sequencing and routing in multiclass queueing networks part ii workload relaxations part ii continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model the following are shown a relaxation of the fluid model based on workload leads to an optimization problem of lower dimension an analogous workload relaxation is introduced for the stochastic model these relaxed control problems admit pointwise optimal solutions in many instances a translation to the original fluid model is almost optimal with vanishing relative error as the network load rho approaches one it is pointwise optimal after a short transient period provided a pointwise optimal solution exists for the relaxed control problem a translation of the optimal policy for the fluid model provides a policy for the stochastic network model that is almost optimal in heavy traffic over all solutions to the relaxed stochastic model again with vanishing relative error the regret is of order log 1 rho

on the minimal solutions of max min fuzzy relational equations in this paper the minimal solutions of max min fuzzy relational equations are investigated a sufficient and necessary condition for discriminating whether a given solution is minimal or not is shown furthermore we propose a new algorithm for computing all minimal solutions less than or equal to a given one

verified bounds for the determinant of real or complex point or interval matrices abstract we discuss several methods to compute a verified inclusion of the determinant of a real or complex point or interval matrix for point matrices large condition number 1 0 15 and large dimension n 1000 still highly accurate inclusions are computed for real interval matrices we show that any vertex may be a unique extreme point for wide radii we show that preconditioning may widen an inclusion significantly and hadamard s bound may be much better

monotone eigenspace structure in max min algebra for a given n n matrix a in a max min algebra the set of all increasing eigenvectors in notation f a is studied it is shown that f a is a union of at most 2n 1 intervals and an explicit formula for the intervals is given moreover it is shown that the endpoints of these intervals can be computed in o n2 time or in o n time if an auxiliary n n matrix c a has been previously computed the results enable a complete description of the structure of the whole eigenspace f a

x simple image eigencones of tropical matrices abstract we investigate max algebraic tropical one sided systems a x b where b is an eigenvector and x lies in an interval x a matrix a is said to have x simple image eigencone associated with an eigenvalue if any eigenvector x associated with and belonging to the interval x is the unique solution of the system a y x in x we characterize matrices with x simple image eigencone geometrically and combinatorially and for some special cases derive criteria that can be efficiently checked in practice

solving systems of linear fuzzy equations by parametric functions an improved algorithm buckley and qu proposed a method to solve systems of linear fuzzy equations basically in their method the solutions of all systems of linear crisp equations formed by the a levels are calculated we propose a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system by observing the monotonicity of the parametric functions in each variable i e each fuzzy coefficient in the system we improve the algorithm by calculating less parametric functions and less evaluations of these parametric functions we show that our algorithm is much more efficient than the method of buckley and qu

sufficient regularity conditions for complex interval matrices and approximations of eigenvalues sets abstract in this paper two approaches are described to establish verifiable sufficient regularity conditions of complex interval matrices in the first approach a complex interval matrix is mapped to a real block interval matrix and then its sufficient regularity conditions are obtained in the second approach a necessary condition for the singularity of a complex interval matrix is derived and used to get its sufficient regularity conditions as an application the above derived sufficient regularity conditions are used to investigate the location of the outer approximations of individual eigenvalue sets of complex interval matrices two algorithms are proposed and results obtained are compared with those obtained by earlier methods and monte carlo simulations the advantages of these algorithms are that they can detect gaps in between the approximations of the whole eigenvalue sets the second algorithm is very effective compared to the first algorithm from the computational time point of view several numerical examples and statistical experiments are worked out to validate and demonstrate the efficacy of our work

on solving vague systems of linear equations with pattern shaped columns abstract in j nedoma ann oper res 47 1993 483 496 a method for optimizing a linear function on the solution set of a vague system of linear equations was proposed in this paper we deal with the so called simultaneous version of this method we show that it works provided the system matrix is pattern shaped in this case moreover the iteration process is extremely simple not requiring any matrix operation after being started the application of this method to the mentioned optimization problem is straightforward provided a list of the space orthants covering the solution set is given a way of finding this list is outlined briefly

a new subclass of p matrices a new subclass of p matrices is stated for which it is easy to calculate bounds for the solution of a linear complementarity problem where the input data of the linear complementarity problem are not exactly known but can be enclosed in intervals this new class of matrices properly contains all h matrices with positive diagonal entries and is properly contained in the class of p matrices

