Using data envelopment analysis to measure the efficiency of not-for-profit organizations: A critical evaluation—comment

This note responds to Nunamaker (1985) who supposedly deals with deficiencies in Data Envelopment Analysis (DEA) as an approach for (1) measuring efficiencies of not-for-profit entities identified as Decision Making Units (DMUs) and (2) locating sources and amounts of inefficiencies in each of the inputs used and in each of the outputs produced by each DMU. Corrections and comments are offered with references supplied for interested readers who wish to examine more detailed treatments of the topics covered.     

Stochastic DEA Models With Different Types of Input-Output Disturbances

This paper presents stochasticmodels in data envelopment analysis (DEA) for the possibilityof variations in inputs and outputs. Efficiency measure of adecision making unit (DMU) is defined via joint probabilisticcomparisons of inputs and outputs with other DMUs and can becharacterized by solving a chance constrained programming problem.By utilizing the theory of chance constrained programming, deterministicequivalents are obtained for both situations of multivariatesymmetric random disturbances and a single random factor in theproduction relationships. The linear deterministic equivalentand its dual form are obtained via the goal programming theoryunder the assumption of the single random factor. An analysisof stochastic variable returns to scale is developed using theidea of stochastic supporting hyperplanes. The relationshipsof our stochastic DEA models with some conventional DEA modelsare also discussed.     

Visualizing Efficiency and Reference Relations in Data Envelopment Analysis with an Application to the Branches of a German Bank

The interest in Data Envelopment Analysis (DEA) as a method for analyzing the productivity of homogeneous Decision Making Units (DMUs) has significantly increased in recent years. One of the main goals of DEA is to measure for each DMU its production efficiency relative to the other DMUs under analysis. Apart from a relative efficiency score, DEA also provides reference DMUs for inefficient DMUs. An inefficient DMU has, in general, more than one reference DMU, and an efficient DMU may be a reference unit for a large number of inefficient DMUs. These reference and efficiency relations describe a net which connects efficient and inefficient DMUs. We visualize this net by applying Sammons mapping. Such a visualization provides a very compact representation of the respective reference and efficiency relations and it helps to identify for an inefficient DMU efficient DMUs respectively DMUs with a high efficiency score which have a similar structure and can therefore be used as models. Furthermore, it can also be applied to visualize potential outliers in a very efficient way.JEL Classification: C14, C61, D24, M2     

Performance evaluation using data envelopment analysis in the presence of time lags

Data Envelopment Analysis (DEA) is a methodology that computes efficiency values for decision making units (DMU) in a given period by comparing the outputs with the inputs. In many applications, inputs and outputs of DMUs are monitored over time. There might be a time lag between the consumption of inputs and the production of outputs. We develop an approach that aims to capture the time lag between the outputs and the inputs in assigning the efficiency values to DMUs. We propose using weight restrictions in conjunction with the model. Our computational results on randomly generated problems demonstrate that the developed approach works well under a large variety of experimental conditions. We also apply our approach on a real data set to evaluate research institutions.     

基于DEA模型的我国轻工业经营效率研究

运用DEA模型对我国31个省区市轻工业的经营效率进行了实证研究和分析,结果表明,大部分省份轻工业的经营效率是非DEA有效的,尤其是沿海发达地区的轻工业大省。针对此现象进行了深入分析,并提出了相应的对策。     

Chance Constrained Programming Formulations for Stochastic Characterizations of Efficiency and Dominance in DEA

Pareto-Koopmans efficiency in Data Envelopment Analysis (DEA) is extended to stochastic inputs and outputs via probabilistic input-output vector comparisons in a given empirical production (possibility) set. In contrast to other approaches which have used Chance Constrained Programming formulations in DEA, the emphasis here is on joint chance constraints. An assumption of arbitrary but known probability distributions leads to the P-Model of chance constrained programming. A necessary condition for a DMU to be stochastically efficient and a sufficient condition for a DMU to be non-stochastically efficient are provided. Deterministic equivalents using the zero order decision rules of chance constrained programming and multivariate normal distributions take the form of an extended version of the additive model of DEA. Contacts are also maintained with all of the other presently available deterministic DEA models in the form of easily identified extensions which can be used to formalize the treatment of efficiency when stochastic elements are present.     

