determining the best decision in the presence of imprecise information abstract this paper addresses general aspects of decision making in an imprecise environment here the task of choosing a decision among the available alternatives with imprecisely defined outcomes is analyzed new definitions for the concept of best decision in such an environment are introduced these definitions are based on fuzzy set theoretic representations of decision outcomes

fuzzy shortest paths abstract shortest paths are one of the simplest and most widely used concepts in nonfuzzy networks fuzzy shortest paths also have a variety of applications but are hampered by the possibility that the fuzzy shortest length may not correspond to an actual path in the network in this paper we present new models based on fuzzy shortest paths that circumvent this problem and we also give a general algorithm based on dynamic programming to solve the new models in addition we analyze the fuzzy shortest path algorithms in terms of submodular functions the analysis concepts developed for fuzzy shortest paths are also discussed in terms of general fuzzy mathematical programming

on the application of fuzzy sets theory to the optimal flood control problem arising in water resources systems abstract we first survey the application of fuzzy sets theory to various problems occurring in water resources systems the problem of optimal flood control planning by an appropriate integration of structural and non structural measures with the objective of optimizing the flood damage reduction due to recurrent floods is modeled via fuzzy sets methodologies a two level optimization model appropriate for planning decisions first on a regional or local level and then on a national level is proposed a third phase involving coordination is appended in particular a fuzzy optimization model involving an adroit combination of fuzzy dynamic programming type reasoning and a branch and bound type search procedure is offered as a more utilitarian approach than its crisp equivalent the performance of the algorithm is illustrated with an example

intermittency in random optical layers at total reflection the authors show that the reflection coefficient r of a succession of n random optical layers has a well defined but sample dependent limit when n to infinity the probability distribution or r at total reflection is found to depend only on imr xi 2 3 where xi 2 is the noise variance the analytical expressions appearing in this localisation problem are found to be the same as the universal functions describing intermittency with noise

approaches to multistage one shot decision making abstract in this research multistage one shot decision making under uncertainty is studied in such a decision problem a decision maker has one and only one chance to make a decision at each stage with possibilistic information based on the one shot decision theory approaches to multistage one shot decision making are proposed in the proposed approach a decision maker chooses one state amongst all the states according to his her attitude about satisfaction and possibility at each stage the payoff at each stage is associated with the focus points at the succeeding stages based on the selected states focus points the sequence of optimal decisions is determined by dynamic programming the proposed method is a fundamental alternative for multistage decision making under uncertainty because it is scenario based instead of lottery based as in the other existing methods the one shot optimal stopping problem is analyzed where a decision maker has only one chance to determine stopping or continuing at each stage the theoretical results have been obtained

fuzzy sets modeling and optimization for disaster control systems planning abstract various forms of disasters natural and man made abound although to some extent some of these have been exacerbated by modern civilization various technologies have been invoked to control or mitigate their adverse effects modeling and optimization based techniques have also been employed we outline some efforts based on fuzzy sets theory which are utilized in conjunction with classical methods to enhance the reliability and utility of these efforts in particular disasters caused by floods and storm water pollution are used as vehicles for the exemplification of our models

a note on a fuzzy goal programming algorithm by tiwari dharmar and rao abstract we propose a new algorithm for solving a fuzzy goal programming fgp problem with symmetrically triangular membership functions of fuzzy goals and priority structure the efficiency of the algorithm is justified by showing its computational superiority over that by tiwari et al

forecasting enrollments with fuzzy time series part ii abstract there have been a good many methods to forecast university enrollments in the literature however none of them could be applied when the historical data are linguistic values fuzzy time series is an effective tool to deal with such problems in this paper as an application of fuzzy time series in educational research the forecast of the enrollments of the university of alabama is carried out in so doing a fuzzy time series model is developed using historical data a complete procedure is proposed which includes fuzzifying the historical data developing a fuzzy time series model and calculating and interpreting the outputs to evaluate the forecasting model the robustness of the fuzzy time series model is tested advantages and problems of the forecasting method are also discussed

