A Review of Ranking Models in Data Envelopment Analysis <s> Cross-Efficiency Ranking Techniques <s> In this paper we examine a neglected aspect of Data Envelopment Analysis: cross-efficiency. We develop the concept of cross-efficiency in a number of new directions. We ground an intuitive understanding of cross-efficiency in the concept of peer-appraisal, as opposed to self-appraisal implied by simple efficiency, and discuss the relative merits of each. We also present mathematical formulations of, and intuitive meanings for three possible implementations of aggressive and benevolent cross-efficiency. We have implemented two of these formulations in computer programs; their performance is compared empirically on a real data set. Finally, we suggest practical uses for cross-efficiency, illustrated with reference to the same data set. <s> BIB001 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Cross-Efficiency Ranking Techniques <s> Cook and Kress (1990), using Data Envelopment Analysis (DEA) as their starting point, proposed a procedure to rank order the candidates in a preferential election. Notionally, each candidate is permitted to choose the most favourable weights to be applied to his/her standings (first place, second place, etc. votes) in the usual DEA manner with the additional ‘assurance region’ restriction that the weight for a j place vote should be more than that for a j +1 amount. We consider that this freedom to choose weights is essentially illusory when maximum discrimination between the candidates is sought, in which case the weights used to evaluate and rank the candidates are as if imposed externally at the outset. To avoid this, we present an alternative procedure which retains Cook and Kress' central idea but where, as well as using each candidate's rating of him/herself, we now make use of each candidate's ratings of all the candidates. We regard the so-called cross-evaluation matrix as the summary of a self- and peer-rating process in which the candidates seek to interpret the voters preferences as favourably for themselves, relative to the other candidates, as possible. The problem then becomes one of establishing an overall rating for each candidate from these individual ratings. For this, for each candidate, we use a weighted average of all the candidates ratings of that candidate, where the weights themselves are in proportion to each candidate's overall rating. The overall ratings are therefore proportional to the components of the principal (left-hand) eigenvector of the cross-evaluation matrix. These ideas are then applied to the selection of R & D projects to comprise an R & D program, thus indicating thier wider applicability. <s> BIB002 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Cross-Efficiency Ranking Techniques <s> Abstract Within data envelopment analysis (DEA) is a sub-group of papers in which many researchers have sought to improve the differential capabilities of DEA and to fully rank both efficient, as well as inefficient, decision-making units. The ranking methods have been divided in this paper into six, somewhat overlapping, areas. The first area involves the evaluation of a cross-efficiency matrix, in which the units are self and peer evaluated. The second idea, generally known as the super-efficiency method, ranks through the exclusion of the unit being scored from the dual linear program and an analysis of the change in the Pareto Frontier. The third grouping is based on benchmarking, in which a unit is highly ranked if it is chosen as a useful target for many other units. The fourth group utilizes multivariate statistical techniques, which are generally applied after the DEA dichotomic classification. The fifth research area ranks inefficient units through proportional measures of inefficiency. The last approach requires the collection of additional, preferential information from relevant decision-makers and combines multiple-criteria decision methodologies with the DEA approach. However, whilst each technique is useful in a specialist area, no one methodology can be prescribed here as the complete solution to the question of ranking. <s> BIB003 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Cross-Efficiency Ranking Techniques <s> Data Envelopment Analysis DEA is a method for estimating (in-)efficiencies of Decision Making Units DMUs by means of weighted output - to input - ratios, being the weights optimal virtual prices of such ex-post activities for all units. The cross-efficiency matrix then evaluates these output - to input - relations with respect to all optimal price systems, and hence permits efficiency rankings for the DMUs by aggregating the matrix entries line - and/or columnwise. In this contribution the classical input oriented DEA approach is generalized twofold: its first aim is an optimal efficiency improving input allocation rather than a mere radial input reduction. The second aim is the choice of a peer-DMU, the price system of which is acceptable for the remaining units. As free input allocation permits substitutional effects and so rises productivities in view of possible peers and for all units, it supports such consensual choice. Numerical examples show the positive effects of the new concept. <s> BIB004 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Cross-Efficiency Ranking Techniques <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> BIB005 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Cross-Efficiency Ranking Techniques <s> Research highlights? The cross efficiency evaluation method was reviewed and its main shortcomings were described when the ultimate average cross efficiency scores were used. ? The assumption of average was eliminated and the Shannon entropy was utilized to determine the weights for the ultimate cross efficiency scores. ? An empirical example was illustrated and some future research directions were pointed out. This paper firstly reviews the cross efficiency evaluation method which is an extension tool of data envelopment analysis (DEA), then we describe the main shortcomings when the ultimate average cross efficiency scores are used to evaluate and rank the decision making units (DMUs). In this paper, we eliminate the assumption of average and utilize the Shannon entropy to determine the weights for ultimate cross efficiency scores, and the procedures are introduced in detail. In the end, an empirical example is illustrated to examine the validity of the proposed method. Some future research directions are also pointed out. <s> BIB006 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Cross-Efficiency Ranking Techniques <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> BIB007 </s> A Review of Ranking Models in Data Envelopment Analysis <s> Cross-Efficiency Ranking Techniques <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> BIB008
Sexton et al.  provided a method for ranking units based on this idea that units are self-and peer-evaluated. For deriving the cross-efficiency of any DMU using weights chosen by DMU , they proposed the following equation: where * , * are optimal weights obtained from the following model for DMU under assessment  Now DMU received the average cross-efficiency score as = ∑ =1 / ; for further details about this averaging see also Green et al. BIB002 . Doyle and Green BIB001 also used crossefficiency matrix for ranking units. According to this method for ranking DMUs, many investigations have been done as reviewed in Adler et al. BIB003 . Rödder and Reucher BIB004 presented a consensual peerbased DEA model for ranking units. As the authors said, this method is generalized twofold. The first is an optimal efficiency improving input allocation; the second aim is the choice of a peer DMU whose corresponding price is acceptable for the other units. Consider Journal of Applied Mathematics The higher the degree of input variation is, the better the chance to be efficient will be. Orkcü and Bal BIB005 provided a goal programming technique to be used in the second stage of the cross-evaluation. Their modified model is as follows: As the authors noted, there exist alternative optimal solutions. Wu et al. BIB006 described the main suffering of crossefficiency when the ultimate average cross-efficiency utilized for ranking units. For removing this shortcoming they eliminated the assumption of average and utilized the Shannon entropy in order to obtain the weights for ultimate crossefficiency scores. Jahanshahloo et al. BIB007 provided a method for selecting symmetric weights to be used in DEA crossefficacy. Step 1. Efficiency of DMUs needs to be computed. Step 2. Choose the solutions, in accordance with the secondary goal for each DMU, as follows: Step 3. The cross-efficiency for any DMU , using the weights that DMU has chosen in the previous model, is then used as follows: Wang et al. BIB008 provided a cross-efficiency evaluation based on ideal and anti-ideal units for ranking. As the authors mentioned, a DMU could choose a unique set of input and output weights to make its distance from ideal DMU as small as possible, or the distance from anti-ideal DMU as large as possible, or the both. Thus, according to this idea they proposed the following procedure for cross-efficiency evaluation.