multiple objective programming problems with possibilistic coefficients abstract we consider the problem of incorporating possibilistic data into a multiple objective linear programming framework a solution concept that is attractive from the standpoints of feasibility and efficiency is specified a necessary and sufficient condition for such a solution is established and ways to single out some satisfying alternative in this context are described a numerical example is also included for the sake of illustration throughout the paper we indicate topics for further research

a simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension it has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex type algorithms for linear programming in this paper the average number of steps performed by a simplex algorithm the so called self dual method is analyzed the algorithm is not started at the traditional point 1 l t but points of the form 1 e e 2 t with e sufficiently small are used the result is better in two respects than those of the previous analyses first it is shown that the expected number of steps is bounded between two quadratic functions c 1 min m n 2 and c 2 min m n 2 of the smaller dimension of the problem this should be compared with the previous two major results in the field borgwardt proves an upper bound of o n 4 m 1 n 1 under a model that implies that the zero vector satisfies all the constraints and also the algorithm under his consideration solves only problems from that particular subclass smale analyzes the self dual algorithm starting at 1 1 t he shows that for any fixed m there is a constant c m such the expected number of steps is less than c m ln n m m 1 megiddo has shown that under smale s model an upper bound c m exists thus for the first time a polynomial upper bound with no restrictions except for nondegeneracy on the problem is proved and for the first time a nontrivial lower bound of precisely the same order of magnitude is established both borgwardt and smale require the input vectors to be drawn from spherically symmetric distributions in the model in this paper invariance is required only under certain

fuzzy non linear integer program by parametric programming approach in this paper we propose a heuristic algorithm to analyze a differentiable nonlinear integer program when the fuzzy inequality constraint is considered by defining a membership function the fuzzy inequality constraint can be transformed into a parametric inequality of which a hybrid solution procedure incorporating a genetic algorithm is developed theoretical analysis and experimental investigation are presented

interpretability of linguistic fuzzy rule based systems an overview of interpretability measures linguistic fuzzy modelling developed by linguistic fuzzy rule based systems allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings linguistic fuzzy modelling comes with two contradictory requirements interpretability and accuracy in recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown whereas the measures of accuracy are straightforward and well known interpretability measures are difficult to define since interpretability depends on several factors mainly the model structure the number of rules the number of features the number of linguistic terms the shape of the fuzzy sets etc moreover due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem in this paper we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule based systems to this end we will propose a taxonomy based on a double axis complexity versus semantic interpretability considering the two main kinds of measures and rule base versus fuzzy partitions considering the different components of the knowledge base to which both kinds of measures can be applied the main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works

multi period mixed production planning with uncertain demands fuzzy and interval fuzzy sets approach this paper shows a general model of a mixed production planning problem with fuzzy demands the main focus is the development of a model for production planning using fuzzy sets in order to use classical mathematical programming techniques to reach an optimal solution over a multiple criteria context the classical fuzzy linear programming model namely the soft constraints model is used to involve flexibility in the problem moreover an interval fuzzy set approach is used to involve uncertainty in the problem

bi criteria multi facility location problem in fuzzy environment abstract this paper presents a fuzzy goal programming model for locating multiple new facilities on a plane bounded by a convex polygon under the criteria 1 minimize the sum of all the transportation costs and 2 minimize the maximum distances from the facilities to the demand points it has also been proved that the methodology given will always give a nondominated solution recti linear distances have been taken as the scenario may be thought of in an urban setting a numerical example has been given to illustrate the solution methodology

a nonlinear programming algorithm for solving semidefinite programs via low rank factorization in this paper we present a nonlinear programming algorithm for solving semidefinite programs sdps in standard form the algorithm s distinguishing feature is a change of variables that replaces the symmetric positive semidefinite variable x of the sdp with a rectangular variable r according to the factorization x rr t the rank of the factorization i e the number of columns of r is chosen minimally so as to enhance computational speed while maintaining equivalence with the sdp fundamental results concerning the convergence of the algorithm are derived and encouraging computational results on some large scale test problems are also presented

