只有产出(投入)BC~2模型中决策单元效率的几何刻画

在DEA方法中,DEA有效和弱DEA有效的决策单元位于生产前沿面上,非弱DEA有效的DEA无效决策单元位于生产可能集的内部而非生产前沿面上.通过引入生产可能集与生产前沿面移动的思想,证明只有产出(投入)的BC²模型评价下的决策单元的最优值与相应的生产前沿面的移动值存在倒数关系,以双产出(投入)情形图示说明,明确了决策单元在生产可能集中所处的位置.     

The interval efficiency based on the optimistic and pessimistic points of view

Data envelopment analysis (DEA) is a data-oriented approach for evaluating the performances of a set of peer entities called decision-making units (DMUs), whose performance is determined based on multiple measures. The traditional DEA, which is based on the concept of efficiency frontier (output frontier), determines the best efficiency score that can be assigned to each DMU. Based on these scores, DMUs are classified into DEA-efficient (optimistic efficient) or DEA-non-efficient (optimistic non-efficient) units, and the DEA-efficient DMUs determine the efficiency frontier. There is a comparable approach which uses the concept of inefficiency frontier (input frontier) for determining the worst relative efficiency score that can be assigned to each DMU. DMUs on the inefficiency frontier are specified as DEA-inefficient or pessimistic inefficient, and those that do not lie on the inefficient frontier, are declared to be DEA-non-inefficient or pessimistic non-inefficient. In this paper, we argue that both relative efficiencies should be considered simultaneously, and any approach that considers only one of them will be biased. For measuring the overall performance of the DMUs, we propose to integrate both efficiencies in the form of an interval, and we call the proposed DEA models for efficiency measurement the bounded DEA models. In this way, the efficiency interval provides the decision maker with all the possible values of efficiency, which reflect various perspectives. A numerical example is presented to illustrate the application of the proposed DEA models.     

基于复合DEA模型高校办学效益评价方法研究

传统DEA只能在固定投入(或产出)的情况下,将产出(或投入)尽量扩大(或缩小),而造成决策单元非DEA有效的原因,既有投入方面的因素,也有产出方面的因素,是由投入和产出两方面的原因共同造成的,而不仅仅是一方面的原因.本文考虑投入和产出两方面的因素,构造了复合DEA模型,并研究了基于复合DEA模型高校办学效益评价方法.     

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.     

Finding common weights based on the DM's preference information

Data Envelopment Analysis (DEA) is basically a linear programming-based technique used for measuring the relative performance of organizational units, referred to as Decision Making Units (DMUs). The flexibility in selecting the weights in standard DEA models deters the comparison among DMUs on a common base. Moreover, these weights are not suitable to measure the preferences of a decision maker (DM). For dealing with the first difficulty, the concept of common weights was proposed in the DEA literature. But, none of the common weights approaches address the second difficulty. This paper proposes an alternative approach that we term as ‘preference common weights’, which is both practical and intellectually consistent with the DEA philosophy. To do this, we introduce a multiple objective linear programming model in which objective functions are input/output variables subject to the constraints similar to the equations that define production possibility set of standard DEA models. Then by using the Zionts–Wallenius method, we can generate common weights as the DM's underlying value structure about objective functions.     

Data Envelopment Analysis with Preference Structure

It is important to consider the decision making unit (DMU)'s or decision maker's preference over the potential adjustments of various inputs and outputs when data envelopment analysis (DEA) is employed. On the basis of the so-called Russell measure, this paper develops some weighted non-radial CCR models by specifying a proper set of ‘preference weights’ that reflect the relative degree of desirability of the potential adjustments of current input or output levels. These input or output adjustments can be either less or greater than one; that is, the approach enables certain inputs actually to be increased, or certain outputs actually to be decreased. It is shown that the preference structure prescribes fixed weights (virtual multiplier bounds) or regions that invalidate some virtual multipliers and hence it generates preferred (efficient) input and output targets for each DMU. In addition to providing the preferred target, the approach gives a scalar efficiency score for each DMU to secure comparability. It is also shown how specific cases of our approach handle non-controllable factors in DEA and measure allocative and technical efficiency. Finally, the methodology is applied with the industrial performance of 14 open coastal cities and four special economic zones in 1991 in China. As applied here, the DEA/preference structure model refines the original DEA model's result and eliminates apparently efficient DMUs.     

