Cross-efficiency Evaluation Research Articles

DEA cross-efficiency evaluation and ranking method based on interval data

Data envelopment analysis (DEA) is an important method of efficiency evaluation. Cross-efficiency evaluation is one of the main aspects of research in the field of DEA that has been applied in various fields. In the traditional cross-efficiency evaluation model, the variable data of decision-making units is exact. Dynamic information is frequently unable to reflect the whole characteristic when determining the exact data. In this study, we select interval data to represent the dynamic information of some variables in the evaluation process. We then build a solution method based on interval efficiency and DEA cross-efficiency. This method retains the reflection of interval data on uncertain variable properties. Finally, the stochastic multi-criteria acceptability analysis 2 (SMAA2) is introduced to solve the whole sequence problem of interval efficiency. We present a case study from a set of 19 reservoir dams suffered from Wenchuan earthquake in Luojiang County, Sichuan Province to demonstrate the applicability of the proposed model.

Pareto approach for DEA cross efficiency evaluation based on interval programming

Pareto approach for DEA cross efficiency evaluation based on interval programming

Improving the evaluation of cross efficiencies: A method based on Shannon entropy weight

Data envelopment analysis (DEA) is a non-parametric statistical method used to assess the production frontiers of decision-making units (DMUs) and evaluate their relative efficiencies. However, using traditional DEA models to evaluate efficiency has certain deficiencies. For example, some DMUs cannot be ranked fully using traditional DEA models. To solve such problems, the cross-efficiency evaluation method has been proposed to replace the self-evaluation system. Nevertheless, this method, which uses a mutual evaluation system to overcome the ranking issue, still has shortcomings such as non-unique cross efficiency weights, which may result in multiple cross efficiency values. Further, providing adequate performance improvement tools to decision makers is difficult using only the average efficiency values. To address the problems of uniqueness and aggregation, this study proposes two cross efficiency models, designated MAX and MIN models. The self-evaluated optimal weight of a certain DMU derived from these MAX and MIN models can maximize or minimize the efficiency of the DMU to form two cross efficiency matrices, which can partially solve the problem that results from multiple optimal weights. To solve the aggregation problem of cross efficiency, the study also applies Shannon entropy, which classifies all cross efficiency values into one group of acquired common objective weights to avoid subjective factors. Finally, the present study confirms an improvement when using the proposed method by examining production data on 15 thermoelectric enterprises in China.

A DEA ranking method based on cross-efficiency intervals and signal-to-noise ratio

Cross-efficiency evaluation is a useful approach to ranking decision making units (DMUs) in data envelopment analysis (DEA). The possible existence of multiple optimal weights for the DEA may reduce the usefulness of the cross-efficiency evaluation since the ranking is according to the choice of weights that different DMUs make. Most of existing approaches for cross-efficiency evaluation employ the average cross-efficiency to further discriminate among the DEA efficient units or focus on how to determine input and output weights uniquely, but lay little emphasis on the consideration of the ranges and variances of cross-efficiencies as alternative ranking factors. In this paper we consider cross-efficiency intervals and their variances for ranking DMUs. The aggressive and benevolent formulations are taken into account at the same time. Consequently, a number of cross-efficiency intervals is obtained for each DMU. The signal-to-noise (SN) ratio, originally designed for optimizing the robustness of a process, is constructed as a numerical index for ranking DMUs. A nonlinear fractional program with bound constraints is formulated to find the optimal value of the SN ratio. By model reduction and variable substitution, this nonlinear fractional program is transformed into a quadratic one for deriving the global optimum solution. With the derived SN ratios, we are able to fully rank all DMUs accordingly. Two examples are given to illustrate the effectiveness of the methodology proposed in this paper.

An aggressive game cross-efficiency evaluation in data envelopment analysis

Cross-efficiency evaluation is an effective method for ranking decision making units (DMUs) in data envelopment analysis, which is performed with peer-evaluation and self-evaluation. From different points of view, various cross-efficiency evaluations have been proposed with different secondary goals. Yet they usually lead to different average cross-efficiencies and different rankings. In this paper, we develop a concept of the aggressive game cross-efficiency, and propose an aggressive secondary model to minimize the cross-efficiencies of other DMUs under the constraints that the aggressive game cross-efficiency of the evaluated DMU is guaranteed. To achieve the aggressive game cross-efficiency, we develop an iterative algorithm. Mathematically, it is proved that all conventional average cross-efficiencies are sure to converge to the same aggressive game cross-efficiency by the iterative algorithm. Finally, numerical examples are presented to show the effectiveness of our approach in evaluating and ranking DMUs.

