Dominance stochastic models in data envelopment analysis

In this paper stochastic models in data envelopment analysis (DEA) are developed by taking into account the possibility of random variations in input-output data, and dominance structures on the DEA envelopment side are used to incorporate the modelbuilder's preferences and to discriminate efficiencies among decision making units (DMUs). The efficiency measure for a DMU is defined via joint dominantly probabilistic comparisons of inputs and outputs with other DMUs and can be characterized by solving a chance constrained programming problem. Deterministic equivalents are obtained for multivariate symmetric random errors and for a single random factor in the production relationships. The goal programming technique is utilized in deriving linear deterministic equivalents and their dual forms. The relationship between the general stochastic DEA models and the conventional DEA models is also discussed.     

A new chance-constrained DEA model with birandom input and output data

The purpose of conventional Data Envelopment Analysis (DEA) is to evaluate the performance of a set of firms or Decision-Making Units using deterministic input and output data. However, the input and output data in the real-life performance evaluation problems are often stochastic. The stochastic input and output data in DEA can be represented with random variables. Several methods have been proposed to deal with the random input and output data in DEA. In this paper, we propose a new chance-constrained DEA model with birandom input and output data. A super-efficiency model with birandom constraints is formulated and a non-linear deterministic equivalent model is obtained to solve the super-efficiency model. The non-linear model is converted into a model with quadratic constraints to solve the non-linear deterministic model. Furthermore, a sensitivity analysis is performed to assess the robustness of the proposed super-efficiency model. Finally, two numerical examples are presented to demonstrate the applicability of the proposed chance-constrained DEA model and sensitivity analysis.     

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.     

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.     

基于白化值的区间DEA建模

传统数据包络分析要求输入输出数据为精确数,然而在某些实际应用中,区间形式的数据相较于精确数更容易获得.将区间数转化为白化值,并基于传统C~2R模型提出了基于白化值的区间C~2R模型.考虑到决策单元的有效性不易通过基于白化值的区间C~2R模型来判断,因此将非阿基米德无穷小概念引入到上述模型,构建了具有非阿基米德无穷小的区间C~2R模型.此外,还给出了用于判断决策单元有效性的区间目标规划方法:分别通过G_(IC~2R)模型和WG_(IC~2R)模型判断决策单元是否为区间DEA有效与区间弱DEA有效.     

Data envelopment analysis with imprecise data

In original data envelopment analysis (DEA) models, inputs and outputs are measured by exact values on a ratio scale. Cooper et al. [Management Science, 45 (1999) 597–607] recently addressed the problem of imprecise data in DEA, in its general form. We develop in this paper an alternative approach for dealing with imprecise data in DEA. Our approach is to transform a non-linear DEA model to a linear programming equivalent, on the basis of the original data set, by applying transformations only on the variables. Upper and lower bounds for the efficiency scores of the units are then defined as natural outcomes of our formulations. It is our specific formulation that enables us to proceed further in discriminating among the efficient units by means of a post-DEA model and the endurance indices. We then proceed still further in formulating another post-DEA model for determining input thresholds that turn an inefficient unit to an efficient one.     

DEA中连续C~2R模型理论的研究

在DEA中的C2R模型的基础上,针对决策单元输入与输出为[0,1]区间上的连续函数,建立了在一个时间区间内评价决策单元间的相对有效性的连续C2R模型以及其对偶模型,同时给出了决策单元的效率定义和弱DEA有效、DEA有效的定义.同时得到了弱对偶定理,从而初步构建了连续C2R模型的理论体系.     

Modeling undesirable factors in data envelopment analysis

Data envelopment analysis is a mathematical programming technique for identifying efficient frontiers for peer decision making 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. In the current paper, we present a data envelopment analysis (DEA) model in which can be used to improve the relative performance via increasing undesirable inputs and decreasing undesirable outputs.     

A variant of radial measure capable of dealing with negative inputs and outputs in data envelopment analysis

Data envelopment analysis (DEA) is a linear programming methodology to evaluate the relative technical efficiency for each member of a set of peer decision making units (DMUs) with multiple inputs and multiple outputs. It has been widely used to measure performance in many areas. A weakness of the traditional DEA model is that it cannot deal with negative input or output values. There have been many studies exploring this issue, and various approaches have been proposed.     

A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making

Data envelopment analysis (DEA) is a methodology for measuring the relative efficiencies of a set of decision making units (DMUs) that use multiple inputs to produce multiple outputs. Crisp input and output data are fundamentally indispensable in conventional DEA. However, the observed values of the input and output data in real-world problems are sometimes imprecise or vague. Many researchers have proposed various fuzzy methods for dealing with the imprecise and ambiguous data in DEA. In this study, we provide a taxonomy and review of the fuzzy DEA methods. We present a classification scheme with four primary categories, namely, the tolerance approach, the α-level based approach, the fuzzy ranking approach and the possibility approach. We discuss each classification scheme and group the fuzzy DEA papers published in the literature over the past 20 years. To the best of our knowledge, this paper appears to be the only review and complete source of references on fuzzy DEA.     

