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.     

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.     

An inverse optimization model for imprecise data envelopment analysis

Inverse data envelopment analysis (InDEA) is a well-known approach for short-term forecasting of a given decision-making unit (DMU). The conventional InDEA models use the production possibility set (PPS) that is composed of an evaluated DMU with current inputs and outputs. In this paper, we replace the fluctuated DMU with a modified DMU involving renewal inputs and outputs in the PPS since the DMU with current data cannot be allowed to establish the new PPS. Besides, the classical DEA models such as InDEA are assumed to consider perfect knowledge of the input and output values but in numerous situations, this assumption may not be realistic. The observed values of the data in these situations can sometimes be defined as interval numbers instead of crisp numbers. Here, we extend the InDEA model to interval data for evaluating the relative efficiency of DMUs. The proposed models determine the lower and upper bounds of the inputs of a given DMU separately when its interval outputs are changed in the performance analysis process. We aim to remain the current interval efficiency of a considered DMU and the interval efficiencies of the remaining DMUs fixed or even improve compared with the current interval efficiencies.     

Nash fuzzy equilibrium: Theory and application to a spatial duopoly

Consider a non-cooperative n-persons game. Each gambler has a set of mixed strategies at his disposal. The payoffs are some physical or immaterial objects. The game is a fuzzy game because (1) gamblers have more or less precise preferences for the payoffs and (2) the outcoming of payoffs is uncertain. The uncertainty can be expressed either by a distribution of possibility or by a distribution of probability. The product set of a gambler's mixed strategies is convex and compact and the payoff functions are continuous. Then a closed and convex fuzzy point-to-set mapping is defined on the product set of strategies and, by using a Butnariu theorem, the existence of a fixed point for this fuzzy point-to-set mapping is proved. The issue allows us to generalize a famous Nash result: a n-persons non-cooperative fuzzy game with mixed strategies has at least one equilibrium point. In the second part of the paper an economic application is devoted to the statement of the equilibrium existence conditions in a spatial duopoly. The model is not only more general than the classical ones, but also more relevant because new results are obtained.     

生产可能集为凸包时的相对有效性评价模型

给出了一个评价决策单元相对有效性的新的DEA模型,它所对应的生产可能集被称为凸包形生产可能集,同时讨论了该模型解的存在性,定义了决策单元技术DEA有效和"上投影"的概念,断定一个决策单元的"上投影"相对于原来的决策单元是技术DEA有效的。最后给出一个应用新模型进行设施农业效率评价的例子。     

Improving envelopment in Data Envelopment Analysis under variable returns to scale

In a Data Envelopment Analysis model, some of the weights used to compute the efficiency of a unit can have zero or negligible value despite of the importance of the corresponding input or output. This paper offers an approach to preventing inputs and outputs from being ignored in the DEA assessment under the multiple input and output VRS environment, building on an approach introduced in Allen and Thanassoulis (2004) for single input multiple output CRS cases. The proposed method is based on the idea of introducing unobserved DMUs created by adjusting input and output levels of certain observed relatively efficient DMUs, in a manner which reflects a combination of technical information and the decision maker’s value judgements. In contrast to many alternative techniques used to constrain weights and/or improve envelopment in DEA, this approach allows one to impose local information on production trade-offs, which are in line with the general VRS technology. The suggested procedure is illustrated using real data.     

基于电能节约的E-DEA模型及其应用

按照全要素能源效率的概念,重点考虑电能投入约束,构造了基于电能节约的E-DEA模型,其目标函数为极大化产出比例和电能投入比例之差,约束条件中除考虑一般投入量约束外,还同时强调电能投入径向节约和产出径向增加。根据模型最优解,给出了相应的有效、非有效、弱有效、用电规模收益状态的判断准则,以及相应于不同有效性情况下决策单元的改进。以合肥市通用制造业规上企业所属21个行业为研究对象,从第二次经济普查中选择年均资产、从业人员、电力、非电力能源、二氧化碳排量为投入指标,主营业务收入为产出指标,对行业电能利用效率进行实证分析,通过分析潜在电能可节约量和主营业务收入可增加量,明确了各行业改进目标。     