on fuzzy random linear programming abstract the fuzzy random linear programming having fuzzy random variable coefficients and the decision vector of fuzzy pseudorandom variables is studied the first results show the fact that a fuzzy pseudorandom resp fuzzy random optimization solution of a fuzzy random linear programmings the subsequent conclusions present methods random optimization solutions of relative random linear programmings the subsequent conclusions present methods that a fuzzy pseudorandom resp fuzzy random optimization solution of a fuzzy random linear programming is structured by a class of pseudorandom resp random optimization solutions of relative random linear programmings finally some theorems are proved which solve the problems finding the fuzzy probability distribution function and fuzzy mathematical expectation of optimization value for a fuzzy random linear programming these results reveal properties of a fuzzy random linear programming and give the solution for a fuzzy random linear programming

a labeling algorithm for the fuzzy assignment problem abstract this paper concentrates on the assignment problem where costs are not deterministic numbers but imprecise ones here the elements of the cost matrix of the assignment problem are subnormal fuzzy intervals with increasing linear membership functions whereas the membership function of the total cost is a fuzzy interval with decreasing linear membership function by the max min criterion suggested by bellman and zadeh the fuzzy assignment problem can be treated as a mixed integer nonlinear programming problem we show that this problem can usually be simplified into either a linear fractional programming problem or a bottleneck assignment problem here we propose an efficient algorithm based on the labeling method for solving the linear fractional programming case the algorithm begins with primal feasibility and proceeds to obtain dual feasibility while maintaining complementary slackness until the primal optimal solution is found the computational results show that the proposed labeling algorithm offers an effective and efficient way for handling the fuzzy assignment problem

uniqueness of solution in flp under parameter perturbations abstract the main purpose of this paper is to establish some conditions which are both necessary and sufficient for a solution to a fuzzy linear programming problem to be unique in the presence of parameter perturbations the basic idea behind our approach is to translate the original problem in deterministic terms via tools provided by fuzzy set theory and investigate uniqueness conditions for the basic case of conventional programming under interval uncertainty in order to obtain the crisp equivalent model we use zimmermann s approach an example is included to illustrate the proposed technique

finding low rank solutions of sparse linear matrix inequalities using convex optimization this paper is concerned with the problem of finding a low rank solution of an arbitrary sparse linear matrix inequality lmi to this end we map the sparsity of the lmi problem into a graph we develop a mathematical framework to relate the rank of the minimum rank solution of the lmi problem to the sparsity of its underlying graph furthermore we propose three graph theoretic convex programs to obtain a low rank solution two of these convex optimization problems are based on a tree decomposition of the sparsity graph the third one does not rely on any computationally expensive graph analysis and is always polynomial time solvable at the cost of offering a milder theoretical guarantee on the rank of the obtained solution compared to the other two methods the results of this work can be readily applied to three separate problems of minimum rank matrix completion conic relaxation for polynomial optimization and affine rank minimization the results are finally illustrated on two applications of opt

study on multiobjective transportation problem with fuzzy numbers abstract in this paper we investigate the efficient solutions and stability of multiobjective transportation problem fmtp with fuzzy coefficient c ij r c r and or fuzzy supply quantities a i and or fuzzy demands quantities b j the concept of fuzzy efficient was introduced in which the ordinary efficient solution is extended based on the level of fuzzy numbers a necessary and sufficient condition for such a solution is established the basic notions as the solvability set and the stability set of the first kind are defined and characterized the parametric analysis is used to characterize the parametric optimal solutions to the auxiliary problems an algorithm for the determination of the stability set is proposed finally a numerical example is given

solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach abstract the solid transportation problem stp arises when bounds are given on three item properties the fuzzy solid transportation problem fstp appears when the nature of the data problem is fuzzy this paper deals with the fstp in the case in which the fuzziness affects the constraint set and a fuzzy solution to the problem is required moreover an arbitrary linear or nonlinear objective function is considered in order to find a fuzzy solution to the problem a parametric approach is used to obtain an auxiliary parametric solid transportation problem pstp associated to the original problem as there are no well known solution methods proposed in literature to solve effectively the pstp in this paper an evolutionary algorithm ea based solution method is proposed to solve it which can finally be applied to find a good fuzzy solution to the fstp comparisons with another conventional method are presented and the results show the ea based approach to be better as a whole

toward probabilistic analysis of interior point algorithms for linear programming we propose an approach based on interior point algorithms for linear programming lp we show that the algorithm solves a class of lp problems in strongly polynomial time o n log n iteration where each iteration solves a system of linear equations with n variables the recent statistical data of the solutions of the netlib lp problems seem to indicate that virtually all of these problems are in this class then we show that some random lp problems with high probability probability converges to one as n approaches infinity are in this class these random lp problems include recent todd s probabilistic models with the standard gauss distribution

