A cross-efficiency fuzzy Data Envelopment Analysis technique for performance evaluation of Decision Making Units under uncertainty

The paper presents a novel cross-efficiency fuzzy Data Envelopment Analysis (DEA) technique for evaluating different elements (Decision Making Units or DMUs) under uncertainty. In order to evaluate the performance of several DMUs while dealing with uncertain input and output data, the presented technique employs triangular fuzzy numbers. A fuzzy triangular efficiency is associated to each DMU through a cross evaluation obtained by a compromise between suitably chosen objectives. Results are then defuzzified to provide a ranking of the DMUs. The proposed method is applied to the performance evaluation of healthcare systems in a region of Southern Italy. The DMU data uncertainty derives from ongoing reforms and the reported assessment is conducted firstly in order to evaluate and rank the efficiency of the considered healthcare systems, and subsequently to assess the evolution of the performance of one of the most affected among these DMUs by the reform plans. The case study demonstrates the model ease of application, its discriminative power among DMUs when compared to a more classical fuzzy DEA approach, and the usefulness in planning and validating targeted reforms in the case of healthcare systems.     

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.     

Fuzzy data envelopment analysis: A discrete approach

Data envelopment analysis (DEA) as introduced by Charnes, Cooper, and Rhodes (1978) is a linear programming technique that has widely been used to evaluate the relative efficiency of a set of homogenous decision making units (DMUs). In many real applications, the input-output variables cannot be precisely measured. This is particularly important in assessing efficiency of DMUs using DEA, since the efficiency score of inefficient DMUs are very sensitive to possible data errors. Hence, several approaches have been proposed to deal with imprecise data. Perhaps the most popular fuzzy DEA model is based on α-cut. One drawback of the α-cut approach is that it cannot include all information about uncertainty. This paper aims to introduce an alternative linear programming model that can include some uncertainty information from the intervals within the α-cut approach. We introduce the concept of “local α-level” to develop a multi-objective linear programming to measure the efficiency of DMUs under uncertainty. An example is given to illustrate the use of this method.     

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.     

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.     

Fuzzy rough DEA model: A possibility and expected value approaches

Uncertainty is certain in the world of uncertainty. Measuring the performance of any entity in such an uncertain environment is unavoidable. Fuzzy rough data envelopment analysis (FRDEA) provides a room to evaluate the relative efficiency of homogenous entities, widely known as decision making units (DMUs) in the data envelopment analysis (DEA) literature. This paper attempts to create a fuzzy rough DEA model by integrating the classical DEA, fuzzy set theory, and rough set theory, which apparently provide a way to accommodate the uncertainty. Moreover, in contrast to the probability approach, this paper provides a pavement to measure the relative efficiency of any given DMUs in line with the possibility approach along with the fuzzy rough expected value operator.     

Positive and normative use of fuzzy DEA‐BCC models: A critical view on NATO enlargement

Data envelopment analysis (DEA) is a widely used mathematical programming approach for comparing the input and output of a set of comparable decision‐making units (DMUs) by evaluating their relative efficiency. The traditional DEA methods require accurate measurement of both the inputs and outputs. However, the real evaluation of the DMUs is often characterized by imprecision and uncertainty in data definitions and measurements. The development of fuzzy DEA (FDEA) with imprecise and ambiguous data has extended the scope of application for efficiency measurement. The purpose of this paper is to develop a fuzzy DEA framework with a BCC model for measuring crisp and interval efficiencies in fuzzy environments. We use an α‐level approach to convert the fuzzy Banker, Charnes, and Cooper (BCC) (variable returns to scale) model into an interval programming model. Instead of comparing the equality (or inequality) of the two intervals, we define a variable in the interval to satisfy our constraints and maximize the efficiency value. We present a numerical example to show the similarities and differences between our solution and the solutions obtained from four fuzzy DEA methods in the literature. In addition, a case study for NATO enlargement is presented to illustrate the applicability of the proposed method.     

A note on simulating weights restrictions in DEA: an improvement of Thanassoulis and Allen's method

Data envelopment analysis (DEA) is a nonparametric technique for measuring and assessing the comparative efficiencies of decision making units (DMUs). Weights restrictions are often imposed on these assessments and create many problems in interpretations of results. This paper focuses on one of those and reviews unobserved DMUs method, introduced by Thanassoulis and Allen, to overcome this problem. This paper clarifies the method proposed by them for specifying full set of unobserved DMUs (FSUD) and reduced set of unobserved DMUs (RSUD) as two sets of unobserved DMUs, too. It is indicated that this method is not suitable for some cases and a simple approach is introduced to identify and eliminate redundant unobserved DMUs and specify a new RSUD convenient for all cases.     

