DEA models for minimizing weight disparity in cross-efficiency evaluation

Cross-efficiency evaluation is a commonly used approach for ranking decision-making units (DMUs) in data envelopment analysis (DEA). The weights used in the cross-efficiency evaluation may sometimes differ significantly among the inputs and outputs. This paper proposes some alternative DEA models to minimize the virtual disparity in the cross-efficiency evaluation. The proposed DEA models determine the input and output weights of each DMU in a neutral way without being aggressive or benevolent to the other DMUs. Numerical examples are tested to show the validity and effectiveness of the proposed DEA models and illustrate their significant role in reducing the number of zero weights.     

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.     

Fuzzy cross-efficiency evaluation: a possibility approach

Cross-efficiency evaluation is an extension of data envelopment analysis (DEA) aimed at ranking decision making units (DMUs) involved in a production process regarding their efficiency. As has been done with other enhancements and extensions of DEA, in this paper we propose a fuzzy approach to the cross-efficiency evaluation. Specifically, we develop a fuzzy cross-efficiency evaluation based on the possibility approach by Lertworasirikul et al. (Fuzzy Sets Syst 139:379–394, 2003a) to fuzzy DEA. Thus, a methodology for ranking DMUs is presented that may be used when data are imprecise, in particular for fuzzy inputs and outputs being normal and convex. We prove some results that allow us to define “consistent” cross-efficiencies. The ranking of DMUs for a given possibility level results from an ordering of cross-efficiency scores, which are real numbers. As in the crisp case, we also develop benevolent and aggressive fuzzy formulations in order to deal with the alternate optima for the weights.     

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.     

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.     

DEA cross-efficiency evaluation under variable returns to scale

Cross-efficiency evaluation in data envelopment analysis (DEA) has been developed under the assumption of constant returns to scale (CRS), and no valid attempts have been made to apply the cross-efficiency concept to the variable returns to scale (VRS) condition. This is due to the fact that negative VRS cross-efficiency arises for some decision-making units (DMUs). Since there exist many instances that require the use of the VRS DEA model, it is imperative to develop cross-efficiency measures under VRS. We show that negative VRS cross-efficiency is related to free production of outputs. We offer a geometric interpretation of the relationship between the CRS and VRS DEA models. We show that each DMU, via solving the VRS model, seeks an optimal bundle of weights with which its CRS-efficiency score, measured under a translated Cartesian coordinate system, is maximized. We propose that VRS cross-efficiency evaluation should be done via a series of CRS models under translated Cartesian coordinate systems. The current study offers a valid cross-efficiency approach under the assumption of VRS—one of the most common assumptions in performance evaluation done by DEA.     

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 consensual peer-based DEA-model with optimized cross-efficiencies - Input allocation instead of radial reduction

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.     

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.     

Data envelopment analysis with common weights: the compromise solution approach

A characteristic of data envelopment analysis (DEA) is to allow individual decision-making units (DMUs) to select the factor weights that are the most advantageous for them in calculating their efficiency scores. This flexibility in selecting the weights, on the other hand, deters the comparison among DMUs on a common base. In order to rank all the DMUs on the same scale, this paper proposes a compromise solution approach for generating common weights under the DEA framework. The efficiency scores calculated from the standard DEA model are regarded as the ideal solution for the DMUs to achieve. A common set of weights which produces the vector of efficiency scores for the DMUs closest to the ideal solution is sought. Based on the generalized measure of distance, a family of efficiency scores called ‘compromise solutions’ can be derived. The compromise solutions have the properties of unique solution and Pareto optimality not enjoyed by the solutions derived from the existing methods of common weights. An example of forest management illustrates that the compromise solution approach is able to generate a common set of weights, which not only differentiates efficient DMUs but also detects abnormal efficiency scores on a common base.     

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.     

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.     