Group division based on common weights in cross efficiency evaluation

In the traditional cross efficiency procedure, each decision making unit (DMU) offers a set of selfish weights to maximise its own efficiency. However, it is often not easy to be accepted by all other decision making units (DMUs). Since the cross efficiency of a DMU not only has a significant relevance with its own evaluation but also with other peer DMUs' measurement, a DMU may choose to cooperate or compete with some other DMUs to obtain a higher cross efficiency. Secondary-goal strategies, such as benevolent and aggressive strategies, have been applied in groups in which the relationship between each DMU is determined. With the benevolent strategy, all of the DMUs are under the situations of cooperation while they will compete with each other with the aggressive strategy. Obviously in reality, there is a situation in which alliances are often uncertain. Pure aggressive or benevolent strategy may be not suitable for the case. To solve this problem, the current paper proposes a new cross efficiency approach based on a series of common weights of each two DMUs. An example is given to illustrate our method.

Group division based on common weights in cross efficiency evaluation

Group division based on common weights in cross efficiency evaluation

An approach to two-sided M&A fits based on a cross-efficiency evaluation with contrasting attitudes

Two-sided mergers and acquisitions (M&A) fits have been regarded as a critical step, which should always be taken by a bidder company when trying to identify suitable target companies prior to an M&A. This paper proposes an approach to two-sided M&A fits based on a cross-efficiency model with contrasting attitudes. In this approach, firstly, feasible M&A fits are screened using a preference function from an M&A fit matrix, according to the preferences of both the bidder companies and target companies in terms of efficiency and return to scale. Secondly, two-sided M&A fits are selected from a feasible M&A fit matrix, according to the value of cross-efficiency with contrasting attitudes. This allows for the existence of contrasting attitudes of peers toward an M&A fit, as opposed to the aggressive, benevolent or neutral cross-efficiency evaluation which consists of just one attitude (either aggressive, benevolent or indifferent). Finally, an illustrative example is given to explain the feasibility and validity of the two-sided M&A fit strategy.

A Revised Model of the Neutral DEA Model and Its Extension

The neutral data envelopment analysis (DEA) model is an alternative way to determine the weights in DEA cross-efficiency evaluation, while avoiding the difficulty in making a choice between the aggressive and benevolent formulations. However, the weights determined by the neutral model merely make the efficiency of part output bigger than other sets of weights. The neutral model is not able to make the efficiency of each output of the DMU biggest among the favorable weights. This neutral model is not purely “neutral” and not most favorable to the DMU. We proposed a revised model for the neutral model. Based on the idea that the DMU should choose a set of weights to maximize its own efficiency, this paper proposes a new cross-efficiency model. The weights determined by the two models are neutral, neither aggressive nor benevolent.

Ranking DMUs by Comparing DEA Cross-Efficiency Intervals Using Entropy Measures

Cross-efficiency evaluation, an extension of data envelopment analysis (DEA), can eliminate unrealistic weighing schemes and provide a ranking for decision making units (DMUs). In the literature, the determination of input and output weights uniquely receives more attentions. However, the problem of choosing the aggressive (minimal) or benevolent (maximal) formulation for decision-making might still remain. In this paper, we develop a procedure to perform cross-efficiency evaluation without the need to make any specific choice of DEA weights. The proposed procedure takes into account the aggressive and benevolent formulations at the same time, and the choice of DEA weights can then be avoided. Consequently, a number of cross-efficiency intervals is obtained for each DMU. The entropy, which is based on information theory, is an effective tool to measure the uncertainty. We then utilize the entropy to construct a numerical index for DMUs with cross-efficiency intervals. A mathematical program is proposed to find the optimal entropy values of DMUs for comparison. With the derived entropy value, we can rank DMUs accordingly. Two examples are illustrated to show the effectiveness of the idea proposed in this paper.

