binary behavior of fuzzy programming with piecewise linear membership functions the nature of vagueness imprecision and uncertainly is fuzzy rather than crisp and or random especially for a multiple objectives decision making problem a key component of fuzzy programming is the membership function that represents a mathematical expression of level function for the decision maker s preference in fact a decision making problem involves the achievement of fuzzy goals some of which are met while others are not because these fuzzy goals are subject to real world constraints to represent this situation the binary piecewise linear membership function is then employed in order to solve the problem we propose a new idea of how to formulate the binary piecewise linear membership function the formulated problem can be easily solved using common integer programming packages in addition an illustrative example is included for demonstrating the usefulness of the proposed model finally the analytical superiority of the proposed method in terms of the execution time can be seen through a computation experiment conducted on a set of generated test examples

fuzzy linear programming and applications abstract this paper presents a survey on methods for solving fuzzy linear programs first lp models with soft constraints are discussed then lp problems in which coefficients of constraints and or of the objective function may be fuzzy are outlined pivotal questions are the interpretation of the inequality relation in fuzzy constraints and the meaning of fuzzy objectives in addition to the commonly applied extended addition based on the min operator and used for the aggregation of the left hand sides of fuzzy constraints and fuzzy objectives a more flexible procedure based on yager s parametrized t norm t p is presented finally practical applications of fuzzy linear programs are listed

comments on fuzzy programming with nonlinear membership functions abstract yang et al in their paper fuzzy programming with nonlinear membership functions published in fuzzy sets and systems 41 1991 declared that their model can solve a fuzzy program with an s shaped membership function by adding only one 0 1 variable this paper indicates that their declaration is correct only for a specific type of s shape membership functions we propose another model to treat the fuzzy programs which cannot be solved effectively by yang et al

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

fuzzy potential solutions in multi criteria and multi constraint level linear programming problems abstract given a multi criteria and multi constraint level mc 2 linear programming problem we can use an mc 2 simplex method to find a set of potential solutions each potential solution optimizes the mc 2 problem under a certain range of decision parameters namely the criteria weight vector and the constraint level weight vector in this paper we extend the potential solution to a fuzzy potential solution for a given fuzzy mc 2 problem with a certain range of the decision parameters a membership function of the mc 2 problem is first constructed then for such a parameter range a fuzzy potential solution is determined in terms of the membership function and a primal potential basis based on the theoretical analysis we propose a heuristic algorithm to systematically and effectively locate all fuzzy potential solutions for all possible changes of the decision parameters a numerical example is used to illustrate the algorithm

membership function as an evaluation abstract after giving an overview of the different kinds of mathematical forms of the membership functions we extract the different demands and determine the rational class of the membership functions from the generalisation of our result we can determine it giving four parameters the internal a b the sharpness and the decision level we show the correspondence of this membership function to the zimmermann zysno model and using their empirical data we also test our mathematical model we also show the connections between the operators of the evaluation and membership functions which gives the generalisation of this concept in a more general form as a special case we can get a linear form

generalization of yang et al s method for fuzzy programming with piecewise linear membership functions li and yu fuzzy sets and systems 101 1999 109 argued that the model of yang et al for fuzzy programming fuzzy sets and systems 41 1991 39 is correct only for a specific type of piecewise linear membership function and proposed their model for other type of membership functions this study generalizes the model of yang et al a numerical example indicates that the model of li and yu is inapplicable to a fuzzy programming problem that involves more than one membership function whereas the proposed model gives the optimal solution

