DATA ENVELOPMENT ANALYSIS (DEA): A TOOL FOR MEASURING EFFICIENCY AND PERFORMANCE

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1 DATA EVELOPET AALYSIS (DEA): A TOOL FOR EASURIG EFFICIECY AD PERFORACE Jorge A. Santos Pd student at: Departamento de atemática Universidade de Évora Évora Portugal jmas@uevora.pt José H. Dulá Scool of Business Administration University of ississippi University, S 8677 jdula@olemiss.edu Abstract Tis paper introduces Data Envelopment Analysis and igligts its potentialities to evaluate te performance of organisations, it describes too te concept of superefficiency an extension of DEA wic increases its performance. Keywords: Data Envelopment Analysis, Efficiency Evaluation. I. ITRODUCTIO Data Envelopment Analysis (DEA) is a matematical programming based tecnique to evaluate te relative performance of organisations. Wile te main applications ave been in te evaluation of notfor-profit organisations, te tecnique can be successfully applied to oter organisations, as a recent evaluation of Russian industry as demonstrated []. Wit tis paper we ave two objectives in mind. Te first one is to present DEA-Data Envelopment Analysis, a tecnique wic may ave useful applications in many evaluation contexts, namely wen assessing not for profit organizations. In addition to allowing te ranking of te organizations traditionally termed decision-making units, DEA also creates te conditions to improve performance troug target setting and rolemodel identification. Te second objective is to describe briefly te tecnique of deleted domain, also known as superefficiency. DEA is suited for tis type of evaluation because it enables results to be compared making allowances for factors [2]. DEA makes it possible to identify efficient and inefficient units in a framework were results are considered in teir particular context. In addition, DEA also provides information tat enables te comparison of eac inefficient unit wit its "peer group", tat is a group of efficient units tat are identical wit te units under analysis. Tese role-model units can ten be studied in order to identify te success factors wic oter comparable units can attempt to follow. Tanassoulis et al [] argue tat DEA is preferable to oter metods, suc as regression analysis, wic also make it possible to contextualize results. Te present paper is structured as follows. Te next section describes te development and fields of application of te tecnique, wile section III introduces te DEA models followed by a numerical example and a final section about superefficiency evaluation, an extension of DEA. Te readersip not familiar wit DEA, a may find te brief introduction to te metod presented below usefull, but for tose wo wis to follow te matter furter tere is a good review of DEA in Boussofiane et al [4]. II. HISTORY AD APPLICATIOS OF DEA DEA is a matematical programming tecnique presented in 978 by Carnes, Cooper and Rodes [5], altoug its roots may be found as early as 957 in Farrel`s seminal work [6]. Tis tecnique is usually introduced as a non-parametric one, but in fact it rests on te assumption of linearity [7] and for te original models even in te more stringent assumption of proportionality. Its application as been focused mainly on te efficiency assessment of not-for-profit organizations, since tese cannot be evaluated on te basis of traditional economic and financial indicators used for commercial companies. Te first application of DEA was in te field of Education, in te analysis of te Program Follow Troug, conducted in te USA, in te late seventies [8]. Since ten it as been used to assess efficiency in areas suc as ealt [9, 0], prisons [], courts [2], universities and many oter not-for-profit sectors. owadays DEA can be seen to ave spread to oter fields suc as Transit [], ining [4], Air Transportation [5], and even Banking [6]. However, many applications belong to te education field and range from primary education [7, 8], to secondary [9, 20, 2] and university levels [22]. In Data Envelopment Analysis te organizational units to be assessed sould be relatively omogeneous and were originally termed Decision aking Units. As te wole tecnique is based on comparison of eac DU wit all te remaining ones a considerable large 42

