Interval efficiency measures in data envelopment analysis with imprecise data

The conventional data envelopment analysis (DEA) measures the relative efficiencies of a set of decision making units (DMUs) with exact values of inputs and outputs. For imprecise data, i.e., mixtures of interval data and ordinal data, some methods have been developed to calculate the upper bound of the efficiency scores. This paper constructs a pair of two-level mathematical programming models, whose objective values represent the lower bound and upper bound of the efficiency scores, respectively. Based on the concept of productive efficiency and the application of a variable substitution technique, the pair of two-level nonlinear programs is transformed to a pair of ordinary one-level linear programs. Solving the associated pairs of linear programs produces the efficiency intervals of all DMUs. An illustrative example verifies the idea of this paper. A real case is also provided to give some interpretation of the interval efficiency. Interval efficiency not only describes the real situation in better detail; psychologically, it also eases the tension of the DMUs being evaluated as well as the persons conducting the evaluation.     

Cost efficiency measures in data envelopment analysis with data uncertainty

This paper extends the classical cost efficiency (CE) models to include data uncertainty. We believe that many research situations are best described by the intermediate case, where some uncertain input and output data are available. In such cases, the classical cost efficiency models cannot be used, because input and output data appear in the form of ranges. When the data are imprecise in the form of ranges, the cost efficiency measure calculated from the data should be uncertain as well. So, in the current paper, we develop a method for the estimation of upper and lower bounds for the cost efficiency measure in situations of uncertain input and output data. Also, we develop the theory of efficiency measurement so as to accommodate incomplete price information by deriving upper and lower bounds for the cost efficiency measure. The practical application of these bounds is illustrated by a numerical example.     

A unified approach to radial,hyperbolic, and directional efficiency measurement in data envelopment analysis

The paper analyses properties of a large class of “path-based” Data Envelopment Analysis models through a unifying general scheme. The scheme includes the well-known oriented radial models, the hyperbolic distance function model, the directional distance function models, and even permits their generalisations. The modelling is not constrained to non-negative data and is flexible enough to accommodate variants of standard models over arbitrary data.Mathematical tools developed in the paper allow systematic analysis of the models from the point of view of ten desirable properties. It is shown that some of the properties are satisfied (resp., fail) for all models in the general scheme, while others have a more nuanced behaviour and must be assessed individually in each model. Our results can help researchers and practitioners navigate among the different models and apply the models to mixed data.     

Stability of efficiency evaluations in data envelopment analysis

Efficiency evaluations in data envelopment analysis are shown to be stable for arbitrary perturbations in the convex hulls of input and output data. Also, the corresponding restricted Lagrange multiplier functions are shown to be continuous. The results are proved using point-to-set mappings and a particular region of stability from input optimization.Research partly supported by National Science Foundation Grants, Office of Naval Research Grant, and by the Natural Sciences and Engineering Council of Canada.     

Diversification-consistent data envelopment analysis with general deviation measures

We propose new efficiency tests which are based on traditional DEA models and take into account portfolio diversification. The goal is to identify the investment opportunities that perform well without specifying our attitude to risk. We use general deviation measures as the inputs and return measures as the outputs. We discuss the choice of the set of investment opportunities including portfolios with limited number of assets. We compare the optimal values (efficiency scores) of all proposed tests leading to the relations between the sets of efficient opportunities. Strength of the tests is then discussed. We test the efficiency of 25 world financial indices using new DEA models with CVaR deviation measures.     

Uncertain random data envelopment analysis for technical efficiency

Fuzzy Optimization and Decision Making - Data envelopment analysis (DEA) is a classical and prevailing tool for estimating relative efficiencies of multiple decision making units (DMUs). However,...     

A distance friction minimization approach in data envelopment analysis: A comparative study on airport efficiency

This paper aims to present a newly developed distance friction minimization (DFM) method in the context of data envelopment analysis (DEA) in order to generate an appropriate (non-radial) efficiency-improving projection model, for both input reduction and output increase. In this approach, a generalized distance function, based on a Euclidean distance metric in weighted spaces, is proposed to assist a decision making unit (DMU) to improve its performance by an appropriate movement towards the efficiency frontier surface. A suitable form of multidimensional projection function for efficiency improvement is given by a Multiple Objective Quadratic Programming (MOQP) model. The paper describes the various steps involved in a systematic manner.     

