rough prime ideals and rough fuzzy prime ideals in semigroups in this paper we shall introduce the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup and give some properties of such ideals also we will discuss the relations between the upper lower rough prime ideals and the upper lower approximations of their homomorphism images

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

a rigorous computational comparison of alternative solution methods for the generalized assignment problem statistical experimental design and analysis is a cornerstone for scientific inquiry that is rarely applied in reporting computational testing this approach is employed to study the relative performance characteristics of the four leading algorithmic and heuristic alternatives to solve the linear cost generalized assignment problem lcgap against a newly developed heuristic variable depth search heuristic vdsh in assessing the relative effectiveness of the prominent solution methodologies and vdsh under the effects of various problem characteristics we devise a carefully designed experimentation of state of the art implementations through a rigorous statistical analysis we identify the most efficient method s for commonly studied lcgaps and determine the effect on solution time and quality of problem class and size

a tabu search approach to the constraint satisfaction problem as a general problem solver many combinatorial problems including a variety of combinatorial optimization problems can be naturally formulated as a constraint satisfaction problem csp we develop in this paper a tabu search based algorithm for the csp as a foundation for a general problem solver in addition to the basic components of tabu search we develop a number of elaborations such as an automatic control mechanism for the tabu tenure modification of the penalty function to handle objective functions and enlargement of the neighborhood by allowing swap operations computational results with our algorithm are reported for various problems selected from a wide range of applications i e graph coloring generalized assignment set covering timetabling and nurse scheduling our results appear to be competitive with those of existing algorithms specially developed for the respective problem domains

the equilibrium generalized assignment problem and genetic algorithm abstract the well known generalized assignment problem gap is to minimize the costs of assigning n jobs to m capacity constrained agents or machines such that each job is assigned to exactly one agent this problem is known to be np hard and it is hard from a computational point of view as well in this paper follows from practical point of view in real systems the gap is extended to the equilibrium generalized assignment problem egap and the equilibrium constrained generalized assignment problem ecgap a heuristic equilibrium strategy based genetic algorithm ga is designed for solving the proposed egap finally to verify the computational efficiency of the designed ga some numerical experiments are performed on some known benchmarks the test results show that the designed ga is very valid for solving egap

lagrangian relaxation guided problem space search heuristics for generalized assignment problems we develop and test a heuristic based on lagrangian relaxation and problem space search to solve the generalized assignment problem gap the heuristic combines the iterative search capability of subgradient optimization used to solve the lagrangian relaxation of the gap formulation and the perturbation scheme of problem space search to obtain high quality solutions to the gap we test the heuristic using different upper bound generation routines developed within the overall mechanism using the existing problem data sets of various levels of difficulty and sizes including the challenging largest instances we observe that the heuristic with a specific version of the upper bound routine works well on most of the benchmark instances known and provides high quality solutions quickly an advantage of the approach is its generic nature simplicity and implementation flexibility

on the exponential decay of the n point correlation functions and the analyticity of the pressure the goal of this paper is to provide estimates leading to a direct proof of the exponential decay of the n point correlation functions for certain unbounded models of kac type the methods are based on estimating higher order derivatives of the solution of the witten laplacian equation on 1 forms associated with the hamiltonian of the system we also provide a formula for the taylor coefficients of the pressure that is suitable for a direct proof of the analyticity

on the structure of generalized rough sets in this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that 1 any reflexive binary relation determines a topology 2 if q is a reflexive and symmetric relation on a set x then o email protected x q a a is a topology such that a is open if and only if it is closed 3 conversely for every topological space x o satisfying the condition that a is open if and only if it is closed there exists a reflexive and symmetric relation r such that o email protected x r a a 4 let q be an equivalence relation on x for any pseudo w closed subset a of x q a is an w closed set if and only if w x x x email protected a for any email protected x moreover we consider properties of generalized rough sets

rough ideals in semigroups abstract we shall introduce the notion of a rough left right two sided bi ideal in a semigroup which is an extended notion of a left right two sided bi ideal in a semigroup and give some properties of such ideals

on the structure of rough prime primary ideals and rough fuzzy prime primary ideals in commutative rings this paper is a continuation of ideas presented by davvaz roughness in rings inform sci 164 2004 147 163 roughness based on fuzzy ideals inform sci 176 2006 2417 2437 we introduce the notions of rough prime primary ideals and rough fuzzy prime primary ideals in a ring and give some properties of such ideals also we discuss the relations between the upper and lower rough prime primary ideals and the upper and lower approximations of their homomorphism images

