A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <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> BIB001 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> Abstract In recent decades, rankings of states on virtually every basis have proliferated. Prominent among these rankings have been those that attempt to measure the economic/business environment of states. These rankings tend to be an indexing based on many types of sometimes unrelated data rendering the rankings of questionable meaning. This paper uses Data Envelopment Analysis (DEA) to provide rankings of state economic performance. By using DEA, we develop rankings that are systematic and have a meaningful interpretation. <s> BIB002 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> In non-parametric methods, many units are evaluated as efficient. Several methods for prioritizing efficient units are reported in the literature. Andersen et al. and thereafter Mehrabian et al. proposed two methods for ranking efficient units, but both methods break down in some cases. This paper describes a new DEA ranking approach that uses L"2-norm. In proposed DEA ranking approach, the difficulties of the existing methods have been removed. <s> BIB003 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> Since in evaluating by traditional data envelopment analysis (DEA) models many decision making units (DMUs) are classified as efficient, a large number of methods for fully ranking both efficient and inefficient DMUs have been proposed. In this paper a ranking method is suggested which basically differs from previous methods but its models are similar to traditional DEA models such as BCC, additive model, etc. In this ranking method, DMUs are compared against an full-inefficient frontier, which will be defined in this paper. Based on this point of view many models can be designed, and we mention a radial and a slacks-based one out of them. This method can be used to rank all DMUs to get analytic information about the system, and also to rank only efficient DMUs to discriminate between them. <s> BIB004 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> Abstract Data Envelopment Analyses (DEA) is a linear programming based method which evaluates relative efficiency of Decision Making Units (DMUs). It can include multiple outputs and inputs without a priori weights and without requiring explicit specification of functional forms between inputs and outputs. It computes a scalar measure of efficiency and determines efficient levels of inputs and outputs for each DMU under evaluation which has a range of zero to “1”; hence, it has the ability to rank DMUs, unless when some DMUs are the same in efficiency score, such as efficient DMUs or inefficient DMUs with the same efficiency score. In many cases, it is necessary to give a full ranking of the DMUs. Hence this paper introduces a new method for complete ranking of decision making units, which it does not need any changes in the models. Also, for presentation of the ability of this method, it is employed to rank the bank branch of a commercial bank branches in an empirical example. <s> BIB005 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> This research proposes a new ranking system for extreme efficient DMUs (Decision Making Units) based upon the omission of these efficient DMUs from reference set of the inefficient DMUs. We state and prove some facts related to our model. A numerical example where the proposed method is compared with traditional ranking approaches is shown. <s> BIB006 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> Data envelopment analysis (DEA) is a body of research methodologies to evaluate overall efficiencies and identify the sources and estimate the amounts of inefficiencies in inputs and outputs. In DEA, the best performers are called DEA efficient and the efficiency score of a DEA efficient unit is denoted by an unity. In the last decade, ranking DEA efficient units has become the interests of many DEA researchers and a variety of models (called super-efficiency models) were developed to rank DEA efficient units. While the models developed in the past are interesting and meaningful, they have the disadvantages of being infeasible or instable occasionally. In this research, we develop a super-efficiency model to overcome some deficiencies in the earlier models. Both theoretical results and numerical examples are provided. <s> BIB007 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> One important issue of aggregating preference rankings is to determine the weights of different ranking places. This paper proposes the use of ordered weighted averaging (OWA) operator weights to aggregate preference rankings, which allows the weights associated with different ranking places to be determined in terms of a decision maker (DM)'s optimism level characterized by an orness degree. By adjusting the DM's optimism level, ties can be avoided and winner can be selected. Two numerical examples are examined using OWA operator weights to show their applications, simplicity and flexibility in aggregating preference rankings. <s> BIB008 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> In data mining applications, it is important to develop evaluation methods for selecting quality and profitable rules. This paper utilizes a non-parametric approach, Data Envelopment Analysis (DEA), to estimate and rank the efficiency of association rules with multiple criteria. The interestingness of association rules is conventionally measured based on support and confidence. For specific applications, domain knowledge can be further designed as measures to evaluate the discovered rules. For example, in market basket analysis, the product value and cross-selling profit associated with the association rule can serve as essential measures to rule interestingness. In this paper, these domain measures are also included in the rule ranking procedure for selecting valuable rules for implementation. An example of market basket analysis is applied to illustrate the DEA based methodology for measuring the efficiency of association rules with multiple criteria. <s> BIB009 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> This