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xivINTRODUCTION TO DATA ENVELOPMENT ANALYSIS AND ITS .79.810.110.210.3Efficiency and Weight of 14 Hospitals by CCR ModelEfficiency and Weight of 14 Hospitals with Assurance RegionMethodEfficiency of 14 Hospitals by CR (Cone-Ratio) and CCR ModelsNumber of Bank Failures (through 10-31-88)Inputs and OutputsCCR and Cone-Ratio Efficiency Scores (1984, 1985)*Printout for Cone-Ratio CCR Model - Interstate Bank of FortWorth, 1985.Scores (Sij) of 10 Sites (A-J) with respect to 18 Criteria (ClC18)Statistics of Weights assigned the 18 Criteria (C1-C18) by 18Council MembersAverages and Medians of Scores of the 10 SitesData for Public Libraries in TokyoEfficiency of Libraries by CCR and NCNData of 12 Japanese Baseball Teams in 1993Projection of Attendance by CCR and Bounded ModelsCategorization of LibrariesNine DMUs with Three Category LevelsComparisons of Stores in Two SystemsComparisons of Two SystemsExample of Bilateral ComparisonsSample Data for Allocative EfficiencyEfficienciesComparison of Traditional and New SchemeData for 12 HospitalsNew Data Set and EfficienciesDecomposition of Actual CostData for a Sensitivity AnalysisInitial SolutionsResults of 5% Data VariationsOLS Regression Estimates without Dummy VariablesStochastic Frontier Regression Estimates without Dummy VariablesOLS Regression Estimates without Dummy Variables on DEAefficient DMUsStochastic Frontier Regression Estimates without Dummy Variables on DEA-efficient DMUsWindow Analysis: 56 DMUs in U.S. Army Recruitment Battalions 3 Outputs - 10 InputsTest DataAndersen-Petersen Ranking*Non-radial 282283284285294303304309

LIST OF TABLES10.410.510.6A.lA.2B.lB.2Data for Super-efficiencySuper-efficiency Scores under Variable RTSSuper-efficiency Scores under Constant RTSSymmetric Primal-Dual ProblemGeneral Form of Duality RelationWindow Analysis by Three Adjacent YearsHeadings to Inputs/OutputsXV312313313319324339342

List of isons of Branch StoresRegression Line vs. Frontier LineImprovement of Store ATwo Inputs and One Output CaseImprovement of Store ylOne Input and Two Outputs CaseImprovementExcel File "HospitalAB.xls"Example 2.1Example 2.2Region PThe Case of DMU AA Scatter Plot and Regression of DEA and Engineering Efficiency RatingsEnvelopment Map for 29 Jet EnginesA Comparative Portrayal and Regression Analysis for 28 Engines (All Data Except for Engine 19)Production Possibility SetExample 3.1Production Frontier of the CCR ModelProduction Frontiers of the BCC ModelThe BCC ModelThe Additive ModelTranslation in the BCC ModelTranslation in the Additive ModelFDH RepresentationTranslation in the CCR ModelReturns to ScaleReturns to Scale: One Output-One InputMost Productive Scale SizeThe IRS Model3457899172730313236373843578484869194, 94107111120122128138xvii

8B.9B.IOB.llB.12B.13B.14INTRODUCTION TO DATA ENVELOPMENT ANALYSIS AND ITS USESThe DRS ModelThe GRS ModelScale EfficiencyMerger is Not Necessarily Efficient.Efficiency Scores with and without Assurance Region ConstraintsAssurance RegionConvex Cone generated by Two VectorsThe Hierarchical Structure for the Capital Relocation ProblemPositives and Negatives of the 10 SitesOne Input Exogenously Fixed, or Non-DiscretionaryDMUs with Controllable Category LevelsComparisons between Stores using Two SystemsBilateral Comparisons-Case 1Bilateral Comparisons-Case 2Case 1Case 2Controllable Input and OutputCategorization (1)Categorization (2)Technical, Allocative and Overall EfficiencyDecomposition of Actual CostStable and Unstable DMUsA Radius of StabilityThe Unit Isoquant Spanned by the Test Data in Table 10.1Sample.xls in Excel SheetSample-AR.xls in Excel SheetSample-ARG.xls in Excel SheetSample-NCN (NDSC).xls in Excel SheetSample-BND.xls in Excel SheetSample-CAT.xls in Excel SheetSample-Cost (New-Cost) .xls in Excel SheetSample-Revenue(New-Revenue).xls in Excel SheetSample-Window.xls in Excel SheetSample-Weighted SBM.xls in Excel SheetA Sample Score SheetA Sample Graph2Variations through WindowVariations by 335336337338339340

