a note on rationality conditions of fuzzy choice functions this note investigates the rationality conditions of fuzzy choice functions for studying the rationality of fuzzy choice functions banerjee introduced many regularity conditions that are defined for revealed preference relations however in the standard framework of economics and decision making theory researchers usually assume people s choice behavior is observable while the preference behind the choice behavior is unobservable it is worthwhile to study rationality conditions that are defined for fuzzy choice functions instead that for revealed preference relations for this purpose we propose a set of new rationality conditions depending only on the fuzzy choice function we study the connection among these rationality conditions and give a relatively complete description for them we also find that banerjee rationality is too weak to capture some kind of consistency of the fuzzy choice function this motivates us to introduce two notions on strong rationality the relationships between our new consistency conditions and the rationality banerjee rationality and strong rationality of the fuzzy choice function are discussed

optimal energy efficient transmit beamforming for multi user miso downlink this paper studies beamforming techniques for energy efficiency maximization eemax in multiuser multiple input single output miso downlink system for this challenging nonconvex problem we first derive an optimal solution using branch and reduce and bound brb approach we also propose two low complexity approximate designs the first one uses the well known zero forcing beamforming zfbf to eliminate inter user interference so that the eemax problem reduces to a concave convex fractional program particularly the problem is then efficiently solved by closed form expressions in combination with the dinkelbach s approach in the second design we aim at finding a stationary point using the sequential convex approximation sca method by proper transformations we arrive at a fast converging iterative algorithm where a convex program is solved in each iteration we further show that the problem in each iteration can also be approximated as a second order cone program socp allowing for exploiting computationally efficient state of the art socp solvers numerical experiments demonstrate that the second design converges quickly and achieves a near optimal performance to further increase the energy efficiency we also consider the joint beamforming and antenna selection jbas problem for which two designs are proposed in the first approach we capitalize on the perspective reformulation in combination with continuous relaxation to solve the jbas problem in the second one sparsity inducing regularization is introduced to approximate the jbas problem which is then solved by the sca method numerical results show that joint beamforming and antenna selection offers significant energy efficiency improvement for large numbers of transmit antennas

rational choice under fuzzy preferences the orlovsky choice function abstract there have been many different suggestions in the literature as to how to generate exact choices on the basis of fuzzy preferences this paper seeks to provide axiomatic support to one of these choice functions due to orlovsky it is shown that under a transitivity restriction independent of max min transitivity this choice function implies and is implied by a set of simple conditions

axiomatic characterizations of threshold choice functions for comparison functions in this paper we characterize the threshold choice functions associated with comparison functions reciprocal fuzzy relations a special of which is the orlovsky choice function with threshold our family of threshold choice functions is somewhat larger than the family of choice functions which select only those alternatives that secure a certain pre specified proportion of the votes against all other alternatives our family of threshold choice functions also include ones which select only those alternatives that exceed a certain pre specified proportion of the votes against all other alternatives this is the price we pay for omitting continuity from our axiomatic characterizations

a method for ranking fuzzy numbers and its application to decision making since fuzzy numbers represent uncertain numeric values it is difficult to rank them according to their magnitude in the paper a method for ranking fuzzy numbers is proposed the method considers the overall possibility distributions of fuzzy numbers in their evaluations for ranking and provides users with a method of changing viewpoints for evaluations users represent their viewpoints with fuzzy sets the method evaluates fuzzy numbers with a satisfaction function and the viewpoint given by users and then ranks the numbers according to their evaluation values the satisfaction function is a measure of comparisons between fuzzy numbers in order to illustrate the ranking method two numeric examples are shown and for the comparative study our method is compared with four existing ranking methods through eight examples as an example of potential applications the proposed method is applied to a decision making problem a two person game with fuzzy profit and loss the ranking method is used to analyze player choices

algorithms for generalized fractional programming a generalized fractional programming problem is specified as a nonlinear program where a nonlinear function defined as the maximum over several ratios of functions is to be minimized on a feasible domain of n the purpose of this paper is to outline basic approaches and basic types of algorithms available to deal with this problem and to review their convergence analysis the conclusion includes results and comments on the numerical efficiency of these algorithms