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

analysis of a class fractional programming problems we propose a solution strategy for fractional programming problems of the form max x x g x u x where the function satisfies certain convexity conditions it is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max x x g x u x where is an exogenous parameter the proposed strategy combines fractional programming andc programming techniques a maximal mean standard deviation ratio problem is solved to illustrate the strategy in action

multiple objective fuzzy dynamic programming problems a survey and some applications many real world problems involve sequential or multistage decision making dynamic programming dp is a powerful optimization technique that is particularly applicable to many complex problems requiring a sequence of interrelated decisions a new methodology multiobjective dynamic programming modp which relies heavily on conventional dynamic programming is developed as a technique for solving problems that involve conflicting objectives that obey dp characteristics in the past two decades a major development of multiobjective dynamic optimization has been made modp one of the most provocative topics within the broader subject a natural extension of dp is its use in conjumction with fuzzy sets this involves the perturbation of its components in a manner that admits of systematic fuzzification fuzzy dynamic programming fdp represents an advance on the theory of dp this paper reviews the major concepts used in modp and fmodp and examins the current progress made in the development of the corresponding theory and methodology

a preference structure on aspiration levels in a goal programming problem a fuzzy approach abstract a method is proposed for comparing different sets of aspiration levels assigned to goals in a goal programming problem this is based on the relative flexibilities that are realised under different sets of aspiration levels the relative flexibility depends upon the relative degree of choosing an objective in presence of other objectives a pair wise comparison method in the sense of shimura has been proposed for measuring the degree of choosing an objective in presence of other objectives this methodology affords to compare different sets of assigned aspiration levels and thereby choosing the best set as the one with least relative flexibility

fuzzy optimal search methods abstract this paper gives analytical approaches and algorithms for decision tree search techniques the decision trees are assumed to have a fixed number of stages and predefined possible states at every stage the costs of traversing the tree are characterized as approximate and defined as fuzzy numbers two search methods each drawing from an existing non fuzzy search algorithm are described the first method is a dynamic programming search where the principle of optimality with fuzzy costs is addressed the second method is an a search for which the notion of a lower bound estimate of costs is utilized to increase the efficiency of the search algorithms for each method are included theorems and proofs which complete the analytical development of these techniques are also included finally a numerical example illustrating the procedure is given

scales interference and identification at the transition to chaos abstract the scaling at the transition to chaos is described by a linear autoregressive model the identification of the universal scales from experimental data generally suffers from the competition of statistical and model errors due to the presence of noise and scaling corrections respectively the dilemma is resolved through a two step least mean squares method as an application the parameter scaling data of hao and zhang j stat phys 28 1982 769 for the period doubling route to chaos of the forced brusselator are reanalysed combined with order detection methods the algorithm also provides a powerful tool to detect and analyse scales interference phenomena

fault tolerance of vertex pancyclicity in alternating group graphs abstract we study fault tolerance of vertex k pancyclicity of the alternating group graph ag n a graph g is vertex k pancyclic if for every vertex p g there is a cycle going through p of every length from k to g xue and liu z j xue s y liu an optimal result on fault tolerant cycle embedding in alternating group graphs inform proc lett 109 2009 1197 1201 put the conjecture that ag n is 2 n 7 fault tolerant vertex pancyclic which means that if the number of faults f 2 n 7 then ag n f is still vertex pancyclic chang and yang j m chang j s yang fault tolerant cycle embedding in alternating group graphs appl math comput 197 2008 760 767 showed that for the shortest cycles the fault tolerance of ag n is much lower they noted that with n 2 faults one can cut all 3 cycles going through a given vertex p it is easy to observe that the same set of faults cuts all 4 and 5 cycles going through p on the other hand they show that ag n is n 3 fault tolerant vertex 3 pancyclic in this paper we show that the cycles of length 6 are much more fault tolerant more precisely we show that ag n is 2 n 6 fault tolerant vertex 6 pancyclic this bound is optimal because every vertex p has 2 n 4 neighbors

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

conditional decision processes with recursive function we consider stochastic optimization of not necessarily additive but recursive functions over multistage decision processes without assuming any monotonicity we optimize a regular process by a direct dynamic programming approach on the regular decision process we propose two related conditional decision processes an a posteriori conditional decision process and an a priori when the markov transition law degenerates into a deterministic dynamics the two conditional processes reduce to the same deterministic decision process the conditional processes with monotonicity are optimized by the usual backward dynamic programming we show that under additional convexity the regular process dominates the a priori in maximum value function and the a priori does the a posteriori we show that the a posteriori process illustrates kreps and porteus s dynamic choice problem the numerical example also verifies the dominance relation in three optimal value functions

