The use of OWA operator weights for cross-efficiency aggregation

Cross-efficiency evaluation is an effective way of ranking decision making units (DMUs) in data envelopment analysis (DEA). Existing approaches for cross-efficiency evaluation are mainly focused on the calculation of cross-efficiency matrix, but pay little attention to the aggregation of the efficiencies in the cross-efficiency matrix. The most widely used approach is to aggregate the efficiencies in each row or column in the cross-efficiency matrix with equal weights into an average cross-efficiency score for each DMU and view it as the overall performance measurement of the DMU. This paper focuses on the aggregation process of the efficiencies in the cross-efficiency matrix and proposes the use of ordered weighted averaging (OWA) operator weights for cross-efficiency aggregation. The use of OWA operator weights for cross-efficiency aggregation allows the decision maker (DM)’s optimism level towards the best relative efficiencies, characterized by an orness degree, to be taken into consideration in the final overall efficiency assessment and particularly in the selection of the best DMU.     

基于证据推理和前景理论的交叉效率排序方法研究

针对由交叉效率评价策略和交叉效率集结方法的多样性而造成评价结果不一致的问题,提出利用证据推理方法和前景理论,综合各个交叉效率评价策略的评价结果,实现对决策单元的统一评价。首先,分别将选用的交叉效率评价策略以及各个评价策略中的他评效率设置成一级指标和二级指标,依据算数平均和前景理论分别确定一、二级指标的权重;其次,依据他评效率确定二级指标置信度,利用证据推理方法将各个交叉效率评价策略的他评效率综合转换成决策单元被评价为有效的置信度。决策者可通过比较决策单元被识别为有效的置信度的大小来判断决策单元交叉效率的大小,进而实现对决策单元的排序;最后,通过案例验证和说明本文提出方法的有效性和实用性。     

DEA game cross-efficiency approach to Olympic rankings

A number of studies have used data envelopment analysis (DEA) to evaluate the performance of the countries in Olympic games. While competition exists among the countries in Olympic games/rankings, all these DEA studies do not model competition among peer decision making units (DMUs) or countries. These DEA studies find a set of weights/multipliers that keep the efficiency scores of all DMUs at or below unity. Although cross efficiency goes a further step by providing an efficiency measure in terms of the best multiplier bundle for the unit and all the other DMUs, it is not always unique. This paper presents a new and modified DEA game cross-efficiency model where each DMU is viewed as a competitor via non-cooperative game. For each competing DMU, a multiplier bundle is determined that optimizes the efficiency score for that DMU, with the additional constraint that the resulting score should be at or above that DMU 's estimated best performance. The problem, of course, arises that we will not know this best performance score for the DMU under evaluation until the best performances of all other DMUs are known. To combat this “chicken and egg” phenomenon, an iterative approach leading to the Nash equilibrium is presented. The current paper provides a modified variable returns to scale (VRS) model that yields non-negative cross-efficiency scores. The approach is applied to the last six Summer Olympic Games. Our results may indicate that our game cross-efficiency model implicitly incorporates the relative importance of gold, silver and bronze medals without the need for specifying the exact assurance regions.     

Exploiting value system multiplicity and preference voting for robust ranking

Under a data envelopment analysis (DEA) framework, full ranking of a group of decision making units (DMUs) can be carried out through an adequate amalgamation of the cross-efficiency (CE henceforth) scores produced for each DMU. In this paper, we propose a ranking procedure that is based on amalgamating the weight profiles selected over the cross-evaluation rather than related CE scores. The new approach builds, for each DMU, a collective weight profile (CWP henceforth) by exploiting the preference voting system embedded within the matrix of weights, which views the assessing DMUs as voters and the input/output factors as candidates. The occurrence of zero votes is discussed as a special case and a two-level aggregation procedure is developed. The CWPs that are produced extend the concept of collective appreciation to the input/output factors of each DMU so that group dynamics is truly reflected, mainly in decision making circumstances where factor prioritization is necessary for making choices or allocating resources. The robustness of the proposed ranking approach is evaluated with three examples drawn from the literature.     

