fuzzy portfolio optimization under downside risk measures this paper presents two fuzzy portfolio selection models where the objective is to minimize the downside risk constrained by a given expected return we assume that the rates of returns on securities are approximated as lr fuzzy numbers of the same shape and that the expected return and risk are evaluated by interval valued means we establish the relationship between those mean interval definitions for a given fuzzy portfolio by using suitable ordering relations finally we formulate the portfolio selection problem as a linear program when the returns on the assets are of trapezoidal form

uncertainty measurement for interval valued information systems abstract interval valued information systems are generalized models of single valued information systems accuracy and roughness are employed to depict the uncertainty of a set under an attribute subset in a pawlak rough set model based on equivalence classes information theoretic measures of uncertainty for rough sets have also been proposed however there are few studies on uncertainty measurements for interval valued information systems this paper addresses the uncertainty measurement problem in interval valued information systems the concept of the similarity degree based on the possible degree is introduced consequently the similarity relation between two interval objects are constructed by a given similarity rate based on the similarity relation similarity classes are defined under this definition accuracy and roughness are given for interval valued information systems which are generalizations of the concepts accuracy and roughness for the equivalence relation based rough set model moreover an alternative uncertainty measure called the rough degree is proposed theoretical studies and numerical experiments show that the proposed measures are effective and suitable for interval valued information systems

convergence rates of the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2002 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc recently we developed a new explicit method in mao 2015 called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and established the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 however due to the page limit there we did not study the convergence rates for the method which is the aim of this paper we will under some additional conditions discuss the rates of l q convergence of the truncated em method for 2 q p and show that the order of l q convergence can be arbitrarily close to q 2

operators and comparisons of hesitant fuzzy linguistic term sets the theory of hesitant fuzzy linguistic term sets hfltss is very useful in objectively dealing with situations in which people are hesitant in providing linguistic assessments the purpose of this paper is to develop comparison methods and study the aggregation theory for hfltss we first define operations on hfltss and give possibility degree formulas for comparing hfltss we then define two aggregation operators for hfltss a hesitant fuzzy lwa operator and a hesitant fuzzy lowa operator in actual application we use these operators and the comparison methods to deal with multicriteria decision making problems with different situations in which importance weights of criteria or experts are known or unknown

min transitivity of fuzzy leftness relationship and its application to decision making abstract we show that the fuzzy leftness relationship left a b between two intervals a and b on a line which we had recently defined for modeling spatial relationships satisfies the min transitivity property the previous definitions of leftness given by krishnapuram et al and miyajima et al do not have this property the min transitivity property plays an important role in fuzzy decision making based on orlovsky s result we give several general schemes for constructing concrete min transitive relations based on our left a b for use in fuzzy decision making

existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay this paper investigates the existence and uniqueness theorem of solutions to neutral stochastic differential equations with infinite delay short for insfdes at a space bc 0 r d under the uniform lipschitz condition linear growth condition is weaken to obtain the moment estimate of the solution for insfdes furthermore the existence uniqueness theorem of the solution for insfdes is derived and the estimate for the error between approximate solution and exact solution is given on the other hand under the linear growth condition the uniform lipschitz condition is replaced by the local lipschitz condition the existence uniqueness theorem is also valid for insfdes on t 0 t moreover the existence uniqueness theorem still holds on interval t 0 where t 0 r is an arbitrary real number

interval number and fuzzy number linear programmings abstract in this paper we focus on two kinds of linear programmings with fuzzy numbers they are called interval number and fuzzy number linear programmings respectively the problems of linear programmings with interval number coefficients are approached by taking maximum value range and minimum value range inequalities as constraint conditions reduced it into two classical linear programmings and obtained an optimal interval solution to it the problems of fuzzy linear programming with fuzzy number coefficients are approached in two ways i e fuzzy decisive set approach and interval number linear programming approach for several membership levels finally we gave the numerical solutions of the illustrative examples

note on some models for deriving the priority weights from interval fuzzy preference relations abstract deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems xu and chen 2008 proposed a number of linear programming models to derive interval weights but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved in this paper a numerical example is given to show how these definitions and models can be improved to increase accuracy a new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

min and max operators for fuzzy intervals and their potential use in aggregation operators this paper aims at expressing min and max operations when triangular fuzzy intervals are taken as inputs that is when zadeh s extension principle is considered first presented approach consists in representing fuzzy intervals by means of their profiles in this context a computation algorithm can be easily derived for implementing the min and max operations another methodology based on interval relations is then proposed for determining a general analytical expression of the min and max operations the potential use of these expressions in the framework of uncertain aggregation operators is illustrated with the two additive choquet integral

