dispersion analysis and element free galerkin solutions of second and fourth order gradient enhanced damage models gradient dependent damage formulations incorporate higher order derivatives of state variables in the constitutive equations different formulations have been derived for this gradient enhancement comparison of which is difficult in a finite element context due to higher order continuity requirements for certain formulations on the other hand the higher order continuity requirements are met naturally by element free galerkin efg shape functions thus the efg method provides a suitable tool for the assessment of gradient enhanced continuum models dispersion analyses have been carried out to compare different gradient enhanced models with the non local damage model the formulation of the additional boundary conditions is addressed numerical examples show the objectivity with respect to the discretization and the differences between various gradient formulations with second and fourth order derivatives it is shown that with the same underlying internal length scale very different results can be obtained copyright 2000 john wiley sons ltd

reducing weight flexibility in fuzzy dea an important outcome of assessing relative efficiencies within a group of decision making units dmus in fuzzy data envelopment analysis is a set of virtual multipliers or weights accorded to each input or output factor taken into account these sets of weights are typically different for each of the participating dmus and in some cases it may be considered unacceptable that the same factor is accorded widely differing weights thus it is important to find a common set of weights csw across the set of dmus in this paper by assessing upper bounds on factor weights and compacting the resulted intervals a csw is determined since resulted efficiencies by the proposed csw are fuzzy numbers rather than crisp values it is more informative for decision maker

a damage model with evolving nonlocal interactions we present a damage model for softening materials with evolving nonlocal interactions the thermodynamic implications and the material stability issue are addressed the proposed nonlocal averaging scheme provides the obtained constitutive models with an evolving nonlocal interaction which is activated only when damage occurs in the analysis of structures made of quasi brittle materials this feature helps not only to overcome some issues with the incorrect initiation of damage but also to better control the evolving size of the active fracture process zone this is an essential feature that is usually not considered in depth in many existing nonlocal approaches to the continuum modelling of quasi brittle fracture numerical examples are given to demonstrate features of the proposed modelling approach

solving the vehicle routing problem with time windows and multiple routes exactly using a pseudo polynomial model in this paper we address a variant of the vehicle routing problem called the vehicle routing problem with time windows and multiple routes it considers that a given vehicle can be assigned to more than one route per planning period we propose a new exact algorithm for this problem our algorithm is iterative and it relies on a pseudo polynomial network flow model whose nodes represent time instants and whose arcs represent feasible vehicle routes this algorithm was tested on a set of benchmark instances from the literature the computational results show that our method is able to solve more instances than the only other exact method described so far in the literature and it clearly outperforms this method in terms of computing time

a generic exact solver for vehicle routing and related problems major advances were recently obtained in the exact solution of vehicle routing problems vrps sophisticated branch cut and price bcp algorithms for some of the most classical vrp variants now solve many instances with up to a few hundreds of customers however adapting and reimplementing those successful algorithms for other variants can be a very demanding task this work proposes a bcp solver for a generic model that encompasses a wide class of vrps it incorporates the key elements found in the best existing vrp algorithms ng path relaxation rank 1 cuts with limited memory path enumeration and rounded capacity cuts all generalized through the new concepts of packing set and elementarity set the concepts are also used to derive a branching rule based on accumulated resource consumption and to generalize the ryan and foster branching rule extensive experiments on several variants show that the generic solver has an excellent overall performance in many problems being better than the best specific algorithms even some non vrps like bin packing vector packing and generalized assignment can be modeled and effectively solved

a taxonomy and review of the fuzzy data envelopment analysis literature two decades in the making data envelopment analysis dea is a methodology for measuring the relative efficiencies of a set of decision making units dmus that use multiple inputs to produce multiple outputs crisp input and output data are fundamentally indispensable in conventional dea however the observed values of the input and output data in real world problems are sometimes imprecise or vague many researchers have proposed various fuzzy methods for dealing with the imprecise and ambiguous data in dea in this study we provide a taxonomy and review of the fuzzy dea methods we present a classification scheme with four primary categories namely the tolerance approach the a level based approach the fuzzy ranking approach and the possibility approach we discuss each classification scheme and group the fuzzy dea papers published in the literature over the past 20 years to the best of our knowledge this paper appears to be the only review and complete source of references on fuzzy dea