Reducing differences between profiles of weights: A “peer-restricted” cross-efficiency evaluation

This paper deals with the selection of the profiles of weights to be used in cross-efficiency evaluations. In an attempt to prevent unrealistic weighting schemes, one of the issues of main interest that we address here is that of the zero weights, since their use implies that some of the variables considered are excluded from the assessments to be made. In the calculation of cross-efficiency scores, we propose to ignore the profiles of weights of the DMUs that cannot make a choice of non-zero weights among their alternate optima. The different units are therefore assessed in a peer-evaluation that does not consider the profiles of weights of some inefficient DMUs. This approach is referred to as “peer-restricted” cross-efficiency evaluation. Aside from avoiding zero weights, the choice of weights that we make also seeks to reduce the differences between the weights profiles selected as much as possible. Thus, in the “peer-restricted” cross-efficiency evaluation in the present paper we also try to avoid that the different DMUs attach very different weights to the same variable. Finally, we extend this approach to derive a common set of weights by exploiting the idea of similarity between profiles of weights.     

评价相对效率的投入—产出型DEA

传统 DEA只能在固定投入或产出的条件下 ,从产出或投入角度测算决策单元相对效率 ,因而不能综合地反映决策单元的投入产出效果 .基于双目标规划 ,本文将提出从投入及产出角度评价决策单元相对效率的投入—产出型 DEA,并研究其相对有效性 .最后以沪市 1 6家高科技上市公司为应用实例 ,研究其相对经营效率     

基于交叉效率和合作博弈的决策单元排序方法

在数据包络分析中，大量的交叉效率模型已被提出。然而选择不同的目标模型将实现不一样的交叉效率评价。本文基于针对单个决策单元实施的对抗型和仁慈型两个交叉效率模型，用合作博弈方法来研究交叉效率模型的选取，并利用Shapley值对决策单元进行排序。最后通过实例分析显示该排序方法充分利用了最小交叉效率和最大交叉效率的信息完全排序了所有决策单元，具有一定的综合性和合理性。     

Eco-efficiency analysis of paper mills along the Huai River: An extended DEA approach

The objective of this paper is to estimate the ecological efficiency of paper mills along the Huai River in China. The main characteristic of the ecological efficiency evaluation problem is that an undesirable output of biochemical oxygen demand (BOD) and a non-discretionary input (BOD emission quota) should be considered simultaneously. By analyzing the impacts of the non-discretionary input on decision-making units’ (DMUs) desirable and undesirable outputs, a non-radial output-oriented DEA model is proposed. In the proposed model, we describe a new approach of defining reference set that requires reference units operate in a similar environment on average. We employ the model to provide efficient inputs/outputs targets for DMU managers to improve DMUs’ efficiencies. Based on the developed model, impacts of the non-discretionary input on DMUs’ returns are also analyzed. We illustrate the proposed model, using real data, for 32 paper mills along the Huai River in China.     

Integrated framework for robustness analysis using ratio-based efficiency model with application to evaluation of Polish airports

We consider a problem of evaluating efficiency of Decision Making Units (DMUs) based on their deterministic performance on multiple consumed inputs and multiple produced outputs. We apply a ratio-based efficiency measure, and account for the Decision Maker׳s preference information representable with linear constraints involving input/output weights. We analyze the set of all feasible weights to answer various robustness concerns by deriving: (1) extreme efficiency scores and (2) extreme efficiency ranks for each DMU, (3) possible and necessary efficiency preference relations for pairs of DMUs, (4) efficiency distribution, (5) efficiency rank acceptability indices, and (6) pairwise efficiency outranking indices. The proposed hybrid approach combines and extends previous results from Ratio-based Efficiency Analysis and the SMAA-D method. The practical managerial implications are derived from the complementary character of accounted perspectives on DMUs׳ efficiencies. We present an innovative open-source software implementing an integrated framework for robustness analysis using a ratio-based efficiency model on the diviz platform. The proposed approach is applied to a real-world problem of evaluating efficiency of Polish airports. We consider four inputs related to the capacities of a terminal, runways, and an apron, and to the airport׳s catchment area, and two outputs concerning passenger traffic and number of aircraft movements. We present how the results can be affected by integrating the weight constraints and eliminating outlier DMUs.     

