bicyclic graphs with small positive index of inertia abstract the positive index of inertia of a graph g denoted by i g is the number of the positive eigenvalues of the adjacency matrix of g in this paper we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n

stochastic data envelopment analysis in measuring the efficiency of taiwan commercial banks conventional data envelopment analysis dea for measuring the efficiency of a set of decision making units dmus requires the input output data to be constant in reality however many observations are stochastic in nature consequently the resulting efficiencies are stochastic as well this paper discusses how to obtain the efficiency distribution of each dmu via a simulation technique the case of taiwan commercial banks shows that firstly the number of replications in simulation analysis has little effect on the estimation of efficiency means yet 1000 replications are recommended to produce reliable efficiency means and 2000 replications for a good estimation of the efficiency distributions secondly the conventional way of using average data to represent stochastic variables results in efficiency scores which are different from the mean efficiencies of the presumably true efficiency distributions estimated from simulation thirdly the interval data approach produces true efficiency intervals yet the intervals are too wide to provide valuable information in conclusion when multiple observations are available for each dmu the stochastic data approach produces more reliable and informative results than the average data and interval data approaches do

expressions for predicting fundamental natural frequencies of non cylindrical helical springs abstract numerical and analytical studies are performed for the free vibration analysis of non cylindrical conical barrel and hyperboloidal types helical springs the stiffness matrix method is used in the numerical analysis a total of 12 degrees of freedom six displacements and six rotations is described for an element the exact element stiffness matrix and the exact concentrated element inertia matrix are used in the formulation the rotary inertia the shear and extensional deformation effects are considered in the analysis comparison of the numerical results with the reported results obtained numerically and experimentally gives satisfactory values after verification of the numerical frequencies the non dimensional fundamental frequencies of fixed fixed non cylindrical helical springs with circular section are expressed in a simple formula with a maximum absolute relative error of 5 using those numerical values for the constant helix pitch angles 5 10 and 15 these expressions restricted to the fundamental frequencies are also verified with ansys results

an enhanced russell measure in dea with interval data abstract traditional data envelopment analysis dea models do not deal with imprecise data and assume that the data for all inputs and outputs are known exactly in real world situations however this assumption may not always be true when some inputs and outputs are unknown decision variables such as interval data ordinal data and ratio bounded data the dea model is called imprecise dea idea in this paper we develop a new approach based upon the enhanced russell measure erm for dealing with interval data in dea

the complementary functions method for the element stiffness matrix of arbitrary spatial bars of helicoidal axes the statical behaviour of a spatial bar of an elastic and isotropic material under arbitrary distributed loads having a non circular helicoidal axis and cross section supported elastically by single and or continuous supports is studied by the stiffness matrix method based on the complementary functions approach by considering the geometrical compatibility conditions together with the constitutive equations and equations of equilibrium a set of 12 first order differential equations having variable coefficients is obtained for spatial elements of helicoidal axes the stiffness matrix and the element load vector of a helicoidal bar with a non circular axis and arbitrary cross section are obtained taking into consideration both the presence of an elastic support and the erects of the axial and shear deformations for helicoidal staircases the significance of both axial and shear deformations and eccentricities existing in wide and shallow sections are also investigated the developed model has been coded in fortran 77 which has been applied to various example problems available in the relevant literature and the results have been compared

sensitivity and stability analysis in dea with interval data in this paper we find radius of stability for all decision making units with interval data in this approach organization classification remains unchanged under perturbations of the interval data some numerical examples for illustration are given

imprecise data envelopment analysis idea a review and improvement with an application abstract the standard data envelopment analysis dea method requires that the values for all inputs and outputs be known exactly when some outputs and inputs are unknown decision variables such as bounded data ordinal data and ratio bounded data the dea model becomes a non linear programming problem and is called imprecise dea idea there are two different approaches in dealing with imprecise outputs and inputs one uses scale transformations and variable alternations to convert the non linear idea model into a linear program the other converts imprecise data into exact data and then uses the standard linear dea model the current paper reviews and compares the two approaches through an efficiency analysis of a set of telephone offices a simplified approach is developed to reduce the computational burden if one uses the first approach the treatment of weight restrictions in idea is discussed it is shown that weight restrictions on imprecise data are redundant new developments and improvements to both approaches are provided