combinatorial algorithms for the minimum interval cost flow problem the aim of minimum the interval cost flow problem micfp is to find the least cost of the shipment of a commodity through a capacitated network in order to satisfy demands at certain nodes from available supplies at other nodes where there exists some vague in vector cost of problem interval cost is a common event in uncertainty environments where statistical data are applied moreover they almost play an essential role in fuzzy programming specially in the case of using their cuts in this paper a complete order on intervals is defined and efficient combinatorial algorithms for micfp are proposed digital simulation results show the performance of the proposed algorithms compared with real scenarios

systematic studies on three way decisions with interval valued decision theoretic rough sets decision theoretic rough sets dtrs are a representative rough set model the loss function is a pivotal ingredient of dtrs which is associated with the decision maker s evaluation considering the value of loss function with the imprecise evaluation interval valued dtrs ivdtrs and its mechanism in this paper are explored first we construct a basic model of ivdtrs the comparison between dtrs and ivdtrs is discussed in the frame of ivdtrs we then focus on deriving three way decisions with the aid of two conventional methods i e a certain ranking method and a degree of possibility ranking method respectively the certain ranking method converts an interval value into single and derives decision rules under a certain risk attitude of decision maker the degree of possibility ranking method assumes the flexibility of interval and utilizes the preference between interval values all the combinations and their prerequisites are summarized in which we obtain two types of decision rules based on the above analysis we further propose an optimization method for three way decisions with ivdtrs which is designed to minimize the overall uncertainty based on the shannon entropy we also compare these methods based on standard data sets finally the criteria for choosing a suitable method to three way decisions with ivdtrs are generated these results can support decision making in the uncertain environment

pricing a contingent claim with random interval or fuzzy random payoff in one period setting this article proposes a method for pricing a contingent claim with random interval and fuzzy random payoff on introduction of the acceptability concept based on classical no arbitrage argument a price interval and a fuzzy price are obtained in random interval market and fuzzy random market respectively new definitions on replicative strategies sub replicative and sup replicative ones in two market setting are given some interesting results similar to those in the classical random market are presented

a new index for bond management in an uncertain environment within the framework of assets liability management we understand that immunization is the main method to assure a certain yield in a future date departing from an initial portfolio although the objective of passive strategies is to design a portfolio that will achieve the performance of a predetermined benchmark active bond management strategies rely on expectations of interest rate movements or changes in yield spread relationships however the variation of the duration increases the risk of a portfolio that why the decision maker will have to choose the combination of expected return and risk which provides the higher utility finally the construction of a fuzzy return risk map will allow the decision maker to know the over risk and the over return as regards immunization strategy for each duration and for each risk aversion of the decision maker the construction of a risk return map presents the results in an appropriate way it will help the decision maker to choose the best duration for the decision maker interest rate forecast finally this methodology is applied to the spanish debt so that the decision maker can select the duration that brings him a greater preference

lower tail probability estimates for subordinators and nondecreasing random walks f xn ex 00 oo and a is the infimum of the support of the distribution of x1 for a xn lk we obtain the asymptotic behavior of log p sn nxn as n oo under the additional assumption of stochastic compactness a stronger result is obtained which gives the asymptotic behavior of p sn nxn itself analogues of these results are given for subordinators when t o oo or t o 0

interactive decision making in fuzzy linear optimization problems abstract this paper presents an interactive method for solving a multi criteria linear program where coefficients of the objective functions and or of the constraints are known exactly but imprecisely assuming fuzzy aspiration levels for each of the goals and basing on a comparison of flat fuzzy numbers the original fuzzy problem is transformed into a crisp satisfactory model to determine a suitable compromise solution we will maximize the minimum of the surpluses over the aspiration levels this induces a problem with a nonlinear objective function which is equivalent to a linear program

ranking function based solutions of fully fuzzified minimal cost flow problem the aim of minimal cost flow problem mcfp is to find the least transportation cost of a single commodity through a capacitated network this paper presents a model to deal with one particular group of such problems in which the supply and demand of nodes and the capacity and cost of edges are represented as fuzzy numbers for easier reference hereafter we refer to this group of problems as fully fuzzified mcfp to represent our model hukuhara s difference and approximated multiplication are used thereafter we sort fuzzy numbers by an order using a ranking function and show that it is a total order i e a reflexive anti symmetric transitive and complete binary relation utilizing the proposed ranking function we transform the fully fuzzified mcfp into three crisp problems solvable in polynomial time from this standpoint combinatorial algorithms are provided to solve the above mentioned problem and find the fuzzy optimal flow furthermore the proposed order is related to the importance weights of the center the left spread and the right spread of each fuzzy number thus this method is capable of handling the decision maker s risk taking by comparing some previous ranking function based works with our method the efficiency of the latter is revealed finally an application of our proposed method to petroleum industry is presented

