Directional distance functions and slacks-based measures of efficiency

In this paper we introduce a SBM (slacks-based measure) of efficiency based on directional distance functions. This measure is contrasted with the SBM due to Professor Tone [Tone, K., 2001. A slacks-based measure of efficiency in data envelopment analysis. European Journal of Operational Research 130, 498–509].     

基于“零和收益”的SBM效率分配方法及应用

在松弛变量度量(slacks-based measure,SBM)效率评价方法的基础上,首先明确投入(产出)要素固定的生产系统中,投入(产出)要素在各决策单元间的竞争性关系;然后采用比例分配策略对SBM无效决策单元的投入(产出)松弛进行效率分配,以构建一个基于零和收益的SBM(zero sum gains SBM,ZSG-SBM)效率分配方法;再通过分析ZSG-SBM模型与SBM模型效率评价结果的关系,给出了比例分配策略ZSG-SBM模型的求解方法;最后应用实例研究验证了本文模型在要素存在竞争性的复杂生产系统效率评价和资源分配中的优势。     

环境效率评价方法的比较研究

文章给出了已有文献对环境效率度量的6种DEA(数据包络分析)模型,比较了它们各自对非期望产出处理的特性和缺陷,并以安徽43家企业为例,对其度量结果存在的差异进行了分析和解释.度量结果显示,只有基于松弛测度的SBM模型对企业环境效率的差异识别性较强.     

A slacks-based measure of efficiency in data envelopment analysis

In this paper, we will propose a slacks-based measure (SBM) of efficiency in Data Envelopment Analysis (DEA). This scalar measure deals directly with the input excesses and the output shortfalls of the decision making unit (DMU) concerned. It is units invariant and monotone decreasing with respect to input excess and output shortfall. Furthermore, this measure is determined only by consulting the reference-set of the DMU and is not affected by statistics over the whole data set. The new measure has a close connection with other measures proposed so far, e.g., Charnes–Cooper–Rhodes (CCR), Banker–Charnes–Cooper (BCC) and the Russell measure of efficiency. The dual side of this model can be interpreted as profit maximization, in contrast to the ratio maximization of the CCR model. Numerical experiments show its validity as an efficiency measurement tool and its compatibility with other measures of efficiency.     

Variations on the theme of slacks-based measure of efficiency in DEA

In DEA, there are typically two schemes for measuring efficiency of DMUs; radial and non-radial. Radial models assume proportional change of inputs/outputs and usually remaining slacks are not directly accounted for inefficiency. On the other hand, non-radial models deal with slacks of each input/output individually and independently, and integrate them into an efficiency measure, called slacks-based measure (SBM). In this paper, we point out shortcomings of the SBM and propose four variants of the SBM model. The original SBM model evaluates efficiency of DMUs referring to the furthest frontier point within a range. This results in the hardest score for the objective DMU and the projection may go to a remote point on the efficient frontier which may be inappropriate as the reference. In an effort to overcome this shortcoming, we first investigate frontier (facet) structure of the production possibility set. Then we propose Variation I that evaluates each DMU by the nearest point on the same frontier as the SBM found. However, there exist other potential facets for evaluating DMUs. Therefore we propose Variation II that evaluates each DMU from all facets. We then employ clustering methods to classify DMUs into several groups, and apply Variation II within each cluster. This Variation III gives more reasonable efficiency scores with less effort. Lastly we propose a random search method (Variation IV) for reducing the burden of enumeration of facets. The results are approximate but practical in usage.     

Efficiency of the strategic style of pension funds: an application of the variants of the slacks-based measure in DEA

A 2010 paper by Kaoru Tone proposes four variants of the slacks-based measure of efficiency (SBM) to overcome the limitations of this well-known Data Envelopment Analysis (DEA) approach when the reference points of the efficient frontier may not be adequate. In this study, we apply these variants for the first time to a real-world problem to evaluate the efficiency of one of the most relevant decisions in pension fund management: The strategic asset allocation. The results highlight the relevance of SBM Variation III, which considers clusters of portfolios with similar characteristics, to appropriately identify the reference set of each portfolio. Therefore, this variant allows for the identification of locally efficient but globally inefficient portfolios. Our results also reject the notion of a positive relation between management resources and efficiency of the strategic investment style.     