Preface1. IntroductionWe earlier coauthored a text, published in 2000, that is described in the following "Preface to Data Envelopment Analysis: A Comprehensive Text withModels, Applications, References and DEA Solver Software." The present textmodifies and revises that book in order to incorporate important new developments. These developments have appeared in papers, books, monographs, etc.,that now number in the thousands and address a wide range of different problems by many different authors in many different countries. See the publicationsthat appear in the disk that accompanies this text. See also Tavares (2003) "ABibliography of Data Envelopment Analysis (1978-2001)" which hsts more than3200 papers, books, monographs, etc. addressing a great variety of problemsby more than 1600 authors in 42 countries.To accommodate all of these developments is now nearly impossible. Wehave therefore directed our attention to an "introduction" — hence the titleof this book. This means that we have focused on approaches that have beenmost widely used or are most likely to be encountered by users of DEA and wehope to include others in a follow-on book.2. What is Data Envelopment Analysis?Data Envelopment Analysis (DEA) was accorded this name because of theway it "envelops" observations in order to identify a "frontier" that is usedto evaluate observations representing the performances of all of the entitiesthat are to be evaluated. Uses of DEA have involved a wide range of differentkinds of entities that include not only business firms but also government andnon-profit agencies including schools, hospitals, military units, police forcesand court and criminal justice systems as well as countries, regions, etc. Theterm "Decision Making Unit" (DMU) was therefore introduced to cover, in aflexible manner, any such entity, with each such entity to be evaluated as partof a collection that utilizes similar inputs to produce similar outputs. Theseevaluations result in a performance score that ranges between zero and unity

XXINTRODUCTION TO DATA ENVELOPMENT ANALYSIS AND ITS USESand represents the "degree of efficiency" obtained by the thus evaluated entity.In arriving at these scores, DEA also identifies the sources and amounts ofinefficiency in each input and output for every DMU. It also identifies theDMUs (located on the "efficiency frontier") that entered actively in arriving atthese results. These evaluating entities are all efficient DMUs and hence canserve as benchmarks en route to effecting improvements in future performancesof the thus evaluated DMUs.The different types of efficiency covered in this text range from "allocative,"or "price," efficiency, and extend through "scale" and "technical" efficiency,as well as "mix" and other kinds of efficiencies. Technical inefficiency, whichrepresents "waste," is the one we focus on in this Preface because it requires theleast information, makes the fewest assumptions, and is the one most likely tobe agreed upon as to what is meant by the term "inefficiency." Uses of DEA toeffect these evaluations are almost entirely "data dependent" and do not requireexplicit characterizations of relations like "linearity," "nonlinearity," etc., whichare customarily used in statistical regressions and related approaches wherethey are assumed to connect inputs to outputs, etc.3. Engineering-Science Definitions of EfficiencyTo illustrate these uses we start with the usual "output-to-input ratio" definitions (and measures) of efficiency used in engineering and science. Consider,for instance, the following definition (quoted from the Encyclopedia Americana(1966) ) where it is illustrated by an example from the field of combustionengineering: "Efficiency is the ratio of the amount of heat liberated in a givendevice to the maximum amount which could be liberated by the fuel [beingused]." In symbols, E Vr/yRj where yji Maximum heat that can be obtained from a given input of fuel and yr Heat obtained from this same fuelinput by the device being rated.In "pure science" the rating would probably be obtained from theoreticalconsiderations, such as are available from the theory of thermodynamics, toobtain a value 0 E 1, with a value of unity not being obtainable. Inapplied science or engineering, alternative approaches are likely be used. Forinstance, the rating of a furnace that is being developed for sale would beevaluated by comparing it with a "reference furnace" with a value of unitybeing obtained when the heat obtained is equal to the heat from the referenceunit.Other approaches are used when ratings for n( 1) devices are to be developed in order to determine relative efficiencies. See Buha et al (2000) foran example involving relative evaluations of 29 jet aircraft engines in whichDEA is compared with the customarily employed engineering measure of efficiency. This engineering measure of efficiency consists of the ratio of "workrate output" to "fuel energy input." Using this same input and output, DEAproduced the same efficiency rankings as the engineering measure. However,other factors (including cost) also need to be considered en route to a decisionon which engine to choose. To accommodate these other measures, DEA was