on fdh efficiency analysis with interval data most of dea models make an assumption that input and output data are exact without any variation cooper et al manage sci 45 1999 597 introduced imprecise data envelopment analysis idea when some inputs and outputs are unknown decision variables such as bounded data ordinal data and ratio bounded data the dea model becomes a non linear programming problem in this paper we consider fdh model with imprecise data first we introduce linear form of fdh model j product anal 16 2001 129 then according to eur j oper res 140 2002 24 we follow efficiency analysis

qualitative factors in data envelopment analysis a fuzzy number approach abstract qualitative factors are difficult to mathematically manipulate when calculating the efficiency in data envelopment analysis dea the existing methods of representing the qualitative data by ordinal variables and assigning values to obtain efficiency measures only superficially reflect the precedence relationship of the ordinal data this paper treats the qualitative data as fuzzy numbers and uses the dea multipliers associated with the decision making units dmus being evaluated to construct the membership functions based on zadeh s extension principle a pair of two level mathematical programs is formulated to calculate the cuts of the fuzzy efficiencies fuzzy efficiencies contain more information for making better decisions a performance evaluation of the chemistry departments of 52 uk universities is used for illustration since the membership functions are constructed from the opinion of the dmus being evaluated the results are more representative and persuasive

fuzzy data envelopment analysis and its application to location problems in this paper fuzzy dea data envelopment analysis models are proposed for evaluating the efficiencies of objects with fuzzy input and output data the obtained efficiencies are also fuzzy numbers that reflect the inherent ambiguity in evaluation problems under uncertainty an aggregation model for integrating fuzzy attribute values is provided in order to rank objects objectively using the proposed method a case study involving a restaurant location problem is analyzed in detail rent of establishment traffic amount level of security consumer consumption level and competition level are identified as the primary factors in determining an ideal location for a japanese style rotisserie restaurant based on field investigation the uncertain information on primary factors is represented by fuzzy numbers using the fuzzy aggregation model the best location of restaurant is determined the case study shows that fuzzy dea models can be quite useful for solving business problems under uncertainty

assumed strain formulation for triangular c0 plate elements based on a weak form of the kirchhoff constraints application of a weak form of the kirchhoff constraints in the formulation of triangular assumed shear strain c0 plate element is described it is shown that compared to a similar formulation of the quadrilateral elements formulation of the triangles requires some additional considerations which lead to additional terms in the energy expression with those terms performance of the assumed shear strain triangular elements is comparable to that of the quadrilateral elements

new efficiency conditions for multiobjective interval valued programming problems abstract in this paper we focus on necessary and sufficient efficiency conditions for optimization problems with multiple objectives and a feasible set defined by interval valued functions a new concept of fritz john and karush kuhn tucker type points is introduced for this mathematical programming problem based on the gh derivative concept the innovation and importance of these concepts are presented from a practical and computational point of view the problem is approached directly without transforming it into a real valued programming problem thereby attaining theoretical results that are more powerful and computationally more efficient under weaker hypotheses we also provide necessary conditions for efficiency which have been inexistent in the relevant literature to date the identification of necessary conditions is important for the development of future computational optimization techniques in an interval valued environment we introduce new generalized convexity notions for gh differentiable interval valued problems which are a generalization of previous concepts and we prove a sufficient efficiency condition based on these concepts finally the efficiency conditions for deterministic programming problems are shown to be particular instances of the results proved in this paper the theoretical developments are illustrated and justified through several numerical examples

