existence of multiple periodic solutions for an sir model with seasonality abstract we study an sir model with a seasonal contact rate and a staged treatment strategy which is an extension of our previous work z bai y zhou existence of two periodic solutions for a non autonomous sir epidemic model appl math model 35 2011 382 391 it is proved that the persistence and extinction of the disease are determined by the basic reproductive number r 0 and a threshold parameter r c it is obtained that the model exhibits two different bistable behaviors under certain conditions the stable disease free state coexists with a stable endemic periodic solution and three endemic periodic solutions coexist with two of them being stable numerical simulations are presented to illustrate theoretical results

bifurcations in an epidemic model with constant removal rate of the infectives an epidemic model with a constant removal rate of infective individuals is proposed to understand the effect of limited resources for treatment of infectives on the disease spread it is found that it is unnecessary to take such a large treatment capacity that endemic equilibria disappear to eradicate the disease it is shown that the outcome of disease spread may depend on the position of the initial states for certain range of parameters it is also shown that the model undergoes a sequence of bifurcations including saddle node bifurcation subcritical hopf bifurcation and homoclinic bifurcation

a polynomial time approximation scheme for the two stage multiprocessor flow shop problem in this paper we investigate the two stage multiprocessor flow shop scheduling problem f2 p cmax where the numbers m1 and m2 of machines available in the two stages are part of the input we demonstrate the existence of a polynomial time approximation scheme for this problem this result solves the simplest case of an open problem that has been posed by leslie hall in a recent paper hall 1995 an extension of our algorithm yields an approximation scheme for the closely related two stage multiprocessor job shop problem

a fuzzy solution to a majority voting game abstract in the fiorina plott majority voting game experiment several puzzling phenomena were discovered 1 in the experiment series where the payoffs were small in general the predictive performance of the core was markedly worse than in the high payoff series 2 the core outcome was found to be defeatable by another proposal 3 some of the majority winning social outcomes were pareto suboptimal 4 even when there was no theoretical equilibrium point there was a clear clustering of outcomes the paper outlines a fuzzy decision making procedure which seems to be capable of explaining all the puzzling features 1 4 in the absence of data on membership functions the explanation provided is of non predictive nature it renders the observed outcomes plausible without precisely predicting each one of them

a periodic epidemic model in a patchy environment an epidemic model in a patchy environment with periodic coefficients is investigated in this paper by employing the persistence theory we establish a threshold between the extinction and the uniform persistence of the disease further we obtain the conditions under which the positive periodic solution is globally asymptotically stable at last we present two examples and numerical simulations

approaches to collective decision making with fuzzy preference relations abstract recent experimental studies show that the predictive accuracy of many of the solution concepts derived from the collective decision making theory leaves much to be desired in a previous paper the author attempted to explain some of the inaccuracies in terms of the fuzzy indifference regions of the individuals participating in the voting game this paper gives straightforward generalizations of the solutions concepts in terms of the fuzzy social or individual preference relations it turns out that some of these new solution concepts cotain their nonfuzzy counterparts as subsets others in turn are subsets of their nonfuzzy counterparts we also discuss a method of aggregating individual nonfuzzy preferences so as to get a fuzzy social preference relation and furthermore a nonfuzzy social choice set

job shop scheduling with unit processing times we consider randomized algorithms for the preemptive job shop problem or equivalently the case in which all operations have unit length we give an approximation for the case of two machines where 1 45 an improved approximation ratio of o log m log log m for an arbitrary number m of machines and the first 2 e approximation for a constant number of machines the first result is via an approximation algorithm for a string matching problem that is of independent interest

group decision making with a fuzzy linguistic majority abstract the determination of solutions in group decision making is considered the point of departure is a collection of individual fuzzy preference relations a solution is derived either directly from the individual fuzzy preference relations or by constructing first a social fuzzy preference relation as opposed to conventional approaches in which a crisp threshold type majority rule is used we employ a fuzzy majority rule specified by a fuzzy linguistic quantifier e g most much more than 50 etc a calculus of linguistically quantified propositions is applied using the fuzzy majority various solution concepts are derived mainly of the type of core minimax opposition set and consensus winner