an integral test for the rate of escape of d dimensional random walk let x 1 x 2 cdots be a sequence of independent identically distributed random variables taking values in mathbb r d and s n x 1 cdots x n for a large class of random variables which includes all of those in the domain of attraction of a type a stable law an integral test is given which determines whether p s n leq gamma n mathrm i o 0 quad text or quad 1 for any increasing sequence gamma n this result generalizes the dvoretzky erdos test for simple random walk and the takeuchi and taylor test for stable random walks

portfolios with fuzzy returns selection strategies based on semi infinite programming this paper provides new models for portfolio selection in which the returns on securities are considered fuzzy numbers rather than random variables the investor s problem is to find the portfolio that minimizes the risk of achieving a return that is not less than the return of a riskless asset the corresponding optimal portfolio is derived using semi infinite programming in a soft framework the return on each asset and their membership functions are described using historical data the investment risk is approximated by mean intervals which evaluate the downside risk for a given fuzzy portfolio this approach is illustrated with a numerical example

the influence of extremes on the law of the iterated logarithm we investigate the law of the iterated logarithm for the partial sum of independent identically distributed random variables when the r n largest summands are excluded

on solutions of fuzzy random multiobjective quadratic programming with applications in portfolio problem in this paper a multiobjective quadratic programming problem fuzzy random coefficients matrix in the objectives and constraints and the decision vector are fuzzy variables is considered first we show that the efficient solutions fuzzy quadratic multiobjective programming problems series optimal solutions of relative scalar fuzzy quadratic programming some theorems are to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients having decision vectors as fuzzy variables an application fuzzy portfolio optimization problem as a convex quadratic programming approach is discussed and an acceptable solution to such problem is given at the end numerical examples are illustrated in the support of the obtained results

genetic algorithm for asymmetric traveling salesman problem with imprecise travel times this paper presents a variant of the asymmetric traveling salesman problem atsp in which the traveling time between each pair of cities is represented by an interval of values wherein the actual travel time is expected to lie instead of a fixed deterministic value as in the classical atsp here the atsp with interval objective is formulated using the usual interval arithmetic to solve the interval atsp i atsp a genetic algorithm with interval valued fitness function is proposed for this purpose the existing revised definition of order relations between interval numbers for the case of pessimistic decision making is used the proposed algorithm is based on a previously published work and includes some new features of the basic genetic operators to analyze the performance and effectiveness of the proposed algorithm and different genetic operators computational studies of the proposed algorithm on some randomly generated test problems are reported

interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming abstract the modern trend in operations research methodology deserves modelling of all relevant vague or uncertain information involved in a real decision problem generally vagueness is modelled by a fuzzy approach and uncertainty by a stochastic approach in some cases a decision maker may prefer using interval numbers as coefficients of an inexact relationship as a coefficient an interval assumes an extent of tolerance or a region that the parameter can possibly take however its use in the optimization problems is not much attended as it merits this paper defines an interval linear programming problem as an extension of the classical linear programming problem to an inexact environment on the basis of a comparative study on ordering interval numbers inequality constraints involving interval coefficients are reduced in their satisfactory crisp equivalent forms and a satisfactory solution of the problem is defined a numerical example is also given

fuzzy optimization an appraisal abstract this paper takes a general look at core ideas that make up the burgeoning body of fuzzy mathematical programming emphasizing the methodological view although fuzzy mathematical programming has enjoyed a rapidly increasing acceptance within the scientific community some technical hurdles exist to hinder a unanimity reasons for this as well as possible ways for improvement are also discussed

dominance degrees for intervals and their application in multiple attribute decision making abstract since intervals are used to express decision information in various decision making problems it is necessary to give a feasible method to rank them the most commonly used method is to employ the possibility degree to compare intervals in pairs then construct a fuzzy preference relation generally the constructed fuzzy preference relation does not keep additive multiplicative consistency in order to ensure the consistency of decision information optimization models are constructed by using the logical relationship between the elements of the consistent fuzzy preference relation and their weights then by solving the optimization models consistent fuzzy preference relations are obtained the consistent fuzzy preference relations are the proxy of the original inconsistent fuzzy preference relation constructed by the possibility degree however such treatment is flawed the inconsistent fuzzy preference relation generated from the possibility degree can t guarantee the reliability of the result while the proxy consistent fuzzy preference relation would cause bias to solve this problem the additive dominance degree and multiplicative dominance degree for intervals are proposed as a result of indirect comparison the fuzzy preference relations are constructed from the dominance degrees of intervals to ensure the consistency of decision information in the process of ranking intervals this can avoid decision making bias furthermore a new approach to interval multiple attribute decision making is proposed based on the dominance degrees

