Finding strong defining hyperplanes of Production Possibility Set

Production Possibility Set (PPS) is defined as the set of all inputs and outputs of a system in which inputs can produce outputs. Data Envelopment Analysis models implicitly use PPS to evaluate relative efficiency of Decision Making Units (DMUs). Although DEA models can determine the efficiency of a DMU, they cannot present efficient frontiers of PPS. In this paper, we propose a method for finding all Strong Defining Hyperplanes of PPS (SDHP). They are equations that form efficient surfaces. These equations are useful in Sensitivity and Stability Analysis, the status of Returns to Scale of a DMU, incorporating performance information into the efficient frontier analysis and so on.     

Computation of efficient targets in DEA models with production trade-offs and weight restrictions

Recently new models of data envelopment analysis (DEA) were introduced that incorporate production trade-offs between inputs and outputs or based on them weight restrictions. In this paper, we develop a computational procedure suitable for the practical application of such models. We show that the standard two-stage optimisation procedure used in DEA to test the full efficiency of units and identify their efficient targets may work incorrectly in the new models. The modified procedure consists of three stages: the first evaluates the radial efficiency of the unit, the second identifies its efficient target, and the third its reference set of efficient peers. Each stage requires solving one linear program for each unit.     

The measurement of relative efficiency using data envelopment analysis with assurance regions that link inputs and outputs

The most popular weight restrictions are assurance regions (ARs), which impose ratios between weights to be within certain ranges. ARs can be categorized into two types: ARs type I (ARI) and ARs type II (ARII). ARI specify bounds on ratios between input weights or between output weights, whilst ARII specify bounds on ratios that link input to output weights. DEA models with ARI successfully maximize relative efficiency, but in the presence of ARII the DEA models may under-estimate relative efficiency or may become infeasible. In this paper we discuss the problems that can occur in the presence of ARII and propose a new nonlinear model that overcomes the limitations discussed. Also, the dual model is described, which enables the assessment of relative efficiency when trade-offs between inputs and outputs are specified. The application of the model developed is illustrated in the efficiency assessment of Portuguese secondary schools.     

Centralized DEA models with the possibility of downsizing

While traditional data envelopment analysis (DEA) models assess the relative efficiency of similar, independent decision making units (DMUs) centralized DEA models aim at reallocating inputs and outputs among the units setting new input and output targets for each one. This system point of view is appropriate when the DMUs belong to a common organization that allocates their inputs and appropriates their outputs. This intraorganizational perspective opens up the possibility that greater technical efficiency for the organization as a whole might be achieved by closing down some of the existing DMUs. In this paper, we present three centralized DEA models that take advantage of this possibility. Although these models involve some binary variables, we present efficient solution approaches based on Linear Programming. We also present some numerical results of the proposed models for a small problem from the literature.     

Data envelopment analysis with nonlinear virtual inputs and outputs

An underlying assumption in DEA is that the weights coupled with the ratio scales of the inputs and outputs imply linear value functions. In this paper, we present a general modeling approach to deal with outputs and/or inputs that are characterized by nonlinear value functions. To this end, we represent the nonlinear virtual outputs and/or inputs in a piece-wise linear fashion. We give the CCR model that can assess the efficiency of the units in the presence of nonlinear virtual inputs and outputs. Further, we extend the models with the assurance region approach to deal with concave output and convex input value functions. Actually, our formulations indicate a transformation of the original data set to an augmented data set where standard DEA models can then be applied, remaining thus in the grounds of the standard DEA methodology. To underline the usefulness of such a new development, we revisit a previous work of one of the authors dealing with the assessment of the human development index on the light of DEA.     