short shop schedules we consider the open shop job shop and flow shop scheduling problems with integral processing times we give polynomial time algorithms to determine if an instance has a schedule of length at most 3 and show that deciding if there is a schedule of length at most 4 is x1d4a9 x1d4ab complete the latter result implies that unless x1d4ab x1d4a9 x1d4ab there does not exist a polynomial time approximation algorithm for any of these problems that constructs a schedule with length guaranteed to be strictly less than 5 4 times the optimal length this work constitutes the first nontrivial theoretical evidence that shop scheduling problems are hard to solve even approximately

performance evaluation an integrated method using data envelopment analysis and fuzzy preference relations in a multi attribute decision making madm context the decision maker needs to provide his preferences over a set of decision alternatives and constructs a preference relation and then use the derived priority vector of the preference to rank various alternatives this paper proposes an integrated approach to rate decision alternatives using data envelopment analysis and preference relations this proposed approach includes three stages first pairwise efficiency scores are computed using two dea models the ccr model and the proposed cross evaluation dea model second the pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation third by use of the row wise summation technique we yield a priority vector which is used for ranking decision making units dmus for the case of a single output and a single input the preference relation can be directly obtained from the original sample data the proposed approach is validated by two numerical examples

persistence and extinction of disease in non autonomous sirs epidemic models with disease induced mortality abstract in this paper non autonomous sirs epidemic models with bilinear incidence and disease induced mortality are studied under the quite weak assumptions the sufficient and necessary conditions on the permanence and strong persistence of the disease and the sufficient condition on the extinction of the disease are established some new threshold values of the integral form r 0 r 1 and r 2 are obtained we prove that the disease is permanent if and only if r 0 0 and if r 1 0 or r 2 0 then the disease is extinct as applications of the main results we discuss the periodic and almost periodic models the corresponding basic reproductive numbers r 0 are obtained we show that if r 0 1 then the disease is permanent and if r 0 1 then the disease is extinct

social choice and lukasiewicz logic abstract an approach to the social choice problem based on multiple valued logic is suggested it is assumed that individual preference are classical binary relations whereas the social preference is a valued binary relation the role of lukasiewicz logic in modelling valued transitive relations is discussed we argue that this logic is a natural choice for contructing valued social preference the paper is concerned with arrow s theory of social welfare functions standard conditions of unrestricted domain pareto independence of irrelevant alternatives and nondictatorship are employed we show that certain means in binary relations spaces are social welfare functions satisfying conditions of the classical arrow s possibility theorem it is also shown that collective choice rules given by means are completely characterized by the conditions of unrestrict domain anomymity neutrality and positive responsiveness

fuzzy intersection graphs abstract this paper extends the use of graphs to the modelling of systems under unceertainty when random assumptions do not adequately hold the paper includes the theoretical and computational aspects which are derived from the matrix representation of fuzzy intersection graphs their ability for applications is also discussed

choosing owa operator weights in the field of social choice one of the most important issues in the theory of owa operators is the determination of associated weights this matter is essential in order to use the best suited owa operator in each aggregation process given that some aggregation processes can be seen as extensions of majority rules to the field of gradual preferences it is possible to determine the owa operator weights by taking into account the class of majority rule that we want to obtain when individuals do not grade their pairwise preferences however a difficulty with this approach is that the same majority rule can be obtained through a wide variety of owa operators for this reason a model for selecting the best suited owa operators is proposed in this paper

