A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> The reformulation of the envelopment linear programs in DEA to score a decision making unit (DMU) by excluding the corresponding column from the technology matrix raises several theoretical and computational questions. This idea appears natural and has been proposed before, but only recently has it appeared in published form. Several issues associated with this modification to standard DEA analysis remain to be formally and comprehensively explored. One is the feasibility of the resultant LPs which, as it turns out, is not guaranteed under this modification in some of the DEA models. Therefore, the status of the DMU for which the LP is infeasible needs to be resolved. Other issues include the formal equivalence relation between the modified and standard formulations; the impact on the problem of pervasive degeneracy in the LPs and the interpretation of the optimal solutions. This paper investigates these issues and others so as to provide both a firm theoretical basis for the proposed modification in all four convexified free-disposability DEA models and a thorough analysis of the associated effects. <s> BIB001 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> Abstract This paper addresses ranked voting systems to determine an ordering of candidates in terms of the aggregate vote by rank for each candidate. It is shown that specifying nothing arbitrary, we can obtain a total ordering of candidates by using a DEA/AR (Data Envelopment Analysis/Assurance Region) exclusion model. Explaining the evaluation criterion used to rank candidates, it is concluded that we may consider the system proposed at least as an alternative to determine an ordering of candidates. <s> BIB002 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> AbstractThe paper investigates the infeasibility of super-efficiency data envelopment analysis (DEA) models in which the unit under evaluation is excluded from the reference set. Necessary and sufficient conditions are provided for infeasibility of the super-efficiency DEA measures. By the returns to scale (RTS) classifications obtained from the standard DEA model, we can further locate the position of the unit under evaluation when infeasibility occurs. It is shown that the ranking of the total set of efficient DMUs is impossible because of the infeasibility of super-efficiency DEA models. Also we are able to identify the endpoint positions of the extreme efficient units. The results are useful for sensitivity analysis of efficiency classifications. <s> BIB003 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> The efficiency measures provided by DEA can be used for ranking Decision Making Units (DMUs), however, this ranking procedure does not yield relative rankings for those units with 100% efficiency. Andersen and Petersen have proposed a modified efficiency measure for efficient units which can be used for ranking, but this ranking breaks down in some cases, and can be unstable when one of the DMUs has a relatively small value for some of its inputs. This paper proposes an alternative efficiency measure, based on a different optimization problem that removes the difficulties. <s> BIB004 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <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> BIB005 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> Abstract Using radial super-efficiency data envelopment analysis (DEA) has improved the discriminating performance across efficient decision-making units (DMUs). This paper extends the super-efficiency approach to a non-radial super-efficiency DEA (NRSE-DEA) index. NRSE-DEA is shown to be invariant to units of input (output) measurement. NRSE-DEA is illustrated here via an application to NATO burden-sharing assessment in which the DMUs are the member nations of NATO. The NRSE-DEA provides additional insights into the ranking of efficient countries, suggesting which are absorbing a particularly large share of NATO responsibilities. The NRSE-DEA generates a smaller set of efficient DMUs. This, in turn, provides more discriminatory power, a more accurate measure of super-efficiency, a more meaningful ranking of the efficient burden sharing countries, and a more reliable assessment of contributions by NATO members, amongst other policy issues. <s> BIB006 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> Abstract This paper addresses “super-efficiency” issue based on improved outputs. Although the method is similar to an output-oriented version of Andersen and Petersen’s model [P. Andersen, N.C. Petersen, A procedure for ranking efficient units in data envelopment analysis, Management Science 39 (10) (1993) 1261–1264] (called AP model) under variable returns to scale, it is not limited to use the input amounts of evaluating DMU, therefore, it differs from AP approach. We can obtain a total ordering of DMUs by using this method. The proposed method will be particularly useful when getting an input combination for improving outputs is suitable for decision makers. <s> BIB007 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> Conventional super-efficiency data envelopment analysis (DEA) models require the exact information of inputs or outputs. However, in many real world applications this simple assumption does not hold. Stochastic super-efficiency is one of recent methods which could handle uncertainty in data. Stochastic super-efficiency DEA models are normally formulated based on chance constraint