solutions of fuzzy relation equations based on continuous t norms this study is concerned with fuzzy relation equations with continuous t norms in the form atr b where a and b are the fuzzy subsets of x and y respectively email protected xxy is a fuzzy relation and t is a continuous t norm the problem is how to determine a from r and b first an if and only if condition of being solvable is presented novel algorithms are then presented for determining minimal solutions when x and y are finite the proposed algorithms generate all minimal solutions for the equations making them efficient solving procedures

steady states in the scheduling of discrete time systems abstract steady solutions to a two sided max min linear system of equations with real coefficients are considered the problem is motivated by the demand to schedule cyclically repeated activities with deterministic processing times the maximum steady solutions are characterized as min eigenvectors of a special matrix q this condition is a necessary but not a sufficient one it has also been proven that the general problem of the recognition of solvability of this two sided system is np complete

numerical solution of fuzzy max min systems abstract for solvable systems of linear equations of the form a x b over the max min algebra b 0 1 max min we propose efficient methods for finding fuzzy determinant and fuzzy solution we formulate a condition for linear systems of equations over b to have not a unique solution by using fuzzy determinant

max algebraic attraction cones of nonnegative irreducible matrices abstract it is known that the max algebraic powers ar of a nonnegative irreducible matrix are ultimately periodic this leads to the concept of attraction cone attr a t by which we mean the solution set of a two sided system t a ar x ar t x where r is any integer after the periodicity transient t a and a is the maximum cycle geometric mean of a a question which this paper answers is how to describe attr a t by a concise system of equations without knowing t a this study requires knowledge of certain structures and symmetries of periodic max algebraic powers which are also described we also consider extremals of attraction cones in a special case and address the complexity of computing the coefficients of the system which describes attraction cone

interval systems of max separable linear equations for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations we define and characterize several types of solutions and solvability of interval systems of max separable linear equations

positively regular vague matrices abstract positive regularity is a common attribute of inaccurate square matrices which can be used in linear equation systems that provide only nonnegative solutions it is studied within the framework of vague matrices which can be considered as a generalization of interval matrices criteria of positive regularity are derived and a method of verifying them is outlined the exposition concludes with a characterization of the radius of positive regularity

reachability of eigenspaces for interval circulant matrices in max algebra abstract a nonnegative matrix a is said to be strongly robust if its max algebraic eigencone is universally reachable i e if the orbit of any initial vector ends up with a max algebraic eigenvector of a consider the case when the initial vector is restricted to an interval and a can be any matrix from a given interval of nonnegative circulant matrices the main aim of this paper is to classify and characterize the six types of interval robustness in this situation this naturally leads us also to study the max algebraic spectral theory of circulant matrices and the relation of inclusion between attraction cones of circulant matrices in max algebra

unique solvability of max min fuzzy equations and strong regularity of matrices over fuzzy algebra a matrix is said to have strongly linearly independent columns or in the case of square matrices to be strongly regular if for some vector b the system a x b has a unique solution we formulate a necessary and sufficient condition for a linear system of equations over a fuzzy algebra to have a unique solution and prove the equivalence of strong regularity and trapezoidal property moreover an algorithm for testing these properties is reviewed

linear optimization with bipolar max min constraints we consider a generalization of the linear optimization problem with fuzzy relational in equality constraints by allowing for bipolar max min constraints i e constraints in which not only the independent variables but also their negations occur a necessary condition to have a non empty feasible domain is given the feasible domain if not empty is algebraically characterized a simple procedure is described to generate all maximizers of the linear optimization problem considered and is applied to various illustrative example problems

solvability and unique solvability of max min fuzzy equations abstract the problem of solvability and the problem of unique solvability of a fuzzy relation equation in an arbitrary max min algebra are considered and corresponding necessary and sufficient conditions are presented the results allow to solve both problems by an o mnp algorithm where m n p are the dimensions of the corresponding relations in the equation the existence of the greatest and of the least solution is also considered

unattainable solutions of a fuzzy relation equation in a fuzzy relation equation the properties of the solution set have been investigated especially when the basic spaces are all finite sets however when the basic spaces are infinite a few properties are known in this paper we use the concepts of attainability and unattainability of a solution to clarify some properties of the solution set and show the condition for existence of a partially attainable solution and an unattainable one