generating an interpretable family of fuzzy partitions from data in this paper we propose a new method to construct fuzzy partitions from data the procedure generates a hierarchy including best partitions of all sizes from n to two fuzzy sets the maximum size n is determined according to the data distribution and corresponds to the finest resolution level we use an ascending method for which a merging criterion is needed this criterion is based on the definition of a special metric distance suitable for fuzzy partitioning and the merging is done under semantic constraints the distance we define does not handle the point coordinates but directly their membership degrees to the fuzzy sets of the partition this leads to the introduction of the notions of internal and external distances the hierarchical fuzzy partitioning is carried independently over each dimension and to demonstrate the partition potential they are used to build fuzzy inference system using a simple selection mechanism due to the merging technique all the fuzzy sets in the various partitions are interpretable as linguistic labels the tradeoff between accuracy and interpretability constitutes the most promising aspect in our approach well known data sets are investigated and the results are compared with those obtained by other authors using different techniques the method is also applied to real world agricultural data the results are analyzed and weighed against those achieved by other methods such as fuzzy clustering or discriminant analysis

fuzzy optimization for supply chain planning under supply demand and process uncertainties in today s global marketplace individual firms do not compete as independent entities rather as an integral part of a supply chain this paper proposes a fuzzy mathematical programming model for supply chain planning which considers supply demand and process uncertainties the model has been formulated as a fuzzy mixed integer linear programming model where data are ill known and modelled by triangular fuzzy numbers the fuzzy model provides the decision maker with alternative decision plans for different degrees of satisfaction this proposal is tested by using data from a real automobile supply chain

sensitivity analysis in fuzzy linear fractional programming problem abstract in this paper a linear fractional programming problem where the numerator or denominator or both of the objective function are fuzzy in nature is first converted to an equivalent fuzzy linear programming problem sensitivity analysis of the problem thus formulated is illustrated with the help of a numerical example

duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables linear programming problems with trapezoidal fuzzy variables fvlp have recently attracted some interest some methods have been developed for solving these problems by introducing and solving certain auxiliary problems here we apply a linear ranking function to order trapezoidal fuzzy numbers then we establish the dual problem of the linear programming problem with trapezoidal fuzzy variables and hence deduce some duality results in particular we prove that the auxiliary problem is indeed the dual of the fvlp problem having established the dual problem the results will then follow as natural extensions of duality results for linear programming problems with crisp data finally using the results we develop a new dual algorithm for solving the fvlp problem directly making use of the primal simplex tableau this algorithm will be useful for sensitivity or post optimality analysis when using primal simplex tableaus

fuzzy association rules general model and applications the theory of fuzzy sets has been recognized as a suitable tool to model several kinds of patterns that can hold in data in this paper we are concerned with the development of a general model to discover association rules among items in a crisp set of fuzzy transactions this general model can be particularized in several ways each particular instance corresponds to a certain kind of pattern and or repository of data we describe some applications of this scheme paying special attention to the discovery of fuzzy association rules in relational databases

generalized fuzzy linear programming for decision making under uncertainty feasibility of fuzzy solutions and solving approach in this study a generalized fuzzy linear programming gflp method is developed for dealing with uncertainties expressed as fuzzy sets the feasibility of fuzzy solutions of the gflp problem is investigated a stepwise interactive algorithm sia based on the idea of design of experiment is then advanced to solve the gflp problem this sia method was implemented through i discretizing membership grade of fuzzy parameters into a finite number of a cut levels ii converting the gflp model into an interval linear programming ilp submodel under every a cut level iii solving the ilp submodels through an interactive algorithm and obtaining the associated interval solutions iv acquiring the membership functions of fuzzy solutions through statistical regression methods a simple numerical example is then proposed to illustrate the solution process of the gflp model through sia a comparison between the solutions obtained though sia and monte carlo method is finally conducted to demonstrate the robustness of the sia method the results indicate that the membership functions for decision variables and objective function are reasonable and robust