linear programming with fuzzy variables in practice there are many problems in which all decision parameters are fuzzy numbers and such problems are usually solved by either possibilistic programming or multiobjective programming methods unfortunately all these methods have shortcomings in this note using the concept of comparison of fuzzy numbers we introduce a very effective method for solving these problems then we propose a new method for solving linear programming problems with fuzzy variables

fuzzy dea a perceptual evalution method abstract dea data envelopment analysis is a non parametric technique for measuring and evaluating the relative efficiencies of a set of entities with common crisp inputs and outputs in fact in a real evaluation problem input and output data of entities evaluated often fluctuate these fluctuating data can be represented as linguistic variables characterized by fuzzy numbers for reflecting a kind of general feeling or experience of experts based on the fundamental ccr model a fuzzy dea model is proposed to deal with the efficiency evaluation problem with the given fuzzy input and output data furthermore an extension of the fuzzy dea model to a more general form is also proposed with considering the relationship between dea and ra regression analysis using the proposed fuzzy dea models the crisp efficiency in ccr model is extended to be a fuzzy number to reflect the inherent uncertainty in real evaluation problems

a fuzzy approach to select the location of the distribution center abstract the location selection of distribution center dc is one of the most important decision issues for logistics managers owing to vague concept frequently represented in decision data a new multiple criteria decision making method is proposed to solve the distribution center location selection problem under fuzzy environment in the proposed method the ratings of each alternative and the weight of each criterion are described by linguistic variables which can be expressed in triangular fuzzy numbers the final evaluation value of each dc location is also expressed in a triangular fuzzy number by calculating the difference of final evaluation value between each pair of dc locations a fuzzy preference relation matrix is constructed to represent the intensity of the preferences of one plant location over another and then a stepwise ranking procedure is proposed to determine the ranking order of all candidate locations finally a numerical example is solved to illustrate the procedure of the proposed method at the end of this paper

fuzzy scores of technical efficiency abstract this paper suggests to extend scores of technical efficiency as used in for example data envelopment analysis dea to efficiency scores defined as so called fuzzy intervals fuzzy scores allow the decision maker to use scores of technical efficiency in combination with other sources of available performance information e g expert opinions key figures etc in this sense fuzzy scores become formal images of the evaluation otherwise done implicitly by the decision maker and may therefore enhance a more consistent evaluation it is shown how to rank a set of fuzzy scores

twisted orlicz algebras and complete isomorphism to operator algebras abstract let g be a locally compact group let g g c be a 2 cocycle and let denote the twisted convolution associated to given a complementary pair of young functions we prove that if is continuous and strictly increasing l g l 2 g and 0 1 s t u s v t s t g for some u v s g then l g with the maximal operator space structure is completely isomorphic to an operator algebra we also present various classes of 2 cocycles for which one could obtain such algebras generalizing in part the results of 15 we apply our methods to compactly generated groups of polynomial growth and demonstrate that our results could be applied to variety of cases

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

mixed stabilized finite element methods in nonlinear solid mechanics part i formulation abstract this paper exploits the concept of stabilized finite element methods to formulate stable mixed stress displacement and strain displacement finite elements for the solution of nonlinear solid mechanics problems the different assumptions and approximations used to derive the methods are exposed the proposed procedure is very general applicable to 2d and 3d problems implementation and computational aspects are also discussed showing that a robust application of the proposed formulation is feasible numerical examples show that the results obtained compare favorably with those obtained with the corresponding irreducible formulation

finite element analysis of bifurcation in nonlocal strain softening solids progressive damage in brittle heterogeneous materials produces at the macroscopic level strain softening from which theoretical difficulties arise e g ill posedness and multiple bifurcation points this characteristics behavior favours spurious strain localization in numerical analyses and calls for the implementation of localization limiters for instance nonlocal damage constitutive relations the issue of possible stable or unstable equilibrium paths multiple localization zones and of the detection of bifurcation points has however never been addressed in the context of nonlocal constitutive laws we extend here the eigenmode analysis and perturbation method proposed by de borst to the study of the bifurcation and post bifurcation response of discrete nonlocal strain softening solids numerical applications on beams show that bifurcation and instability may occur in the post peak regime as opposed to the case of local constitutive relations the number of possible solutions at a bifurcation point is restricted due to the constraint introduced by the localization limiter and these solutions are shown to be mesh independent