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

long term inventory policy making through fuzzy decision making models abstract the inventory policy meant as a replenishment rule has a considerable impact on most firms the paper considers the determination of optimal inventory policy of firms from a global viewpoint of top management the inventory is represented as a fuzzy system with the fuzzy inventory level as the output the fuzzy replenishment as the input and fuzzy demand the control problem is formulated in terms of decision making in a fuzzy environment with fuzzy constraints imposed on replenishments a fuzzy goal for preferable inventory levels to be attained and the fuzzy decision as the intersection of fuzzy constraints and the fuzzy goal at subsequent stages the planning horizon is infinite the problem is to find an optimal time invariant strategy relating the optimal replenishments to the current inventory levels maximizing the membership function of fuzzy decision the existence of such a strategy is proved and an algorithm for its determination is given the optimal time invariant strategy obtained is represented as a fuzzy conditional statement equated with a fuzzy relation which is the firm s optimal fuzzy replenishment rule

on the decomposition problem of fuzzy sets in this paper we consider an important problem that is frequently encountered and yet almost universally ignored both in the application and in the theoretic studies of fuzzy sets we refer to this problem as the decomposition of fuzzy sets in nearly all applications of fuzzy sets theory e g fuzzy decision making fuzzy control fuzzy forecasting etc the output is in general a fuzzy set this is then defuzzified before it is applied although defuzzification may be necessary if a crisp quantity is definitely needed when a fuzzy output is appropriate desired or required in an application then the defuzzification step becomes unnecessary yet in this case one finds that it is some times difficult to correctly interpret a fuzzy quantity or identify the true meaning of this fuzzy output using a family of predefined fuzzy sets thus how to match the fuzzy output and the predefined fuzzy sets is the problem of concern in this paper to address this problem we first present and review some definitions and then proceed with an investigation of the preliminary properties of these concepts complementing them with illustrative examples finally we propose some algorithms for the solution of this problem

the relationship between goal programming and fuzzy programming abstract goal programming gp and fuzzy programming fp are two approaches for solving the vector optimization problem by reducing it to a single or sequential objective one these two approaches have some similarities and both of them have more than one form to handle the multiobjective problem this paper highlights the similarities between gp and fp and answers the question how each one can lead to the other however we will consider in this paper the min operator to transform the linear fp to a crisp program since this approach is more popular and applied than the others in addition some new forms of fuzzy programs are presented in this paper these new forms use the concept of deviational variables of gp and not the min operator to transform the fp to a crisp one finally a numerical example to illustrate the relationship between the two approaches is given

fault tolerant cycle embedding in alternating group graphs abstract cycle embedding is an important issue in evaluating the efficiency of interconnection networks and is also an extension of the theoretical research on hamiltonicity in this paper we study the fault tolerant hamiltonicity of alternating group graphs which were recently proposed as interconnection topologies for parallel and distributed systems a cycle of length l is said to be an l cycle let f be a set of faulty vertices in an n dimensional alternating group graph ag n and let m be the number of vertices in ag n f a previous result in j m chang j s yang y l wang y cheng panconnectivity fault tolerant hamiltonicity and hamiltonian connectivity in alternating group graphs networks 44 2004 302 310 showed that for n 4 ag n f is hamiltonian i e ag n f possesses a hamiltonian cycle if f n 2 and ag n f is hamiltonian connected i e every two vertices of ag n f are connected by a hamiltonian path if f n 3 in this paper we show the following results i ag n f is pancyclic i e ag n f contains an l cycle for each l with 3 l m if f n 2 ii ag n f is vertex pancyclic i e every vertex of ag n f is contained in an l cycle for each l with 3 l m if f n 3 and iii ag n f is edge 4 pancyclic i e every edge of ag n f is contained in an l cycle for each l with 4 l m if f n 4 the first result is an improvement over the previous one and the last two results are new characterization of fault tolerant pancyclicity on ag n all the results we achieved are the best possible in the sense that the number of faulty vertices cannot be increased