a new index for bond management in an uncertain environment within the framework of assets liability management we understand that immunization is the main method to assure a certain yield in a future date departing from an initial portfolio although the objective of passive strategies is to design a portfolio that will achieve the performance of a predetermined benchmark active bond management strategies rely on expectations of interest rate movements or changes in yield spread relationships however the variation of the duration increases the risk of a portfolio that why the decision maker will have to choose the combination of expected return and risk which provides the higher utility finally the construction of a fuzzy return risk map will allow the decision maker to know the over risk and the over return as regards immunization strategy for each duration and for each risk aversion of the decision maker the construction of a risk return map presents the results in an appropriate way it will help the decision maker to choose the best duration for the decision maker interest rate forecast finally this methodology is applied to the spanish debt so that the decision maker can select the duration that brings him a greater preference

comparison of fuzzy values on a continuous domain abstract a comparison measure called the satisfaction function sf is extended in this paper the sf estimates the degree to which arithmetic comparisons between two fuzzy values are satisfied the previously proposed sf was defined on a discrete domain sfd so in order to compare continuous fuzzy values by the sfd the fuzzy values should be converted into discrete ones in order to directly compare continuous fuzzy values the sf is extended to a continuous domain sfc the sfc is defined based on the sfd and its properties are also developed since the sfc is extended from the sfd its properties includes those of the sfd and it has the same interpretation as the sfd thus it can be used in any applications instead of the sfd the sfc requires less computing time than the sfd under some conditions for comparisons between a fuzzy and a crisp value and between two crisp values the sfc are also extended

an iteratively weighted mmse approach to distributed sum utility maximization for a mimo interfering broadcast channel consider the multiple input multiple output mimo interfering broadcast channel whereby multiple base stations in a cellular network simultaneously transmit signals to a group of users in their own cells while causing interference to each other the basic problem is to design linear beamformers that can maximize the system throughput in this paper we propose a linear transceiver design algorithm for weighted sum rate maximization that is based on iterative minimization of weighted mean square error mse the proposed algorithm only needs local channel knowledge and converges to a stationary point of the weighted sum rate maximization problem furthermore the algorithm and its convergence can be extended to a general class of sum utility maximization problem the effectiveness of the proposed algorithm is validated by numerical experiments

robust transceiver optimization for downlink coordinated base station systems distributed algorithm this paper considers the joint transceiver design for downlink multiuser multiple input single output miso systems with coordinated base stations bss where imperfect channel state information csi is available at the bss and mobile stations mss by incorporating antenna correlation at the bss and taking channel estimation errors into account we solve two robust design problems 1 minimizing the weighted sum of mean square error mse with per bs antenna power constraint and 2 minimizing the total power of all bss with per user mse target and per bs antenna power constraints these problems are solved as follows first for fixed receivers we propose centralized and novel computationally efficient distributed algorithms to jointly optimize the precoders of all users our centralized algorithms employ the second order cone programming socp approach whereas our novel distributed algorithms use the lagrangian dual decomposition modified matrix fractional minimization and an iterative method second for fixed bs precoders the receivers are updated by the minimum mean square error mmse criterion these two steps are repeated until convergence is achieved in all of our simulation results we have observed that the proposed distributed algorithms achieve the same performance as that of the centralized algorithms moreover computer simulations verify the robustness of the proposed robust designs compared to the nonrobust naive designs

a dynamic consensus scheme based on a nonreciprocal fuzzy preference relation modeling in this paper we propose a dynamical consensus scheme which makes use of an index of comparability and of an index of concordance to regulate the information flow in the discussion among the experts in addition to the traditional use of a concordance index to identify the discordant expert who can be invited to review his opinion or to explain his judgment to the other experts here an index of comparability is utilized to identify the experts that have faced difficulties in expressing their preferences with this information at hand a moderator can invite the experts that are confident on their opinions to support the experts that are hesitating in the construction of the fuzzy preference relations

on the manipulability of the fuzzy social choice functions in many social decision making contexts a manipulator has incentives to change the social choice in his favor by strategically misrepresenting his preference gibbard manipulation of voting schemes a general result econometrica 41 4 1973 587 601 and satterthwaite strategy proofness and arrow s conditions existence and correspondence theorems for voting procedures and social welfare functions j econom theory 10 1975 187 217 have shown that any non dictatorial voting choice procedure is vulnerable to strategic manipulation this paper extends their result to the case of fuzzy weak preference relations for this purpose the best alternative set is defined in three ways and consequently three generalizations of the gibbard satterthwaite theorem to the fuzzy context are provided