Super efficiencies or super inefficiencies? Insights from a joint computation model for slacks-based measures in DEA

The slacks-based measure (SBM) can incorporate input and output slacks that would otherwise be neglected in the classical DEA model. In parallel, the super-efficiency model for SBM (S-SBM) has been developed for the purpose of ranking SBM efficient decision-making units (DMUs). When implementing SBM in conjunction with S-SBM, however, several issues can arise. First, unlike the standard super-efficiency model, S-SBM can only solve for super-efficiency scores but not SBM scores. Second, the S-SBM model may result in weakly efficient reference points. Third, the S-SBM and SBM scores for certain DMUs may be discontinuous with a perturbation to their inputs and outputs, making it hard to interpret and justify the scores in applications and the efficiency scores may be sensitive to small changes/errors in data. Due to this discontinuity, the S-SBM model may overestimate the super-efficiency score. This paper extends the existing SBM approaches and develops a joint model (J-SBM) that addresses the above issues; namely, the J-SBM model can (1) simultaneously compute SBM scores for inefficient DMUs and super-efficiency for efficient DMUs, (2) guarantee the reference points generated by the joint model are Pareto-efficient, and (3) the J-SBM scores of a firm are continuous in the input and output space. Interestingly, the radial DEA efficiency and super-efficiency scores for a DMU are continuous in the input–output space. The J-SBM model combines the merits of the radial and SBM models (i.e., continuity and Pareto-efficiency).     

Input–output substitutability and strongly monotonic p-norm least distance DEA measures

In DEA, there are two frameworks for efficiency assessment and targeting: the greatest and the least distance framework. The greatest distance framework provides us with the efficient targets that are determined by the farthest projections to the assessed decision making unit via maximization of the p-norm relative to either the strongly efficient frontier or the weakly efficient frontier. Non-radial measures belonging to the class of greatest distance measures are the slacks-based measure (SBM) and the range-adjusted measure (RAM). Whereas these greatest distance measures have traditionally been utilized because of their computational ease, least distance projections are quite often more appropriate than greatest distance projections from the perspective of managers of decision-making units because closer efficient targets may be attained with less effort. In spite of this desirable feature of the least distance framework, the least distance (in) efficiency versions of the additive measure, SBM and RAM do not even satisfy weak monotonicity. In this study, therefore, we introduce and investigate least distance p-norm inefficiency measures that satisfy strong monotonicity over the strongly efficient frontier. In order to develop these measures, we extend a free disposable set and introduce a tradeoff set that implements input–output substitutability.     

Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations

The efficient and accurate calculation of sensitivities of the price of financial derivatives with respect to perturbations of the parameters in the underlying model, the so-called ‘Greeks’, remains a great practical challenge in the derivative industry. This is true regardless of whether methods for partial differential equations or stochastic differential equations (Monte Carlo techniques) are being used. The computation of the ‘Greeks’ is essential to risk management and to the hedging of financial derivatives and typically requires substantially more computing time as compared to simply pricing the derivatives. Any numerical algorithm (Monte Carlo algorithm) for stochastic differential equations produces a time-discretization error and a statistical error in the process of pricing financial derivatives and calculating the associated ‘Greeks’. In this article we show how a posteriori error estimates and adaptive methods for stochastic differential equations can be used to control both these errors in the context of pricing and hedging of financial derivatives. In particular, we derive expansions, with leading order terms which are computable in a posteriori form, of the time-discretization errors for the price and the associated ‘Greeks’. These expansions allow the user to simultaneously first control the time-discretization errors in an adaptive fashion, when calculating the price, sensitivities and hedging parameters with respect to a large number of parameters, and then subsequently to ensure that the total errors are, with prescribed probability, within tolerance.     

