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

partial ordering in l underdeterminate sets abstract the purpose of this paper is to improve results on fuzzy partial orderings obtained by zadeh in 9 we overcome the difficulties connected with the axioms of antisymmetry and linearity moreover if the underlying lattice l is a complete residuated lattice we establish a szpilrajn theorem i e any l fuzzy partial ordering has a linear extension in opposition to zadeh s our point of view is that an axiom of antisymmetry without a reference to a concept of equality is meaningless therefore we first introduce the category lus cf 2 which can be considered as a mathematical model of fuzzy equality and subsequently we specify the axioms of l fuzzy partial orderings with respect to the frame given by lus the axioms we use clearly display the usefulness of having a zadeh like complementation and as a consequence the usefulness of a positivistic and nonintuitionistic frame of study an example concerning l rn which we give clearly shows that the lus version of the szpilrajn theorem cannot be reduced to a fuzzification of an already existing theorem but provides us with additional information

fuzzy control on the basis of equality relations with an example from idle speed control the way engineers use fuzzy control in real world applications is often not coherent with an understanding of the control rules as logical statements or implications in most cases fuzzy control can be seen as an interpolation of a partially specified control function in a vague environment which reflects the indistinguishability of measurements or control values in this paper the authors show that equality relations turn out to be the natural way to represent such vague environments and they develop suitable interpolation methods to obtain a control function as a special case of our approach the authors obtain mamdani s model and can justify the inference mechanism in this model and the use of triangular membership functions not only for the reason of simplified computations and they can explain why typical fuzzy partitions are preferred the authors also obtain a criterion for reasonable defuzzification strategies the fuzzy control methodology introduced in this paper has been applied successfully in a case study of engine idle speed control for the volkswagen golf gti

on strict lower and upper sections of weakly complete fuzzy pre orders based on co implication fono and gwet on strict lower and upper sections of fuzzy orderings fuzzy sets and systems 139 2003 583 599 introduced and studied strict lower section strict upper section and strict order interval of a given weakly complete fuzzy pre order r when the minimal regular fuzzy strict component of r is defined by the residual coimplicator of the lukasiewicz t conorm and they gave an example of application of fuzzy strict lower sections in economics in this paper we generalize their framework for the minimal regular fuzzy strict component of r defined by the residual complicator of a continuous t conorm we show that fuzzy strict sections and fuzzy strict order intervals defined in this general case generate fuzzy topologies

fuzzy option prioritization for the graph model for conflict resolution abstract a fuzzy option prioritization technique is developed to efficiently model uncertain preferences of dms in strategic conflicts as fuzzy preferences by using the decision makers dms fuzzy truth values of preference statements at feasible states within the framework of the graph model for conflict resolution the preference statements of a dm express desirable combinations of options or courses of action and are listed in order of importance a fuzzy truth value is a truth degree expressed as a number between 0 and 1 capturing uncertainty in the truth of a preference statement at a feasible state a fuzzy preference formula is introduced based on the fuzzy truth values of preference statements and it is established that the output of this formula is a fuzzy preference relation it is shown that fuzzy option prioritization can also be used when the truth values of preference statements at feasible states are completely based on boolean logic thereby generating a crisp preference over feasible states that is the same as would be found by employing the existing crisp option prioritization making the crisp option prioritization technique a special case of the fuzzy option prioritization methodology to demonstrate how this methodology can be employed to represent fuzzy preferences in real world decision problems fuzzy option prioritization is applied to an actual dispute over groundwater contamination that took place in elmira ontario canada

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

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

fuzzy sets and vague environments abstract in this paper we propose a natural approach to handle imprecise numbers as they arise for example from measurements fuzzy sets turn out to be a canonical representation for such imprecise numbers that are induced by taking different tolerance or error bounds into account fuzzy sets are induced by scaling factors that describe the magnitude of the imprecision on the other the scaling factors can be derived from given fuzzy sets so that we have a correspondence between scaling factors and fuzzy sets when these concepts are applied to control problems the max min rule is rediscovered as an interpolation technique viewing fuzzy control as an interpolation technique in vague environments enables us to validate various concepts for the design and tuning of fuzzy controllers and also suggests new methods based on clear semantics

