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.     

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.     

Evaluation and ranking DMUs in the presence of both undesirable and ordinal factors in data envelopment analysis

In the last decade,ranking units in data envelopment analysis(DEA) has become the interests of many DEA researchers and a variety of models were developed to rank units with multiple inputs and multiple outputs.These performance factors(inputs and outputs) are classified into two groups:desirable and undesirable.Obviously,undesirable factors in production process should be reduced to improve the performance.Also,some of these data may be known only in terms of ordinal relations.While the models developed in the past are interesting and meaningful,they didn t consider both undesirable and ordinal factors at the same time.In this research,we develop an evaluating model and a ranking model to overcome some deficiencies in the earlier models.This paper incorporates undesirable and ordinal data in DEA and discusses the efficiency evaluation and ranking of decision making units(DMUs) with undesirable and ordinal data.For this purpose,we transform the ordinal data into definite data,and then we consider each undesirable input and output as desirable output and input,respectively.Finally,an application that shows the capability of the proposed method is illustrated.     

Optimal direct mailing modelling based on data envelopment analysis

Data envelopment analysis (DEA) is a mathematical programming technique that is frequently used for measuring and benchmarking efficiency of the homogenous decision‐making units (DMUs). This paper proposes a new use of DEA for customers scoring and particularly their direct mailing modelling. Moreover, because DEA models suffer from some weaknesses, that is, unrealistic weighting scheme of the inputs and outputs and incomplete ranking among efficient DMUs, the present paper compares different ways of solving these problems and concludes that common set of weights method, as a result of some advantages, outperforms other procedures.     

Improving the discrimination power and weights dispersion in the data envelopment analysis

Data envelopment analysis (DEA) has been a very popular method for measuring and benchmarking relative efficiency of peer decision making units (DMUs) with multiple input and outputs. Beside of its popularity, DEA has some drawbacks such as unrealistic input–output weights and lack of discrimination among efficient DMUs. In this study, two new models based on a multi-criteria data envelopment analysis (MCDEA) are developed to moderate the homogeneity of weights distribution by using goal programming (GP). These goal programming data envelopment analysis models, GPDEA-CCR and GPDEA-BCC, also improve the discrimination power of DEA.     

New analytical hierarchical process/data envelopment analysis methodology for ranking decision‐making units

The analytical hierarchical process/data envelopment analysis (AHP/DEA) methodology for ranking decision‐making units (DMUs) has some problems: it illogically compares two DMUs in a DEA model; it is not compatible with DEA ranking in the case of multiple inputs/multiple outputs; and it leads to weak discrimination in cases where the number of inputs and outputs is large. In this paper, we propose a new two‐stage AHP/DEA methodology for ranking DMUs that removes these problems. In the first stage, we create a pairwise comparison matrix different from AHP/DEA methodology; the second stage is the same as AHP/DEA methodology. Numerical examples are presented in the paper to illustrate the advantages of the new AHP/DEA methodology.     

A cross-dependence based ranking system for efficient and inefficient units in DEA

Decision making units (DMUs) can be ranked using data envelopment analysis (DEA) technologies. This paper develops a new ranking system under the condition of variable returns to scale (VRS) based on a measure of cross-dependence efficiency (MCDE), where the evaluation for an efficient DMU is dependent of the efficiency changes of all inefficient units due to its absence in the reference set, while the appraisal of inefficient DMUs depends on the influence of the exclusion of each efficient unit from the reference set. The infeasibility problem that arises from the conventional super-efficiency models is eliminated. A new approach to ranking inefficient units is embedded in the proposed ranking system. Through an example, advantages of our proposal are demonstrated. A real application illustrates the effectiveness of the cross-dependence based ranking system.     

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.     

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.     

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 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.     

Common sets of weights as summaries of DEA profiles of weights: With an application to the ranking of professional tennis players

In this paper, we propose a DEA approach aimed at deriving a common set of weights (CSW) to be used to the ranking of decision making units (DMUs). The idea of this approach is to minimize the deviations of the CSW from the DEA profiles of weights without zeros of the efficient DMUs. This minimization reduces in particular the differences between the DEA profiles of weights that are chosen, so the CSW proposed is a representative summary of such DEA weights profiles. We use several norms to the measurement of such differences. As a result, the CSWs derived are actually different summaries of the chosen DEA profiles of weights like their average profile of their median profile. This approach is illustrated with an application to the ranking of professional tennis players.     

