Cross-efficiency Matrix Research Articles

Ranking ranges, performance maverick and diversity for decision-making units with interval cross-efficiency matrix

ABSTRACTAn alternative strategy in cross-efficiency evaluation is to take into consideration all the possible choices of data envelopment analysis weights that all decision-making units (DMUs) can make, thus eliminate the non-uniqueness issue of cross-efficiency scores, and also avoid the need of the decision maker’s preference in choosing aggressive or benevolent strategy among various alternative secondary goals. We are interested in this alternative strategy and continue investigating it. With the derived interval cross-efficiency matrix for all the DMUs, we determine cross-efficiency intervals, ranking ranges, performance maverick and diversity for each DMU, which could provide more insightful information in cross-efficiency evaluation for decision makers. Our developed cross-efficiency interval and ranking ranges are useful for assessing the stability of the cross-efficiency scores. Besides, our developed interval performance maverick and diversity extend previous study to consider all possible weights space, which make results more acceptable for supplier rationalisation. Two examples of data on 18 Chinese cities and 23 suppliers are presented for illustrations.

A novel three-stage distance-based consensus ranking method

In this study, we propose a three-stage weighted sum method for identifying the group ranks of alternatives. In the first stage, a rank matrix, similar to the cross-efficiency matrix, is obtained by computing the individual rank position of each alternative based on importance weights. In the second stage, a secondary goal is defined to limit the vector of weights since the vector of weights obtained in the first stage is not unique. Finally, in the third stage, the group rank position of alternatives is obtained based on a distance of individual rank positions. The third stage determines a consensus solution for the group so that the ranks obtained have a minimum distance from the ranks acquired by each alternative in the previous stage. A numerical example is presented to demonstrate the applicability and exhibit the efficacy of the proposed method and algorithms.

Integrated approach for computing aggregation weights in cross-efficiency evaluation

Cross-efficiency evaluation is an extension of data envelopment analysis that has been widely used in many different applications aimed at producing a ranking of the set of decision making units. Besides the traditional self-appraisal of units, cross-evaluation methods also take into account peer-appraisals, which are then summarized into an overall performance measure. The standard approach for this aggregation process relies on an equally-weighted average that disregards that some cross-efficiency scores might be considered more relevant or reliable than others. This paper focuses on the aggregation process of cross-efficiency scores and proposes a new approach for deriving meaningful aggregation weights for a more comprehensive evaluation of the units. Our method integrates two complementary perspectives that weights should reflect: the discriminatory ability of the information contained in the cross-efficiency matrix and the relative importance that can be attributed to each of the peer-appraisals. In this sense, the approach presented here provides a more accurate evaluation of the units than previous approaches and therefore it is likely to produce more meaningful rankings. Some numerical examples are provided that validate the approach proposed and examine the results obtained in comparison with previous known methods.

Hybrid cluster analyzing and data envelopment analysis with interval data

Data envelope analysis (DEA) is an approach to estimate the relative efficiency of decision making units (DMUs). Several studies were conducted in order to prioritize efficient units and some useful models such as cross-efficiency matrix (CEM) were presented. Besides, a number of DEA models with interval data have been developed and ranking DMUs with such data was solved. However, presenting an obtained crisp data derived interval data is a critical problem, so that many researches were implemented so as to compute weights and averaging the interval data. In this paper we propose the new algorithm to find more suitable weight applying a data mining approach of DMU’s data. For this purpose, we employed clustering and pair-wise comparison matrix on given relative efficiency from CEM. Results indicate there is meaningful different between efficiency of DMUs with lower bound and that of DMUs with upper bound.

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.

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.

Cross Efficiency Matrix in Non-Radial Models

Abstract. In data envelopment analysis each decision making unit is analyzed in best situation and in CCR models for the unit under analysis the best weights are considered for inputs and outputs. So, if the decision making unit in its best situation is less than one, it is inefficient, otherwise it is efficient. Therefore, the present article is an attempt to achieve the cross efficiency matrix by using SBM and then rank the decision making units by using the cross efficiency matrix. Overall, selecting the optimal portfolio is done by cross efficiency matrix. In order to find the efficient firms the mean, column variance and covariance of data of cross efficiency scale are utilized. Finally, the cross efficiency matrix of 22 firms with two inputs and one output has been calculated using non-radial models.