a dynamic approach to multiple objective resource allocation problem a new approach using dynamic programming is developed for solving the multiple objective resource allocation problem there are two key issues being addressed in this approach the first one is to develop a methodology of fuzzy evaluation and fuzzy optimization for multiple objective systems the procedure of getting the marginal evaluation for each objective and aggregating them synthetically into a global evaluation is presented in this paper the second one is to design a dynamic optimization algorithm by incorporating the method of fuzzy evaluation and fuzzy optimization with the conventional dynamic programming technique a characteristic feature of the approach presented is that various objectives are synthetically considered by the fuzzy systematic technique instead of the frequently employed weighted average method numeric examples are also given to clarify the developed approach and to demonstrate its effectiveness

a fuzzy dynamic approach to the multicriterion resource allocation problem abstract in this paper a new approach using fuzzy dynamic programming is proposed for solving the multiple criteria resource allocation problems it has been concluded that solutions obtained by this approach are always efficient hence an optimal compromise solution can be introduced a characteristic feature of the presented approach is that the objective function scalarization can be taken into account by using fuzzy set notations a numerical example is given to clarify the developed approach

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

one to one disjoint path covers on alternating group graphs the alternating group graph denoted by ag n is one of the popular interconnection networks which has many attractive properties in this paper we prove that for any two distinct nodes and there exist m node disjoint paths for any integer n 3 with 1 m 2 n 4 whose union covers all the nodes of ag n for any node of ag n has exactly 2 n 4 neighbors 2 n 4 is the maximum number of node disjoint paths can be constructed in ag n

combining fuzzy iteration model with dynamic programming to solve multiobjective multistage decision making problems this paper aims to develop a fuzzy dynamic programming approach for multiobjective multistage decision making problems fuzzy dynamic programming usually convert the problems into corresponding single objective problems by an aggregate of hybrid objective values that expresses the performance of the particular stage decision weight assessment is an important problem arising from the evaluation of multiple objectives therefore a fuzzy iteration model is first developed to simultaneously provide the objective weights and the evaluation of alternatives with multiple objectives then a fuzzy dynamic programming is derived by applying the fuzzy iteration model to classic dynamic programming to evaluate the decisions at each stage in the dynamic process of decision makings where the objective weights are initially elicited from the information implicit in alternatives and can be interactively adjusted to reflect the changes of decision situations by using a non structured decision making analysis method if necessary finally the fuzzy dynamic programming is validated through a problem of water resource allocation

fuzzy dynamic programming fuzzy adaptive neuro control and the general medical diagnosis problem abstract the fuzzy medical diagnosis decision models of esogbue and elder employed fuzzy sets theory to directly and more correctly model all the information nets useful in reaching a scientific understanding of a patient s health status this knowledge was utilized in constructing a sensible medical hypothesis decision problem in this paper we show how fuzzy dynamic programming and neural networks can be used to extend the process to other phases by linking the four stages of medical hypothesis physician s observation preliminary diagnosis and final diagnosis in an adroit manner

a stable model based fuzzy predictive control based on fuzzy dynamic programming a stable model based fuzzy predictive controller based on fuzzy dynamic programming is introduced the objective of the fuzzy predictive controller is to drive the state of the system to a terminal region where a local stabilizing controller is invoked leading to a dual mode strategy the prediction horizon is fixed and specified the stability of the controlled system is studied using the value function as a lyapunov function guaranteed stability is obtained under conditions on the terminal region the local control law and the membership functions of fuzzy goal and constraints therein the solution procedure is based on dynamic programming with branch and bound

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs the arrangement graph a sub n k is a generalization of the star graph there are some results concerning fault hamiltonicity and fault hamiltonian connectivity of the arrangement graph however these results are restricted in some particular cases and thus are less completed we improve these results and obtain a stronger and simpler statement let n k spl ges 2 and f spl sube v a sub n k spl cup e a sub n k we prove that a sub n k f is hamiltonian if f spl les k n k 2 and a sub n k f is hamiltonian connected if f spl les k n k 3 these results are optimal