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

algorithms for computing the min transitive closure and associated partition tree of a symmetric fuzzy relation abstract based on a previously derived weight driven algorithm for the computation of the t transitive closure of an arbitrary binary fuzzy relation on a finite universe with t a triangular norm we establish three algorithms for the computation of the min transitive closure of a symmetric matrix with elements in 0 1 as a by product these algorithms enable to generate the partition tree associated with the min transitive closure of the given matrix in descending order of the cutting parameter

on an idempotent transformation of aggregation functions and its application on absolutely continuous archimedean copulas in this paper we isolate an interesting transformation of binary aggregation functions from a result on the propagation of transitivity in additive preference structures this transformation is based on a projection technique and results in a smaller or equal binary aggregation function the transformation is idempotent and acts internally on the classes of conjunctors semi copulas and quasi copulas hence also copulas are transformed into quasi copulas but in general not into copulas remarkably the frank copulas are mapped to new copulas

on the compositional characterization of complete fuzzy pre orders complete pre orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations in this paper we investigate this characterization in a fuzzy setting we consider two types of completeness weak completeness and strong completeness and decompose a fuzzy pre order by means of an indifference generator in particular a frank t norm in the weakly complete case we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them in the strongly complete case we lay bare a stronger type of transitivity of the strict preference relation we conclude the paper with a rather negative result there is no hope to obtain a compositional characterization of weakly complete fuzzy pre orders and hence also not of fuzzy pre orders in general

a compendium of fuzzy weak orders representations and constructions the present paper gives a state of the art overview of representation and construction results for fuzzy weak orders we do not assume that the underlying domain is finite instead we concentrate on results that hold in the most general case that the underlying domain is possibly infinite this paper presents three fundamental representation results each of which also provides a construction method i score function based representations ii inclusion based representations iii representations by decomposition into crisp linear orders and fuzzy equivalence relations which also facilitates a pseudo metric based construction

factoring fuzzy transitivity transitivity of a precise weak preference relation implies that its asymmetric component strict preference and its symmetric component indifference are transitive and conversely the transitivity of strict preference and indifference implies that the underlying weak preference relation is transitive this paper examines to what extent analogous results may be obtained for the different transitivity conditions and different methods for deriving strict preference and indifference being used in the literature of choice with fuzzy preferences

matrix representation and extension of coalition analysis in group decision support coalition stability is extended to include uncertain preference using matrix representation within the framework of the graph model for conflict resolution coalition stability analysis within a graph model provides guidance for decision makers and analysts by providing an assessment of whether states that are individually stable are unstable for coalitions in using the graph model method for conflict resolution one carries out stability analysis of a graph model and then follows up with post stability analysis an important component of which is coalition analysis although basic coalition stabilities have been defined for the graph model they are based on a transitive graph which is inconsistent with the standard restriction in the graph model algorithms for implementing these coalition stabilities have not been developed because the nature of their logical representations makes coding difficult additionally existing coalition stabilities apply only to simple preference which limits their utility for exploring more complicated applications in practice in this paper coalition stabilities are extended to include uncertain preference and represented using matrices which are more effective and convenient for computer implementation and for adaptation to new analysis techniques

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

a top down algorithm for generating the hasse tree of a fuzzy preorder closure this paper describes a new top down algorithm for the stepwise generation of the different levels or hasse diagrams of the hasse tree associated to the fuzzy preorder closure min transitive closure of a given reflexive binary fuzzy relation the algorithm is based upon a recently established weight driven method for computing the min transitive closure of a reflexive binary fuzzy relation the way in which this method gradually establishes the fuzzy preorder closure implies that for the generation of a specific level of the hasse tree the newly proposed algorithm does not require the complete computation of this closure

on the structure of pythagorean fields abstract in this paper we study in detail the relationship between fans and valuations first uncovered by brocker see 2 our techniques are somewhat different as our valuations are constructed by brute force from the properties of the fields involved this approach has the advantage in that the valuation theory it yields can be directly related to the structure theory for spaces of orderings as developed by murray marshall see 11 14 we also discuss several applications which include results in k theory

consistent models of transitivity for reciprocal preferences on a finite ordinal scale in this paper we consider a decision maker who shows his her preferences for different alternatives through a finite set of ordinal values we analyze the problem of consistency taking into account some transitivity properties within this framework these properties are based on the very general class of conjunctors on the set of ordinal values each reciprocal preference relation on a finite ordinal scale has both a crisp preference and a crisp indifference relation associated to it in a natural way taking this into account we have started by analyzing the problem of propagating transitivity from the preference relation on a finite ordinal scale to the crisp preference and indifference relations after that we carried out the analysis in the opposite direction we provide some necessary and sufficient conditions for that propagation and therefore we characterize the consistent class of conjunctors in each direction