2 set of units is necessary for te assessment to be meaningful. We will assume tat eac DU produces outputs by means of inputs. III. DEA FORULATIOS In DEA, efficiency (j ) of a specific decision making unit (DU j ) is defined as te ratio between a weigted sum of its outputs Ynj and a weigted sum of its inputs Xmj, a natural extension of te concept of efficiency used in te fields of pysics and engineering [2]: ' nj' n= nj' = v y mj m= mj' ' Wen assessing a set of J organisations, were Xmj stands for te m t input of te j t DU, wit a similar meaning for Ynj, te weigts µmj and νnj, in Eq (), are cosen for eac j DU under evaluation as tose tat maximize its efficiency as defined by j. Several constraints ave to be added to te maximization problem: Te strict positivity [24] of te weigts µmj,νnj (also known as virtual multipliers). For scaling purposes, all J DUs under analysis must ave efficiencies not exceeding an agreed value, typically one or 00%, as is usual in engineering definitions of efficiency. A tird kind of restriction as to be included since oterwise tis linear fractional program would yield an infinite number of solutions. In fact, if a set of weigts µmj,νnj returns te optimal solution, so would kµmj, kνnj. aking te denominator, in Eq (), equal to one or 00%, circumvents tis situation. So, we ave to solve te following maximization problem for eac one of te J DUs under analysis: ax nj' n nj ' = = j' m mj' = µmj ε > 0 m=... (0) () Eq.. νnj ε > 0 n=... () (2) s.t. µmj > 0 m=... () νnj > 0 n=... (4) nj n nj = = ' ' j m mj' = m= mj' j=...j (5) = (6) Tis Fractional Linear Program can be solved by means of te Carnes and Cooper transformation [25] wic yields te following Linear Program: ax = j' nj' n= nj ' m mj' = (7) s.t. = (8) nj' n= nj m= mj' mj j=...j (9) Te problem above is known as te multiplier problem, since its unknowns are te weigts, wic are usually lower bounded by a small quantity ε (Eq: 0-) so tat all Inputs and Outputs are considered in te evaluation [24], even if wit a minor weigt ε, set in all te following formulations equal to 0-6. Te dual of tis problem, wic we sall call te envelopment problem, provides important information about economies tat could be acieved in all te inputs; it also indicates wic efficient units te inefficient unit being assessed sould emulate. Tose efficient units are usually referred to as te reference set or peer group of te unit under evaluation. IV. UERICAL EXAPLE To illustrate te Data Envelopment Analysis tecnique, an example is introduced in Table I, wit 2 DUs producing two Outputs Y and Y2 from a single Input X, under te assumption of constant returns to scale, wic simply means tat if one doubles te Inputs of any unit it would be expected tat its Outputs would also double. In algebraic form tis can be stated as: if x j yields Outputs y j tan Inputs kx j sould produce Outputs ky j. Table I- Outputs normalised by Input X DU x y y2 y/x y2/x

3 In tis simple example we can normalise te Outputs by te only Input and plot tem in te plane. From Figure it is easy to understand te reason for naming tis tecnique Data Envelopment Analysis; in fact eac DU is analysed against te envelope of te most efficient units. For instance, te efficiency of DU 8 is (see Table II). Tis means tat it could reduce its input to 85.7% of its current value reacing its target, Ci 8 (were Ci stands for Composite unit under minimisation of inputs), wic is te same as for DU except tat, for te later, a reduction to 28.6% of X s current level of inputs would be necessary for tis DU to become efficient, since its efficiency is only C i Ci8;Ci C i y/x Figure - Efficiency frontier and radial projections for inefficient units. Altougt te results could be obtained grapically we present in table II te results obtained by a standard linear optimisation software Table II- Results for te multiplier problem DU µ ν ν 2 Efficiency We minimised te consumption of X, te only Input, so tat DU 8 (or ) become efficient. Tis contrasts clearly wit Eq (2) were we ave a maximisation problem. Te reason for tis is simply tat we ave been using te dual of our original problem (for a generic introduction to duality in Linear Programming see [26] or [27]). Tis related problem can be written as follows: + min: Z j -ε Snj' + Smj' n= m= s.t. J λ j xmj + j= J j= λ j y nj S mj + nj (2) ' = Z j xmj m=... () S ' = y nj n=... (4) λ j 0 j=...j (5) S + nj' 0 n=... (6) S mj' 0 m=... (7) Tis formulation can be interpreted as follows: given DU j find te Composite unit wic as no smaller outputs tan tis one and wose inputs are smaller tan tose of DU j scaled down by a factor Z j as little as possible. Tis is wy tis formulation is known as Input minimisation; since we are minimising Z j, we are seeking te minimal inputs tat, based on best acieved performance, could still produce te same amount of Outputs as DU j is currently doing. Tese Composite units are linear combinations of efficient units wic lie on te efficiency frontier. Tese efficient DUs are known as te peer group, te role model units tat inefficient DU j sould try to emulate. However, some DUs cannot be expressed as a linear combination (wit positive weigts) of te efficient units. In our example tese are DUs 2, 7 and 9 (see Figure ). Te goal for tese inefficient units is an equally proportional contraction of teir Input and te increase in Output or 2; S j + or S 2j +, known as Slacks, teir name in standard Linear Programming terminology. Te Input minimisation for te above envelopment problem would return te values presented in Table III (all te absent λs are null). Table III- Dual Variables(λs), Slacks and Efficiency DU λ λ4 λ0 S + S2 + Efficiency