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

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

solving fuzzy inequalities with piecewise linear membership functions this paper deals with systems of fuzzy inequalities it shows that a system of fuzzy inequalities with piecewise linear membership functions can be converted to a one constraint nonlinear programming problem by employing the concepts of surrogate constraints and maximum entropy an augmented lagrangean algorithm is then applied to solve the resulting problem some computational results are included

pareto optimal solutions in fuzzy multi objective linear programming the problem of solving multi objective linear programming problems by assuming that the decision maker has fuzzy goals for each of the objective functions is addressed several methods have been proposed in the literature in order to obtain fuzzy efficient solutions to fuzzy multi objective programming problems in this paper we show that in the case that one of our goals is fully achieved a fuzzy efficient solution may not be pareto optimal and therefore we propose a general procedure to obtain a non dominated solution which is also fuzzy efficient two numerical examples illustrate our procedure

linear programming with fuzzy objectives abstract this paper presents a new method for solving linear programming problems with fuzzy parameters in the objective function to determine a compromise solution the authors do not reduce the set of infinitely many objective functions to a single compromise objective function as done in classical deterministic or stochastic procedures here the problem is only reduced to a few extreme objective functions the information contained in the membership functions can be used to any extent by a method called level related pair formation moreover if different level sets are considered the stability of the optimal solution can be tested

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

binary fuzzy goal programming goal programming is an important technique for solving many decision management problems fuzzy goal programming involves applying the fuzzy set theory to goal programming thus allowing the model to take into account the vague aspirations of a decision maker using preference based membership functions we can define the fuzzy problem through natural language terms or vague phenomena in fact decision making involves the achievement of fuzzy goals some of them are met and some not because these goals are subject to the function of environment resource constraints thus binary fuzzy goal programming is employed where the problem cannot be solved by conventional goal programming approaches this paper proposes a new idea of how to program the binary fuzzy goal programming model the binary fuzzy goal programming model can then be solved using the integer programming method finally an illustrative example is included to demonstrate the correctness and usefulness of the proposed model

taylor series approach to fuzzy multiobjective linear fractional programming this paper presents the use of a taylor series for fuzzy multiobjective linear fractional programming problems fmolfp the taylor series is a series expansion that a representation of a function in the proposed approach membership functions associated with each objective of fuzzy multiobjective linear fractional programming problem transformed by using a taylor series are unified thus the problem is reduced to a single objective practical applications and numerical examples are used in order to show the efficiency and superiority of the proposed approach

fuzzy goal programming a parametric approach narasimhan incorporated fuzzy set theory within goal formulation in 1980 since then much research has been performed in this field and various models for solving fuzzy goal programming have been proposed one of the well known models was proposed by tiwari et al in 1987 19 where an additive model was proposed this paper is an extension to the tiwari et al model that deals with the sum of weighted negative deviations between the desirable achievement degree and the common target here properties of the model are proposed a numerical example is also given to illustrate the approach

bector chandra type duality in fuzzy linear programming with exponential membership functions in this paper we study a pair of fuzzy primal dual linear programming problems and calculate duality results using an aspiration level approach we use an exponential membership function which is in contrast to the earlier works that relied on a linear membership function as the fuzzy environment causes a duality gap we investigate how choosing the exponential membership function impacts this gap this issue is particularly important for fuzzy linear programming where in general the primal and dual objective values may not be bounded

an interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters abstract this paper presents an interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy numbers on the basis of the level sets of the fuzzy numbers the concept of multiobjective nonlinear programming and local m pareto optimality is introduced through interaction with the decision maker dm the fuzzy goals of the dm for each of the objective functions in multiobjective nonlinear programming are quantified by eliciting the corresponding membership functions after determining the membership functions in order to generate a candidate for the satisficing solution which is also locally m pareto optimal if the dm specified the degree of the level sets and the reference membership values the augmented minimax problem is solved and the dm is supplied with the corresponding local m pareto optimal solution together with the trade off rates among the values of the membership functions and the degree then by considering the current values of the membership functions and as well as the trade off rates the dm responds by updating his reference membership values and or the degree in this way the local satisfying solution for the dm can be derived efficiently from among an m pareto optimal solution set based on the proposed method an interactive computer program has been written and an illustrative numerical example is given along with the corresponding computer outputs

