A Review of Ranking Models in Data Envelopment Analysis <s> Introduction <s> Abstract A nonlinear (nonconvex) programming model provides a new definition of efficiency for use in evaluating activities of not-for-profit entities participating in public programs. A scalar measure of the efficiency of each participating unit is thereby provided, along with methods for objectively determining weights by reference to the observational data for the multiple outputs and multiple inputs that characterize such programs. Equivalences are established to ordinary linear programming models for effecting computations. The duals to these linear programming models provide a new way for estimating extremal relations from observational data. Connections between engineering and economic approaches to efficiency are delineated along with new interpretations and ways of using them in evaluating and controlling managerial behavior in public programs. <s> BIB001 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Introduction <s> Abstract This paper develops theory and algorithms for a “multiplicative” Data Envelopment Analysis (DEA) model employing virtual outputs and inputs as does the CCR ratio method for efficiency analysis. The frontier production function results here are of piecewise log-linear rather than piecewise linear form. <s> BIB002 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Introduction <s> In management contexts, mathematical programming is usually used to evaluate a collection of possible alternative courses of action en route to selecting one which is best. In this capacity, mathematical programming serves as a planning aid to management. Data Envelopment Analysis reverses this role and employs mathematical programming to obtain ex post facto evaluations of the relative efficiency of management accomplishments, however they may have been planned or executed. Mathematical programming is thereby extended for use as a tool for control and evaluation of past accomplishments as well as a tool to aid in planning future activities. The CCR ratio form introduced by Charnes, Cooper and Rhodes, as part of their Data Envelopment Analysis approach, comprehends both technical and scale inefficiencies via the optimal value of the ratio form, as obtained directly from the data without requiring a priori specification of weights and/or explicit delineation of assumed functional forms of relations between inputs and outputs. A separation into technical and scale efficiencies is accomplished by the methods developed in this paper without altering the latter conditions for use of DEA directly on observational data. Technical inefficiencies are identified with failures to achieve best possible output levels and/or usage of excessive amounts of inputs. Methods for identifying and correcting the magnitudes of these inefficiencies, as supplied in prior work, are illustrated. In the present paper, a new separate variable is introduced which makes it possible to determine whether operations were conducted in regions of increasing, constant or decreasing returns to scale in multiple input and multiple output situations. The results are discussed and related not only to classical single output economics but also to more modern versions of economics which are identified with "contestable market theories." <s> BIB003 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Introduction <s> Abstract : There are four basic questions related to efficiency and capability which are of particular interest to officials in the military services who are interested in better ways of evaluating military capability and efficiency: (1) What level of military capability can the services achieve with available resources? (2) What capability is required, and where are the shortfalls? (3) What resource acquisitions or redistributions are needed to achieve maximum improvement in efficiency and effectiveness? and (4) How can management systems be changed to improve the identification and correction of factors which limit the readiness and efficiency of our military operations? The last question, which differs in its emphasis from the other three, provides an opening to the topics that will be addressed in this report. In particular, reported are results from a study of DEA (Data Envelopment Analysis) as a method for evaluating the efficiency of Air Force Wings--or, more precisely, their maintenance operations--as elements in Numbered Units in the U.S. Air Force. <s> BIB004 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Introduction <s> The construction and analysis of Pareto-efficient frontier production functions by a new Data Envelopment Analysis method is presented in the context of new theoretical characterizations of the inherent structure and capabilities of such empirical production functions. Contrasts and connections with other developments, including solutions of some remaining problems, are made re aspects such as informatics, economies of scale, isotonicity and non-concavity, discretionary and non-discretionary inputs, piecewise linearity, partial derivatives and Cobb-Douglas properties of the functions. Non-Archimedean constructs are nor required. <s> BIB005 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Introduction <s> Data Envelopment Analysis DEA evaluates the relative efficiency of decision-making units DMUs but does not allow for a ranking of the efficient units themselves. A modified version of DEA based upon comparison of efficient DMUs relative to a reference technology spanned by all other units is developed. The procedure provides a framework for ranking efficient units and facilitates comparison with rankings based on parametric methods. <s> BIB006 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Introduction <s> Abstract This paper deals with the evaluation of decision making units which have multiple inputs and outputs. A new method (CCA/DEA) is developed where the Canonical Correlation Analysis (CCA) is utilized to provide a full rank scaling for all the units rather than a categorical classification (for efficient and inefficient units) as done by the Data Envelopment Analysis (DEA). The CCA/DEA approach is an attempt to bridge the gap between the frontier approach of DEA and the average tendencies of statistics (econometrics). Nonparametric statistical tests are employed to validate the consistency between the classification from the DEA and the postclassification that was generated by the CCA/DEA. <s> BIB007 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Introduction <s> Abstract US experience shows that deregulation of the airline industry leads to the formation of hub-and-spoke (HS) airline networks. Viewing potential HS networks as decision-making units, we use data envelopment analysis (DEA) to select the most efficient networks configurations from the many that are possible in the deregulated European Union airline market. To overcome the difficulties that DEA encounters when there is an excessive number of inputs or outputs, we employ principal component analysis (PCA) to aggregate certain, clustered data, whilst ensuring very similar results to those achieved under the original DEA model. The DEA–PCA formulation is then illustrated with real-world data gathered from the West European air transportation industry. <s> BIB008
Data envelopment analysis as a mathematical tool was initiated by Farrell  and Charnes et al. BIB001 . They formulated a linear programming problem with which it is possible to evaluate decision-making units (DMUs) with multiple inputs and outputs. Note that in this technique it is not necessary to know the production function. In this technique, an LP problem is solved for each DMU, and the relative efficiency of each unit obtained as a linear combination of corresponding optimal weights. In this problem, weights are free to get their value to show the under evaluation unit in optimistic viewpoint. Those units with optimal objective function equal to one are called "best practice. " These units are located onto the efficient frontier, and those far away from this frontier are called inefficient. As proved in DEA literature, at least one of the units is located onto this frontier. Note that units located unto this frontier can be considered as benchmarks for inefficient units. Based on what Charnes et al. BIB001 provided, many extensions to DEA Models are presented in the literature. As the example of the most important one Banker et al. BIB003 can be mentioned. Also, multiplicative and additive models developed in the literature by Charnes et al. BIB002 BIB004 BIB005 . In that time Thrall  provided a complete comparison of all classic DEA models. One of the important issues discussed in DEA literature is ranking efficient units since the efficient units obtained in the efficiency score of one cannot be compared with each other on the basis of this criterion any more. Therefore, it seems necessary to provide models for further discrimination among these units. Many papers are presented in the literature review for ranking the efficient units. Note that Adler and Golany BIB008 used principle component analysis for improving the discrimination of DEA. But this attempt was not sufficient be discussed. In this method first suggested by Sexton et al.  , the DMUs are self-and peer-assessed. In Section 4, some of the ranking methods based on optimal weights obtained from DEA models, common set of weights are briefly reviewed. Super-efficiency methods first introduced by Andersen and Petersen BIB006 will be reviewed in Section 5. The basic idea is based on the idea of leaving out one unit and assessing by the remaining units. Section 6 discusses the evaluation of DMUs through benchmarking, an approach originating in Torgersen et al.  . Section 7 will review the papers which use the statistical tools for ranking units first suggested by Friedman and Sinuany-Stern BIB007 such as canonical correlation analysis and discriminant analysis. Section 8 discusses the ranking of units based on multicriteria decision-making (MCDM) methodologies and DEA. Section 9 discusses some different ranking methods existing in the DEA literature. Section 10 presents the results of the various methodologies applied to an example.