A Review of Ranking Models in Data Envelopment Analysis <s> Data Envelopment Analysis <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> Data Envelopment Analysis <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> BIB002 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Data Envelopment Analysis <s> In 1984, the Texas Legislature funded a four-university, interdisciplinary effort to identify feasible sites for location of a very high-energy physics lab in Texas and to evaluate the comparative advantages of one site versus another. Six feasible sites were identified and a comparative site analysis was made by applying Data Envelopment Analysis (DEA), incorporating project cost, user time delay, and environmental impact data. In addition, for the efficient sites, the price weights for user time delay and environmental impact, given normalization on project cost, were analyzed and arguments were developed to bracket these pairs of price weights into an “assurance region” for the preferred site(s). The South Dallas site was found to be preferred for a wide range of conditions, while the North Houston site was sensitive to the method of indexing the impact of the environment. The method appears to be applicable to a wide range of siting problems faced by not only government, but also industry. <s> BIB003 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Data Envelopment Analysis <s> Abstract As long recognized, the problem of efficiency involves both technical and economic facets. Determination of the technically efficient firms provides the base for economic analysis. Values in terms of prices or costs must be introduced into the problem to work towards finding firms which might be regarded as overall efficient. That problem came to the forefront in a 1984 study to find the best site for location of a Superconducting Super Collider (SSC) in the state of Texas. Application of a modern value-free frontier method called Data Envelopment Analysis (DEA) to the data, which was primarily engineering and geological in character, showed five of the six feasible sites were technically efficient. However, additional socioeconomic/environmental data provided ‘price-cost’ inequality bounds for the mathematical multipliers in the DEA problem. Including those bounds in the analysis, which was called an Assurance Region (AR), reduced the number of efficient sites from five to one. In 1988, the U.S. Department of Energy in national competitionactually selected the site identified by the bounding method for location of the SSC. In this paper, the AR concept is defined for efficiency analysis of the linear production possibility set. As applied here to 83 farms, we use only the special case of AR consisting of separate linear homogeneous restrictions on the input and output multipliers. When applied to the technically efficient farms, the AR principles reduced the number of candidates for overall efficiency from 23 to 8 in one case (Ratio Model) and from 44 to 13 in another case (Convex Model). <s> BIB004 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Data Envelopment Analysis <s> The productive efficiencies of 45 randomly sampled oil/gas independent firms were analyzed year-by-year by application of data envelopment analysis (DEA) methods for the 7 years 1980–1986. This approach differs fundamentally from a “window” analysis, where structural differences in the efficiency distributions over time may not be easily discerned. Both DEA Ratio and Convex models were applied; also, bounds were placed on the modeled prices (multipliers) by the use of assurance region (AR) principles. The distributions of DEA-efficiency measures were found to have significantly different characteristics in the 1980–1982 period than in the 1983–1986 period. Constant returns-to-scale prevailed in each of the 7 years. The AR-efficiencies, in the presence of multiplier bounds, refined appreciably the candidate set of firms for overall efficiency. The AR-efficiency distributions exhibited significantly different levels and spreads in the 1980–1982 period than in the 1983–1986 period. Very few (5 at most) of the candidates for overall efficiency were viable economic firms in 1985 and 1986. Strikingly different energy policy conjectures follow from this micro analysis than from the U.S. Department of Energy's macro analysis in its Energy Security study. <s> BIB005
Data envelopment analysis is a mathematical programming technique for performance evaluation of a set of decisionmaking units. Let a set consists of homogeneous decision-making units to be evaluated. Assume that each of these units uses inputs ( = 1, . . . , ) to produce outputs ( = 1, . . . , ). Moreover, ∈ and ∈ are considered to be nonnegative vectors. We define the set of production possibility as = {( , ) | can produce }. When variable returns to scale form of technology is assumed we have = BCC  and when constant returns to scale form of technology is assumed we have =  Journal of Applied Mathematics 3 The two-phase enveloping problem with constant returns to scale form of technology, first provided by Charnes et al. BIB001 , is as follows:  The two-phase enveloping problem with variable returns to scale form of technology, first provided by Banker et al. BIB002 , is as follows:  As regards the above-mentioned problems and due to corresponding feasible region it is evident that * CCR ≤ * BCC . According to the definition of CCR and BCC , an envelop constructed through units called best practice or efficient. Considering mentioned problems if a DMU is not CCR (BCC) efficient, it is possible to project this DMU onto the CCR (BCC) efficiency frontier considering the following formulas:̂= * The dual model corresponding to the following model is as follows: For overcoming the problem of zero weights, and variability of weights use of assurance region is suggested by Thompson et al. BIB003 BIB004 BIB005 . As mentioned in the literature usually there exist more than one efficient units, and these units cannot be further compared to each other on basis of efficiency scores. Thus, it felt necessary to provide new models for ranking these units. There exist a variety of ranking models in context of data envelopment analysis. In the remaining of this paper we review some of these models.