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

decomposition with branch and cut approaches for two stage stochastic mixed integer programming decomposition has proved to be one of the more effective tools for the solution of large scale problems especially those arising in stochastic programming a decomposition method with wide applicability is benders decomposition which has been applied to both stochastic programming as well as integer programming problems however this method of decomposition relies on convexity of the value function of linear programming subproblems this paper is devoted to a class of problems in which the second stage subproblem s may impose integer restrictions on some variables the value function of such integer subproblem s is not convex and new approaches must be designed in this paper we discuss alternative decomposition methods in which the second stage integer subproblems are solved using branch and cut methods one of the main advantages of our decomposition scheme is that stochastic mixed integer programming smip problems can be solved by dividing a large problem into smaller mip subproblems that can be solved in parallel this paper lays the foundation for such decomposition methods for two stage stochastic mixed integer programs

brief paper finite time multi agent deployment a nonlinear pde motion planning approach the systematic flatness based motion planning using formal power series and suitable summability methods is considered for the finite time deployment of multi agent systems into planar formation profiles along predefined spatial temporal paths thereby a distributed parameter setting is proposed where the collective leader follower agent dynamics is modeled by two boundary controlled nonlinear time varying pdes governing the motion of an agent continuum in the plane the discretization of the pde model directly induces a decentralized communication and interconnection structure for the multi agent system which is required to achieve the desired spatial temporal paths and deployment formations

a solution algorithm for fuzzy linear programming with piecewise linear membership functions abstract this paper deals with the fuzzy linear program with continuous piecewise linear membership functions a fuzzy linear program with piecewise linear membership functions has been considered by hannan and nakamura hannan s method can solve the problems only when all membership functions are concave in the range 0 1 and nakamura s method needs to use the linear programming technique repetitionally we propose a technique to solve the problem using a standard linear programming when membership functions are strictly quasiconcave and the minimum operator is adopted for aggregating fuzzy goals the repetitional use of linear programming technique is not required in our method

satisfying optimization method based on goal programming for fuzzy multiple objective optimization problem this paper proposes a satisfying optimization method based on goal programming for fuzzy multiple objective optimization problem the aim of this presented approach is to make the more important objective achieving the higher desirable satisfying degree for different fuzzy relations and fuzzy importance the reformulated optimization models based on goal programming is proposed not only the satisfying results of all the objectives can be acquired but also the fuzzy importance requirement can be simultaneously actualized the balance between optimization and relative importance is realized we demonstrate the efficiency flexibility and sensitivity of the proposed method by numerical examples

a fuzzy goal programming approach to multi objective optimization problem with priorities abstract in goal programming problem the general equilibrium and optimization are often two conflicting factors this paper proposes a generalized varying domain optimization method for fuzzy goal programming fgp incorporating multiple priorities according to the three possible styles of the objective function the varying domain optimization method and its generalization are proposed this method can generate the results consistent with the decision maker dm s expectation that the goal with higher priority may have higher level of satisfaction using this new method it is a simple process to balance between the equilibrium and optimization and the result is the consequence of a synthetic decision between them in contrast to the previous method the proposed method can make that the higher priority achieving the higher satisfactory degree to get the global solution of the nonlinear nonconvex programming problem resulting from the original problem and the varying domain optimization method the co evolutionary genetic algorithms gas called genocopiii is used instead of the sqp method in this way the dm can get the optimum of the optimization problem we demonstrate the power of this proposed method by illustrative examples

an approach for solving fuzzy implicit variational inequalities with linear membership functions in this paper we consider a class of fuzzy implicit variational inequalities with linear membership functions by using the tolerance approach we show that solving such problems can be reduced to a semi infinite programming problem a version of the method of centres with entropic regularization techniques only used a quasi newton line search using matlab software is required in our implementation we also give a numerical example to illustrate the validity of our approach

fuzzy multicriteria integer programming via fuzzy generalized networks recent advances in network based mathematical programming methods have given rise to practical economical and computationally efficient solution methodologies for truly large scale problems today a wide variety of real world problems have been solved in such a manner even though on the surface they do not have an apparent network structure one type of network the generalized network has been successfully applied to solve large scale zero one or mixed integer mathematical programming models however one drawback to such approaches is their typical focus on but a single objective the purpose of this paper then is to document the results of an ongoing research effort which combines the generalized network concept with the technique of fuzzy programming the resultant hybrid approach has been found to provide a computationally efficient approach to multi objective zero one or integer programming problems

