A Review of Ranking Models in Data Envelopment Analysis <s> Application <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> BIB001 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <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> BIB002 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> This article describes a new DEA (data envelopment analysis) ranking approach that combines efficiency analysis (by DEA) with index measurement (by DEA sensitivity analysis). The sensitivity analysis, incorporated into the index measurement, omits an efficient DMU (decision making unit) to obtain the index numbers of all the DMUs. The proposed DEA ranking approach is theoretically linked to a non-parametric rank sum test. Using the rank sum test, this article statistically examines whether two groups of DMUs have different distribution functions of efficiency. As an illustrative case study, this article applies the proposed approach to Japanese Agriculture Cooperatives. <s> BIB003 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Abstract In most models of Data Envelopment Analysis (DEA), the best performers have the full efficient status denoted by unity (or 100%), and, from experience, we know that usually plural Decision Making Units (DMUs) have this “efficient status”. To discriminate between these efficient DMUs is an interesting subject. This paper addresses this “super-efficiency” issue by using the slacks-based measure (SBM) of efficiency, which the author proposed in his previous paper [European Journal of Operational Research 130 (2001) 498]. The method differs from the traditional one based on the radial measure, e.g. Andersen and Petersen model, in that the former deals directly with slacks in inputs/outputs, while the latter does not take account of the existence of slacks. We will demonstrate the rationality of our approach by comparing it with the radial measure of super-efficiency. The proposed method will be particularly useful when the number of DMUs are small compared with the number of criteria employed for evaluation. <s> BIB004 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> In this paper a method, based on using gradient line, for ranking DMUs is proposed. The advantage of this method is its stability and robustness, where data have special structure. Some numerical examples are presented to show the results. <s> BIB005 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Efficiency evaluation in Data Envelopment Analysis (DEA) depends on different factors. The most important factors arc the values of input and output. In this paper, we present an alternative proof that, if one component of output or input vectors of a DMU dominates the corresponding component of other DMUs whatever the value of other components of this DMU may be, then that DMU is efficient in some of the DEA models. An important outcome of such an analysis is a set of virtual multipliers or weights accorded to each factor taken into account. These sets of weights are, typically, different for each of the participating DMUs. In this paper, by means of solving only one problem, we can determine common set of weights (CSW) for all DMUs and their efficiencies. Finally, a method for ranking DMUs, is presented. In this method by solving only two problems, efficient DMUs are ranked. <s> BIB006 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> For ranking efficient DMUs some methods have been developed. These methods are not able to rank non-extreme efficient DMUs. In this paper, using Monte Carlo method, a method has been developed which is able to rank all (extreme and non-extreme) efficient DMUs. <s> BIB007 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Abstract Efficiency Data Envelopment Analysis (DEA) models assessing decision making units (DMUs) are unable to discriminate between efficient DMUs. From experience, we know that usually plural DMUs are efficient. To discriminate between these efficient DMUs is an interesting subject. To this aim, super-efficiency models are proposed. In this paper we study a super-efficiency model, namely MAJ. Each time an efficient DMU is excluded from the set of the observed DMUs, a new production possibility set (PPS) is obtained. In this model, ranking is done based on the position of each excluded efficient DMU in relation to its corresponding new PPS. If the efficient DMUs and the new PPSs remain unchanged, it is expected that the ranking, also, remain unchanged. But, in this paper, we show that the technique used for rendering this model unit-invariant causes the ranking to change when some inputs of some inefficient DMUs change, without causing any change in the new PPS. In this paper, we modify this model so that this problem will not occur. <s> BIB008 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Conventional data envelopment analysis (DEA) assists decision makers in distinguishing between efficient and inefficient decision-making units (DMUs) in a homogeneous group. However, DEA does not provide more information about the efficient DMUs. This research proposes a methodology to determine one common set of weights for the performance indices of only DEA efficient DMUs. Then, these DMUs are ranked according to the efficiency score weighted by the common