multi phase dynamic constraint aggregation for set partitioning type problems dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems this speed up is mostly due to the use at each iteration of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model this aggregation is updated when needed to ensure the exactness of the overall approach in this paper we propose a new version of this method called the multi phase dynamic constraint aggregation method which essentially adds to the original method a partial pricing strategy that involves multiple phases this strategy helps keeping the size of the aggregated problem as small as possible yielding a faster average computation time per iteration and fewer iterations we also establish theoretical results that provide some insights explaining the success of the proposed method tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2 000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4 5

a branch and cut and price algorithm for one dimensional stock cutting and two dimensional two stage cutting abstract the one dimensional cutting stock problem 1d csp and the two dimensional two stage guillotine constrained cutting problem 2d 2cp are considered in this paper the gilmore gomory models of these problems have very strong continuous relaxations providing a good bound in an lp based solution approach in recent years there have been several efforts to attack the one dimensional problem by lp based branch and bound with column generation called branch and price and by general purpose c hv a tal g omory cutting planes in this paper we investigate a combination of both approaches i e the lp relaxation at each branch and price node is strengthened by c hv a tal g omory and g omory mixed integer cuts the branching rule is that of branching on variables of the gilmore gomory formulation tests show that for 1d csp general purpose cuts are useful only in exceptional cases however for 2d 2cp their combination with branching is more effective than either approach alone and mostly better than other methods from the literature

an exact algorithm for a single vehicle routing problem with time windows and multiple routes this paper describes an exact algorithm for solving a problem where the same vehicle performs several routes to serve a set of customers with time windows the motivation comes from the home delivery of perishable goods where vehicle routes are short and must be combined to form a working day a method based on an elementary shortest path algorithm with resource constraints is proposed to solve this problem the method is divided into two phases in the first phase all non dominated feasible routes are generated in the second phase some routes are selected and sequenced to form the vehicle workday computational results are reported on euclidean problems derived from benchmark instances of the classical vehicle routing problem with time windows

size effect and localization in j2 plasticity this paper studies the phenomenon of structural size effect and strain localization in j2 plasticity size effect is here understood as the change in the response of a given structure when the spatial dimensions are scaled up or down while the geometry and other relevant properties of the structure are preserved the work exploits the advantages of the mixed displacement pressure formulation in incompressible or quasi incompressible situations elasto j2 plastic constitutive behaviour with regularized softening is considered stability issues are discussed to ensure existence and uniqueness of the solution of the corresponding discrete finite element formulation numerical examples show that the formulation derived is able to solve a wide range of structural scales including real life engineering applications the results obtained do not suffer from spurious mesh dependence furthermore the formulation includes the classical theories of perfect plasticity and linear fracture mechanics as limit cases

fuzzy data envelopment analysis dea a possibility approach abstract evaluating the performance of activities or organizations by traditional data envelopment analysis dea models requires crisp input output data however in real world problems inputs and outputs are often imprecise this paper develops dea models using imprecise data represented by fuzzy sets i e fuzzy dea models it is shown that fuzzy dea models take the form of fuzzy linear programming which typically are solved with the aid of some methods to rank fuzzy sets as an alternative a possibility approach is introduced in which constraints are treated as fuzzy events the approach transforms fuzzy dea models into possibility dea models by using possibility measures of fuzzy events fuzzy constraints we show that for the special case in which fuzzy membership functions of fuzzy data are of trapezoidal types possibility dea models become linear programming models a numerical experiment is used to illustrate the approach and compare the results with those obtained with alternative approaches

bidimensional packing by bilinear programming we consider geometric problems in which rectangles have to be packed into identical squares these problems turn out to be very hard in practice and ilp formulations in which variables specify the coordinates in the packing perform very poorly while most methods developed so far are based on simple geometric considerations a recent landmark result of fekete and schepers suggests to put these geometric aspects aside and use the most advanced tools for the one dimensional case in this paper we make additional progress in this direction especially on the basic question does a given set of rectangles fit into a square which turns out to be the bottleneck of all the approaches known given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles respectively in addition we illustrate how to solve the associated separation problem as a bilinear program for which we develop a solution method that turns out to be fast in practice and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature the same tools are used to derive lower bounds for the two dimensional bin packing problem corresponding to the determination of an optimal pair of so called dual feasible functions these lower bounds in many cases equal those obtained by the customary set covering formulation for which column generation is very hard but are computable within a time that is smaller by some orders of magnitude this allows us to close a few of the benchmark instances in the literature

