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.     

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.     

Selecting symmetric weights as a secondary goal in DEA cross-efficiency evaluation

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.     

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.     

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.     

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.     

Some clarifications on the DEA clustering approach

This paper clarifies the role of alternative optimal solutions in the clustering of multidimensional observations using data envelopment analysis (DEA). The paper shows that alternative optimal solutions corresponding to several units produce different groups with different sizes and different decision making units (DMUs) at each class. This implies that a specific DMU may be grouped into different clusters when the corresponding DEA model has multiple optimal solutions.     

An epsilon-free approach for finding the most efficient unit in DEA

Data envelopment analysis (DEA), considering the best condition for each decision making unit (DMU), assesses the relative efficiency and partitions DMUs into two sets: efficient and inefficient. Practically, in traditional DEA models more than one efficient DMU are recognized and these models cannot rank efficient DMUs. Some studies have been carried out aiming at ranking efficient DMUs, although in some cases only discrimination of the most efficient unit is desirable. Furthermore, several investigations have been done for finding the most CCR-efficient DMU. The basic idea of the majority of them is to introduce an integrated model which achieves an optimal common set of weights (CSW). These weights help us identify the most efficient unit in an identical condition.     

Outlier detection in two-stage semiparametric DEA models

In the use of peer group data to assess individual, typical or best practice performance, the effective detection of outliers is critical for achieving useful results, particularly for two-stage analyses. In the DEA-related literature, prior work on this issue has focused on the efficient frontier as a basis for detecting outliers. An iterative approach for dealing with the potential for one outlier to mask the presence of another has been proposed but not demonstrated. This paper proposes using both the efficient frontier and the inefficient frontier to identify outliers and thereby improve the accuracy of second stage results in two-stage nonparametric analysis. The iterative outlier detection approach is implemented in a leave-one-out method using both the efficient frontier and the inefficient frontier and demonstrated in a two-stage semi-parametric bootstrapping analysis of a classic data set. The results show that the conclusions drawn can be different when outlier identification includes consideration of the inefficient frontier.     

Super-efficiency DEA in the presence of infeasibility: One model approach

A two-stage procedure is developed by Lee et al. (2011) [European Journal of Operational Research doi:10.1016/j.ejor.2011.01.022] to address the infeasibility issue in super-efficiency data envelopment analysis (DEA) models. We point out that their two-stage procedure can be solved in a single DEA-based model.     

Measurement of returns to scale using non-radial DEA models

There are some specific features of the non-radial data envelopment analysis (DEA) models which cause some problems for the returns to scale measurement. In the scientific literature on DEA, some methods were suggested to deal with the returns to scale measurement in the non-radial DEA models. These methods are based on using Strong Complementary Slackness Conditions from optimization theory. However, our investigation and computational experiments show that such methods increase computational complexity significantly and may generate as optimal, solutions contradicting optimization theory. In this paper, we propose and substantiate a direct method for the returns to scale measurement in the non-radial DEA models. Our computational experiments documented that the proposed method works reliably and efficiently on the real-life data sets.     

Measurement of returns to scale using a non-radial DEA model: A range-adjusted measure approach

This research theoretically explores the measurement of returns to scale (RTS), using a non-radial DEA (data envelopment analysis) model. A range-adjusted measure (RAM) is used as a representative of such non-radial models. Historically, a type of RTS has been discussed within an analytical framework of radial models. The radial-based RTS measurement is replaced by the non-radial RAM/RTS measurement in this study. When discussing the non-radial RAM/RTS measurement, this study finds a problem of multiple projections that cannot be found in the radial measurement. In this research, a new linear programming approach is proposed to identify all efficient DMUs (decision making units) on a reference set. The important feature of the proposed approach is that it can deal with a simultaneous occurrence of (a) multiple reference sets, (b) multiple supporting hyperplanes and (c) multiple projections. All of the three difficulties are handled by the proposed RAM/RTS measurement. In particular, we discuss both when the three different types of multiple solutions occur on the RAM/RTS measurement and how to deal with such difficulties. Our research results make it possible to measure not only the type of RTS but also the magnitude of RTS in the RAM measurement.     