An interactive benchmark model ranking performers — Application to financial holding companies

This study aims to resolve the problems associated with ranking fairly for both efficient and inefficient decision making units (DMUs) by proposing an ‘interactive benchmark’ model. The main concept is derived as a result of taking one certain DMU as a fixed benchmark and estimating the efficiency scores of the remaining DMUs based on that benchmark pair by pair. The process is repeated until all of the DMUs have served as the fixed benchmark. The DMUs can then be evaluated on a fair basis by computing the average efficiency scores using the above rolling procedure. The model is applied to fourteen financial holding companies (FHCs) in Taiwan using a production transfer model that is adopted by Seiford and Zhu [L.M. Seiford, J. Zhu, Profitability and marketability of the top 55 US commercial banks, Management Science 45 (9) (1999) 1270–1288]. The empirical results can serve as valuable reference to both policy-makers and investors.     

A closer look at the economic-environmental disparities for regional development in China

Although China has harvested the fruits of its rapid economic growth over a period of several decades, it has encountered serious environmental problems, an important one being air pollution in the form of soot, dust, and sulfur dioxide. In considering the concept of ‘green-GDP’, this paper analyzes China’s regional development by examining its economic performance while taking into account various environmental factors. In addition to computing technical efficiency for 31 regions in China, a cross-efficiency measure is applied to differentiate the genuine DMUs. ‘Overall’ efficient regions and ‘false positive’ ones are recognized by a false positive index (FPI). It is found that the coastal regions perform on average better than the inland regions both economically and environmentally. For inefficient regions, the benchmark should be those regions with high cross-efficiency mean scores (e.g., Guangdong) rather than those with high self-appraisal scores (e.g., Shanghai). A cross-tabulation illustrating the difference between GDP-oriented performance and Pollution-oriented performance shows that the coastal regions make up the dominant proportion in terms of the benchmarks for economic-environmental optimization.     

Evaluation of credit risk based on firm performance

This paper investigates whether productive inefficiency measured as the distance from the industry’s ‘best practice’ frontier is an important ex-ante predictor of business failure. We use samples of French textiles, wood and paper products, computers and R&D companies to obtain efficiency estimates for individual firms in each industry. These efficiency measures are derived from a directional technology distance function constructed empirically using non-parametric data envelopment analysis (DEA) methods. Estimating binary and ordered logit regression models we find that productive efficiency has significant explanatory power in predicting the likelihood of default over and above the effect of standard financial indicators.     

A slacks-based measure of super-efficiency in data envelopment analysis

In most models of Data Envelopment Analysis (DEA), the best performers have the full efficient status denoted by unity (or 100%), and, from experience, we know that usually plural Decision Making Units (DMUs) have this “efficient status”. To discriminate between these efficient DMUs is an interesting subject. This paper addresses this “super-efficiency” issue by using the slacks-based measure (SBM) of efficiency, which the author proposed in his previous paper [European Journal of Operational Research 130 (2001) 498]. The method differs from the traditional one based on the radial measure, e.g. Andersen and Petersen model, in that the former deals directly with slacks in inputs/outputs, while the latter does not take account of the existence of slacks. We will demonstrate the rationality of our approach by comparing it with the radial measure of super-efficiency. The proposed method will be particularly useful when the number of DMUs are small compared with the number of criteria employed for evaluation.     

Network DEA: A slacks-based measure approach

Traditional DEA models deal with measurements of relative efficiency of DMUs regarding multiple-inputs vs. multiple-outputs. One of the drawbacks of these models is the neglect of intermediate products or linking activities. After pointing out needs for inclusion of them to DEA models, we propose a slacks-based network DEA model, called Network SBM, that can deal with intermediate products formally. Using this model we can evaluate divisional efficiencies along with the overall efficiency of decision making units (DMUs).     

Citizen coproduction and efficient public good provision: Theory and evidence from local public libraries

In both public administration and economics, efficiency is brought forward as an important criterion for evaluating administrative actions. Clearly, its value as an assessment principle depends on our ability to adequately measure efficiency. This article argues that citizen’s coproduction in public services requires a careful reassessment of how we approach the measurement of productive efficiency in public service delivery. Theoretically, we illustrate that using observable outcomes (e.g., library circulation, school results, health outcomes, fires extinguished, and crimes solved) as output indicators is inappropriate and leads to biased estimates of public service providers’ productive efficiency. This bias arises because citizens co-determine final outputs, leaving them at least partly beyond the service providers’ control. Empirically, we find supportive evidence of both the existence and importance of such ‘demand-induced’ bias.     