an effective vns for the capacitated p median problem in the capacitated p median problem cpmp a set of n customers is to be partitioned into p disjoint clusters such that the total dissimilarity within each cluster is minimized subject to constraints on maximum cluster capacity dissimilarity of a cluster is the sum of the dissimilarities between each customer who belongs to the cluster and the median associated with the cluster an effective variable neighbourhood search heuristic for this problem is proposed the heuristic is characterized by the use of easily computed lower bounds to assess whether undertaking computationally expensive calculation of the worth of moves within the neighbourhood search is necessary the small proportion of moves that need to be assessed fully are then evaluated by an exact solution of a relatively small subproblem computational results on five standard sets of benchmark problem instances show that the heuristic finds all the best known solutions for one instance the previously best known solution is improved if only marginally

principles of scatter search scatter search is an evolutionary method that has been successfully applied to hard optimization problems the fundamental concepts and principles of the method were first proposed in the 1970s based on formulations dating back to the 1960s for combining decision rules and problem constraints in contrast to other evolutionary methods like genetic algorithms scatter search is founded on the premise that systematic designs and methods for creating new solutions afford significant benefits beyond those derived from recourse to randomization it uses strategies for search diversification and intensification that have proved effective in a variety of optimization problems this paper provides the main principles and ideas of scatter search and its generalized form path relinking we first describe a basic design to give the reader the tools to create relatively simple implementations more advanced designs derive from the fact that scatter search and path relinking are also intimately related to the tabu search ts metaheuristic and gain additional advantage by making use of ts adaptive memory and associated memory exploiting mechanisms capable of being tailored to particular contexts these and other advanced processes described in the paper facilitate the creation of sophisticated implementations for hard problems that often arise in practical settings due to their flexibility and proven effectiveness scatter search and path relinking can be successfully adapted to tackle optimization problems spanning a wide range of applications and a diverse collection of structures as shown in the papers of this volume

a set partitioning heuristic for the generalized assignment problem this paper discusses a heuristic for the generalized assignment problem gap the objective of gap is to minimize the costs of assigning j jobs to m capacity constrained machines such that each job is assigned to exactly one machine the problem is known to be np hard and it is hard from a computational point of view as well the heuristic proposed here is based on column generation techniques and yields both upper and lower bounds on a set of relatively hard test problems the heuristic is able to find solutions that are on average within 0 13 from optimality

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

a short note on algebraic t rough sets our aim in this paper is to introduce the concept of a set valued homomorphism for groups which is a generalization of ordinary homomorphism then using the definitions of lower inverse and upper inverse we introduce the definition of uniform set valued homomorphism and prove that every set valued homomorphism is uniform finally the concept of probabilistic t rough set is defined

rough set theory for the interval valued fuzzy information systems the notion of a rough set was originally proposed by pawlak z pawlak rough sets international journal of computer and information sciences 11 5 1982 341 356 later on dubois and prade d dubois h prade rough fuzzy sets and fuzzy rough sets international journal of general system 17 2 3 1990 191 209 introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets this paper deals with an interval valued fuzzy information system by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory and discusses the basic rough set theory for the interval valued fuzzy information systems in this paper we firstly define the rough approximation of an interval valued fuzzy set on the universe u in the classical pawlak approximation space and the generalized approximation space respectively i e the space on which the interval valued rough fuzzy set model is built secondly several interesting properties of the approximation operators are examined and the interrelationships of the interval valued rough fuzzy set models in the classical pawlak approximation space and the generalized approximation space are investigated thirdly we discuss the attribute reduction of the interval valued fuzzy information systems finally the methods of the knowledge discovery for the interval valued fuzzy information systems are presented with an example

roughness in rings in this paper basic notions of the rough set theory will be given in fact the paper concerns a relationship between rough sets and ring theory we shall introduce the notion of rough subring resp ideal with respect to an ideal of a ring which is an extended notion of a subring resp ideal in a ring and we shall give some properties of the lower and the upper approximations in a ring