on finding compromise solutions in multicriteria problems using the fuzzy min operator abstract this paper presents an application of fuzzy approaches to the linear vectormaximum problem it shows that using the fuzzy min operator together with linear as well as special nonlinear membership functions the obtained solutions are always compromise solutions of the original multicriteria problem

flatness based control of a nonlinear chemical reactor model the nonlinear model of a continuous stirred tank reactor is shown to be flat the flatness permits the design of suitable trajectories on the basis of the explicit stationary solution and the tracking of these trajectories asymptotically using quasi static state feedback linearization a nonlinear local observer with a time varying gain is designed allowing the realisation of the state feedback in the case of partial measurement of the state simulation results illustrate the tracking behavior of the closed loop with the observer

fuzzy goal programming with different importance and priorities abstract this paper formulates fuzzy goal programming fgp incorporating different importance and preemptive priorities by using an additive model to maximize the sum of achievement degrees of all fuzzy goals in contrast to previous works the proposed approach allows the decision maker to determine a desirable achievement degree for each fuzzy goal to reflect explicitly the relative importance of these goals this approach can generate a set of achievement degrees consistent with the decision maker s expectations even though the relative importance of the goals may change furthermore in this paper we incorporate the decision maker s preemptive priority structure into a single formulation the resulting solutions satisfy both the preemptive priority structure and have the maximum achievement degrees in sum the proposed approaches effectiveness and computational superiority over the existing approaches are demonstrated and compared with examples from the literature

spline membership function and its application in multiple objective fuzzy control problem abstract in fuzzy decision theory first introduced by bellman and zadeh in 1970 the use of linear membership functions is well established the rate of change in the degree of membership need not in general be constant particularly in time dependent problems or situations where it is considered as a criterion the idea of normalized spline membership function and normalized basis spline membership function is introduced here it is observed that the fuzzy decision operator corresponding to the fuzzy alternatives can be expressed in terms of these proposed membership functions a new method of solution in a multistage form is discussed for the general non linear multiple objective fuzzy control problem the application of normalized spline membership function in the multiple objective fuzzy control problem is illustrated with a numerical example

an interactive fuzzy satisficing method for multiobjective linear fractional programming problems abstract we present a new interactive fuzzy decision making method for solving multiobjective linear fractional programming problems by assuming that the decision maker dm has fuzzy goals for each of the objective functions through the interaction with the dm the fuzzy goals of the dm are quantified by eliciting the corresponding membership functions including nonlinear functions after determining the membership functions if the dm specifies reference membership values the minimax problem is solved by combined use of the bisection method and the linear programming method and the dm is supplied with the corresponding pareto optimal solution together with the trade off rates between the membership functions then by considering the current values of the membership functions as well as the trade off rates the dm responds by updating his her reference membership values in this way the compromise or satisficing solution for the dm can be derived efficiently from among a pareto optimal solution set by updating his her reference membership values on the basis of the proposed method a time sharing computer program is written and an illustrative numerical example is demonstrated along with the computer outputs

flatness based trajectory planning for diffusion reaction systems in a parallelepipedon a spectral approach spectral analysis is considered for the flatness based solution of the trajectory planning problem for a boundary controlled diffusion reaction system defined on a 1 r dimensional parallelepipedon by exploiting the riesz spectral properties of the system operator it is shown that a suitable reformulation of the resolvent operator allows a systematic introduction of a basic output which yields a parametrization of both the system state and the boundary input in terms of differential operators of infinite order their convergence is verified for both infinite dimensional and finite dimensional actuator configurations by restricting the basic output to certain gevrey classes involving non analytic functions with this a systematic approach is introduced for basic output trajectory assignment and feedforward tracking control towards the realization of finite time transitions between stationary profiles