selective majority additive ordered weighting averaging operator abstract usually in order to summarize various opinions about a particular situation mainly product or service valuation on internet a process called aggregation is used this process basically consists of determining the appropriate value to represent the majority s opinion and many strategies and operators can be used for this purpose simple arithmetic mean is widely used to resume several opinions in a single value but this value is generally not representative or it is affected by the extreme values an alternative to aggregate opinions are the ordered weighting averaging owa operators nevertheless they have distribution problems when applied to aggregates with cardinalities these problems may be solved by using majority additive owa ma owa operator a sort of arithmetic mean of arithmetic means ma owa operator works adequately but in some cases discards the minority s opinion specifically when it does not coincide with the largest cardinality value in order to generalize the usage of ma owa operator the rest of opinions are taken into account using a cardinality relevance factor this paper introduces a selective majority additive owa sma owa which manages the significance of all opinions varying the cardinality relevance factor mathematical extension of sma owa its properties and some illustrative examples are presented in this article

improved approximation algorithms for shop scheduling problems in the job shop scheduling problem there are m machines and n jobs a job consists of a sequence of operations each of which must be processed on a specified machine and the aim is to complete all jobs as quickly as possible this problem is strongly cal np hard even for very restrictive special cases the authors give the first randomized and deterministic polynomial time algorithms that yield polylogarithmic approximations to the optimal length schedule these algorithms also extend to the more general case where a job is given not by a linear ordering of the machines on which it must be processed but by an arbitrary partial order comparable bounds can also be obtained when there are m types of machines a specified number of machines of each type and each operation must be processed on one of the machines of a specified type as well as for the problem of scheduling unrelated parallel machines subject to chain precedence constraints

approximation schemes for job shop scheduling problems with controllable processing times in this paper we study the job shop scheduling problem under the assumption that the jobs have controllable processing times the fact that the jobs have controllable processing times means that it is possible to reduce the processing time of the jobs by paying a certain cost we consider two models of controllable processing times continuous and discrete for both models we present polynomial time approximation schemes when the number of machines and the number of operations per job are fixed

stability and bifurcation of an sir epidemic model with nonlinear incidence and treatment the dynamical behaviors of an sir epidemic model with nonlinear incidence and treatment is investigated it is assumed that treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity it is found that a backward bifurcation occurs if the capacity is small it is also found that there exist bistable endemic equilibria if the capacity is low theoretical and numerical results suggest that decreasing the basic reproduction number below one is insufficient for disease eradication

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

approximating disjoint path problems using packing integer programs in a packing integer program we are given a matrix a and column vectors b c with nonnegative entries we seek a vector x of nonnegative integers which maximizes c t x subject to ax leq b the edge and vertex disjoint path problems together with their unsplittable flow generalization are np hard problems with a multitude of applications in areas such as routing scheduling and bin packing these two categories of problems are known to be conceptually related but this connection has largely been ignored in terms of approximation algorithms we explore the topic of approximating disjoint path problems using polynomial size packing integer programs motivated by the disjoint paths applications we introduce the study of a class of packing integer programs called column restricted we develop improved approximation algorithms for column restricted programs a result that we believe is of independent interest additional approximation algorithms for disjoint paths are presented that are simple to implement and achieve good performance when the input has a special structure

simple and absolute special majorities generated by owa operators abstract simple and absolute special majorities are decision procedures used very often in real life however these rules do not allow individuals to express the intensity with which they prefer some alternatives to others in order to consider this situation individual preferences can be represented by fuzzy preferences through values located between 0 and 1 then the collective preference is obtained by means of aggregation functions in this paper we use ordered weighted averaging owa operators in order to aggregate individual preferences and we generalize simple and absolute special majorities by means of owa operators

spectral bound and reproduction number for infinite dimensional population structure and time heterogeneity spectral bounds of quasi positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ordinary differential equations the sign of the spectral bound of the variational matrix at 0 decides whether at low density the population becomes extinct or grows another important threshold parameter is the reproduction number mathcal r which is the spectral radius of a related positive matrix as is well known the spectral bound and mathcal r 1 have the same sign provided that the matrices have a particular form the relation between spectral bound and reproduction number extends to models with infinite dimensional state space and then holds between the spectral bound of a resolvent positive closed linear operator and the spectral radius of a positive bounded linear operator we also extend an analogous relation between the spectral radii of two positive linear operators which is relevant for discrete time models we illustrate the general theory