rough approximation operators on r0 algebras nilpotent minimum algebras with an application in formal logic l an abstract axiomatization to pawlak rough set theory in the context of r0 algebras equivalently nm algebras has been proposed in the present paper more precisely by employing the conjunction operator and the disjunction operator in r0 algebras the notions of rough upper approximation operator u and rough lower approximation operator l on r0 algebras are proposed respectively owing to the logical properties of and any r0 algebra equipped with l and u forms an abstract approximation space in the sense of g cattaneo a duality relationship between the set of lower crisp elements and the set of upper crisp elements is established and some important properties are examined moreover its connection with tarski closure interior approximation space and halmos closure interior approximation is studied such a pair of rough approximations on r0 algebras can naturally induce a pair of rough upper lower truth degrees for formulae in l some uncertainty measures such as roughness degree and accuracy degree are subsequently presented and two kinds of approximate reasoning methods merging rough approximation and fuzzy logic are eventually established

some properties of generalized rough sets some algebraic and topological properties of generalized rough sets associated with generalized lower and upper approximations are studied generalized definable sets give rise to two topological spaces degree of accuracy dag for generalized rough sets induces two types of equivalence relations one is defined on x whereas the other is defined on p y both of these relations strictly manage an order in classes of x and p y it has been shown that dag induces two types of fuzzy subsets one of the set x and the other of p y finally some properties of these fuzzy sets have been discussed

a path relinking approach with ejection chains for the generalized assignment problem the generalized assignment problem is a classical combinatorial optimization problem known to be np hard it can model a variety of real world applications in location allocation machine assignment and supply chains the problem has been studied since the late 1960s and computer codes for practical applications emerged in the early 1970s we propose a new algorithm for this problem that proves to be more effective than previously existing methods the algorithm features a path relinking approach which is a mechanism for generating new solutions by combining two or more reference solutions it also features an ejection chain approach which is embedded in a neighborhood construction to create more complex and powerful moves computational comparisons on benchmark instances show that the method is not only effective in general but is especially effective for types d and e instances which are known to be very difficult

roughness in mv algebras in this paper by considering the notion of an mv algebra we consider a relationship between rough sets and mv algebra theory we introduce the notion of rough ideal with respect to an ideal of an mv algebra which is an extended notion of ideal in an mv algebra and we give some properties of the lower and the upper approximations in an mv algebra

roughness based on fuzzy ideals the theory of rough set proposed by pawlak and the theory of fuzzy set proposed by zadeh are complementary generalizations of classical set theory many sets are naturally endowed with two binary operations addition and multiplication one concept which does this is a ring this paper concerns a relationship between rough sets fuzzy sets and ring theory it is a continuation of ideas presented by kuroki and wang n kuroki p p wang the lower and upper approximations in a fuzzy group inform sci 90 1996 203 220 we consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal in fact we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring some characterizations of the above approximations are made and some examples are presented

rough sets induced by ideals in lattices this paper is to study the rough sets within the context of lattices we study the special properties of the rough sets which can be constructed by means of the congruences determined by ideals of lattice also the properties of the generalized rough sets with respect to ideals of lattice are investigated finally we give an example of their application in formal concept analysis

roughness and fuzziness in quantales abstract rough set theory and fuzzy set theory are two useful mathematical tools for dealing with uncertainty and granularity in information systems motivated by the studies of roughness and fuzziness in algebraic systems and partially ordered sets such as semigroups rings and lattices in this paper we introduce first the notions of fuzzy prime semi prime primary ideals of quantales and investigate their properties several characterizations of such ideals are presented then we introduce the concepts of weak fuzzy prime ideals and strong fuzzy prime ideals of quantales and establish relationships between these ideals and fuzzy prime ideals of quantales by applying rough set theory to fuzzy ideals of quantales we furthermore define rough fuzzy prime semi prime primary ideals of quantales generalizing yang and xu s work on quantales to the fuzzy environment finally relationships between the upper resp lower rough fuzzy ideals of quantales and the upper resp lower approximations of their homomorphic images are also discussed

roughness in quantales rough set theory is a useful mathematical tool for dealing with the uncertainty and granularity in information systems in this paper rough approximations are introduced into quantales a kind of partially ordered algebraic structure with an associative binary multiplication the notions of upper lower rough prime semi prime primary ideals of quantales are proposed and verified to be the extended notions of usual prime semi prime primary ideals of quantales respectively global order properties of all lower upper rough ideals of a quantale are also investigated the rough radical of an upper rough ideal in a quantale is defined and the prime radical theorem of rings is generalized to upper rough ideals of quantales some results about homomorphic images of rough ideals and rough prime ideals of semigroups and rings are also generalized and improved in the field of quantales