Selecting the Best of Portfolio Using OWA Operator Weights in Cross Efficiency-Evaluation

The present study is an attempt toward evaluating the performance of portfolios and asset selection using cross-efficiency evaluation. Cross-efficiency evaluation is an effective way of ranking decision making units (DMUs) in data envelopment analysis (DEA). The most widely used approach is to evaluate the efficiencies in each row or column in the cross-efficiency matrix with equal weights into an average cross-efficiency score for each DMU and consider it as the overall performance measurement of the DMU. This paper focuses on the evaluation process of the efficiencies in the cross-efficiency matrix and proposes the use of ordered weighted averaging (OWA) operator weights for cross-efficiency evaluation. The OWA operator weights are generated by the minimax disparity approach and allow the decision maker (DM) or investor to select the best assets that are characterized by an orness degree. The problem consists of choosing an optimal set of assets in order to minimize the risk and maximize return. This method is illustrated by application in mutual funds and weights are obtained via OWA operator for making the best portfolio. The finding could be used for constructing the best portfolio in stock companies, in various finance organization, and public and private sector companies.

Cross-Efficiency Aggregation by OWA Operator Weights with Fuzzy Data

Cross-efficiency evaluation is an effective way of ranking decision-making units (DMUs) in data envelopment analysis (DEA) and can be performed with different formulations (aggressive or benevolent), secondary goals and models. In this paper we use neutral formulation for cross-efficiency aggregation. The neutral formulation determines one set of input and output weights for each DMU from its own point of view without being aggressive or benevolent to the other DMUs. Existing approaches for cross-efficiency evaluation are mainly focused on the calculation of cross-efficiency matrix, but pay little attention to the aggregation of the efficiencies in the cross-efficiency matrix. This paper focuses on the use of ordered weighted averaging (OWA) operator weights for cross-efficiency aggregation. The use of OWA operator weights allows the decision maker (DM)’s optimism level towards the best relative efficiencies. But in real world, we are often conformed to ambiguous and uncertain data. So, there is an undeniable need for fuzzy logic to evaluate the efficiency unit.

Optimising proportional weights as a secondary goal in DEA cross-efficiency evaluation

Cross-efficiency evaluation is an alternative approach to rank of decision making units (DMUs) in data envelopment analysis (DEA). This approach introduces a cross-efficiency matrix, in which the units are self and peer evaluated. However, the cross-efficiency scores may not be unique due to the presence of alternate optima. To overcome this shortcoming, the secondary goals have been introduced in the cross-efficiency evaluation. In this paper, we propose the proportional weight assignment technique that makes a selection of weight proportional to its corresponding input or output as a secondary goal in DEA cross-efficiency evaluation. A numerical example is solved by use of the proposed method and its results are compared with other approaches. The results show that our proposed method can be effective and practical.

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.

A novel fuzzy data envelopment analysis model with double frontiers for supplier selection

Selecting the best suppliers is a significant yet complex decision which requires careful consideration of various performance criteria. Hence, supplier selection is an important issue and has a strategic significance for every company. One of the techniques that can be used for selecting suppliers is data envelopment analysis (DEA). In this paper, to handle ambiguity and fuzziness in supplier selection problems, a novel fuzzy DEA approach with double frontiers for selecting suppliers is developed. Compared with the traditional DEA, the DEA approach with double frontiers can identify the best supplier appropriately and easily without the need to impose any weight restriction or the need to calculate the cross-efficiency matrix, which requires a large number of computations and may also result in inconsistent conclusions. A numerical example demonstrates the application of the proposed model.

A Review of Ranking Models in Data Envelopment Analysis

In the course of improving various abilities of data envelopment analysis (DEA) models, many investigations have been carried out for ranking decision-making units (DMUs). This is an important issue both in theory and practice. There exist a variety of papers which apply different ranking methods to a real data set. Here the ranking methods are divided into seven groups. As each of the existing methods can be viewed from different aspects, it is possible that somewhat these groups have an overlapping with the others. The first group conducts the evaluation by a cross-efficiency matrix where the units are self- and peer-evaluated. In the second one, the ranking units are based on the optimal weights obtained from multiplier model of DEA technique. In the third group, super-efficiency methods are dealt with which are based on the idea of excluding the unit under evaluation and analyzing the changes of frontier. The fourth group involves methods based on benchmarking, which adopts the idea of being a useful target for the inefficient units. The fourth group uses the multivariate statistical techniques, usually applied after conducting the DEA classification. The fifth research area ranks inefficient units through proportional measures of inefficiency. The sixth approach involves multiple-criteria decision methodologies with the DEA technique. In the last group, some different methods of ranking units are mentioned.