ranking fuzzy numbers with preference weighting function expectations based on the recent research developments of fuzzy number ranking the paper extends the centroid expectation approach and proposes a preference weighting function expectation method to rank fuzzy numbers the weighting function gives different emphasis on the possible values of the fuzzy number which can be used to represent the decision maker s pessimistic neutral or optimistic attitude an optimistic measure to represent this preference information is proposed some properties of the weighting function and optimistic measure are systematically analyzed then the paper proposes a special class weighting function with maximum entropy principle and analyzes some of its properties with this parameterized maximum entropy weighting function we can get preference expectation of any fuzzy number with given optimistic level next the weighting function is extended to represent the confidential attitude of the decision maker by giving different emphasis on the cut level set of the fuzzy number some results about confidential attitude that parallel to the valuation attitude are attained we especially extended the badd basic defuzzification distribution approach with preference expectation of fuzzy numbers and another confidential measure with badd operator is proposed finally the expected valuations of fuzzy numbers are used to ranking the fuzzy numbers compared with other fuzzy number ranking methods the method of this paper is easily interpreted robust flexible and invariant with translating and scaling for real numbers

a new multi section based technique for constrained optimization problems with interval valued objective function in this paper an efficient optimization technique is proposed for constrained optimization problems with interval valued objective function at first the significance of interval valued objective function and the meaning of the interval valued solution of the proposed problem have been explained with graphical interpretation generally this type of problems has infinitely many compromise solutions the aim of this approach is to obtain one of such solutions with higher accuracy and lower computational cost the proposed technique is mainly based on the splitting criterion of the accepted prescribed search region calculation of the interval inclusion functions and the selection of subregion depending on the modified interval order relations in the context of the decision makers point of view novel interval oriented constraint satisfaction rules are used for non interval equality and inequality constraints clearly the proposed technique is nothing but an imitation of well known interval analysis based branch and bound b b optimization approach the modified multi section division criterion with some new interval oriented constraint satisfaction rules for non interval valued equality and inequality constraints and some novel interval order relations in the context of the decision makers point of view have been applied to increase the efficiency of the proposed algorithm finally the technique is applied for solving some test problems and the results are compared with the same obtained from the existing methods

random walks driven by low moment measures we study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition for any locally compact unimodular group g and any positive function g 0 we introduce a function g which describes the fastest possible decay of n 2n e when is a symmetric continuous probability density such that is finite we estimate g for a variety of groups g and functions when is of the form with 0 0 a fixed increasing function and g 0 a natural word length measuring the distance to the identity element in g g can be thought of as a group invariant

the shortest path problem on networks with fuzzy parameters shortest path problem where the costs have vague values is one of the most studied problems in fuzzy sets and systems area however due to its high computational complexity previously published algorithms present peculiarities and problems that need to be addressed e g they find costs without an existing path they determine a fuzzy solution set but do not give any guidelines to help the decision maker choose the best path they can only be applied in graphs with fuzzy non negative parameters etc in this paper one proposes an iterative algorithm that assumes a generic ranking index for comparing the fuzzy numbers involved in the problem in such a way that each time in which the decision maker wants to solve a concrete problem s he can choose or propose the ranking index that best suits that problem this algorithm that solves the above remarked drawbacks is based on the ford moore bellman algorithm for classical graphs and in concrete it can be applied in graphs with negative parameters and it can detect whether there are negative circuits for the sake of illustrating the performance of the algorithm in the paper it has been here developed using only certain order relations but it is not restricted at all to use these comparison relations exclusively the proposed algorithm is easy of understanding as the theoretical base of a decision support system oriented to solving this kind of problems

implicit numerical methods for highly nonlinear neutral stochastic differential equations with time dependent delay abstract this paper represents the continuation of the analysis from papers milosevic 2011 10 and milosevic 2013 11 the main aim of this paper is to establish certain results for the backward euler method for a class of neutral stochastic differential equations with time dependent delay for that purpose the split step backward euler method which represents an extension of the backward euler method is introduced for this class of equations conditions under which the split step backward euler method and thus the backward euler method is well defined are revealed moreover the convergence in probability of the backward euler method is proved under certain nonlinear growth conditions including the one sided lipschitz condition this result is proved using the technique which is based on the application of the continuous time approximation for this reason the discrete forward backward euler method is involved since it allows its continuous version to be well defined from the aspect of measurability the convergence in probability is established for the continuous forward backward euler solution which is essential for proving the same result for both discrete forward backward and backward euler methods additionally it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without the linear growth condition on the drift coefficient of the equation as usual the whole consideration is affected by the presence and properties of the delay function