A flexible cross-efficiency fuzzy data envelopment analysis model for sustainable sourcing

Sustainable sourcing is a recent priority for firms considering customer behavior and societal norms with respect to the supply chain. Customer attitudes, particularly in the developed countries, are affected by the perceived sustainability of products or services regarding environmental, social and economic aspects. Seeking to maximize their market shares, firms frequently require an effective sourcing approach in supply chain management (SCM) by selecting sustainable suppliers (sourcing) and by enforcing standards through continuous supplier evaluations (monitoring) as well as by contract adjustments (retention). Most existing sourcing methodologies are either cost-oriented or ad hoc, without the tools and techniques necessary to deal with sustainability. In this paper, we propose a product-based framework for sustainable supplier sourcing considering different sustainability, operational and organizational criteria based on the type of outsourced products in the evaluation process. We develop a flexible cross-efficiency evaluation methodology based on data envelopment analysis (DEA) for identifying supplier performance. This research also uses fuzzy set theory to tackle the vagueness of information that is often present in the information-gathering step. We present a case study from the semiconductor industry to demonstrate the applicability of the proposed model and the efficacy of the procedures and algorithms.

An iterative method for determining weights in cross efficiency evaluation

Cross efficiency evaluation in data envelopment analysis (DEA) is a commonly used skill for ranking decision making units (DMUs). However, the presence of multiple optimal weights in traditional DEA models leads to different cross efficiency scores for a DMU, which seriously affects the validity of this skill for ranking DMUs. Although many secondary-goal techniques have been proposed to address this issue, sometimes the weight sets determined by these techniques are still not unique. On the other hand, since zero weights make much information on inputs and outputs be ignored in cross efficiency evaluation, the number of them should be reduced as many as possible. The current approaches have not fully solved the above two problems. This paper proposes an iterative method for determining weights in cross efficiency evaluation, which not only ensures a unique weight set for positive input and output data but also reduces the number of zero weights maximally without imposing any prior weight restriction. Numerical examples are used to show the validity and superiorities of the proposed method in choosing a unique weight set and reducing the number of zero weights.

DEA cross-efficiency evaluation considering undesirable output and ranking priority: a case study of eco-efficiency analysis of coal-fired power plants

Data envelopment analysis cross-efficiency evaluation has been widely accepted as a useful tool for performance evaluation and ranking of decision making units. However, the non-uniqueness of optimal weights is a problem that has reduced the usefulness of this powerful method. In addition, current studies generally have not considered the situation in which undesirable outputs appear in the data envelopment analysis cross-efficiency evaluation. To solve these problems, firstly, we present an equitable model for efficiency evaluation of decision-making units with undesirable outputs and introduce a technique for cross-efficiency evaluation considering undesirable outputs. Then, a ranking priority model is proposed considering the decision making units' intentions of pursuing the best ranking positions. In addition, an aggressive model is given to guarantee the uniqueness of the optimal solution. The proposed approach can not only solve the problem of non-uniqueness of optimal weights in data envelopment analysis cross-efficiency evaluation but also considers undesirable outputs and ranking preferences of the decision-making units. Finally, the proposed approach is applied for eco-efficiency analysis of coal-fired power plants in a big data environment, and the results show that most coal-fired power plants in China have not performed well.

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.

DEA cross-efficiency evaluation based on satisfaction degree: an application to technology selection

Data envelopment analysis (DEA) has been extended to cross-efficiency evaluation to provide better discrimination and ranking of decision-making units (DMUs). However, the non-uniqueness of optimal weights in the traditional DEA models (CCR and BCC models) has reduced the usefulness of the DEA cross-efficiency evaluation method. To solve this problem, we introduce the concept of the satisfaction degree of a DMU towards a set of optimal weights for another DMU. Then, a new DEA cross-efficiency evaluation approach, which contains a maxmin model and two algorithms, is proposed based on the satisfaction degrees of the DMUs. Our maxmin model and algorithm 1 can obtain for each DMU an optimal set of weights that maximises the least satisfaction degrees among all the other DMUs. Further, our algorithm 2 can then be used to guarantee the uniqueness of the optimal weights for each DMU. Finally, our approach is applied to a real-world case study of technology selection.

An improved cross-ranking method in data envelopment analysis

ABSTRACTCross-efficiency evaluation is an extension of data envelopment analysis methodology for fully ranking decision-making units (DMUs). Based on the idea that ranking order is much more meaningful than the individual efficiency score in several circumstances, a cross-ranking matrix was recently introduced. The matrix was built by replacing the efficiency score in the conventional cross-efficiency matrix with the ranking order of that efficiency score in each row. However, the non-uniqueness issue in the cross-efficiency scores in the cross-efficiency evaluation may result in different ranking orders in the matrix and may thus limit the usefulness of cross-ranking method. This study improves the cross-ranking method by introducing an interval cross-efficiency matrix in cross-efficiency evaluation. The interval cross-efficiency matrix can reveal all possible cross-efficiency scores through self-evaluation and peer evaluation. Therefore, the derived cross-ranking matrix based on the interval cross-efficiency matrix can show all information on possible ranking orders for all DMUs. The improved method is illustrated by two numerical examples.