strict vs weak ordinal relations for multipliers in data envelopment analysis in a recent paper golany 1988 proposes an interesting extension of data envelopment analysis to the situation where ordinal relations among weights corresponding to certain dimensions exist however the development of the extended model contains mathematical errors and the proposed equivalence is incorrect we establish the correct model equivalence and prove that weak ordinal relations require a nonstandard dea model we also show that the case of strict ordinal relations can be handled with the standard dea model these results hold for the ccr bcc additive and multiplicative models and for relations involving both input and output multipliers

on return to scale of fully efficient dmus in data envelopment analysis under interval data recently cooper et al manage sci 45 1999 597 proposed imprecise data envelopment analysis idea a situation in which some or all data are unknown the data in idea could be interval data ordinal data or data with bounded ratio usually in such cases the corresponding data envelopment analysis models becomes non linear when data assume to be interval decision making units dmus can divide to three classes as follows i efficient in any cases ii efficient in maximal sense iii always inefficient in this paper we investigate to the return to scale of dmus that are locate in first class also we introduce models that use exact data and present a combination if any which via that the dmu becomes a most productive scale size

integer interval dea an axiomatic derivation of the technology and an additive slacks based model abstract the present paper studies the efficiency assessment of decision making units dmus when their inputs and outputs are described under uncertainty of the type of integer interval data an axiomatic derivation of the production possibility set pps is presented an additive slacks based data envelopment analysis dea model is formulated consisting of two phases this has required the use of adequate arithmetic and lu partial orders for integer intervals this novel integer interval dea approach is the first step towards dea models under fuzzy integer intervals with the extension of the corresponding arithmetic and lu partial orders to fuzzy integer intervals the proposed method is applied on a dataset taken from the literature that involves both continuous and integer interval variables

relations between the inertia indices of a mixed graph and those of its underlying graph abstract given a graph g the mixed graph d g is obtained from g by orienting some of its edges where g is called the underlying graph of d g let p d g n d g resp p g n g be the positive inertia index and negative inertia index of d g resp g in this paper we first establish the inequalities d g p d g p g d g and d g n d g n g d g where d g is the dimension of cycle space of g furthermore all the corresponding extremal graphs are characterized

a linear optimization problem to derive relative weights using an interval judgement matrix abstract in this paper we propose a new decision making model enabling to assess a finite number of alternatives according to a set of bounds on the preference ratios for the pairwise comparisons between alternatives that is an interval judgement matrix in the case in which these bounds cannot be achieved by any assessment vector we analyze the problem of determining of an efficient or pareto optimal solution from a multi objective optimization problem this multi objective formulation seeks for assessment vectors that are near to simultaneously fulfil all the bound requirements imposed by the interval judgement matrix our new model introduces a linear optimization problem in order to define a consistency index for the interval matrix by solving this optimization problem it can be associated a weakly efficient assessment vector to the consistency index in those cases in which the bound requirements are infeasible otherwise this assessment vector fulfils all the bound requirements and has geometrical properties that make it appropriate as a representative assessment vector of the set of feasible weights

the mixed finite element formulation for three dimensional bars abstract in this study mixed finite element equations which arc based on a new functional are obtained by gateaux differential this formulation is applicable to three dimensional bars with arbitrary geometry and variable cross sections boundary conditions are included in the element equations known nodal variable values are imposed by the lagrange multiplier method this newly suggested full functional method ffm approach gives very accurate results using a few elements

positive and negative inertia index of a graph abstract in this paper the positive and negative inertia index of trees unicyclic graphs and bicyclic graphs are discussed the methods of calculating them are obtained and an inequality about the difference between positive and negative inertia index is proved moreover a conjecture is proposed

the maximum and minimum number of efficient units in dea with interval data in original data envelopment analysis dea models inputs and outputs are measured by exact values on a ratio scale cooper et al manage sci 45 1999 597 recently addressed the problem of imprecise data in dea in its general form in this case units do not have constant efficiency score and their efficiency score depend on the choice of data of units therefore it is attractive to know upper and lower bounds for the number of efficient units in this paper we concentrate on dea with interval data and describe which data settings of units produce the maximum number of efficient units then we discuss about the minimum number of efficient units