grey dynamic input output analysis abstract a system whose information is partially known and partially unknown is named as a grey system and it is general situation in reality on the basis of grey number matrix and their covered operation some definitions of grey functions are enumerated combining grey system theory with traditional dynamic input output analysis we propose the grey dynamic input output analysis it has two types i e the continuous and the disperse the most important results are the computational formulas of the matrix covered set of inverse grey investment coefficient matrix by using them we can forecast and control the economic system under the uncertain situation for the calculation of the matrix covered set of inverse grey matrices we obtain the desired formulas which can be applied to general grey matrices the formulas can greatly promote the development of grey theory and enrich the applied fields of uncertain mathematics the modified case illustrates our method

numerical methods for interval and fuzzy number comparison based on the probabilistic approach and dempster shafer theory there are many different methods for interval and fuzzy number comparison proposed in the literature which provide the results of comparison in the form of a real or boolean value in this paper we use the dempster shafer theory of evidence with its probabilistic interpretation to justify and construct the method which provides the result of comparison in the form of an interval or fuzzy number the complete and consistent set of expressions for inequality and equality relations between intervals and fuzzy numbers has been obtained in the framework of the probabilistic approach these relations make it possible to compare intervals and fuzzy numbers with real values and may be considered as an asymptotic limit of the results we obtain using the dempster shafer theory a natural fuzzy extension of the proposed approach is considered and discussed using some illustrative examples

interval weight generation approaches based on consistency test and interval comparison matrices a simple yet pragmatic method of consistency test is developed to check whether an interval comparison matrix is consistent or not linear programming method is used to derive consistent interval weights from consistent interval comparison matrices and to aggregate local interval weights to generate global interval weights in cases where an interval comparison matrix is inconsistent an eigenvector method based nonlinear programming nlp approach is developed to generate interval weights that can meet pre determined consistency requirements a simple and effective preference ranking method is utilized to compare the interval weights of criteria or rank alternatives three numerical examples including a hierarchical ahp decision problem are provided to illustrate the validity and practicality of the proposed methods

a general model for fuzzy linear programming abstract in the current literature there are several models of fuzzy linear programming problems the aim of this paper is first to describe the more important problems in fuzzy linear programming and second to give a general model of fuzzy linear programming problems involving all of the above ones finally a resolution method for that general model is proposed

the class of limit laws for stochastically compact normed sums khintchine showed that every infinitely divisible law can be obtained as the limit of a subsequence of normed sums of independent identically distributed random variables here we restrict the summands to be in a class which makes the normed sums stochastically compact i e so that every subsequence has a further subsequence which converges to a nondegenerate limit a nice analytic condition for stochastic compactness was obtained by feller our result is an analogous characterization of the class of limit laws of subsequences of stochastically compact normed sums one consequence is that they have c infty densities

multistep methods for sdes and their application to problems with small noise in this article the numerical approximation of solutions of ito stochastic differential equations is considered in particular for equations with a small parameter epsilon in the noise coefficient we construct stochastic linear multistep methods and develop the fundamental numerical analysis concerning their mean square consistency numerical stability in the mean square sense and mean square convergence for the special case of two step maruyama schemes we derive conditions guaranteeing their mean square consistency further for the small noise case we obtain expansions of the local error in terms of the step size and the small parameter epsilon simulation results using several explicit and implicit stochastic linear k step schemes k 1 2 illustrate the theoretical findings

enhanced interval linear programming abstract an enhanced interval linear programming eilp model and its solution algorithm have been developed that incorporate enhanced interval uncertainty e g a b and c in a linear optimization framework as a new extension of linear programming the eilp model has the following advantages its solution space is absolutely feasible compared to that of interval linear programming ilp which helps to achieve insight into the expected value oriented trade off between system benefits and risks of constraint violations the degree of uncertainty of its enhanced interval objective function eiof would be lower than that of ilp model when the solution space is absolutely feasible and the eiof s expected value could be used as a criterion for generating the appropriate alternatives which help decision makers obtain non extreme decisions moreover because it can be decomposed into two submodels eilp s computational requirement is lower than that of stochastic and fuzzy lp models the results of a numeric example further indicated the feasibility and effectiveness of eilp model in addition ei nonlinear programming models hybrid stochastic or fuzzy eilp models as well as risk based trade off analysis for ei uncertainty within decision process can be further developed to improve its applicability

on fuzzy random multiobjective quadratic programming in this paper a multiobjective quadratic programming problem having fuzzy random coefficients matrix in the objective and constraints and the decision vector are fuzzy pseudorandom variables is considered first we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series optimal solutions of relative scalar fuzzy quadratic programming some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients having decision vectors as fuzzy variables at the end numerical examples are illustrated in the support of the obtained results