study aims to resolve the problems associated with ranking fairly for both efficient and inefficient decision making units (DMUs) by proposing an 'interactive benchmark' model. The main concept is derived as a result of taking one certain DMU as a fixed benchmark and estimating the efficiency scores of the remaining DMUs based on that benchmark pair by pair. The process is repeated until all of the DMUs have served as the fixed benchmark. The DMUs can then be evaluated on a fair basis by computing the average efficiency scores using the above rolling procedure. The model is applied to fourteen financial holding companies (FHCs) in Taiwan using a production transfer model that is adopted by Seiford and Zhu [L.M. Seiford, J. Zhu, Profitability and marketability of the top 55 US commercial banks, Management Science 45 (9) (1999) 1270-1288]. The empirical results can serve as valuable reference to both policy-makers and investors. <s> BIB010 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <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> BIB011 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <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> BIB012 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> Decision making units (DMUs) can be ranked using data envelopment analysis (DEA) technologies. This paper develops a new ranking system under the condition of variable returns to scale (VRS) based on a measure of cross-dependence efficiency (MCDE), where the evaluation for an efficient DMU is dependent of the efficiency changes of all inefficient units due to its absence in the reference set, while the appraisal of inefficient DMUs depends on the influence of the exclusion of each efficient unit from the reference set. The infeasibility problem that arises from the conventional super-efficiency models is eliminated. A new approach to ranking inefficient units is embedded in the proposed ranking system. Through an example, advantages of our proposal are demonstrated. A real application illustrates the effectiveness of the cross-dependence based ranking system. <s> BIB013 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <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> BIB014 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <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> BIB015 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> In data envelopment analysis (DEA), the cross-efficiency evaluation method introduces a cross-efficiency matrix, in which the units are self and peer evaluated. A problem that possibly reduces the usefulness of the cross-efficiency evaluation method is that the cross-efficiency scores may not be unique due to the presence of alternate optima. So, it is recommended that secondary goals be introduced in cross-efficiency evaluation. In this paper we propose the symmetric weight assignment technique (SWAT) that does not affect feasibility and rewards decision making units (DMUs) that make a symmetric selection of weights. A numerical example is solved by our proposed method and its solution is compared with those of alternative approaches. <s> BIB016 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> In this paper, a new method for aggregating the opinions of experts in a preferential voting system is proposed. The method, which uses fuzzy concept in handling crisp data, is computationally efficient and is able to completely rank the alternatives. Through this method, the number of votes for certain rank position that each alternative receives are first grouped together to form fuzzy numbers. The nearest point to a fuzzy number concept is then used to introduce an artificial ideal alternative. Data envelopment analysis is next used to find the efficiency scores of the alternatives in a pair-wise comparison with the artificial ideal alternative. Alternatives are rank based on these efficiency scores. If the alternatives are not completely ranked, a weight restriction method also based on fuzzy concept is used on the un-discriminated alternatives until they are completely ranked. Two examples are given for illustration of the method. <s> BIB017 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> Data envelopment analysis models usually split decision making units into two basic groups, efficient and inefficient. Efficiency score of inefficient units allows their ranking but efficient units cannot be ranked directly because of their maximum efficiency. That is why there are formulated several models for ranking of efficient units. The paper presents two original models for ranking of efficient units in data envelopment analysis—they are based on multiple criteria decision making techniques—goal programming and analytic hierarchy process. The first model uses goal programming methodology and minimizes either the sum of undesirable deviations or maximal undesirable deviation from the efficient frontier. The second approach is analytic hierarchy process model for ranking of efficient units. The two presented models are compared with several super-efficiency models and other approaches for ranking decision making units in DEA models including definitions based on distances from optimistic and pessimistic envelopes and cross efficiency evaluation models. The results of the analysis by all presented models are illustrated on a real data set—evaluation of 194 bank branches of one of the Czech commercial banks. <s> BIB018 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> In this paper, the cross efficiency evaluation method, regarded as a DEA extension tool, is firstly reviewed for its utilization in identifying the Decision Making Unit (DMU) with the best practice and ranking the DMUs by their respective cross-efficiency scores. However, we then point out that the main drawback of the method lies in non-uniqueness of cross-efficiency scores resulted from the presence of alternate optima in traditional DEA models, obviously making it become less effective. Aiming at the research gap, a weight-balanced DEA model is proposed to lessen large differences in weighted data (weighted inputs and weighted outputs) and to effectively reduce the number of zero weights for inputs and outputs. Finally, we