flatness based boundary control of a class of quasilinear parabolic distributed parameter systems motion planning and control of a boundary controlled quasilinear parabolic partial differential equation in one spatial variable is considered the approach relies on a flatness property of the system namely that the system solution can be differentially parameterized in terms of a flat output which in the case considered is a boundary value such a parameterization allows straightforward motion planning and computation of a control law the approach is based on power series in the spatial variable and the convergence of these series is ensured by choosing the flat output to be a nonanalytic smooth function of appropriate gevrey class

an efficient method combining active learning kriging and monte carlo simulation for profust failure probability abstract for more and more complicated engineering structures it is a challenge to efficiently estimate the profust failure probability based on the probability inputs and fuzzy state assumption by combining active learning kriging with monte carlo simulation ak mcs an efficient method is proposed to estimate the profust failure probability firstly the profust failure probability is transformed into an integral of the classical failure probability by introducing a variable related to the fuzzy state assumption this integral is further reorganized as a weighted sum of a series of classical failure probabilities by gaussian quadrature and the series of the classical failure probabilities have the similar limit state function constructions constrained by different thresholds secondly mcs is used according to the probability input distribution to generate the sample pool in which the active learning kriging is used to establish the surrogates of the series of similar limit state functions with different thresholds an improved learning function is proposed by minimizing the u learning function minima corresponding to all limit state functions so that the candidate with the largest effect on the surrogating quality of all limit states can be selected as a training point to update the kriging model once the updating process of the kriging model converges all limit state functions can be identified by the kriging model and the profust failure probability can be estimated by using the kriging model without any extra model evaluation several examples are used to demonstrate the feasibility of the proposed strategy for estimating the profust failure probability

multi choice goal programming with utility functions abstract goal programming gp has been and still is the most widely used technique for solving multiple criteria decision problems and multiple objective decision problems by finding a set of satisfying solutions however the major limitation of goal programming is that can only use aspiration levels with scalar value for solving multiple objective problems in order to solve this problem multi choice goal programming mcgp was proposed by chang 2007a following the idea of mcgp this study proposes a new concept of level achieving in the utility functions to replace the aspiration level with scalar value in classical gp and mcgp for multiple objective problems according to this idea it is possible to use the skill of mcgp with utility functions to solve multi objective problems the major contribution of using the utility functions of mcgp is that they can be used as measuring instruments to help decision makers make the best appropriate policy corresponding to their goals with the highest level of utility achieved in addition the above properties can improve the practical utility of mcgp in solving more real world decision management problems

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

a closed form representation of mixed integer program value functions we identify a class of formulas computable in polynomial time such that the functions defined by these formulas are precisely the value functions of mixed integer programs with rational constraint coefficients

genetic algorithms for portfolio selection problems with minimum transaction lots abstract conventionally portfolio selection problems are solved with quadratic or linear programming models however the solutions obtained by these methods are in real numbers and difficult to implement because each asset usually has its minimum transaction lot methods considering minimum transaction lots were developed based on some linear portfolio optimization models however no study has ever investigated the minimum transaction lot problem in portfolio optimization based on markowitz model which is probably the most well known and widely used based on markowitz model this study presents three possible models for portfolio selection problems with minimum transaction lots and devises corresponding genetic algorithms to obtain the solutions the results of the empirical study show that the portfolios obtained using the proposed algorithms are very close to the efficient frontier indicating that the proposed method can obtain near optimal and also practically feasible solutions to the portfolio selection problem in an acceptable short time one model that is based on a fuzzy multi objective decision making approach is highly recommended because of its adaptability and simplicity

fuzzy goal programming with multiple priorities via generalized varying domain optimization method this paper proposes a generalized varying domain optimization method for fuzzy goal programming incorporating multiple priorities according to the three possible styles of the objective function the varying domain optimization method and its generalization are corresponding proposed in contrast to the previous method the proposed method can make that the higher priority achieving the higher satisfaction degree in this way the decision maker can get the optimal solution as well as guarantee the priorities of the multiple objective optimization problem we demonstrate the power of this proposed method by three illustrative examples and a practice application

