Reply to comments by Soleimani-damaneh and Mostafaee (2006) and Zhang (2006)

In Fukuyama [Fukuyama, H., 2000. Returns to scale and scale elasticity in data envelopment analysis. European Journal of Operational Research 125, 93–112], I investigated some mathematical structure on scale elasticity and returns to scale. Soleimani-damaneh and Mostafaee [Soleimani-damaneh, M., Mostafaee, A., in press. A comment on “Returns to scale and scale elasticity in data envelopment analysis”. European Journal of Operational Research. doi:10.1016/j.ejor.2006.11.042] and Zhang [Zhang, B., in press. A Note on Fukuyama (2000). European Journal of Operational Research. doi:10.1016/j.ejor.2006.11.040] claim that some results, which are related to homogeneity, are incorrect. This note replies to their comments by demonstrating that Fukuyama (2000) results are still valid.     

A note on Fukuyama (2000)

This brief note discusses Fukuyama’s paper [Fukuyama, H., 2000. Returns to scale and scale elasticity in data envelopment analysis. European Journal of Operational Research 125, 93–112]. We prove that the Proof of Lemma 1 by Fukuyama (2000) is incorrect due to incorrect substitution. In addition, we exemplify our conclusion by a counterexample taken from Fukuyama (2000). Thus all conclusions, based on Lemma 1, of Fukuyama (2000) cannot be reached and are left for further research.     

A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation

In this paper we focus on scale elasticity measure based on directional distance function for multi-output–multi-input technologies, explore its fundamental properties and show its equivalence with the input oriented and output oriented scale elasticity measures. We also establish duality relationship between the scale elasticity measure based on the directional distance function with scale elasticity measure based on the profit function. Finally, we discuss the estimation issues of the scale elasticity based on the directional distance function via the DEA estimator.     

Calculating scale elasticity in DEA models

In the data envelopment analysis (DEA) efficiency literature, qualitative characterizations of returns to scale (increasing, constant, or decreasing) are most common. In economics it is standard to use the scale elasticity as a quantification of scale properties for a production function representing efficient operations. Our contributions are to review DEA practices, apply the concept of scale elasticity from economic multi-output production theory to DEA piecewise linear frontier production functions, and develop formulas for scale elasticity for radial projections of inefficient observations in the relative interior of fully dimensional facets. The formulas are applied to both constructed and real data and show the differences between scale elasticities for the two valid projections (input and output orientations). Instead of getting qualitative measures of returns to scale only as was done earlier in the DEA literature, we now get a quantitative range of scale elasticity values providing more information to policy-makers.     

Topological sensitivity analysis of a multi‐scale constitutive model considering a cracked microstructure

This paper deals with the sensitivity analysis of the macroscopic elasticity tensor to topological microstructural changes of the underlying material. In particular, the microstucture is topologicaly perturbed by the nucleation of a small circular inclusion. The derivation of the proposed sensitivity relies on the concept of topological derivative, applied within a variational multi‐scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. We consider that the RVE can contain a number of voids, inclusions and/or cracks. It is assumed that non‐penetration conditions are imposed at the crack faces, which do not allow the opposite crack faces to penetrate each other. The derived sensitivity leads to a symmetric fourth‐order tensor field over the unperturbed RVE domain, which measures how the macroscopic elasticity parameters estimated within the multi‐scale framework changes when a small circular inclusion is introduced at the micro‐scale level. Copyright © 2009 John Wiley & Sons, Ltd.     

Re-examining scale elasticity in DEA

We propose a new scheme for measuring scale elasticity of production based on a new cost efficiency model developed in Tone (2002). Comparing our model with classical model we establish the superiority of our model over the latter based on the premise that the classical estimates of cost efficiency and scale elasticity can be illusory.     

Evaluation of the degenerate scale in antiplane elasticity using null field BIE

This paper provides a numerical solution for the degenerate scale in antiplane elasticity using the null field boundary integral equation (BIE). With coordinate transformation, the BIE can be formulated in a normal scale, and a basic solution for the BIE in the normal scale is illustrated. The degenerate scale is easily obtained from the basic solution. Several numerical examples are given.     