interval oriented multi section techniques for global optimization this paper deals with two different optimization techniques to solve the bound constrained nonlinear optimization problems based on division criteria of a prescribed search region finite interval arithmetic and interval ranking in the context of a decision maker s point of view in the proposed techniques two different division criteria are introduced where the accepted region is divided into several distinct subregions and in each subregion the objective function is computed in the form of an interval using interval arithmetic and the subregion containing the best objective value is found by interval ranking the process is continued until the interval width for each variable in the accepted subregion is negligible in this way the global optimal or close to global optimal values of decision variables and the objective function can easily be obtained in the form of an interval with negligible widths both the techniques are applied on several benchmark functions and are compared with the existing analytical and heuristic methods

matrix normalized sums of independent identically distributed random vectors on considere x 1 x 2 une suite de vecteurs aleatoires i i d et s n x 1 x n on donne des conditions necessaires et suffisantes pour qu il existe des matrices b n et des vecteurs n tels que b n s n n soit stochastiquement compact

an application of genetic algorithm in a marketing oriented inventory model with interval valued inventory costs and three component demand rate dependent on displayed stock level the objective of this research is to investigate an inventory model for a single item with imprecise inventory costs by considering the impact of marketing strategies such as pricing and advertising on three component demand rate this rate is dependent on selling price frequency of advertisement and displayed stock level dsl in a show room shop here the impreciseness of inventory costs like carrying cost purchase cost ordering cost and advertisement cost has been represented by interval valued numbers analyzing the relative size of the storage capacity of the show room shop and the stock level dependency parameters of demand different scenarios with sub scenarios of each have been mentioned then for each sub scenario the model has been formulated as a constrained optimization problem with interval objective to solve these problems an advanced genetic algorithm ga for mixed integer non linear programming has been developed with interval valued fitness function in this developed ga the order relations of interval valued numbers have been used with respect to pessimistic decision maker s point of view this approach has been used in the ranked based selection process for selecting better chromosomes individuals for the next generation and also for finding the best chromosomes individuals in each generation finally the model has been illustrated with some numerical examples and the performance of the ga has been tested by computing appropriate statistical measures and the computing time

linguistic computational model based on 2 tuples and intervals herrera and martinez initiated a 2 tuple fuzzy linguistic representation model for computing with words moreover wang and hao further developed a new 2 tuple fuzzy linguistic representation model to deal with the linguistic term sets that are not uniformly and symmetrically distributed this study proposes another linguistic computational model based on 2 tuples and intervals which we call an interval version of the 2 tuple fuzzy linguistic representation model the proposed model possesses three steps 1 interval numerical scale 2 computation based on interval numbers and 3 a generalized inverse operation of the interval numerical scale the first step transforms linguistic terms into interval numbers based on which the second step is executed with output as an interval number finally this number is then mapped into the interval of the linguistic 2 tuples by the generalized inverse operation this study also generalizes the numerical scale approach presented in the wang and hao model to set the interval numerical scale by considering the context where semantics of linguistic terms are defined by interval type 2 fuzzy sets it2 fss in order to compare the proposed model with the existing linguistic computational model based on it2 fss we have conducted extensive simulations the simulations demonstrate that the results obtained by our proposal are consistent with the results of the linguistic computational model based on it2 fss in some sense in a vast majority of cases

convergence and stability of the semi tamed euler scheme for stochastic differential equations with non lipschitz continuous coefficients recently explicit tamed schemes were proposed to approximate the sdes with the non lipschitz continuous coefficients this work proposes a semi tamed euler scheme which is also explicit to solve the sdes with the drift coefficient equipped with the lipschitz continuous part and non lipschitz continuous part it is shown that the semi tamed euler converges strongly with the standard order one half to the exact solution of the sde we also investigate the stability inheritance of the semi tamed euler schemes and reveal that this scheme does have advantage in reproducing the exponential mean square stability of the exact solution numerical experiments confirm the theoretical analysis

a ga based penalty function technique for solving constrained redundancy allocation problem of series system with interval valued reliability of components in this paper we have considered the problem of constrained redundancy allocation of series system with interval valued reliability of components for maximizing the overall system reliability under limited resource constraints the problem is formulated as an unconstrained integer programming problem with interval coefficients by penalty function technique and solved by an advanced ga for integer variables with interval fitness function tournament selection uniform crossover uniform mutation and elitism as a special case considering the lower and upper bounds of the interval valued reliabilities of the components to be the same the corresponding problem has been solved the model has been illustrated with some numerical examples and the results of the series redundancy allocation problem with fixed value of reliability of the components have been compared with the existing results available in the literature finally sensitivity analyses have been shown graphically to study the stability of our developed ga with respect to the different ga parameters

maximum excess dominance identifying impractical solutions in linear problems with interval coefficients abstract in this paper we propose the concept of maximum excess dominance med and illustrate how it can be used to compare solutions in linear optimization problems with interval objective coefficients when a solution dominates another solution with med the expected outcome of the former is guaranteed to be better than that of the latter across a wide range of probability distributions hence med can be used to eliminate dominated solutions from consideration furthermore we provide an efficient way to check if a solution is dominated by another feasible solution in binary optimization problems and illustrate how dominated solutions can be pruned away by introducing med constraints to the original binary optimization formulation we also propose an algorithm to find the best non dominated solution and conduct computational experiments to evaluate its performance