the value function of an integer program we consider integer programs in which the objective function and constraint matrix are fixed while the right hand side varies the value function gives for each feasible right hand side the criterion value of the optimal solution we provide a precise characterization of the closed form expression for any value function

some extensions of fuzzy linear programming abstract in the formulation of fuzzy linear programming proposed by zimmermann it is assumed that fuzzy goals and fuzzy constraints are explicitly given as fuzzy sets on appropriate real lines respectively and are actually characterized by membership functions in linear form for relaxing those assumptions two kinds of extensions are derived in this paper in the first extension it is shown that fuzzy goals and fuzzy constraints expressed as fuzzy relations with fuzzy parameters can be treated as fuzzy sets on different real lines under some assumptions in the second extension it is shown that optimization in the case where the membership functions of the fuzzy goals and the fuzzy constraints are given in piecewise linear form can be achieved by using a standard linear programming technique repetitionally

an approximation approach for representing s shaped membership functions in general to formulate a fuzzy linear programming problem with n s shaped utility membership functions traditional methods require n or more extra binary variables because s shaped curves are neither convex nor concave in all places adding binary variables does not improve the bound of the linear programming relaxation on the contrary added binary variables increase the computational burden in the solution process if problems get large therefore a formulation without binary variables should be more efficient accordingly this study proposes a piecewise linear approach to formulate an s shaped membership function mf without adding any extra binary variables which improves the efficiency of fuzzy linear programming in solving decision management problems with s shaped mfs finally a computational experiment is provided to demonstrate the superiority of the proposed models an illustrative example is also provided to show the usefulness of the proposed method

nearest piecewise linear approximation of fuzzy numbers abstract the problem of the nearest approximation of fuzzy numbers by piecewise linear 1 knot fuzzy numbers is discussed by using 1 knot fuzzy numbers one may obtain approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study they might be also perceived as a generalization of the trapezoidal approximations moreover these approximations possess some desirable properties apart from theoretical considerations approximation algorithms that can be applied in practice are also given

a fuzzy multiobjective program with quasiconcave membership functions and fuzzy coefficients abstract a method is proposed for solving a fuzzy multi objective linear programming problem fmp with quasiconcave membership functions and fuzzy coefficients the proposed method first expresses a piecewise function as the summation of absolute terms then we search for the interval where the optimal solution is allocated by finding the corresponding points with same value of membership functions after that the problem is solved by goal programming techniques comparing with other fmp methods the proposed method does not need to add extra zero one variables to divide the original problem into several sub problems or transforming all original quasiconcave functions into concave functions in addition the proposed method could solve a fmp problem with fuzzy coefficients to obtain a solution closing to a global optimum

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

a weighted max min model for fuzzy goal programming abstract after narasimhan s pioneering study of applying fuzzy set theory to goal programming in 1980 many achievements in the field have been recorded most of them followed the max min approach however when objectives have different levels of importance only the weighted additive model of tiwari et al seems to be applicable however the shortcoming of the additive model is that the summation of quasiconcave functions may not be quasiconcave this study proposes a novel weighted max min model for fuzzy goal programming fgp and for fuzzy multiple objective decision making the proposed model adapts well to even the most complicated membership functions numerical examples demonstrate that the proposed model can be effectively incorporated with other approaches to fgp and is superior to the weighted additive approach

a goal programming procedure for solving problems with multiple fuzzy goals using dynamic programming abstract this paper describes how the preemptive priority based goal programming gp can be used to solve a class of fuzzy programming fp problems with the characteristics of dynamic programming dp in the proposed approach the membership functions of the objective goals of a problem with fuzzy aspiration levels are defined first then under the framework of preemptive priority based gp a multi stage dp model of the problem for achievement of the highest degree unity of each of the membership functions is developed in the decision process the goal satisficing philosophy of gp is used recursively to arrive at the most satisfactory solution two numerical examples are provided to illustrate the approach