the graded witt ring and galois cohomology ii a primary problem in the theory of quadratic forms over a field f of characteristic different from two is to prove that the rings hq f and gwf are isomorphic here hqf h gal fq f z 2z where fq is the quadratic closure of f and gwf is the graded witt nrng associated to the fundamental ideal of even dimensional forms in the witt ring wf of f in this paper we assume we are given a field extension k of f such that wk is close to wf or hq k is close to hq f a method is developed to obtain information about these graded rings over f and its 2 extensions from information about the corresponding graded ring of k this relative theory extends and includes the previously developed absolute case where k fq applications are also given to show that hq f and gwf are isomorphic for a collection of fields anrsing naturally from the theory of abstract witt rings

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

rigid elements valuations and realization of witt rings to the theory over the residue class field of the valuation this theorem generalizes to any 2 henselian valuation cf kl sect 12 21 let v be the valuation let a e k f o 1 be such that v a is not divisible by two the key idea behind the proof is that the quadratic form 1 a only represents elements in u a such an element a is called

demuskin fields with valuations we study the arithmetic structure of fields f of characteristic neq p containing a primitive pth root of unity for which the maximal pro p galois group of f is a finitely generated demuskin group

witt rings and almost free pro 2 groups let f be a field of characteristic not 2 wf the witt ring of quadratic forms over f if the maximal ideal consisting of even dimensional forms and i f its square let g 2 gal f 2 f where f 2 is the maximal 2 extension quadratic closure of f in this note we prove a decomposition theorem for wf in the case that z f is torsion free and use it to obtain a new proof of a theorem of ershov on the structure of almost free pro a groups e theorem 41 in what follows w f denotes the torsion subgroup of wf and wredf wf w f is the reduced witt ring the set of orderings on f will be denoted by x and will be given the harrison topology with subbasis consisting of the sets h a e x 1 0 a f is called a sap field if every open and closed clopen subset of x is equal to some h a if r r are abstract witt rings then r x r denotes their product in the category of witt rings ml all unexplained notations and terminology about quadratic forms and witt rings can be found in l and m for elementary facts about profinite groups the reader is referred to ri following kijima and nishi kn the field f is called quasi pythugorean if z f is torsion free i e u f d 2 equivalently if the kaplansky radical of f consists of all nonzero sums of two squares ell corollary 2 91 compare el2 classification theorem 23 examples of these fields include formally real generalized hilbert fields k pseudo real closed fields p and fields satisfying the strong hasse principle h4 every totally indefinite quadratic form of dimension 4 over f is isotropic i e ii f 2 a formally real field satisfies h if and only if it is a quasi pythagorean sap field elp in ko koziol gives a criterion characterizing the class of quasi pythagorean algebraic extensions of q such fields necessarily satisfy h the best known examples of fields satisfying h are formally real

duality in fuzzy multi criteria and multi constraint level linear programming a parametric approach this paper presents a parametric approach for duality in fuzzy multi criteria and multi constraint level linear programming mc2lp which extends fuzzy linear programming approaches first the mc2 simplex method is used to solve the crisp prima dual mc2lp pair and then through these crisp formulations separate membership functions are constructed for fuzzy primal and dual program by considering the corresponding primal and dual decisions for each program a set of fuzzy potential solutions is determined in terms of the membership function and the related optimal solution with a certain range of decision parameters finally using the primal and dual membership functions a fuzzy weak duality function is obtained for any pair of primal and dual fuzzy potential solutions

numerical representation of transitive fuzzy relations we show that there exists a continuous fuzzy quasi order r on the set of real numbers r such that for any fuzzy quasiorder r on a finite set x there is a mapping f x r such that r x y r f x f y for all x y x

a fuzzy programming approach for solving a multiple criteria and multiple constraint level linear programming problem abstract this paper presents a solution method to the multiple criteria and multiple constraint level linear program mc2lp using a fuzzy linear programming approach this approach finds a potential solution to the problem by solving a fuzzy linear program which is different than the usual approach of mc2 simplex method