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

post optimality analysis on the membership functions of a fuzzy linear programming problem models of linear programming problems with fuzzy constraints are very well known in the current literature in almost all cases to solve these problems linear membership functions are used because they have very good properties and are very easy to manipulate in some cases however because of the knowledge that the decision maker has such membership functions could be modeled as nonlinear although the complexity of the problem could increase this paper considers the use of nonlinear membership functions in fuzzy linear programming problems to show that the corresponding solution to be obtained can be derived from a parallel linear model moreover it is easier to solve than the nonlinear model making use of a similar procedure to that of post optimal analysis in classical linear programming the case in which these membership functions are defined by means of piecewise linear approximations is also considered and analyzed

fuzzy programming in multiobjective linear programming a parametric approach abstract zimmermann s fuzzy approach to the compromise solution concept in the multi objective linear programming problem is considered it is shown that given a class of membership functions of fuzzy goals assigned to objective functions in the problem wider than primarily proposed by zimmermann the use of classical linear programming methods to solve and analyse the problem is also possible as a result of application of the method based on the parametric programming technique one can obtain a fuzzy solution of the problem this solution is a certain fuzzy subset of the set of weakly efficient solutions of the problem the solution which belongs to this set to the highest degree is a maximizing one in bellman and zadeh s terminology it may also be obtained by use of the usual non parametric simplex method

sensitivity analysis in fuzzy multiobjective linear fractional programming problem in this paper we study measurement of sensitivity for changes of violations in the aspiration level for the fuzzy multiobjective linear fractional programming problem

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

river assimilative capacity analysis via fuzzy linear programming abstract the assimilative capacity of a river that is the so called allowable pollution loading in a river basin is a problem of water quality management involving optimization subject to constraints of standards of water quality and their models this study is an up to date evaluation of assimilative capacity in a river basin with consideration of the imprecision of the water environment i e water quality and river flowrate the purposes of this study in comparison with crisp linear programming lp is to select and evaluate four well known fuzzy linear programming flp approaches and to apply them to the assimilative vapacity of a river basin tou chen river basin in taiwan the results show the models of assimilative capacity of flp are better than those of a crisp one which reflect the flexible characteristics of the former approaches the emphases in this paper are on how to formulate problems of assimilative vapacity of a river according to flp models and on how to solve them according to algorithms of crisp lp furthermore the results of river analysis serve as criteria for regulatory agencies to implement the control of pollution sources and to accomplish finally the sustainable use of water resources

on solutions and distribution problems of the linear programming with fuzzy random variable coefficients abstract this paper studies the solution method and distribution problems for two kinds of linear programming with fuzzy random variable coefficients some concepts of solutions for example the fuzzy pseudo random optimization solution are introduced for these new programming problems we prove some equivalent theorems that transform the fuzzy random programming problems into a series of random programming problems by using the simplex method of linear programming with random variable coefficients we give the solution method of linear programming with fuzzy random variable coefficients the formulas of probability distribution function projective distribution function and expectation on these new programmings are presented

complete solutions of multiple objective transportation problems with possibilistic coefficients abstract this paper deals with the complete solutions of multiple objective transportation problems with possibilistic coefficients we consider the problem by incorporating possibilistic data into the coefficients of objectice functions a solution concept that is attractive from the standpoint of efficiency is specified a necessary and sufficient condition for such a solution is established a relationship between solutions of possibilistic levels is constructed the parametric analysis is used to decompose the parametric space of the equivalent problem a numerical example is given to illustrate the aspects of the developed results

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

performance evaluation of nonlinear optimization methods via pairwise comparison and fuzzy numbers in this paper we extend the deterministic performance evaluation of nonlinear optimization methods we carry out a pairwise comparison using fuzzy estimates of the performance ratios to obtain fuzzy final scores of the methods under consideration the key instrument is the concept of fuzzy numbers with triangular membership functions the algebraic operations on them are simple extensions of the operations on real numbers they are exact in the parameters lower modal and upper values not necessarily exact in the shape of the membership function we illustrate the fuzzy performance evaluation by the ranking and rating of five methods geometric programming and four general methods for solving geometric programming problems using the results of recent computational studies some general methods appear to be leading an outcome which is not only due to their performance under subjective criteria like domain of applications and conceptual simplicity of use they also score higher under more objective criteria like robustness and efficiency

optimality for fuzzified mathematical programming problems a parametric approach from a conventional mathematical programming model and in accordance with which fuzzification is used several models of fuzzy mathematical programming problems can be obtained this paper deals with the study of the optimality concept for g p fuzzified mathematical programming problems an auxiliary parametric mathematical programming problem is presented which allows the above model to be solved in a straightforward way in addition some results about the g p fuzzified mathematical programming problem are obtained using the parametric mathematical programming problem