makespan minimization in open shops a polynomial time approximation scheme in this paper we demonstrate the existence of a polynomial time approximation scheme for makespan minimization in the open shop scheduling problem with an arbitrary fixed numberm of machines for the variant of the problem where the number of machines is part of the input it is known that the existence of an approximation scheme would implyp np hence our result draws a precise separating line between approximable cases i e withm fixed and non approximable cases i e withm part of the input of this shop problem 1998 the mathematical programming society inc published by elsevier science b v

a fuzzy multicriteria group decision making approach to select configuration items for software development selection of configuration items in software configuration management is important to determine the software quality and reduce the development time and cost the objective of this research is to develop a multicriteria group decision making model based on fuzzy set theory to improve the configuration items selection process since most information available in this stage is not numerical fuzzy set theory is used to represent the evaluation ratings of candidate items the developed model ranks candidate items into partial or complete orders that can assist decision makers in selecting more proper set of configuration items the consensus measures are also developed to determine the group acceptability of the obtained ranking orders in addition sensitivity analysis can be performed to examine the solution robustness an example of flight simulator development project is used to illustrate the concept developed

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

hardness of approximating flow and job shop scheduling problems we consider several variants of the job shop problem that is a fundamental and classical problem in scheduling the currently best approximation algorithms have worse than logarithmic performance guarantee but the only previously known inapproximability result says that it is np hard to approximate job shops within a factor less than 5 4 closing this big approximability gap is a well known and long standing open problem this article closes many gaps in our understanding of the hardness of this problem and answers several open questions in the literature it is shown the first nonconstant inapproximability result that almost matches the best known approximation algorithm for acyclic job shops the same bounds hold for the general version of flow shops where jobs are not required to be processed on each machine similar inapproximability results are obtained when the objective is to minimize the sum of completion times it is also shown that the problem with two machines and the preemptive variant with three machines have no ptas

an approach to group choice with fuzzy preference relations abstract this paper presents an approach to group choice with fuzzy preference relations the experts judgements are presented as fuzzy preorder relations as group choice we assume the nearest relation to these relations in the sense of the hamming distance in this paper an algorithm for finding a preorder relation as an expression of group choice is given and a way is pointed out to find the best alternative on the basis of this group choice furthermore the connection between the impossibility theorem of arrow and this group choice is considered

minimizing average completion time in the presence of release dates a natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time in this paper we consider the problem of schedulingn jobs that are released over time in order to minimize the average completion time of the set of jobs in contrast to the problem of minimizing average completion time when all jobs are available at time 0 all the problems that we consider are np hard and essentially nothing was known about constructing good approximations in polynomial time we give the first constant factor approximation algorithms for several variants of the single and parallel machine models many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both many of the algorithms generalize to the minimization of averageweighted completion time as well 1998 the mathematical programming society inc published by elsevier science b v

approximability of flow shop scheduling shop scheduling problems are notorious for their intractability both in theory and practice in this paper we construct a polynomial approximation scheme for the flow shop scheduling problem with an arbitrary fixed number of machines for the three common shop models open flow and job this result is the only known approximation scheme since none of the three models can be approximated arbitrarily closely in the general case unless p np the result demonstrates the approximability gap between the models in which the number of machines is fixed and those in which it is part of the input of the instance the result can be extended to flow shops with job release dates and delivery times and to flow shops with a fixed number of stages where the number of machines at any stage is fixed 1998 the mathematical programming society inc published by elsevier science b v