an interactive satisfying method based on alternative tolerance for multiple objective optimization with fuzzy parameters an interactive satisfying method based on alternative tolerance is presented for the multiple objective optimization problem with fuzzy parameters using the alpha level sets of the fuzzy numbers all the objectives are modeled as the fuzzy goals and the tolerances of the objectives are iteratively changed according to a decision maker for a satisfying solution via a specific attainable point programming model the membership functions can be modified and then a lexicographic two phase programming procedure is constructed correspondingly to find the final solution in a special case the objective constraint is added instead of changing the membership functions therefore the dissatisfying objectives for the decision maker can be improved step by step the presented method not only acquires the alpha pareto optimal or weak alpha pareto optimal solution of the fuzzy multiple objective optimization but also satisfies the progressive preference of the decision maker a numerical example shows its power

a method for solving fuzzy goal programming problems based on minmax approach abstract narasimhan incorporated fuzzy set theory within goal programming formulation in 1980 since then numerous research has been carried out in this field one of the well known models for solving fuzzy goal programming problems was proposed by hannan in 1981 in this paper the conventional minmax approach in goal programming is applied to solve fuzzy goal programming problems it is proved that the proposed model is an extension to hannan model that deals with unbalanced triangular linear membership functions in addition it is shown that the new model is equivalent to a model proposed in 1991 by yang et al moreover a weighted model of the new approach is introduced and is compared with kim and whang s model presented in 1998 a numerical example is given to demonstrate the validity and strengths of the new models

two step interactive satisfactory method for fuzzy multiple objective optimization with preemptive priorities this paper presents a two step interactive satisfactory optimization method for fuzzy multiple objective optimization with preemptive priorities in contrast to previous works this proposed approach guarantees the order of satisfactory degrees consistent with priorities the decision maker not only acquires satisfactory solution of all the objectives but also realizes the preemptive priority requirement among them the originally complex optimization problem is simplified and divided into two subproblems that are solved in sequence numerical examples and actual application show the proposed method s effectiveness flexibility and efficiency in comparison with results from the literature

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

solving a system of infinitely many fuzzy inequalities with piecewise linear membership functions abstract a cutting plane type algorithm for solving a system of infinitely many fuzzy inequalities with piecewise linear membership functions is proposed in each iteration we solve a finite nonlinear optimization problem and add one or two more constraints the iterative process ends when an optimal solution is identified a convergence proof under some mild conditions is given an efficient implementation based on the concepts of constraint surrogation and maximum entropy is included some computational results are also reported

flatness of semilinear parabolic pdes a generalized cauchy kowalevski approach a generalized cauchy kowalevski approach is proposed for flatness based trajectory planning for boundary controlled semilinear systems of partial differential equations pdes in a one dimensional spatial domain for this the ansatz presented in trajectory planning for boundary controlled parabolic pdes with varying parameters on higher dimensional spatial domains t meurer and a kugi ieee trans autom control vol 54 no 8 pp 1854 1868 aug 2009 using formal integration is generalized towards a unified design framework which covers linear and semilinear pdes including rather broad classes of nonlinearities arising in applications in addition an efficient semi numerical solution of the implicit state and input parametrizations is developed and evaluated in different scenarios simulation results for various types of nonlinearities and a tubular reactor model described by a system of semilinear reaction diffusion convection equations illustrate the applicability of the proposed method

an additive fuzzy programming model for multiobjective transportation problem abstract all existing methods generate a set of nondominated solutions or construct a single compromise solution for the linear multiobjective transportation problem in which objectives are equally important in this paper an additive fuzzy programming model for the multiobjective transportation problem is presented the method aggregates the membership functions of the objectives to construct the relevant decision function weights and priorities for nonequivalent objectives are also incorporated in the method this model gives a nondominated solution which is nearer to the best compromise solution a fortran program has been developed based on simple and weighted additive algorithms the method is illustrated with a numerical example

