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.     

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.     

Validating absolute weight bounds in Data Envelopment Analysis (DEA) models

The Charnes, Cooper and Rhodes (CCR) DEA model and its linear forms maximise the efficiency of the assessed decision making unit (DMU) and, at the same time, the ratio of this efficiency to the maximum efficiency taken across all the DMUs, the latter naturally always being equal to one. It has been shown recently that, in the presence of absolute weight bounds, these models may not maximise the ratio of these efficiencies, a fact that may cause problems with the interpretation and use of the optimal primal and dual solutions. For example, an inefficient DMU may have greater efficiency than its target unit for some weights. This paper investigates the problem in greater detail; it shows that, in the linear DEA model maximising the total virtual output of the assessed DMU, the problem occurs only if upper bounds are imposed on the output weights. A similar result is established for the model that minimises the total virtual input.     

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.     

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.     

基于SMAA和公共权重的决策单元效率评价与排序

在数据包络分析(DEA)中,公共权重模型是决策单元效率评价与排序的常用方法之一。与传统DEA模型相比,公共权重模型用一组公共的投入产出权重评价所有决策单元,评价结果往往更具有区分度且更为客观。本文考虑决策单元对排序位置的满意程度,提出了基于最大化最小满意度和最大化平均满意度两类新的公共权重模型。首先,基于随机多准则可接受度分析(SMAA)方法,计算出每个决策单元处于各个排名位置的可接受度;然后,通过逆权重空间分析,分别求得使最小满意度和平均满意度最大化的一组公共权重;最后,利用所求的公共权重,计算各决策单元的效率值及相应的排序。算例分析验证了本文提出的基于SMAA的公共权重模型用于决策单元效率评价与排序的可行性。     

Super-efficiency and DEA sensitivity analysis

This paper discusses and reviews the use of super-efficiency approach in data envelopment analysis (DEA) sensitivity analyses. It is shown that super-efficiency score can be decomposed into two data perturbation components of a particular test frontier decision making unit (DMU) and the remaining DMUs. As a result, DEA sensitivity analysis can be done in (1) a general situation where data for a test DMU and data for the remaining DMUs are allowed to vary simultaneously and unequally and (2) the worst-case scenario where the efficiency of the test DMU is deteriorating while the efficiencies of the other DMUs are improving. The sensitivity analysis approach developed in this paper can be applied to DMUs on the entire frontier and to all basic DEA models. Necessary and sufficient conditions for preserving a DMU’s efficiency classification are developed when various data changes are applied to all DMUs. Possible infeasibility of super-efficiency DEA models is only associated with extreme-efficient DMUs and indicates efficiency stability to data perturbations in all DMUs.     

Measuring super-efficiency in DEA in the presence of infeasibility

Super-efficiency data envelopment analysis (DEA) model is obtained when a decision making unit (DMU) under evaluation is excluded from the reference set. Because of the possible infeasibility of super-efficiency DEA model, the use of super-efficiency DEA model has been restricted to the situations where constant returns to scale (CRS) are assumed. It is shown that one of the input-oriented and output-oriented super-efficiency DEA models must be feasible for a any efficient DMU under evaluation if the variable returns to scale (VRS) frontier consists of increasing, constant, and decreasing returns to scale DMUs. We use both input- and output-oriented super-efficiency models to fully characterize the super-efficiency. When super-efficiency is used as an efficiency stability measure, infeasibility means the highest super-efficiency (stability). If super-efficiency is interpreted as input saving or output surplus achieved by a specific efficient DMU, infeasibility does not necessary mean the highest super-efficiency.     

Allocating fixed costs and resources via data envelopment analysis

In this paper we show that data envelopment analysis (DEA) can be viewed as maximising the average efficiency of the decision-making units (DMUs) in an organisation. Building upon this we present DEA based models for: (a) allocating fixed costs to DMUs and (b) allocating input resources to DMUs. Simultaneous to allocating input resources output targets are also decided for each DMU. Numeric results are presented for a number of example problems taken from the literature.     