PREFACEXXIexpanded to three inputs and two outputs and produced rankings that weremarkedly different from the engineering measure.To accompHsh this kind of result, DEA expands the ratio definition of efficiency to accommodate multiple outputs and multiple inputs. This expansion to multiple inputs and outputs is accomplished without using preassignedweights or other devices such as are commonly employed in index number (orlike) constructions and no a priori identification of reference units, etc., arerequired. Proceeding in this manner DEA identifies a set of "best performers"from a collection of DMUs and uses them to generate "efficiency scores" obtained by evaluating each of the n DMUs. This is accomplished by identifyinga point on this frontier that assigns a value of 0 1 to each of the nDMUs. This score is obtained by comparing the performance of the DMU tobe evaluated relative to the performances of all DMUs.As obtained by DEA, this score reflects a series of weights as determinedfrom the data by DEA — one for each output and one for each input - thatproduces the highest efficiency score that the data will allow relative to thevalues 0 Ej 1 for each of the several DMUj j 1 , . . . , n, being considered.Moreover, these weights (called "multipliers" to distinguish them from customary weighting processes) have additional uses. A multiplier can be used, forinstance, to evaluate changes in the efficiency score for a DMU that will accompany a change in the corresponding input or output. Since these multipliers areapplicable to all DMUs, they can also be used to conduct "sensitivity analyses"to determine changes in all data (considered simultaneously) that will resultin a change of classification from "efficient" to "inefficient," or vice versa forany DMU, and also identify the ranges of data variation that can be allowedbefore such a change occurs. Unlike statistical and mathematical programminganalyses, in which sensitivity analyses apply to only one data point at a time,these sensitivity analysis can allow simultaneous variation in all of the data.How this is done is described in this book. Also described in this book aremethods for undertaking a one-data-point-at-a-time analysis (as in statistics,say) that can also be used to determine the allowable range of stability for thedata changes associated with any individual DMU.4. Different Definitions of Efficiency—and DualityThe above discussion is based on generalizations (and additional uses) of engineering-science definitions of efficiency. There are, however, other definitions ofefficiency that are used in other disciplines. One such is the very general definition of efficiency, called "Pareto optimaHty," as used in "welfare economics,"which was formulated by the Swiss-Italian economist, Vilfredo-Pareto. Thedefinition, as given on page 319 in Pearce (1986)" is as follows: "A Pareto optimum is a welfare maximum defined as a position [in an economy] from whichit is impossible to improve anyone's welfare by altering production or exchangewithout impairing someone else's welfare." This definition forms a basis for"welfare economics" in which proposed changes in policy (e.g., enactment of atariff) are to be judged according to whether the proposed change can increase

