Optimizing the rank position of the DMU as secondary goal in DEA cross-evaluation

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.     

On the DEA total weight flexibility and the aggregation in cross-efficiency evaluations

This paper discusses the DEA total weight flexibility in the context of the cross-efficiency evaluation. The DMUs in DEA are often assessed with unrealistic weighting schemes in their attempt to achieve the best ratings in their self-evaluation. We claim here that in a peer-appraisal like the cross-efficiency evaluation the cross-efficiencies provided by such weights cannot play the same role as those obtained with more reasonable weights. To address this issue, we propose to calculate the cross-efficiency scores by means of a weighted average of cross-efficiencies, instead of with the usual arithmetic mean, so the aggregation weights reflect the disequilibrium in the profiles of DEA weights that are used. Thus, the cross-efficiencies provided by profiles with large differences in their weights, especially those obtained with zero weights, would be attached lower aggregation weights (less importance) than those provided by more balanced profiles of weights.     

Use of DEA cross-efficiency evaluation in portfolio selection: An application to Korean stock market

We propose a way of using DEA cross-efficiency evaluation in portfolio selection. While cross efficiency is an approach developed for peer evaluation, we improve its use in portfolio selection. In addition to (average) cross-efficiency scores, we suggest to examine the variations of cross-efficiencies, and to incorporate two statistics of cross-efficiencies into the mean-variance formulation of portfolio selection. Two benefits are attained by our proposed approach. One is selection of portfolios well-diversified in terms of their performance on multiple evaluation criteria, and the other is alleviation of the so-called “ganging together” phenomenon of DEA cross-efficiency evaluation in portfolio selection. We apply the proposed approach to stock portfolio selection in the Korean stock market, and demonstrate that the proposed approach can be a promising tool for stock portfolio selection by showing that the selected portfolio yields higher risk-adjusted returns than other benchmark portfolios for a 9-year sample period from 2002 to 2011.     

Fixed cost and resource allocation based on DEA cross-efficiency

In many managerial applications, situations frequently occur when a fixed cost is used in constructing the common platform of an organization, and needs to be shared by all related entities, or decision making units (DMUs). It is of vital importance to allocate such a cost across DMUs where there is competition for resources. Data envelopment analysis (DEA) has been successfully used in cost and resource allocation problems. Whether it is a cost or resource allocation issue, one needs to consider both the competitive and cooperative situation existing among DMUs in addition to maintaining or improving efficiency. The current paper uses the cross-efficiency concept in DEA to approach cost and resource allocation problems. Because DEA cross-efficiency uses the concept of peer appraisal, it is a very reasonable and appropriate mechanism for allocating a shared resource/cost. It is shown that our proposed iterative approach is always feasible, and ensures that all DMUs become efficient after the fixed cost is allocated as an additional input measure. The cross-efficiency DEA-based iterative method is further extended into a resource-allocation setting to achieve maximization in the aggregated output change by distributing available resources. Such allocations for fixed costs and resources are more acceptable to the players involved, because the allocation results are jointly determined by all DMUs rather than a specific one. The proposed approaches are demonstrated using an existing data set that has been applied in similar studies.     

Ranking ranges in cross-efficiency evaluations

The existence of alternate optima for the DEA weights may reduce the usefulness of the cross-efficiency evaluation, since the ranking provided depends on the choice of weights that the different DMUs make. In this paper, we develop a procedure to carry out the cross-efficiency evaluation without the need to make any specific choice of DEA weights. The proposed procedure takes into consideration all the possible choices of weights that all the DMUs can make, and yields for each unit a range for its possible rankings instead of a single ranking. This range is determined by the best and the worst rankings that would result in the best and the worst scenarios of each unit across all the DEA weights of all the DMUs. This approach might identify good/bad performers, as those that rank at the top/bottom irrespective of the weights that are chosen, or units that outperform others in all the scenarios. In addition, it may be used to analyze the stability of the ranking provided by the standard cross-efficiency evaluation.     

Preference voting and project ranking using DEA and cross-evaluation

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.     