control solvability of interval systems of max separable linear equations abstract for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations several types of solvability of interval systems are known we define control solvability weak control solvability and weak universal solvability and give necessary and sufficient condition for an interval system to be solvable according to each definition

minimal join decompositions and their applications to fuzzy relation equations over complete brouwerian lattices this paper deals with the existence of minimal solutions of fuzzy relation equations over complete brouwerian lattices from the viewpoint of join decompositions first the concept of a minimal join decomposition is introduced then the existence conditions and properties of such decomposition are discussed as their applications the conditions for the existence of minimal solutions of fuzzy relation equations over complete brouwerian lattices and complete lattices are given respectively

ae solutions to interval linear systems over max plus algebra abstract this paper introduces a concept of ae solutions to interval max plus linear systems a rather general concept which includes many known concepts of solutions to interval systems weak solutions strong solutions tolerance solutions and control solutions as its special cases we state full characterizations of ae solutions for the interval max plus systems including both linear inequalities and linear equations moreover a kind of dependency characterized by double appearance of some sub matrix of the constraint matrix is discussed and we prove that which can be relaxed with no change in ae solvability

stability of interval matrices the real eigenvalue case c v hollot and a c bartlett 1987 showed that testing at most 2 to the n sup 2 certain matrices for stability is sufficient for verifying stability of an n n interval matrix with real eigenvalues it is proven that this upper bound can be reduced to 2 sup 2n 1 and a special case where testing only two matrices is needed is considered

sign solvable linear complementarity problems abstract this paper presents a connection between qualitative matrix theory and linear complementarity problems lcps an lcp is said to be sign solvable if the set of the sign patterns of the solutions is uniquely determined by the sign patterns of the given coefficients we provide a characterization for sign solvable lcps such that the coefficient matrix has nonzero diagonals which can be tested in polynomial time this characterization leads to an efficient combinatorial algorithm to find the sign pattern of a solution for these lcps the algorithm runs in o time where is the number of the nonzero coefficients

on the control of fork join networks networks in which the processing of jobs occurs both sequentially and in parallel are prevalent in many application domains such as computer systems healthcare manufacturing and project managem

new characterizations of tolerance control and localized solutions to interval system of linear equations abstract in this paper we propose new characterizations of tolerance control and localized solutions to interval system of linear equations and apply our proposed results to represent their solution sets in terms of level sets

x robustness of interval circulant matrices in fuzzy algebra abstract fuzzy algebra is an algebraic structure in which classical addition and multiplication are replaced by and where a b max a b a b min a b a matrix is x robust if its eigenspace is achieved starting at each vector from a given interval vector x the possible and universal x robustness of interval matrices over fuzzy algebra are defined polynomial algorithms for checking the possible and universal x robustness of interval circulant matrices are described

an algorithm for solving fuzzy relation equations with max t composition operator this paper studies the problem of solving a max t composite finite fuzzy relation equation where t is a special class of pseudo t norms if the equation is solvable then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value under these conditions some minimal solutions of the system can be obtained easily some procedures are also proposed in order to simplify the original system the simplified system is then decomposed if possible into several subsystems with smaller dimensions which are very easy to solve furthermore a method is presented to solve each subsystem by combining the method and those procedures an efficient algorithm is proposed to obtain the set of feasible solutions of the original system two examples are also given to illustrate the algorithm

strong regularity of matrices in a discrete bounded bottleneck algebra abstract the results concerning strong regularity of matrices over bottleneck algebras are reviewed we extend the known conditions to the discrete bounded case and modify the known algorithms for testing strong regularity

the general trapezoidal algorithm for strongly regular max min matrices abstract the problem of the strong regularity for square matrices over a general max min algebra is considered an o n 2 log n algorithm for recognition of the strong regularity of a given n n matrix is proposed the algorithm works without any restrictions on the underlying max min algebra concerning the density or the boundedness

strong regularity of matrices in general max min algebra abstract the problem of the strong regularity of a square matrix in a general max min algebra is considered and a necessary and sufficient condition using the trapezoidal property is described the results are valid without any restrictions on the underlying max min algebra concerning the density or the boundedness previous results on this topic are special cases of the theorems presented in this paper

on matrix powers in max algebra abstract let a a ij r n n n 1 n and da be the digraph n i j a ij the matrix a is called irreducible if da is strongly connected and strongly irreducible if every max algebraic power of a is irreducible a is called robust if for every x with at least one finite component a k x is an eigenvector of a for some natural number k we study the eigenvalue eigenvector problem for powers of irreducible matrices this enables us to characterise robust irreducible matrices in particular robust strongly irreducible matrices are described in terms of eigenspaces of matrix powers