ranking fuzzy numbers based on decomposition principle and signed distance by using the decomposition principle and the crisp ranking system on r we construct a new ranking system for fuzzy numbers which is very realistic and also matching our intuition as in r

a new ranking method to fuzzy data envelopment analysis due to its wide practical use data envelopment analysis dea has been adapted to many fields to deal with problems that have occurred in practice one adaptation has been in the field of ranking decision making units dmus most methods of ranking dmus assume that all input and output data are exactly known but in real life the data cannot be precisely measured thus this paper will carry out some researches to dea under fuzzy environment a fuzzy comparison of fuzzy variables is defined and the ccr model is extended to be a fuzzy dea model based on credibility measure in order to rank all the dmus a full ranking method will be given since the ranking method involves a fuzzy function a fuzzy simulation is designed and embedded into the genetic algorithm to establish a hybrid intelligent algorithm however it is shown to be possible to avoid some of the need for dealing with these nonlinear problems by identifying conditions under which they can be replaced by linear problems finally we will provide a numerical example to illustrate the fuzzy dea model and the ranking method

non local damage model based on displacement averaging continuum damage models describe the changes of material stiffness and strength caused by the evolution of defects in the framework of continuum mechanics in many materials a fast evolution of defects leads to stress strain laws with softening which creates serious mathematical and numerical problems to regularize the model behaviour various generalized continuum theories have been proposed integral type non local damage models are often based on weighted spatial averaging of a strain like quantity this paper explores an alternative formulation with averaging of the displacement field damage is assumed to be driven by the symmetric gradient of the non local displacements it is demonstrated that an exact equivalence between strain and displacement averaging can be achieved only in an unbounded medium around physical boundaries of the analysed body both formulations differ and the non local displacement model generates spurious damage in the boundary layers the paper shows that this undesirable effect can be suppressed by an appropriate adjustment of the non local weight function alternatively an implicit gradient formulation could be used issues of algorithmic implementation computational efficiency and smoothness of the resolved stress fields are discussed copyright 2005 john wiley sons ltd

example of a weighted algebra lpw g on an uncountable discrete group abstract we construct examples of weighted algebras l p w g with 1 p 2 on uncountable free groups for p 2 no weighted algebras exist on these groups on the other hand we prove that for any amenable locally compact group g if l p w g is an algebra for some weight w and p 1 then g is compact

a fuzzy systems approach in data envelopment analysis abstract the use of fuzzy set theoretic measures is explored here in the context of data envelopment analysis which utilizes a nonparametric approach to measure efficiency three types of fuzzy statics are employed here e g fuzzy mathematical programme fuzzy regression and fuzzy entropy to illustrate the types of decisions and solutions that are achievable when the data are vague and prior information is inexact and imprecise

measuring efficiency by a fuzzy statistical approach methods of measuring economic efficiency of input output systems by employing a fuzzy statistical approach are explored here in the context of the recent technique of data envelopment analysis which utilizes a nonparametric approach to measure efficiency three applied fuzzy measures are illustrated here e g fuzzy linear programs fuzzy linear regressions and finally the fuzzy entropy measures in the context of a two person game theory model

lp models for bin packing and cutting stock problems abstract we review several linear programming lp formulations for the one dimensional cutting stock and bin packing problems namely the models of kantorovich gilmore gomory onecut models as in the dyckhoff stadtler approach position indexed models and a model derived from the vehicle routing literature we analyse some relations between the corresponding lp relaxations and their relative strengths and refer how to derive branching schemes that can be used in the exact solution of these problems using branch and price

branch and price algorithms for the solution of the multi trip vehicle routing problem with time windows we investigate the exact solution of the vehicle routing problem with time windows where multiple trips are allowed for the vehicles in contrast to previous works in the literature we specifically consider the case in which it is mandatory to visit all customers and there is no limitation on duration we develop two branch and price frameworks based on two set covering formulations a traditional one where columns variables represent routes that is a sequence of consecutive trips and a second one in which columns are single trips one important difficulty related to the latter is the way mutual temporal exclusion of trips can be handled it raises the issue of time discretization when solving the pricing problem our dynamic programming algorithm is based on concept of groups of labels and representative labels we provide computational results on modified small sized instances 25 customers from solomon s benchmarks in order to evaluate and compare the two methods results show that some difficult instances are out of reach for the first branch and price implementation while they are consistently solved with the second