roughness in modules the theory of rough sets was proposed by pawlak in 1982 the relations between rough sets and algebraic systems have been already considered by many mathematicians important algebraic structures are groups rings and modules rough groups and rough rings have been investigated by biswas and nanda kuroki and wang and davvaz in this paper we consider an r module as a universal set and we introduce the notion of rough submodule with respect to a submodule of an r module which is an extended notion of a submodule in an r module we also give some properties of the lower and the upper approximations in an r module

tabu search for the multilevel generalized assignment problem abstract the multilevel generalized assignment problem mgap differs from the classical gap in that agents can perform tasks at more than one efficiency level important manufacturing problems such as lot sizing can be formulated as mgaps however the large number of variables in the related 0 1 integer program makes the use of commercial optimization packages impractical in this paper we present a heuristic approach to the solution of the mgap which consists of a novel application of tabu search ts our ts method employs neighborhoods defined by ejection chains that produce moves of greater power without significantly increasing the computational effort

around a stationary phase theorem in large dimension in this article we want to present a stationary phase theorem appearing in a recent article by j sjostrand ann inst h poincare phys theor in press and explain the role it can play in the context of statistical mechanics this theorem is also technically connected to the semi classical determination of eigenvalues for a schrodinger operator in large dimension and also in the same spirit with small or large temperature expansions

generalized rough sets based on reflexive and transitive relations in this paper we investigate the relationship between generalized rough sets induced by reflexive and transitive relations and the topologies on the universe which is not restricted to be finite it is proved that there exists a one to one correspondence between the set of all reflexive and transitive relations and the set of all topologies which satisfy a certain kind of compactness condition

bzmv dm algebras and stonian mv algebras applications to fuzzy sets and rough approximations the natural algebraic structure of fuzzy sets suggests the introduction of an abstract algebraic structure called de morgan bzmv algebra bzmv dm algebra we study this structure and sketch its main properties a bzmv dm algebra is a system endowed with a commutative and associative binary operator and two unusual orthocomplementations a kleene orthocomplementation and a brouwerian one as expected every bzmv dm algebra is both an mv algebra and a distributive de morgan bz lattice the set of all closed elements which coincides with the set of all idempotent elements turns out to be a boolean algebra the boolean algebra of sharp or crisp elements by means of and t wo modal like unary operators for necessity and for possibility can be introduced in such a way that a resp a can be regarded as the sharp approximation from the bottom resp top of a this gives rise to the rough approximation a a ofa finally we prove that bzmv dm algebras which are equationally characterized are the same as the stonian mv algebras and a rst representation theorem is proved c 1999 elsevier science b v all rights reserved

a hybrid tabu search branch bound approach to solving the generalized assignment problem a new approach for solving the generalized assignment problem gap is proposed that combines the exact branch bound approach with the heuristic strategy of tabu search ts to produce a hybrid algorithm for solving gap the algorithm described uses commercial software to solve sub problems generated by the ts guiding strategy the ts approach makes use of the concept of referent domain optimisation and introduces novel add drop strategies in addition the linear programming relaxation of gap that forms part of the branch bound approach is itself helpful in suggesting which variables might take binary values computational results on benchmark test instances are presented and compared with results obtained by the standard branch bound approach and also several other heuristic approaches from the literature the results show the new algorithm performs competitively against the alternatives and is able to find some new best solutions for several benchmark instances

on rough set and fuzzy sublattice let l be a lattice with the least element 0 and the greatest element 1 and let be a full congruence relation on l in this paper the notion of upper and lower approximations of a fuzzy subset of l is introduced and some important properties will be studied

a tabu search heuristic for the generalized assignment problem abstract this paper considers the generalized assignment problem gap it is a well known np hard combinatorial optimization problem that is interesting in itself and also appears as a subproblem in other problems of practical importance a tabu search heuristic for the gap is proposed the algorithm uses recent and medium term memory to dynamically adjust the weight of the penalty incurred for violating feasibility the most distinctive features of the proposed algorithm are its simplicity and its flexibility these two characteristics result in an algorithm that compared to other well known heuristic procedures provides good quality solutions in competitive computational times computational experiments have been performed in order to evaluate the behavior of the proposed algorithm