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

a multicriteria fuzzy linear programming method for water supply system development planning abstract a method is presented for solving a multicriteria linear program where the coefficients of the objective functions and the constraints are fuzzy numbers of the l r type assuming the aspiration levels for particular criteria to be fuzzy and basing on comparison of fuzzy numbers the original problem is transformed into a multicriteria linear fractional program the latter is solved using an interactive technique involving a linear programming procedure in the calculation phase this method has been elaborated with a view to the application to the development of a water supply system

on the calculation of a membership function for the solution of a fuzzy linear optimization problem in the present paper the fuzzy linear optimization problem with fuzzy coefficients in the objective function is considered recent concepts of fuzzy solution to the fuzzy optimization problem based on the level cut and the set of pareto optimal solutions of a multiobjective optimization problem are applied chanas and kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem which is based on calculating the sum of lengths of certain intervals the purpose of this paper is to determine a method for realizing this idea we derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered

elitist genetic algorithm for assignment problem with imprecise goal the objective of this research paper is to solve a generalized assignment problem with imprecise cost s time s instead of precise one by elitist genetic algorithm ga here the impreciseness of cost s time s has been represented by interval valued numbers as interval valued numbers are the best representation than others like random variable representation with a known probability distribution and fuzzy representation to solve these types of problems an elitist ga has been developed with interval valued fitness function in this developed ga the existing ideas about the order relations of interval valued numbers have been modified from the point of view of two types of decision making viz optimistic decision making and pessimistic decision making this modified approach has been used in the selection process for selecting better chromosomes individuals for the next generation and in finding the best as well as the worst chromosomes individuals in each generation here two new crossover schemes and two new mutation schemes have been introduced in order to maintain the feasibility with crossover operations a repair algorithm has been suggested extensive comparative computational studies based on different parameters of our developed algorithm on one illustrative example have also been reported

on sufficiency and duality for a class of interval valued programming problems abstract in this paper we are concerned with an interval valued programming problem sufficient optimality conditions are established under generalized convex functions for a feasible solution to be an efficient solution appropriate duality theorems for mond weir and wolfe type duals are discussed in order to relate the efficient solutions of primal and dual programs

the solution and duality of imprecise network problems duality properties have been investigated by many researchers in the recent literature they are introduced in this paper for a fully fuzzified version of the minimal cost flow problem which is a basic model in network flow theory this model illustrates the least cost of the shipment of a commodity through a capacitated network in terms of the imprecisely known available supplies at certain nodes which should be transmitted to fulfil uncertain demands at other nodes first we review on the most valuable results on fuzzy duality concepts to facilitate the discussion of this paper by applying hukuhara s difference approximated and exact multiplication and wu s scalar production we exhibit the flow in network models then we use combinatorial algorithms on a reduced problem which is derived from fully fuzzified mcfp to acquire fuzzy optimal flows to give duality theorems we utilize a total order on fuzzy numbers due to the level of risk and realize optimality conditions for providing some efficient combinatorial algorithms finally we compare our results with the previous worthwhile works to demonstrate the efficiency and power of our scheme and the reasonability of our solutions in actual decision making problems

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

a fuzzy approach to deriving priorities from interval pairwise comparison judgements abstract in this paper we study the problem of priority elicitation in the analytic hierarchy process and propose a new approach to deriving crisp priorities from interval pairwise comparison judgements by introducing linear or non linear membership functions representing the decision maker s degree of satisfaction with various crisp priority vectors the interval judgements are transformed into fuzzy inequality constraints the interval prioritisation problem is then formulated as a fuzzy mathematical programming problem for obtaining an optimal crisp priority vector that maximises the overall degree of satisfaction the proposed approach yields linear or non linear mathematical programs capable of deriving priorities from consistent and inconsistent interval judgements the presence of a consistency index that measures the level of inconsistency of interval judgements is an attractive feature of our approach another feature which does not exist in the known prioritisation methods is the opportunity for additional prioritisation of the initial judgements numerical examples are given and comparisons with other interval prioritisation methods are carried out

the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2003 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc in this paper we will develop a new explicit method called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and establish the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 the type of convergence specifically addressed in this paper is strong l q convergence for 2 q p and p is a parameter in the khasminskii type condition