Multiobjective data envelopment analysis

Data envelopment analysis (DEA) is popularly used to evaluate relative efficiency among public or private firms. Most DEA models are established by individually maximizing each firm's efficiency according to its advantageous expectation by a ratio. Some scholars have pointed out the interesting relationship between the multiobjective linear programming (MOLP) problem and the DEA problem. They also introduced the common weight approach to DEA based on MOLP. This paper proposes a new linear programming problem for computing the efficiency of a decision-making unit (DMU). The proposed model differs from traditional and existing multiobjective DEA models in that its objective function is the difference between inputs and outputs instead of the outputs/inputs ratio. Then an MOLP problem, based on the introduced linear programming problem, is formulated for the computation of common weights for all DMUs. To be precise, the modified Chebychev distance and the ideal point of MOLP are used to generate common weights. The dual problem of this model is also investigated. Finally, this study presents an actual case study analysing R&D efficiency of 10 TFT-LCD companies in Taiwan to illustrate this new approach. Our model demonstrates better performance than the traditional DEA model as well as some of the most important existing multiobjective DEA models.     

DEA model with shared resources and efficiency decomposition

Data envelopment analysis (DEA) has proved to be an excellent approach for measuring performance of decision making units (DMUs) that use multiple inputs to generate multiple outputs. In many real world scenarios, DMUs have a two-stage network process with shared input resources used in both stages of operations. For example, in hospital operations, some of the input resources such as equipment, personnel, and information technology are used in the first stage to generate medical record to track treatments, tests, drug dosages, and costs. The same set of resources used by first stage activities are used to generate the second-stage patient services. Patient services also use the services generated by the first stage operations of housekeeping, medical records, and laundry. These DMUs have not only inputs and outputs, but also intermediate measures that exist in-between the two-stage operations. The distinguishing characteristic is that some of the inputs to the first stage are shared by both the first and second stage, but some of the shared inputs cannot be conveniently split up and allocated to the operations of the two stages. Recognizing this distinction is critical for these types of DEA applications because measuring the efficiency of the production for first-stage outputs can be misleading and can understate the efficiency if DEA fails to consider that some of the inputs generate other second-stage outputs. The current paper develops a set of DEA models for measuring the performance of two-stage network processes with non splittable shared inputs. An additive efficiency decomposition for the two-stage network process is presented. The models are developed under the assumption of variable returns to scale (VRS), but can be readily applied under the assumption of constant returns to scale (CRS). An application is provided.     

Production trade-offs and weight restrictions in data envelopment analysis

In this paper we suggest two equivalent ways in which the information about production trade-offs between the inputs and outputs can be incorporated into the models of data envelopment analysis (DEA). Firstly, this can be implemented by modifying envelopment DEA models. Secondly, the same information can be captured using weight restrictions in multiplier DEA models. Unlike other methods used for the assessment of weight restrictions, for example those based on value judgements or monetary considerations, the trade-off approach developed in this paper ensures that the radial target of any inefficient unit is technologically realistic and, therefore, the efficiency measure retains its traditional meaning of the extreme radial improvement factor. In other words, this paper suggests that ‘technology thinking’ could be used instead of ‘value thinking’ in the construction of weight restrictions, which offers real practical advantages. The method is equally applicable to the models under constant and variable returns-to-scale assumptions.     

The explicit role of weight bounds in models of data envelopment analysis

In this paper, we suggest that weight bounds used in models of data envelopment analysis (DEA) can be assessed using production trade-offs between inputs and outputs. This development is based on a transformation of the envelopment DEA model to a special form, which exhibits an explicit link between weight bounds and production trade-offs. This allows us to specify the guidelines to the construction of weight bounds, which ensure that the envelopment form has a clear economic meaning. In this development, the radial efficiency measure retains its standard meaning as a technologically realistic improvement factor. This contrasts with the methods based on value judgements for which this traditional meaning of efficiency becomes void. We show that the exact economic meaning of the same weight bounds depends on the unit under the assessment and the orientation (input-minimization or output-maximization) of the model. The suggested approach is valid under the assumptions of constant and variable returns to scale.     

