DEA models for the explicit maximisation of relative efficiency

It has recently been demonstrated that incorporating weight bounds and other non-homogeneous restrictions in DEA models may lead to underestimation of the maximum relative efficiency of decision making units. This paper suggests a way of avoiding this by replacing the objective function in DEA models by the relative efficiency of the assessed unit and converting the resulting models to linear forms. An alternative approach based on incorporating weight restrictions in the recently introduced maximin DEA model is also considered. It is shown that imposing weight bounds in the maximin model is equivalent to imposing bounds on ratios of individual weights.     

Some remarks on the two-level DEA model

The two-level DEA model was introduced to increase the discriminational power of Data Envelopment Analysis (DEA) models. This nonlinear model was presented by Meng et al. (2008) [3], and then converted into a linear model by Kao (2008) [4].In this paper two subjects will be discussed: first, we show that the two-level DEA model is a special case of DEA models where weight restrictions are applied. Then, we express that the nonlinear model is equivalent to the conventional DEA model.     

Zero weights and non-zero slacks: Different solutions to the same problem

This paper re-assesses three independently developed approaches that are aimed at solving the problem of zero-weights or non-zero slacks in Data Envelopment Analysis (DEA). The methods are weights restricted, non-radial and extended facet DEA models. Weights restricted DEA models are dual to envelopment DEA models with restrictions on the dual variables (DEA weights) aimed at avoiding zero values for those weights; non-radial DEA models are envelopment models which avoid non-zero slacks in the input-output constraints. Finally, extended facet DEA models recognize that only projections on facets of full dimension correspond to well defined rates of substitution/transformation between all inputs/outputs which in turn correspond to non-zero weights in the multiplier version of the DEA model. We demonstrate how these methods are equivalent, not only in their aim but also in the solutions they yield. In addition, we show that the aforementioned methods modify the production frontier by extending existing facets or creating unobserved facets. Further we propose a new approach that uses weight restrictions to extend existing facets. This approach has some advantages in computational terms, because extended facet models normally make use of mixed integer programming models, which are computationally demanding.     

Assessing the relative efficiency of decision making units using DEA models with weight restrictions

In models of data envelopment analysis (DEA), an optimal set of input and output weights is generally assumed to represent the assessed decision making unit (DMU) in the best light in comparison to all the other DMUs. The paper shows that this may not be correct if absolute weight bounds or some other weight restrictions are added to the model. A consequence may be that the model will underestimate the relative efficiency of DMUs. The incorporation of weight restrictions in a maximin DEA model is suggested. This model can be further converted to more operational forms, which are similar to the classical DEA models.     

A bi-objective weighted model for improving the discrimination power in MCDEA

Lack of discrimination power and poor weight dispersion remain major issues in Data Envelopment Analysis (DEA). Since the initial multiple criteria DEA (MCDEA) model developed in the late 1990s, only goal programming approaches; that is, the GPDEA-CCR and GPDEA-BCC were introduced for solving the said problems in a multi-objective framework. We found GPDEA models to be invalid and demonstrate that our proposed bi-objective multiple criteria DEA (BiO-MCDEA) outperforms the GPDEA models in the aspects of discrimination power and weight dispersion, as well as requiring less computational codes. An application of energy dependency among 25 European Union member countries is further used to describe the efficacy of our approach.     

Ranking decision making units by imposing a minimum weight restriction in the data envelopment analysis

It has been widely recognized that data envelopment analysis (DEA) lacks discrimination power to distinguish between DEA efficient units. This paper proposes a new methodology for ranking decision making units (DMUs). The new methodology ranks DMUs by imposing an appropriate minimum weight restriction on all inputs and outputs, which is decided by a decision maker (DM) or an assessor in terms of the solutions to a series of linear programming (LP) models that are specially constructed to determine a maximin weight for each DEA efficient unit. The DM can decide how many DMUs to be retained as DEA efficient in final efficiency ranking according to the requirement of real applications, which provides flexibility for DEA ranking. Three numerical examples are investigated using the proposed ranking methodology to illustrate its power in discriminating between DMUs, particularly DEA efficient units.     