coordinated beamforming for energy efficient transmission in multicell multiuser systems in this paper we study energy efficient joint power allocation and beamforming for coordinated multicell multiuser downlink systems the considered optimization problem is in a non convex fractional form and hard to tackle we propose to first transform the original problem into an equivalent optimization problem in a parametric subtractive form by which we reach its solution through a two layer optimization scheme the outer layer only involves one dimension search for the energy efficiency parameter which can be addressed using the bi section search the key issue lies in the inner layer where a non fractional sub problem needs to tackle by exploiting the relationship between the user rate and the mean square error we then develop an iterative algorithm to solve it the convergence of this algorithm is proved and the solution is further derived in closed form our analysis also shows that the proposed algorithm can be implemented in parallel with reasonable complexity numerical results illustrate that our algorithm has a fast convergence and achieves near optimal energy efficiency it is also observed that at the low transmit power region our solution almost achieves the optimal sum rate and the optimal energy efficiency simultaneously while at the middle high transmit power region a certain sum rate loss is suffered in order to guarantee the energy efficiency

a further study on rationality conditions of fuzzy choice functions the investigation of rationality plays a central role in the study of fuzzy choice functions in this paper we deal with some important rationality conditions of fuzzy choice functions including weak strong fuzzy congruence axiom weak strong axiom of fuzzy revealed preference fuzzy versions of famous crisp conditions a b c and d etc in the framework of the banerjee choice function we systematically investigate the relationships between these conditions under the assumption that every involved choice set is normal and verify some equivalence results for arbitrary t norms as a result a fuzzy version of the arrow sen theorem is presented the research makes a significant difference compared with the current investigation in the framework of the georgescu choice function

joint design of tx rx beamformers in mimo downlink channel we consider a single cell multiple input multiple output mimo downlink channel where linear transmission and reception strategy is employed the base station bs transmitter is equipped with a scheduler using a simple opportunistic beamforming strategy which associates an intended user for each of the transmitted data streams for the case when the channel of the scheduled users is available at the bs we propose a general method for joint design of the transmit and the receive beamformers according to different optimization criteria including weighted sum rate maximization weighted sum mean square error minimization minimum signal to interference plus noise ratio sinr maximization and sum power minimization under a minimum sinr constraint the proposed method can handle multiple antennas at the bs and at the mobile user with single and or multiple data streams per scheduled user the optimization problems encountered in the beamformer design e g covariance rank constraint are not convex in general therefore the problem of finding the global optimum is intrinsically nontractable however by exploiting the uplink downlink sinr duality we decompose the original optimization problem as a series of simpler optimization problems which can be efficiently solved by using standard convex optimization tools even though each subproblem is optimally solved there is no guarantee that the global optimum has been found due to the nonconvexity of the problem however the simulations show that the algorithms converge fast to a solution which can be a local optimum but is still efficient

fuzzy preference and orlovsky choice procedure abstract this paper provides an axiomatic characterization of the orlovsky choice procedure for a decision maker whose preference are fuzzy in contrast to the result in banerjee 1993 our characterization does not require that in a pairwise choice the choice coincides with the orlovsky set moreover our characterization result holds for a larger domain of preferences than that in banerjee 1993

numerical implementation of a space transformation approach for solving the three dimensional flow equation this work is concerned with the numerical implementation of the space transformation approach in porous media modeling space transformation is a mathematical approach that simplifies the study of multidimensional problems by reducing them to unidimensional ones in particular the implementation aspects considered in this work include i the development and testing of numerical space transformation algorithms for three dimensional space dependent functions that can be used to model flow within porous media and ii the use of these algorithms to solve partial differential equations governing groundwater flow we also establish the competitiveness of the space transformation based numerical procedures in terms of accuracy and speed further improvements in computational speed can be accomplished by parallel implementation of the space transformation calculations the space transformation method is shown to provide a computationally efficient approach for obtaining three dimensional flow solutions that honor the boundary conditions this approach can be also used in stochastic flow modeling to generate multiple realizations of the random hydraulic head field