Theory of integer-valued data envelopment analysis

Conventional data envelopment analysis (DEA) models assume real-valued inputs and outputs. In many occasions, some inputs and/or outputs can only take integer values. In some cases, rounding the DEA solution to the nearest whole number can lead to misleading efficiency assessments and performance targets. This paper develops the axiomatic foundation for DEA in the case of integer-valued data, introducing new axioms of “natural disposability” and “natural divisibility”. We derive a DEA production possibility set that satisfies the minimum extrapolation principle under our refined set of axioms. We also present a mixed integer linear programming formula for computing efficiency scores. An empirical application to Iranian university departments illustrates the approach.     

On relations between DEA-risk models and stochastic dominance efficiency tests

In this paper, several concepts of portfolio efficiency testing are compared, based either on data envelopment analysis (DEA) or the second-order stochastic dominance (SSD) relation: constant return to scale DEA models, variable return to scale (VRS) DEA models, diversification-consistent DEA models, pairwise SSD efficiency tests, convex SSD efficiency tests and full SSD portfolio efficiency tests. Especially, the equivalence between VRS DEA model with binary weights and the SSD pairwise efficiency test is proved. DEA models equivalent to convex SSD efficiency tests and full SSD portfolio efficiency tests are also formulated. In the empirical application, the efficiency testing of 48 US representative industry portfolios using all considered DEA models and SSD tests is presented. The obtained efficiency sets are compared. A special attention is paid to the case of small number of the inputs and outputs. It is empirically shown that DEA models equivalent either to the convex SSD test or to the SSD portfolio efficiency test work well even with quite small number of inputs and outputs. However, the reduced VRS DEA model with binary weights is not able to identify all the pairwise SSD efficient portfolios.     

非期望产出的DEA效率评价

将非期望产出作为投入应用到传统DEA模型上,解决了非期望产出生产活动的效率评价问题.结合生产可能集,将非期望产出直接反映到生产可能集中,建立了基于投入导向的径向和非径向两种DEA模型.并对两种DEA模型效率值的大小关系、相对有效性的等价性问题进行了证明,指出非径向DEA模型更能准确的实现效率定量评价.     

Uncertain data envelopment analysis with imprecisely observed inputs and outputs

Data envelopment analysis (DEA) is a powerful analytical tool in operations research and management for measuring and estimating the efficiency of decision-making units. Both the inputs and the outputs are assumed to be known constants in the classical DEA models. However, in many cases, those data (e.g., carbon emissions and social benefit) cannot be measured in a precise way. Therefore, in this article, the inputs and outputs are considered as uncertain variables and a new uncertain DEA model is introduced. The sensitivity and stability of the new model are also analyzed. Finally, a numerical example of the new model is documented.     

Methods of critical value reduction for type-2 fuzzy variables and their applications

A type-2 fuzzy variable is a map from a fuzzy possibility space to the real number space; it is an appropriate tool for describing type-2 fuzziness. This paper first presents three kinds of critical values (CVs) for a regular fuzzy variable (RFV), and proposes three novel methods of reduction for a type-2 fuzzy variable. Secondly, this paper applies the reduction methods to data envelopment analysis (DEA) models with type-2 fuzzy inputs and outputs, and develops a new class of generalized credibility DEA models. According to the properties of generalized credibility, when the inputs and outputs are mutually independent type-2 triangular fuzzy variables, we can turn the proposed fuzzy DEA model into its equivalent parametric programming problem, in which the parameters can be used to characterize the degree of uncertainty about type-2 fuzziness. For any given parameters, the parametric programming model becomes a linear programming one that can be solved using standard optimization solvers. Finally, one numerical example is provided to illustrate the modeling idea and the efficiency of the proposed DEA model.     