Optimal decisions in combining the SOM with nonlinear projection methods

Visual data mining is an efficient way to involve human in search for a optimal decision. This paper focuses on the optimization of the visual presentation of multidimensional data.A variety of methods for projection of multidimensional data on the plane have been developed. At present, a tendency of their joint use is observed. In this paper, two consequent combinations of the self-organizing map (SOM) with two other well-known nonlinear projection methods are examined theoretically and experimentally. These two methods are: Sammon’s mapping and multidimensional scaling (MDS). The investigations showed that the combinations (SOM_Sammon and SOM_MDS) have a similar efficiency. This grounds the possibility of application of the MDS with the SOM, because up to now in most researches SOM is applied together with Sammon’s mapping. The problems on the quality and accuracy of such combined visualization are discussed. Three criteria of different nature are selected for evaluation the efficiency of the combined mapping. The joint use of these criteria allows us to choose the best visualization result from some possible ones.Several different initialization ways for nonlinear mapping are examined, and a new one is suggested. A new approach to the SOM visualization is suggested.The obtained results allow us to make better decisions in optimizing the data visualization.     

DEA models for resource reallocation and production input/output estimation

This paper discusses the “inverse” data envelopment analysis (DEA) problem with preference cone constraints. An inverse DEA model can be used for a decision making unit (DMU) to estimate its input/output levels when some or all of its input/output entities are revised, given its current DEA efficiency level. The extension of introducing additional preference cones to the previously developed inverse DEA model allows the decision makers to incorporate their preferences or important policies over inputs/outputs into the production analysis and resource allocation process. We provide the properties of the inverse DEA problem through a discussion of its related multi-objective and weighted sum single-objective programming problems. Numerical examples are presented to illustrate the application procedure of our extended inverse DEA model. In particular, we demonstrate how to apply the model to the case of a local home electrical appliance group company for its resource reallocation decisions.     

On functions whose stationary points are global minima

In this paper, a characterization of functions whose stationary points are global minima is studied. By considering the level sets of a real function as a point-to-set mapping, and by examining its semicontinuity properties, we obtain the result that a real function, defined on a subset ofR ⁿ and satisfying some mild regularity conditions, belongs to the above family iff the point-to-set mapping of its level sets is strictly lower semicontinuous. Mathematical programming applications are also mentioned.The authors are thankful to an anonymous referee of an earlier version of this paper for his valuable comments. This research was partially supported by the Office of Naval Research, Contract No. N-00014-75-C-0267, by the National Science Foundation, Grant No. MPS-71-03341-A03, and by the US Energy Research and Development Administration, Contract No. E(04-3)-326 PA-18. An earlier version of this paper appeared as CORE Discussion Paper No. 7502. Part of this research was carried out while the first author was at CORE, Louvain, Belgium.     

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.     

Evaluating public school district performance via DEA gain functions

Existing measures of input allocative efficiency may be biased when estimated via data envelopment analysis (DEA) because of the possibility of slack in the constraints defining the reference technology. In this paper we derive a new measure of input allocative efficiency and compare it to existing measures. We measure efficiency by comparing the actual outputs of a decision-making unit relative to Koopmans’ efficient subset of the direct and indirect output possibility sets. We estimate the existing measures and our new measure of input allocative efficiency for a sample of public school districts operating in Texas.     

A new robust DEA model and super-efficiency measure

Applications of traditional data envelopments analysis (DEA) models require knowledge of crisp input and output data. However, the real-world problems often deal with imprecise or ambiguous data. In this paper, the problem of considering uncertainty in the equality constraints is analyzed and by using the equivalent form of CCR model, a suitable robust DEA model is derived in order to analyze the efficiency of decision-making units (DMUs) under the assumption of uncertainty in both input and output spaces. The new model based on the robust optimization approach is suggested. Using the proposed model, it is possible to evaluate the efficiency of the DMUs in the presence of uncertainty in a fewer steps compared to other models. In addition, using the new proposed robust DEA model and envelopment form of CCR model, two linear robust super-efficiency models for complete ranking of DMUs are proposed. Two different case studies of different contexts are taken as numerical examples in order to compare the proposed model with other approaches. The examples also illustrate various possible applications of new models.     

Measuring environmental efficiency of thermoelectric power plants: a common equilibrium efficient frontier DEA approach with fixed-sum undesirable output

China’s rapid development in economy has intensified many problems. One of the most important issues is the problem of environmental pollution. In this paper, a new DEA approach is proposed to measure the environmental efficiency of thermoelectric power plants, considering undesirable outputs. First, we assume that the total amount of undesirable outputs of any particular type is limited and fixed to current levels. In contrast to previous studies, this study requires fixed-sum undesirable outputs. In addition, the common equilibrium efficient frontier is constructed by using different input/output multipliers (or weights) for each different decision making unit (DMU), while previous approaches which considered fixed-sum outputs assumed a common input/output multiplier for all DMUs. The proposed method is applied to measure the environmental efficiencies of 30 thermoelectric power plants in mainland China. Our empirical study shows that half of the plants perform well in terms of environmental efficiency.