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

control of a deterministic system es a fuzzy environment over infinite planning horizon the multistage control of a time invariant finite state deterministic system in a fuzzy environment over infinite planning horizon is considered the fuzzy environment is meant as fuzzy constraints imposed on controls and fuzzy goals to be attained the fuzzy constraints are assumed to be state dependent the fuzzy decision is assumed to he the intersection of fuzzy constraints and fuzzy goals the problem is to find an optimal strategy i e a sequence of policies relating the control to the state maximizing the membership function of fuzzy decision for the problem a functional equation is formulated an algorithm of policy iteration type is given the existence of a lime invariant optimal strategy and the convergence of the algorithm are proved a numerical example is shown

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

job search and employment abstract a multistage search method based on fuzzy preference is used to analyze the processes of job search an employment decision based on fuzzy preference results in an exact choice the incorporation of fuzzy interval preference into employment decision however may lead to multiple choices which provides a theoretical justification for the existence of fuzzy supply of labor thus fuzziness in addition to randomness offers an explanation to wage dispersion

fuzzy approach to multilevel knapsack problems this study proposes a fuzzy approach for solving the multiobjective and multilevel knapsack problems kps the problem was first formulated as a multilevel programming problem with multiple decision makers dms then the degree of satisfaction of each dm was established and represented by their individual membership functions the recursive formulation of dynamic programming was used to solve the decisions of the interrelated stages the overall satisfaction of the decision was obtained through this stage wise operation on the hierarchical structure capacity allocation was developed and a step by step solution procedure was illustrated a detailed comparison between multiobjective and multilevel kps was also carried out finally the possible use of turnpike theorem in kps was scrutinized in the fuzzy domain

effect of tolerance in fuzzy linear fractional programming abstract recently fuzzy linear programming flp is evoking greater interest since it can in many ways be well adopted to different decision problems in fuzzy constraints and objective functions of flp tolerance plays an important role in this paper the effect of tolerance in flp is investigated a similar effect is also observed in fuzzy linear fractional programming flfp the presence of singular points is investigated for the flfp

fuzzy dynamic programming approach to hybrid multiobjective multistage decision making problems the purpose of this paper is to develop a new fuzzy dynamic programming approach for solving hybrid multiobjective multistage decision making problems we first present a methodology of fuzzy evaluation and fuzzy optimization for hybrid multiobjective systems in which the qualitative and quantitative objectives are synthetically considered the qualitative objectives are evaluated by decision makers with linguistic variables and the quantitative objectives are converted into proper dimensionless indices after getting the marginal evaluations for each objective a new aggregation method based on the principle of fuzzy pattern recognition is developed to get a global evaluation for all objectives with the global evaluation obtained a fuzzy optimization process is performed then we present a dynamic optimization algorithm by incorporating the fuzzy optimization process with the conventional dynamic programming technique to solve hybrid multiobjective multistage decision making problems a characteristic feature of the approach proposed is that various objectives are synthetically considered by the fuzzy systematic technique instead of the frequently employed weighted average method finally an illustrative example is also given to clarify the developed approach and to demonstrate its effectiveness

goal programming with linear fractional criteria abstract in this paper we present an extension of goal programming to include linear fractional criteria the extension forms a natural link between goal programming gp and multiple objective linear fractional programming molfp

hierarchy of chaotic bands results of a detailed numerical study on the structure of chaotic bands in a forced limit cycle oscillator the brusselator are presented embedded in the chaotic bands of primary bifurcation sequence there are many secondary sequences with both direct and inverse segments within secondary chaotic bands tertiary sequences of similar structure exist this has been shown by using subharmonic stroboscopic sampling combined with power spectra analysis

scaling exponents relations and order dependence for circle maps abstract scaling relations describing devil s staircase structures obtained by iterations of orientation preserving circle maps with minimal slope converging to zero have been investigated in the limit the circle map has a zero slope inflection point and the influence of the order at this point on the scaling exponents is analyzed we study the behavior of the step sizes and the minimal distance to an integer from a non zero cycle point for the super stable orbit and in particular we treat the average behavior this leads to a definition of an average crossover scale exponent and an average measure exponent besides the earlier defined average dimension d we find and d d moreover 1 d finally as the order increases we observe that hourglass structures evolve in the arnold tongues