modeling rationality in a linguistic framework in classical decision theory there exists a large class of rationality models which try to capture different kinds of behavior when individuals compare by pairs a set of alternatives all these models assume that decision makers have dichotomous preferences however in real decisions individuals feel different degrees of preference in this paper we have checked the mentioned models in a real case where different kinds of linguistic preferences are allowed after the empirical analysis the main conclusion is that the fulfillment of rational conditions decreases when individuals have non extreme preferences based on the obtained empirical evidences we propose some classes of transitivity conditions in the framework of linguistic preferences

a hitchhiker s guide to quasi copulas abstract this paper presents a review of the concept of an n quasi copula starting from the characterization of functions that can be derived from operations on random variables we recall several other characterizations and properties that have been proven in the literature highlighting the differences between the bivariate case and the higher dimensional case additionally we recall several applications of n quasi copulas such as their role in the study of bounds on sets of n copulas we also recall various classes of n quasi copulas that have been studied in the literature finally we present some questions that have not yet been answered in the literature

a fuzzy approach to hierarchical multiobjective programming problems and its application to an industrial pollution control problem in this paper we focus on hierarchical multiobjective programming problems where multiple decision makers in a hierarchical organization have their own multiple objective functions and propose an interactive algorithm based on the dual decomposition method to obtain the satisfactory solution which reflects not only the hierarchical relationships between multiple decision makers but also their own preferences for their objective functions in the proposed algorithm assuming that each of the decision makers has fuzzy goals for his her objective functions corresponding membership functions are elicited from the decision makers in their subjective manner in order to deal with hierarchical multiobjective programming problems a new kind of pareto optimality concept in membership spaces is defined and the concept of decision powers of multiple decision makers in a hierarchical decision structure is introduced the proposed algorithm is applied to the industrial pollution control problem in osaka city in japan in order to show the efficiency of the proposed method

bell type inequalities for quasi copulas we present all bell type inequalities regarding at most four random events of which not more than two are intersected at the same time reformulating these inequalities in the context of fuzzy probability calculus leads to related inequalities on commutative conjunctors we prove that some of these inequalities hold for quasi copulas or copulas additionally for the important parametric family of frank t norms which are also copulas we identify for each of the inequalities the parameter values for which it is fulfilled thereby completing the results of pykacz and d hooghe

axiomatizing logics of fuzzy preferences using graded modalities abstract the aim of this paper is to propose a many valued modal framework to formalize reasoning with both graded preferences and propositions in the style of van benthem et al s classical modal logics for preferences to do so we start from bou et al s minimal modal logic over a finite and linearly ordered residuated lattice we then define appropriate extensions on a multi modal language with graded modalities both for weak and strict preferences and with truth constants actually the presence of truth constants in the language allows us to show that the modal operators and of the minimal modal logic are inter definable finally we propose an axiomatic system for this logic in an extended language where the preference modal operators are definable and prove completeness with respect to the intended graded preference semantics

strength of preference in graph models for multiple decision maker conflicts abstract models of strength of preference are incorporated into the graph model for conflict resolution to study realistically multi objective decision making situations in disputes with any finite number of participants a decision maker s preference is expressed using a triplet of binary relations that allows preference of one state over another to be strong or weak and also allows indifference four basic stability definitions used in the graph model to represent human behavior in conflicts are extended to the context of the new triplet preference structure producing strong and weak stabilities theorems governing the relationships among standard strong and weak stabilities are presented finally application of the new preference representation to an international water resource management conflict confirms the practical utility of the triplet structure and the significance of the strategic insights it can provide

representations and constructions of similarity based fuzzy orderings this contribution advocates the dissemination of similarity based fuzzy orderings a generalized concept of fuzzy orderings which appears to be unavoidable as soon as the classical correspondences between preorderings equivalence relations and orderings are taken into account in contrast to earlier papers on the topic this paper is focused on representation and construction results which once more demonstrate the feasibility of the proposed concept

additive decomposition of fuzzy pre orders fuzzy pre orders reflexive and min transitive fuzzy relations constitute an important class of fuzzy relations by means of an indifference generator a fuzzy pre order can be decomposed additively into two parts an indifference relation and a strict preference relation when using a frank t norm as indifference generator we fully characterize the transitivity of these parts only in case the minimum operator is used as generator both parts are min transitive the transitivity of the indifference relation is determined by the frank t norms while the transitivity of the strict preference relation is determined by transforms of the nilpotent minimum t norm

