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.     

Multi-period efficiency and Malmquist productivity index in two-stage production systems

Conventional two-stage data envelopment analysis (DEA) models measure the overall performance of a production system composed of two stages (processes) in a specified period of time, where variations in different periods are ignored. This paper takes the operations of individual periods into account to develop a multi-period two-stage DEA model, which is able to measure the overall and period efficiencies at the same time, with the former expressed as a weighted average of the latter. Since the efficiency of a two-stage system in a period is the product of the two process efficiencies, the overall efficiency of a decision making unit (DMU) in the specified period of time can be decomposed into the process efficiency of each period. Based on this decomposition, the sources of inefficiency in a DMU can be identified. The efficiencies measured from the model can also be used to calculate a common-weight global Malmquist productivity index (MPI) between two periods, in that the overall MPI is the product of the two process MPIs. The non-life insurance industry in Taiwan is used to verify the proposed model, and to explain why some companies performed unsatisfactorily in the specified period of time.     

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.     

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 .     

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.     

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.     

Identifying the efficiency status in network DEA

Data envelopment analysis (DEA) is commonly employed to evaluate the efficiency performance of a decision making unit (DMU) that transforms exogenous inputs into final outputs. In such a black-box DEA approach, details of an internal production process of the DMU are typically ignored and hence the locations of inefficiency are not adequately provided. In view of this, DEA researchers have recently developed various network approaches by looking into the black box, where the inputs that enter the box and the outputs that come out of it are only considered. However, most of these network approaches evaluate divisional efficiency by using an optimal solution of their respective optimization problem. If such an optimal solution is used in the case when there are multiple optima, then managerial guidance based on this solution alone may be inappropriate because more appropriate targets from the viewpoint of management may be ignored. Taking this fact into account, therefore, we propose a network approach for identifying the efficiency status of each DMU and its divisions. This approach provides a practical computational procedure.     

Additive DEA based on MCDA with imprecise information

This work exploits links between Data Envelopment Analysis (DEA) and multicriteria decision analysis (MCDA), with decision making units (DMUs) playing the role of decision alternatives. A novel perspective is suggested on the use of the additive DEA model in order to overcome some of its shortcomings, using concepts from multiattribute utility models with imprecise information. The underlying idea is to convert input and output factors into utility functions that are aggregated using a weighted sum (additive model of multiattribute utility theory), and then let each DMU choose the weights associated with these functions that minimize the difference of utility to the best DMU. The resulting additive DEA model with oriented projections has a clear rationale for its efficiency measures, and allows meaningful introduction of constraints on factor weights.     

On some methods for performance ranking and correspondence analysis in the DEA context

Two novel methods named performance baseline and performance correspondence matrices are proposed to evaluate the performance of decision making units (DMUs) based on the techniques of singular value decomposition (SVD). The performance baseline matrix can be used to rank all the DMUs because it provides a common basis for performance comparison. The performance correspondence matrix can be used to conduct performance cluster analysis, with which to explore the structure of input/output variables that are associated with DMUs. The analysis can reveal the performance difference of the DMUs and the key input/output variables determining the efficiency of a certain DMU, and provides valuable quantitative information for adjusting variables to improve efficiency of the DMU. Three case studies are presented to demonstrate that the proposed methods in this work are effective and easy to use and can provide insights into proper selection of input/output variables for performance comparison to avoid over manipulating DEA models in practice.     

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.     

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.     

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.     

Unoriented two-stage DEA: The case of the oscillating intermediate products

Data envelopment analysis (DEA) allows us to evaluate the relative efficiency of each of a set of decision-making units (DMUs). However, the methodology does not permit us to identify specific sources of inefficiency because DEA views the DMU as a “black box” that consumes a mix of inputs and produces a mix of outputs. Thus, DEA does not provide a DMU manager with insight regarding the internal source of the organization’s inefficiency.     

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.     

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.     

指标可取负值的基于输入与输出的DEA模型

有关基于输入与输出的DEA模型,本文与现有文献的不同之处,一是模型中的评价指标可取负值,二是被评的决策单元可以不是所给的n个决策单元之一,三是模型并非由多目标规划模型推得.此外,给出了有关此模型的三个定理.因此,可知有关此模型的最优解存在的充分条件;在求解此模型后就能在判断决策单元的DEA有效性的同时计算出其相对效率,并能计算出其在DEA相对有效面上的"投影".     

Dual models of interval DEA and its extension to interval data

The purpose of this paper is to develop a new DEA with an interval efficiency. An original DEA model is to evaluate each DMU optimistically. There is another model called “Inverted DEA” to evaluate each DMU pessimistically. But, there are no relations essentially between DEA and inverted DEA. Thus, we formulate a DEA model with an interval efficiency which consists of efficiencies obtained from the optimistic and pessimistic viewpoints. Thus, two end points can construct an interval efficiency. With the same idea, we deal with the interval inefficiency model which is inverse to interval efficiency. Finally, we extend the proposed DEA model to interval data and fuzzy data.     

Sensitivity and stability analysis in data envelopment analysis

One of the topics of interest in data envelopment analysis (DEA) is sensitivity and stability and stability analysis of the specific decision making unit (DMU), which is under evaluation. In DEA, efficient DMUs are of primary importance as they define the efficient frontier. In this paper, we develop a new sensitivity analysis approach for the CCR, BCC and Additive models, when variations in the data are considered for a specific efficient DMU and the data for the remaining DMUs are assumed fixed.     

链式网络DEA模型

数据包络分析(DEA)是评价决策单元(DMU)相对有效性的一种工具,现已得到广泛的应用.传统的DEA不考虑系统内部结构,而是将系统作为一个"黑箱"来度量效率.针对多阶段网络结构提出一个新的网络DEA模型—链式网络DEA模型.研究网络决策单元的网络DEA有效性及各个阶段的弱DEA有效性之间的关系,给出了网络DEA有效的充分必要条件.若网络决策单元不是网络DEA有效的,根据模型可以指出系统在哪些阶段是无效的.     

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.