A robust data envelopment analysis model with different scenarios

Input and output data, under uncertainty, must be taken into account as an essential part of data envelopment analysis (DEA) models in practice. Many researchers have dealt with this kind of problem using fuzzy approaches, DEA models with interval data or probabilistic models. This paper presents an approach to scenario-based robust optimization for conventional DEA models. To consider the uncertainty in DEA models, different scenarios are formulated with a specified probability for input and output data instead of using point estimates. The robust DEA model proposed is aimed at ranking decision-making units (DMUs) based on their sensitivity analysis within the given set of scenarios, considering both feasibility and optimality factors in the objective function. The model is based on the technique proposed by Mulvey et al. (1995) for solving stochastic optimization problems. The effect of DMUs on the product possibility set is calculated using the Monte Carlo method in order to extract weights for feasibility and optimality factors in the goal programming model. The approach proposed is illustrated and verified by a case study of an engineering company.     

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.     

Optimal system design series-network DEA models

There is an urgent need in a wide range of fields such as logistics and supply chain management to develop effective approaches to measure and/or optimally design a network system comprised of a set of units. Data envelopment analysis (DEA) researchers have been developing network DEA models to measure decision making units’ (DMUs’) network systems. However, to our knowledge, there are no previous contributions on the DEA-type models that help DMUs optimally design their network systems. The need to design optimal systems is quite common and is sometimes necessary in practice. This research thus introduces a new type of DEA model termed the optimal system design (OSD) network DEA model to optimally design a DMUs (exogenous and endogenous) input and (endogenous and final) output portfolios in terms of profit maximization given the DMUs total available budget. The resulting optimal network design through the proposed OSD network DEA models is efficient, that is, it lies on the frontier of the corresponding production possibility set.     

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.     

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.     

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.     

Evaluating the performances of decision-making units based on interval efficiencies

Efficiency could be not only the ratio of weighted sum of outputs to that of inputs but also that of weighted sum of inputs to that of outputs. When the previous efficiency measures the best relative efficiency within the range of no more than one, the decision-making units (DMUs) who get the optimum value of one perform best among all the DMUs. If the previous efficiency is measured within the range of no less than one, the DMUs who get the optimum value of one perform worst among all the DMUs. When the later efficiency is measured within the range of no more than one, the DMUs who get the optimum value of one perform worst among all the DMUs. If the later efficiency is measured within the range of no less than one, the DMUs who get the optimum value of one perform best among all the DMUs. This paper mainly studies an interval DEA model with later efficiency, in which efficiency is measured within the range of an interval, whose upper bound is set to one and the lower bound is determined by introducing a virtual ideal DMU, whose performance is definitely superior to any DMUs. The efficiencies, obtained from interval DEA model, turn out to be all intervals and are referred to as interval efficiencies, which combine the best and the worst relative efficiency in a reasonable manner to give an overall assessment of performances for all DMUs. Assessor's preference information on input and output weights is also incorporated into interval DEA model reasonably and conveniently. Through an example, some differences are found from the ranking results obtained from interval DEA model and bounded DEA model using the Hurwicz criterion approach to rank the interval efficiencies.     

DEA with efficiency classification preserving conditional convexity

We propose to relax the standard convexity property used in Data Envelopment Analysis (DEA) by imposing additional qualifications for feasibility of convex combinations. We specifically focus on a condition that preserves the Koopmans efficiency classification. This yields an efficiency classification preserving conditional convexity property, which is implied by both monotonicity and convexity, but not conversely. Substituting convexity by conditional convexity, we construct various empirical DEA approximations as the minimal sets that contain all DMUs and are consistent with the imposed production assumptions. Imposing an additional disjunctive constraint to standard convex DEA formulations can enforce conditional convexity. Computation of efficiency measures relative to conditionally convex production set can be performed through Disjunctive Programming (DP).     

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.     