     

Variations on the theme of slacks-based measure of efficiency in DEA

In DEA, there are typically two schemes for measuring efficiency of DMUs; radial and non-radial. Radial models assume proportional change of inputs/outputs and usually remaining slacks are not directly accounted for inefficiency. On the other hand, non-radial models deal with slacks of each input/output individually and independently, and integrate them into an efficiency measure, called slacks-based measure (SBM). In this paper, we point out shortcomings of the SBM and propose four variants of the SBM model. The original SBM model evaluates efficiency of DMUs referring to the furthest frontier point within a range. This results in the hardest score for the objective DMU and the projection may go to a remote point on the efficient frontier which may be inappropriate as the reference. In an effort to overcome this shortcoming, we first investigate frontier (facet) structure of the production possibility set. Then we propose Variation I that evaluates each DMU by the nearest point on the same frontier as the SBM found. However, there exist other potential facets for evaluating DMUs. Therefore we propose Variation II that evaluates each DMU from all facets. We then employ clustering methods to classify DMUs into several groups, and apply Variation II within each cluster. This Variation III gives more reasonable efficiency scores with less effort. Lastly we propose a random search method (Variation IV) for reducing the burden of enumeration of facets. The results are approximate but practical in usage.     

在DEA中有关输出与输入的比值的模型的探讨

对以决策单元的输出与输入的比值为目标函数的多目标规划模型,证明了有关它与(弱)DEA有效(C2R)关系的三个定理.     

Generalized sensitivity indices based on vector projection for multivariate output

Analyzing the sensitivity of model outputs to inputs is important to assess risk and make decisions in engineering application. However, for model with multiple outputs, it is difficult to interpret the sensitivity index since the effect of the dimension and the correlation between multiple outputs are often ignored in the existing methods. In this paper, a new kind of sensitivity analysis method is proposed by use of vector projection and dimension normalization for multiple outputs. Through the dimension normalization, the space of multiple outputs can be unified into a dimensionless one to eliminate the effect of the dimension of the different output. After an affine coordinate system is constructed by considering the correlation of the multiple normalized outputs, a total variance vector for the multiple outputs can be composed by the individual variance of each output. Then, by projecting the variance contribution vector composed by the individual variance contribution of the input to each output on the total variance vector, the new sensitivity indices are proposed for measuring the comprehensive effect of the input on the total variance vector of multiple outputs, it is defined as the ratio of the projection of the variance contribution vector to the norm of the total variance vector. We derive that the Sobol’ indices for a scalar output and the covariance decomposition based indices for multiple outputs are special cases of the proposed vector projection based indices. Then, the mathematical properties and geometric interpretation of the proposed method are discussed. Three numerical examples and a rotating shaft model of an aircraft wing are used to validate the proposed method and show their potential benefits.     

Using Zionts–Wallenius method to improve estimate of value efficiency in DEA

Halme et al. [M. Halme, T. Joro, P. Korhonen, S. Salo, J. Wallenius, A value efficiency approach to incorporating preference information in data envelopment analysis, Management Science 45 (1) (1999) 103-115] proposed value efficiency analysis as an approach to incorporate preference information in Data Envelopment Analysis (DEA). In this paper, we develop some related concepts, and present a refinement to Halme et al.’s approach to measure value efficiency scores more precisely. For do this, we will introduce an MOLP model which its objective functions are input/output variables subject to the defining constraints of Production Possibility Set (PPS) of DEA models. Then by using the so-called Zionts–Wallenius method, we aid the Decision Maker (DM) in searching for the Most Preferred Solution (MPS) and generating input/output weights as the DM’s underlying value structure about objective functions. Finally, value efficiency scores are calculated by comparing the inefficient units to units having the same value as the MPS.     

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.     

Characteristics and structures of weak efficient surfaces of production possibility sets

This paper studies the characteristics and structure of the weak surface of the production possibility set. We apply techniques and methods of transferring a polyhedral cone from its intersection form to its sum form, identify an intersection representation of the production possibility set. We give the structure theorem of weak surface of the production possibility set, which includes three complementary slackness conditions. We define the input weak efficiency and output weak efficiency for different DEA models according to the representation of the intersection form. It investigates the characteristics of the weak surfaces, and proves the structure theorems of input weak DEA efficiency and output weak DEA efficiency. The structure theorems establish weighted combination of inputs and outputs that are weak DEA efficient. Numerical examples are provided for illustration.