Measurement of the worst practice of decision-making units in the presence of non-discretionary factors and imprecise data

The objective of the present paper is to propose a novel pair of data envelopment analysis (DEA) models for measurement of relative efficiencies of decision-making units (DMUs) in the presence of non-discretionary factors and imprecise data. Compared to traditional DEA, the proposed interval DEA approach measures the efficiency of each DMU relative to the inefficiency frontier, also called the input frontier, and is called the worst relative efficiency or pessimistic efficiency. On the other hand, in traditional DEA, the efficiency of each DMU is measured relative to the efficiency frontier and is called the best relative efficiency or optimistic efficiency. The pair of proposed interval DEA models takes into account the crisp, ordinal, and interval data, as well as non-discretionary factors, simultaneously for measurement of relative efficiencies of DMUs. Two numeric examples will be provided to illustrate the applicability of the interval DEA models.     

带有随机因素的逆DEA模型

本文讨论含有随机因素的逆 DEA模型 .逆 DEA模型解决的问题是 :对于某个决策单元 (DMU ) ,若增加其输入 ,在保持相对效率水平不变的情况下 ,估计 (预测 )输出应增加多少 .因此逆 DEA模型可用于短期预测问题 .带有随机因素的逆 DEA模型 ,是将该问题转化成机会约束的多目标规划问题 ,在某些特殊情况下 ,成为机会约束的线性规划问题 .     

有关判断决策单元的DEA有效性的新方法的探讨

为了判断决策单元是否(弱)DEA有效并克服现有的模型及[1]中模型在解决上述问题时的不足之处,本文将讨论的新模型是由CCR模型与CCGSS模型变来的,且定理的证明不同于[1]．还讨论了文中新模型的最优解的存在性,此外,研究了所有决策单元的输入输出的变化对某决策单元有效性的影响．     

广义与传统BC~2模型效率评价差异及其几何刻画

传统DEA方法相对于决策单元全体对决策单元进行评价,广义DEA方法相对于样本单元全体对决策单元进行评价.由于参照系的不同,对不同决策单元的相对效率评价结果可能不同.针对这种情况,对基于BC2模型的只有投入或只有产出的传统和广义DEA模型进行说明,并通过样本前沿面的移动对广义DEA模型中相对效率值进行几何刻画.     

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.     

Cost,Revenue and Profit Efficiency Models in Generalized Fuzzy Data Envelopment Analysis

One of the most important information given by data envelopment analysis models is the cost, revenue and profit efficiency of decision making units (DMUs). Cost efficiency is defined as the ratio of minimum costs to current costs, while revenue efficiency is defined as the ratio of maximum revenue to current revenue of the DMU. This paper presents a framework where data envelopment analysis (DEA) is used to measure cost, revenue and profit efficiency with fuzzy data. In such cases, the classical models cannot be used, because input and output data appear in the form of ranges. When the data are fuzzy, the cost, revenue and profit efficiency measures calculated from the data should be uncertain as well. Fuzzy DEA models emerge as another class of DEA models to account for imprecise inputs and outputs for DMUs. Although several approaches for solving fuzzy DEA models have been developed, numerous deficiencies including the $\alpha$-cut approaches and types of fuzzy numbers must still be improved. This scheme embraces evaluation method based on vector for proposed fuzzy model. This paper proposes generalized cost, revenue and profit efficiency models in fuzzy data envelopment analysis. The practical application of these models is illustrated by a numerical example.     

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 generalized DEA model for inputs/outputs estimation

In this paper we discuss the question: among a group of decision making units (DMUs), if a DMU changes some of its input (output) levels, to what extent should the unit change outputs (inputs) such that its efficiency index remains unchanged? In order to solve this question we propose a solving method based on Data Envelopment Analysis (DEA) and Multiple Objective Linear Programming (MOLP). In our suggested method, the increase of some inputs (outputs) and the decrease due to some of the other inputs (outputs) are taken into account at the same time, while the other offered methods do not consider the increase and the decrease of the various inputs (outputs) simultaneously. Furthermore, existing models employ a MOLP for the inefficient DMUs and a linear programming for weakly efficient DMUs, while we propose a MOLP which estimates input/output levels, regardless of the efficiency or inefficiency of the DMU. On the other hand, we show that the current models may fail in a special case, whereas our model overcomes this flaw. Our method is immediately applicable to solve practical problems.     

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.     

关于求解DEA原始CCR模型中最优输入输出权重的方法

本文给出了求解DEA原始CCR模型中最优输入输出权重的简便方法：首先将原始CCR模型化为线性规划模型，然后从该线性规划模型的对偶模型入手，运用单纯形法，在得到决策单元最优效率评价指数时，根据线性规划的对偶理论，得到决策单元最优输入输出权重。该权重可用在逆DEA新算法中。     

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.