4 In tis model DUs and 8 can be tougt of as combinations of efficient DUs 4 and 0. Or, in oter words, te later being te peers wit wic te former sould be compared. Tis reflects te basic idea of DEA - eac unit is evaluated by comparison wit oter similar ones; if oter DUs wit similar inputs can acieve iger outputs so sould te one being evaluated. ote tat inputs and outputs sould include all relevant variables. For a discussion of te coice of variables see orman and Stoker [28]. As a wole te interpretation of te DEA tecnique is straigtforward and can be put in te following terms: ultiplier problem Evaluate eac DU wit te set of weigts wic maximises its efficiency, provided tat all oter DUs, rated wit tat set of weigts, ave efficiency not greater tan unity. Envelopment problem Find te smallest proportion of Inputs tat would bring te current DUj to te enveloping surface of all DUs. Te model presented above, named CCR after te autors: Carnes Cooper and Rodes [2], assumes constant returns to scale. Tis means tat wen te input of an efficient unit is multiplied by a given factor its output level is also multiplied by te same factor. In many situations tis is not te case. Te scale of operations may ave an impact on te outputs creating "economies" or "diseconomies" of scale. Te BCC model, developed by Banker, Carnes and Cooper [29] can deal wit variable returns to scale. Te example presented above assumes constant returns to scale. Great care sould be taken wen applying DEA, since it is an extreme point tecnique, very sensitive to errors on outliers. Since DEA permits substitution between Outputs or Inputs, tese sould be very carefully cosen and kept to as few as possible since oterwise units deemed as efficient are merely comparing temselves wit temselves. In extreme cases one can be faced wit te situation of all units being efficient, especially wen te ratio: [number of variables]/[number of units] increases. Tis weakness is overcome by te judicious selection of Inputs and Outputs. In addition many applications also limit te range of variation of te weigts in order to overcome tis problem as will be discussed later. V. SUPEREFFICIECY We arrive at te concept of superefficiency by allowing te efficiency of te DU being assessed, to be greater tan unity. Tis is acieved by removing te corresponding constraint from te set of J constraints in Eq (9) Superefficiency only affects units deemed as efficient as te removed constraint is not binding for te inefficient units since teir efficiency is, by definition, less tan unity. Tis extension to DEA was first suggested by Andersen and Petersen [0] and its use is strongly recommended by te autors as a consequence of its simplicity and usefulness. By using superefficiency, it is possible to rank all units, even te efficient ones, tat by standard DEA tecniques would all be rated as equal - teir efficiency aving reaced te top value of 00%. To exemplify te benefits of using superefficiency let us look at two examples of te CCR model represented in Figures 2 and. Wile DU C in Figure 2 is clearly efficient and robust, since it would still continue to be efficient if te output were to decrease until OC/OCr, te same does not appen wit te case represented in Figure were a small variation in OC'/OC'r could bring it under 00% wic would mean tat DU C' would become inefficient. Out2 inp A Cr O Out inp Figure 2- Example of a DU, efficient and robust Tis ratio (superefficiency) is in fact a measure of te robustness of te efficient units tat can be used to rank tese units. Out2' inp' A' C'r O Out' inp' Figure - Example of a DU, efficient but not robust For te example presented in te previous section te superefficiency for te efficient DUs would be as presented in table IV. C' B C B' 426