the signature of line graphs and power trees abstract let g be a graph and let a g be the adjacency matrix of g the signature s g of g is the difference between the positive inertia index and the negative inertia index of a g ma et al 2013 10 conjectured that c 3 g s g c 5 g where c 3 g and c 5 g respectively denote the number of cycles in g which have length 4 k 3 and 4 k 5 for some integers k 0 and proved the conjecture holds for trees unicyclic or bicyclic graphs it is known that s g 0 if g is bipartite and the signature is closely related to the odd cycles or nonbipartiteness of a graph from the existing results in this paper we show that the conjecture holds for line graphs and power trees

interval efficiency assessment using data envelopment analysis this paper studies how to conduct efficiency assessment using data envelopment analysis dea in interval and or fuzzy input output environments a new pair of interval dea models is constructed on the basis of interval arithmetic which differs from the existing dea models handling interval data in that the former is a linear ccr model without the need of extra variable alternations and uses a fixed and unified production frontier i e the same constraint set to measure the efficiencies of decision making units dmus with interval input and output data while the latter is usually a nonlinear optimization problem with the need of extra variable alternations or scale transformations and utilizes variable production frontiers i e different constraint sets to measure interval efficiencies ordinal preference information and fuzzy data are converted into interval data through the estimation of permissible intervals and a level sets respectively and are incorporated into the interval dea models the proposed interval dea models are developed for measuring the lower and upper bounds of the best relative efficiency of each dmu with interval input and output data which are different from the interval formed by the worst and the best relative efficiencies of each dmu a minimax regret based approach mra is introduced to compare and rank the efficiency intervals of dmus two numerical examples are provided to show the applications of the proposed interval dea models and the preference ranking approach

a discussion of testing dmus returns to scale abstract the issue of using banker s most productive scale size mpss concept to characterize dmus returns to scale rts is studied in this paper it is shown that the mpss concept can always be used to estimate rts without any adjustments unless a set of efficient dmus exhibit linear dependency i e it is the dmu itself that causes the mpss concept not to work the mpss concept itself is independent of assuming a linear production function in the ccr model as a result chang and ghu s conclusions are erroneous also several other errors in chang and ghu s proof and illustration of proof are listed the linear dependency condition in the current paper corresponds to the nonunique lambda solution situation in banker and thrall meanwhile a remedy is developed for the mpss concept so that it can still work under linear dependency it is also shown that those dmus which naturally exhibit increasing and decreasing returns to scale have nothing to do with linear dependency

data envelopment analysis with imprecise data abstract in original data envelopment analysis dea models inputs and outputs are measured by exact values on a ratio scale cooper et al management science 45 1999 597 607 recently addressed the problem of imprecise data in dea in its general form we develop in this paper an alternative approach for dealing with imprecise data in dea our approach is to transform a non linear dea model to a linear programming equivalent on the basis of the original data set by applying transformations only on the variables upper and lower bounds for the efficiency scores of the units are then defined as natural outcomes of our formulations it is our specific formulation that enables us to proceed further in discriminating among the efficient units by means of a post dea model and the endurance indices we then proceed still further in formulating another post dea model for determining input thresholds that turn an inefficient unit to an efficient one

interval efficiency measures in data envelopment analysis with imprecise data the conventional data envelopment analysis dea measures the relative efficiencies of a set of decision making units dmus with exact values of inputs and outputs for imprecise data i e mixtures of interval data and ordinal data some methods have been developed to calculate the upper bound of the efficiency scores this paper constructs a pair of two level mathematical programming models whose objective values represent the lower bound and upper bound of the efficiency scores respectively based on the concept of productive efficiency and the application of a variable substitution technique the pair of two level nonlinear programs is transformed to a pair of ordinary one level linear programs solving the associated pairs of linear programs produces the efficiency intervals of all dmus an illustrative example verifies the idea of this paper a real case is also provided to give some interpretation of the interval efficiency interval efficiency not only describes the real situation in better detail psychologically it also eases the tension of the dmus being evaluated as well as the persons conducting the evaluation