paths of least resistance in possibilistic production systems abstract we introduce the idea of possibilistic production systems we provide a general algorithm for finding a path from an initial node to a goal node we show that the use of a heuristic function can help in the efficient implementation of the search process we see that a significant role in determining potential optimal paths is played by concentrating on the most difficult activity to accomplish

fault tolerant edge and vertex pancyclicity in alternating group graphs abstract in j m chang j s yang fault tolerant cycle embedding in alternating group graphs appl math comput 197 2008 760 767 the authors claim that every alternating group graph agn is n 4 fault tolerant edge 4 pancyclic which means that if the number of faults f n 4 then every edge in agn f is contained in a cycle of length l for every 4 l n 2 f they also claim that agn is n 3 fault tolerant vertex pancyclic which means that if f n 3 then every vertex in agn f is contained in a cycle of length l for every 3 l n 2 f their proofs are not complete they left a few important things unexplained in this paper we fulfill these gaps and present another proofs that agn is n 4 fault tolerant edge 4 pancyclic and n 3 fault tolerant vertex pancyclic

fuzzy dynamic programming main developments and applications abstract we survey major developments and applications of fuzzy dynamic programming which is here advocated as a promising attempt at making dynamic programming models more realistic by a relaxation of often artificial assumptions of precision as to the constraints goals states and their transitions termination time etc issues related to numerical efficiency are considered applications in r d planning health care and medicine socioeconomic regional development energy systems water and environmental systems chemical engineering etc are surveyed

an optimal stopping problem in dynamic fuzzy systems with fuzzy rewards abstract this paper deals with an optimal stopping problem in dynamic fuzzy systems with fuzzy rewards and shows that the optimal discounted fuzzy reward is characterized by a unique solution of a fuzzy relational equation we define a fuzzy expectation with a density given by fuzzy goals and we estimate discounted fuzzy rewards by the fuzzy expectation this paper characterizes the optimal fuzzy expected value and gives an optimal stopping time

corrections to scaling for period doubling we have studied a multiple scaling which describes corrections to scaling for the period doubling in one dimensional dissipative maps two dimensional areapreserving maps and four dimensional symplectic maps the multiple scaling is seen to be well obeyed and new scaling factors have been found the multiple scaling is also seen to be a very powerful tool for searching for scaling behavior

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

an optimistic decision making in fuzzy environment bellman and zadeh have originated three systems of multistage decision processes in a fuzzy environment deterministic stochastic and fuzzy systems in this article we consider an optimization problem with an optimistic criterion on a fuzzy system by making use of minimization maximization expectation in a fuzzy environment we derive a recursive equation for the fuzzy decision process through invariant imbedding approach by illustrating a three state two decision and two stage model we give an optimal solution through dynamic programming the optimal solution is also verified by the method of multistage fuzzy decision tree table

multifractality of prime numbers abstract the multifractal formalism is applied to prime numbers the spectrum of critical indices is found to be contained in the interval min 1 where min tends to 1 for increasing sets of numbers besides the scaling of moments with respect to the length of intervals the scaling with respect to the sizes of subsets of natural numbers is also considered we have found the cusps in the plots of the functions and we claim that they are not caused by numerical roundings but they are a real effect besides the computer method some analytical calculations are presented

an interactive algorithm for decomposing the parametric space in fuzzy multiobjective dynamic programming problem the aim of this chapter is to study the stability of multi objective dynamic programming modp problems with fuzzy parameters in the objective functions and in the constraints these fuzzy parameters are characterized by fuzzy numbers for such problems the concept and notion of the stability set of the first kind in parametric nonlinear programming problems are redefined and analyzed qualitatively under the concept of pareto optimality an interactive fuzzy decision making algorithm for the determination of any subset of the parametric space that has the same corresponding pareto optimal solution is proposed a numerical example is given to illustrate the method developed in the chapter

fuzzy approaches for multiple objective linear fractional optimization abstract the problem of finding a solution to a multiple objective linear fractional program arises in several real world situations in this paper we advocate that fuzzy sets theory provides a basis for solving this problem with sufficient consistency and rigorousness after representing imprecise aspirations of the decision maker by structured linguistic variables or converting the original problem via approximations or change of variables into a multiple objective linear program techniques of fuzzy linear programming may be used to reach a satisfactory solution it is shown that under reasonable restrictions this solution is efficient pareto optimal for the original problem numerical examples are also included for illustration