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.     

A Meta heuristic approach for performance assessment of production units

There have been many efficiency frontier analysis methods reported in the literature. However, each of these methodologies has its strength as well as major limitations. This study proposes a Meta heuristic approach based on adaptive neural network (ANN) technique, fuzzy C-means and numerical taxonomy (NT) for measuring efficiency as a complementary tool for the common techniques of the efficiency studies in the previous studies. Homogenous test is done by NT. It is used to determine if the DMUs are homogenous or not. The proposed computational methods are able to find a stochastic frontier based on a set of input–output observational data and do not require explicit assumptions about the functional structure of the stochastic frontier. In this algorithm, for calculating the efficiency scores, a similar approach to za has been used. Moreover, the effect of the return to scale of decision making unit (DMU) on its efficiency is included and the unit used for the correction is selected by notice of its scale (under constant return to scale assumption). Also in non homogenous situation, for increasing DMUs’ homogeneousness, fuzzy C-means method is used to cluster DMUs. Two examples using real data are presented for illustrative purposes. Homogenous test result is positive in the first example, which deals with power generation sectors, and is negative in the second example dealing auto industries of various developed countries. Overall, we find that the proposed integrated algorithm based on ANN, fuzzy C-means and numerical taxonomy provides more robust results and identifies more efficient units than the conventional methods since better performance patterns are explored.     

Modeling fuzzy data envelopment analysis by parametric programming method

Data envelopment analysis (DEA) is a methodology for measuring the relative efficiency of decision making units (DMUs) consuming the same types of inputs and producing the same types of outputs. This paper studies the DEA models with type-2 data variations. In order to deal with the existed type-2 fuzziness, we propose the mean reduction methods for type-2 fuzzy variables. Based on the mean reductions of the type-2 fuzzy inputs and outputs, we formulate a new class of fuzzy generalized expectation DEA models. When the inputs and outputs are mutually independent type-2 triangular fuzzy variables, we discuss the equivalent parametric forms for the constraints and the generalized expectation objective, where the parameters characterize the degree of uncertainty of the type-2 fuzzy coefficients so that the information cannot be lost via our reduction method. For any given parameters, the proposed model becomes nonlinear programming, which can be solved by standard optimization solvers. To illustrate the modeling idea and the efficiency of the proposed DEA model, we provide one numerical example.     

A DEA‐based method of allocating the fixed cost as a complement to the original input

This paper focuses on the problem of how to divide a fixed cost as a complement to an original input among decision‐making units (DMUs) equitably. Using the data envelopment analysis (DEA) technique, this paper concerns the problem from the perspective of efficiency analysis. It is found that not all DMUs can become efficient under common weights if a low enough fixed cost is assigned. Therefore, the global modified additive DEA (MAD) model is introduced. By optimizing the global MAD‐efficiency, a new allocation method and the corresponding algorithm to ensure the uniqueness of the allocation result is designed. The proposed method can be used under both constant returns to scale and variable returns to scale for nonnegative data; it is suitable for the situation where the costs play a great role in the production of DMUs. Numerical results show the validity and advantages of our method.     

A fuzzy DEA model with undesirable fuzzy outputs and its application to the banking sector in India

Data envelopment analysis (DEA) is a widely used technique for measuring the relative efficiencies of decision making units (DMUs) with multiple inputs and multiple outputs. However, in real life applications, undesirable outputs may be present in the production process which needs to be minimized. The present study endeavors to propose a DEA model with undesirable outputs and further to extend it in fuzzy environment in view of the fact that input/output data are not always available in exact form in real life problems. We propose a fuzzy DEA model with undesirable fuzzy outputs which can be solved as crisp linear program for each α in (0, 1] using α-cut approach. Further, cross-efficiency technique is applied to increase the discrimination power of the proposed models and to rank the efficient DMUs at every α in (0, 1]. Moreover, for better understanding of the proposed methodology, we present a numerical illustration followed by an application to the banking sector in India. This is the first study which attempts to measure the performance of public sector banks (PuSBs) in India using fuzzy input/output data for the period 2009–2011. The results obtained from the proposed methodology not only depict the impact of undesirable output on the performance of PuSBs but also analyze efficiently the influence of the presence of uncertainty in the data over the efficiency results. The findings show that the efficiency results of many PuSBs vary with the variation in α during the selected period.     