Sensitivity Analysis of an Efficient DMU in DEA Model with Variable Returns to Scale (VRS)

In this paper we consider the Variable Returns to Scale (VRS) Data Envelopment Analysis (DEA) model. In a DEA model each Decision Making Unit (DMU) is classified either as efficient or inefficient. Changes in inputs or outputs of any DMU can alter its classification, i.e. an efficient DMU can become inefficient and vice versa. The goal of this paper is to assess changes in inputs and outputs of an extreme efficient DMU that will not alter its efficiency status, thus obtaining the region of efficiency for that DMU. Namely, a DMU will remain efficient if and only if after applying changes this DMU stays in that region. The representation of this region is done using an iterative procedure. In the first step an extended DEA model, whereby a DMU under evaluation is excluded from the reference set, is used. In the iterative part of the procedure, by using the obtained optimal simplex tableau we apply parametric programming, thus moving from one facet to the adjacent one. At the end of the procedure we obtain the complete region of efficiency for a DMU under consideration.     

Centralized Resource Allocation Using Data Envelopment Analysis

While conventional DEA models set targets separately for each DMU, in this paper we consider that there is a centralized decision maker (DM) who “owns” or supervises all the operating units. In such intraorganizational scenario the DM has an interest in maximizing the efficiency of individual units at the same time that total input consumption is minimized or total output production is maximized. Two new DEA models are presented for such resource allocation. One type of model seeks radial reductions of the total consumption of every input while the other type seeks separate reductions for each input according to a preference structure. In both cases, total output production is guaranteed not to decrease. The two key features of the proposed models are their simplicity and the fact that both of them project all DMUs onto the efficient frontier. The dual formulation shows that optimizing total input consumption and output production is equivalent to finding weights that maximize the relative efficiency of a virtual DMU with average inputs and outputs. A graphical interpretation as well as numerical results of the proposed models are presented.     

A new methodology to measure efficiencies of inputs (outputs) of decision making units in Data Envelopment Analysis with application to agriculture

This paper aims at developing a new methodology to measure and decompose global DMU efficiency into efficiency of inputs (or outputs). The basic idea rests on the fact that global DMU's efficiency score might be misleading when managers proceed to reallocate their inputs or redefine their outputs. Literature provides a basic measure for global DMU's efficiency score. A revised model was developed for measuring efficiencies of global DMUs and their inputs (or outputs) efficiency components, based on a hypothesis of virtual DMUs. The present paper suggests a method for measuring global DMU efficiency simultaneously with its efficiencies of inputs components, that we call Input decomposition DEA model (ID-DEA), and its efficiencies of outputs components, that we call output decomposition DEA model (OD-DEA). These twin models differ from Supper efficiency model (SE-DEA) and Common Set Weights model (CSW-DEA). The twin models (ID-DEA, OD-DEA) were applied to agricultural farms, and the results gave different efficiency scores of inputs (or outputs), and at the same time, global DMU's efficiency score was given by the Charnes, Cooper and Rhodes (Charnes et al., 1978) [1], CCR78 model. The rationale of our new hypothesis and model is the fact that managers don't have the same information level about all inputs and outputs that constraint them to manage resources by the (global) efficiency scores. Then each input/output has a different reality depending on the manager's decision in relationship to information available at the time of decision. This paper decomposes global DMU's efficiency into input (or output) components' efficiencies. Each component will have its score instead of a global DMU score. These findings would improve management decision making about reallocating inputs and redefining outputs. Concerning policy implications of the DEA twin models, they help policy makers to assess, ameliorate and reorient their strategies and execute programs towards enhancing the best practices and minimising losses.     

Projections Onto Efficient Frontiers: Theoretical and Computational Extensions to DEA

Data Envelopment Analysis (DEA) has been widely studied in the literature since its inception in 1978. The methodology behind the classical DEA, the oriented method, is to hold inputs (outputs) constant and to determine how much of an improvement in the output (input) dimensions is necessary in order to become efficient. This paper extends this methodology in two substantive ways. First, a method is developed that determines the least-norm projection from an inefficient DMU to the efficient frontier in both the input and output space simultaneously, and second, introduces the notion of the observable frontier and its subsequent projection. The observable frontier is the portion of the frontier that has been experienced by other DMUs (or convex combinations of such) and thus, the projection onto this portion of the frontier guarantees a recommendation that has already been demonstrated by an existing DMU or a convex combination of existing DMUs. A numerical example is used to illustrate the importance of these two methodological extensions.     