Relationship between DEA models without explicit inputs and DEA-R models

Data envelopment analysis (DEA) is one of often used modeling tools for efficiency and performance evaluation of decision making units. Ratio DEA (DEA-R) is a group of novel mathematical models that combines standard DEA methodology and ratio analysis. The efficiency score given by standard DEA CCR model is less than or equal to that given by DEA-R model. In case of single input or single output the efficiency scores in CCR and DEA-R models are identical. The paper deals with DEA-R models without explicit inputs, i.e. models where only pure outputs or index data are taken into account. A basic DEA-R model without explicit inputs is formulated and a relation between output-oriented DEA models without explicit inputs and output-oriented DEA-R models is analyzed. Central resource allocation and slack-based measure models within DEA-R framework are examined. Finally they are used for projections of decision making units on the efficient frontier. The results of the proposed models are applied for efficiency evaluation of 15 units (Chinese research institutes) and they are discussed.     

Efficiency analysis in multi-period systems: an application to performance evaluation in Czech higher education

Data envelopment analysis (DEA) is a non-parametric method for efficiency and performance analysis of decision making units. The paper deals with production systems where decision making units are described by their inputs and outputs in several consecutive periods. The paper presents (Park and Park in Eur J Oper Res 193(2):567–580, 2009) multi-period DEA model that is oriented on the “best” period of the unit under evaluation only. This aim of this paper is to overcome the disadvantage of this model and formulate new models of this class that allow evaluation the efficiency of decision making units within the whole production chain. The presented efficiency and super-efficiency multi-period DEA models are illustrated on a case study. The study consists in analysis of research and teaching performance of 19 Czech economic faculties in four years period from 2009 until 2012. The model considers two inputs (number of academic employees and labour costs) and two outputs for teaching efficiency (number of students and number of graduated). Research efficiency is expressed using the number of publications in various important categories and the number of so called RIV points that describe the quality of publications.     

Radial DEA models without inputs or without outputs

In this paper we consider radial DEA models without inputs (or without outputs), and radial DEA models with a single constant input (or with a single constant output). We demonstrate that (i) a CCR model without inputs (or without outputs) is meaningless; (ii) a CCR model with a single constant input (or with a single constant output) coincides with the corresponding BCC model; (iii) a BCC model with a single constant input (or a single constant output) collapses to a BCC model without inputs (or without outputs); and (iv) all BCC models, including those without inputs (or without outputs), can be condensed to models having one less variable (the radial efficiency score) and one less constraint (the convexity constraint).     

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.     

Sensitivity and stability analysis on the first and second levels of efficiency score relative to data error

In data envelopment analysis (DEA), efficient decision making units (DMUs) are of primary importance as they define the efficient frontier. The current paper develops a new sensitivity analysis approach for a category DMUs and finds the stability radius for all efficient DMUs. By means of combining some classic DEA models and with the condition that the efficiency scores of efficient DMUs remain unchanged, we are able to determine what perturbations of the data can be tolerated before efficient DMUs become inefficient. Our approach generalizes the conventional sensitivity analysis approach in which the inputs of efficient DMUs increase and their outputs decrease, while the inputs of inefficient DMUs decrease and their outputs increase. We find the maximum quantity of perturbations of data so that all first level efficient DMUs remain at the same level.     

Selecting DEA specifications and ranking units via PCA

Data envelopment analysis (DEA) model selection is problematic. The estimated efficiency for any DMU depends on the inputs and outputs included in the model. It also depends on the number of outputs plus inputs. It is clearly important to select parsimonious specifications and to avoid as far as possible models that assign full high-efficiency ratings to DMUs that operate in unusual ways (mavericks). A new method for model selection is proposed in this paper. Efficiencies are calculated for all possible DEA model specifications. The results are analysed using Principal Component Analysis. It is shown that model equivalence or dissimilarity can be easily assessed using this approach. The reasons why particular DMUs achieve a certain level of efficiency with a given model specification become clear. The methodology has the additional advantage of producing DMU rankings. These rankings can always be established independently of whether the model is estimated under constant or under variable returns to scale.