DEA efficiency analysis: A DEA approach with double frontiers

Data envelopment analysis (DEA) is a method for measuring efficiency of peer decision-making units (DMUs). Conventional DEA evaluates the performance of each DMU using a set of most favourable weights. As a result, traditional DEA models can be considered methods for the analysis of the best relative efficiency or analysis of the optimistic efficiency. DEA efficient DMUs obtained from conventional DEA models create an efficient production frontier. Traditional DEA can be used to identify units with good performance in the most desirable scenarios. There is a similar approach that evaluates the performance indicators of each DMU using a set of most unfavourable weights. Accordingly, such models can be considered models for analysing the worst relative efficiency or pessimistic efficiency. This approach uses the inefficient production frontier for determining the worst relative efficiency that can be assigned to each DMU. DMUs lying on the inefficient production frontier are referred to as DEA inefficient while those neither on the efficient frontier nor on the inefficient frontier are declared DEA inefficient. It can be argued that both relative efficiencies should be considered simultaneously and any approach with only one of them would be biased. This paper proposed the integration of both efficiencies as an interval so that the overall performance score would belong to this interval. It was shown that efficiency interval provided more information than either of the two efficiencies, which was illustrated using two numerical examples.     

A full ranking methodology in data envelopment analysis based on a set of dummy decision making units

In this paper, we propose a new methodology for ranking decision making units in data envelopment analysis (DEA). Our approach is a benchmarking method, seeks a common set of weights using a proposed linear programming model and is based on the TOPSIS approach in multiple attribute decision making (MADM). To this end, five artificial or dummy decision making units (DMUs) are defined, the ideal DMU (IDMU), the anti-ideal DMU (ADMU), the right ideal DMU (RIDMU), the left anti-ideal DMU (LADMU) and the average DMU (AVDMU). We form two comprehensive indexes for the AVDMU called the Left Relative Closeness (LRC) and the Right Relative Closeness (RRC) with respect to the RIDMU and LADMU. The LRC and RRC indexes will be used in the new proposed linear programming model to estimate the common set of weights, the new efficiency of DMUs and finally an overall ranking for all the DMUs. The change of the ratio between LRC and RRC indexes is capable to be provoked alternative rankings. One of the best advantages of this model is that we can make a rationale ranking which is demonstrated by the realized correlation analysis. Also, the new proposed efficiency score of the DMUs is close to the efficiency score of the DEA (CCR) methodology. Three numerical examples are provided to illustrate the applicability of the new approach and the effectiveness of the new approach in DEA ranking in comparison with other conventional ranking methods. Also, an "error" analysis proves the robustness of the proposed methodology.     

Exploration of efficiency underestimation of CCR model: Based on medical sectors with DEA-R model

Data Envelopment Analysis (DEA) is one of the best-known efficiency evaluation methods due to its advantages in selection of weights. Many research papers have extensively discussed the issue of weight restrictions, rather than those implied in the model itself. However, this often leads to a failure to represent the relations of certain weights, as well as underestimation of the efficiency of Decision Making Units (DMUs). When analyzing the medical sectors of Taiwan with the developed models and CCR, it is found that efficiency underestimation by efficient DMUs is more serious than that of inefficient DMUs. In addition, underestimation occurs when weights are concentrated in the same output, however, every output of referenced DMU is the same times of corresponding output of targeted DMU.     

Measurement of overall performances of decision-making units using ideal and anti-ideal decision-making units

Data envelopment analysis (DEA) is a method for evaluating relative efficiencies of decision-making units (DMUs) which perform similar functions in a production system, consuming multiple inputs to produce multiple outputs. The conventional form of DEA evaluates performances of DMUs only from the optimistic point of view. In other words, it chooses the most favorable weights for each DMU. There is another approach that measures efficiency of a DMU from the pessimistic point of view. This approach chooses the most unfavorable weights for evaluation of each DMU. In this paper, we propose to integrate both efficiencies in the form of an interval in order to measure the overall performance of a DMU. The proposed DEA models for evaluation of efficiencies are called bounded DEA models. The proposed approach will be compared using a numerical example. Another example regarding performance evaluation of 50 bank branches in Iranian cities will be presented to demonstrate the advantages, simplicity, and utility of this approach in real-life situations.     