set of weights. For the decision maker, this ranking is based on the optimization of the group's efficiency. <s> BIB009 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Making optimal use of available resources has always been of interest to humankind, and different approaches have been used in an attempt to make maximum use of existing resources. Limitations of capital, manpower, energy, etc., have led managers to seek ways for optimally using such resources. In fact, being informed of the performance of the units under the supervision of a manager is the most important task with regard to making sensible decisions for managing them. Data envelopment analysis (DEA) suggests an appropriate method for evaluating the efficiency of homogeneous units with multiple inputs and multiple outputs. DEA models classify decision making units (DMUs) into efficient and inefficient ones. However, in most cases, managers and researchers are interested in ranking the units and selecting the best DMU. Various scientific models have been proposed by researchers for ranking DMUs. Each of these models has some weakness(es), which makes it difficult to select the appropriate ranking model. This paper presents a method for ranking efficient DMUs by the voting analytic hierarchy process (VAHP). The paper reviews some ranking models in DEA and discusses their strengths and weaknesses. Then, we provide the method for ranking efficient DMUs by VAHP. Finally we give an example to illustrate our approach and then the new method is employed to rank efficient units in a real world problem. <s> BIB010 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Abstract Cross efficiency evaluation has long been proposed as an alternative method for ranking the decision making units (DMUs) in data envelopment analysis (DEA). This study proposes goal programming models that could be used in the second stage of the cross evaluation. Proposed goal programming models have different efficiency concepts as classical DEA, minmax and minsum efficiency criteria. Numerical examples are provided to illustrate the applications of the proposed goal programming cross efficiency models. <s> BIB011 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Cross-efficiency evaluation is an effective approach to ranking decision making units (DMUs) that utilize multiple inputs to produce multiple outputs. Its models can usually be developed in a way that is either aggressive or benevolent to other DMUs, depending upon the decision maker (DM)'s subjective preference to the two extreme cases. This paper proposes several new data envelopment analysis (DEA) models for cross-efficiency evaluation by introducing a virtual ideal DMU (IDMU) and a virtual anti-ideal DMU (ADMU). The new DEA models determine input and output weights from the point of view of distance from IDMU or ADMU without the need to be aggressive or benevolent to any DMUs. As a result, the cross-efficiencies measured by these new DEA models are neutral and more logical. Numerical examples are provided to illustrate the potential applications of these new DEA models and their effectiveness in ranking DMUs. <s> BIB012 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Problem statement: Conventional Data Envelopment Analysis (DEA) helps decision makers to discriminate between efficient and inefficient Decision Making Units (DMUs). However, DEA does not provide more information about the efficient DMUs. Super-efficiency DEA model can be used in ranking the performance of efficient DMUs. Because of the possible infeasibility of radial super-efficiency DEA model, the ranking has been limited to the model under the assumption of Constant Returns to Scale (CRS). Approach: This study proposes a super-efficiency model based on the Enhanced Russell Measure (ERM) of efficiency. This is a non-radial measure and appropriate for ranking the efficient DMUs when inputs and outputs may change non-proportionally. Results: Theoretical results show that the new super-efficiency model is always feasible under the assumption of non-CRS. Also, numerical examples from the literature are provided to test the new super-efficiency approach. Conclusion: This study provides a non-radial measure of super-efficiency based on the ERM model to discriminate among the efficient DMUs resulting different efficiency scores greater than one. Unlike the traditional radial super-efficiency models, the proposed method is always feasible. <s> BIB013 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Super-efficiency data envelopment analysis (SE-DEA) models have been developed and applied in many situations. However, under the condition of variable returns to scale (VRS), infeasibility of the SE-DEA model may occur and restrict its application. A modified SE-DEA measure based on simultaneous input-output projection is proposed as a way to systematically characterize the super-efficiency in both inputs and outputs. The modified measure overcomes the infeasibility problem while providing ease of computation and interpretation. The practicability of the proposed measure in real applications and its comparison to other super-efficiency measures are illustrated empirically using an example. <s> BIB014 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Abstract