computing a fuzzy shortest path in a network with mixed fuzzy arc lengths using α cuts we are concerned with the design of a model and an algorithm for computing a shortest path in a network having various types of fuzzy arc lengths first we develop a new technique for the addition of various fuzzy numbers in a path using a cuts by proposing a linear least squares model to obtain membership functions for the considered additions then using a recently proposed distance function for comparison of fuzzy numbers we present a dynamic programming method for finding a shortest path in the network examples are worked out to illustrate the applicability of the proposed model

probability estimates for the small deviations of d dimensional random walk let x 1 x 2 cdots be a sequence of independent identically distributed random variables taking values in mathbb r d and s n x 1 cdots x n for a large class of distributions we obtain estimates for the probability that s n is in a ball centered at the origin such an estimate would follow from a local limit theorem if x 1 were in the domain of attraction of a stable law

exponential stability of impulsive stochastic functional differential equations in this paper we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using lyapunov method several stability theorems of impulsive stochastic functional differential equations with finite delay are derived these new results are employed to impulsive stochastic equations with bounded time varying delays and stochastically perturbed equations meanwhile an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay

the law of the iterated logarithm for local time of a levy process let x t be a one dimensional levy process with local timel t x andl t sup l t x x under an assumption which is more general than being a symmetric stable process with index 1 we obtain a lil forl t also with an additional condition of symmetry a lil for range is proved

selection of programme slots of television channels for giving advertisement a graph theoretic approach the present paper aims at developing a linear time algorithm to find a solution to the maximum weight 1 colouring problem for an interval graph with interval weight this algorithm has been applied to solve the problem that involves selecting different programme slots telecast on different television channels in a day so as to reach the maximum number of viewers it is shown that all programmes of all television channels can be modelled as a weighted interval graph with interval numbers as weights the programme slots are taken as the vertices of the graph and if the time durations of two programme slots have non empty intersection the corresponding vertices are considered to be connected by an edge the number of viewers of a programme is taken as the weight of the vertex in reality the number of viewers of a programme is not a fixed one generally it lies in an interval thus the weights of the vertices are taken as interval numbers we assume that a company sets the objective of selecting the popular programme in different channels so as to make its commercial advertisement reach the maximum number of viewers however the constraint imposed is that all selected programmes are mutually exclusive in respect of time scheduling the objective of the paper is therefore to helps the companies to select the programme slots which are mutually exclusive with respect to the time schedule of telecasting time in such a way that the total number of viewers of the selected programme slots rises to the optimum level it is shown that the solution of this problem is obtained by solving the maximum weight colouring problem on an interval graph an algorithm is designed to solve this optimization problem using o n time where n represents the total number of programmes of all channels

acceptable consistency analysis of interval reciprocal comparison matrices when a decision maker expresses his her opinions by means of an interval reciprocal comparison matrix the study of consistency becomes a very important aspect in decision making in order to avoid a misleading solution in the present paper an acceptably consistent interval reciprocal comparison matrix is defined which can be reduced to an acceptably consistent crisp reciprocal comparison matrix when the intervals become exact numbers an interval reciprocal comparison matrix with unacceptable consistency can be easily adjusted such that the revised matrix possesses acceptable consistency utilizing a convex combination method a family of crisp reciprocal comparison matrices with acceptable consistency can be obtained whose weights are further found to exhibit a style of convex combination and aggregated to obtain interval weights from an acceptably consistent interval reciprocal comparison matrix a novel simple yet effective formula of possibility degree is presented to rank interval weights numerical results are calculated to show the quality and quantity of the proposed approaches and compare with other existing procedures

mean square convergence of explicit two step methods for highly nonlinear stochastic differential equations abstract in this paper we propose the projected two step euler maruyama method and the projected two step milstein method for highly nonlinear stochastic differential equations under a global monotonicity condition we first examine the strong convergence in mean square sense for these two explicit schemes based on the notions of stochastic stability and b consistency for two step methods we prove that the convergence rates of the projected two step euler maruyama method and the projected two step milstein method are 1 2 and 1 respectively in particular our results can be applied to equations with super linearly growing drift and diffusion coefficients finally we numerically verify the optimal mean square convergence orders of these two schemes by a series of examples

multiobjective programming in optimization of the interval objective function abstract in this paper the mathematical programming problem whose objective function has interval coefficients is investigated to maximize the interval objective function the order relations which represent the decision maker s preference between interval profits are defined by the right limit the left limit the center and the width of an interval the maximization problem with the interval objective function is converted into a multiobjective problem using the order relations the minimization problem is also converted into a multiobjective problem using the order relations which represent the decision maker s preference between interval costs