Comments on finding the most efficient DMUs in DEA: An improved integrated model

This paper improves the integrated DEA model proposed for finding the most efficient DMUs introduced by Amin and Toloo, [Amin, Gholam R., Toloo, M. (2007). Finding the most efficient DMUs in DEA: An improved integrated model. Computers & Industrial Engineering, 52(2), 71–77]. The paper shows the problem of using the integrated DEA model and presents an improved integrated DEA model for determining a single efficient unit. Also the paper indicates the property of the improvements mathematically and a numerical example shows the usefulness and intelligibility of the study.     

Alternative mixed integer linear programming models for identifying the most efficient decision making unit in data envelopment analysis

A mixed integer linear model for selecting the best decision making unit (DMU) in data envelopment analysis (DEA) has recently been proposed by Foroughi [Foroughi, A. A. (2011a). A new mixed integer linear model for selecting the best decision making units in data envelopment analysis. Computers and Industrial Engineering, 60(4), 550–554], which involves many unnecessary constraints and requires specifying an assurance region (AR) for input weights and output weights, respectively. Its selection of the best DMU is easy to be affected by outliers and may sometimes be incorrect. To avoid these drawbacks, this paper proposes three alternative mixed integer linear programming (MILP) models for identifying the most efficient DMU under different returns to scales, which contain only essential constraints and decision variables and are much simpler and more succinct than Foroughi’s. The proposed alternative MILP models can make full use of input and output information without the need of specifying any assurance regions for input and output weights to avoid zero weights, can make correct selections without being affected by outliers, and are of significant importance to the decision makers whose concerns are not DMU ranking, but the correct selection of the most efficient DMU. The potential applications of the proposed alternative MILP models and their effectiveness are illustrated with four numerical examples.     

Finding the most efficient DMUs in DEA: An improved integrated model

In many applications of DEA finding the most efficient DMUs is desirable. This paper presents an improved integrated DEA model in order to detect the most efficient DMUs. The proposed integrated DEA model does not use the trial and error method in the objective function. Also, it is able to find the most efficient DMUs without solving the model n times (one linear programming (LP) for each DMU) and therefore allows the user to get faster results. It is shown that the improved integrated DEA model is always feasible and capable to rank the most efficient one. To illustrate the model capability the proposed methodology is applied to a real data set consisting of the 19 facility layout alternatives.     

In the determination of the most efficient decision making unit in data envelopment analysis

In recent years, several mixed integer linear programming (MILP) models have been proposed for determining the most efficient decision making unit (DMU) in data envelopment analysis. However, most of these models do not determine the most efficient DMU directly; instead, they make use of other less related objectives. This paper introduces a new MILP model that has an objective similar to that of the super-efficiency model. Unlike previous models, the new model’s objective is to directly discover the most efficient DMU. Similar to the super-efficiency model, the aim is to choose the most efficient DMU. However, unlike the super-efficiency model, which requires the solution of a linear programming problem for each DMU, the new model requires that only a single MILP problem be solved. Consequently, additional terms in the objective function and more constraints can be easily added to the new model. For example, decision makers can more easily incorporate a secondary objective such as adherence to a publicly stated preference or add assurance region constraints when determining the most efficient DMU. Furthermore, the proposed model is more accurate than two recently proposed models, as shown in two computational examples.     

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.     

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.     

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 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.     

The worst-practice DEA model with slack-based measurement

An original data envelopment analysis (DEA) model is to evaluate each decision-making unit (DMU) with a set of most favorable weights of performance indices. The efficient DMUs obtained from the original DEA construct an efficient (best-practice) frontier. The original DEA can be considered to identify good (efficient) performers in the most favorable scenario. For the purpose of identifying bad performers such as bankrupt firms in the most unfavorable (worst-case) scenario, radial worst-practice frontier DEA (WPF–DEA) model in which the “worst efficient” DMUs construct a worst-practice frontier has been proposed. To identify bad performers together with the slack values we formulate another model called WPF–SBM. Then we develop the HypoSBM model to distinguish the worst performers from the bad ones. Finally, a solution approach is suggested to fully rank worst efficiencies in the worst-case scenario.     

