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

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

prediction based stabilization of linear systems subject to input dependent input delay of integral type in this paper it is proved that a predictor based feedback controller can effectively yield asymptotic convergence for a class of linear systems subject to input dependent input delay this class is characterized by the delay being implicitly related to past values of the input via an integral model this situation is representative of systems where transport phenomena take place as is frequent in the process industry the sufficient conditions obtained for asymptotic stabilization bring a local result and require the magnitude of the feedback gain to be consistent with the initial conditions scale arguments of proof for this novel result include general halanay inequalities for delay differential equations and build on recent advances of backstepping techniques for uncertain or varying delay systems

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

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

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

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

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

from lines to epipoles through planes in two views this paper addresses the computation of the fundamental matrix between two views when camera motion and 3d structure are unknown but planar surfaces can be assumed we use line features which are automatically matched in two steps firstly with image based parameters a set of matches are obtained to secondly compute homographies which allows to reject wrong ones and to grow good matches in a final stage the inclusion of projective transformations gives much better results to match features with short computing overload when two or more planes are observed different homographies can be computed segmenting simultaneously the corresponding planar surfaces these can be used to obtain the fundamental matrix which gives constraints for the whole scene the results show that the global process is robust enough turning out stable and useful to obtain matches and epipolar geometry from lines in man made environments

multiview constraints on homographies the image motion of a planar surface between two camera views is captured by a homography a 2d projective transformation the homography depends on the intrinsic and extrinsic camera parameters as well as on the 3d plane parameters while camera parameters vary across different views the plane geometry remains the same based on this fact we derive linear subspace constraints on the relative homographies of multiple spl ges 2 planes across multiple views the paper has three main contributions 1 we show that the collection of all relative homographies homologies of a pair of planes across multiple views spans a 4 dimensional linear subspace 2 we show how this constraint can be extended to the case of multiple planes across multiple views 3 we show that for some restricted cases of camera motion linear subspace constraints apply also to the set of homographies of a single plane across multiple views all the results derived are true for uncalibrated cameras the possible utility of these multiview constraints for improving homography estimation and for detecting nonrigid motions are also discussed

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

a robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry abstract this paper proposes a robust approach to image matching by exploiting the only available geometric constraint namely the epipolar constraint the images are uncalibrated namely the motion between them and the camera parameters are not known thus the images can be taken by different cameras or a single camera at different time instants if we make an exhaustive search for the epipolar geometry the complexity is prohibitively high the idea underlying our approach is to use classical techniques correlation and relaxation methods in our particular implementation to find an initial set of matches and then use a robust technique the least median of squares lmeds to discard false matches in this set the epipolar geometry can then be accurately estimated using a meaningful image criterion more matches are eventually found as in stereo matching by using the recovered epipolar geometry a large number of experiments have been carried out and very good results have been obtained regarding the relaxation technique we define a new measure of matching support which allows a higher tolerance to deformation with respect to rigid transformations in the image plane and a smaller contribution for distant matches than for nearby ones a new strategy for updating matches is developed which only selects those matches having both high matching support and low matching ambiguity the update strategy is different from the classical winner take all which is easily stuck at a local minimum and also from loser take nothing which is usually very slow the proposed algorithm has been widely tested and works remarkably well in a scene with many repetitive patterns

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

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

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

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

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 fundamental matrix theory algorithms and stability analysis in this paper we analyze in some detail the geometry of a pair of cameras i e a stereo rig contrarily to what has been done in the past and is still done currently for example in stereo or motion analysis we do not assume that the intrinsic parameters of the cameras are known coordinates of the principal points pixels aspect ratio and focal lengths this is important for two reasons first it is more realistic in applications where these parameters may vary according to the task active vision second the general case considered here captures all the relevant information that is necessary for establishing correspondences between two pairs of images this information is fundamentally projective and is hidden in a confusing manner in the commonlyused formalism of the essential matrix introduced by longuet higgins 1981 this paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 x 3 matrix of rank 2 which we propose to call the fundamental matrix after this theoretical analysis we embark on the task of estimating the fundamental matrix from point corre spondences a task which is of practical importance we analyze theoretically and compare experimentally using synthetic and real data several methods of estimation the problem of the stability of the estimation is studied from two complementary viewpoints first we show that there is an interesting relationship between the fundamental matrix and three dimensional planes which induce homographies between the images and create unstabilities in the estimation procedures second we point to a deep relation between the unstability of the estimation procedure and the presence in the scene of so called critical surfaces which have been studied in the context of motion analysis finally we conclude by stressing the fact that we believe that the fundamental matrix will play a crucial role in future applications of three dimensional computer vision by greatly increasing its versatility robustness and hence applicability to real difficult problems

in defense of the eight point algorithm the fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras and the eight point algorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches it has the advantage of simplicity of implementation the prevailing view is however that it is extremely susceptible to noise and hence virtually useless for most purposes this paper challenges that view by showing that by preceding the algorithm with a very simple normalization translation and scaling of the coordinates of the matched points results are obtained comparable with the best iterative algorithms this improved performance is justified by theory and verified by extensive experiments on real images

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

determining straight line correspondences from intensity images abstract this paper presents an algorithm for straight edge extraction from intensity images and an algorithm for straight line matching in the first part of the paper straight edge extraction is described image data are first processed by the operation of edge support focusing to remove unnecessary image details an edge support is formed in this process then straight edges are extracted from line support regions which are segmented from the edge support in the second part of this paper we describe straight line matching using a matching function which characterizes the similarity of edge lines of two images and is based on not only the geometrical relations of the lines but also the information from the intensity images a technique of two step matching is applied to reduce the cost of computation the motivation behind this paper is our work on motion estimation from line correspondences of sequential images where straight edge extraction and matching is an essential first step the output data of the algorithm can also be used for calibration of stereo camera setups the results of experiments using indoor and outdoor scene images are shown

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

linear multi view reconstruction and camera recovery using a reference plane this paper presents a linear algorithm for simultaneous computation of 3d points and camera positions from multiple perspective views based on having a reference plane visible in all views the reconstruction and camera recovery is achieved in a single step by finding the null space of a matrix built from image data using singular value decomposition contrary to factorization algorithms this approach does not need to have all points visible in all views this paper investigates two reference plane configurations finite reference planes defined by four coplanar points and infinite reference planes defined by vanishing points a further contribution of this paper is the study of critical configurations for configurations with four coplanar points by simultaneously reconstructing points and views we can exploit the numerical stabilizing effect of having wide spread cameras with large mutual baselines this is demonstrated by reconstructing the outside and inside courtyard of a building on the basis of 35 views in one single singular value decomposition

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

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

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

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

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

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

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

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