rank order data in dea a general framework in data envelopment analysis dea performance evaluation is generally assumed to be based upon a set of quantitative data in many real world settings however it is essential to take into account the presence of qualitative factors when evaluating the performance of decision making units dmus very often rankings are provided from best to worst relative to particular attributes such rank positions might better be presented in an ordinal rather than numerical sense the paper develops a general frame work for modeling and treating qualitative data in dea and provides a unified structure for embedding rank order data into the dea framework the existing techniques are discussed and their equivalence is demonstrated both continuous and discrete projection models are provided it is shown that qualitative data can be treated in conventional dea methodology

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

a decision model for technology selection in the existence of both cardinal and ordinal data technology selection is an important part of management of technology to select the best technologies in the existence of both cardinal and ordinal data this paper proposes an innovative approach which is based on a mathematical programming a numerical example demonstrates the application of the proposed method

on the nullity of graphs with pendant trees the nullity of a graph is defined as the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph we investigate a class of graphs with pendant trees and express the nullity of such graph in terms of that of its subgraphs as an application of our results we characterize unicyclic graphs with a given nullity

data envelopment analysis with missing values an interval dea approach missing values in inputs outputs cannot be handled by the original data envelopment analysis dea models in this paper we introduce an approach based on interval dea that allows the evaluation of the units with missing values along with the other units with available crisp data the missing values are replaced by intervals in which the unknown values are likely to belong the constant bounds of the intervals depending on the application can be estimated by using statistical or experiential techniques for the units with missing values the proposed models are able to identify an upper and a lower bound of their efficiency scores the efficiency analysis is further extended by estimating new values for the initial interval bounds that may turn the unit to an efficient one the proposed methodology is illustrated by an application which evaluates the efficiency of a set of secondary public schools in greece a number of which appears to have missing values in some inputs and outputs

multi component efficiency measurement with imprecise data data envelopment analysis dea evaluates the efficiency of decision making units with multiple inputs and outputs in most applications of dea presented in literature the models presented are designed to obtain a single measure of efficiency where all inputs and outputs are known exactly in many instances however the decision making units involved may perform several different functions or can be separated into different components and some inputs and outputs are unknown decision variables such as bounded data and ordinal data in such situations inputs are often shared among those components and all components play an important role in producing some outputs in this case the standard dea model becomes a non linear program we develop in this paper an alternative approach for dealing with imprecise data in multi component efficiency measurement in dea that preserves the linearity of dea model

multiple criteria modelling and ordinal data evaluation in terms of subsets of criteria in an earlier article an ordinalmultiplecriteria model was presented in which each member of a set of alternatives was given an evaluation on each member of a set k of criteria in this paper we extend this concept to the situation where each alternative i can be assessed in terms of only a subsetki of k various models are presented for dealing with this partial criteria case and the pros and cons of these models are discussed

the inertia of unicyclic graphs and the implications for closed shells the inertia of a graph is an integer triple specifying the number of negative zero and positive eigenvalues of the adjacency matrix of the graph a unicyclic graph is a simple connected graph with an equal number of vertices and edges this paper characterizes the inertia of a unicyclic graph in terms of maximum matchings and gives a linear time algorithm for computing it chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is properly closed shell having exactly half of its eigenvalues greater than zero because this designates a stable electron configuration the inertia determines whether a graph is closed shell and hence the reported result gives a linear time algorithm for determining this for unicyclic graphs

bounds for the positive and negative inertia index of a graph abstract let g be a graph and let a g be adjacency matrix of g the positive inertia index respectively the negative inertia index of g denoted by p g respectively n g is defined to be the number of positive eigenvalues respectively negative eigenvalues of a g in this paper we present the bounds for p g and n g as follows m g c g p g m g c g m g c g n g m g c g where m g and c g are respectively the matching number and the cyclomatic number of g furthermore we characterize the graphs which attain the upper bounds and the lower bounds respectively