A simple derivation of scale elasticity in data envelopment analysis

In this paper, we suggest a simple derivation of the formulae for the scale elasticity in the variable returns-to-scale technology as used in data envelopment analysis. Our development is consistent with the existing literature but the proof is much shorter and applies to the general case without any simplifying conditions.     

Improving the identification of returns to scale in data envelopment analysis

In order to assess the efficiencies of a set of production units, it is necessary to identify the nature of returns to scale which characterise efficient production. Some methods have been developed to test the nature of the scale elasticity across the full range of scale sizes. However these tests are heavily weighted by the majority of the units and may not identify small ranges of scale size where different returns to scale hold. This paper develops a procedure based on a combination of Data Envelopment Analysis and regression analysis to identify the ranges of scale size where the returns to scale may differ from those in other ranges for the single-output, multi-input case. We also develop a measure of scale size across different input mixes.     

Numerical solution for degenerate scale problem arising from multiple rigid lines in plane elasticity

This paper studies the degenerate scale problem arising from multiple rigid lines in plane elasticity. In the first step, the problem should be formulated on a degenerate scale by distribution of body force densities along rigid lines. The condition of vanishing displacement along lines is also assumed. The coordinate transform with a reduced factor “h” is performed in the next step. The new obtained BIE is a particular non-homogenous BIE defined in the transformed coordinates with normal scale. In the normal scale, the integral operator is invertible. By using two fundamental solutions that are formulated in the normal scale, the new obtained BIE can be reduced to an equation for finding the factor “h”. Finally, the degenerate scale is obtained. It is proved from computed results that the degenerate scale only depends on the configuration of rigid lines, and does not depend on the initial normal scale used. In addition, the degenerate scale is invariant with respect to the rotation of rigid lines. Many examples are carried out.     

On the nano-structural dependence of nonlocal dynamics and its relationship to the upper limit of nonlocal scale parameter

Nonlocal elasticity theory is one of the most popular theoretical approaches to investigate the intrinsic scale effect of nano-materials/structures. The coupling of an internal characteristic length and a material parameter can be regarded as a nonlocal scale parameter in nano-meters. The range of this non-dimensional scale parameter is from zero up to different values previously. There is no doubt that the zero nonlocal scale parameter corresponds to a situation without any nonlocal effect. However, the determination of a peak value for the scale parameter is still uncertain. In fact, we frequently ask a simple but unresolved question, i.e., how strong is the nonlocal scale effect? This question is equivalent to what the maximum value of the nonlocal scale parameter is, since it was introduced to characterize the scale effect theoretically. Until now, various maximum values have been selected without rigorous verifications. In this paper, the nano-structural dependence of nonlocal dynamical behavior is investigated to present the existence of an upper limit for the scale parameter. Through three typical examples, the size-dependent behavior of nonlocal dynamics for various nano-structures is analyzed. The upper limit of the scale parameter can be determined accordingly. It is shown that an interval for the scale parameter in the illustrative examples can be found on the basis of the nonlocal softening physical mechanism, in which the equivalent stiffness of nano-structures is weakened than that predicted by the classical continuum theory. The present study contributes to a fuzzy zone in nonlocal elasticity where people are puzzled over the question how to select the upper limit of the nonlocal scale parameter. It is not only beneficial to the refinement of the nonlocal theory of elasticity, and also useful for the exploration of similar theories in nano-mechanics.     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

Scale elasticity and returns to scale in the presence of alternative solutions

Scale elasticity (SE) and returns to scale (RTS) are important topics in performance analysis, which help managers to make decisions about the expansion or contraction of the operation of decision making units under assessment. In this paper, some new results about these topics in the presence of alternative solutions, regarding the concept of multifunction, are provided.At first, some properties of some multifunctions (functions), defined with respect to the optimal solutions of DEA models, are established which help us in what follows. In turn, the relationships between the considered multifunctions and the concept of RTS and SE are studied. Finally an approach for the estimation of the RTS classification of units is obtained, which leads to an important corollary as an interesting result which introduces a connection between two concepts, RTS and SE. This is important from an applied point of view. Also, from a technical point of view, the proofs of theorems which give this corollary use a main lemma of convex analysis literature and give a constructive proof about RTS.     