Avoiding infeasibility in DEA models with weight restrictions

The flexibility of weights assigned to inputs and outputs is a key aspect of DEA modeling. However, excessive weight variability and implausible weight values have led to the development of DEA models that incorporate weight restrictions, reflecting expert judgment. This in turn has created problems of infeasibility of the corresponding linear programs. We provide an existence theorem that establishes feasibility conditions for DEA multiplier programs with weight restrictions. We then propose a linear model that tests for feasibility and a nonlinear model that provides minimally acceptable adjustments to the original restrictions that render the program feasible. The analysis can be applied to restrictions on weight ratios, or to restrictions on virtual inputs or outputs.     

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.     

具有偏好锥的面向输出的超效率DEA模型分析

鉴于传统DEA模型无法区分有效决策单元,超效率DEA模型未考虑决策者的偏好,现提出面向输出的权重受限的综合超效率DEA模型及其投影概念,并讨论该模型与其他超效率DEA模型之间的关系.接着,分析模型的最优目标函数值与决策单元有效性之间的关系,并讨论面向输出的权重受限的综合超效投影与多目标规划问题的非支配解之间的关系.最后,通过对中国西部12个地区工业企业科技创新效率综合评价,并与原有方法进行比较研究,得出本文方法更具优势和合理性.     

A generalized model for weight restrictions in data envelopment analysis

Data envelopment analysis (DEA) is designed to maximize the efficiency of a given decision-making unit (DMU) relative to all other DMUs by the choice of a set of input and output weights. One strength of the original models is the absence of any need of a priori information about the process of transforming inputs into outputs. However, in the practical application of DEA models, this strength has also become a weakness. Incorporation of process knowledge is more a norm than an exception in practice, and typically involves placing constraints on the input and/or output weights. New DEA formulations have evolved to address this issue. However, existing formulations for weight restrictions may underestimate relative efficiency or even render a problem infeasible. A new model formulation is introduced to address this issue. This formulation represents a significant improvement over existing DEA models by providing a generalized, comprehensive treatment for weight restrictions.     

Congestion analysis in DEA inputs under weight restrictions

Data envelopment analysis (DEA), which is used to determine the efficiency of a decision-making unit (DMU), is able to recognize the amount of input congestion. Moreover, the relative importance of inputs and outputs can be incorporated into DEA models by weight restrictions. These restrictions or a priori weights are introduced by the decision maker and lead to changes in models and efficiency interpretation. In this paper, we present an approach to determine the value of congestion in inputs under the weight restrictions. Some discussions show how weight restrictions can affect the congestion amount.     

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.     

In the determination of weight sets to compute cross-efficiency ratios in DEA

Data envelopment analysis (DEA) measures the production performance of decision-making units (DMUs) which consume multiple inputs and produce multiple outputs. Although DEA has become a very popular method of performance measure, it still suffers from some shortcomings. For instance, one of its drawbacks is that multiple solutions exist in the linear programming solutions of efficient DMUs. The obtained weight set is just one of the many optimal weight sets that are available. Then why use this weight set instead of the others especially when this weight set is used for cross-evaluation? Another weakness of DEA is that extremely diverse or unusual values of some input or output weights might be obtained for DMUs under assessment. Zero input and output weights are not uncommon in DEA. The main objective of this paper is to develop a new methodology which applies discriminant analysis, super-efficiency DEA model and mixed-integer linear programming to choose suitable weight sets to be used in computing cross-evaluation. An advantage of this new method is that each obtained weight set can reflect the relative strengths of the efficient DMU under consideration. Moreover, the method also attempts to preserve the original classificatory result of DEA, and in addition this method produces much less zero weights than DEA in our computational results.     