derivation and application of an expert fuzzy optimal control system abstract this paper presents derivation of a fuzzy optimal controller which may be used to develop a knowledge based expert control system for supervising and tuning of conventional feedback control loops a general theory based on the conceptual framework for optimization in a fuzzy environment is presented for the development of linear and nonlinear fuzzy optimal control systems this technique is based on integration of the optimal control theorem and fuzzy variables the resulting expert controller is expressed in the form of linguistic fuzzy rules representing the knowledge base for tuning of the feedback control loops tuning inferences are made from the response of local feedback control loops by applying the compositional rule of inference to the knowledge base

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

quadratic programs with hollows let mathcal f be a quadratically constrained possibly nonconvex bounded set and let mathcal e 1 ldots mathcal e l denote ellipsoids contained in mathcal f with non intersecting interiors we prove that minimizing an arbitrary quadratic q cdot over mathcal g mathcal f setminus cup k 1 ell mathrm int mathcal e k is no more difficult than minimizing q cdot over mathcal f in the following sense if a given semidefinite programming sdp relaxation for min q x x in mathcal f is tight then the addition of l linear constraints derived from mathcal e 1 ldots mathcal e l yields a tight sdp relaxation for min q x x in mathcal g we also prove that the convex hull of x xx t x in mathcal g equals the intersection of the convex hull of x xx t x in mathcal f with the same l linear constraints inspired by these results we resolve a related question in a seemingly unrelated area mixed integer nonconvex quadratic programming

a fuzzy transportation algorithm this paper proposes an algorithm for solving transportation problems with fuzzy constraints and investigates the relationship between the algebraic structure of the optimum solution of the deterministic problem and its fuzzy equivalent

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

a concept of the optimal solution of the transportation problem with fuzzy cost coefficients abstract in the paper a definition of the optimal solution of the transportation problem with fuzzy cost coefficients as well as an algorithm determining this solution are proposed

an exponential membership function for fuzzy multiple objective linear programming abstract the operator min is one of the most frequently used aggregation operators in fuzzy decision however this operator is the softest operator and no allowance is made for any compensation the product and other operators some of them may be compensatory are seldom used because of the nonlinearity of the resulting problem in this paper an exponential instead of linear membership function is proposed the advantages of using exponential membership are two fold first the resulting problems can be transformed to linear ones when the product and several other nonlinear aggregate operators are used secondly exponential representation is more realistic than the linear ones usually used for some practical applications

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

fuzzy integer transportation problem an algorithm has been proposed which solves the transportation problem with fuzzy supply and demand values and the integrality condition imposed on the solution this algorithm is exact and computationally effective although the problem is formulated in the general way i e its fuzzy supply and demand values can differ from each other and be fuzzy numbers of any type

second order cone constraints for extended trust region subproblems the classical trust region subproblem trs minimizes a nonconvex quadratic objective over the unit ball in this paper we consider extensions of trs having extra constraints when two parallel cuts are added to trs we show that the resulting nonconvex problem has an exact representation as a semidenite program with additional linear and second order cone constraints for the case where an additional ellipsoidal constraint is added to trs resulting in the two trust region subproblem ttrs we provide a new relaxation including second order cone constraints that strengthens the usual sdp relaxation

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

a multi objective transportation problem under fuzziness this paper presents a fuzzy programming approach to determine the optimal compromise solution of a multi objective transportation problem motp its fuzzy membership function is defined the approach performance is tested by measuring the degree of closeness of the compromise solution to the ideal solution using a family of distance functions in addition the paper shows that the fuzzy approach outperforms the interactive procedure as the number of objectives and constraints increases two illustrative numerical examples are provided to illustrate the approach

a fuzzy approach to the transportation problem abstract the transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed for the solution of the problem the technique of parametric programming is used this makes it possible to obtain not only the maximizing solution according to the bellman zadeh criterion but also other alternatives close to the optimal solution

