Measuring the relative balance of DMUs

In basic Data Envelopment Analysis (DEA) models, the weights of performance criteria are endogenously determined, assigning to each decision-making unit (DMU) its best possible efficiency score. This property is widely considered to be a major strength of the method, but it is also a source of considerable problems concerning performance assessment and performance control. Such problems result from the possibility of zero-value weights that exclude criteria from the evaluation. Unlike approaches that deal with this issue by incorporating value judgments into the analysis, we suggest a complementary balance score that is derived from the given data set. This score evaluates the extent to which a DMU avoids concentration on only some of the crucial performance criteria. One of the possible decisions resulting from a balance analysis is to reduce the set of DMUs considered to serve as benchmarks. For this case, a modified CCR–O model is presented.     

A demonstration of consistency of an entropy balance with balance of energy

It is shown in this note that a general balance of entropy postulated previously with only a limited motivation (based on the form of the energy equation for an inviscid fluid) is consistent with, and can be derived from, a general balance of energy. In this derivation, an early form of entropy balance does not make use of invariance conditions under superposed rigid body motions. However, with the help of the latter invariance conditions, additional results are also derived which provide some insight on the structure of the basic equations in thermomechanics.     

Nonlinear transversely vibrating beams by the improved energy balance method and the global residue harmonic balance method

Mathematical modeling of many engineering systems such as beam structures often leads to nonlinear ordinary or partial differential equations. Nonlinear vibration analysis of the beam structures is very important in mechanical and industrial applications. This paper presents the high order frequency-amplitude relationship for nonlinear transversely vibrating beams with odd and even nonlinearities using the improved energy balance method and the global residue harmonic balance method. The accuracy of the energy balance method is improved based on combining features of collocation method and Galerkin–Petrov method, and an improved harmonic balance method is proposed which is called the global residue harmonic balance method. Unlike other harmonic balance methods, all the former global residual errors are introduced in the present approximation to improve the accuracy. Finally, preciseness of the present analytic procedures is evaluated in contrast with numerical calculations methods, giving excellent results.     

On the convergence of the heat balance integral method

Convergence properties are established for the piecewise linear heat balance integral solution of a benchmark moving boundary problem, thus generalising earlier results [Numer. Heat Transfer 8 (1985) 373]. A convergence rate of O(n⁻¹) is identified with minor effects at large values of the Stefan number β (slow interface movement). The correct O(n^−1/2) behaviour for incident heat flux is recovered for β → 0 (pure heat conduction) as previously found [Numer. Heat Transfer 8 (1985) 373–382]. Numerical illustrations support the theoretical findings.     

Forms of Lepage type and the balance systems

In this work we apply infinitesimal variational calculus to the systems of balance equations. We determine a class of the exterior $n+(n+1)$-forms Θ on the jet bundle of infinite order over a configurational bundle $\pi :Y^{n+m}\to X^n$ similar to the class of Lepage n-forms. Systems of differential equations obtained in the way similar to one used in the Lagrangian field theory, include the Euler–Lagrange equations corresponding to a Lagrangian functions as well as arbitrary regular systems of balance equations. For a balance system with a symmetry group G we present the Noether balance laws corresponding to the generators of the Lie algebra of the group G.     

Numerical calculation of the moments of the population balance equation

The combined CFD–PBM (population balance models) are computationally intensive, so a possibility is to calculate only a few moments of the probability density function (PDF) of the PBM minimizing the computational costs. However, this formulation results in an excess of unknowns with respect to equations which is referred to as a closure problem. One approach for dealing with this closure problem is to apply a numerical quadrature approximation. On the other hand, a different possibility is to compute the PDF and from this, the moments of interest if required.In this work, the two mentioned approaches are discussed and numerical experiments are used to show the capability of the methods for predicting the moments of the PBE. In particular, the quadrature method of moments and a time–space least squares spectral method will be discussed.     

On the lower Markov spectrum

C. Hightower found two infinite sequences of gaps in the Markov spectrum, ( _n , _n ) and ( _n , _n ) with _n and _n both Markov elements, converging to . This paper exhibits Markov elements _n ^* and _n ^* such that, for alln 1, ( _n ^* , _n ) and ( _{n n} ^* ) are gaps in the Markov spectrum. Other results include showing that, for alln 1, _n is completely isolated, while the other endpoints of the gaps are limit points in the Markov spectrum.     