a new fuzzy multiple attributive group decision making methodology and its application to propulsion manoeuvring system selection problem in this paper a new fuzzy multiple attribute decision making fmadm method which is suitable for multiple attributive group decision making gdm problems in fuzzy environment is proposed to deal with the problem of ranking and selection of alternatives since the subjectivity imprecision and vagueness in the estimates of a performance rating enter into multiple attribute decision making madm problems fuzzy set theory provides a mathematical framework for modelling vagueness and imprecision in the proposed approach an attribute based aggregation technique for heterogeneous group of experts is employed and used for dealing with fuzzy opinion aggregation for the subjective attributes of the decision problem the propulsion manoeuvring system selection as a real case study is used to demonstrate the versatility and potential of the proposed method for solving fuzzy multiple attributive group decision making problems the proposed method is a generalised model which can be applied to great variety of practical problems encountered in the naval architecture from propulsion manoeuvring system selection to warship requirements definition

inequality relation between fuzzy numbers and its use in fuzzy optimization the inequality relation between two fuzzy numbers is investigated a certain type of such a relation motivated by practical interpretation is proposed and its correspondence with the usual lattice type relation generated by the extended maximum and minimum operators and its possible interpretation are discussed the concept of r l fuzzy number is introduced the class of all r l fuzzy numbers covering practically the whole set of normal convex fuzzy numbers comparing two r l fuzzy numbers of the same type the relation introduced in the paper may be replaced by four ordinary inequalities this fact may be taken advantage of in optimization problems with linear fuzzy constraints

efficient and reliable nonlocal damage models we present an efficient and reliable approach for the numerical modelling of failure with nonlocal damage models the two major numerical challenges the strongly nonlinear highly localized and parameter dependent structural response of quasi brittle materials and the interaction between non adjacent finite elements associated to nonlocality are addressed in detail efficiency is achieved with a suitable combination of load stepping control technique and nonlinear solver for equilibrium equations reliability of the numerical results is ensured by an h adaptive strategy based on error estimation we use a residual type error estimator for nonlinear fe analysis based on local computations which at the same time accounts for the nonlocality of the damage model the proposed approach is illustrated by means of three application examples the three point bending test the single edge notched beam test and the brazilian test

a fuzzy multicriteria group decision making approach to select configuration items for software development selection of configuration items in software configuration management is important to determine the software quality and reduce the development time and cost the objective of this research is to develop a multicriteria group decision making model based on fuzzy set theory to improve the configuration items selection process since most information available in this stage is not numerical fuzzy set theory is used to represent the evaluation ratings of candidate items the developed model ranks candidate items into partial or complete orders that can assist decision makers in selecting more proper set of configuration items the consensus measures are also developed to determine the group acceptability of the obtained ranking orders in addition sensitivity analysis can be performed to examine the solution robustness an example of flight simulator development project is used to illustrate the concept developed

dispersion and isogeometric analyses of second order and fourth order implicit gradient enhanced plasticity models implicit gradient plasticity models incorporate higher order spatial gradients via an additional helmholtz type equation for the plastic multiplier so far the enrichment has been limited to second order spatial gradients resulting in a formulation that can be discretised using c0 continuous finite elements herein an implicit gradient plasticity model is formulated that includes a fourth order gradient term as well a comparison between the localisation properties of both the implicit gradient plasticity formulations and the explicit second order gradient plasticity model is made using a dispersion analysis the higher order continuity requirement for the fourth order implicit gradient plasticity model has been met by exploiting the higher order continuity property of isogeometric analysis which uses nonuniform rational b splines as shape functions instead of lagrange polynomials the discretised variables displacements and plastic multiplier may require different orders of interpolation an issue that is also addressed numerical results show that both formulations can be used as a localisation limiter but that quantitative differences occur and a different evolution of the localisation band is obtained for 2 dimensional problems