A new data envelopment analysis method for ranking decision making units: an application in industrial parks

Relative efficiency of decision‐making units (DMUs) is assessed by classical data envelopment analysis (DEA) models. DEA is a popular technique for efficiency evaluation. There might be a couple of efficient DMUs. Classical DEA models cannot fully rank efficient DMUs. In this paper, a novel technique for fully ranking all DMUs based on changing reference set using a single virtual inefficient DMU is proposed. To this end, the first concept of virtual DMU is defined as average of all inefficient DMUs. Virtual DMU is a proxy of all inefficient DMUs. This new method proposes a new ranking method that takes into account impact of efficient DMUs on virtual DMU and impact of efficient DMUs on influences of other efficient DMUs. A case study is given to show applicability of the proposed approach.     

Additive centralized and Stackelberg DEA models for two‐stage system with shared resources

Data envelopment analysis (DEA) is a nonparametric programming method for evaluating the efficiency performance of decision making units (DMUs) with multiple inputs and outputs. The classic DEA model cannot provide accurate efficiency measurement and inefficiency sources of DMUs with complex internal structure. The network DEA approach opens the “black box” of DMU by taking its internal operations into consideration. The complexities of DMU's internal structure involve not only the organization of substages, but also the inputs allocation and the operational relations among the individual stages. This paper proposes a set of additive DEA models to evaluate and decompose the efficiency of a two‐stage system with shared inputs and operating in cooperative and Stackelberg game situations. Under the assumptions of cooperative and noncooperative gaming, the proposed models are able to highlight the effects of strategic elements on the efficiency formation of DMUs by calculating the optimal proportion of the shared inputs allocated to each stage. The case of information technology in the banking industry at the firm level, as discussed by Wang, is revisited using the developed DEA approach.     

Common weights for fully ranking decision making units by regression analysis

Existing methods for generating common weights in data envelopment analysis (DEA) are either very complicated or unable to produce a full ranking for decision making units (DMUs). This paper proposes a new methodology based on regression analysis to seek a common set of weights that are easy to estimate and can produce a full ranking for DMUs. The DEA efficiencies obtained with the most favorable weights to each DMU are treated as the target efficiencies of DMUs and are best fitted with the efficiencies determined by common weights. Two new nonlinear regression models are constructed to optimally estimate the common weights. Four numerical examples are examined using the developed new models to test their discrimination power and illustrate their potential applications in fully ranking DMUs. Comparisons with a similar compromise approach for generating common weights are also discussed.     

Expansion of the cross-efficiency evaluation by using the equations forming the efficient frontier in DEA

In this paper, we propose an algorithm to calculate cross-efficiency scores which used the equations forming the efficient frontier in data envelopment analysis (DEA). In many standard DEA models, each decision-making unit (DMU) is evaluated by using the advantageous weight for itself. Then, many DMUs are evaluated as efficient, and those efficient DMUs are not ranked by the models. The cross-efficiency evaluation is a method to rank DMUs by using the advantageous weights for all DMUs. Previously, the cross-efficiency scores based on different ideas are calculated by solving multiple linear or nonlinear programming problems. However, it is often hard to solve such a nonlinear programming problem. Therefore, by analysing the efficient frontier, we construct an algorithm to calculate alternative cross-efficiency scores.     

An ideal-seeking fuzzy data envelopment analysis framework

Data envelopment analysis (DEA) is a widely used mathematical programming approach for evaluating the relative efficiency of decision making units (DMUs) in organizations. Crisp input and output data are fundamentally indispensable in traditional DEA evaluation process. However, the input and output data in real-world problems are often imprecise or ambiguous. In this study, we present a four-phase fuzzy DEA framework based on the theory of displaced ideal. Two hypothetical DMUs called the ideal and nadir DMUs are constructed and used as reference points to evaluate a set of information technology (IT) investment strategies based on their Euclidean distance from these reference points. The best relative efficiency of the fuzzy ideal DMU and the worst relative efficiency of the fuzzy nadir DMU are determined and combined to rank the DMUs. A numerical example is presented to demonstrate the applicability of the proposed framework and exhibit the efficacy of the procedures and algorithms.     

Ranking units in DEA based on efficiency intervals and decision‐maker's preferences

Data Envelopment Analysis (DEA) uses the best favorable weight set for the inputs and outputs of each decision‐making unit (DMU) to obtain its best possible score. Hence, this score can be considered as an upper bound of the real efficiency score. If we also use the least favorable weight set of each DMU, then a lower bound of the efficiency score can also be obtained. So, instead of one score, we can find an interval that gives all possible values of the efficiency score for each DMU. The aim of this paper is to propose an approach for determining efficiency intervals and setting up a full ranking of DMUs based on these intervals. We incorporate explicitly the decision‐maker's preferences in two phases. The first phase is for obtaining efficiency intervals, by introducing some restrictions on the input and output weights. The second one is for ranking the intervals based on the combination of the lower and the upper bounds of the efficiency intervals. The developed formulations will be illustrated through some numerical examples.     