programming. The method is used to estimate the efficiency of various decision making units (DMUs). In stochastic chance constraint super-efficiency DEA, the distinction of probability distribution function for input/output data is difficult and also, in several cases, there is not enough data for estimating of distribution function. We present a new method which incorporates the robust counterpart of super-efficiency DEA. The perturbation and uncertainty in data is assumed as ellipsoidal set and the robust super-efficiency DEA model is extended. The implementation of the proposed method of this paper is applied for ranking different gas companies in Iran. <s> BIB008 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> One of the most important approaches for ranking decision-making units in data envelopment analysis is the cross-efficiency method. The main idea of cross-efficiency is to use data envelopment analysis in peer evaluation, instead of only a self evaluation. However, in the cross-efficiency method, optimal weights corresponding to evaluation of decision-making units may not be unique. In this research with modifying the cross-efficiency method we are going to overcome this problem. Then with regard to the changes and using the TOPSIS method we present a new super-efficient method to rank all decision-making units. Furthermore, we extend the proposed method to the case that data are intervals. <s> BIB009 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> Abstract One of the most important topics in management science is determining the efficiency of Decision Making Units (DMUs). The Data Envelopment Analysis (DEA) technique is employed for this purpose. In many DEA models, the best performance of a DMU is indicated by an efficiency score of one. There is often more than one DMU with this efficiency score. To rank and compare efficient units, many methods have been introduced under the name of super-efficiency methods. Among these methods, one can mention Andersen and Petersen’s (1993) [1] super-efficiency model, and the slack-based measure introduced by Tone (2002) [4] . Each of the methods proposed for ranking efficient DMUs has its own advantages and shortcomings. In this paper, we present a super-efficiency method by which units that are more effective and useful in society have better ranks. In fact, in order to determine super-efficiency by this method, the effectiveness of each unit in society is considered rather than the cross-comparison of the units. To do so, we divide the inputs and outputs into two groups, desirable and undesirable, at the discretion of the manager, and assign weights to each input and output. Then we determine the rank of each DMU according to the weights and the desirability of inputs and outputs. <s> BIB010 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <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> BIB011 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <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> BIB012 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> Data envelopment analysis (DEA) evaluates the performance of decision making units (DMUs). When DEA models are used to calculate efficiency of DMUs, a number of them may have the equal efficiency 1. In order to choose a winner among DEA efficient candidates, some methods have been proposed. But most of these methods are not able to rank non-extreme efficient DMUs. Since, the researches performed about ranking of non-extreme efficient units are very limited, incomplete and with some difficulties, we are going to develop a new method to rank these DMUs in this paper. Therefore, we suppose that DMUo is a non-extreme efficient under evaluating DMU. In continue, by using “Representation Theorem”, DMUo can be represented as a convex combination of extreme efficient DMUs. So, we expect the performance of DMUo be similar to the performance of convex combination of these extreme efficient DMUs. Consequently, the ranking score of DMUo is calculated as a convex combination of ranking scores of these extreme efficient DMUs. So, the rank of this unit will be determined. <s> BIB013 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Super-Efficiency Ranking Techniques <s> The problem of infeasibility arises in conventional radial super-efficiency data envelopment analysis (DEA) models under variable returns to scale (VRS). To tackle this issue, a Nerlove–Luenberger (N–L) measure of super-efficiency is developed based on a directional distance function. Although this N–L super-efficiency model does not suffer infeasibility problem as in the conventional radial super-efficiency DEA models, it can produce an infeasible solution in two special situations. The current paper proposes to modify the directional distance function by selecting proper feasible reference bundles so that the resulting N–L measure of super-efficiency is always feasible. As a result, our modified VRS super-efficiency model successfully addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. Numerical examples are used to demonstrate our approach and compare results obtained from various super-efficiency measures. <s> BIB014