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

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

a fuzzy mathematical programming approach to the assessment of efficiency with dea models in many real applications the data of production processes cannot be precisely measured this is particularly worrying when assessing efficiency with frontier type models such as data envelopment analysis dea models since they are very sensitive to possible data errors for this reason the possibility of having available a methodology that allows the analyst to deal with imprecise data becomes an issue of great interest in these contexts to that end we develop some fuzzy versions of the classical dea models in particular the bcc model by using some ranking methods based on the comparison of cuts the resulting auxiliary crisp problems can be solved by the usual dea software we show in a numerical example how our models become specially useful for detecting sensitive decision making units our approaches can be seen as an extension of the dea methodology that provides users and practitioners with models which represent some real life processes more appropriately

dynamic aggregation of set partitioning constraints in column generation column generation is often used to solve problems involving set partitioning constraints such as vehicle routing and crew scheduling problems when these constraints are in large numbers and the columns have on average more than 8 12 nonzero elements column generation often becomes inefficient because solving the master problem requires very long solution times at each iteration due to high degeneracy to overcome this difficulty we introduce a dynamic constraint aggregation method that reduces the number of set partitioning constraints in the master problem by aggregating some of them according to an equivalence relation to guarantee optimality this equivalence relation is updated dynamically throughout the solution process tests on the linear relaxation of the simultaneous vehicle and crew scheduling problem in urban mass transit show that this method significantly reduces the size of the master problem degeneracy and solution times especially for larger problems in fact for an instance involving 1 600 set partitioning constraints the master problem solution time is reduced by a factor of 8

measure of efficiency in dea with fuzzy input output levels a methodology for assessing ranking and imposing of weights restrictions in this paper a fuzzy comparison of fuzzy numbers is defined and a slack based measure sbm model in data envelopment analysis dea is extended to be a fuzzy dea model using it proposed measure is employed for evaluation and ranking of all decision making units using a fuzzy concept called fuzzy profit also it is shown that the introduced model is convenient for using weights restrictions furthermore we compare the results of proposed model with guo and tanaka s results fuzzy sets syst 119 2001 149 by representing a numerical example introduced by them

a new approach to some possibilistic linear programming problems abstract in practice the unit costs profits of new products or new projects the lending and borrowing interest rates and cash flows are always imprecise we provide an auxiliary multiple objective linear programming model to solve a linear programming problem with imprecise objective and or constraint coefficients our strategy is to maximize the most possible value of the imprecise profit at the same time we would like to minimize the risk of obtaining lower profit and maximize the possibility of obtaining higher profit this strategy is equivalent to the practical considerations of financial problems in this paper a numeric investment problem is solved for illustrating the new approach

worst case analyses linear programming and the bin packing problem in this paper we consider the familiar bin packing problem and its associated set partitioning formulation we show that the optimal solution to the bin packing problem can be no larger than 4 3 zlp wherezlp is the optimal solution value of the linear programming relaxation of the set partitioning formulation an example is provided to show that the bound is tight a by product of our analysis is a new worst case bound on the performance of the well studied first fit decreasing and best fit decreasing heuristics

a simple approximation of productivity scores of fuzzy production plans this paper suggests a simple approximation procedure for the assessment of productivity scores with respect to fuzzy production plans the procedure has a clear economic interpretation and all the necessary calculations can be performed in a spreadsheet making it highly operational

mixed stabilized finite element methods in nonlinear solid mechanics part ii strain localization this paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining a standard local isotropic rankine damage model with strain softening is used as exemplary constitutive model both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated from this analysis the issue of local discretization error in the pre localization regime is deemed as the main difficulty to be overcome in the discrete problem focus is placed on low order finite elements with continuous strain and displacement fields triangular p1p1 and quadrilateral q1q1 although the presented approach is very general numerical examples show that the resulting procedure is remarkably robust it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh bias dependence

a parametric approach to fuzzy linear programming in this paper we discuss how to deal with decision problems that are described with lp models and formulated with elements of imprecision and uncertainty more precisely we will study lp models in which the parameters are not fully known but only with some degree of precision even with incomplete information the model builder or model user is normally able to give a realistic interval for the parameters of an lp model for the constraint vector this is combined with some wishes or some leeway on the constraints even with ambiquity in the objective function there is normally some preference ordering to be found among alternative ways of action we will demonstrate that these modelling complications can be handled with the help of some results developed in the theory of fuzzy sets after an overview of some central contributions to fuzzy linear programming we will develop an lp model in which the parameters are not fully known only with some degree of precision and show that the model can be parametrised in such a way that the optimal solution becomes a function of the degree of precision the fuzzy lp model derived in this way appears to be fairly easy to handle computationally which is demonstrated with a numerical example

dual models of interval dea and its extension to interval data abstract the purpose of this paper is to develop a new dea with an interval efficiency an original dea model is to evaluate each dmu optimistically there is another model called inverted dea to evaluate each dmu pessimistically but there are no relations essentially between dea and inverted dea thus we formulate a dea model with an interval efficiency which consists of efficiencies obtained from the optimistic and pessimistic viewpoints thus two end points can construct an interval efficiency with the same idea we deal with the interval inefficiency model which is inverse to interval efficiency finally we extend the proposed dea model to interval data and fuzzy data