generalizing topsis for fuzzy multiple criteria group decision making in this paper we generalize topsis to fuzzy multiple criteria group decision making fmcgdm in a fuzzy environment topsis is one of the well known methods for multiple criteria decision making mcdm most of the steps of topsis can be easily generalized to a fuzzy environment except max and min operations in finding the ideal solution and negative ideal solution thus we propose two operators up and lo which satisfy the partial ordering relation on fuzzy numbers to the generalization of topsis in generalized topsis these two operations up and lo are employed to find ideal and negative ideal solutions under a fuzzy environment then the fmcgdm problem can be solved effectively and efficiently

a study on several combination problems of classic shop scheduling and shortest path in this paper we study several combinatorial optimization problems which combine the classic open shop or job shop scheduling problem and the shortest path problem our goal is to select a subset of jobs that constitutes a feasible solution of the shortest path problem and then execute the selected jobs on the shop machines to minimize the makespan i e the last completion time of all the jobs we prove that these problems are np hard even if there are two machines if the number of machines is an input we show that it is unlikely to find approximation algorithms with performance ratios better than 2 unless p np we present an intuitive approximation algorithm when the number of machines is an input and an improved approximation algorithm when the number of machines is fixed in addition we propose a polynomial time approximation scheme for the open shop case when the number of machines is fixed

homoclinic bifurcation in an siqr model for childhood diseases abstract we consider a system of odes which describes the transmission dynamics of childhood diseases a center manifold reduction at a bifurcation point has the normal form x y y axy bx2y o 4 indicating a bifurcation of codimension greater than two a three parameter unfolding of the normal form is studied to capture possible complex dynamics of the original system which is subjected to certain constraints on the state space due to biological considerations it is shown that the perturbed system produces homoclinic bifurcation

tight bounds for permutation flow shop scheduling in flow shop scheduling there are m machines and n jobs such that every job has to be processed on the machines in the fixed order 1 m in the permutation flow shop problem it is also required that each machine process the set of all jobs in the same order formally given n jobs along with their processing times on each machine the goal is to compute a single permutation of the jobs n n that minimizes the maximum job completion time makespan of the schedule resulting from the previously best known approximation guarantee for this problem was o m log m 1 2 sviridenko m 2004 a note on permutation flow shop problem ann oper res 129 247 252 in this paper we obtain an improved o min m1 2 n1 2 approximation algorithm for the permutation flow shop scheduling problem by finding a connection between the scheduling problem and the longest increasing subsequence problem our approximation ratio is relative to the lower bounds of maximum job length and maximum machine load and is the best possible such result this also resolves an open question from potts et al potts c d shmoys d williamson 1991 permutation vs nonpermutation flow shop schedules oper res lett 10 281 284 by algorithmically matching the gap between permutation and nonpermutation schedules we also consider the weighted completion time objective for the permutation flow shop scheduling problem using a natural linear programming relaxation and our algorithm for the makespan objective we obtain an o min m1 2 n1 2 approximation algorithm for minimizing the total weighted completion time improving on the previously best known guarantee of em for any constant e 0 smutnicki c 1998 some results of the worst case analysis for flow shop scheduling eur j oper res 109 66 87 we give a matching lower bound on the integrality gap of our linear programming relaxation

a method for fuzzy aggregation based on group expert evaluations this paper introduces an approach to group decision making based on the coordination index and similarity of the finite collections of fuzzy sets the mathematical apparatus of this approach based on the concept of maximal coordination is constructed in the metric lattice of fuzzy sets a specific kind of fuzzy aggregation operator is proposed a method for fuzzy aggregation is introduced thoroughly discussed and its algorithm determined the practical example of application of the introduced method is offered as well

on the structure of fuzzy social welfare functions we investigate the structure of fuzzy aggregation rules which for every permissible profile of individual preferences specify a fuzzy social ordering it is shown that all fuzzy aggregation rules which have non narrow domains and which satisfy the fuzzy counterparts of independence of irrelevant alternatives and pareto criterion are characterized by a distribution of veto power which would be generally considered undesirable

