A Review of Ranking Models in Data Envelopment Analysis <s> Ranking Techniques Based on Finding Optimal Weights in DEA Analysis <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> BIB001 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Ranking Techniques Based on Finding Optimal Weights in DEA Analysis <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> BIB002 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Ranking Techniques Based on Finding Optimal Weights in DEA Analysis <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> BIB003 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Ranking Techniques Based on Finding Optimal Weights in DEA Analysis <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> BIB004 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Ranking Techniques Based on Finding Optimal Weights in DEA Analysis <s> It has been widely recognized that data envelopment analysis (DEA) lacks discrimination power to distinguish between DEA efficient units. This paper proposes a new methodology for ranking decision making units (DMUs). The new methodology ranks DMUs by imposing an appropriate minimum weight restriction on all inputs and outputs, which is decided by a decision maker (DM) or an assessor in terms of the solutions to a series of linear programming (LP) models that are specially constructed to determine a maximin weight for each DEA efficient unit. The DM can decide how many DMUs to be retained as DEA efficient in final efficiency ranking according to the requirement of real applications, which provides flexibility for DEA ranking. Three numerical examples are investigated using the proposed ranking methodology to illustrate its power in discriminating between DMUs, particularly DEA efficient units. <s> BIB005 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Ranking Techniques Based on Finding Optimal Weights in DEA Analysis <s> A common weight multi criteria decision analysis (MCDA)-data envelopment analysis (DEA) approach is proposed to construct composite indicators (CIs). The proposed MCDA-DEA model enables the construction of CIs among all entities via a set of common weights. The model is capable to discriminate entities which receive CI score of 1, i.e., the efficient entities leading to determination of a single best entity. Common weights structure of the proposed model has more discriminating power when compared to those obtained by previous DEA-like models. In order to validate the proposed MCDA-DEA model, it is applied to two case studies taken from the literature to construct two well-known CIs, i.e., Sustainable Energy Index (SEI) and Human Development Index (HDI). Both the robustness and discriminating power of the proposed method are studied through these case studies and tested by Spearman's rank correlation coefficient. The results reveal several merits of the proposed method in constructing CIs. <s> BIB006 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Ranking Techniques Based on Finding Optimal Weights in DEA Analysis <s> Existing methods for generating common weights in data envelopment analysis (DEA) are either very complicated or unable to produce a full ranking for decision making units (DMUs). This paper proposes a new methodology based on regression analysis to seek a common set of weights that are easy to estimate and can produce a full ranking for DMUs. The DEA efficiencies obtained with the most favorable weights to each DMU are treated as the target efficiencies of DMUs and are best fitted with the efficiencies determined by common weights. Two new nonlinear regression models are constructed to optimally estimate the common weights. Four numerical examples are examined using the developed new models to test their discrimination power and illustrate their potential applications in fully ranking DMUs. Comparisons with a similar compromise approach for generating common weights are also discussed. <s> BIB007 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Ranking Techniques Based on Finding Optimal Weights in DEA Analysis <s> In this paper, we propose a DEA approach aimed at deriving a common set of weights (CSW) to be used to the ranking of decision making units (DMUs). The idea of this approach is to minimize the deviations of the CSW from the DEA profiles of weights without zeros of the efficient DMUs. This minimization reduces in particular the differences between the DEA profiles of weights that are chosen, so the CSW proposed is a representative summary of such DEA weights profiles. We use several norms to the measurement of such differences. As a result, the CSWs derived are actually different summaries of the chosen DEA profiles of weights like their average profile of their median profile. This approach is illustrated with an application to the ranking of professional tennis players. <s> BIB008
Jahanshahloo et al. BIB001 gave a note on some of the DEA models for complete ranking using common set of weights. They proved that by solving only one problem, it is possible to determine the common set of weights max s.t. where is the set of efficient units of the following model: , , ≥ , 0 free. Note that DMUs can be ranked based on the evaluation of their efficiencies. Wang et al. BIB002 provided an aggregating preference ranking. In this paper use of ordered weighted averaging (OWA) operator is proposed for aggregating preference rankings. Let be the relative importance weight given to the jth ranking place and V the vote candidate i receives in the jth ranking place. The total score of each candidate is defined as Alirezaee and Afsharian BIB003 discussed multiplier model, in which the variables are considered as shadow prices; note that ∑ =1 and ∑ =1 V are total revenue and cost of DMU which are considered in optimization problem. The authors claimed that ∑ =1 − ∑ =1 V ≤ 0, = 1, . . . , is the profit restriction for DMU . If ( , ) = 0 is considered to be the efficient production function, then They mentioned that the connected profit of the DMU is zero when shadow prices are derived from the technology and called this situation as balance situation. As the authors mentioned therefore in the case that DMU 1 is efficient but DMU 2 is inefficient or the efficiency score of both DMUs is the same, and it obtains more negative quantity in balance index, it can be concluded that DMU 1 has a better rank than DMU 2 . Liu and Hsuan Peng BIB004 in their paper proposed a method for determining the common set of weights for ranking units. In common weights analysis methodology they provided the following model: The mentioned ratio form the linear equations. Wang et al. BIB005 proposed a paper for ranking decisionmaking units by imposing a minimum weight restriction in DEA. The authors noted that using data envelopment analysis, it is not possible to distinguish between DEA efficient units. Thus, they presented a method for ranking units using imposing minimum weight restriction for the inputoutput data. As they mentioned, these weights restrictions are decided by a decision maker (DM) or an assessor as regards the solutions to a series of LP models considered for determining a maximin weight for each efficient DMU. Hatefi and Torabi BIB006 proposed a common weight multi criteria decision analysis (MCDA)-data envelopment analysis (DEA) for constructing composite indicators (CIs). As the authors proved the presented model can discriminate between efficient units. The obtained common weights have discriminating power more than those obtained from Journal of Applied Mathematics 7 previous models. Finally they studied the robustness and discriminating power of the proposed method by Spearman's rank correlation coefficient. Hosseinzadeh Lotfi et al.  proposed one DEA ranking model based on applying aggregate units. In doing so artificial units called aggregate units are defined as follows. The aggregate unit is shown by DMU where = { | DMU ∈ }, = {DMU }. Note that is the set of efficient units. First, it is tried to maximize the efficiency score of the DMU and then to maximize the efficiency score of the DMU . For resolving the existence of alternative solutions the authors presented an approach comprising ( + ) simple linear problems to achieve the most appropriate optimal solutions among all alternative optimal solutions. Finally, they proposed the RI index for ranking all efficient DMUs. Let ,̃and̃,̃be the optimal solutions of the multiplier model for assent DMU and DMU . Consider BIB007 presented two nonlinear regression models for deriving common set of weights for fully rank units Ramón et al. BIB008 aimed at deriving a common set of weights for ranking units. As the authors mentioned the idea is based upon minimization of the deviations of the common weights from the nonzero weights obtained from DEA. Furthermore, several norms are used for measuring such differences.