exactness of semidefinite relaxations for nonlinear optimization problems with underlying graph structure this work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems including quadratic and polynomial optimization problems the main objective of this paper is to investigate how the hidden structure of a given real complex valued optimization problem makes it easy to solve to this end three conic relaxations are proposed necessary and sufficient conditions are derived for the exactness of each of these relaxations and it is shown that these conditions are satisfied if the optimization problem is highly structured more precisely the structure of the optimization problem is mapped into a generalized weighted graph where each edge is associated with a weight set extracted from the coefficients of the optimization problem in the real valued case it is shown that the relaxations are all exact if each weight set is sign definite and in addition a condition is satisfied for each cycle of the graph it is also proved that if some of these conditi

fuzzy total transportation cost measures for fuzzy solid transportation problem abstract the solid transportation problem considers the supply the demand and the conveyance to satisfy the transportation requirement in a cost effective manner this paper develops a method that is able to derive the fuzzy objective value of the fuzzy solid transportation problem when the cost coefficients the supply and demand quantities and conveyance capacities are fuzzy numbers based on the extension principle the fuzzy solid transportation problem is transformed into a pair of mathematical programs that is employed to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level from different values of the membership function of the objective value is approximated since the objective value is fuzzy the values of the decision variables derived in this paper are fuzzy as well an example is illustrated for the whole idea

process planning in a fuzzy environment abstract mixed integer optimization models for chemical process planning typically assume that model parameters can be accurately predicted as precise forecasts are difficult to obtain process planning usually involves uncertainty and ambiguity in the data this paper presents an application of fuzzy programming to process planning the forecast parameters are assumed to be fuzzy with a linear or triangular membership function the process planning problem is then formulated in terms of decision making in a fuzzy environment with fuzzy constraints and fuzzy net present value goals the model is transformed to a deterministic mixed integer linear program or mixed integer nonlinear program depending on the type of uncertainty involved in the problem for the nonlinear case a global optimization algorithm is developed for its solution this algorithm is applicable to general possibilistic programs and can be used as an alternative to the commonly used bisection method illustrative examples and computational results for a petrochemical complex with 38 processes and 24 products illustrate the applicability of the developed models and algorithms

fuzzy linear programming based on statistical confidence interval and interval valued fuzzy set abstract in this paper we use statistical data and statistical confidence interval to derive level 1 fuzzy numbers 0 1 1 interval valued fuzzy numbers 0

linear programming with fuzzy random variable coefficients abstract linear programming with fuzzy random variable coefficients is introduced by discussing a practical engineering problem in order to study the solution of the linear programming with fuzzy random variable coefficients we discuss the simplex algorithm for linear programming with random variable coefficients furthermore the solution and distribution problem of this new fuzzy random programming are studied

new method for solving fuzzy transportation problems with lr flat fuzzy numbers transportation problems have various applications in logistics and supply chains for reducing costs in real life situations the parameters of transportation problems may not be known precisely because of uncontrollable factors herein we propose a new method for solving fuzzy transportation problems ftps in which the transportation costs and supply and demand are represented by non negative lr flat fuzzy numbers the ftp is converted into four transportation problems which are solved using standard transportation simplex algorithms the advantages of the proposed method over existing methods are discussed in the context of two application examples the results show that the proposed method is simpler and computationally more efficient than existing methods in the literature

the use of parametric programming in fuzzy linear programming in this paper the possibility of the identification of a complete fuzzy decision not only the maximizing alternative in fuzzy linear programming by use of the parametric programming technique is presented also it is shown that this fact can be useful in the zimmermann approach to multiple objective linear programming the presented remarks are illustrated by some numerical examples

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

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

solving fuzzy transportation problems based on extension principle abstract transportation models play an important role in logistics and supply chain management for reducing cost and improving service this paper develops a procedure to derive the fuzzy objective value of the fuzzy transportation problem in that the cost coefficients and the supply and demand quantities are fuzzy numbers the idea is based on the extension principle a pair of mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level from different values of the membership function of the objective value is constructed two different types of the fuzzy transportation problem are discussed one with inequality constraints and the other with equality constraints it is found that the membership function of the objective value of the equality problem is contained in that of the inequality problem since the objective value is expressed by a membership function rather than by a crisp value more information is provided for making decisions