Centralized resource allocation based on efficiency analysis for step-by-step improvement paths

The existing centralized resource allocation models often assume that all of the DMUs are efficient after resource allocation. For the DMU with a very low efficiency score, it means the dramatic reduction of the resources, which can cause the organizational resistance. In addition, in reality, it is particularly difficult for the DMUs to achieve their target efficiencies in a single step, especially when they are far from the efficient frontier. Thus, gradual progress towards benchmarking targets is gaining importance. In this paper, we present a new approach for resource allocation based on efficiency analysis under a centralized decision-making environment. Through our approach, the central decision-maker can obtain a sequence of intermediate benchmark targets based on efficiency analysis, which provide a level-wise improvement path to direct the DMUs to reach their ultimate targets on the efficient frontier in an implementable and realistic manner. Numerical examples are presented to illustrate the application procedure of the proposed approach.     

评价相对效率的投入-产出型DEA

传统DEA只能在固定投入或产出的条件下,从产出或投入角度测算决策单元相对效率,因而不能综合地反映决策单元的投入产出效果.基于双目标规划,本文将提出从投入及产出角度评价决策单元相对效率的投入-产出型DEA,并研究其相对有效性.最后以沪市16家高科技上市公司为应用实例,研究其相对经营效率.     

一种新的基于双前沿面的交叉效率方法及其应用

本文将双前沿面效率评价的思想引入到传统交叉效率模型中，同时，针对双前沿面交叉效率方法中仁慈型和激进型交叉效率策略无法抉择，以及这两种交叉效率策略的应用范围有限的不足，提出了一种新的基于双前沿面的交叉效率方法。该方法的基本思想是选取一个理想决策单元和负理想决策单元，使用被评价决策单元的权重来计算理想决策单元和负理想决策单元的效率，并使被评价决策单元的效率尽可能接近理想解的效率，同时，尽可能远离负理想解的效率。根据该思想，分别在乐观前沿面和悲观前沿面下求解交叉效率值并进行集结，避免了由于前沿面的选择不同导致的差异以及决策者对仁慈型和激进型交叉效率策略进行抉择的困难。最后，将本文方法与现有方法进行对比分析，并将本文方法应用于我国东部地区10个省(直辖市)的创新效率评价中，以验证方法的有效性。     

A slacks-based measure model for calculating cross efficiency in data envelopment analysis

Cross efficiency is a concept for solving the problem of incomparability among the efficiencies of a set of decision making units (DMUs) calculated from different weights in data envelopment analysis, and is helpful for ranking. Conventional cross efficiency is a radial measure, and the radial measure has some weaknesses. First, it is not able to provide appropriate efficiency scores for weakly efficient DMUs. Second, the efficiencies measured from the input and output sides under variable returns to scale are different which may cause difficulties in subsequent analyses. Third, negative values may appear when the efficiency is measured from the input side. To solve these problems, this paper proposes a slacks-based measure to calculate efficiency. The basic idea is the same as that of the radial measure using the frontier corresponding to a DMU to measure the efficiency of all DMUs, except that the efficiency measure is slacks-based instead of radial. Due to the nature of the slacks-based efficiency measure, the problems caused by weak efficiency and difference in input and output radial measures do not exist. More importantly, negative efficiencies are not produced. The proposed method is applied to a real case involving the selection of the most efficient robot to use for production. The results help identify the top-ranked robot.     

A slack-based measure of efficiency in context-dependent data envelopment analysis

Data envelopment analysis (DEA) has been proven as an excellent data-oriented performance evaluation method when multiple inputs and outputs are present in a set of peer decision-making units (DMUs). In the DEA literature, a context-dependent DEA is developed to provide finer evaluation results by examining the efficiency of DMUs in specific performance levels based upon radial DEA efficiency scores. In DEA, non-zero input and output slacks are very likely to present after the radial efficiency score improvement. Often, these non-zero slack values represent a substantial amount of inefficiency. Therefore, in order to fully measure the inefficiency in DMU's performance, it is very important to also consider the inefficiency represented by the non-zero slacks in the context-dependent DEA. This study proposes a slack-based context-dependent DEA which allows a full evaluation of inefficiency in a DMUs performance. By using slack-based efficiency measure, we obtain different frontier levels and more appropriate performance benchmarks for inefficient DMUs.     