Additive super-efficiency in integer-valued data envelopment analysis

Conventional data envelopment analysis (DEA) methods assume that input and output variables are continuous. However, in many real managerial cases, some inputs and/or outputs can only take integer values. Simply rounding the performance targets to the nearest integers can lead to misleading solutions and efficiency evaluation. Addressing this kind of integer-valued data, the current paper proposes models that deal directly with slacks to calculate efficiency and super-efficiency scores when integer values are present. Compared with standard radial models, additive (super-efficiency) models demonstrate higher discrimination power among decision making units, especially for integer-valued data. We use an empirical application in early-stage ventures to illustrate our approach.     

随机逆DEA模型的输出估计研究

逆DEA模型讨论了在保持决策单元的效率指数(即最优值)不变的情况下,当输入水平给定时估计输出值.在逆DEA模型的基础上研究了效率指数提高的输出估计,讨论了带有随机因素的情况,将该问题转化成机会约束的线性规划问题,并用数值算例加以说明.     

A new proposed model of restricted data envelopment analysis by correlation coefficients

The concept of efficiency in data envelopment analysis (DEA) is defined as weighted sum of outputs/weighted sum of inputs. In order to calculate the maximum efficiency score, each decision making unit (DMU)’s inputs and outputs are assigned to different weights. Hence, the classical DEA allows the weight flexibility. Therefore, even if they are important, the inputs or outputs of some DMUs can be assigned zero (0) weights. Thus, these inputs or outputs are neglected in the evaluation. Also, some DMUs may be defined as efficient even if they are inefficient. This situation leads to unrealistic results. Also to eliminate the problem of weight flexibility, weight restrictions are made in DEA. In our study, we proposed a new model which has not been published in the literature. We describe it as the restricted data envelopment analysis ((ARIII(COR))) model with correlation coefficients. The aim for developing this new model, is to take into account the relations between variables using correlation coefficients. Also, these relations were added as constraints to the CCR and BCC models. For this purpose, the correlation coefficients were used in the restrictions of input–output each one alone and their combination together. Inputs and outputs are related to the degree of correlation between each other in the production. Previous studies did not take into account the relationship between inputs/outputs variables. So, only with expert opinions or an objective method, weight restrictions have been made. In our study, the weights for input and output variables were determined, according to the correlations between input and output variables. The proposed new method is different from other methods in the literature, because the efficiency scores were calculated at the level of correlations between the input and/or output variables.     

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.     

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.     

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.     

A one-model approach based on relaxed combinations of inputs for evaluating input congestion in DEA

This paper provides a one-model approach of input congestion based on input relaxation model developed in data envelopment analysis (e.g. [G.R. Jahanshahloo, M. Khodabakhshi, Suitable combination of inputs for improving outputs in DEA with determining input congestion — Considering textile industry of China, Applied Mathematics and Computation (1) (2004) 263–273; G.R. Jahanshahloo, M. Khodabakhshi, Determining assurance interval for non-Archimedean ele improving outputs model in DEA, Applied Mathematics and Computation 151 (2) (2004) 501–506; M. Khodabakhshi, A super-efficiency model based on improved outputs in data envelopment analysis, Applied Mathematics and Computation 184 (2) (2007) 695–703; M. Khodabakhshi, M. Asgharian, An input relaxation measure of efficiency in stochastic data analysis, Applied Mathematical Modelling 33 (2009) 2010–2023]. This approach reduces solving three problems with the two-model approach introduced in the first of the above-mentioned reference to two problems which is certainly important from computational point of view. The model is applied to a set of data extracted from ISI database to estimate input congestion of 12 Canadian business schools.     

Integrated data envelopment analysis: Global vs. local optimum

Chiou et al. (2010) (A joint measurement of efficiency and effectiveness for non-storable commodities: integrated data envelopment analysis approaches. European Journal of Operational Research 201, 477–489) propose an integrated data envelopment analysis model in measuring decision making units (DMUs) that have a two-stage internal network structure with multiple inputs, outputs, and consumptions. They claim that any optimal solutions determined by their DEA model are a global optimum, not a local optimum. We show that such a conclusion is a false statement due to their misuse of Hessian matrix in examining the concavity of the objective function, and their DEA model is actually a non-convex optimization problem. As a result, their DEA model is unusable in practice due to a lack of efficient algorithm for this particular non-convex DEA model. We further show that Chiou et al.’s (2010) model is a special case of a well-known two-stage network DEA model, and it can be transformed into a parametric linear program for which an approximate global optimal solution can be obtained by solving a sequence of linear programs in combination with a simple search algorithm.     

Data envelopment scenario analysis with imprecise data

In the existing DEA models, we have a centralized decision maker (DM) who supervises all the operating units. In this paper, we solve a problem in which the centralized DM encounters limited or constant resources for total inputs or total outputs. We establish a DEA target model that solves and deals with such a situation. In our model, we consider the decrease of total input consumption and the increase of total output production; however, in the existing DEA models, total output production is guaranteed not to decrease. Considering the importance of imprecise data in organizations, we define our model so as to deal with interval and ordinal data. A numerical illustration is provided to show the application of our model and the advantages of our approach over the previous one.