A novel method for computing single output for DEA with application in hospital efficiency

There are two main methods for measuring the efficiency of decision-making units (DMUs): data envelopment analysis (DEA) and stochastic frontier analysis (SFA). Each of these methods has advantages and disadvantages. DEA is more popular in the literature due to its simplicity, as it does not require any pre-assumption and can be used for measuring the efficiency of DMUs with multiple inputs and multiple outputs, whereas SFA is a parametric approach that is applicable to multiple inputs and a single output. Since many applied studies feature multiple output variables, SFA cannot be used in such cases. In this research, a unique method to transform multiple outputs to a virtual single output is proposed. We are thus able to obtain efficiency scores from calculated virtual single output by the proposed method that are close (or even the same depending on targeted parameters at the expense of computation time and resources) to the efficiency scores obtained from multiple outputs of DEA. This will enable us to use SFA with a virtual single output. The proposed method is validated using a simulation study, and its usefulness is demonstrated with real application by using a hospital dataset from Turkey.     

Hierarchies and Groups in DEA

Conventional applications of data envelopment analysis (DEA) presume the existence of a set of similar decision making units, wherein each unit is evaluated relative to other members of the set. Often, however, the DMUs fall naturally into groupings, giving rise first to the problem of how to view the groups themselves as DMUs, and second to the issue of how to deal with several different ratings for any given DMU when groupings can be formed in different ways. In the present paper we introduce the concept of hierarchical DEA, where efficiency can be viewed at various levels. We provide a means for adjusting the ratings of DMUs at one level to account for the ratings received by the groups (into which these DMUs fall) at a higher level. We also develop models for aggregating different ratings for a DMU arising from different possible groupings. An application of these models to a set of power plants is given.     

Sensitivity and Stability Analysis in DEA: Some Recent Developments

This papersurveys recently developed analytical methods for studying thesensitivity of DEA results to variations in the data. The focusis on the stability of classification of DMUs (Decision MakingUnits) into efficient and inefficient performers. Early workon this topic concentrated on developing solution methods andalgorithms for conducting such analyses after it was noted thatstandard approaches for conducting sensitivity analyses in linearprogramming could not be used in DEA. However, some of the recentwork we cover has bypassed the need for such algorithms. Evolvingfrom early work that was confined to studying data variationsin only one input or output for only one DMU at a time, the newermethods described in this paper make it possible to determineranges within which all data may be varied for any DMU beforea reclassification from efficient to inefficient status (or vice versa) occurs. Other coverage involves recent extensionswhich include methods for determining ranges of data variationthat can be allowed when all data are varied simultaneously for all DMUs. An initial section delimits the topics to be covered.A final section suggests topics for further research.     

Assessment and budget allocation of Iranian natural gas distribution company- A CSW DEA based model

Regarding the importance of budgeting in organizations, this research proposes an empirical approach to budget allocation problems. The methodological instrument utilized is data envelopment analysis (DEA) which is a nonparametric mathematical programming technique. In the DEA methodology a standard DEA model should be independently solved to evaluate each decision making unit (DMU). Consequently, it is hard to find the magnitude of budget for each DMU by applying a budget allocation model based on standard DEA models because identifying the DMU under evaluation is problematic. Also, to overcome problems of evaluation using standard DEA models, common set of weights (CSW) DEA models were suggested. These models can be developed for use in budget allocation DEA models that lead to finding a single magnitude of budget for each DMU. Moreover, the opinion of the decision maker can be incorporated into the model using budgetary constraints. As a result, a restricted linear budget allocation CSW DEA model is proposed in which the central authority would like to plan for improving the total efficiency scores of all DMUs. In essence, the proposed model is used to reallocate the available budget and, thus, the results obtained will be a suggestion for budget allocation in subsequent periods. Finally, the proposed model is applied to budget allocation in the Iranian gas industry in which the available budget is reallocated to increase the total efficiency scores of Iranian gas distribution branches.     

Marginal Rates and Elasticities of Substitution with Additive Models in DEA

Marginal rates and elasticities of substitution are derived from the optimal slack values obtained from modified versions of additive DEA models. Projection formulas are used to ensure that all points are on the efficient frontier as required for conformance with assumptions in micro-economics. The models used differ from standard versions in that slack values are allowed to be negative as well as positive in these additive models. This makes movement possible on efficiency frontiers, where improvement in some inputs or outputs requires worsening other inputs or outputs. A new definition is therefore introduced in which efficiency is attained only if the value of the worsenings is exactly offset by the value of the improvements. This includes, but is not restricted to, the case in which all slacks must be zero for full attainment of efficiency—as in standard versions of additive models.     