xxiiINTRODUCTION TO DATA ENVELOPMENT ANALYSIS AND ITS USESthe welfare of some without worsening the welfare of others. This approach,we might note, avoids the need for making interpersonal comparisons such aswould take the form of weights, or similar devices, that assign relative valuesto different individuals in making such decisions.Tjalling Koopmans (1951), a Dutch-American economist, subsequentlyextended this definition to "production economics" by introducing "efficiencyprices" ( multipliers) to guide production and exchange to positions that aresimilar to a Pareto optimum. This approach is further extended and exploitedin DEA by reference to what is referred to as the "Pareto-Koopmans" definitionof eflSciency.As shown in the present book, these seemingly different definitions involvedin engineering-science and Pareto-Koopmans definitions of efficiency are mathematically related to each other by the "duality theorem" of linear programming.See Appendix A in this book for a description and discussion of duality andother parts of linear programming. To state this differently, there is a problemthat is dual to the one used to implement the engineering definition of efficiencywhich employs the same data in a different manner to arrive at a measure ofPareto-Koopmans efficiency. At their corresponding minimum and maximumvalues the Pareto-Koopmans and engineering science scores are then equal.This dual problem utilizes the following variant of a "welfare optimum" indefining efficiency:Definition 1 (Pareto-Koopmans Definition of Efficiency) The performance ofa DMU is efficient if and only if it is not possible to improve any input or outputwithout worsening any other input or output.From this we also haveDefinition 2 {Definition of Inefficiency) The performance of a DMU is inefficient if and only if it is possible to improve some input or output withoutworsening some other input and output.Thus a characterization of being inefficient holds if and only if the observation— i.e., the input-output coordinates — for the DMU being evaluated is not onthe efficiency frontier of the set of production possibilities that can be generatedby DEA from the data on all observations. Conversely, the geometric pointsassociated with the performance of the DMU to be evaluated by DEA will befound to be efficient if and only if it is on the efficiency frontier.5. Extensions and Further Uses of DEAThe above generalization of the engineering-science definition of efficiency involves maximizing a ratio of inputs to outputs subject to a collection of nratios that also take the form of inequality constraints with a bound of unityon their possible values. However, as shown in this book, this nonlinear (andnon-convex) problem can be replaced by a simple (ordinary) linear programming problem that makes the solution easily obtainable from any of numerouscomputer codes that are used to solve linear programming problems. It also

PREFACEXXlllopens other possibilities for use from now available computer codes that arespecifically designed for use in DEA. See, for instance, the DEA-Solver codethat is described in Appendix B of this book.Using the duality relation of linear programming to comprehend these twodifferent definitions of efficiency and giving them implementable form greatlyextends the ability of DEA to deal with a wide range of problems. For instance,the study recommending a relocation of the Japanese capital to a new site (nowbeing considered by the Japanese Diet) had to deal with "inputs" like "susceptibility to earthquakes" and with "outputs" like "ability to recover from earthquakes." These are not "inputs" or "outputs" in the ordinary senses of theseterms and are therefore referred to as "goods" or "bads" according to whetherthey should be placed in the numerator or denominator of an engineeringscience type of "efficiency ratio" in accordance with whether an increase in thisitem should result in an increase or a decrease in the performance score. Thisgreatly simplifies matters that would otherwise be difficult to accomplish enroute to identifying inputs as distinguished from outputs (in borderline cases)that could be difficult to do with the Pareto-Koopmans definition of efficiency.This illustrates one way that the "conceptual power" of the engineering-sciencedefinition of efficiency can be used to obtain characterizations that also satisfythe Pareto-Koopmans definition. The thus identified inputs and outputs thathad to be considered in the Japan relocation study were then incorporated ina DEA model that was directed to "effectiveness" rather than "efficiency" inorder to determine the best location (two "equally best" locations were identified). As defined in this book, these two concepts differ as follows:Effectiveness- I ' ' " ' '( Ability to achieve goals.goaBenefits securedEfficiency:Resources used.Thus effectiveness refers to goal achievement and, in contrast to efficiency,effects its evaluations without reference to the resources used. (See Chapter 6 inthis book for further discussion of how the committee appointed to conduct thisstudy developed its reports to produce recommendations that were submittedto the Prime Minister who has transmitted them to the Japanese Diet forconsideration).These and the other developments in the use of DEA are all given implementable f

X INTRODUCTION TO DATA ENVELOPMENT ANALYSIS AND ITS USES 6.10 Related DEA-Solver Models for Chapter 6 194 6.11 Problem Supplement for Chapter 6 195 7. NON-DISCRETIONARY AND CATEGORICAL VARIABLES 203 7.1 Introduction 203 7.2 Examples 205 7.3 Non-contro