Right and left returns to scales in data envelopment analysis: Determining type and measuring value

This study identifies types and values of right and left returns to scales (RTSs) of efficient decision making units (DMUs) in data envelopment analysis (DEA). In this research, we first introduce a new approach to estimate types of right and left returns to scales of efficient DMUs and then, values of right and left returns to scales of these DMUs are measured by presenting two new DEA models.     

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.     

A new ranking method to fuzzy data envelopment analysis

Due to its wide practical use, data envelopment analysis (DEA) has been adapted to many fields to deal with problems that have occurred in practice. One adaptation has been in the field of ranking decision-making units (DMUs). Most methods of ranking DMUs assume that all input and output data are exactly known, but in real life the data cannot be precisely measured. Thus this paper will carry out some researches to DEA under fuzzy environment. A fuzzy comparison of fuzzy variables is defined and the CCR model is extended to be a fuzzy DEA model based on credibility measure. In order to rank all the DMUs, a full ranking method will be given. Since the ranking method involves a fuzzy function, a fuzzy simulation is designed and embedded into the genetic algorithm to establish a hybrid intelligent algorithm. However, it is shown to be possible to avoid some of the need for dealing with these nonlinear problems by identifying conditions under which they can be replaced by linear problems. Finally we will provide a numerical example to illustrate the fuzzy DEA model and the ranking method.     

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.     

A robust super-efficiency data envelopment analysis model for ranking of provincial gas companies in Iran

Conventional super-efficiency data envelopment analysis (DEA) models require the exact information of inputs or outputs. However, in many real world applications this simple assumption does not hold. Stochastic super-efficiency is one of recent methods which could handle uncertainty in data. Stochastic super-efficiency DEA models are normally formulated based on chance constraint programming. The method is used to estimate the efficiency of various decision making units (DMUs). In stochastic chance constraint super-efficiency DEA, the distinction of probability distribution function for input/output data is difficult and also, in several cases, there is not enough data for estimating of distribution function. We present a new method which incorporates the robust counterpart of super-efficiency DEA. The perturbation and uncertainty in data is assumed as ellipsoidal set and the robust super-efficiency DEA model is extended. The implementation of the proposed method of this paper is applied for ranking different gas companies in Iran.     

A concept of fuzzy input mix-efficiency in fuzzy DEA and its application in banking sector

Data envelopment analysis (DEA) is a linear programming based non-parametric technique for evaluating the relative efficiency of homogeneous decision making units (DMUs) on the basis of multiple inputs and multiple outputs. There exist radial and non-radial models in DEA. Radial models only deal with proportional changes of inputs/outputs and neglect the input/output slacks. On the other hand, non-radial models directly deal with the input/output slacks. The slack-based measure (SBM) model is a non-radial model in which the SBM efficiency can be decomposed into radial, scale and mix-efficiency. The mix-efficiency is a measure to estimate how well the set of inputs are used (or outputs are produced) together. The conventional mix-efficiency measure requires crisp data which may not always be available in real world applications. In real world problems, data may be imprecise or fuzzy. In this paper, we propose (i) a concept of fuzzy input mix-efficiency and evaluate the fuzzy input mix-efficiency using α – cut approach, (ii) a fuzzy correlation coefficient method using expected value approach which calculates the expected intervals and expected values of fuzzy correlation coefficients between fuzzy inputs and fuzzy outputs, and (iii) a new method for ranking the DMUs on the basis of fuzzy input mix-efficiency. The proposed approaches are then applied to the State Bank of Patiala in the Punjab state of India with districts as the DMUs.     

A new DEA‐based method for fully ranking all decision‐making units

Abstract: Decision makers always lay great emphasis on performance evaluation upon a group of peer business units to pick out the best performer. Standard data envelopment analysis models can evaluate the relative efficiency of decision‐making units (DMUs) and distinguish efficient ones from inefficient ones. However, when there are more than one efficient DMU, it is impossible to rank all of them solely according to standard efficiency scores. In this paper, a new method for fully ranking all DMUs is proposed, which is based on the combination of each efficient DMU's influence on all the other DMUs and the standard efficiency scores. This method is effective in helping decision makers differentiate all units' performance thoroughly and select the best performer.     

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.