interactive fuzzy goal programming approach for bilevel programming problem abstract this paper presents an interactive fuzzy goal programming fgp approach for bilevel programming problems with the characteristics of dynamic programming dp in the proposed approach the tolerance membership functions in fuzzy sets for measuring the degree of satisfactions of the objectives of both the decision makers and degree of achievement of the decision vector controlled by the leader is defined first then the dynamic fuzzy goal programming model for achievement of highest degree unity to the extent possible to each of the defined membership functions for the fuzzy goal is developed to solve the decisions of the interrelated stages a satisfactory solution is derived efficiently by updating the satisfactory degrees of the decision makers with the consideration of overall satisfactory balance between both the levels illustrative numerical examples are provided to demonstrate the feasibility of the proposed method

a generalization of fuzzy multistage decision making and control via linguistic quantifiers abstract multistage incision making control under fuzziness is considered at each control stage a fuzzy constraint is imposed on the control applied and a fuzzy goal is imposed on the state attained an optimal sequence of controls is sought which maximizes the membership function of the fuzzy decision which is assumed to be the intersection of fuzzy constraints and fuzzy goals thin is basically equivalent to the determination of an optimal sequence of controls which best satisfies the fuzzy constraints and fuzzy goals at nil the control stages the generalization proposed in the paper replaces the strict all the something milder for example most thus we seek an optimal sequence of controls which best satisfies the fuzzy constraints and fuzzy goals at for example most of the control stages a calculus of linguistically quantified statements based upon fuzzy sets and possibility theory is used first the algebraic method is employed the resulting control problem is solved by dy

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

stochastic optimization of forward recursive functions this note solves a finite horizon stochastic optimization problem with forward recursive criterion through dynamic programming the forward recursive criterion is wide it includes additive discounted multiplicative discounted risk sensitive minimum and terminal criteria the basic idea is to apply invariant imbedding method for the stochastic optimization the method incorporates recursive accumulation process into dynamics by expanding the original state space

a generalization of discounted multistage decision making andcontrol through fuzzy linguistic quantifiers an attempt to introduce commonsense knowledge abstract this paper is an extension of work published by kacprzyk in 1983 in which by using a fuzzy logic based calculus of linguistically quantified propositions some generalizations of multistage decision making and control have been proposed namely a new class of models has been developed that make it possible roughly speaking to find an optimal sequence of controls best satisfying fuzzy constraints and goals at say most or any other suitable linguistic quantifier for example almost all at least a few of the control stages in this paper we further extend the above mentioned approach by introducing discounting a variable importance of particular control stages in principle the earlier the more important we propose a new model that can find an optimal sequence of controls best satisfying the fuzzy constraints and goals at say most of the earlier control stages we employ a fuzzy logic based calculus of linguistically quantified propositions and develop a branch and bound algo

multiple objective linear fractional programming a fuzzy set theoretic approach abstract a method for optimizing a multiobjective linear fractional programming problem is developed which yields always an efficient solution

sequential decision making in fuzzy environment in this paper we reexamine bellman and zadeh s seminal paper on multistage decision making in a fuzzy environment we propose a correction to their functional equation associated with a stochastic system and formulate a model for deterministic processes governed by fuzzy dynamics

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

a multicriteria fuzzy linear programming method for water supply system development planning abstract a method is presented for solving a multicriteria linear program where the coefficients of the objective functions and the constraints are fuzzy numbers of the l r type assuming the aspiration levels for particular criteria to be fuzzy and basing on comparison of fuzzy numbers the original problem is transformed into a multicriteria linear fractional program the latter is solved using an interactive technique involving a linear programming procedure in the calculation phase this method has been elaborated with a view to the application to the development of a water supply system

dynamic programming for fuzzy systems with fuzzy environment abstract a dynamic programming functional equation for a multi stage decision problem with fuzzy dynamics and environment is formulated and solved by a process of fuzzy interpolation this is an extension of the bellman zadeh model

panconnectivity and pancyclicity of the 3 ary n cube network under the path restrictions the k ary n cube denoted by q n k is one of the most important interconnection networks for parallel computing in this paper we consider the problem of embedding paths and cycles into 3 ary n cubes under the path restrictions let p be a path in q n 3 we show that when v p 2n 3 there exists a path of any length from n 1 to v q n 3 p 1 between two arbitrary nodes in q n 3 p we also prove that when e p 2n 1 there exists a cycle of any length from e p n to v q n 3 in q n 3 passing through p our results are best possible in some sense