Measurement of returns to scale in radial DEA models

A general approach is proposed in order to measure returns to scale and scale elasticity at projections points in the radial data envelopment analysis (DEA) models. In the first stage, a relative interior point belonging to the optimal face is found using a special, elaborated method. In previous work it was proved that any relative interior point of a face has the same returns to scale as any other interior point of this face. In the second stage, we propose to determine the returns to scale at the relative interior point found in the first stage.     

Properties of nonstationary boundary operators in a full scale of Sobolev-type spaces

Properties of boundary operators are studied for a system of equations of elasticity theory in a full scale of Sobolev-type spaces.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 962–965, May, 1994.     

The emergence of the Planck scale

In this paper, we first observe some interesting parallels between Planck scale considerations and elementary particle Compton wavelength scale considerations, particularly in the context of Wheeler's space-time foam and a space-time arising out of a stochastic random heap of elementary particles discussed in previous papers. These parallels lead to a semi-qualitative picture which shows how the short-lived Planck scale arises from the Compton wavelength considerations. Finally all this is quantified.     

中国乡镇企业增长的随机前沿生产函数分析

论文在运用中国1990-2005年间省际平衡面板数据的基础上,采用考虑非效率项的非中性技术进步随机前沿生产函数模型,分析了影响我国各省区乡镇企业增长的规模报酬、技术效率和技术进步因素。主要得出结论如下:在乡镇企业的发展过程中,资本产出弹性不断接近甚至有超过劳动产出弹性的趋势,呈现出一定的"资本深化"过程;整个规模报酬略大于1,这表明适当扩大乡镇企业经营规模存在着一定的规模经济效应;而整体平均技术效率水平逐年递增,技术进步率则逐年下降。在这些实证工作的基础上,论文还得出一些政策建议,希冀能够对实现我国乡镇企业的可持续发展有所裨益。     

Effects of a small length scale on vibrations of an embedded double-walled carbon nanotube

Extensive continuum analyses are carried out to estimate the influence of matrix stiffness, a small length scale, and intertubular radial displacements on free vibrations of an individual double-walled carbon nanotybe. The analyses are based on both local and classical Euler–Bernoulli and Timoshenko elasticity theories with concentricity and nonconcentricity assumptions. The effect of a small length scale is incorporated in the formulations. New intertubular resonant frequencies are calculated based on these theories. Detailed results are demonstrated for the resonant frequencies as functions of matrix stiffness and the small length scale. The results indicate that the internal radial displacement and the stiffness of the surrounding matrix can greatly affect the resonant frequencies, especially at higher frequencies, and thus the latter does not keep the otherwise concentric structure at ultrahigh frequencies. More over, at high frequencies and small aspect ratios, the effect of the small length scale be comes more significant.     

The emergence of large‐scale coherent structure under small‐scale random bombardments

We provide mathematical justification of the emergence of large‐scale coherent structure in a two‐dimensional fluid system under small‐scale random bombardments with small forcing and appropriate scaling assumptions. The analysis shows that the large‐scale structure emerging out of the small‐scale random forcing is not the one predicted by equilibrium statistical mechanics. But the error is very small, which explains earlier successful prediction of the large‐scale structure based on equilibrium statistical mechanics. © 2005 Wiley Periodicals, Inc.     

基于推测弹性的双寡头研发AJ模型最优成本缩减决策

文章通过将推测弹性引入AJ模型,首先考察一致性推测弹性对双寡头企业研发策略的影响,从而得出有推测的双寡头市场结构下的研发均衡结果,然后将零一致性推测弹性的均衡成本缩减量与AJ模型的结果进行对比,最后给出一致性推测弹性和溢出系数之间的关系对研发绩效的影响效果.