Computation of efficient targets in DEA models with production trade-offs and weight restrictions

Recently new models of data envelopment analysis (DEA) were introduced that incorporate production trade-offs between inputs and outputs or based on them weight restrictions. In this paper, we develop a computational procedure suitable for the practical application of such models. We show that the standard two-stage optimisation procedure used in DEA to test the full efficiency of units and identify their efficient targets may work incorrectly in the new models. The modified procedure consists of three stages: the first evaluates the radial efficiency of the unit, the second identifies its efficient target, and the third its reference set of efficient peers. Each stage requires solving one linear program for each unit.     

基于前景理论的两阶段DEA交叉效率评价模型

针对两阶段系统交叉效率值不唯一的问题,现有解决方法大多假设决策者完全理性,忽略了决策者心理因素对评价过程的影响。本文构建以平均值为新的参考点,分别在集中和分散决策环境下建立基于前景理论的两阶段DEA交叉效率评价模型,通过实际算例验证模型的可行性和有效性。     

A DEA approach for fair allocation of common revenue

An issue of considerable importance, how to allocate a common revenue in an equitable manner across a set of competing entities. This paper introduces a new approach to obtaining allocation common revenue on all decision making units (DMUs) in such a way that the relative efficiency is not changed. In this method for determining allocation common revenue dose not need to solving any linear programming. A numerical example is provided to illustrate the results of the analysis.     

On the boundedness of the SORM DEA models with negative data

Emrouznejad et al. (2010) proposed a Semi-Oriented Radial Measure (SORM) model for assessing the efficiency of Decision Making Units (DMUs) by Data Envelopment Analysis (DEA) with negative data. This paper provides a necessary and sufficient condition for boundedness of the input and output oriented SORM models.     

An MOLP based procedure for finding efficient units in DEA models

In this paper a multiple objective linear programming (MOLP) problem whose feasible region is the production possibility set with variable returns to scale is proposed. By solving this MOLP problem by multicriterion simplex method, the extreme efficient Pareto points can be obtained. Then the extreme efficient units in data envelopment analysis (DEA) with variable returns to scale, considering the specified theorems and conditions, can be obtained. Therefore, by solving the proposed MOLP problem, the non-dominant units in DEA can be found. Finally, a numerical example is provided.     

Preference aggregation and DEA: An analysis of the methods proposed to discriminate efficient candidates

There are different ways to allow the voters to express their preferences on a set of candidates. In ranked voting systems, each voter selects a subset of the candidates and ranks them in order of preference. A well-known class of these voting systems are scoring rules, where fixed scores are assigned to the different ranks and the candidates with the highest score are the winners. One of the most important issues in this context is the choice of the scoring vector, since the winning candidate can vary according to the scores used. To avoid this problem, Cook and Kress [W.D. Cook, M. Kress, A data envelopment model for aggregating preference rankings, Management Science 36 (11) (1990) 1302–1310], using a DEA/AR model, proposed to assess each candidate with the most favorable scoring vector for him/her. However, the use of this procedure often causes several candidates to be efficient, i.e., they achieve the maximum score. For this reason, several methods to discriminate among efficient candidates have been proposed. The aim of this paper is to analyze and show some drawbacks of these methods.     

A two-phase approach for setting targets in DEA with negative data

Conventional DEA models have been introduced to deal with non-negative data. In the real world, in some occasions, we have outputs and/or inputs, which can take negative data. In DEA literature some approaches have been presented for evaluating performance of units, which operate with negative data. In this paper, firstly, we give a brief review of these works, then we present a new additive based approach in this framework. The proposed model is designed to provide a target with non-negative value associated with negative components for each observed unit, failed by other methods. An empirical application in banking is then used to show the applicability of the proposed method and make a comparison with the other approaches in the literature.