Data Envelopment Analysis Based on Choquet Integral

Data envelopment analysis (DEA) is a widely used technique in decision making. The existing DEA models always assume that the inputs (or outputs) of decision‐making units (DMUs) are independent with each other. However, there exist positive or negative interactions between inputs (or outputs) of DMUs. To reflect such interactions, Choquet integral is applied to DEA. Self‐efficiency models based on Choquet integral are first established, which can obtain more efficiency values than the existing ones. Then, the idea is extended to the cross‐efficiency models, including the game cross‐efficiency models. The optimal analysis of DEA is further investigated based on regret theory. To estimate the ranking intervals of DMUs, several models are also established. It is founded that the models considering the interactions between inputs (or outputs) can obtain wider ranking intervals.     

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.     

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.     

A modified slacks‐based ranking method handling negative data in data envelopment analysis

Performance ranking for a set of comparable decision‐making units (DMUs) with multiple inputs and outputs is an important and often‐discussed topic in data envelopment analysis (DEA). Conventional DEA models distinguish efficient units from inefficient ones but cannot further discriminate the efficient units, which all have a 100% efficiency score. Another weakness of these models is that they cannot handle negative inputs and/or outputs. In this paper, a new modified slacks‐based measure is proposed that works in the presence of negative data and provides quantitative data that helps decision makers obtain a full ranking of DMUs in situations where other methods fail. In addition, the new method has the properties of unit invariance and translation invariance, and it can give targets for inefficient DMUs to guide them to achieve full efficiency. Two numerical examples are analysed to demonstrate the usefulness of the new method.     

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.     

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.     

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.     

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.     

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.     

Recognizing strong and weak congestion slack based in data envelopment analysis

The common concept of congestion is that a decrease (increase) in one or more inputs of a decision making unit (DMU) causes an increase (decrease) in one or more outputs (Cooper, Gu, & Li, 2001a). So far several congestion approaches have been proposed in DEA (data envelopment analysis) literature by many authors, such as Färe’s et al. (FGL), Brockett’s et al. (BCSW), and Tone and Sahoo’s congestion approaches (Färe et al., 1985, Färe et al., 1994, Brockett et al., 1998, Tone and Sahoo, 2004). Tone and Sahoo’s approach (Tone & Sahoo, 2004) is one of the most robust congestion approaches in DEA literature. Moreover, Tone and Sahoo’s approach has some advantages with respect to FGL and BSCW congestion approaches. However, the proposed approaches have many difficulties to treat congestion. For instance, in the presence of alternative optimal solutions, the approach proposed by Tone and Sahoo is unable to detect congestion (strong and weak). Moreover, in Tone and Sahoo’s approach, all inputs and outputs of decision making units (DMUs) have been considered positive, while in real world, data is often non-negative.In this research, a slack-based DEA approach is proposed to recognize congestion (strong and weak) for the target DMUs. One of the advantages of our proposed approach is capable of detecting congestion (strong and weak) for evaluating the DMUs in the presence of alternative optimal solutions. Other advantage of our research is capable of identifying congesting (strong and weak) DMUs with non-negative inputs and outputs. However in these situations, Tone and Sahoo’s congestion approach is incapable of identifying congestion. Lastly, we apply the approach to the data sets for making comparisons between the proposed approach and Tone and Sahoo’s approach then some conclusions are drawn and directions for future research are suggested.     

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.     

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.     

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.     

The upper and lower bound evaluation based on the quantile efficiency in stochastic data envelopment analysis

Data envelopment analysis (DEA) has been extended to handle random inputs and outputs by using chance constrained programming. In this paper, for DMUs with random inputs and outputs, we aim to measure a kind of relative efficiency, and achieve it from the optimistic viewpoint and the pessimistic viewpoint respectively. Considering the quantile of the distribution of the weighted output-input ratio of each DMU, we develop two stochastic DEA models to obtain the upper and lower bounds of the quantile efficiency under a constraint, and then achieve an interval efficiency evaluation. The best quantile efficiency and the worst quantile efficiency achieved by our models are closely similar to the CCR efficiency and belong to relative efficiencies. Further, the deterministic equivalents of our models are developed when the input and output vector of each DMU follows a multivariate joint normal distribution. Finally, three examples are presented to illustrate the performance of our approach.     

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.     

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.     

Allocating fixed resources and setting targets using a common-weights DEA approach

Data envelopment analysis (DEA) is a data-driven non-parametric approach for measuring the efficiency of a set of decision making units (DMUs) using multiple inputs to generate multiple outputs. Conventionally, DEA is used in ex post evaluation of actual performance, estimating an empirical best-practice frontier using minimal assumptions about the shape of the production space. However, DEA may also be used prospectively or normatively to allocate resources, costs and revenues in a given organization. Such approaches have theoretical foundations in economic theory and provide a consistent integration of the endowment-evaluation-incentive cycle in organizational management. The normative use, e.g. allocation of resources or target setting, in DEA can be based on different principles, ranging from maximization of the joint profit (score), combinations of individual scores or game-theoretical settings. In this paper, we propose an allocation mechanism that is based on a common dual weights approach. Compared to alternative approaches, our model can be interpreted as providing equal endogenous valuations of the inputs and outputs in the reference set. Given that a normative use implicitly assumes that there exists a centralized decision-maker in the organization evaluated, we claim that this approach assures a consistent and equitable internal allocation. Two numerical examples are presented to illustrate the applicability of the proposed method and to contrast it with earlier work.     

The voting analytic hierarchy process method for discriminating among efficient decision making units in data envelopment analysis

Making optimal use of available resources has always been of interest to humankind, and different approaches have been used in an attempt to make maximum use of existing resources. Limitations of capital, manpower, energy, etc., have led managers to seek ways for optimally using such resources. In fact, being informed of the performance of the units under the supervision of a manager is the most important task with regard to making sensible decisions for managing them. Data envelopment analysis (DEA) suggests an appropriate method for evaluating the efficiency of homogeneous units with multiple inputs and multiple outputs. DEA models classify decision making units (DMUs) into efficient and inefficient ones. However, in most cases, managers and researchers are interested in ranking the units and selecting the best DMU. Various scientific models have been proposed by researchers for ranking DMUs. Each of these models has some weakness(es), which makes it difficult to select the appropriate ranking model. This paper presents a method for ranking efficient DMUs by the voting analytic hierarchy process (VAHP). The paper reviews some ranking models in DEA and discusses their strengths and weaknesses. Then, we provide the method for ranking efficient DMUs by VAHP. Finally we give an example to illustrate our approach and then the new method is employed to rank efficient units in a real world problem.     

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.