Estimating most productive scale size with imprecise-chance constrained input–output orientation model in data envelopment analysis

Varieties of data envelopment analysis (DEA) models have been formulated to assess performance of decision making units (DMUs) in various fields with different data such as: deterministic, interval, fuzzy, etc. Classic DEA requires that values of all inputs and outputs are known exactly. However, this assumption may not be true, since in practice, data can not be precisely measured. Furthermore, a realistic situation is no longer realistic when imprecise and uncertain information are neglected to analyze efficiency of DMUs and measurement errors and data entry errors, etc.     

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.     

Type-2 fuzzy multi-objective DEA model: An application to sustainable supplier evaluation

In the last 10 years, sustainable supply chain management (SSCM) has become one of the important topics in business and academe. Sustainable supplier performance evaluation and selection play a significant role in establishing an effective SSCM. One of the techniques that can be used for evaluating sustainable supplier performance is data envelopment analysis (DEA). The conventional DEA methods require accurate measurement of both input and output variables present in the problem. In practice, the observed values of the input and output data present in real-world problems are often imprecise. To cope with this situation, fuzzy DEA models were constructed for expressing relative fuzzy efficiencies of decision-making units (DMUs). However, fuzzy DEA is still limited to fuzzy input/output data while some inputs and outputs might be affected by various factors of uncertainty and information granularity, meaning that they could be better modeled in terms of fuzzy sets of type-2. In this paper, we develop a multi-objective DEA model in a setting of type-2 fuzzy modeling to evaluate and select the most appropriate sustainable suppliers. In the proposed model, both efficiency and effectiveness are considered to describe the integrated productivity of suppliers. In sequel, chance constrained programming, critical value-based reduction methods and equivalent transformations are considered to solve the problem. A detailed case study is employed to show the advantages of the proposed model in terms of measuring effectiveness, efficiency and productivity in an uncertain environment expressed at different confidence levels. At the same time, the results demonstrate that the model is capable of helping decision makers to balance economic, social, and environmental factors when selecting sustainable suppliers.     

A new method based on the dispersion of weights in data envelopment analysis

One of the drawbacks of the data envelopment analysis (DEA) is the problem of lack of discrimination among efficient decision making units (DMUs) and hence yielding many numbers of DMUs as efficient. The main purpose of this study is to overcome this inability. In the case in which the minimization of the coefficient of variation (CV) for input–output weights is added to the DEA model, more reasonable and more homogeneous input–output weights are obtained. For this new proposed model based on the CV it is observed that the number of efficient DMUs is reduced, improving the discrimination power. When this new approach is applied to two well-known examples in the literature, and a real-world data of OECD countries, it has been seen that the new model yielded a more balanced dispersion of input–output weights and reduced the number of efficient DMUs. In addition, the applicability of the new model is tested by a simulation study.     

Fuzzy stochastic data envelopment analysis with application to base realignment and closure (BRAC)

Data envelopment analysis (DEA) is a non-parametric method for evaluating the relative efficiency of decision-making units (DMUs) on the basis of multiple inputs and outputs. Conventional DEA models assume that inputs and outputs are measured by exact values on a ratio scale. However, the observed values of the input and output data in real-world problems are often vague or random. Indeed, decision makers (DMs) may encounter a hybrid uncertain environment where fuzziness and randomness coexist in a problem. Several researchers have proposed various fuzzy methods for dealing with the ambiguous and random data in DEA. In this paper, we propose three fuzzy DEA models with respect to probability-possibility, probability-necessity and probability-credibility constraints. In addition to addressing the possibility, necessity and credibility constraints in the DEA model we also consider the probability constraints. A case study for the base realignment and closure (BRAC) decision process at the U.S. Department of Defense (DoD) is presented to illustrate the features and the applicability of the proposed models.     

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.     

Data envelopment analysis with classification and regression tree – a case of banking efficiency

Abstract: Data envelopment analysis (DEA) is a non‐parametric method for measuring the efficiency and productivity of decision‐making units (DMUs). On the other hand data mining techniques allow DMUs to explore and discover meaningful, previously hidden information from large databases. Classification and regression (C&R) is the commonly used decision tree in data mining. DEA determines the efficiency scores but cannot give details of factors related to inefficiency, especially if these factors are in the form of non‐numeric variables such as operational style in the banking sector. This paper proposes a framework to combine DEA with C&R for assessing the efficiency and productivity of DMUs. The result of the combined model is a set of rules that can be used by policy makers to discover reasons behind efficient and inefficient DMUs. As a case study, we use the proposed methodology to investigate factors associated with the efficiency of the banking sector in the Gulf Cooperation Council countries.