the lower and upper approximations in a quotient hypermodule with respect to fuzzy sets this paper provides a continuation of ideas presented by davvaz and mahdavipour b davvaz m mahdavipour roughness in modules inform sci 176 2006 3658 3674 the notion of hypermodule is a generalization of the notion of module in this paper we consider the quotient hypermodule m a and interpret the lower and upper approximations as subsets of the quotient hypermodule m a then we introduce the concept of quotient rough sub hypermodule also using the concept of fuzzy sets we introduce and discuss the concept of fuzzy rough hypermodules and then we obtain the relation between fuzzy rough sub hypermodules and level rough sets this relation is characterized as a necessary and sufficient condition

witten laplacian method for the decay of correlations the aim of this paper is to apply direct methods to the study of integrals that appear naturally in statistical mechanics and euclidean field theory we provide weighted estimates leading to the exponential decay of the two point correlation functions for certain classical convex unbounded models the methods involve the study of the solutions of the witten laplacian equations associated with the hamiltonian of the system

correlation at low temperature i exponential decay abstract the present paper generalizes the analysis in ann h poincare 1 2000 59 math j ams 8 1997 123 of the correlations for a lattice system of real valued spins at low temperature the gibbs measure is assumed to be generated by a fairly general hamiltonian function with pair interaction the novelty as compared to 2 20 is that the single site self energies of the spins are not required to have only a single local minimum and no other extrema our derivation of exponential decay of correlations goes through the spectral analysis of a deformed laplacian closely related to the witten laplacian studied in 2 20 we prove that this laplacian has a spectral gap above zero and argue that this implies exponential decay of the correlations

generalized rough sets based on relations rough set theory has been proposed by pawlak as a tool for dealing with the vagueness and granularity in information systems the core concepts of classical rough sets are lower and upper approximations based on equivalence relations this paper studies arbitrary binary relation based generalized rough sets in this setting a binary relation can generate a lower approximation operation and an upper approximation operation but some of common properties of classical lower and upper approximation operations are no longer satisfied we investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations

semi classical analysis for the transfer operator formal wkb constructions in large dimension this paper is devoted to problems coming from statistical mechanics the transfer matrix or transfer operator approach consists in reducing the analysis of asymptotic properties of statistical systems to the analysis of the spectral properties of their transfer operator sometimes the new problem appears to have a semi classical nature although the problem is similar to the semiclassical study of kac s operator presented in our paper with m brunaud bruhe which was devoted to the study of for h small new features appear for the model our first results concern semi classical analysis of the ground state for this second operator we then analyze the two models in the large dimension situation one basic technique is sjostrand s formalism of the 0 standard functions as introduced in sj5 the one dimensional case was presented in he11

roughness bounds in rough set operations this paper presents some roughness bounds for rough set operations the results show that a bound of the set operation can be determined from their operand s roughnesses we prove also that this bound is not determined under some special operations

θ t fuzzy rough approximation operators and the tl fuzzy rough ideals on a ring in this paper we consider a ring as a universal set and study t fuzzy rough approximation operators with respect to a tl fuzzy ideal of a ring first some new properties of generalized t fuzzy rough approximation operators are obtained then a new fuzzy algebraic structure tl fuzzy rough ideal is defined and its properties investigated and finally the homomorphism of t fuzzy rough approximation operators is studied

a survey of algorithms for the generalized assignment problem abstract this paper surveys algorithms for the well known problem of finding the minimum cost assignment of jobs to agents so that each job is assigned exactly once and agents are not overloaded all approaches seem to be based on branch and bound with bound supplied through heuristics and through relaxations of the primal problem formulation from the survey one can select building blocks for the design of one s own tailor made algorithm the survey also reveals that although just about every mathematical programming technique was tried on this problem there is still a lack of a representative set of test problems on which competing enumeration algorithms can be compared as well as a shortage of effective heuristics

minimization of axiom sets on fuzzy approximation operators axiomatic characterization of approximation operators is an important aspect in the study of rough set theory in this paper we examine the independence of axioms and present the minimal axiom sets characterizing fuzzy rough approximation operators and rough fuzzy approximation operators