technologies ranking in the presence of both cardinal and ordinal data ranking of technologies is an important phase for technology transfer to rank the best technologies in the presence of both cardinal and ordinal data this paper proposes an innovative approach which is based on imprecise data envelopment analysis idea the objective of this paper is to propose a comprehensive reference that discusses the use of idea in technology ranking a numerical example demonstrates the application of the proposed method

a note on including ordinal relations among multipliers in data envelopment analysis data envelopment analysis dea evaluates the efficiency of decision making units without imposing a priori weights on the input and output dimensions involved this note extends dea to situations where ordinal relations among weights corresponding to certain dimensions exist the extended model is reduced to the original dea formulation where some of the input and output factors are accumulated an illustrative example is provided and potential areas of applications are pointed out

duality efficiency computations and interpretations in imprecise dea data envelopment analysis dea has proven to be a useful tool for assessing efficiency or productivity of organizations which is of vital practical importance in managerial decision making while dea assumes exact input and output data the development of imprecise dea idea broadens the scope of applications to efficiency evaluations involving imprecise information which implies various forms of ordinal and bounded data possibly or often occurring in practice the primary purpose of this article is to characterize the variable efficiency in idea since dea describes a pair of primal and dual models also called envelopment and multiplier models we can basically consider two idea models one incorporates imprecise data into envelopment model and the other includes the same imprecise data in multiplier model the issues of rising importance are thus the relationships between the two models and how to solve them the groundwork we will make includes a duality study which makes it possible to characterize the efficiency solutions from the two models and link with the efficiency bounds and classifications that some of the published idea studies have done the other purposes are to present computational aspects of the efficiency bounds and how to interpret the efficiency solutions the computational method developed here extends the previous idea method to effectively incorporate a more general form of strict ordinal data and partial orders in its framework which in turn overcomes some drawbacks of the previous approaches the interpretation of the resulting efficiency is also important but we have never seen it before

the positive and the negative inertia index of line graphs of trees abstract the positive resp negative inertia index of g denoted by p g resp n g is defined to be the number of positive resp negative eigenvalues of its adjacency matrix the nullity of g denoted by g is defined to be the multiplicity of the eigenvalue zero in the adjacency spectrum of g in 2001 gutman and sciriha 11 proved that the nullity of the line graph of a tree is either 0 or 1 in this paper we consider the positive and the negative inertia index for the line graph l t of a tree t showing that e t 1 2 p l t e t 1 where e t denotes the number of internal non pendant edges contained in t the extremal trees for which p l t attains the upper bound and the lower bound are respectively characterized it is shown that l t is nonsingular if p l t attains the upper bound and l t is singular if p l t attains the lower bound

cost efficiency measures in data envelopment analysis with data uncertainty this paper extends the classical cost efficiency ce models to include data uncertainty we believe that many research situations are best described by the intermediate case where some uncertain input and output data are available in such cases the classical cost efficiency models cannot be used because input and output data appear in the form of ranges when the data are imprecise in the form of ranges the cost efficiency measure calculated from the data should be uncertain as well so in the current paper we develop a method for the estimation of upper and lower bounds for the cost efficiency measure in situations of uncertain input and output data also we develop the theory of efficiency measurement so as to accommodate incomplete price information by deriving upper and lower bounds for the cost efficiency measure the practical application of these bounds is illustrated by a numerical example

optimality conditions for generalized differentiable interval valued functions in this paper we study the optimal solutions set for a generalized differentiable interval valued function necessary and sufficient optimality conditions are established for gh differentiable functions convexity assumptions that are necessary or required to ensure the characterization of the optimal solutions are weaker or less strict than those presented in previous works these convexity assumptions are the weakest to characterize the optimal solutions set known results for classical non interval valued optimization are particular cases of the ones proved here