Extended secondary goal models for weights selection in DEA cross-efficiency evaluation

Data Envelopment Analysis (DEA) has been extended to cross-efficiency evaluation for ranking decision making units (DMUs) and eliminating unrealistic weighting schemes. Unfortunately, the non-unique optimal weights problem in DEA has reduced the usefulness of this extended method. Aiming at solving this problem, we first incorporate a target identification model to get reachable targets for all DMUs. Then, several secondary goal models are proposed for weights selection considering both desirable and undesirable cross-efficiency targets of all the DMUs. Compared with the traditional secondary goal models, the cross-efficiency targets are improved in that all targets are always reachable for the DMUs. In addition, the proposed models considered the DMUs’ willingness to get close to their desirable cross-efficiency targets and to avoid their undesirable cross-efficiency targets simultaneously while the traditional secondary goal models considered only the ideal targets of the DMUs. Finally, the calculation results of our proposed models are compared with those of some other traditional methods for two published examples: an efficiency evaluation of six nursing homes and an R&D project selection.

Ranking of DMUs with interval cross-efficiencies based on absolute dominance

This paper follows an alternative approach for ranking decision-making units (DMUs) based upon cross-efficiency evaluation in data envelopment analysis (DEA). The approach determines the DMUs' peer-evaluation efficiencies with interval numbers and derives an interval cross-efficiency matrix (CEM), avoiding the need to make a choice of weights according to a specific secondary goal among various alternatives for the decision makers. Absolute dominance is defined to deal with the DMUs with interval efficiencies in the interval CEM. The ranking for all the DMUs is based upon the overall absolute dominance values and any preference information of decision makers are not needed. A numerical example of five enterprise self-innovation projects selection is used for illustration. We also apply the method to performance ranking of 30 National Basketball Association (NBA) teams as a real application and the results show that the western conference has better performance and the southwest division has the best division performance in the 2012-13 regular season.

DEA cross-efficiency evaluation based on Pareto improvement

Cross-efficiency evaluation, as an extension tool of data envelopment analysis (DEA), has been widely applied in evaluating and ranking decision making units (DMUs). Unfortunately, the cross-efficiency scores generated may not be Pareto optimal, which has reduced the effectiveness of this method. To solve this problem, we propose a cross-efficiency evaluation approach based on Pareto improvement, which contains two models (Pareto optimality estimation model and cross-efficiency Pareto improvement model) and an algorithm. The Pareto optimality estimation model is used to estimate whether the given set of cross-efficiency scores are Pareto-optimal solutions. If these cross-efficiency scores are not Pareto optimal, the Pareto improvement model is then used to make cross-efficiency Pareto improvement for all the DMUs. In contrast to other cross-efficiency approaches, our approach always obtains a set of Pareto-optimal cross efficiencies under the predetermined weight selection principles for these DMUs. In addition, if the proposed algorithm terminates at its step 3, the evaluation results generated by our approach unify self-evaluation, peer-evaluation, and common-weight-evaluation in DEA cross-efficiency evaluation. Specifically, the self-evaluated efficiency and the peer-evaluated efficiency converge to the same common-weight-evaluated efficiency when the algorithm stops. This will make the evaluation results more likely to be accepted by all the DMUs.

A modification of a mixed integer linear programming (MILP) model to avoid the computational complexity

Having multiple optimal solutions to weights affects to a great extent the consistency of operations related to weights. The cross efficiency method is the most frequently studied topic in data envelopment analysis (DEA) literature. Originally, the cross efficiency method included the efficiency evaluations that were obtained for a decision making unit (DMU) by the classical DEA for the reuse of optimal weights in other DMUs. As the optimal weights in classical DEA solutions usually have multiple solutions, this reduces the usefulness of the cross evaluation. Lam (J Oper Res Soc 61:134–143, 2010) proposed a mixed-integer linear programming (MILP) formulation based on linear discriminant analysis and super efficiency method to choose suitable weight sets to be used in cross efficiency evaluation. In this study, Lam’s MILP model has been modified to reduce the steps during the solution process. The model also becomes a linear programming model after the modification to make it easier to use and to reduce the computational complexity. Numerical examples indicate that the proposed weight determination model both reduces the steps and minimizes computational complexity. Furthermore, it has similar performance with Lam’s MILP model for the cross efficiency evaluation.