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

contributions to max min convex geometry i segments abstract we give some contributions to the theory of max min convex geometry that is convex geometry in the semimodule r max min n over the max min semiring r max min r we introduce elementary segments that generalize from n 2 the horizontal vertical or oblique segments contained in the main bisector of r max min 2 we show that every segment in r max min n is a concatenation of a finite number of elementary subsegments at most 2 n 1 respectively at most 2 n 2 in the case of comparable respectively incomparable endpoints x y in this first part we study max min segments and in the subsequent second part submitted we study max min semispaces and some of their relations to max min convex sets

strong regularity of matrices in a discrete bottleneck algebra abstract systems of linear equations of the form a x b over a discrete bottleneck algebra b where max and min are studied a square matrix a is said to be strongly regular if for some vector b the system a x b is uniquely solvable a necessary and sufficient condition for strong regularity is proved together with an o n 2 log n method for testing this property

new techniques for the analysis of linear interval equations abstract the basic properties of interval matrix multiplication and several well known solution algorithms for linear interval equations are abstracted by the concept of a sublinear map the new concept coupled with a systematic use of ostrowski s comparison operator in a form generalized to interval matrices is used to derive quantitative information about the result of interval gauss elimination andthe limit of various iterative schemes for the solution of linear interval equations moreover optimality results are prove for 1 the use of the midpoint inverse as a preconditioning matrix and 2 gauss seidel iteration with componentwise intersection this extends and improves results by scheu krawczyk and alefeld and herzberger

conservatism of the circle criterion solution of a problem posed by a megretski in the collection of open problems in mathematical systems and control theory blondel et al 1999 megretski poses a problem related to the degree of conservativism of the well known circle criterion follows we solve this problem

linear matrix period in max plus algebra abstract linear periodicity of matrices in max plus algebra is studied it is proved that the linear period of a matrix can be computed in o n 3 time if the digraph of the matrix is strongly connected the general problem of deciding whether a given matrix is almost linear periodic is shown to be np complete

the robustness of interval matrices in max plus algebra abstract robust interval matrices in max plus algebra are studied and equivalent conditions for interval matrices to be robust are presented the necessary and sufficient conditions for robustness of interval matrices are introduced in addition efficient algorithms for verifying the robustness of interval circulant and interval zero one matrices are described

on general conditions for nestedness of the solution set of fuzzy interval linear systems abstract this study addresses the problem of providing general conditions for nestedness of the solution set of fuzzy interval linear systems of equalities and inequalities relying on the fact that they are a direct extension of the interval linear systems some illustrative examples demonstrate an applicable viewpoint of the results which connects our obtained results with the optimization theory

nonsingularity under data rounding abstract for a square floating point matrix a we give a test for the existence of a singular real matrix which rounds to a

on the spectrum of convex sets of matrices let k be an arbitrary compact convex set of square matrices this paper presents necessary and sufficient conditions for the spectrum of k to have no eigenvalues in a prespecified closed convex subset of the complex plane the obtained result implies different criteria for analysis of the spectral set of k in particular we have formulated criteria for nonsingularity inertia and hurwitz stability of k which can be used in the robustness analysis of linear control systems with uncertain parameters

on pre solution matrices of fuzzy relation equations over complete brouwerian lattices abstract each solution of a system of fuzzy relation equations over a complete brouwerian lattice can be decomposed into a matrix which we call pre solution matrix every pre solution matrix can induce at least one solution in this paper we discuss connections between solutions and pre solution matrices of a system of fuzzy relation equations over a complete brouwerian lattice then we explore properties of pre solution matrices and characterize lower pre solution matrices moreover we discuss constructions of pre solution matrices

