mrp with flexible constraints a fuzzy mathematical programming approach in manufacturing environments with complex product structures and multiple production stages material requirements planning mrp systems are the most commonly used for production planning and material supply decision making however in practice diverse difficulties arise such as uncertainty in market demand resources with limited available capacities uncertainty in capacity data or uncertain costs likewise classical procedures of resolution applied in mrp environments do not optimize production decisions this paper provides a new linear programming model for medium term production planning in a capacity constrained mrp multi product multi level and multi period manufacturing environment then this model is transformed into three fuzzy models with flexibility in the objective function in the market demand and in the available capacity of resources the main goal is to determine the master production schedule the mrp for each raw component in each period stock levels delayed demand and capacity usage levels over a given planning horizon in such a way as to hedge against uncertainty finally the model is tested using real data from an automobile seat manufacturer

optimal birth control for an age dependent n dimensional food chain model abstract in this paper we investigate optimal policies for an age dependent n dimensional food chain model which is controlled by fertility by using dubovitskii milyutin s general theory the maximum principles are obtained for problems with free terminal states infinite horizon and target sets respectively

the lagrangian structure of long time torsional dynamics leading to rna folding within the context of biopolymer renaturation in vitro a principle of maximization in the economy of the folding process has been previously formulated as the principle of sequential or stepwise minimization of conformational entropy loss smel when specialized to the rna folding context this principle leads to a predictive folding algorithm under the assumption that an adiabatic approximation is valid this approximation requires that conformational microstates be lumped up into base pairing patterns bpps which are treated as quasiequilibrium states while folding pathways are coarsely represented as sequences of bpp transitions in this work we develop a semiempirical microscopic treatment aimed at validating the adiabatic approximation and its underlying smel principle we start by coarse graining the conformation torsional space x 3n torus with n length of the chain representing it as the lattice z2 3n where z2 integers modulo 2 this is done so that each point in the lattice represents a complete set of local torsional isomeric states coarsely specifying the chain conformation then a coarse lagrangian governing the long time dynamics of chain torsions is identified as the variational counterpart of the smel principle to prove this statement the lagrangian computation of the coarse shannon information entropy associated to the specific partition of x into bpps is performed at different times and contrasted with the adiabatic computation revealing a the subordination of torsional microstate dynamics to bpp transitions within time scales relevant to folding and b the coincidence of both plots in the range of folding time scales

optimal harvesting problems for age dependent interacting species with diffusion we investigate the optimal harvesting problems of a predator prey system for two age dependent populations with diffusion the existence and uniqueness of solutions for the system are proven using the fixed point theorem the existence of optimal control and maximum principles for problems are obtained

a simplex based algorithm for the lexicographically extended linear maxmin problem abstract let f x cx c where c is a real m n matrix c a constant m vector and x a variable n vector g x denotes the vector with components f i x in order of ascending magnitude the problem is to maximize g x over x in the lexicographic sense where x is the set of all points such that ax b x 0 a a real r n matrix it is known that this problem can be reduced to a sequence of m maximum problems where the maxmin problems can be solved by the well known simplex method of linear optimization a prerequisite such a procedure is the knowledge of at least one common index per stage it is shown that these common indices can readily be obtained from the respective optimal simplex tableaux some simple tests help to obtain more than just one common index per stage

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

on the controllability of the lotka mckendrick model of population dynamics abstract in this work we consider the problem of exact controllability of the linear lotka mckendrick model of population dynamics we establish an observability inequality and show that any initial condition can be steered into any quasi steady state by a distributed control except for a small interval of ages near zero in contrast with that we observe that the boundary control is more efficient in the sense that it controls the entire population the biological implication of our conclusions is that a closed population is more efficiently controlled through birth control as opposed to migration and eradication in particular this supports the trap alter release method for controlling a population of stray cats as opposed to the trap and kill method

