the empirical Pareto-efficient function is always isotone.Themultiple output situation, however, may only satisfy a weaker functionproperty which we shall call "c-d-isotone" or "cone-directional-isotone",i.e. there is a cone of directions in output space on which the outputsprojection is isotone.Consider first the "facets" of an empirical Pareto-efficientfunction for our reference set.These consist of convex combinations ofthe Pareto-efficient sample input-output points with respect to thisreference set.1The mathematical term "isotone" is synonymous with the expression"monotonically increasing".7.
17Suppose we run the C2 -test with (as the point being tested.the optimal dual variables corresponding to input(IF)andiThenand output y are respectivelyThus, the rate of change of output Y4 with.respect to input xi is simply the negative of the ratio of the optimal dualvariable! 1xi constraint variable to the optimal dual y. constraintMore specifically, all Pareto-efficient (XjYj) of the facet for thepoint(x,y) satisfy(5.3)where (p*T, V*T)vx0,Ty0 -are the dual evaluators at an optimal basic solution,since they do not depend on the C2 -test right hand sides.F(x,y). (5.4)C l F/P*TyV*Tx-, -v* -Thereby our0as already stated.aF/ax iIt should be borne in mind, of course, that these rates of change arevalid only for changes which keep one within the facet.1 Seealso p. 439 in 15) for a discussion which can now relate thisdeve-lopment to the ordinary conditions of economic theory for equalitybetween ratios of marginal productivities and marginal rates of substitution. -.-.,-.-.-, -,
16INFORMATICS AND FUNCTION PROPERTIES(A) Partial Derivatives:The guidanceprovided by the CCR, BCC, C2 S2 formulations does notinclude convenient access to the rates of change of the outputs with changein the inputs.The optimal dual variables in the DEA side linear programmingproblems give rates of change of the efficiency measure with changes in inputsor outputs.The non-Archimedean formulations further may give infinitesimalrates., which are not easily employed.And, for most of the efficient pointsone has non-differentiability because they are extreme points rather than(relative) interior points.Nevertheless, because of the informatics, e.g.,computational tactics, we employ in testing via C2 for Pareto-efficiency,the following constructive method can be employed.On reaching a non-Pareto efficient point, our software discovers allthe optimal observed points in its facet, hence, implicitly, all the convex combinations which form the facet.Since the Pareto-efficient facet is a linearsurface it is not only differentiable everywhere in its relative interiorbut all its partial derivatives are constant throughout the facet.Thus,we need only obtain these for any relative interior point of the facet tohave them for the whole facet.Let(5.1)F(xl,.,xm,y1,. ,ys)0be the linear equation of the facet.Since we have sufficient differentia-bility in the neighborhood of an interior point (.3y), we knowI(5.2I.7--xywhere the right side partial derivatives are also evaluated at . . . . . 1. .,. .1. . . . . .- . . 1.:.-.-. 1--.-,.--.'-1-.- .-.1. .,.,-,,.'.1.-.il
15The CCR efficient DMU's are also among the new Pareto-efficient DMU's.Projection of a non-optimal DMU onto its Pareto-efficient facet is renderedbyxo xo s*"(4.6)(4.)0X 0 -sYoYo0.0:ii" S*sTo achieve a convenient efficiency measure, we modify the functionalby multiplying it by a a 0 and dividing the s the entries in Y and X0 , e.g.,-aeT D 1 (Y) s (4.7)where D( (Yo), D 16 eTD1and s- by respectively(Xo )X)0 are diagonal matrices whose diagonals are thereciprocals of the entries in Yo, Xo respectively. This achieves a unitsinvariant measure which may be thought of as the logarithm of the efficiencymeasure.A S 10/(m s) will yield a logarithm between 0 and -10.Thismeasure might then be called the "efficiency pH" by analogy with the pH ofchemistry.Our new measure relates to the units invariant multiplicative measureof Charnes, Cooper, Seiford and Stutz which, as shown in [19], is necessaryand sufficient that the DEA envelopments be piecewise Cobb-Douglas, by considering the entries in the Xj., Y to be logarithms of the entries in Xj, Y.which we employ in the multiplicative formulation.