strong linear independence in bottleneck algebra abstract let b be a dense linearly ordered set without maximum and minimum and max min we say that a matrix a has strongly linearly independent sli columns if for some b the system a x b is uniquely solvable an n n matrix a a ij is said a to be strongly regular if it has sli columns b to have a strong permanent if the equality per a i 1 n a i i holds for unique p n per a is p n n i 1 a i i and p n is the set all permutations of the set 1 2 n we prove i that an m n matrix has sli columns if and only if it contains an n n submatrix which is strongly regular we derive an o mn log n algorithm for checking this property ii that every matrix with strong permanent is strongly regular and iii that a solution to the bottleneck assignment problem for strongly regular n n matrices can be found using o n 2 log n operations

some properties of minimal solutions for a fuzzy relation equation abstract some properties of the solution set and minimal solutions of a fuzzy relation equation are considered in this paper we show the necessary and sufficient condition for existence of a minimal solution of a finite fuzzy relation equation defined on infinite index sets

exact bounds for the static response set of structures with uncertain but bounded parameters abstract the numerical estimation of the static displacement bounds of structures with uncertain but bounded parameters is considered in this paper by representing each uncertain but bounded parameter as an interval number or vector a static response analysis problem for the structure can be expressed in the form of a system of linear interval equations in which the coefficient matrix and the right hand side term are respectively the interval matrix and the interval vector in this study we present two new simple mathematical proofs of the vertex solution theorem using cramer s rule for solving linear interval equations different from the other proof methods to find the upper and lower bounds on the set of solutions the first takes advantage of optimization theory while the second is based on interval extension by means of a typical example considered first by hansen it can be seen that the result calculated by the vertex solution theorem is the same as one predicted by the oettli prager criterion examples of a three stepped beam and a 10 bar truss are presented to illustrate the computational aspects of the vertex solution theorem in comparison with the interval perturbation method

the robustness of interval fuzzy matrices abstract robust interval matrices over max min algebra fuzzy matrices are studied and equivalent conditions for interval fuzzy matrices to be robust are presented polynomial algorithms for checking the necessary and sufficient conditions for robustness of interval fuzzy matrices are introduced in addition more efficient algorithms for verifying the robustness of interval circulant matrices are described

linear independence in bottleneck algebras abstract let r be a linearly ordered set a bottleneck algebra is the set r together with two operations a b max a b and a b min a b in this paper we suggest various definitions of linear independence of sets of n tuples over r study their characteristics and interrelations

solution of the fully fuzzy linear systems using the decomposition procedure abstract this paper tries to extend the adomian decomposition method for solving fully fuzzy linear systems shown as ffls for finding a positive fuzzy vector x that satisfies a x b where a and b are respectively a fuzzy matrix and a fuzzy vector we employ dubois and prade s approximate arithmetic operators on lr fuzzy numbers we also transform the ffls and use the adomian decomposition method for solving an ffls then we compare this method with the jacobi iterative technique additionally we give some numerical examples

on the possibilistic approach to linear regression models involving uncertain indeterminate or interval data abstract we consider linear regression models where both input data the observations of independent variables and output data the observations of the dependent variable are affected by loss of information caused by uncertainty indeterminacy rounding or censoring instead of real valued crisp data only intervals are available we study a possibilistic generalization of the least squares estimator so called ols set for the interval model investigation of the ols set allows us to quantify whether the replacement of real valued crisp data by interval values can have a significant impact on our knowledge of the value of the ols estimator we show that in the general case very elementary questions about properties of the ols set are computationally intractable assuming p np we also focus on restricted versions of the general interval linear regression model to the crisp input case taking the advantage of the fact that in the crisp input interval output model the ols set is a zonotope we design both exact and approximate methods for its description we also discuss special cases of the regression model e g a model with repeated observations

solving systems of linear fuzzy equations by parametric functions buckley and qu proposed a method to solve systems of linear fuzzy equations basically in their method the solutions of all systems of linear crisp equations formed by the alpha levels are calculated we propose in this paper a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system we show that our algorithm is much more efficient than the method of buckley and qu

iteration algorithm for solving ax b in max min algebra abstract for solving the linear system a x b we introduce some iteration algorithms where a aij and b bi a given matrix and a column vector with elements from a brouwerian lattice and is a max min composite operation

a linear complementarity problem with a p matrix in this paper we present an application of a linear complementarity problem where m is a p matrix but in general is neither an h matrix nor a positive definite matrix this application occurs originally in j rohn linear algebra appl 126 1989 pp 39 78 which is less known to the lcp community its focus is in computing the exact interval enclosures of the components of the solution set of an interval linear system we extend the idea such that the calculations can be done by a computer with rigorous error control