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

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

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

continuous time fuzzy decision processes with discounted rewards a continuous time decision process with fuzzy transitions which is termed fuzzy decision process is discussed from a viewpoint of a dynamic fuzzy system by the authors the discounted total reward is given by a fuzzy number and is discussed by the fuzzy max order this paper gives the optimality equation for the fuzzy decision model by a fuzzy differential equation the optimization problem of the discounted total rewards is also discussed by a fuzzy expectation generated from a fuzzy goal and an optimal policy and the permissible range of the decision maker s optimal expected payoffs are given for this problem also an illustrated example is given to explain the idea of the paper

fuzzy multi criteria facility location problem this paper presents a fuzzy goal programming model for locating a single facility on a plane bounded by a convex polygon under the triple criteria maximin minimax and minisum location rectilinear distances are assumed as the scenario may be considered in an urban setting a numerical example is given to illustrate the solution procedure

an approach to stopping problems of a dynamic fuzzy system we formulate a stopping problem for dynamic fuzzy systems concerning with fuzzy decision environment it could be regarded as a natural fuzzification of non fuzzy stopping problem with a deterministic dynamic system the validity of the approach by cuts of fuzzy sets will be discussed in constructing one step look ahead policy of an optimal fuzzy stopping time a numerical example is given to illustrate the theoretical results and to show a merit to use fuzzy stopping times in dynamic fuzzy systems

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

fuzzy stopping in continuous time dynamic fuzzy systems in a continuous time dynamic fuzzy system a stopping problem regarding fuzzy rewards is discussed using fuzzy stopping times the fuzzy rewards are evaluated by an expectation generated from a linear ranking function the optimality equations for the optimal fuzzy rewards are given by variational inequalities and an optimal fuzzy stopping time is constructed under assumptions of monotonicity and regularity for stopping rules a numerical example is given to discuss that fuzzy stopping times are better than non fuzzy ones in comparison between the fuzzy stopping models and the non fuzzy ones

a class of dual fuzzy dynamic programs in this paper we propose a large class of fuzzy dynamic programs by use of the notion of dual binary relation we define a dual fuzzy dynamic program in the class we establish two duality theorems between primal and dual fuzzy dynamic programs one is for the two parametric recursive equations the other is for the nonparametric we specify maximum minimum process and minimum minimum process in fuzzy environment and multiplicative multiplicative process in quasi stochastic environment it is shown that the duality theorems hold between primal and dual programs

markov type fuzzy decision processes with a discounted reward on a closed interval abstract we formulate a new multi stage decision process with markov type fuzzy transition which is termed markov type fuzzy decision process in the general framework of the decision process both of state and action are assumed to be fuzzy itself the transition of states is defined using the fuzzy relation with markov property and the discounted total reward is described as a fuzzy number on a closed bounded interval to discuss the optimization problem a partial order of convex fuzzy numbers is introduced in this paper the discounted total reward associated with an admissible stationary policy is characterized by a unique fixed point of the contractive mapping moreover the optimality equation for the fuzzy decision model is derived under some continuity conditions also an illustrated example is given to explain the theoretical results and the computation in the paper

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

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

associative dynamic programs abstract in this paper we consider finite stage stochastic dynamic programs with associative reward systems an invariant imbedding technique yields a dynamic program with a terminal reward system on an augmented state space according to the associativity we clarify the validity of three possible recursive equations one parametrized equation and two non parametrized ones each of the additive multiplicative and multiplicative additive dynamic programs admits the three recursive equations further the corresponding composite relations among three optimum value functions hold however minimum maximum and fractional dynamic programs admit only the parametrized recursive equation they do not admit the two non parametrized equations further the corresponding relations among the three optimum value functions do not hold