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

on a distributed control problem arising in dynamic optimization of a fixed size population the practical motivation for this paper is provided by the recruitment problem faced by many organizations of fixed size to keep the average age young and thus keep innovation and productivity high while at the same time keeping levels of recruitment high a typical example is an academy of sciences the problem is formalized by an infinite horizon optimal control model for a first order pde with nonlocal dynamics a mckendrick type equation based on the nonstandard necessary optimality condition proved in the paper the following results are established i stationarity of the optimal recruitment density ii strong ergodicity of the optimal solution iii principle of bipolar recruitment in the case where the productivity of the organization is measured by the average age of the members the analysis involves a new type of transversality condition for the costate system the stability with respect to perturbations of the optimal solution of a noncoercive bang bang type problem boundedness and stability of the solution of a specific volterra integral equation of the second kind

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

optimal harvesting for nonlinear size structured population dynamics a harvesting problem is considered for a size structured population model with separable mortality rate and nonlinear fertility rate we transform the model to a system of equations and show the existence of a unique solution we also establish the existence of an optimal harvesting rate which maximizes the total harvest in a given time interval

selecting preferred solutions in the minimax approach to dynamic programming problems under flexible constraints abstract dynamic programming is a powerful approach to the optimization of sequential or multistage decision processes e g in planning or in system control in this paper we consider both theoretical and algorithmic issues in sequential decision processes under flexible constraints such processes must attain a given goal within some tolerance tolerances or preferences also apply to the values the decision variables may take or on the action chosen at each step such problems boil down to maximin optimization unfortunately this approach suffers from the so called drowning effect lack of discrimination and the optimality principle of dynamic programming is not always verified in this context we introduce a general framework for refined minimax optimization procedures in order to compare and select preferred alternatives this framework encompasses already introduced methods such as leximin and discrimin but it allows their extension to the comparison of vectors of unequal lengths we show that these refined comparisons restore compatibility with the optimality principle and that classical algorithms can be adapted to compute such preferred solutions by exploiting existing results on idempotent semirings

optimal birth control of age dependent competitive species abstract this work is concerned with optimal policies for two age structured biological populations in a competing system which is controlled by fertilities the maximum principles for problems with free terminal infinite horizon and target sets are obtained respectively via dubovitskii milyutin s general extremal theory

optimal control for age structured population dynamics of incomplete data we are concerned with the control question for linear age structured population dynamics of incomplete initial data more precisely the initial population age distribution is supposed to be unknown we here generalize the notion of no regret control of lions 1992 10 to such singular population dynamics following the method by nakoulima omrane and velin 2000 16 we prove that the problem we are considering has a unique no regret control that we characterize by a singular optimality system

optimal control for a class of nonlinear age distributed population systems this paper deals with an optimal control problem for a kind of age dependent biological population systems the well posedness of the state system is treated by means of characteristics line and fixed point principle necessary optimality conditions are obtained via tangent normal cone technique in nonlinear functional analysis the existence and uniqueness of the optimal controller are established by the use of ekeland s principle

optimal birth control of population dynamics abstract we study optimal birth control policies for an age structured population of mckendrick type which is a distributed parameter system involving first order partial differential equations with nonlocal bilinear boundary control the functional analytic approach of dubovitskii and milyutin is adopted in the investigation maximum principles for problems with a free end condition a fixed final horizon are developed and the time optimal control problem the problem with target sets and the infinite planning horizon case are investigated

renormalized hamiltonian for a peptide chain digitalizing the protein folding problem a renormalized hamiltonian for a flexible peptide chain is derived to generate the long time limit dynamics compatible with a coarsening of torsional conformation space the renormalization procedure is tailored taking into account the coarse graining imposed by the backbone torsional constraints due to the local steric hindrance and the local backbone side group interactions thus the torsional degrees of freedom for each residue are resolved modulo basins of attraction in its so called ramachandran map this ramachandran renormalization rr procedure is implemented so that the chain is energetically driven to form contact patterns as their respective collective topological constraints are fulfilled within the coarse description in this way the torsional dynamics are digitalized and become codified as an evolving pattern in a binary matrix each accepted monte carlo step in a canonical ensemble simulation is correlated with the real mean first passage time it takes to reach the destination coarse top

internal stabilizability of some diffusive models abstract we consider a single species population dynamics model with age dependence spatial structure and a nonlocal birth process arising as a boundary condition we prove that under a suitable internal feedback control one can improve the stabilizability results given in kubo and langlais j math biol 29 1991 363 378 this result is optimal our proof relies on an identical stabilizability result of independent interest for the heat equation that we state and prove in section 3