Does knowledge management matter? Assessing the performance of electricity distribution districts based on slacks-based data envelopment analysis

This study uses slacks-based data envelopment analysis (DEA) to examine the performance of electricity distribution districts (EDDs) in Taiwan in 2004. It also explores the relationship between a knowledge management system (KMS) and variations in the efficiency of EDDs in the Taiwan Power Company from 2000 to 2004. The findings show that the EDDs performed well in terms of managerial efficiency in 2004, and 75% of the districts exhibited an increasing performance variation when using cross-period analysis for that period. We show that there is a positive relationship between a KMS and variations in organizational efficiency. In addition, we present a decision-making matrix and conceptual map to help EDD managers improve their managerial efficiency and skills. This study contributes to the field of knowledge management research by applying the DEA model to link the performance between KMS and an organization. Several significant findings emerge from the study.     

Auditors’ evidence evaluation and aggregation using beliefs and probabilities

This paper examines possible differences in auditors’ performance when they make belief-based versus probability-based risk assessments by focusing on two phases of the financial statement audit process: the assessment of two attributes of audit evidence (‘strength’ and ‘direction’) and the aggregation of evidence. Based on an experiment in which 48 experienced auditors participated, three important findings were observed. First, there was no significant difference in the mean assessment of strength of evidence measured using the likelihood ratio. However, the difference in the assessed direction of evidence, that is whether the evidence is interpreted as being confirming or disconfirming, is significant for one of the cases examined. This result shows that auditors making belief-based assessments are able to assess the direction of the evidence more accurately than auditors making probability-based assessments. Third, the auditors’ aggregation of evidence was not in accordance with ‘AND’ logic for either auditors making belief-based or probability-based assessments. These empirical results raise issues which need to be addressed in practice and in future research.     

监督力度对地方政府财政效率存在空间效应吗?

基于1999-2012年我国省级政府财政投入产出数据,采用考虑非期望产出的非径向非角度SBM模型对地方政府财政效率进行测度,研究表明我国地方政府财政效率存在区域差异性.采用动态空间计量研究发现财政违规资金和行政腐败程度抑制了我国地方政府财政效率,同时存在着负的空间外溢性特征.因此,促进我国地方政府财政效率的关键点在于需要考虑空间因素下的监督工作合理实施.     

A dichotomy for the convex spaces of probability measures

We show that every nonempty compact and convex space M of probability Radon measures either contains a measure which has ‘small’ local character in M or else M contains a measure of ‘large’ Maharam type. Such a dichotomy is related to several results on Radon measures on compact spaces and to some properties of Banach spaces of continuous functions.     

An iterated Radau method for time-dependent PDEs

This paper is concerned with the time integration of semi-discretized, multi-dimensional PDEs of advection-diffusion-reaction type. To cope with the stiffness of these ODEs, an implicit method has been selected, viz., the two-stage, third-order Radau IIA method. The main topic of this paper is the efficient solution of the resulting implicit relations. First, a modified Newton process has been transformed into an iteration process in which the 2 stages are decoupled and, moreover, can exploit the same LU-factorization of the iteration matrix. Next, we apply a so-called Approximate Matrix Factorization (AMF) technique to solve the linear systems in each Newton iteration. This AMF approach is very efficient since it reduces the ‘multi-dimensional’ system to a series of ‘one-dimensional’ systems. The total amount of linear algebra work involved is reduced enormously by this approach. The idea of applying AMF to two-dimensional problems is quite old and goes back to Peaceman and Rachford in the early fifties. The situation in three space dimensions is less favourable and will be analyzed here in more detail, both theoretically and experimentally. Furthermore, we analyze a variant in which the AMF-technique has been used to really solve (‘until convergence’) the underlying Radau IIA method so that we can rely on its excellent stability and accuracy characteristics. Finally, the method has been tested on several examples. Also, a comparison has been made with the existing codes VODPK and IMEXRKC, and the efficiency (CPU time versus accuracy) is shown to be at least competitive with the efficiency of these solvers.