Interval cross efficiency for fully ranking decision making units using DEA/AHP approach

Data envelopment analysis (DEA) is a popular technique for measuring the relative efficiency of a set of decision making units (DMUs). Fully ranking DMUs is a traditional and important topic in DEA. In various types of ranking methods, cross efficiency method receives much attention from researchers because it evaluates DMUs by using self and peer evaluation. However, cross efficiency score is usual nonuniqueness. This paper combines the DEA and analytic hierarchy process (AHP) to fully rank the DMUs that considers all possible cross efficiencies of a DMU with respect to all the other DMUs. We firstly measure the interval cross efficiency of each DMU. Based on the interval cross efficiency, relative efficiency pairwise comparison between each pair of DMUs is used to construct interval multiplicative preference relations (IMPRs). To obtain the consistency ranking order, a method to derive consistent IMPRs is developed. After that, the full ranking order of DMUs from completely consistent IMPRs is derived. It is worth noting that our DEA/AHP approach not only avoids overestimation of DMUs’ efficiency by only self-evaluation, but also eliminates the subjectivity of pairwise comparison between DMUs in AHP. Finally, a real example is offered to illustrate the feasibility and practicality of the proposed procedure.     

Evaluating the performances of decision-making units based on interval efficiencies

Efficiency could be not only the ratio of weighted sum of outputs to that of inputs but also that of weighted sum of inputs to that of outputs. When the previous efficiency measures the best relative efficiency within the range of no more than one, the decision-making units (DMUs) who get the optimum value of one perform best among all the DMUs. If the previous efficiency is measured within the range of no less than one, the DMUs who get the optimum value of one perform worst among all the DMUs. When the later efficiency is measured within the range of no more than one, the DMUs who get the optimum value of one perform worst among all the DMUs. If the later efficiency is measured within the range of no less than one, the DMUs who get the optimum value of one perform best among all the DMUs. This paper mainly studies an interval DEA model with later efficiency, in which efficiency is measured within the range of an interval, whose upper bound is set to one and the lower bound is determined by introducing a virtual ideal DMU, whose performance is definitely superior to any DMUs. The efficiencies, obtained from interval DEA model, turn out to be all intervals and are referred to as interval efficiencies, which combine the best and the worst relative efficiency in a reasonable manner to give an overall assessment of performances for all DMUs. Assessor's preference information on input and output weights is also incorporated into interval DEA model reasonably and conveniently. Through an example, some differences are found from the ranking results obtained from interval DEA model and bounded DEA model using the Hurwicz criterion approach to rank the interval efficiencies.     

Approaches to determining the relative importance weights for cross-efficiency aggregation in data envelopment analysis

Cross-efficiency evaluation has been widely used for identifying the most efficient decision making unit (DMU) or ranking DMUs in data envelopment analysis (DEA). Most existing approaches for cross-efficiency evaluation are focused on how to determine input and output weights uniquely, but pay little attention to the aggregation process of cross-efficiencies and simply aggregate them equally without considering their relative importance. This paper focuses on aggregating cross-efficiencies by taking into consideration their relative importance and proposes three alternative approaches to determining the relative importance weights for cross-efficiency aggregation. Numerical examples are examined to show the importance and necessity of the use of relative importance weights for cross-efficiency aggregation and the most efficient DMU can be significantly affected by taking into consideration the relative importance weights of cross-efficiencies.     

DEA Cobb–Douglas frontier and cross-efficiency

The current paper examines the cross-efficiency concept in data envelopment analysis (DEA). While cross-efficiency has appeal as a peer evaluation approach, it is often the subject of criticism, due mainly to the use of DEA weights that are often non-unique. As a result, cross-efficiency scores are routinely viewed as arbitrary in that they depend on a particular set of optimal DEA weights generated by the computer code in use at the time. While imposing secondary goals can reduce the variability of cross-efficiency scores, such approaches do not completely solve the problem of non-uniqueness, and meaningful secondary goals can lead to computationally intractable non-linear programs. The current paper proposes to use the units-invariant multiplicative DEA model to calculate the cross-efficiency scores. This allows one to calculate the maximum cross-efficiency score for each DMU in a converted linear model, and eliminates the need for imposing secondary goals.     