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

characterizations bounds and probabilistic analysis of two complexity measures for linear programming problems this note studies a a condition number used in the linear programming algorithm of vavasis and ye 14 whose running time depends only on the constraint matrix a m n and a a variant of another condition number due to ye 17 that also arises in complexity analyses of linear programming problems we provide a new characterization of a and relate a and a furthermore we show that if a is a standard gaussian matrix then e ln a o min mlnn n thus the expected running time of the vavasis ye algorithm for linear programming problems is bounded by a polynomial in m and n for any right hand side and objective coefficient vectors when a is randomly generated in this way as a corollary of the close relation between a and a we show that the same bound holds for e ln a

applications of fuzzy set theory to mathematical programming mathematical programming is one of the areas to which fuzzy set theory has been applied extensively primarily based on bellman and zadeh s model of decision in fuzzy environments models have been suggested which allow flexibility in constraints and fuzziness in the objective function in traditional linear and nonlinear programming in integer and fractional programming and in dynamic programming these models in turn have been used to offer computationally efficient approaches for solving vector maximum problems this paper surveys major models and theories in this area and offers some indication on future developments which can be expected

distribution planning decisions using interactive fuzzy multi objective linear programming this work develops an interactive fuzzy multi objective linear programming i fmolp method for solving the fuzzy multi objective transportation problems with piecewise linear membership function the proposed i fmolp method aims to simultaneously minimize the total distribution costs and the total delivery time with reference to fuzzy available supply and total budget at each source and fuzzy forecast demand and maximum warehouse space at each destination additionally the proposed method provides a systematic framework that facilitates the fuzzy decision making process enabling a decision maker dm to interactively modify the fuzzy data and related parameters until a set of satisfactory solutions is obtained an industrial case is presented to demonstrate the feasibility of applying the proposed method to real transportation problems consequently the proposed method yields an efficient solution and overall degree of dm satisfaction with the determined objective values especially several significant management implications regarding the practical application of the proposed i fmolp method are presented

a new graph parameter related to bounded rank positive semidefinite matrix completions the gram dimension mathrm gd g of a graph g is the smallest integer k ge 1 such that any partial real symmetric matrix whose entries are specified on the diagonal and at the off diagonal positions corresponding to edges of g can be completed to a positive semidefinite matrix of rank at most k assuming a positive semidefinite completion exists for any fixed k the class of graphs satisfying mathrm gd g le k is minor closed hence it can be characterized by a finite list of forbidden minors we show that the only minimal forbidden minor is k k 1 for k le 3 and that there are two minimal forbidden minors k 5 and k 2 2 2 for k 4 we also show some close connections to euclidean realizations of graphs and to the graph parameter nu g of van der holst combinatorica 23 4 633 651 2003 in particular our characterization of the graphs with mathrm gd g le 4 implies the forbidden minor characterization of the 3 realizable graphs of belk discret comput geom 37 139 162 2007 and belk and connelly discret comput geom 37 125 137 2007 and of the graphs with nu g le 4 of van der holst combinatorica 23 4 633 651 2003

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

the complete decision set of the generalized symmetrical fuzzy linear programming problem abstract the generalized symmetrical fuzzy linear programming sflp problem is defined in this paper as involving both fuzzy and nonfuzzy constraints of which fuzzy constraints are represented by linear and nonlinear membership functions fuzzy intervals are proposed to represent fuzzy equality constraints the general form of fuzzy constraints is proposed by defining the membership function d as an explicit function of min z rather than an explicit function of the solution set x the complete solution set and its existence criterion are studied the improvement of objective values based on the obtained complete solution set is also discussed a truck fleet problem presented in the literature is cited to explain the proposed method and to prove the correctness of the solution

interactive fuzzy linear programming abstract in this study an interactive fuzzy linear programming approach is investigated to improve the flexibility and robustness of linear programming lp technique this paper provides a practical modelling method which is a symmetric integration of zimmermann s werners s vergegay s and chanas s fuzzy lp approaches and additionally provides a decision support system for solving a specific domain of practical lp problems the interactive concept provides a learning process about the system whereby the decision maker can learn to recognize good solutions relative importance of factors in the system and then design a high productivity system instead of optimizing a given system this interactive fuzzy lp system provides integration oriented adaptation and learning features by considering all the possibilities of a specific domain of lp problems which are integrated in logical order using an if then rule