optimal harvesting control problem for an age dependent competing system of n species this paper is concerned with optimal harvesting policy for an age dependent n dimension competing species the existence and uniqueness of nonnegative solution of the system are proved using the fixed point theorem the existence of optimal control strategy is discussed optimality conditions are derived by means of normal cone and dubovitskii milyutin s general theory our results extend some known criteria

exact and approximate controllability of the age and space population dynamics structured model we prove exact and approximate controllability for a linear age dependent and spatially structured population dynamics problem the birth process is nonlocal and the demographic functions depends on age time and space our method relies on the derivation of a carleman inequality for the backward nonlocal adjoint system

optimal birth control problems for a nonlinear vermin population model with size structure abstract in this paper we investigate theoretically the least cost size problem and the least cost deviation problem for a nonlinear size structured vermin population model with separable mortality which takes fertility rate as the control variable we obtain the existence of a unique non negative solution by considering the separable solution of the model which is crucial to obtain the optimal control strategy we also illustrate the continuous dependence of the population density on initial value and control parameter finally we study the least cost size problem and the least cost deviation problem our conclusion would be useful for managing the vermin

short rational generating functions for solving some families of fuzzy integer programming problems in this paper we present new complexity results for solving single and multiobjective fuzzy integer programs with soft constraints and fuzzy objective function coefficients our method is based on the use of barvinok s short rational generating functions of convenient transformations of the fuzzy problems to crisp ones this analysis allows us to provide new insights on the structure of these families of fuzzy programs

optimal control of heterogeneous systems basic theory a general model of a heterogeneous control system is introduced in the form of a first order distributed system with nonlocal dynamics and exogenous side conditions the heterogeneity is represented by a parameter taking values in an abstract measurable space so that both continuous and discrete heterogeneity as well as probabilistic heterogeneity without density are included a distributed and a lumped controls are involved the latter appearing also in the side conditions an existence theorem is proved for the uncontrolled system and the sensitivity of the solution with respect to the control variables is estimated the main result is an optimality condition in the form of the pontryagin local maximum principle a global maximum principle holds for the distributed control under an additional condition that rules out discrete measurable heterogeneity spaces a number of possible applications are outlined age structured systems size structured systems nonlocal advection reaction equations static parametric heterogeneity in epidemiology and two stage control systems with uncertain switching time

optimal control of a nonlinear population dynamics with diffusion abstract in this paper we study the optimal control for some population dynamics problems we determine the necessary conditions and present an iterative scheme for the optimal control problems

an interactive fuzzy programming system abstract an interactive system is introduced which supports a decision maker in solving programming models with crisp or fuzzy constraints and crisp or fuzzy goals one part of the system is the determination of membership functions representing goals to this purpose fuzzy extreme solutions are computed and are presented to the decision maker these and each of the proposed compromise solutions are fuzzy efficient

nonlinear age dependent population growth under harvesting we consider the nonlinear age dependent population growth model introduced by gurtin maccamy arch rat mech anal 54 281 300 1974 to which is added a harvest of members at a rate which is constant in time but may depend on the age of members being harvested this partial differential equation may be transformed by the method of characteristics into a pair of functional equations for the total population size and the birth rate we discuss the behaviour of solutions when the birth and death moduli are functions of a single variable either the age or the total population size

a degenerate population system carleman estimates and controllability abstract in this paper we consider a system in non divergence form which models the interaction between two different species u and v both of them depend on time on age and on space moreover the diffusion coefficients degenerate at the boundary of domain we study in particular null controllability of the system via observability inequalities and carleman estimates for the associated non homogeneous adjoint problem

exact null controllability of the lobesia botrana model with diffusion abstract this paper is devoted to analyze the exact null controllability of the diffusive lobesia botrana model with nonlocal boundary condition we study the null controllability of butterfly population by acting on eggs larvas and female moths in a small age interval we assume that the control of female moths is in a certain given region of the vineyard the main result is established by combining some estimations and the carleman inequality for the backward system related to an optimal control problem a fixed point theorem is then used to draw the conclusion