Super-efficiency DEA in the presence of infeasibility

It is well known that super-efficiency data envelopment analysis (DEA) approach can be infeasible under the condition of variable returns to scale (VRS). By extending of the work of Chen (2005), the current study develops a two-stage process for calculating super-efficiency scores regardless whether the standard VRS super-efficiency mode is feasible or not. The proposed approach examines whether the standard VRS super-efficiency DEA model is infeasible. When the model is feasible, our approach yields super-efficiency scores that are identical to those arising from the original model. For efficient DMUs that are infeasible under the super-efficiency model, our approach yields super-efficiency scores that characterize input savings and/or output surpluses. The current study also shows that infeasibility may imply that an efficient DMU does not exhibit super-efficiency in inputs or outputs. When infeasibility occurs, it can be necessary that (i) both inputs and outputs be decreased to reach the frontier formed by the remaining DMUs under the input-orientation and (ii) both inputs and outputs be increased to reach the frontier formed by the remaining DMUs under the output-orientation. The newly developed approach is illustrated with numerical examples.     

Measuring the performances of decision-making units using interval efficiencies

Efficiency is a relative measure because it can be measured within different ranges. The traditional data envelopment analysis (DEA) measures the efficiencies of decision-making units (DMUs) within the range of less than or equal to one. The corresponding efficiencies are referred to as the best relative efficiencies, which measure the best performances of DMUs and determine an efficiency frontier. If the efficiencies are measured within the range of greater than or equal to one, then the worst relative efficiencies can be used to measure the worst performances of DMUs and determine an inefficiency frontier. In this paper, the efficiencies of DMUs are measured within the range of an interval, whose upper bound is set to one and the lower bound is determined through introducing a virtual anti-ideal DMU, whose performance is definitely inferior to any DMUs. The efficiencies turn out to be all intervals and are thus referred to as interval efficiencies, which combine the best and the worst relative efficiencies in a reasonable manner to give an overall measurement and assessment of the performances of DMUs. The new DEA model with the upper and lower bounds on efficiencies is referred to as bounded DEA model, which can incorporate decision maker (DM) or assessor's preference information on input and output weights. A Hurwicz criterion approach is introduced and utilized to compare and rank the interval efficiencies of DMUs and a numerical example is examined using the proposed bounded DEA model to show its potential application and validity.     

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.     

Benchmark optimization and attribute identification for improvement of container terminals

The aim of this paper is to optimize the benchmarks and prioritize the variables of decision-making units (DMUs) in data envelopment analysis (DEA) model. In DEA, there is no scope to differentiate and identify threats for efficient DMUs from the inefficient set. Although benchmarks in DEA allow for identification of targets for improvement, it does not prioritize targets or prescribe level-wise improvement path for inefficient units. This paper presents a decision tree based DEA model to enhance the capability and flexibility of classical DEA. The approach is illustrated through its application to container port industry. The method proceeds by construction of multiple efficient frontiers to identify threats for efficient/inefficient DMUs, provide level-wise reference set for inefficient terminals and diagnose the factors that differentiate the performance of inefficient DMUs. It is followed by identification of significant attributes crucial for improvement in different performance levels. The application of this approach will enable decision makers to identify threats and opportunities facing their business and to improve inefficient units relative to their maximum capacity. In addition, it will help them to make intelligent investment on target factors that can improve their firms’ productivity.     

A new method for evaluating decision making units in DEA

This paper provides a new structure in data envelopment analysis (DEA) for assessing the performance of decision making units (DMUs). It proposes a technique to estimate the DEA efficient frontier based on the Arash Method in a way different from the statistical inferences. The technique allows decisions in the target regions instead of points to benchmark DMUs without requiring any more information in the case of interval/fuzzy DEA methods. It suggests three efficiency indexes, called the lowest, technical and highest efficiency scores, for each DMU where small errors occur in both input and output components of the Farrell frontier, even if the data are accurate. These efficiency indexes provide a sensitivity index for each DMU and arrange both inefficient and technically efficient DMUs together while simultaneously detecting and benchmarking outliers. Two numerical examples depicted the validity of the proposed method.