roughness in n ary hypergroups this paper is aimed at studying the rough sets within the context of the commutative n ary hypergroups for this approach the presentation of a detailed study of the various types of n ary subhypergroups is of the greatest importance this study was initiated in v leoreanu fotea several types of n ary subhypergroups italian journal of pure and applied mathematics in press subsequently we shall use the findings in this study to define the upper and lower approximation of a subset with respect to an invertible n ary subhypergroup of a commutative n ary hypergroup in the final part the concept of n ary rough subhypergroup is introduced

topological approaches to covering rough sets rough sets a tool for data mining deal with the vagueness and granularity in information systems this paper studies covering based rough sets from the topological view we explore the topological properties of this type of rough sets study the interdependency between the lower and the upper approximation operations and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation lastly axiomatic systems for the lower approximation operation and the upper approximation operation are constructed

rough set theory applied to lattice theory in this paper we intend to study a connection between rough sets and lattice theory we introduce the concepts of upper and lower rough ideals filters in a lattice then we offer some of their properties with regard to prime ideals filters the set of all fixed points compact elements and homomorphisms

a branch and price algorithm for the generalized assignment problem the generalized assignment problem examines the maximum profit assignment of jobs to agents such that each job is assigned to precisely one agent subject to capacity restrictions on the agents a new algorithm for the generalized assignment problem is presented that employs both column generation and branch and bound to obtain optimal integer solutions to a set partitioning formulation of the problem

technical note an improved dual based algorithm for the generalized assignment problem the generalized assignment problem gap determines the minimum cost assignment of n jobs to m agents such that each job is assigned to exactly one agent subject to an agent s capacity existing solution algorithms have not solved problems with more than 100 decision variables this paper designs an optimization algorithm for the gap that effectively solves problems with up to 500 variables compared with existing procedures this algorithm requires fewer enumeration nodes and shorter running times improved performance stems from an enhanced lagrangian dual ascent procedure that solves a lagrangian dual at each enumeration node adding a surrogate constraint to the lagrangian relaxed model and an elaborate branch and bound scheme an empirical investigation of various problem structures not considered in existing literature is also presented

information theoretic measures associated with rough set approximations in rough set theory some information theoretic measures of uncertainty and granularity have been proposed a common feature of these measures is that they are only dependent on the partitions and the cardinality of a universe which means that they are independent of the lower and upper approximations of rough sets this seems somewhat unreasonable since the basic idea of rough set theory is to describe incomplete or inexact concepts by the lower and upper approximations in light of this we develop a new pair of information theoretic entropy and co entropy functions associated to partitions and approximations in this paper such functions are used to measure the uncertainty and granularity of an approximation space after introducing the novel notions of entropy and co entropy we then examine their properties in particular we disclose the relationship of co entropies between different universes the theoretical development is accompanied by illustrative numerical examples

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

a systematic study on attribute reduction with rough sets based on general binary relations attribute reduction is considered as an important preprocessing step for pattern recognition machine learning and data mining this paper provides a systematic study on attribute reduction with rough sets based on general binary relations we define a relation information system a consistent relation decision system and a relation decision system and their attribute reductions furthermore we present a judgment theorem and a discernibility matrix associated with attribute reduction in each type of system based on the discernibility matrix we can compute all the reducts finally the experimental results with uci data sets show that the proposed reduction methods are an effective technique to deal with complex data sets

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

relaxation heuristics for a generalized assignment problem abstract we propose relaxation heuristics for the problem of maximum profit assignment of n tasks to m agents n m such that each task is assigned to only one agent subject to capacity constraints on the agents using lagrangian or surrogate relaxation the heuristics perform a subgradient search obtaining feasible solutions relaxation considers a vector of multipliers for the capacity constraints the resolution of the lagrangian is then immediate for the surrogate the resulting problem is a multiple choice knapsack that is again relaxed for continuous values of the variables and solved in polynomial time relaxation multipliers are used with an improved heuristic of martello and toth or a new constructive heuristic to find good feasible solutions six heuristics are tested with problems of the literature and random generated problems best results are less than 0 5 from the optimal with reasonable computational times for an at 386 computer it seems promising even for problems with correlated coefficients

bees algorithm for generalized assignment problem bees algorithm ba is a new member of meta heuristics ba tries to model natural behavior of honey bees in food foraging honey bees use several mechanisms like waggle dance to optimally locate food sources and to search new ones this makes them a good candidate for developing new algorithms for solving optimization problems in this paper a brief review of ba is first given afterwards development of a ba for solving generalized assignment problems gap with an ejection chain neighborhood mechanism is presented gap is a np hard problem many meta heuristic algorithms were proposed for its solution so far ba is generally applied to continuous optimization in order to investigate the performance of ba on a complex integer optimization problem an attempt is made in this paper an extensive computational study is carried out and the results are compared with several algorithms from the literature