A slacks-based measure of efficiency in data envelopment analysis

In this paper, we will propose a slacks-based measure (SBM) of efficiency in Data Envelopment Analysis (DEA). This scalar measure deals directly with the input excesses and the output shortfalls of the decision making unit (DMU) concerned. It is units invariant and monotone decreasing with respect to input excess and output shortfall. Furthermore, this measure is determined only by consulting the reference-set of the DMU and is not affected by statistics over the whole data set. The new measure has a close connection with other measures proposed so far, e.g., Charnes–Cooper–Rhodes (CCR), Banker–Charnes–Cooper (BCC) and the Russell measure of efficiency. The dual side of this model can be interpreted as profit maximization, in contrast to the ratio maximization of the CCR model. Numerical experiments show its validity as an efficiency measurement tool and its compatibility with other measures of efficiency.     

Fuzzy efficiency ranking in fuzzy two-stage data envelopment analysis

Data envelopment analysis (DEA) is a useful tool for efficiency measurement of firms and organizations. Many production systems in the real world are composed of two processes connected in series. Measuring the system efficiency without taking the operation of each process into consideration will obtain misleading results. Two-stage DEA models show the performance of individual processes, thus is more informative than the conventional one-stage models for making decisions. When input and output data are fuzzy numbers, the derived efficiencies become fuzzy as well. This paper proposes a method to rank the fuzzy efficiencies when the exact membership functions of the overall efficiencies derived from fuzzy two-stage model are unknown. By incorporating the fuzzy two-stage model with the fuzzy number ranking method, a pair of nonlinear program is formulated to rank the fuzzy overall efficiency scores of DMUs. Solving the pair of nonlinear programs determines the efficiency rankings. An example of the ranking of the 24 non-life assurance companies in Taiwan is illustrated to explain how the proposed method is applied.     

A Euclidean distance-based measure of efficiency in data envelopment analysis

Data envelopment analysis (DEA) has been proven as an excellent data-oriented performance evaluation method when multiple inputs and outputs are present in a set of peer decision-making units (DMUs). Several efficiency measures have been proposed in the DEA literature, see, for instances, radial efficiency measure of Charnes et al. (CCR)(A. Charnes. W.W. Cooper, and E. Rhodes, 1978. Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2, 429–444), Russell graph measure (J.T. Russell, and R. Sirvant. 1999. An enhanced DEA Russell graph efficiency measure. Eur. J. Oper. Res. 115, pp. 596–607) and slack-based measure of Tone (K. Tone, 2001. A slack-based measure of efficiency in DEA. Eur. J. Oper. Res. 130, p. 498–509). In this article, we will propose an Euclidean distance-based measure of efficiency. Then, in order to discriminate the performance of efficient DMUs, an alternative super-efficiency DEA model is proposed. The applicability of the models developed is illustrated in the context of the analysis of gas companies performance.     

Goal programming approaches for data envelopment analysis cross efficiency evaluation

Cross efficiency evaluation has long been proposed as an alternative method for ranking the decision making units (DMUs) in data envelopment analysis (DEA). This study proposes goal programming models that could be used in the second stage of the cross evaluation. Proposed goal programming models have different efficiency concepts as classical DEA, minmax and minsum efficiency criteria. Numerical examples are provided to illustrate the applications of the proposed goal programming cross efficiency models.     

Incorporating performance measures with target levels in data envelopment analysis

Data envelopment analysis (DEA) is a technique for evaluating relative efficiencies of peer decision making units (DMUs) which have multiple performance measures. These performance measures have to be classified as either inputs or outputs in DEA. DEA assumes that higher output levels and/or lower input levels indicate better performance. This study is motivated by the fact that there are performance measures (or factors) that cannot be classified as an input or output, because they have target levels with which all DMUs strive to achieve in order to attain the best practice, and any deviations from the target levels are not desirable and may indicate inefficiency. We show how such performance measures with target levels can be incorporated in DEA. We formulate a new production possibility set by extending the standard DEA production possibility set under variable returns-to-scale assumption based on a set of axiomatic properties postulated to suit the case of targeted factors. We develop three efficiency measures by extending the standard radial, slacks-based, and Nerlove–Luenberger measures. We illustrate the proposed model and efficiency measures by applying them to the efficiency evaluation of 36 US universities.     