a plea for the use of lukasiewicz triplets in the definition of fuzzy preference structures ii the identity case abstract it has been shown in the first part of this paper that the concept of a fuzzy preference structure is only meaningful provided that the de morgan triplet involved contains a continuous archimedean triangular norm having zero divisors or hence a transform of the ukasiewicz triangular norm in this second part additional arguments for this statement are supplied in what we call the identity case is the identity mapping and the involutive negator is the standard negator first it is shown that the use of a continuous non archimedean triangular norm having zero divisors in the definition of a fuzzy preference structure indeed is possible secondly using such a triangular norm implies that at least in the square 0 4 5 2 it actually behaves as the ukasiewicz triangular norm furthermore an important transformation theorem indicates that any fuzzy preference structure with respect to a de morgan triplet containing such a continuous non archimedean triangular norm having zero divisors can be transformed into a fuzzy preference structure with respect to the standard ukasiewicz triplet these additional arguments conclude in a convincing way our plea for the use of ukasiewicz triplets in the definition of fuzzy preference structures

an integrated algebraic approach to conflict resolution with three level preference an integrated algebraic approach is developed to calculate stabilities in multiple decision maker graph models with three levels of preference the algebraic approach establishes an integrated paradigm for stability analysis and status quo analysis under different preference structures such as two level preference and three level preference difficulties in coding algorithms to analyze stabilities rooted in their logical representation led to the development of matrix representations of preference and explicit matrix calculations to determine stabilities here the algebraic approach is used to represent graph models with three levels of preference and to conduct stability analysis for such models the algebraic approach facilitates the development of new stability concepts and algorithms to calculate them and reveals an inherent link between status quo analysis and traditional stability analysis hence it will facilitate the design of an integrated decision support system for the graph model for conflict resolution

fuzzy preference structures without incomparability abstract in this paper we establish important relationships between the basic properties of the components of a fuzzy preference structure without incomparability this study is carried out for the fuzzy preference structures introduced recently by de baets van de walle and kerre a set of remarkable theorems gives detailed insight in the relationships between the sup t transitivity of the fuzzy preference and indifference relations and the sup t transitivity of the fuzzy large preference relation several paths of thought involving t norms with or without zero divisors are explored and where required illustrative counterexamples confirm the falsity of certain implications finally we introduce the t n ferrers property of a binary fuzzy relation and show that the fuzzy preference and fuzzy large preference relations share certain types of this ferrers property

a formal study of linearity axioms for fuzzy orderings abstract this contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings different existing concepts are evaluated with respect to three fundamental correspondences from the classical case linearizability of partial orderings intersection representation and one to one correspondence between linearity and maximality as a main result we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case if we do not require a one to one correspondence between linearity and maximality however we obtain that an implication based definition appears to constitute a sound compromise in particular if ukasiewicz type logics are considered

using analytic hierarchy process and fuzzy set theory to rate and rank the disability abstract any discrimination against qualified disabled people in all employment practices is violating the americans with disabilities act ada and it must be prohibited assigning qualified disabled people to a proper job or making reasonable accommodations for disabled employees is not only ethical but also necessary for reducing the social economic burden therefore in order to place qualified disabled people in appropriate job positions a multidimensional assessment method should be developed for measuring and interpreting the disabled people s residual capabilities with respect to their strengths weaknesses and their compatibility to job requirements and work environments thus the primary purpose of this study is to discuss how to structure a hierarchy related to the problem of deriving an overall disability index odi for measuring the individual s disability the analytic hierarchy process along with the entropy theory and fuzzy set theory is applied for eliciting the weights among the attributes and aggregating the multiple attributes into a single measurement we believe that the resultant odi is meaningful and useful for both the evaluator who supervises a rehabilitation program and the human resource director who engages in a hiring process

fuzzy levels of preference strength in a graph model with multiple decision makers abstract a new hybrid preference structure combining multiple level strength of preference and fuzzy preference is proposed to facilitate the modeling and analysis of strategic conflicts involving multiple decision makers dms with complex preferences using the graph model for conflict resolution gmcr the new preference structure named fuzzy strength of preference provides a more comprehensive and flexible representation of dms relative preferences among states a key contribution of this paper is to redefine four graph model stability definitions for fuzzy preferences fuzzy nash stability fnash fuzzy general metarationality fgmr fuzzy symmetric metarationality fsmr and fuzzy sequential stability fseq as general stabilities strong stabilities and weak stabilities at each level permitting deeper analysis of graph models with the new preference structure the resulting methodology can be utilized to model and analyze complex multiple dm conflicts thereby enhancing the capability of the graph model to provide strategic insights a graph model of the zhanghe river water allocation conflict in china demonstrates that the method can find the evolution path of a conflict and give new strategic insights for both practitioners and researchers the fuzzy strength of preference framework makes gmcr more capable for addressing a wider range of practical conflicts