Are all Scales Optimal in DEA? Theory and Empirical Evidence

Policy recommendations concerning optimal scale of production units may have serious implications for the restructuring of a sector. The piecewise linear frontier production function framework (DEA) is becoming the most popular one for assessing not only technical efficiency of operations, but also for scale efficiency and calculation of optimal scale sizes. The main purpose of the present study is to investigate if neoclassical production theory gives any guidance as to the nature of scale properties in the DEA model, and empirically explore such properties. Theoretical results indicate that the DEA model may have more irregular properties than usually assumed in neoclassical production theory, concerning shape of optimal scale curves and the M-locus. The empirical results indicate that optimal scale may be found over almost the entire size variations in outputs and inputs, thus making policy recommendations about efficient scale difficult. It seems necessary to establish the nature of optimal scale before any practical conclusions can be drawn. Proposals for indexes characterizing the nature of optimal scale are provided.     

基于DEA的安徽省可持续发展技术创新评价

阐述了DEA方法及区域可持续发展的技术创新的定义与内涵,提出了以区域技术创新为投入,以整个区域包括经济、社会、资源、环境四个方面的复合系统为产出的决策单元(DMU)概念,建立了旨在区别同为有效的DMU的相对效率指数DEA方法。运用该方法,对安徽省近六年来可持续发展的技术创新进行了评价比较。     

Using data envelopment analysis to evaluate efficiency in the economic performance of Chinese cities

This paper studies the use of DEA (data envelopment analysis) as a tool for possible use in evaluating and planning the economic performance of China's cities (28 in all) which play a critical role in the government's program of economic development. DEA promises advantages which include the absence of any need for the assignment of weights on an a priori basis (to reflect the supposed relative importance of various outputs or inputs) when evaluating technical efficiency. It is also unnecessary to explicitly specify underlying functions that are intended to prescribe the analytical form of the relations between inputs and outputs. Finally, as is illustrated in the paper, DEA can be used to identify sources, and estimate amounts of inefficiencies in each city's performance as well as to identify returns-to-scale possibilities in ways that seem well-suited to the mixture of centralized and decentralized planning and performance that China is currently trying to use.     

Efficiency and Global Scale Characteristics on the “No Free Lunch” Assumption Only

In a production technology, the type of returns to scale (RTS) associated with an efficient decision making unit (DMU) is indicative of the direction of marginal rescaling that the DMU should undertake in order to improve its productivity. In this paper a concept of global returns to scale (GRS) is developed as an indicator of the direction in which the most productive scale size (MPSS) of an efficient DMU is achieved. The GRS classes are useful in assisting strategic decisions like those involving mergers of units or splitting into smaller firms. The two characterisations, RTS and GRS, are the same in a convex technology but generally different in a non-convex one. It is shown that, in a non-convex technology, the well-known method of testing RTS proposed by Färe et al. is in fact testing for GRS and not RTS. Further, while there are three types of RTS: constant, decreasing and increasing (CRS, DRS and IRS, respectively), the classification according to GRS includes the fourth type of sub-constant GRS, which describes a DMU able to achieve its MPSS by both reducing and increasing the scale of operations. The notion of GRS is applicable to a wide range of technologies, including the free disposal hull (FDH) and all polyhedral technologies used in data envelopment analysis (DEA).     

Some extensions of techniques to handle non-discretionary factors in data envelopment analysis

Data Envelopment Analysis (DEA) assumes, in most cases, that all inputs and outputs are controlled by the Decision Making Unit (DMU). Inputs and/or outputs that do not conform to this assumption are denoted in DEA asnon-discretionary (ND) factors. Banker and Morey [1986] formulated several variants of DEA models which incorporated ND with ordinary factors. This article extends the Banker and Morey approach for treating nondiscretionary factors in two ways. First, the model is extended to allow for thesimultaneous presence of ND factors in both the input and the output sets. Second, a generalization is offered which, for the first time, enables a quantitative evaluation ofpartially controlled factors. A numerical example is given to illustrate the different models.The editor for this paper was Wade D. Cook.