fuzzy goal programming an additive model abstract an additive model to solve fuzzy goal programming fgp is formulated the method uses arithmetic addition to aggregate the fuzzy goals to construct the relevant decision function cardinal and ordinal weights for nonequivalent fuzzy goals are also incorporated in the method the solution procedure is illustrated with a numerical example

solving fuzzy inequalities with concave membership functions abstract solving systems of fuzzy inequalities could lead to the solutions of fuzzy mathematical programs it is shown that a system of fuzzy inequalities with concave membership functions can be converted to a regular convex programming problem a method of centres with entropic regularization techniques is proposed for solving such a problem some computational results are included

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability

trajectory planning for boundary controlled parabolic pdes with varying parameters on higher dimensional spatial domains the flatness based design of a feedforward tracking control is considered for the solution of the trajectory planning problem for a boundary controlled diffusion convection reaction system with spatially and temporally varying parameters defined on a 1 les m dimensional parallelepipedon with the nonlinear input being restricted to a m 1 dimensional hyperplane for this an implicit state and input parametrization in terms of a basic output is determined via a volterra type integral equation with operator kernel by recursively computing successive series coefficients a series solution of the integral equation is obtained whose absolute and uniform convergence is verified by restricting the system parameters and the basic output to a certain but broad gevrey class hence prescribing an admissible desired trajectory for the basic output directly yields the feedforward control by evaluating the input parametrization this results in a systematic procedure for trajectory planning and feedforward control design for boundary controlled parabolic distributed parameter systems defined on higher dimensional domains

fuzzy programming with nonlinear membership functions piecewise linear approximation one promising approach that has been proposed for dealing with multicriteria i e multiple objective programming models has been fuzzy linear programming flp in essence the flp approach involves the replacement of the multiple objectives with goals by means of assigning an aspiration level to each objective fuzzy membership functions are then introduced to represent the measure of the achievement of the actual solution for each goal relative to its aspired level it has been shown that if the flp problem is linear and if linear membership functions are employed to reflect goal achievement then the flp problem may be transformed into a conventional i e single objective linear programming model as such one may use any conventional linear programming algorithm to solve the transformed model however in the more general case of nonlinear membership functions the transformation process becomes considerably more involved and has in the past typically led to less desirable formulations in this paper we present what we believe to be a straightforward and computationally efficient procedure for dealing with the flp problem with any general class of nonlinear membership functions conversion of such a flp will result in either a regular linear programming model or a linear integer programming model depending upon the specific characteristics of the membership function in either case existing commercial software is readily available to solve such models

non l r type fuzzy parameters in mathematical programming problems the triangular norm based operations in fuzzy logic usually lead to non l r type fuzzy sets this study considers mathematical programming problems with non l r type fuzzy parameters it shows that the fuzzy solutions to such problems can be obtained by solving an optimization problem on a mixed domain the necessary and sufficient conditions for solving the resulting optimization problems are investigated by employing the theory of convex optimization on mixed domains this is the first attempt to solve the fuzzy optimization problem with non l r type membership functions in view of optimization problems on a mixed domain

method for solving quasi concave and non concave fuzzy multi objective programming problems abstract this paper proposes a method based on linear programming techniques to treat quasi concave and non concave fuzzy multi objective programming fmop problems the proposed method initially presents a piecewise linear expression to interpreting a quasi concave membership function then we find the convex type break points and transform all quasi concave membership functions into concave functions after that the converted program is solved by linear programming techniques to obtain a global optimum in addition to not containing any of the zero one variables the proposed method does not require dividing the quasi concave fmop problem into large sub problems as in conventional methods the extension of the proposed method can treat general non concave fmop problems by merely adding less number of zero one variables

integer programming duality price functions and sensitivity analysis recently a duality theory for integer programming has been developed here we examine some of the economic implications of this theory in particular the necessity of using price functions in place of prices and the possibility of carrying out sensitivity analysis of optimal solutions in addition we consider the form of price functions that are generated by known algorithms for integer programming