14and "managerial efficiency" in their analysis of programs Follow-Through andnon-Follow-Through.It also shows quantitatively what improvements in inputsand outputs will (ceteris paribus) bring a DMU to efficient operation. IThus, although the relative efficiency measure of an inefficient DMU willinvolve the infinitesimal e, non-infinitesimal changes for improvement aresuggested.Both Farrell and Shepard knew that ratio measures required adjustmentsto correctly exhibit inefficiency of the second DMU in examples like thefollowing 2 input, 1 output, 2 DMU case:DMUxlx2y11212141Farrell added geometric points at infinity; Shephard simply excluded suchcases without giving a method for their'exclusion. The non-Archimedeanextension in the CCR formulation was introduced to have an algebraicallyclosed system of linear programming type.Linear programming theory holdsfor non-Archimedean as well as Archimedean entries in the vector and matrixproblem data. 2Our new Pareto-efficient DEA method like C2 S2 [19] associates facets withnon-optimal ( non-Pareto-efficient) DMU's.Clearly, by the C2-test, DMU ois Pareto-efficient (Pareto-optimal) iff -eTs* 2l-distanceT*e sfrom (X ,Y ) to the farthest "northwesterly"0, i.e., iff theQEpoint is zero.1The analysis in [8 ] shows how one might take account of the possibleeffects on other DMUs when one or more of the efficient DMUs is altered.2 Seethe discussion on p. 756. in [Il].
.13constraints for an empirical production possibility set of Farrell, Shephard,U B8EU C, and, sinceetc. cone type(4.4)e-YeTyXeTxX]is an equivalent form for the functional, as being a Charnes-Cooper Paretooptimality test for (eX ,Yo ) over the cone on the (Xj,Y.), j 1,.,n, withpre-emption on the intensity 6 of input Xo . As mentioned above and shown,0for example, in l4],DMUo is efficient iff* e 1, s -0, s* o.-Re informatics, which are particularly important since all nefficiency evaluations must be made (i.e., n linear programs must be solved),the dual problem can be computed exactly (in the base field) as shown in [11],e.g., with the code NONARC of Dr. I.Ali (Center for Cybernetic Studies, TheUniversity of Texas at Austin), or approximately by using a sufficientlysmall numerical value for e. A typical efficient point is designated by(x,y) in Figure 1.If a DMU is inefficient, the optimal x 0 in its DEA problem( Charnes-Cooper test) designate efficient DMU's, as do alternate optima.Thus, a "proper" subset of the efficient DMU's determines the efficiencyvalue of an inefficient DMU. The convex combinations of this subset arealso efficient. Thereby to each inefficient DMU a "facet" of efficientDMU's is associated.(4.5)Xoe0The transformation-so.Y o Y00where the asterisk designates optimality, projects DMU o , i.e., (Xo,Yo), onto0 00its efficiency facet.This projection was introduced by Charnes, Cooper and Rhodes [6] to correct.for differences 'inmanagerial ability in order to distinguish between "program".
12Employing the Charnes-Cooper transformation of fractional programming IT(4.2)TTo,T{TTX 10".V0we obtain the dual non-Archimedean linear programsTYmaxI,tVT-mineet X, S , ssubject toT -ces ce sYX -s TX1Y00]TY Tx(4.3)TTTUwhereX[X I1'*.,X,nY: -' s )0-.X, s,s0 -eTV-s 0- XX0eX' -eT[YJ,. ,Y].1'nThe problem on the right is associated with the origin of the term"Data Envelopment Analysis" since the minimization (a) envelopsvector Y from above and (b) envelops00the outputthe input vector X from below viathe minimizing choice of the scalar value of the intensity e* min e. Theproblem on the left is said to be in efficiency analysis form with the maximization oriented toward the choice of V and v(called virtual multipliersor transformation rates) which produces the greatest rate of virtual outputper unit virtual input allowed by the first constraint together with the re*quirements (a) vi.rtual output cannot exceed virtual input and (b) all virtual*transformation rates must be positive.Although, clearly, no assumptions have been made concerning thetype of functional relations for the input-output pairs (X.,Y.), the minimization program may be recognized as having the Data Envelopment Analysis.See Charnes and Cooper [9] and S. Schaible [29].". .'