approximate controllability of the semilinear heat equation this article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain when the control acts on any open and nonempty subset of or on a part of the boundary in the case of both an internal and a boundary control the approximate controllability in l p for 1 p l in the case of the interior control we also prove approximate controllability in c 0 the proof combines a variational approach to the controllability problem for linear equations and a fixed point method we also prove that the control can be taken to be of quasi bang bang form

a pontryagin maximum principle for infinite dimensional problems a basic idea of the classical approach for obtaining necessary optimality conditions in optimal control is to construct suitable needle like control variations we use this idea to prove the main result of the present paper a pontryagin maximum principle for infinite dimensional optimal control problems with pointwise terminal constraints in arbitrary banach state space by refining the classical variational technique we are able to replace the differentiability of the norm of the state space guaranteed by the strict convexity of its dual norm which is assumed in the known results by a separation argument we also drop another key assumption which is common in the existing literature on infinite dimensional control problems that the set of variations in the state space of the state trajectory s endpoint resulting from the control variations be finite codimensional instead we require only that it has nonempty interior in its closed affine hull as an application of the abstract result we present an illustrative example an optimal control problem for an age structured system with pointwise terminal state constraints

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

optimal birth control of age dependent competitive species ii free horizon problems we study optimal birth policies for two age dependent populations in a competing system which is controlled by fertilities new results on problems with free final time and integral phase constraints are presented and the approximate controllability of system is discussed

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

optimal harvesting for a nonlinear age dependent population dynamics here we investigate an optimal harvesting problem for a nonlinear age dependent population dynamics existence and uniqueness of a positive solution for the model are demonstrated the structure of the solution is also investigated we establish the existence of the optimal control and the convergence of a certain fractional step scheme for some approximating problems we obtain the optimal controllers in feedback form via the dynamic programming method

overtaking optimal control problem for an age dependent competition system of n species abstract in this paper the overtaking optimal control problem for an age dependent competition model of n species with unbounded time interval is considered the existence of overtaking optimal policy is proved and a maximum principle is carefully derived under less restrictive conditions weak and strong turnpike properties of optimal trajectories are established

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

a two phase approach for multi objective programming problems with fuzzy coefficients in this study a two phase procedure is introduced to solve multi objective fuzzy linear programming problems the procedure provides a practical solution approach which is an integration of fuzzy parametric programming fpp and fuzzy linear programming flp for solving real life multiple objective programming problems with all fuzzy coefficients the interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision maker dm the procedure can be also performed to obtain lexicographic optimal and or additive solutions if it is needed in the first phase of the procedure a family of vector optimization models is constructed by using fpp then in the second phase each model is solved by flp the solutions are optimal and each one is an alternative decision plan for the dm

on a population dynamics control problem with age dependence and spatial structure we consider the control problem for a population dynamics model with age dependence spatial structure and a nonlocal birth process arising as a boundary condition we examine the controllability at a given time t and show that approximate controllability holds for every fixed finite time t as a consequence a new uniqueness condition continuation result is proved

analysis of structure in fuzzy linear programs abstract fuzzy linear programming problems are analyzed within the fuzzy set context to uncover redundancies infeasibilities variables whose values are fixed and implied bounds on rows and columns applications include pre analysis of perturbed constraint matrices and post optimization analysis fuzzy linear programming problems

solving mathematical programs with fuzzy equilibrium constraints this work deals with mathematical programs with fuzzy equilibrium constraints it shows that solving the fuzzy mpec is equivalent to solving a fuzzy complementarity constrained optimization problem by using the tolerance approach we show that the fuzzy complementarity constrained optimization problem can be converted to a regular nonlinear programming problem a new smoothing approach based on entropic regularization is developed for solving the resulting optimization problem numerical examples are also included to illustrate the solution procedure

two phase approach for solving the fuzzy linear programming problems we propose a two phase approach to solve the fuzzy linear programming problem although several methods in the literature have been proposed to treat this problem the two phase approach has the merit stated as if the decision maker is seeking an efficient solution which can improve the max min operators solution so that each membership degree should be improved then the two phase method automatically attains this desire if there is room to improve an interpretation of this result is that the two phase method not just pursue the highest membership degree in the objective but also pursue a better utilization of each constrained resource