Slacks-based measures of efficiency in imprecise data envelopment analysis: An approach based on data envelopment analysis with double frontiers

Data envelopment analysis (DEA) is a mathematical approach for evaluating the efficiency of decision-making units (DMUs) that convert multiple inputs into multiple outputs. Traditional DEA models assume that all input and output data are known exactly. In many situations, however, some inputs and/or outputs take imprecise data. In this paper, we present optimistic and pessimistic perspectives for obtaining an efficiency evaluation for the DMU under consideration with imprecise data. Additionally, slacks-based measures of efficiency are used for direct assessment of efficiency in the presence of imprecise data with slack values. Finally, the geometric average of the two efficiency values is used to determine the DMU with the best performance. A ranking approach based on degree of preference is used for ranking the efficiency intervals of the DMUs. Two numerical examples are used to show the application of the proposed DEA approach.     

Evaluation method based on ranking in data envelopment analysis

Data envelopment analysis (DEA) has been developed as a method to evaluate efficiency of Decision Making Unit (DMU). In order to analyze DMU in detail, each DEA model is formulated as a mathematical programming problem utilizing the values of inputs and outputs of all DMUs as coefficients. Each DMU is evaluated by a different weight. Then, the efficiency score of each DMU is determined by using an advantageous weight for itself. In general, the efficiency score is obtained by selecting the most advantage weight. In some real cases, seeking the best ranking is sometimes more important than maximizing the efficiency score.In this paper, we propose a model called rank-based measure (RBM) to evaluate DMU from a different standpoint. We suggest a method to obtain a weight which gives the best ranking, and calculates a weight between maximizing the efficiency score and keeping the best ranking. In order to calculate an efficiency score and the best ranking, we repeatedly solve linear programming problems. Moreover, we apply RBM model to the cross efficiency evaluation. Furthermore, a numerical experiment is shown to compare the rankings and scores with traditional evaluations.     

一种考虑DMU间交叉竞争的博弈效率DEA评价方法

针对绩效评价过程中一般只考虑DMU与评价者之间的合作竞争而忽视DMU间的非合作竞争的博弈,引入交叉竞争的博弈理念,将评价问题界定为评价者与DMU间合作竞争与博弈、DMU间交叉竞争的博弈两大类;考虑到在交叉竞争的博弈情境下,DMU的指标值不再是固定不变,而是随之动态调整的特点,设计交叉竞争的博弈规则,并运用决策树法描述考虑交叉竞争博弈下的DEA评价与选择过程;变评价过程中效用值改变的途径由“基于权重的交换”转化为“基于交叉竞争博弈的指标值调整”,实施对DEA模型的改进,设计交叉竞争的博弈效率DEA评价方法,得出确定型、风险型和不确定型DEA方法的分类和交叉竞争的博弈效率评价过程;从经济学的博弈论和管理学的决策分析来解释DEA,实现更加直观的DMU评价过程和更符合客观实际的评价情景.最后通过算例验证所提出方法的可行性、有效性和保序性.     

Some properties of Pareto efficiency under the framework of data envelopment analysis

Data envelopment analysis (DEA) has been widely applied to measure the Pareto efficiency of multiple-input and multiple-output decision making units (DMUs). In this paper it is shown that under linear production frontiers DMU efficiency is a weighted arithmetic mean of the efficiencies of the outputs; whereas under loglinear production frontiers DMU efficiency is a weighted geometric mean of the output efficiencies. Furthermore, DMU efficiency can be decomposed with respect to input factors as well, and some results are derived. As a consequence, a modified DEA model is devised, whereby the efficiency of each output (or input) in addition to DMU efficiency is able to be measured in one linear programming solution.     

DEA nonradial efficiency based on vector properties

Classic data envelopment analysis (DEA) models determine the efficiency of productive units, called decision making units (DMUs). DEA uses as its methodology the equiproportional reduction of inputs or increase of outputs and the finding of a single target for each DMU. This target does not incorporate the preference of the decision maker. Later works propose obtaining alternative targets based on nonradial projections on the efficiency frontier that are obtained through nonproportional variations of inputs or outputs. However, the efficiencies are not calculated for these alternative targets. This impedes a comparison among the DMUs. Thus, diverse nonradial efficiency indexes have been proposed based on mathematical averages or weighted averages that do not consider the vectorial characteristics of the efficiency. In this work, we present a nonradial efficiency index based on the initial concept of efficiency associated with each alternative (nonradial) target obtained through a multiobjective model of an inefficient DMU.     