generalized lower and upper approximations in a ring in this paper the concepts of set valued homomorphism and strong set valued homomorphism of a ring are introduced and related properties are investigated the notions of generalized lower and upper approximation operators constructed by means of a set valued mapping which is a generalization of the notion of lower and upper approximation of a ring are provided we also propose the notion of generalized lower and upper approximations with respect to an ideal of a ring which is an extended notation of rough ideal introduced lately by davvaz b davvaz roughness in rings information science 164 2004 147 163 in a ring and discuss some significant properties of them

a weighted rough set based method developed for class imbalance learning in this paper we introduce weights into pawlak rough set model to balance the class distribution of a data set and develop a weighted rough set based method to deal with the class imbalance problem in order to develop the weighted rough set based method we design first a weighted attribute reduction algorithm by introducing and extending guiasu weighted entropy to measure the significance of an attribute then a weighted rule extraction algorithm by introducing a weighted heuristic strategy into lem2 algorithm and finally a weighted decision algorithm by introducing several weighted factors to evaluate extracted rules furthermore in order to estimate the performance of the developed method we compare the weighted rough set based method with several popular methods used for class imbalance learning by conducting experiments with twenty uci data sets comparative studies indicate that in terms of auc and minority class accuracy the weighted rough set based method is better than the re sampling and filtering based methods and is comparable to the decision tree and svm based methods it is therefore concluded that the weighted rough set based method is effective for class imbalance learning

on definable concepts of rough set models in various generalized rough set models definable concepts are used to approximate other concepts based on a unified abstract rough set theory this paper investigates the mathematical structure of the set of definable concepts in several generalized rough set models such as relation based models covering based models and fuzzy based models it is shown that there exist two kinds of interesting structures for the set of definable concepts which are often used in rough set models the complementary structure and the equational structure in addition relations between different rough set models are discussed

universal estimate of the gap for the kac operator in the convex case the aim of this paper is to prove that ifv is a strictly convex potential with quadratic behavior at then the quotient 2 1 between the largest eigenvalue and the second eigenvalue of the kac operator defined on l2 m by exp v x 2 exp x exp v x 2 where x is the laplacian on m satisfies the condition mu 2 mathord left vphantom mu 2 mu 1 leqslant exp cosh 1 sigma 1 mathord left vphantom leqslant exp cosh 1 sigma 1 2 right kern nulldelimiterspace 2 right kern nulldelimiterspace mu 1 leqslant exp cosh 1 sigma 1 mathord left vphantom leqslant exp cosh 1 sigma 1 2 right kern nulldelimiterspace 2 where is such that hessv x 0

a new lagrangian relaxation approach to the generalized assignment problem abstract in this paper we present a new lagrangian relaxation approach to the generalized assignment problem gap the new approach is based on a reformulation of gap into an equivalent problem which is then relaxed by traditional lagrangian relaxation techniques the reformulation is created by introducing a set of auxiliary variables and a number of coupling constraints by relaxing the coupling constraints we get subproblems where both types of constraint structures present in the gap are active the lower bound from the lagrangian dual of this approach is shown to be at least as strong as that from the best of the traditional lagrangian relaxation approaches an important feature is also the possibility of generating upper bounds quite easily the new approach has been tested on 10 problems of size m 4 n 25 i e 100 0 1 variables and the results have been compared with the traditional approaches

a tabu search heuristic for the vehicle routing problem with stochastic demands and customers this paper considers a version of the stochastic vehicle routing problem where customers are present at locations with some probabilities and have random demands a tabu search heuristic is developed for this problem comparisons with known optimal solutions on problems whose sizes vary from 6 to 46 customers indicate that the heuristic produces an optimal solution in 89 45 of cases with an average deviation of 0 38 from optimality

the variety of modular basic algebras generated by mv chains and horizontal sums of three element chain basic algebras basic algebras or lattices with section antitone involutions turned out to be appropriate structures when axiomatizing both many valued and quantum logics see e g 11 the aim of the present paper is to describe the variety of basic algebras generated by mv chains and horizontal sums of three element chain basic algebras