One problem that has been discussed frequently in data envelopment analysis (DEA) literature has been lack of discrimination in DEA applications, in particular when there are insufficient DMUs or the number of inputs and outputs is too high relative to the number of units. This is an additional reason for the growing interest in complete ranking techniques. In this paper a method for ranking extreme efficient decision making units (DMUs) is proposed. The method uses L ∞ (or Tchebycheff) Norm, and it seems to have some superiority over other existing methods, because this method is able to remove the existing difficulties in some methods, such as Andersen and Petersen [2] (AP) that it is sometimes infeasible. The suggested model is always feasible. <s> BIB015 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Abstract Ranking DMUs is an important issue in DEA studies. Traditional DEA model usually leads to the failure to ranking all DMUs when DMUs get the same efficiency score, such as efficient DMUs or inefficient DMUs with the same efficiency score. For the sake of comparing and improving capabilities of DMUs, many researchers have sought to fully rank both efficient, as well as inefficient DMUs. In this paper, an extended DEA model considering undesirable outputs using restrictions is presented to realize a unique ranking of DMUs through the new “Maximal Balance Index” based on the optimal shadow prices. Meanwhile, a numerical example and an empirical example are provided to illustrate the difference from previous work and the application of the extended model. In the empirical study, the complete ranking of DMUs incorporating undesirable outputs is proved to be useful and unique. <s> BIB016 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Application <s> Abstract Data Envelopment Analysis (DEA) is a mathematical model that evaluates the relative efficiency of Decision Making Units (DMUs) with multiple input and output. In some applications of DEA, ranking of the DMUs are important. For this purpose, a number of approaches have been introduced. Among them is the cross-efficiency method. The method utilizes the result of the cross-efficiency matrix and averages the cross-efficiency scores of each DMU. Ranking is then performed based on the average efficiency scores. In this paper, we proposed a new way of handling the information from the cross-efficiency matrix. Based on the notion that the ranking order is more important than individual efficiency score, the cross-efficiency matrix is converted to a cross-ranking matrix . A cross-ranking matrix is basically a cross-efficiency matrix with the efficiency score of each element being replaced with the ranking order of that efficiency score with respect to the other efficiency scores in a column. By so doing, each DMU assume the role of a decision maker and how they voted or ranked the other DMUs are reflected in their respective column of the cross-ranking matrix. These votes are then aggregated using a preference aggregation method to determine the overall ranking of the DMUs. Comparison with an existing cross-efficiency method indicates a relatively better result through usage of the proposed method. <s> BIB017
In this section, some of the reviewed models are applied on the example used in Soltanifar and Hosseinzadeh Lotfi BIB010 . Consider twenty commercial banks of Iran with input-output data tabulated in Table 1 and summarized as follows. Also the results of CCR model are listed in this table. As it can be seen seven units are efficient, DMUS 1, BIB001  BIB011 BIB012 BIB016 BIB017 . Inputs are staff, computer terminal, and space. Outputs are deposits, loans granted, and charge. Consider some of the important ranking methods in literature as follows: R.M1: A.P. model BIB002 is based upon the idea of leave unit evaluation out and measuring the distance of the unit under evaluation from the production possibility set constructed by the remaining DMUs. R.M1 -- R.M6: Slack-based model BIB004 is based upon the input and output variables at the same time. R.M7: SA DEA model BIB003 overcomes the problem of infeasibility which existed in A.P. model. R.M8: Cross-efficiency  is provided based on using weights of each unit under evaluation in optimality for other units. R.M9: A model based on finding common set of weights BIB009 which determine the common set of weights for DMUs and ranked DMUs based this idea. R.M10: A model based on finding common set of weights BIB006 BIB007 for ranking efficient units. R.M11: L1-norm model   BIB005 , the idea is based upon the leave-one-out efficient unit and l1-norm which is always feasible and stable. R.M12: ∞ -norm model Rezai Balf et al. BIB015 provided a method with more ability over other existing methods, based on Tchebycheff norm. R.M13: An enhanced Russel measure of superefficiency model for ranking units BIB013 . R.M14: A ranking model which considers the distance of unit from the full inefficient frontier BIB008 R.M15: A modified super-efficiency model BIB014 which overcomes the infeasibility that may happen in problem. This model is based on simultaneous projection of input output. In accordance with properties of different ranking models in order to rank efficient units, consider In Tables 3 and 4 