Ranking efficient DMUs using cooperative game theory

The problem of ranking Decision Making Units (DMUs) in Data Envelopment Analysis (DEA) has been widely studied in the literature. Some of the proposed approaches use cooperative game theory as a tool to perform the ranking. In this paper, we use the Shapley value of two different cooperative games in which the players are the efficient DMUs and the characteristic function represents the increase in the discriminant power of DEA contributed by each efficient DMU. The idea is that if the efficient DMUs are not included in the modified reference sample then the efficiency score of some inefficient DMUs would be higher. The characteristic function represents, therefore, the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient units is dropped from the sample. Alternatively, the characteristic function of the cooperative game can be defined as the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient DMUs are the only efficient DMUs that are included in the sample. Since the two cooperative games proposed are dual games, their corresponding Shapley value coincide and thus lead to the same ranking. The more an efficient DMU impacts the shape of the efficient frontier, the higher the increase in the efficiency scores of the inefficient DMUs its removal brings about and, hence, the higher its contribution to the overall discriminant power of the method. The proposed approach is illustrated on a number of datasets from the literature and compared with existing methods.     

Erratum to “Finding the most efficient DMUs in DEA: An improved integrated model” [Comput. Indus. Eng. 52 (2007) 71–77]

There are two errors existing in “Finding the most efficient DMUs in DEA: An improved integrated model” (Amin & Toloo, 2007), one exists in the process of proving the theorem that “Theorem:model (3) is feasible (p. 75)”, while the other is that model (3) and model (4) are incorrectly built.     

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 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 note on “A new method for ranking discovered rules from data mining by DEA”, and a full ranking approach

In a recent paper by Toloo et al. [Toloo, M., Sohrabi, B., & Nalchigar, S. (2009). A new method for ranking discovered rules from data mining by DEA. Expert Systems with Applications, 36, 8503–8508], they proposed a new integrated data envelopment analysis model to find most efficient association rule in data mining. Then, utilizing this model, an algorithm is developed for ranking association rules by considering multiple criteria. In this paper, we show that their model only selects one efficient association rule by chance and is totally depended on the solution method or software is used for solving the problem. In addition, it is shown that their proposed algorithm can only rank efficient rules randomly and will fail to rank inefficient DMUs. We also refer to some other drawbacks in that paper and propose another approach to set up a full ranking of the association rules. A numerical example illustrates some contents of the paper.     

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.     

A new DEA approach to fully rank DMUs with an application to MBA programs

Data envelopment analysis (DEA) can be used to evaluate the efficiencies of decision‐making units (DMUs) in various areas like education, healthcare, and energy. Several DEA methods are proposed for this purpose; however, some of these methods cannot provide a full ranking and others often overlook some considerations that arise with special characteristics of DMUs. We propose a new DEA‐based approach to achieve a full ranking of DMUs. Our approach takes various issues into account such as the initial efficiency score of the DMU, the DMUs that should be removed from the set for it to become efficient (if any) and its effects on the efficiency scores of other DMUs. We demonstrate the shortcomings of several other DEA methods and discuss how our approach overcomes these. We apply our approach to evaluate 50 MBA programs from Financial Times 2018 rankings and compare the results with the evaluations of other methods. As opposed to some methods, our approach has the advantage of differentiating between all efficient DMUs as well as inefficient ones. In addition, the results demonstrate that we can achieve a consistent ranking that considers different aspects of the problem setting. The generated scores are also used to sort DMUs in classes of preference.     

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.     

Improved data envelopment analysis models for evaluating interval efficiencies of decision-making units

In a recent article, Wang et al. [Wang, N. S., Yi, R. H., & Wang, W. (2008). Evaluating the performances of decision-making units based on interval efficiencies. Journal of Computational and Applied Mathematics, 216, 328–343] proposed a pair of interval data envelopment analysis (DEA) models for measuring the overall performances of decision-making units (DMUs) with crisp data. In this paper, we demonstrate that interval DEA models face problems in determining the efficiency interval for each DMU when there are zero values for every input. To remedy this drawback, we propose a pair of improved interval DEA models which make it possible to perform a DEA analysis using the concepts of the best and the worst relative efficiencies. Two numerical examples will be examined using the improved interval DEA models. One of the examples is a real-world application about 42 educational departments in one of the branches of the Islamic Azad University in Iran that shows the advantages and applicability of the improved approach in real-life situations.     

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.