基于最近距离投影的DEA两阶段效率评价方法研究

数据包络分析是进行效率评价最重要的方法之一。传统的数据包络分析理论主要寻找有效前沿面上的最远距离投影，在极大化无效性指数的同时也面临着效率改进的巨大难度和高额成本。对于具有两阶段内部生产结构的决策单元，本文从考虑最小改进难度的视角出发，提出了最近距离投影的两阶段效率评价方法。该方法首先得到所有强有效决策单元的线性组合，且这些组合均占优于被评价的两阶段决策单元。然后建立了两阶段范围调整效率评价模型，在确定具有最近投影距离的占优组合的同时，得到了两阶段评价效率。最后，本文运用我国32家上市银行的年度数据对所提出方法进行了应用验证。     

An integrated model for slack-based measure of super-efficiency in additive DEA

As a non-radial approach, a super-efficiency model, Super SBM, was proposed by Tone [15] to rank efficient DMUs. Du et al. [7] extends the Super SBM model to the additive (slacks-based) DEA model. To obtain the super-efficiencies of the DMUs, one needs to identify the efficient DMUs first and then apply the additive super-efficiency model to those efficient DMUs. In this paper, we propose an integrated model so that the efficiencies of the inefficient DMUs and the super-efficiencies of the efficient DMUs can be obtained by a single model. The efficiency scores obtained by our integrated model are the same as those obtained by Du et al. [7] and the additive DEA model.     

Two-stage DEA models with undesirable input-intermediate-outputs

Data envelopment analysis (DEA) is a non-parametric approach for measuring the relative efficiencies of peer decision making units (DMUs). Many studies have examined DEA efficiencies of two-stage systems, where all the outputs from the first stage are the only inputs to the second stage. Although single-stage DEA models with undesirable input-outputs have been extensively studied, there still lacks of more systematical investigation on two-stage DEA with undesirable variables. For instance, depending on its operating model, even whether an intermediate variable is desirable or undesirable can be questionable for a particular two-stage system. Furthermore, most of the existing studies on two-stage systems focus on the case where only the final outputs are undesirable. In this work, we try to systematically examine two-stage DEA models with undesirable input-intermediate-outputs. Particularly, we utilize the free-disposal axioms to construct the production possibility sets (PPS) and the corresponding DEA models with undesirable variables. The proposed models are then used to illustrate some theoretical perspectives by using the data of China׳s listed banks.     

Ranking efficient units in DEA

Super-efficiency data envelopment analysis (DEA) model can be used in ranking the performance of efficient decision making units (DMUs). Because of the infeasibility problem associated with the super-efficiency DEA model, ranking has been restricted to the model where constant returns to scale and proportional changes in all inputs or all outputs are assumed. In fact, when super-efficiency is used as an efficiency stability measure, infeasibility means the highest super-efficiency. However, if super-efficiency is interpreted as input saving or output surplus achieved by a specific efficient DMU, infeasibility does not necessarily mean the highest super-efficiency. In order to obtain a complete ranking of efficient DMUs when the two assumptions are relaxed, a modified super-efficiency DEA model is proposed to overcome the infeasibility problem and to correctly capture the super-efficiency represented by the input saving or the output surplus. The current paper suggests using both input- and output-oriented super-efficiency models to characterize the super-efficiency when infeasibility occurs. As a result, we can rank the efficient DMUs if infeasibility occurs. The approach is applied to 20 largest Japanese companies and 15 US cities, respectively.     

基于可变权重的DEA效率评价模型

传统DEA模型将投入、产出权重视为固定变量。为了更合理地对决策单元的效率进行评价,本文研究了如何将有关投入、产出变量的权重的更多信息融入效率评价模型,提出了可变权重的概念,给出了一种基于可变权重的DEA效率评价模型。本文模型是CCR模型的推广。在有关权重信息可得的前提下,本文模型较之CCR模型中效率评价上更为合理。但是,如何准确度量投入、产出权重并将其函数化,是本文方法应用的难点。一个算例分析演示了本文模型。     

基于交叉评价策略的DEA全局协调相对效率排序模型

在DEA(数据包络分析)研究领域,建构在交叉效率概念基础上的现有决策单元排序方法仅以定义的方式给出了用于决策单元排序的交叉效率评价值。对于这种方法构建方式,分别基于管理学的效率概念和多属性决策理论,分析指出其中的交叉效率评价值从本质上讲既与效率的管理学概念不符,也与决策单元的优劣不存在理性逻辑联系。为克服现有决策单元排序方法所存在的上述问题,基于交叉评价策略和效率的管理学概念内涵给出了DEA全局协调相对效率的新概念,在此基础上利用优化理论给出了可以用于决策单元优劣排序的DEA全局协调相对效率测度模型,并通过理论分析和数值案例验证解释了该模型相对于现有决策单元排序方法所拥有的比较优势。