Super efficiency models introduced in DEA technique are based upon the idea of leave one out and assessing this unit trough the remanding units. (27) Although this idea is useful for further discriminating efficient units, it has been shown in the literature that it may be infeasible and nonstable. Thrall  mentioned the infeasibility of super-efficiency CCR model. Also a condition under which infeasibility occurred in super-efficiency DEA models is mentioned by Zhu  , Seiford and Zhu BIB003 , and Dulá and Hickman BIB001 . Hashimoto BIB002 provided a model based on the idea of one leave out and assurance region for ranking units. Mehrabian et al. BIB004 presented a complete ranking for efficiency units in DEA context. As the authors mentioned this model does not have difficulties of A.P. model (28) With this method it is not possible to rank nonextreme efficient units. Tone BIB005 presented super efficiency of SBM model. This model has the advantages of nonradial models, and it is always feasible and stable Journal of Applied Mathematics Jahanshahloo et al.  added some ratio constraints to the multiplier form of A.P. model and introduced a new method for ranking DMUs DMU is efficient if the optimal objective function of the previous model is greater than or equal one. Jahanshahloo et al.  presented a method for ranking efficient units on basis of the idea of one leave out and 1 norm. As the authors proved this model is always feasible and stable where = ∑ =1 − ∑ =1 . In their paper Chen and Sherman BIB006 presented a nonradial super-efficiency method and discussed the advantage of it. They verified that this model is invariant to units of input/output measurement. Let = /{DMU }. Step 1. Solve the following model to find the extreme efficient units in : Consider as the set of efficient units of . Step 2. Solve the following model: Consider * as the optimal solution of the previous model and solve the following model for ∈ {1, . . . , }: According to the obtained optimal solution of the previous model for = 1, . . . , let  , − * (0) = − * , and Step 3. Solve the following model: Now according to the optimal solution of the previous model, solve the following model for each ∈ ( ): Step 4. Journal of Applied Mathematics 9 let = + 1 and go to Step 3. Now define as the average RNSE-DEA index In this case extreme efficient units cannot be ranked. In a paper Khodabakhshi BIB007 addresses super efficiency on improved outputs. He mentioned that as A.P. model may be infeasible under variable returns to scale technology, using the presented model gives a complete ranking when getting an input combination for improving outputs is suitable. Sadjadi et al. BIB008 presented a robust super-efficiency DEA for ranking efficient units. They noted that as in most of the times exact data do not exist and the stochastic super-efficiency model presented in their paper incorporates the robust counterpart of super-efficiency DEA where , are input-output data. This model does not rank nonextreme efficient units. It may be unstable and infeasible. Gholam Abri et al. BIB013 proposed a model for ranking efficient units. They used representation theory and represented the DMU under assessment as a convex combination of extreme efficient units. As the authors noted it is expected that the performance of DMU is the same as the performance of convex combination of extreme efficient units. Thus, it is possible to represent DMU as follows: As regards representation theorem, this system has + − 1 constraints and variables, ( 1 , . . . , ). If this system has a unique solution we will have * = ∑ =1 = 1, * , otherwise two models should be considered Consider 1 and 2 as the optimal solution of the abovementioned models, respectively. If 1 = 2 , this is the same as what has been mentioned previously. If 1 < 2 , then mentioned models provide an interval which helps rank units from the worst to the best. Sometimes it will obtain the ranking score with a bounded interval [ 1 , 2 ]. Jahanshahloo et al. BIB009 presented models for ranking efficient units. The presented models are somehow a modification of cross-efficiency model that overcomes the difficulty of alternative optimal weights. The authors in their paper with regard to the changes and also utilizing TOPSIS technique presented a new super-efficient method for ranking units. The presented model is as follows: where is the efficiency score of DMU using corresponding weights. Also is the efficiency of DMU using optimal weights of DMU . Noura et al. BIB010 provided a method for ranking efficient units based on this idea that more effective and useful units in society should have better rank. Step 1. For each efficient unit choose the lower and upper limit for each inputs and outputs. Let be the set of efficient units * = Max ∈ , * = Min ∈ , = 1, . . . , , * = Max ∈ , * = Min ∈ , = 1, . . . , . Step 2. In accordance with the previous step, here, the utility inputs and outputs are as follows: Step 3. Consider dimensionless ( , ) introduced as follows for each efficient unit belonging to : where is representative of a small and nonzero number used for not dividing by zero. Now consider − as follows which shows that more successful DMU will be if the larger value of the is where Ashrafi et al. BIB011 introduced an enhanced Russell measure of super efficiency for ranking efficient units in DEA. The linear counterpart of the proposed model is as follows:  Rezai Balf et al. BIB012 provide a model for ranking units based on Tchebycheff norm. As proved that this model is always feasible and stable, it seems to have superiority over other models  Chen et al. BIB014 for overcoming the infeasibility problem that occurred in variable returns to scale super-efficiency DEA model according to a directional distance function developed Nerlove-Luenberger (N-L) measure of super-efficiency.