coarse semiempirical solution to the protein folding problem abstract we introduce a semiempirical theory leading to the ab initio prediction of conducive folding pathways and coarsely resolved native backbone geometries of proteins suddenly exposed to in vitro renaturation conditions the underlying model incorporates a discrete codification of local steric hindrances of the peptide backbone we first determine a time evolving finite set of local torsional constraints upon which large scale organization is built thus the torsional state of the chain is topologically represented by viewing the state of each residue modulo the basin of attraction to which it belongs in the ramachandran plot a grammar to combine such coarsely defined torsional states topologies and translate them into meaningful patterns of long range interactions is developed an algorithm for structure prediction is shown to emerge once this grammar is combined with prescriptions for the time evolution of topological patterns this algorithm is rooted in the fact that local contributions to the potential energy may be subsumed into time evolving conformational constraints coarsely defining sets of restricted backbone geometries responsible for framing the patterns of nonbonded interactions the predictive power of the algorithm is established by obtaining stable topologies of small proteins which prove to be compatible with their native folds and computing ab initio folding pathways for mammalian ubiquitin that ultimately yield a stable structural pattern reproducing its native features

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

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

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

optimal birth control for predator prey system of three species with age structure in this paper we investigate optimal policies for three age dependent populations in a predator prey system which is controlled by fertility by using dubovitskii milyutin s general theory the maximum principles are obtained for problems with free terminal states infinite horizon and target sets respectively

a class of nonlinear diffusion problems in age dependent population dynamics in this paper we study the existence the uniqueness and the asymptotic behaviour of a class of nonlinear diffusion problems these problems are motivated by models of age dependent population dynamics which originate in the article by gurtin maccamy ll only a few special cases of these models have been analyzed to date with the most recent published work being that of maccamy 17 gurtin maccamy 12 and langlais 15 161 the cases treated in 12 171 are restricted to special simple forms of the birth and death moduli that occur in these models and to a specific form of the nonlinear diffusion mechanism on the other hand langlais 16 considers a different nonlinear diffusion mechanism and obtains a type of generalized solution for general forms of the birth and death moduli in this work we consider a general form of the nonlinear diffusion mechanism and we develop a method for establishing the existence and uniqueness of solutions of the resulting equations our method allows us to obtain classical solutions even with general forms of the birth and death moduli we also describe the asymptotic behaviour of the solutions when the time variable becomes large and apply our results to some specific examples the equations that we study are motivated as follows consider a population that can disperse in a spatial domain r for simplicity we take q j where j is an open interval in r but it will be clear from the context that more general regions q of r can be considered let a t x denote the number of individuals per unit length and unit age that are of age a at time t and are located at the position x e j the total population per unit length at time t and at position x is given by

optimal control of free horizon problems for an age dependent n dimensional food chain model this paper is concerned with optimal birth policies for an age dependent n dimensional food chain model new results on problems with free final time and integral phase constraints are presented the approximate controllability of system is discussed

optimal harvesting for periodic age dependent population dynamics here we investigate the optimal harvesting problem for some periodic age dependent population dynamics namely we consider the linear lotka mckendrick model with periodic vital rates and a periodic forcing term that sustains oscillations existence and uniqueness of a positive periodic solution are demonstrated and the existence and uniqueness of the optimal control are established we also state necessary optimality conditions a numerical algorithm is developed to approximate the optimal control and the optimal harvest some numerical results are presented

optimal birth control of population dynamics ii problems with free final time phase constraints and mini max costs abstract we study optimal birth control of population systems of mckendrick type which is a distributed parameter system involving first order partial differential equations with nonlocal bilinear boundary control new results on problems with free final time phase constraints and mini max costs are presented

decisions and evaluations by hierarchical aggregation of information usually in preparing a decision a hierarchy of criteria for evaluating alternatives can he agreed upon by the decision makers the problem addressed in this paper is the appropriate modelling of such evaluations and their aggregation in order to arrive at a proper overall evaluation function by which the decision alternatives can be ordered easily representing the criteria subjective categories by fuzzy sets it is possible to use aggregation operators as models for the amalgamation of those criteria in the framework of an empirical research project a hierarchy of criteria was developed for the evaluation of credit worthiness of customers 50 credit applications were evaluated by a number of credit managers of banks and their judgement could be appropriately modelled by a four level hierarchy of criteria

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