fuzzy efficiency measures in data envelopment analysis abstract the existing data envelopment analysis dea models for measuring the relative efficiencies of a set of decision making units dmus using various inputs to produce various outputs are limited to crisp data to deal with imprecise data the notion of fuzziness has been introduced this paper develops a procedure to measure the efficiencies of dmus with fuzzy observations the basic idea is to transform a fuzzy dea model to a family of conventional crisp dea models by applying the cut approach a pair of parametric programs is formulated to describe that family of crisp dea models via which the membership functions of the efficiency measures are derived since the efficiency measures are expressed by membership functions rather than by crisp values more information is provided for management by extending to fuzzy environment the dea approach is made more powerful for applications

convolution weights on l2 spaces abstract it is known that the weighted l p space on a locally compact group is stable with respect to convolution if the weight function satisfies a certain type of convolution inequality there are several counterexamples showing that this sufficient condition is not necessary however for one class of groups namely discrete abelian groups no such counterexample is known thus there remains the possibility that the convolution inequality truly characterizes stability of convolution for weighted l p spaces on these groups in this paper we investigate this inequality and in the case p 2 reinterpret it in the light of operator theory and in the context of the theory of reproducing kernel hilbert spaces

iterative aggregation and disaggregation algorithm for pseudo polynomial network flow models with side constraints this paper develops a general solution framework based on aggregation techniques to solve np hard problems that can be formulated as a circulation model with specific side constraints the size of the extended mixed integer linear programming formulation is generally pseudo polynomial to efficiently solve exactly these large scale models we propose a new iterative aggregation and disaggregation algorithm at each iteration it projects the original model onto an aggregated one producing an approximate model the process iterates to refine the current aggregated model until the opti mality is proved the computational experiments on two hard optimization problems a variant of the vehicle routing problem and the cutting stock problem show that a generic implementation of the proposed framework allows us to out perform previous known methods

mathematical formulations and exact algorithm for the multitrip cumulative capacitated single vehicle routing problem abstract this paper addresses the multitrip cumulative capacitated single vehicle routing problem mt ccsvrp in this problem inspired by disaster logistics a single vehicle can perform successive trips to serve a set of affected sites and minimize an emergency criterion the sum of arrival times two mixed integer linear programs a flow based model and a set partitioning model are proposed for small instances with 20 sites an exact algorithm for larger cases transforms the mt ccsvrp into a resource constrained shortest path problem where each node corresponds to one trip and the sites to visit become resources the resulting problem can be solved via an adaptation of bellman ford algorithm to a directed acyclic graph with resource constraints and a cumulative objective function seven dominance rules two upper bounds and five lower bounds speed up the procedure computational results on instances derived from classical benchmark problems for the capacitated vrp indicate that the exact algorithm outperforms a commercial mip solver on small instances and can solve cases with 40 sites to optimality

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

computational and theoretical pitfalls in some current performance measurement techniques and a new approach this paper addresses some pitfalls of some fuzzy data envelopment analysis models which have been provided in recent published papers this study deals with these papers from both computational and theoretical points of view moreover a new approach to deal with fuzzy data in dea framework is provided which does not have the mentioned pitfalls

aggregation and disaggregation techniques and methodology in optimization a fundamental issue in the use of optimization models is the tradeoff between the level of detail and the ease of using and solving the model aggregation and disaggregation techniques have proven to be valuable tools for manipulating data and determining the appropriate policies to employ for this tradeoff furthermore aggregation and disaggregation techniques offer promise for solving large scale optimization models supply a set of promising methodologies for studying the underlying structure of both univariate and multivariate data sets and provide a set of tools for manipulating data for different levels of decision makers in this paper we develop a general framework for aggregation and disaggregation methodology survey previous work regarding aggregation and disaggregation techniques for optimization problems illuminate the appropriate role of aggregation and disaggregation methodology for optimization applications and propose future research directions