rationality and aggregation of preferences in an ordinally fuzzy framework abstract the paper explores the problem of aggregating ordinally fuzzy individual preferences into ordinally fuzzy social preferences using goguen s ordinal formulation of fuzziness it is shown that given certain plausible conditions the requirement that the society s preferences should satisfy any of several alternative transitivity conditions creates a dilemma either power in the society is rather unevenly distributed or the society tends to be indecisive the problem increases as weaker transitivity conditions are replaced by stronger ones

a study on the transitivity of probabilistic and fuzzy relations given a set of alternatives we consider a fuzzy relation and a probabilistic relation defined on such a set we investigate the relation between the t transitivity of the fuzzy relation and the cycle transitivity of the associated probabilistic relation we provide a general result valid for any t norm and we later provide explicit expressions for important particular cases we also apply the results obtained to explore the transitivity satisfied by the probabilistic relation defined on a set of random variables we focus on uniform continuous random variables

backward bifurcation of an epidemic model with saturated treatment function an epidemic model with saturated incidence rate and saturated treatment function is studied here the treatment function adopts a continuous and differentiable function which can describe the effect of delayed treatment when the number of infected individuals is getting larger and the medical condition is limited the global dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when such effect is weak however it is shown that a backward bifurcation will take place when this delayed effect for treatment is strong therefore driving the basic reproduction number below the unity is not enough to eradicate the disease and a critical value at the turning point is deduced as a new threshold some sufficient conditions for the disease free equilibrium and the endemic equilibrium being globally asymptotically stable are also obtained mathematical results in this paper suggest that giving the patients timely treatment improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease

generalized ordered weighted utility averaging hyperbolic absolute risk aversion operators and their applications to group decision making abstract this paper develops a new class of aggregation operator based on utility function which introduces the risk attitude of decision makers dms in the aggregation process first under the general framework of utility function we provide a new operator called the generalized ordered weighted utility averaging gowua operator and study its properties which are suitable for any utility function then under the hyperbolic absolute risk aversion hara utility function we propose another new operator named as the generalized ordered weighted utility averaging hyperbolic absolute risk aversion gowua hara operator and further investigate its families including a wide range of aggregation operators to determine the gowua hara operator weights we put forward an orness measure of the gowua hara operator and analyze its properties considering that different dms may have different opinions toward decision making and their opinions can be characterized by different orness measures we construct a new optimization model to determine the optimal weights which can aggregate all the individual sets of weights into an overall set of weights furthermore based on the gowua hara operator a method for the multiple attribute group decision making magdm is developed finally an example is given to illustrate the application of the gowua hara operator to the magdm

a fuzzy multi criteria decision making model by associating technique for order preference by similarity to ideal solution with relative preference relation generally classical multi criteria decision making mcdm methods were extended to encompass uncertainty and vagueness of messages under fuzzy environment for solving decision making problems especially for technique for order preference by similarity to ideal solution topsis in the fuzzy extension of topsis fuzzy numbers comparison and aggregation based on fuzzy preference relation are important issues to compute distance values between alternatives and ideal or anti ideal solution or rank feasible alternatives because lots of messages are reserved by fuzzy preference relation however fuzzy preference relation on pair wise comparison is commonly too complex to calculate to avoid the drawback we use a relative preference relation improved from fuzzy preference relation in the fuzzy extension of topsis for computing distance values between alternatives and ideal or anti ideal solution or obtaining relative closeness coefficients of alternatives thus the relative preference relation on fuzzy numbers will be associated with topsis under fuzzy environment to develop a fuzzy multi criteria decision making fmcdm model through the association above fmcdm problems can be easily solved by the model further we compare the proposed model with other methods to demonstrate the model s feasibility and rationality

the w reciprocity rule abstract given a small number of choices a matrix is a convenient way to represent a fuzzy preference relation usually this matrix is reciprocal the w reciprocity rule is presented here as a way to relax the rigidity of the reciprocal assumption and to embody linguistic meanings for indifference and for the representation of the converse preference

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