Measuring ecological efficiency with data envelopment analysis (DEA)

The measurement of ecological efficiency provides some important information for the companies’ environmental management. Ecological efficiency is usually measured by comparing environmental performance indicators. Data envelopment analysis (DEA) shows a high potential to support such comparisons, as no explicit weights are needed to aggregate the indicators. In general, DEA assumes that inputs and outputs are ‘goods’, but from an ecological perspective also ‘bads’ have to be considered. In the literature, ‘bads’ are treated in different and sometimes arbitrarily chosen ways. This article aims at the systematic derivation of ecologically extended DEA models. Starting from the assumptions of DEA in production theory and activity analysis, a generalisation of basic DEA models is derived by incorporating a multi-dimensional value function f. Extended preference structures can be considered by different specifications of f, e.g. specifications for ecologically motivated applications of DEA.     

A note on efficiency decomposition in two-stage data envelopment analysis

Data envelopment analysis (DEA) is a useful tool of efficiency measurement for firms and organizations. Kao and Hwang (2008) take into account the series relationship of the two sub-processes in a two-stage production process, and the overall efficiency of the whole process is the product of the efficiencies of the two sub-processes. To find the largest efficiency of one sub-process while maintaining the maximum overall efficiency of the whole process, Kao and Hwang (2008) propose a solution procedure to accomplish this purpose. Nevertheless, one needs to know the overall efficiency of the whole process before calculating the sub-process efficiency. In this note, we propose a method that is able to find the sub-process and overall efficiencies simultaneously.     

A full-scale investigation of the directional returns to scale in data envelopment analysis

Returns to scale is considered as one of the important concepts in data envelopment analysis (DEA) which can be useful for deciding to increase or decrease the size of a particular decision making unit. Traditional returns to scale on the efficient surface of the production possibility set with variable returns to scale (VRS) technology is introduced as a ratio of proportional changes of output components to proportional changes of input components. However, a problem which may arise in the real world is the impossibility or undesirability of proportional change in the input or output components. One of the attempts which is made to solve the aforementioned problem is the work of Yang et al., 2014. They have introduced the “directional returns to scale” in the DEA framework and have proposed some procedures to estimate and measure it. In this paper, the introduced directional returns to scale is investigated from a new perspective based on the defining hyperplanes of the production possibility set with VRS technology. We propose some algebraic equations and linear programming models which in addition to measuring the directional returns to scale, they enable us to analyse it. Moreover, we introduce the concepts of the best input and output direction vectors for expansion of input components or compression of output components, respectively, and propose two linear programming models in order to obtain these directions. The presented equations and models are demonstrated using a case study and numerical examples.     

An input relaxation measure of efficiency in stochastic data envelopment analysis

We introduce stochastic version of an input relaxation model in data envelopment analysis (DEA). The input relaxation model, recently developed in DEA, is useful to resource management [e.g. G.R. Jahanshahloo, M. Khodabakhshi, Suitable combination of inputs for improving outputs in DEA with determining input congestion, Appl. Math. Comput. 151(1) (2004) 263–273]. This model allows more changes in the input combinations of decision making units than those in the observed inputs of evaluating decision making units. Using this extra flexibility in input combinations we can find better outputs. We obtain a non-linear deterministic equivalent to this stochastic model. It is shown that under fairly general conditions this non-linear model can be replaced by an ordinary deterministic DEA model. The model is illustrated using a real data set.     

Improving weak efficiency frontiers in the fuzzy data envelopment analysis models

Evaluating the performance of activities or organization by common data envelopment analysis models requires crisp input/output data. However, the precise inputs and outputs of production processes cannot be always measured. Thus, the data envelopment analysis measurement containing fuzzy data, called “fuzzy data envelopment analysis”, has played an important role in the evaluation of efficiencies of real applications. This paper focuses on the fuzzy CCR model and proposes a new method for determining the lower bounds of fuzzy inputs and outputs. This improves the weak efficiency frontiers of the corresponding production possibility set. Also a numerical example illustrates the capability of the proposed method.