Slack-based ranking method: an interpretation to the cross-efficiency method in DEA

The cross-efficiency method is generally utilized to rank decision-making units (DMUs) in data envelopment analysis (DEA) based on peer-evaluation logic. This brief note provides a method of using the available information from the linear program outputs to calculate the ranking of all DMUs with fewer computations and offers an alternative interpretation to the cross-efficiency method based on slack analysis in DEA.     

Cross-ranking of Decision Making Units in Data Envelopment Analysis

Data Envelopment Analysis (DEA) is a mathematical model that evaluates the relative efficiency of Decision Making Units (DMUs) with multiple input and output. In some applications of DEA, ranking of the DMUs are important. For this purpose, a number of approaches have been introduced. Among them is the cross-efficiency method. The method utilizes the result of the cross-efficiency matrix and averages the cross-efficiency scores of each DMU. Ranking is then performed based on the average efficiency scores. In this paper, we proposed a new way of handling the information from the cross-efficiency matrix. Based on the notion that the ranking order is more important than individual efficiency score, the cross-efficiency matrix is converted to a cross-ranking matrix. A cross-ranking matrix is basically a cross-efficiency matrix with the efficiency score of each element being replaced with the ranking order of that efficiency score with respect to the other efficiency scores in a column. By so doing, each DMU assume the role of a decision maker and how they voted or ranked the other DMUs are reflected in their respective column of the cross-ranking matrix. These votes are then aggregated using a preference aggregation method to determine the overall ranking of the DMUs. Comparison with an existing cross-efficiency method indicates a relatively better result through usage of the proposed method.     

The interval efficiency based on the optimistic and pessimistic points of view

Data envelopment analysis (DEA) is a data-oriented approach for evaluating the performances of a set of peer entities called decision-making units (DMUs), whose performance is determined based on multiple measures. The traditional DEA, which is based on the concept of efficiency frontier (output frontier), determines the best efficiency score that can be assigned to each DMU. Based on these scores, DMUs are classified into DEA-efficient (optimistic efficient) or DEA-non-efficient (optimistic non-efficient) units, and the DEA-efficient DMUs determine the efficiency frontier. There is a comparable approach which uses the concept of inefficiency frontier (input frontier) for determining the worst relative efficiency score that can be assigned to each DMU. DMUs on the inefficiency frontier are specified as DEA-inefficient or pessimistic inefficient, and those that do not lie on the inefficient frontier, are declared to be DEA-non-inefficient or pessimistic non-inefficient. In this paper, we argue that both relative efficiencies should be considered simultaneously, and any approach that considers only one of them will be biased. For measuring the overall performance of the DMUs, we propose to integrate both efficiencies in the form of an interval, and we call the proposed DEA models for efficiency measurement the bounded DEA models. In this way, the efficiency interval provides the decision maker with all the possible values of efficiency, which reflect various perspectives. A numerical example is presented to illustrate the application of the proposed DEA models.     

基于参数设计的信噪比交叉效率评价方法

针对DEA交叉效率评价过程中没有考虑自评与互评效率的作用而主观赋予相同权重导致交叉效率评价值不准确的问题.文章基于参数设计的思想,依据试验设计中可控与不可控因素的作用机理区分自评权重和互评权重对所评价决策单元交叉效率的影响与作用,将其界定为可控与不可控因素的管理学属性,明确不同权重作用机理;引入信噪比作为衡量决策单元交叉效率评价时的性能指标,实施DEA交叉效率评价方法的改进,设计出DEA信噪比交叉效率集结方法,从而实现交叉效率的集结方式由单一考虑交叉效率波动的均值转化为综合考虑交叉效率波动情况(均值与方差),交叉效率评价值用信噪比交叉效率替代交叉效率平均值更具有统计学意义并可从管理学角度解释,评价结果也具有更高的可区分性;最后通过算例分析验证了交叉效率评价理论上的必要性和该方法的合理性与可行性,同时发现了交叉效率评价中存在CCR有效DMU序位超出了有效DMU范围现象,建议应实施同质DMU检验和评价值归一化.文章的研究也为提高DEA交叉效率测算的准确性提供一种新思路.