A procedure for large-scale DEA computations

Data envelopment analysis (DEA), a performance evaluation method, measures the relative efficiency of a particular decision making unit (DMU) against a peer group. Most popular DEA models can be solved using standard linear programming (LP) techniques and therefore, in theory, are considered as computationally easy. However, in practice, the computational load cannot be neglected for large-scale—in terms of number of DMUs—problems. This study proposes an accelerating procedure that properly identifies a few “similar” critical DMUs to compute DMU efficiency scores in a given set. Simulation results demonstrate that the proposed procedure is suitable for solving large-scale BCC problems when the percentage of efficient DMUs is high. The computational benefits of this procedure are significant especially when the number of inputs and outputs is small, which are most widely reported in the literature and practices.     

A learning performance evaluation with benchmarking concept for English writing courses

This paper adopts data envelopment analysis (DEA), a robust and reliable evaluation method widely applied in various fields to explore the key indicators contributing to the learning performance of English freshmen writing courses in a university of Taiwan from the academic year 2004 to 2006. The results of DEA model applied in learning performance change our original viewpoint and reveal that some decision-making units (DMUs) with higher actual values of inputs and outputs have lower efficiency because the relative efficiency of each DMU is measured by their distance to the efficiency frontier. DMUs may refer to different facet reference sets according to their actual values located in lower or higher ranges. In the managerial strategy of educational field, the paper can encourage inefficient DMUs to always compare themselves with efficient DMUs in their range and make improvement little by little. The results of DEA model can also give clear indicators and the percentage of which input and output items to improve. The paper also demonstrates that the benchmarking characteristics of the DEA model can automatically segment all the DMUs into different levels based on the indicators fed into the performance evaluation mechanism. The efficient DMUs on the frontier curve can be considered as the boundaries of the classification which are systematically defined by the DEA model according to the statistic distribution.     

A method of stepwise benchmarking for inefficient DMUs based on the proximity-based target selection

DEA is a useful nonparametric method of measuring the relative efficiency of a DMU and yielding a reference target for an inefficient DMU. However, it is very difficult for inefficient DMUs to be efficient by benchmarking a target DMU which has different input use. Identifying appropriate benchmarks based on the similarity of input endowment makes it easier for an inefficient DMU to imitate its target DMUs. But it is rare to find out a target DMU, which is both the most efficient and similar in input endowments, in real situation. Therefore, it is necessary to provide an optimal path to the most efficient DMU on the frontier through several times of a proximity-based target selection process. We propose a dynamic method of stepwise benchmarking for inefficient DMUs to improve their efficiency gradually.The empirical study is conducted to compare the performance between the proposed method and the prior methods with a dataset collected from Canadian Bank branches. The comparison result shows that the proposed method is very practical to obtain a gradual improvement for inefficient DMUs while it assures to reach frontier eventually.     

Determination of the weights for the ultimate cross efficiency using Shapley value in cooperative game

This paper firstly revisits the cross efficiency evaluation method which is an extension tool of data envelopment analysis (DEA), then analyzes the potential flaws which happens when the ultimate average cross efficiency scores are used. In this paper, we consider the DMUs as the players in a cooperative game, where the characteristic function values of coalitions are defined to compute the Shapley value of each DMU, and the common weights associate with the imputation of the Shapley values are used to determine the ultimate cross efficiency scores. In the end, an empirical example is illustrated to examine the validity of the proposed method, and we also point out some further research directions in future.     

Negative data in DEA: Recognizing congestion and specifying the least and the most congested decision making units

One of the important concepts of data envelopment analysis (DEA) is congestion. A decision making unit (DMU) has congestion if an increase (decrease) in one or more input(s) of the DMU leads to a decrease (increase) in one or more its output(s). The drawback of all existing congestion DEA approaches is that they are applicable only to technologies specified by non-negative data, whereas in the real world, it may exist negative data, too. Moreover, specifying the strongly and weakly most congested DMUs is a very important issue for decision makers, however, there is no study on specifying these DMUs in DEA. These two facts are motivations for creating this current study. Hence, in this research, we first introduce a DEA model to determine candidate DMUs for having congestion and then, a DEA approach is presented to detect congestion status of these DMUs. Likewise, we propose two integrated mixed integer programming (MIP)-DEA models to specify the strongly and weakly most congested DMUs. Note that the proposed approach permits the inputs and outputs that can take both negative and non-negative magnitudes. Also, a ranking DEA approach is introduced to rank the specified congested DMUs and identify the least congested DMU. Finally, a numerical example and an empirical application are presented to highlight the purpose of this research.