inverse engineering preferences in simple games a method for inverse engineering decision makers preferences based on observable behaviour is designed this technique allows analysts to narrow down the list of potential preference rankings of possible states in a conflict for each decision maker using probabilities and expected values during the inverse engineering procedure the list of all possible preference rankings is narrowed as decision makers move and counter move accurate preference information is key to building quality conflict and game models however preference rankings for decision makers are often difficult to obtain directly a simple two decision maker four state game is used to demonstrate the applicability of the method and to illustrate the insights it provides

relations in fuzzy class theory this paper studies fuzzy relations in the graded framework of fuzzy class theory fct this includes i rephrasing existing work on graded properties of binary fuzzy relations in the framework of fuzzy class theory and ii generalizing existing crisp results on fuzzy relations to the graded framework our particular aim is to demonstrate that fuzzy class theory is a powerful and easy to use instrument for handling fuzzified properties of fuzzy relations this paper does not rephrase the whole theory of fuzzy relations instead it provides an illustrative introduction showing some representative results with a strong emphasis on fuzzy preorders and fuzzy equivalence relations

linear optimal designs and optimal contingency plans in optimal design problems we try to select a subset from the opportunity set for commitment for a period of time as the external parameters can vary with time optimal contingency plans must be constructed to cope with the external conditions we describe the formulation and methods for locating potentially good designs and their corresponding optimal contingency plans using a multiple criteria and multiple constraint level mc2 simplex method to solve the optimal design problems which have been shown to become amenable to the techniques of decision making under uncertainty

kendall distribution functions and associative copulas in this paper we prove that any kendall distribution function is the kendall distribution function of some associative copula we use this result to show that each equivalence class of the relation to have the same kendall distribution function as defined on the set of copulas contains a unique associative representative

a plea for the use of lukasiewicz triplets in the definition of fuzzy preference structures i general argumentation abstract the generalization of the concept of a classical or crisp preference structure to that of a fuzzy preference structure expressing degrees of strict preference indifference and incomparability among a set of alternatives requires the choice of a de morgan triplet i e of a triangular norm and an involutive negator the resulting concept is only meaningful provided that this choice allows the representation of truly fuzzy preferences more specifically one of the degrees of strict preference indifference or incomparability should always be unconstrained to the preference modeller this intuitive requirement is violated when choosing a triangular norm without zero divisors since in that case fuzzy preference structures reduce to classical preference structures and hence none of the degrees can be freely assigned furthermore it is shown that the choice of a continuous non archimedean triangular norm having zero divisors is not compatible with our basic requirement the sets of degrees of strict preference indifference and incomparability in 0 1 are always bounded from above by a value strictly smaller than 1 these fundamental results imply that when working with a continuous triangular norm only archimedean ones having zero divisors are suitable candidates these arguments sufficiently support our plea for the use of ukasiewicz triplets in the definition of fuzzy preference structures

quasi copulas and signed measures we study the relationship between multivariate quasi copulas and measures that they may or may not induce on 0 1 n we first study the mass distribution of the pointwise best possible lower bound for the set of n quasi copulas for n 3 as a consequence we show that not every n quasi copula induces a signed measure on 0 1 n

on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations in this paper we study systems of fuzzy relation inequalities and equations of the form u v i v i u i i where u is an unknown and v i i i are given fuzzy relations the dual systems v i u u v i i i their conjunctions the systems of the form u v i v i u i i and certain special types of these systems we call them weakly linear systems for each weakly linear system with a complete residuated lattice as the underlying structure of truth values we prove the existence of the greatest solution and we provide an algorithm for computing the greatest solution which works whenever the underlying complete residuated lattice is locally finite otherwise we determine some sufficient conditions under which the algorithm works the algorithm is iterative and each its single step can be viewed as solving of a particular linear system weakly linear systems emerged from the fuzzy automata theory but we show that they also have important applications in other fields e g in the concurrency theory and social network analysis