EFFICIENCY ANALYSISAs mentioned, managerial and program comparison aspects ofefficiency analysis were initiated by Charnes, Cooper and Rhodes in [15],(16), and [12], through a generalization of the single input, single outputabsolute efficiency determination of classical engineering and science tomulti-input, multi-output relative efficiencies of a finite number ofdecision-making units "DMU's" (sometimes called i"productive" units or "response"units).The multi-input, multi-output situations were reduced to "virtual"single input single output ones through use of virtual multipliers and sums.Explicitly, the CCR ratio measure of efficiency of the DMU designated "o"is given by the non-linear, non-convex, non-Archimedean fractional program(see [141).MaxnTYOnTysubject toX41n nTT Ee T,j ,., n,.x(4.1)'-iTxoTirwhere the entries of the X. and Y. are assumed positive, e is a nonArchimedean infinitesimal, eT is a row vector of ones and, by abuse ofTTnotation, has s entries for n m entries for T (XY is one of then input-output pairs.-7. .-. .l".
10Since -eT(yX-Yo) eT(X'- Xo) is an equivalent functional (it differs from00the above one only be a constant), we can re
In efficiency analysis, observations are generated by a finite number of "DMU"s, or "productive," or "response" units, all of which have the same inputs and outputs. A relative efficiency rating is to be obtained for each unit. Typically, observations over time will be made of each unit and the results of efficiency analyses will be employed to
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In contrast, pile-supported foundations transmit design loads into the adjacent soil mass through pile friction, end bearing, or both. This chapter addresses footing foundations. Pile foundations are covered in Chapter 5, Pile Foundations-General. Each individual footing foundation must be sized so that the maximum soil-bearing pressure does not exceed the allowable soil bearing capacity of .
It is an honour for Assifero to present this guide to community foundations in Italy. The community philanthropy movement is growing rapidly all over the world. In Italy, the establishment of community foundations began in 1999 with foundations in Lecco and Como. There are now 37 registered Italian community foundations (based on the atlas of
239 Batman Hush Deluxe 240 Batman Hush Joker 241 242 1st Appearance Wonder Woman (NYCC) 243 Shirtless Arthur Curry (NYCC 18) 244 Arthur Curry 244 Patina Arthur Curry (Walmart) 244 Gold Chrome Arthur Curry (Target) 245 Aquaman 246 Mera 247 Orm 248 Black Manta 248 Glossy Black Manta (Walmart) 248 Matte Black Manta (Target)
travel abroad with the prior consent” of the Department of Homeland Security (DHS). Section 244(f)(3) of the Act. However, by name and definition, TPS is temporary in nature and merely defers removal from the United States during the period in which a country or area is designated under the statute. sections 244(a)(1) and 244(f)(1) of the Act.
PUBLIC LAW 105–244—OCT. 7, 1998 112 STAT. 1581 Public Law 105–244 105th Congress An Act To extend the authorization of programs under the Higher Education Act of 1965, and for other purposes. Be it enacted by the Senate and House of Representatives of the United States of America in
Chelsea PTO 242/244 Parts Manual Pro Gear Chelsea 242/244 Series PTO parts manual to assist in id
8th Grade Writing and Speaking/Listening Scope and Sequence 1 s t Q u a r te r 2 n d Q u a r te r 3 r d Q u a r te r 4 th Q u a r te r Writing N a rra t i ve I n t ro d u ce ch a ra ct e rs a n d o rg a n i ze a n e ve n t se q u e n ce (W . 8 . 3 a ) U se n a rra t i ve t e ch n i q u e s i n cl u d i n g