DEA cross-efficiency aggregation method based upon Shannon entropy

Existing approaches for DEA cross-efficiency evaluation are mainly focused on the calculation of cross-efficiency matrix but pay little attention to the aggregation of the efficiencies in the cross-efficiency matrix. The most widely used approach is to aggregate the efficiencies in each row or column in the cross-efficiency matrix with equal weights into an average cross-efficiency score for each Decision Making Unit (DMU) and view it as the overall performance measurement of the DMU. This paper focuses on the aggregation process of the efficiencies in the cross-efficiency matrix and proposes the use of Shannon entropy for cross-efficiency aggregation. In the study, we propose an entropy model to generate a set of weights for aggregating and determining the ultimate cross-efficiency instead of the traditional average cross-efficiency. We prove that the set of weight is a unique global optimal solution which can reflect the goodness of this method. Finally, two examples of a flexible manufacturing system and 27 industrial robots are illustrated to examine the validity of the proposed method.

Ranking DMUs by using interval DEA cross efficiency matrix with acceptability analysis

Cross efficiency method is an extension of data envelopment analysis (DEA), and has been widely used for ranking performance of decision making units (DMUs). To eliminate the non-uniqueness of cross efficiency scores, the aggressive and benevolent strategies have been proposed as secondary goals to determine the unique cross efficiency score. The current paper aims to propose an alternative strategy which does not consider the preference of the decision maker in choosing aggressive or benevolent strategy. Instead, the paper considers all possible weight sets in weight space when computing the cross efficiency and each DMU is given an interval cross efficiency. By using the stochastic multicriteria acceptability analysis (SMAA-2) method, all DMUs in the interval cross efficiency matrix (CEM) could be fully ranked according to the acceptability indices. A numerical example about efficiency evaluation to seven academic departments in a university is illustrated.

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.

Determination of Weights for the Ultimate Cross Efficiency: A Use of Principal Component Analysis Technique

Data Envelopment Analysis (DEA) has becoming more and more important in evaluating the performance of homogenous Decision Making Units (DMUs). Cross efficiency evaluation method, a DEA extension technique, can be utilized to identify efficient DMUs and to rank DMUs in a peer appraisal mode, instead of a pure self-evaluation of traditional DEA models. Traditionally, the ultimate cross efficiency is determined based on the average assumption. However it cannot ensure this result contains the most information of the cross-efficiency matrix (CEM). In the current paper, we use principal component analysis (PCA) to determine the ultimate cross-efficiency of each DMU and then rank them. Compared with the tradition average cross efficiency evaluation method, the method proposed in this paper can contain the most of the information of CEM. Finally, an empirical example is illustrated to examine the validity of the proposed method.

Advanced X-efficiencies for CCR- and BCC-models – towards Peer-based DEA controlling

Classical CCR and BCC DEA-models follow a general concept: they allow each DMU to evaluate its (in-) efficiency in the most favorable way, and then propose input reduction and/or output raise so as to follow its best practice units. A first step beyond this ‘self-appraisal’ is the consideration of X-efficiencies thus evaluating DMUs with optimal weights of a peer. Doing this for all possible peers yields a cross-efficiency matrix, either for CCR or for BCC models. This matrix might help to find a fair peer for the remaining DMUs. In a second step recent contributions analyze for CCR-models how such X-evaluated DMUs might improve their efficiency with respect to a peer’s weight system. In these models even free variation of inputs/outputs is possible rather than reduction and/or raise. Such models will be portrayed here and generalized for variable returns to scale. The remaining discomfort which a DMU might feel with the choice for peer among business rivals, leads to the concept of a ‘virtual peer’ VP. This paper proposes such a peer as a consensual option for all DMUs. Now for either return to scale – CCR and BCC – for an input or output oriented focus and by free variation of inputs and outputs they can meet the requirements of VP. The DMUs pay a heavy price, however: the peer controls their respective weights and even their activities; he is a dictator.

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.

The use of OWA operator weights for cross-efficiency aggregation

Cross-efficiency evaluation is an effective way of ranking decision making units (DMUs) in data envelopment analysis (DEA). Existing approaches for cross-efficiency evaluation are mainly focused on the calculation of cross-efficiency matrix, but pay little attention to the aggregation of the efficiencies in the cross-efficiency matrix. The most widely used approach is to aggregate the efficiencies in each row or column in the cross-efficiency matrix with equal weights into an average cross-efficiency score for each DMU and view it as the overall performance measurement of the DMU. This paper focuses on the aggregation process of the efficiencies in the cross-efficiency matrix and proposes the use of ordered weighted averaging (OWA) operator weights for cross-efficiency aggregation. The use of OWA operator weights for cross-efficiency aggregation allows the decision maker (DM)’s optimism level towards the best relative efficiencies, characterized by an orness degree, to be taken into consideration in the final overall efficiency assessment and particularly in the selection of the best DMU.