A new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process

The DEAHP method for weight deviation and aggregation in the analytic hierarchy process (AHP) has been found flawed and sometimes produces counterintuitive priority vectors for inconsistent pairwise comparison matrices, which makes its application very restrictive. This paper proposes a new data envelopment analysis (DEA) method for priority determination in the AHP and extends it to the group AHP situation. In this new DEA methodology, two specially constructed DEA models that differ from the DEAHP model are used to derive the best local priorities from a pairwise comparison matrix or a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent. The new DEA method produces true weights for perfectly consistent pairwise comparison matrices and the best local priorities that are logical and consistent with decision makers (DMs)’ subjective judgments for inconsistent pairwise comparison matrices. In hierarchical structures, the new DEA method utilizes the simple additive weighting (SAW) method for aggregation of the best local priorities without the need of normalization. Numerical examples are examined throughout the paper to show the advantages of the new DEA methodology and its potential applications in both the AHP and group decision making.     

Measuring the performance of nations at the Olympic Games using DEA models with different preferences

It is well known that Olympic Games often use Lexicographic preference to rank the nations that won the medals. However, Lexicographic preference is not the underlying preference that is used in the standard DEA models. Hence, we discuss the issue of the underlying preferences in DEA models for measuring the performance of nations at the Olympic Games, and then propose new DEA models with Lexicographic preference to measure the performance of the nations.     

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.     

Network DEA pitfalls: Divisional efficiency and frontier projection under general network structures

Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). Recently network DEA models been developed to examine the efficiency of DMUs with internal structures. The internal network structures range from a simple two-stage process to a complex system where multiple divisions are linked together with intermediate measures. In general, there are two types of network DEA models. One is developed under the standard multiplier DEA models based upon the DEA ratio efficiency, and the other under the envelopment DEA models based upon production possibility sets. While the multiplier and envelopment DEA models are dual models and equivalent under the standard DEA, such is not necessarily true for the two types of network DEA models. Pitfalls in network DEA are discussed with respect to the determination of divisional efficiency, frontier type, and projections. We point out that the envelopment-based network DEA model should be used for determining the frontier projection for inefficient DMUs while the multiplier-based network DEA model should be used for determining the divisional efficiency. Finally, we demonstrate that under general network structures, the multiplier and envelopment network DEA models are two different approaches. The divisional efficiency obtained from the multiplier network DEA model can be infeasible in the envelopment network DEA model. This indicates that these two types of network DEA models use different concepts of efficiency. We further demonstrate that the envelopment model’s divisional efficiency may actually be the overall efficiency.     

Value-based DEA models: application-driven developments

Data envelopment analysis (DEA) is an approach based on linear programming to assess the relative efficiency of peer decision-making units (DMUs). Typically, each DMU is free to choose the weights of the factors used in its evaluation. However, the evaluator's preferences may not warrant so much freedom. Several approaches have been proposed to allow the incorporation of managerial preferences in DEA, but few address the additive DEA model specifically. This paper presents additive DEA models that use multi-criteria decision analysis concepts to incorporate managerial preferences, and presents the corresponding preference elicitation protocols. The models developed allow the incorporation of preferences at different levels: on valuing performance improvements, on introducing weight restrictions, and on finding adequate targets. These were application-driven developments, resulting from discussing modelling options and preliminary results with the top-level management of a retail chain in the context of an assessment of stores’ performance, also described in this paper.     

Linear transformations to decrease computational requirements of solving some known linear programming models

Imposing additional weight restrictions increases the degeneracy and computational complexity of solving Data Envelopment Analysis (DEA) models, a known class of linear programming models. In this paper some linear transformations to reduce these problems are provided.     

On some generalization of the DEA models

Inadequate results may arise in some instances of DEA model applications. For example, a data envelopment analysis (DEA) model may show ‘a notoriously inefficient unit’ as an efficient one. In addition, too many efficient units may appear in some DEA models. An elegant and subtle approach was proposed to deal with these problems, which is based on incorporating domination cones in DEA models. Yu, Wei and Brockett suggested the generalized DEA (GDEA) model that unifies and extends most of the well-known DEA models based on using domination cones. In this paper, we propose a model that is more general than the GDEA model, on the one hand, as it covers situations that the GDEA model cannot describe. On the other hand, our model enables one to construct step-by-step any model from the family of the GDEA models by incorporating artificial units and rays in the space of inputs and outputs in the Banker, Charnes, Cooper (BCC) model, which makes the process of model construction visible and more understandable. Moreover, we show that any GDEA model can be approximated by some BCC model.