a soft computing approach to big data summarization abstract the added value of a dataset lies in the knowledge a domain expert can extract from it considering the continuously increasing volume and velocity of these datasets efficient tools have to be defined to generate meaningful condensed and human interpretable representations of big datasets in the proposed approach soft computing techniques are used to define an interface between the numerical and categorical space of data definition and the linguistic space of human reasoning based on the expert s own vocabulary about the data a personal summary composed of linguistic terms is efficiently generated and graphically displayed as a term cloud offering a synthetic view of the data properties using dedicated indexing strategies linking data and their subjective linguistic rewritings exploration functionalities are provided on top of the summary to let the user browse the data experimentations confirm that the space change operates in linear time wrt the size of the dataset making the approach tractable on large scale data

a gentle geometric introduction to copositive optimization this paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way we focus on examples having just a few variables or a few constraints for which the quadratic problem can be formulated as a copositive style problem which itself can be recast in terms of linear second order cone and semidefinite optimization a particular highlight is the role played by the geometry of the feasible set

a new approach to uncertain parameter linear programming abstract a novel method for solving uncertain linear programming is put forward in this paper in contrast to stochastic linear programming and most fuzzy linear programming models the parameters of the problem discussed here are neither random variables with known probability distributions nor fuzzy parameters with known possibility distributions all we know about the parameters is merely their intervals of existence to solve this kind of uncertain linear programming might be very attractive since many actual decision problems could be formulated as such if we do not have enough information about the parameters in this paper the concept of a confidence function for inequalities is introduced based on this definition the problem is solved by an iterative algorithm then the convergence of the algorithm is proved and the decision risk grades for different confidence functions are discussed finally an illustrative example is given to demonstrate the actual application of thismethod

the fuzzy shortest path problem and its most vital arcs abstract the shortest path problem is to find the shortest distance between two specified nodes in a network an arc is called a single most vital arc in the network if its removal from the network results in the greatest increase in the shortest distance the availability of arcs plays a key role in network problems the most vital arcs problems provide a means by which the importance of arc s availability can be measured in the traditional shortest path problem the arc lengths are assumed to be crisp numbers in this paper we consider the case that the arc lengths are fuzzy numbers in particular we derive the membership function of the shortest distance by using a fuzzy linear programming approach based on this result we then propose an algorithm for finding the single most vital arc in a network

polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems we show that a particular pivoting algorithm which we call the lexicographic lemke algorithm takes an expected number of steps that is bounded by a quadratic inn when applied to a random linear complementarity problem of dimensionn we present two probabilistic models both requiring some nondegeneracy and sign invariance properties the second distribution is concerned with linear complementarity problems that arise from linear programming in this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem and independent of the larger similar results have been obtained by adler and megiddo

approximating quadratic programming with bound and quadratic constraints we consider the problem of approximating the global maximum of a quadratic program qp subject to bound and simple quadratic constraints based on several early results we show that a 4 7 approximate solution can be obtained in polynomial time

fuzzy partitions a way to integrate expert knowledge into distance calculations this work proposes a new pseudo metric based on fuzzy partitions fps this pseudo metric allows for the introduction of expert knowledge into distance computations performed on numerical data and can be used in various types of statistical clustering or other applications the knowledge is formalized by a fp in which each fuzzy set represents a linguistic concept the pseudo metric is designed to respect the fp semantics the univariate case is first studied and the pseudo metric behavior is discussed using synthetic experiments then a multivariate version is proposed as a minkowski like combination of univariate distances or semi distances the value of the proposal is illustrated with two real world case studies in the fields of biology and precision agriculture

linear programming with multiple fuzzy goals abstract this paper describes the use of fuzzy set theory in goal programming gp problems in particular it is demonstrated how fuzzy or imprecise aspirations of the decision maker dm can be quantified through the use of piecewise linear and continuous functions models are then presented for the use of fuzzy goal programming with preemptive priorities with archimedean weights and with the maximization of the membership function corresponding to the minimum goal examples are also provided

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc

a method to defuzzify the fuzzy number transportation problem application in many problems of transportation engineering and planning we encounter situations in which the observed or derived values of the variables are approximate yet the variables themselves must satisfy a set of rigid relationships dictated by physical principle when the observed values do not satisfy the relationships each value is adjusted until they satisfy the relationship we propose a simple adjustment method that finds the most appropriate set of crisp numbers the method assumes that each observed value is an approximate number or a fuzzy number and the true value is found in the support of the membership function for each of many possible sets of values that satisfy the relationships the lowest membership grade is checked and the set whose lowest membership grade is the highest is chosen as the best set of values for the problem this process is performed using the fuzzy linear programming method the paper presents the model the computational process and applications