a sharp version of bauer fike s theorem in this paper we present a sharp version of bauer fike s theorem we replace the matrix norm with its spectral radius or sign complex spectral radius for diagonalizable matrices 1 norm and norm for non diagonalizable matrices we also give the applications to the pole placement problem and the singular system

conditions for the existence of the least solution and minimal solutions to fuzzy relation equations over complete brouwerian lattices this paper discusses properties and existence conditions of minimal solutions to sup inf fuzzy relation equations over complete brouwerian lattices the relationship between minimal solutions to a system of fuzzy relation equations and minimal solutions to a fuzzy relation equation in the system is illustrated by two examples necessary and sufficient conditions for the existence of the least solution and minimal solutions to a system of fuzzy relation equations are given

on the dimension of max min convex sets we introduce a notion of dimension of max min convex sets following the approach of tropical convexity we introduce a max min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope we describe the relation between this rank and the notion of strong regularity in max min algebra which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property

solving piecewise linear systems in abs normal form abstract with the ultimate goal of iteratively solving piecewise smooth ps systems we consider the solution of piecewise linear pl equations as shown in 7 pl models can be derived in the fashion of automatic or algorithmic differentiation as local approximations of ps functions with a second order error in the distance to a given reference point the resulting pl functions are obtained quite naturally in what we call the abs normal form a variant of the state representation proposed by bokhoven in his dissertation 27 apart from the tradition of pl modelling by electrical engineers which dates back to the master thesis of thomas stern 26 in 1956 we take into account more recent results on linear complementarity problems and semi smooth equations originating in the optimization community 3 25 5 we analyze simultaneously the original pl problem opl in abs normal form and a corresponding complementary system cpl which is closely related to the absolute value equation ave studied by mangasarian and meyer 14 and a corresponding linear complementarity problem lcp we show that the cpl like kkt conditions and other simply switched systems cannot be open without being injective hence some of the intriguing pl structure described by scholtes in 25 is lost in the transformation from opl to cpl to both problems one may apply newton variants with appropriate generalized jacobians directly computable from the abs normal representation alternatively the cpl can be solved by bokhoven s modulus method and related fixed point iterations we compile the properties of the various schemes and highlight the connection to the properties of the schur complement matrix in particular its partial contractivity as analyzed by rohn and rump 23 numerical experiments and suitable combinations of the fixed point solvers and stabilized generalized newton variants remain to be realized

fuzzy function as an approximate solution to a system of fuzzy relation equations this paper is mostly focused on the problem of approximate solvability of a system of fuzzy relation equations however we put a new light on this problem connecting it with the interpolation of a fuzzy function we introduce a notion of fuzzy function and its representation by fuzzy relation we demonstrate how problems of interpolation and approximation of fuzzy functions are connected with solvability of systems of fuzzy relation equations first we explain the general framework and then we prove some results related to the problem of the best approximation in particular we have shown that fuzzy relations introduced by sanchez and mamdani are the best approximations in certain approximation spaces

system of fuzzy relation equations as a continuous model of if then rules we claim that a fuzzy relation which models a system of fuzzy if then rules has the property of continuity only if the underlying system of fuzzy relation equations is solvable in view of this connection it is important to consider the problem of solvability of a system of fuzzy relation equations with respect to an unknown fuzzy relation this problem is analyzed in the present paper we propose several new necessary and sufficient criteria for the so called mamdani relation to be a solution to the system we prove that in the general case these same criteria are sufficient but not always necessary for the solvability of the system a new easy to check criterion is proposed for the solvability of the system with special fuzzy parameters

universal grey system theory for analysis of uncertain structural systems the uncertainty present in many structural systems is commonly modeled using probabilistic fuzzy or interval approaches depending on the nature of uncertainty present the probabilistic approach

gaussian limits for a fork join network with nonexchangeable synchronization in heavy traffic we study a fork join network of stations with multiple servers and nonexchangeable synchronization in heavy traffic under the first come first served fcfs discipline tasks are only synchronized if all the tasks associated with the same job are completed service times of parallel tasks of each job can be correlated we jointly consider the number of tasks in each waiting buffer for synchronization with the number of tasks in each parallel service station and the number of synchronized jobs we develop a new approach to show a functional central limit theorem for these processes in the quality driven regime under general assumptions on arrival and service processes specifically we represent these processes as functionals of a sequential empirical process driven by the sequence of service vectors for each job s parallel tasks all of the limiting processes are functionals of two independent processes i e the limiting arrival process and a generalized kiefer process driven by the service vector of