Ranking decision-making units using common weights in DEA

Data envelopment analysis (DEA) is a linear programming technique that is used to measure the relative efficiency of decision-making units (DMUs). Liu et al. (2008) [13] used common weights analysis (CWA) methodology to generate a CSW using linear programming. They classified the DMUs as CWA-efficient and CWA-inefficient DMUs and ranked the DMUs using CWA-ranking rules. The aim of this study is to show that the criteria used by Liu et al. are not theoretically strong enough to discriminate among the CWA-efficient DMUs with equal efficiency. Moreover, there is no guarantee that their proposed model can select one optimal solution from the alternative components. The optimal solution is considered to be the only unique optimal solution. This study shows that the proposal by Liu et al. is not generally correct. The claims made by the authors against the theorem proposed by Liu et al. are fully supported using two counter examples.     

Cross-efficiency aggregation in DEA models using the evidential-reasoning approach

Cross-efficiency in data envelopment analysis (DEA) models is an effective way to rank decision-making units (DMUs). The common methods to aggregate cross-efficiency do not consider the preference structure of the decision maker (DM). When a DM’s preference structure does not satisfy the “additive independence” condition, a new aggregation method must be proposed. This paper uses the evidential-reasoning (ER) approach to aggregate the cross-efficiencies obtained from cross-evaluation through the transformation of the cross-efficiency matrix to pieces of evidence. This paper provides a new method for cross-efficiency aggregation and a new way for DEA models to reflect a DM’s preference or value judgments. Additionally, this paper presents examples that demonstrate the features of cross-efficiency aggregation using the ER approach, including an empirical example of the evaluation practice of 16 basic research institutes in Chinese Academy of Sciences (CAS) in 2010 that illustrates how the ER approach can be used to aggregate the cross-efficiency matrix produced from DEA models.     

Selecting non-zero weights to evaluate effectiveness of basketball players with DEA

In this paper, we show how DEA may be used to identify component profiles as well as overall indices of performance in the context of an application to assessments of basketball players. We go beyond the usual uses of DEA to provide only overall indexes of performance. Our focus is, instead, on the multiplier values for the efficiently rated players. For this purpose we use a procedure that we recently developed that guarantees a full profile of non-zero weights, or “multipliers.” We demonstrate how these values can be used to identify relative strengths and weaknesses in individual players. Here we also utilize the flexibility of DEA by introducing bounds on the allowable values to reflect the views of coaches, trainers and other experts on the basketball team for which evaluations are being conducted. Finally we show how these combinations can be extended by taking account of team as well as individual considerations.     

On the choice of weights profiles in cross-efficiency evaluations

In this paper, we propose a new approach to cross-efficiency evaluation that focuses on the choice of the weights profiles to be used in the calculation of the cross-efficiency scores. It has been claimed in the literature that cross-efficiency eliminates unrealistic weighting schemes in the sense that their effects are cancelled out in the summary that the cross-efficiency evaluation makes. The idea of our approach here is to try to avoid these unreasonable weights instead of expecting that their effects are cancelled out in the amalgamation of weights that is made. To do it, we extend the ideas of the multiplier bound approach to the assessment of efficiency without slacks in Ramón et al. (2010) to its use in cross-efficiency evaluations. The models used look for the profiles with the least dissimilar weights, and also guarantee non-zero weights. In particular, this approach allows the inefficient DMUs to make a choice of weights that prevent them from using unrealistic weighting schemes. We use some examples of the literature to illustrate the performance of this approach and discuss some issues of interest regarding the choice of weights in cross-efficiency evaluations.     

A revised and generalized model with improved discrimination for finding most efficient DMUs in DEA

In this paper we first propose a generalized model which in its special case revises a recently proposed model for finding most BCC-efficient DMU. Then, by explaining about the drawbacks of existing approaches, an algorithm will be developed to consider all possible alternative optimal solutions and determine the set of most efficient DMUs. Another model also will be proposed to provide more discrimination which can be used to select a single most efficient DMU, when it is desirable. The proposed approaches of this paper are applicable for all assumptions about returns-to-scale, and can be utilized to select a DMU or provide a full ranking of DMUs. The contents of the paper are illustrated through numerical examples.     

A new DEA ranking system based on changing the reference set

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.     

对DEA中有效单元排序的一种新方法

文章对数据包络分析(DEA)的强有效性问题提出了一种新的研究方法.利用有效值和负有效值来构造复合输入和输出这种方法可以实现有效决策单元的完全排序.文章还给出了新方法中模型的一些性质.最后,用两个例子来检验此方法并和其他模型的计算结果进行了比较.     