5 Table IV- Superefficiency scores for te efficient units Unit Superefficienc y Unit % Unit 25.00% Unit % Units and 0 are efficient and robust wile any small increase in te Input or decrease in te Outputs of Unit 4 may make it inefficient. An important additional benefit from tis extension to te DEA model is tat te set of weigts, is uniquely, determined for te efficient units in all practical applications [2]. REFERECES [] Paterson, I., Data envelopment analysis of enterprises in te Russian metallurgy sector, IHS (Institute of Advanced Studies, Vienna), Working Paper. [2] Tanassoulis, E. and P. Dunstan, (994), Guiding scools to improved performance using data envelopment analysis: An illustration wit data from a local education autority, Journal of te Operational Researc Society, 45, (), [] Tanassoulis, E., (99), Comparison of regression analysis and data envelopment analysis as alternative metods for performance assessments, Journal of te Operational Researc Society, 44, (), [4] Boussofiane, A., R. G. Dyson, Tanassoulis, E., (99), Applied data envelopment analysis, European Journal of Operational Researc, 52, (), -5. [5] Carnes, A., Cooper, W. W., Rodes, E., (978), easuring te efficiency of decision making units, European Journal of Operations Researc, 2, (6), [6] Farrell,. J., (957), Te measurement of productive efficiency, Journal of Royal Statistical Society A, 20, [7] Cang, K.-P. and Y.-Y. Gu, (99), Linear production functions and te data envelopment analysis, European Journal of Operational Researc, 52, (2), 25-2 [8] Rodes, E., (978), Data envelopment analysis and related approaces for measuring te efficiency of decision-making units wit an application to program follow troug in U.S. education, unpublised doctoral tesis, Scool of Urban and Public Affairs, Carnegie - ellon University. [9] unamaker, T. R. and A. Y. Lewin, (98), easuring routine nursing service efficiency: A comparison of cost per patient day and data envelopment analysis models, Healt Services Researc, 8, (2 (Part )), [0] Tanassoulis, E., A. Boussofiane, R. G. Dyson,(995), Exploring output quality targets in te provision of prenatal care in England using data envelopment analysis, European Journal of Operational Researc, 80, (), [] Ganley, J. A., Cubbin, J. S., (992), Public Sector Efficiency easurement - Applications of Data Envelopment Analysis, ort - Holland, [2] Lewin, A. Y. and R. C. orey, Cook, T.J., (982), Evaluating te Administrative Efficiency of Courts, Omega, 0,(4), 40-. [] Cu, X. and G. J. Fielding, (992), easuring transit performance using data envelopment analysis, Transportation Researc Part A (Policy and Practice), 26A, (), [4] Tompson, R. G., Darmapala, P. S., Trall, R., (995), Sensitivity analysis of efficiency measures wit applications to Kansas farming and Illinois coal mining, Data Envelopment Analysis: Teory, etodology and Applications,. A. Carnes, W. W. Cooper, A. Lewin and L.. Seiford. Boston, Kluwer Academic Publisers, [5] Scefczyk,., (99), Operational performance of airlines: an extension of traditional measurement paradigms, Strategic anagement Journal, 4, 0-7. [6] Vassiloglou,. and D. Giokas, (990), A study of te relative efficiency of bank brances: an application of data envelopment analysis, Journal of te Operational Researc Society, 4, ( 7), [7] Bessent, A., Bessent, W., Kennington, J., e Regan, B., (982), An application of atematical Programming to assess productivity in te Houston independent scool district, anagement Science, Vol. 28, o. 2, [8] Carnes, A., Cooper, W.W., Rodes, E., (98), Evaluating program and managerial efficiency: an application of Data Envelopment Analysis to Program Follow Troug, anagement Science, 27, (6), [9] Carnes, A., Cooper, W.W., e Rodes, E., (978), easuring te efficiency of decision making units, European Journal of Operational Researc, 2, (6), [20] Castro, R., (99), O método DEA - Aplicação à avaliação da eficiência comparativa das escolas secundárias do distrito do Porto, unpublised sc. tesis, Faculdade de Economia, University of Oporto. [2] Santos, J., (994), Ordenação de unidades eficientes por técnicas de data envelopment analysis, 427

6 unpublised sc. tesis, Instituto Superior Técnico, Tecnical University of Lisbon. [22] Beasley, J.E., (990), Comparing university departments, Omega, 8, 7-8. [2] Carnes, A., Cooper, W.W., Rodes, E., (978), easuring te efficiency of decision making units, European Journal of Operational Researc, 2, (6), [24] Carnes, A., Cooper, W. W., Rodes, E., (979), Sort communication: measuring te efficiency of decision making units, European Journal of Operational Researc,, (4), 9. [25] Carnes, A., Cooper, W.W., (962), Programming wit linear fractional functionals, aval Researc Logistics Quarterly, 9, [26] Gass, S. I. & Harris, C.., Encyclopedia of Operations Researc and anagement Science, 996 [27] Frederick S. Hillier, Gerald J. Lieberman, Introduction to Operations Researc, c Graw Hill, 990. [28] orman,., Stoker,B., Data Envelopment Analysis: te Assessment of Performance, Jon Wiley, Cicester, 99. [29] Banker, R. D., Carnes, A., Cooper, W. W., (984), Some models for estimating tecnical and scale inefficiencies in data envelopment analysis, anagement Science, 0, (9), [0] Andersen, P., Petersen,.C., (99), A procedure for ranking efficient units in data envelopment analysis, anagement Science, 9,(0),

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