error estimation and adaptivity for nonlocal damage models abstract nonlocal damage models are typically used to model failure of quasi brittle materials due to brittleness the choice of a particular model or set of parameters can have a crucial influence on the structural response to assess this influence it is essential to keep finite element discretization errors under control if not the effect of these errors on the result of a computation could be erroneously interpreted from a constitutive viewpoint to ensure the quality of the fe solution an adaptive strategy based on error estimation is proposed here it is based on the combination of a residual type error estimator and quadrilateral h remeshing another important consequence of brittleness is that it leads to structural responses of the snap through or snap back type this requires the use of arc length control with a definition of the arc parameter that accounts for the localized nature of quasi brittle failure all these aspects are discussed for two particular nonlocal damage models mazars and modified von mises and for two tests the brazilian tensile splitting test and the single edge notched beam test for the latter test the capability of the mazars model to capture the curved crack pattern observed in experiments a topic of debate in the literature is confirmed

an exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles the vehicle routing problem with multiple use of vehicles is a variant of the classical vehicle routing problem it arises when each vehicle performs several routes during the workday due to strict time limits on route duration e g when perishable goods are transported the routes are defined over customers with a revenue a demand and a time window given a fixed size fleet of vehicles it might not be possible to serve all customers thus the customers must be chosen based on their associated revenue minus the traveling cost to reach them we introduce a branch and price approach to address this problem where lower bounds are computed by solving the linear programming relaxation of a set packing formulation using column generation the pricing subproblems are elementary shortest path problems with resource constraints computational results are reported on euclidean problems derived from well known benchmark instances for the vehicle routing problem with time windows

friendly bin packing instances without integer round up property it is well known that the gap between the optimal values of bin packing and fractional bin packing if the latter is rounded up to the closest integer is almost always null known counterexamples to this for integer input values involve fairly large numbers specifically the first one was derived in 1986 and involved a bin capacity of the order of a billion later in 1998 a counterexample with a bin capacity of the order of a million was found in this paper we show a large number of counterexamples with bin capacity of the order of a hundred showing that the gap may be positive even for numbers which arise in customary applications the associated instances are constructed starting from the petersen graph and using the fact that it is fractionally but not integrally 3 edge colorable

comparison of fuzzy numbers using a fuzzy distance measure a new approach for ranking fuzzy numbers based on a distance measure is introduced a new class of distance measures for interval numbers that takes into account all the points in both intervals is developed first and then it is used to formulate the distance measure for fuzzy numbers the approach is illustrated by numerical examples showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches

fuzzy outranking for environmental assessment case study iron and steel making industry recent years have seen a world wide change in the environmental policy towards integrated pollution prevention taking into account all environmental media consequently the environmental assessment of production techniques has to consider multiple criteria which cannot be aggregated to one single index the concepts of multi criteria decision making mcdm seem a suitable means in order to implement life cycle assessment lca into integrated decision processes the outranking methods as a special subgroup of mcdm methods are particularly suitable for integral decision making through the notion of weak preference and incomparability which better represent the real decision situation as demonstrated in this paper especially the outranking method promethee brings together flexibility and simplicity for the user and is therefore chosen for the enhancement towards the evaluation of fuzzy data on preferences scores and weights

ranking fuzzy numbers with maximizing set and minimizing set abstract up to now these are five methods of ranking n fuzzy numbers in order but these methods contain some confusions and occasionally conflict with intuition this paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers in addition we give a method for calculating the ordering value of each fuzzy number with triangular trapezoidal and two sided drum like shaped membership functions

an algorithm for gradient regularized plasticity coupled to damage based on a dual mixed fe formulation abstract a thermodynamically consistent theory of gradient regularized plasticity with coupling to isotropic damage was presented by svedberg and runesson 23 its main purpose was to describe the successive development of the localization zone that is pertinent to the softening regime due to excessive damage close to failure explicit constitutive relations were established for a model problem based on von mises plasticity with isotropic hardening of two types local and gradient enhanced the main purpose of this paper is to describe a new numerical algorithm for the gradient enhanced formulation using the backward euler rule leads to a gradient enhanced format of the classical closest point projection method the plastic multiplier increment is solved via a dual mixed method which is a convenient feature in the present context this leads to an unambiguous representation of the constitutive relations in the equilibrium equation moreover the classical local algebraic kuhn tucker conditions are obtained in a straightforward fashion in the limit case of local theory the developed algorithm was implemented in the educational finite element package 5 numerical results are shown for the extension of a plate in plane stress these results demonstrate the convergence characteristics with mesh refinement as well as the salient features of the adopted damage based constitutive model