use two examples of the literature to illustrate the performance of this approach and discuss some issues of interest regarding the choosing of weights in cross-efficiency evaluations. <s> BIB019 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> Abstract Cross-efficiency evaluation is an extension of Data Envelopment Analysis (DEA) that permits not only the determination of a ranking of Decision Making Units (DMUs) but also the elimination of unrealistic weighting schemes, thereby rescinding the necessity for the inclusion of individual judgements in the models. The main deficiency of the procedure is the non-uniqueness of the optimal weights, which results in the peer evaluations dependences, for instance, on the software used to determine DMU’s efficiencies. This shortfall justifies the inclusion of secondary goals in order to determine cross-efficiency values. In this paper a new proposal of a secondary goal is studied. The idea is related with that proposed in Wu et al. (2009), in which the objective is the optimization of the rank position of the DMU under evaluation. In the procedure proposed here, an incentive to break level-pegging ties between alternatives is introduced by considering that efficiency scores induce a weak order of alternatives. The model is illustrated with a preference-aggregation application. <s> BIB020 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> A mixed integer linear model for selecting the best decision making unit (DMU) in data envelopment analysis (DEA) has recently been proposed by Foroughi [Foroughi, A. A. (2011a). A new mixed integer linear model for selecting the best decision making units in data envelopment analysis. Computers and Industrial Engineering, 60(4), 550-554], which involves many unnecessary constraints and requires specifying an assurance region (AR) for input weights and output weights, respectively. Its selection of the best DMU is easy to be affected by outliers and may sometimes be incorrect. To avoid these drawbacks, this paper proposes three alternative mixed integer linear programming (MILP) models for identifying the most efficient DMU under different returns to scales, which contain only essential constraints and decision variables and are much simpler and more succinct than Foroughi's. The proposed alternative MILP models can make full use of input and output information without the need of specifying any assurance regions for input and output weights to avoid zero weights, can make correct selections without being affected by outliers, and are of significant importance to the decision makers whose concerns are not DMU ranking, but the correct selection of the most efficient DMU. The potential applications of the proposed alternative MILP models and their effectiveness are illustrated with four numerical examples. <s> BIB021 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Different Applications in Ranking Units <s> Abstract Scoring rules are an important disputable subject in data envelopment analysis (DEA). Various organizations use voting systems whose main object is to rank alternatives. In these methods, the ranks of alternatives are obtained by their associated weights. The method for determining the ranks of alternatives by their weights is an important issue. This problem has been the subject at hand of some authors. We suggest a three-stage method for the ranking of alternatives. In the first stage, the rank position of each alternative is computed based on the best and worst weights in the optimistic and pessimistic cases, respectively. The vector of weights obtained in the first stage is not a singleton. Hence, to deal with this problem, a secondary goal is used in the second stage. In the third stage of our method, the ranks of the alternatives approach the optimistic or pessimistic case. It is mentionable that the model proposed in the third stage is a multi-criteria decision making (MCDM) model and there are several methods for solving it; we use the weighted sum method in this paper. The model is solved by mixed integer programming. Also, we obtain an interval for the rank of each alternative. We present two models on the basis of the average of ranks in the optimistic and pessimistic cases. The aim of these models is to compute the rank by common weights. <s> BIB022
As discussed formerly there exist a variety of ranking methods and applications in theliterature. Nowadays DEA models are widely used in different areas for efficiency evaluation, benchmarking and target setting, ranking entities, and so forth. Ranking units, as one of the important issues in DEA, has been performed in different areas. Jahanshahloo et al. BIB011 ranked cities in Iran to find the best place for creating a data factory. Hosseinzadeh Lotfi et al.  used a new method for ranking in order to find the best place for power plant location. Ali and Nakosteen BIB002 , Amirteimoori et al. BIB003 and Alirezaee and Afsharian BIB005 , Soltanifar and Hosseinzadeh Lotfi BIB012 , Zerafat Angiz et al. BIB017 , Hosseinzadeh Lotfi et al.  , Jahanshahloo BIB006 , Chen and Deng BIB013 , and Jablonsky BIB018 used different ranking methods in banking system. Sadjadi BIB014 ranked provincial gas companies in Iran. Mehrabian et al. BIB001 , Li et al. BIB007 ,Örkcü and Bal BIB015 , and Wu et al. BIB019 ranked different departments of universities. Jahanshahloo et al.  , Jahanshahloo and Afzalinejad BIB004 utilized the presented models in ranking 28 Chinese cities. Jahanshahloo at al.  provided an application to burden sharing amongst NATO member nations. Jahanshahloo et al. BIB016 utilized the presented model for ranking nursery homes. Contreras BIB020 and Wang et al. BIB008 used the provided ranking techniques in ranking candidates. Lu and Lo BIB010 applied their method on an application to financial holding companies. Hosseinzadeh Lotfi et al. BIB022 consider an empirical example in which voters are asked to rank two out of seven alternatives. Wang and Jiang BIB021 utilized the provided method in facility layout design in manufacturing systems and performance evaluation of 30 OECD countries. Chen BIB009 used an example of market basket data in order to illustrate the provided approach.