Remarks on lower semicontinuity and lower closure

In the context of a recent geometric condition of Cesari, used in the reduction of seminormality requirements in lower closure theorems, this paper shows that the existence of a strongly convergent selection from the sequence of orientor fields, under Kuratowski property (K), is adequate to guarantee lower closure theorems. This generalization is justified through examples. Several related remarks are made.This work was done in the framework of Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan. The author wishes to thank Professor L. Cesari for helpful discussions during the writing of this paper.     

On the equivalence of the max-min transportation lower bound and the time-indexed lower bound for single-machine scheduling problems

New observations are made about two lower bound schemes for single-machine min-sum scheduling problems. We find that the strongest bound of those provided by transportation problem relaxations can be computed by solving a linear program. We show the equivalence of this strongest bound and the bound provided by the LP relaxation of the time-indexed integer programming formulation. These observations lead to a new lower bound scheme that yields fast approximation of the time-indexed bound. Several techniques are developed to facilitate the effective use of the new lower bound in branch-and-bound. Numerical experiments are conducted on 375 benchmark problems of the total weighted tardiness problem from OR-Library. Results obtained with our new method are spectacular; we are able to solve all 125 open problems to optimality.     

Derivation of the material momentum equation from the energy balance

Based on a recent paper of Yavari et al, we show how the equation of material momentum can be obtained from invariance of the energy balance under translations in the material space. This approach further highlights the duality between the physical and material space.      

A statistic for measuring the balance of a sample

Consider a finite population which has many auxiliary variables. A statistic, which is a function of the moments of the auxiliary variables, is proposed to measure the balance of a sample. The mean and variance of this statistic are derived.     

Inner balance of symmetric designs

A triple array is a row-column design which carries two balanced incomplete block designs (BIBDs) as substructures. McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005), Section 8, gave one example of a triple array that also carries a third BIBD, formed by its row-column intersections. This triple array was said to be balanced for intersection, and they made a search for more such triple arrays among all potential parameter sets up to some limit. No more examples were found, but some candidates with suitable parameters were suggested. We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced. As triple arrays in the canonical case correspond to SBIBDs, this in turn yields new existence criteria for triple arrays balanced for intersection. In particular, we prove that the residual design of the related SBIBD with respect to the defining block must be quasi-symmetric, and give necessary and sufficient conditions on the intersection numbers. This, together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005) and many others in a wide search. Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block. We show as a key result that such SBIBDs must possess the quasi-3 property, and we answer the existence question for all known classes of these designs.     

Upper and lower bounds of the valence-functional

For the non-negative integerg let (M, g) denote the closed orientable 2-dimensional manifold of genusg. K-realizationsP of (M, g) are geometric cell-complexes inP with convex facets such that set (P) is homeomorphic toM. ForK-realizationsP of (M, g) and verticesv ofP, val (v,P) denotes the number of edges ofP incident withv and the weighted vertex-number Σ(val(v, P)-3) taken over all vertices ofP is called valence-valuev (P) ofP. The valence-functionalV, which is important for the determination of all possiblef-vectors ofK-realisations of (M, g), in connection with Eberhard's problem etc., is defined byV(g):=min[v(P)|P is aK-realization of (M,g)]. The aim of the note is to prove the inequality 2g+1≦V(g)≦3g+3 for every positive integerg.     

A phase transition model with the entropy balance

This paper deals with a phase transition model in which the energy balance is equivalently rewritten in terms of the entropy balance. The thermodynamical consistence of the model is proved and also under physically meaningful assumptions on the data, existence of a solution is stated for the corresponding initial boundary values problem by a maximum principle. Hence, L¹‐arguments yield the uniqueness of the solution and show that it evolves in accordance with thermodynamics and everyday practical properties. Copyright © 2003 John Wiley & Sons, Ltd.     

Convection caused by Radiation through the Layer

A nonlinear stability threshold is determined for the problemof convection in a layer of non-Boussinesq fluid with prescribedheat flux on the lower boundary and constant-temperature uppersurface. The convection problem is one in which motion can penetratefrom an unstable layer into a gravitationally stable one andthe unconditional nonlinear analysis necessitates utilizationof a spatially weighted energy.