an application of fuzzy goal programming to a multiobjective project network problem abstract this paper presents a practical application of fuzzy goal programming fgp in a real life project network problem with two objectives as minimum completion time and crashing costs wanted to be optimized simultaneously fuzziness in the problem stems from the imprecise aspiration levels attained by the decision maker dm to both objectives these imprecise aspiration levels are quantified through the use of piecewise linear and continuous membership functions the objective function of the fgp is to maximize the membership value of both objectives intersection which form the fuzzy decision the problem is solved by using lindo computer package and the best compromised solution is found comparisons between solutions of fgp fuzzy linear programming flp lexicographic maximization method lmm and linear programming lp are also presented

a new fuzzy multi objective programming entropy based geometric programming and its application of transportation problems in this paper we introduce a fuzzy mathematical programming with generalized fuzzy number as objective coefficients we also examine a transportation problem with additional restriction there is an additional entropy objective function in the transportation problem besides transportation cost objective function using new fuzzy mathematical programming this multi objective entropy transportation problem with generalized trapezoidal fuzzy number costs has been reduced to a primal geometric programming problem pareto optimal solution of the transportation model is found numerical examples have been provided to illustrate the problem

an approach for solving a fuzzy multiobjective programming problem abstract in this paper we present a new approach based on the nearest interval approximation operator for dealing with a multiobjective programming problem with fuzzy valued objective functions by the way we have established a karush kuhn tucker k k t kind of pareto optimality conditions for the resulting interval multiobjective program to this end we made use of gh differentiability of involved interval valued functions two algorithms play a pivotal role in the proposed method the first one returns a nearest interval approximation to a given fuzzy number the other one makes use of k k t conditions to deliver a pareto optimal solution of the above mentioned resulting interval program

fuzzy goal programming approach to multilevel programming problems the purpose of this paper is to propose a procedure for solving multilevel programming problems in a large hierarchical decentralized organization through linear fuzzy goal programming approach here the tolerance membership functions for the fuzzily described objectives of all levels as well as the control vectors of the higher level decision makers are defined by determining individual optimal solution of each of the level decision makers since the objectives are potentially conflicting in nature a possible relaxation of the higher level decision is considered for avoiding decision deadlock then fuzzy goal programming approach is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables sensitivity analysis with variation of tolerance values on decision vectors is performed to present how the solution is sensitive to the change of tolerance values the efficiency of our concept is ascertained by comparing results with other fuzzy programming approaches

a chance constrained fuzzy goal program abstract a chance constrained goal program in the case of having uncertain aspiration levels is considered however these aspiration levels are random variables with known probability distribution functions the goal constraints considered are probabilistic and fuzzy constraints the equivalent deterministic goal program is developed and an illustrative example is given

a fuzzy goal programming method with imprecise goal hierarchy two most widely used approaches to treating goals of different importance in goal programming gp are 1 weighted gp where importance of goals is modelled using weights and 2 preemptive priority gp where a goal hierarchy is specified implying infinite trade offs among goals placed in different levels of importance these approaches may be too restrictive in modelling of real life decision making problems in this paper a novel fuzzy goal programming method is proposed where the hierarchical levels of the goals are imprecisely defined the imprecise importance relations among the goals are modelled using fuzzy relations an additive achievement function is defined which takes into consideration both achievement degrees of the goals and degrees of satisfaction of the fuzzy importance relations examples are given to illustrate the proposed method

solution of optimal control problems with lumped parameters having single or multiple goal objectives in fuzzy environment abstract in this paper optimal control problems ocp of both linear and nonlinear type with lumped parameters are considered continuous ocp with single or multiple objectives are transformed to finite dimensional optimization problems through discretization the formulation of goal optimal control problem gocp using goal programming methodology first given by levary 1986 for equal time subintervals is extended for unequal subintervals by a generalized formulation the concept of local and global goals proposed here is very useful for multicriteria dynamic optimization problems in several physical situations ocp involve fuzziness in their goals priorities parameters or relative weights and associated linguistic variables etc a new formulation of ocp in fuzzy environment is given as a fuzzy goal optimal control problem fgocp and the solution procedure is illustrated by linear and nonlinear examples

solving multi period project selection problems with fuzzy goal programming based on topsis and a fuzzy preference relation project portfolio managers are multi objective decision makers dms who are expected to select the best mix of projects by maximizing profits and minimizing risks over a multi period planning horizon however project portfolio decisions are complex multi objective problems with a high number of projects from which a subset has to be chosen subject to various constraints and a multitude of priorities and preferences we propose a goal programming gp approach for project portfolio selection that embraces conflicting fuzzy goals with imprecise priorities a fuzzy goal with an aspiration level and a predefined membership function is defined for each objective the impreciseness in the priorities of the membership values of the fuzzy goals is modeled with fuzzy relations this leads to type ii fuzzy sets since fuzzy relations are organized between the membership values of the fuzzy goals which are themselves fuzzy sets the proposed model is based on the technique for order preference by similarity to ideal solution topsis and fuzzy preference relations topsis is used to reduce the multi objective problem into a bi objective problem the resulting bi objective problem is solved with fuzzy gp fgp the fuzzy preference relations are used to help dms express their preferences with respect to the membership values of the fuzzy goals the proposed approach is used to solve a real life problem characterized as a fuzzy multi objective project selection with multi period planning horizon mops mpph the performance of the proposed approach is compared with a competing method in the literature we show that our approach generates high quality solutions with minimal computational efforts

a geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices abstract the usefulness of encoding the fuzzy evaluations of alternatives and the importance weights of criteria in a multiple objective decision problem through binary comparison matrices or pairwise judgment matrices is receiving considerable attention the methodology for identifying the best alternative in a given decision problem involves the computation of the principal eigenvectors of the binary comparison matrices the eigenvectors transform the fuzzy evaluations of the importance of the criteria and the ratings of the alternatives into a ratio scale a difficulty that is often experienced in using this approach in practice is the inconsistency of the binary evaluations this paper proposes a simple averaging procedure to construct a supertransitive approximation to a binary comparison matrix where inconsistency is a problem it is further suggested that such an adjustment might be necessary to more closely reflect the inherent fuzziness of the evaluations contained in a binary comparison matrix the procedure is illustrated by means of examples

fuzzy goal programming approach for water quality management in a river basin abstract two fuzzy goal programming fgp approaches are applied to water quality management in a river basin for solving multiobjective optimization problems involving vague and imprecise information several fgp models including equal weight and unequal weight nonpreemptive priority and preemptive priority are proposed to assist water quality management involving multiple conflicting goals optimal water quality management involves obtaining optimal analysis of assimilative capacity also referred to as allowable pollution loading and treatment cost of wastewater based on models and standards of water quality as well as an equitable removal of wastewater in a river basin two fgp schemes that are capable of maximizing achieved membership function and minimizing the deviation from a set of preferred target assimilative capacity and treatment cost are considered moreover methodologies using two fuzzy decision theories in fgp approaches i e max min operator and compensatory operator are also proposed those methodologies are illustrated in a case study of multiobjective water quality management in the tou chen river basin taiwan the case study demonstrates the capability of the two fgp approaches based on variable fuzzy decision theories to work suitably in water quality management in a river basin

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

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

a goal programming procedure for fuzzy multiobjective linear fractional programming problem abstract this paper presents a goal programming gp procedure for fuzzy multiobjective linear fractional programming fmolfp problems in the proposed approach which is motivated by mohamed fuzzy sets and systems 89 1997 215 gp model for achievement of the highest membership value of each of fuzzy goals defined for the fractional objectives is formulated in the solution process the method of variable change on the under and over deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using linear goal programming lgp methodology the approach is illustrated by two numerical examples

example of an order 16 non symplectic action on a k3 surface abstract we exhibit an example of a k3 surface of picard rank 14 with a non symplectic automorphism of order 16 which fixes a rational curve and 10 isolated points this settles the existence problem for the last case of al tabbaa sarti and taki s classification 1

multi item eoq model with hybrid cost parameters under fuzzy fuzzy stochastic resource constraints a geometric programming approach in this paper a multi item economic order quantity eoq model is considered in which the cost parameters are of fuzzy hybrid nature under two types of resources a resources as fuzzy quantities b resources as fuzzy and fuzzy random quantities the unit cost depends on demand rate the time horizon is taken to be infinite we find the average cost for the model which is a function of order quantity and demand rate and also of some hybrid parameters when the resources are fuzzy quantities the problem is transformed into its equivalent unconstrained deterministic form by using a surprise function for the constraints the problem involving hybrid number is again equivalently rewritten as a multi objective minimization of the mean of the objective function and variance function of the distribution inventory problem introducing new variables we transform the terms of the functions into signomial types using fuzzy multi objective solution procedure we solve the problem through geometric programming approach sensitivity analysis has been performed to study the effect of different weights considered for mean objective function and variance function

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

priority structure in fuzzy goal programming abstract this paper introduces priority structure in fuzzy goal programming fgp this approach utilizes the lexicographic order of goal programming gp and yields an efficient computational algorithm for solving fgp fuzzy priorities are also discussed a numerical example is given to highlight the procedure concluding remarks are also made

fuzzy goal programming with nested priorities abstract this paper proposes a new approach to formulating fuzzy priorities in a goal programming problem the proposed methodology remedies certain shortcomings of the composite membership function approach discussed in previous works 7 10 the principal advantage of the proposed method is that it leads to a formulation in which tradeoffs between goals more closely reflect the decision maker s intentions than in other noninteractive approaches 8 9 10 14 in some of which a fixed hierarchy of goals is assumed

multi item imperfect production lot size model with hybrid number cost parameters in this study we have developed a multi item finite production lot size model with reworking of imperfect quality items when production stops defected items are assumed to rework at a constant rate the cost parameters in this model are taken here as hybrid numbers i e some of the costs are fuzzy numbers as well as random variables then the model is converted into a multi objective geometric programming gp problem a numerical example is given to support the problem a sensitivity analysis has given to study the nature of the objective functions

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

perfectly matched layer boundary integral equation method for wave scattering in a layered medium for scattering problems of time harmonic waves the boundary integral equation bie methods are highly competitive since they are formulated on lower dimension boundaries or interfaces and can automatically satisfy outgoing radiation conditions for scattering problems in a layered medium standard bie methods based on green s function of the background medium need to evaluate the expensive sommerfeld integrals alternative bie methods based on the free space green s function give rise to integral equations on unbounded interfaces which are not easy to truncate since the wave fields on these interfaces decay very slowly we develop a bie method based on the perfectly matched layer pml technique the pmls are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space our pml based bie method uses the pml transformed free space green s function to define the boundary integral operators the method is efficient since the pml transformed free space green s func

multi objective fuzzy inventory model with three constraints a geometric programming approach a multi item multi objective inventory model with shortages and demand dependent unit cost has been formulated along with storage space number of orders and production cost restrictions in most of the real world situations the cost parameters the objective functions and constraints of the decision makers are imprecise in nature hence the cost parameters the objective functions and constraints are imposed here in fuzzy environment this model has been solved by geometric programming method the results for the model without shortages are obtained as a particular case the sensitivity analysis has been discussed for the change of the cost parameters the models are illustrated with numerical examples

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

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

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

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

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