Cross-efficiency evaluation with directional distance functions

This paper extends the cross-efficiency evaluation for use with directional distance functions. Cross-efficiency evaluation has been developed with oriented Data Envelopment Analysis (DEA) models, so the extension proposed here is aimed at providing a peer-evaluation of decision making units (DMUs) based on measures that account for the inefficiency both in inputs and in outputs simultaneously. We explore the duality relations regarding the models of directional distance functions and define the cross-efficiencies on the basis of the equivalences with some fractional programming problems. Finally, we address in this new context the problem with the alternate optima for the weights and propose some models that implement different alternative secondary goals.     

Performance ranking of units considering ideal and anti-ideal DMU with common weights

A characteristic of traditional DEA CCR mode is that it allows DMUs to measure their maximum efficiency score with the most favorable weights. Thus, it would have some shortcomings, for example, the efficiencies of different DMUs obtained by different sets of weights may be unable to be compared and ranked on the same basis. Besides, there are always more than one DMU to be evaluated as efficient because of the flexibility in the selection of weights; it would cause the situation that all DMUs cannot be fully discriminated. With the research gaps, in this paper, we propose two models considering ideal and anti-ideal DMU to generate common weights for performance evaluation and ranking. Finally, two examples of Asian lead frame firms and flexible manufacturing systems are illustrated to examine the validity of the proposed methods.     

A modified complete ranking of DMUs using restrictions in DEA models

Alirezaee and Afsharian [1] have proposed a new index, namely, Balance Index, to rank DMUs. In this paper, we will use their examples to illustrate that the proposed index is not stable. As a result, the corresponding rankings are also unstable. Then we analyze where an error occurs in the new method for complete ranking of decision making units and amend it by introducing the Maximal Balance Index. The numeral example reports the reasonability of our methods.     

Choosing weights from alternative optimal solutions of dual multiplier models in DEA

In this paper we propose a two-step procedure to be used for the selection of the weights that we obtain from the multiplier model in a DEA efficiency analysis. It is well known that optimal solutions of the envelopment formulation for extreme efficient units are often highly degenerate and, consequently, have alternate optima for the weights. Different optimal weights may then be obtained depending, for instance, on the software used. The idea behind the procedure we present is to explore the set of alternate optima in order to help make a choice of optimal weights. The selection of weights for a given extreme efficient point is connected with the dimension of the efficient facets of the frontier. Our approach makes it possible to select the weights associated with the facets of higher dimension that this unit generates and, in particular, it selects those weights associated with a full dimensional efficient facet (FDEF) if any. In this sense the weights provided by our procedure will have the maximum support from the production possibility set. We also look for weights that maximize the relative value of the inputs and outputs included in the efficiency analysis in a sense to be described in this article.     

Validating DEA as a ranking tool: An application of DEA to assess performance in higher education

There is a general interest in ranking schemes applied to complex entities described by multiple attributes. Published rankings for universities are in great demand but are also highly controversial. We compare two classification and ranking schemes involving universities; one from a published report, ‘Top American Research Universities’ by the University of Florida's TheCenter and the other using DEA. Both approaches use the same data and model. We compare the two methods and discover important equivalences. We conclude that the critical aspect in classification and ranking is the model. This suggests that DEA is a suitable tool for these types of studies.     

Fuzzy data envelopment analysis (DEA): Model and ranking method

Data Envelopment Analysis (DEA) is a very effective method to evaluate the relative efficiency of decision-making units (DMUs). Since the data of production processes cannot be precisely measured in some cases, the uncertain theory has played an important role in DEA. This paper attempts to extend the traditional DEA models to a fuzzy framework, thus producing a fuzzy DEA model based on credibility measure. Following is a method of ranking all the DMUs. In order to solve the fuzzy model, we have designed the hybrid algorithm combined with fuzzy simulation and genetic algorithm. When the inputs and outputs are all trapezoidal or triangular fuzzy variables, the model can be transformed to linear programming. Finally, a numerical example is presented to illustrate the fuzzy DEA model and the method of ranking all the DMUs.     

A ranking method based on a full-inefficient frontier

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.