ae and ea robustness of interval circulant matrices in max min algebra abstract max min algebra is defined as a linearly ordered set with two binary operations classical addition and multiplication are replaced by maximum and minimum respectively a square matrix is robust if its eigenspace is achieved starting at arbitrary vector and it is x robust if its eigenspace is achieved starting at each vector from a given interval vector x the ea and ae robustness and x robustness of interval circulant matrices over max min algebra are defined polynomial algorithms for checking these types of robustness and x robustness are given

interval eigenproblem in max min algebra the eigenvectors of square matrices in max min algebra correspond to steady states in discrete events system in various application areas such as design of switching circuits medical diagnosis models of organizations and information systems imprecise input data lead to considering interval version of the eigenproblem in which interval eigenvectors of interval matrices in max min algebra are investigated six possible types of an interval eigenvector of an interval matrix are introduced using various combination of quantifiers in the definition the previously known characterizations of the interval eigenvectors were restricted to the increasing eigenvectors see 11 in this chapter the results are extended to the non decreasing eigenvectors and further to all possible interval eigenvectors of a given max min matrix classification types of general interval eigenvectors are studied and characterization of all possible six types is presented

recognizing weakly stable matrices a max plus matrix a is called weakly stable if the sequence orbit x a otimes x a 2 otimes x ldots does not reach an eigenvector of a for any x unless x is an eigenvector this is in contrast to previously studied strongly stable robust matrices for which the orbit reaches an eigenvector with any nontrivial starting vector max plus matrices are used to describe multiprocessor interactive systems for which reachability of a steady regime is equivalent to reachability of an eigenvector by a matrix orbit we prove that an irreducible matrix is weakly stable if and only if its critical graph is a hamiltonian cycle in the associated graph we extend this condition to reducible matrices these criteria can be checked in polynomial time

theorems of perron frobenius type for matrices without sign restrictions abstract the paper attempts to solve a problem which was called a challenge for the future in linear algebra appl we define and investigate a new quantity for real matrices the sign real spectral radius and derive various characterizations bounds and properties of it in certain aspects our quantity shows similar behavior to the perron root of a nonnegative matrix it is shown that our quantity also has intimate connections to the componentwise distance to the nearest singular matrix relations to the perron root of the entrywise absolute value of the matrix and to the number are given as well

approximate linear algebra is intractable abstract it is shown that the problem of computing the optimal solutions of several versions of imprecise linear systems of equations is np hard an imprecise linear system is a linear system ax b where a a 0 p a b b 0 qv b v with unknown coefficients p qv constrained by one of the five relations p q p 2 q p q 2 p q 2 where p q is a vector formed by all values p and qv given such a system we would like to find its optimal solution i e the largest possible x j and the smallest possible x j values of xj our main result is that this problem is np hard in any of the five versions this holds even when the a b v 0 v 0 are restricted to have only a single nonzero entry each in disjoint positions and even if instead of the exact values x j x j we want to compute their approximations for a given accuracy 0 so unless np p algorithms that find the optimal solution of imprecise linear systems require in the worst case exponential time

analysis of uncertain structural systems using interval analysis the imprecision or uncertainty present in many engineering analysis design problems can be modeled using probabilistic fuzzy or interval methods this work considers the modeling of uncertain structural systems using interval analysis by representing each uncertain input parameter as an interval number a static structural analysis problem can be expressed in the form of a system of linear interval equations in addition to the direct and gaussian elimination based solution approaches a combinatorial approach based on an exhaustive combination of the extreme values of the interval numbers and an inequality hased method are presented for finding the solution of interval equations the range or interval of the solution vector response parameters is found to increase with increasing size of the problem in all of the methods an interval truncation approach is proposed to limit the growth of intervals of response parameters so that realistic and accurate solutions can be obtained in the presence of large amounts of uncertainty numerical examples are presented to illustrate the computational aspects of the methods and also to indicate the importance of the truncation approach in practical problems the utility of interval methods in predicting the extreme values of the response parameters of structures is discussed