i fuzzy alexandrov topologies and specialization orders the aim of this paper is to investigate the relationships between certain generalized topological structures and fuzzy order structures on a universe set x in details a fuzzy preordered relation induces an i fuzzy alexandrov topology and an i fuzzy topology induces a fuzzy preordered relation which turns out to be a specialization order in i fuzzy setting in addition the representation theorem of fuzzy preorders by i fuzzy topologies is obtained these results coincide with the conclusions obtained in literature in the version of two valued logic

a matrix based approach to searching colored paths in a weighted colored multidigraph an algebraic approach to finding all edge weighted colored paths within a weighted colored multidigraph is developed generally the adjacency matrix represents a simple digraph and determines all paths between any two vertices and is not readily extendable to colored multidigraphs to bridge the gap a conversion function is proposed to transform the original problem of searching edge colored paths in a colored multidigraph to a standard problem of finding paths in a simple digraph moreover edge weights can be used to represent some preference attribute its potentially wide realm of applicability is illustrated by a case study status quo analysis in the graph model for conflict resolution the explicit matrix function is more convenient than other graphical representations for computer implementation and for adapting to other applications additionally the algebraic approach reveals the relationship between a colored multidigraph and a simple digraph thereby providing new insights into algebraic graph theory

conflict models in graph form solution concepts and their interrelationships abstract a comprehensive approach for modeling conflicts is the recently developed graph form of conflict models within the flexible structure of the graph model solution concepts including nash stability the various metagame techniques the sequential stability method of fraser and hipel and stackelberg equilibrium are defined and characterized these stability principles are compared with the limited move and non myopic stabilities developed previously for the graph model framework and mathematical interrelationships in this very general context are derived as well some steps are taken toward the development of a taxonomy of solution concepts for the graph form

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

building consensus in group decision making with an allocation of information granularity abstract consensus is defined as a cooperative process in which a group of decision makers develops and agrees to support a decision in the best interest of the whole it is a questioning process more than an affirming process in which the group members usually modify their choices until a high level of agreement within the group is achieved given the importance of forming an accepted decision by the entire group the consensus problem has attained a great attention as it is a major goal in group decision making in this study we propose the concept of the information granularity being regarded as an important and useful asset supporting the goal to reach consensus in group decision making by using fuzzy preference relations to represent the opinions of the decision makers we develop a concept of a granular fuzzy preference relation where each pairwise comparison is formed as a certain information granule say an interval fuzzy set rough set and alike instead of a single numeric value as being more abstract the granular format of the preference model offers the required flexibility to increase the level of agreement within the group using the fact that we select the most suitable numeric representative of the fuzzy preference relation

fuzzy preferences in the graph model for conflict resolution a new framework for the graph model for conflict resolution is developed so that decision makers dms with fuzzy preferences can be included in conflict models a graph model is both a formal representation for multiple participant multiple objective decision problems and a set of analysis procedures that add insights into them within the new framework graph models can include and integrate into the analysis both certain and uncertain information about dms preferences one key contribution of this study is to extend the four basic stability definitions for two or more dms to models with fuzzy preferences together fuzzy nash stability fuzzy general metarationality fuzzy symmetric metarationality and fuzzy sequential stability provide anuanced description of human behavior a state is fuzzy stable for a dm if a move to any other state is not sufficiently likely to yield an outcome which the dm prefers where sufficiency is measured according to a fuzzy satisficing threshold that is the characteristic of the dm a fuzzy equilibrium which is an outcome that is fuzzy stable for all dms therefore represents a possible resolution of the strategic conflict the practical application and interpretation of these new stability definitions are illustrated with an example

collective transitivity in majorities based on difference in support a common criticism to simple majority voting rule is the slight support that such rule demands to declare an alternative as a winner among the distinct majority rules used for diminishing this handicap we focus on majorities based on difference in support with these majorities voters are allowed to show intensities of preference among alternatives through reciprocal preference relations these majorities also take into account the difference in support between alternatives in order to select the winner in this paper we have provided some necessary and sufficient conditions for ensuring transitive collective decisions generated by majorities based on difference in support for all the profiles of individual reciprocal preference relations these conditions involve both the thresholds of support and some individual rationality assumptions that are related to transitivity in the framework of reciprocal preference relations

compatible valuations and generalized milnor k theory given a field and a subgroup of there is a minimal group for which there exists an compatible valuation whose units are contained in assuming that has finite index in and contains for prime we describe in computable theoretic terms