weighted sum rate optimization for downlink multiuser mimo coordinated base station systems centralized and distributed algorithms this paper considers the joint linear transceiver design problem for the downlink multiuser multiple input multiple output mimo systems with coordinated base stations bss we consider maximization of the weighted sum rate with per bs antenna power constraint problem we propose novel centralized and computationally efficient distributed iterative algorithms that achieve local optimum to the latter problem these algorithms are described as follows first by introducing additional optimization variables we reformulate the original problem into a new problem second for the given precoder matrices of all users the optimal receivers are computed using minimum mean square error mmse method and the optimal introduced variables are obtained in closed form expressions third by keeping the introduced variables and receivers constant the precoder matrices of all users are optimized by using second order cone programming socp and matrix fractional minimization approaches for the centralized and distributed algorithms respectively finally the second and third steps are repeated until these algorithms converge we have shown that the proposed algorithms are guaranteed to converge we also show that the proposed algorithms require less computational cost than that of the existing linear algorithm all simulation results demonstrate that our distributed algorithm achieves the same performance as that of the centralized algorithm moreover the proposed algorithms outperform the existing linear algorithm in particular when each of the users has single antenna we have observed that the proposed algorithms achieve the global optimum

multiobjective and multiattribute decision making in a fuzzy environment and their power engineering applications two classes of problems requiring the use of a multicriteria approach are defined fuzzy set theory serves for aggregating information arriving from diverse sources combined branches of mathematics of uncertainty serve as a new decision making tool the use of fuzzy set based decision making in power system problems is described benefits of applying fuzzy set theory in problems of power engineering are shown this work presents and generalizes our experience in the use of fuzzy set theory including its combination with another branch of mathematics of uncertainty in developing general approaches and methods for optimization and decision making with considering the uncertainty and multicriteria factors in problems of the design planning operation and control of complex systems two major classes of situations that require the use of a multicriteria approach are identified in accordance with this two general classes of models related to multiobjective x f models and multiattribute x r models problems respectively are considered methods for their analysis based on the use of the bellman zadeh approach to decision making in a fuzzy environment and fuzzy preference modeling techniques respectively are considered although the use of x f and x r models is of an independent character they also serve as parts of a general scheme for multicriteria decision making under conditions of uncertainty this scheme is associated with the use of a generalization of the classic approach to considering the uncertainty of information to multicriteria problems based on analyzing special aggregations of payoff matrices the authors experience in the use of the results indicated above for solving diverse classes of power system planning and operation problems is described this experience convincingly demonstrates diverse advantages of applying fuzzy mathematics in optimization and decision making problems of power engineering

orlovsky s concept of decision making with fuzzy preference relation further results orlovsky s concept of decision making on a finite set of alternatives with a fuzzy preference relation is analysed several theorems which extend the possible effective application of that concept for optimization of many decision problems are formulated and proved new fuzzy relations are developed and their basic properties are investigated

a note on congruence conditions of fuzzy choice functions in order to reflect the fuzziness existing in a choice problem banerjee introduces the concept of fuzzy choice function meanwhile banerjee proposes three congruence conditions in assessing the rationality of a fuzzy choice function in this paper we prove some dependencies among these conditions although banerjee asserts that they are independent

fuzzy preference modeling and its application to multiobjective decision making this paper reflects results of research into the construction and analysis of models within the framework of a general approach to solving optimization problems with fuzzy coefficients this approach involves a modification of traditional mathematical programming methods and is associated with formulating and solving one and the same problem within the framework of mutually interrelated models the use of the approach allows one to maximally cut off dominated alternatives the subsequent contraction of the decision uncertainty region is based on reducing the problem to models of multiobjective choosing alternatives in a fuzzy environment with the use of fuzzy preference relation techniques for analyzing these models three techniques for fuzzy preference modeling are discussed in the paper the first technique is based on the construction of membership functions of subsets of nondominated alternatives with simultaneous considering of all criteria fuzzy preference relations the second technique is of a lexicographic character and consists of step by step introducing of fuzzy preference relations the third technique is based on aggregating membership functions of subsets of nondominated alternatives corresponding to each preference relation these techniques have served for developing a corresponding system for multiobjective decision making mdms c windows of the mdms are presented for input output and some intermediate procedures the results of the paper are of a universal character and are already being used to solve power engineering naval engineering and management problems