区间犹豫梯形模糊Bonferroni Mean算子及其应用

结合犹豫模糊集和区间梯形模糊数,提出了区间犹豫梯形模糊集的概念;针对信息集成过程中需要考虑输入变量之间的相互影响,提出了区间犹豫梯形模糊Bonferroni Mean（IVHTrFBM）算子和区间犹豫梯形模糊几何Bonferroni Mean（IVHTrFGBM）算子,并讨论了它们的性质和特例;根据输入变量的重要程度不同,提出了区间犹豫梯形模糊加权Bonferroni Mean（IVHTrFWBM）算子和区间犹豫梯形模糊加权几何Bonferroni Mean（IVHTrFWGBM）算子。最后,提出了基于IVHTrFWBM算子和IVHTrFWGBM算子的区间犹豫梯形模糊多属性决策方法,并通过实例进行验证。     

Solution of fuzzy matrix games: An application of the extension principle

Conventional game theory is concerned with how rational individuals make decisions when they are faced with known payoffs. This article develops a solution method for the two‐person zero‐sum game where the payoffs are only approximately known and can be represented by fuzzy numbers. Because the payoffs are fuzzy, the value of the game is fuzzy as well. Based on the extension principle, a pair of two‐level mathematical programs is formulated to obtain the upper bound and lower bound of the value of the game at possibility level α. By applying a dual formulation and a variable substitution technique, the pair of two‐level mathematical programs is transformed to a pair of ordinary one‐level linear programs so they can be manipulated. From different values of α, the membership function of the fuzzy value of the game is constructed. It is shown that the two players have the same fuzzy value of the game. An example illustrates the whole idea of a fuzzy matrix game. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 891–903, 2007.     

Correlation Coefficients of Interval‐Valued Hesitant Fuzzy Sets and Their Application Based on the Shapley Function

Interval‐valued hesitant fuzzy sets permit the membership degree of an element to have several possible interval values in [0, 1] rather than real numbers, which can well deal with inherent hesitancy and uncertainty in the human decision‐making process. In this paper, we first point out the issue of the existing correlation coefficients of interval‐valued hesitant fuzzy sets and then define several new ones, which do not have to consider the lengths of interval‐valued hesitant fuzzy elements and the arrangement of their possible interval values. Since the assumption that the elements in a set are independent is usually violated, we further define several Shapley weighted correlation coefficients, which consider their interactions. To deal with the situations where the elements are correlative and the weight formation is incompletely known, models for the optimal fuzzy measures on a feature set and on an attribute set are established, respectively. Finally, a procedure to pattern recognition and multiattribute decision making with incomplete weight information and interactive conditions is developed. Meanwhile, the corresponding examples are provided to show the practicality and feasibility of the proposed procedures.     

New exponential operational laws and their aggregation operators for interval‐valued Pythagorean fuzzy multicriteria decision‐making

In this article, we define two new exponential operational laws about the interval‐valued Pythagorean fuzzy set (IVPFS) and their corresponding aggregation operators. However, the exponential parameters (weights) of all the existing operational laws of IVPFSs are crisp values in IVPFS decision‐making problems. As a supplement, this paper first introduces new exponential operational laws of IVPFS, where bases are crisp values or interval numbers and exponents are interval‐valued Pythagorean fuzzy numbers. The prominent characteristic of these proposed operations is studied. Based on these laws, we develop some new weighted aggregation operators, namely the interval‐valued Pythagorean fuzzy weighted exponential averaging operator and the dual interval‐valued Pythagorean fuzzy weighted exponential averaging. Finally, a decision‐making approach is presented based on these operators and illustrated with some numerical examples to validate the developed approach.     

Static Bayesian games with finite fuzzy types and the existence of equilibrium

This paper studies a special game with incomplete information, in which the payoffs of the players are both random and fuzzy. Such a game is considered in the context of a Bayesian game with the uncertain types characterized as fuzzy variables. A static fuzzy Bayesian game is then introduced and the decision rules for players are given based on credibility theory. We further prove the existence of the equilibrium of the game. Finally, a Cournot competition model with fuzzy efficiency under asymmetric information is investigated as an application and some results are presented.     

基于结构元理论的模糊联盟合作对策

仅考虑局中人参与率模糊的合作对策,称为模糊联盟合作对策。将该模型中的模糊参与率用模糊结构元表示,得到基于结构元理论的具有模糊数Choquet积分表达形式的支付函数和Shapley值的理论框架,继而定义结构元线性生成的模糊支付函数和Shapley值表达式。通过算例与区间数的方法进行对比,结果表明：基于结构元理论的模糊联盟合作对策,模型中的模糊数均由结构元线性生成,模糊数之间的四则运算转化成简单的函数表达式之间的四则运算,避免了模糊数之间运算的遍历性,运算简便。运算结果包括区间及区间上各点的隶属度,结果更加精确。     

Interactive Fuzzy Interval Reasoning for smart Web shopping

In this paper, we propose an Interactive Fuzzy Interval Reasoning (FIR) method by combining fuzzy logic with interval computing to better serve Web users in terms of effectiveness and flexibility. Web users may use convenient interval inputs for online shopping. In order to serve different customers based on their preferences, different personalized fuzzy partitions to meet different needs are provided for the different Web customers. The Interactive Fuzzy Interval Reasoning method is used to design the Web shopping agent. Java servlets and Microsoft Access are used to implement the fuzzy Web shopping system.     

An interval type‐2 fuzzy trust evaluation model in social commerce

Trust is crucial for purchasing decisions in social commerce. However, inexperienced users may not have a direct trust relation to experienced users in practice. Besides, users tend to give their trust degrees to others with linguistic labels rather than crisp values. To evaluate the trust degree for inexperienced users to experienced ones, we propose an interval type‐2 fuzzy trust evaluation model in this paper. Interval type‐2 fuzzy linguistic labels are used to represent trust degree among users. An interval type‐2 fuzzy Algebraic t‐norm is addressed to compute propagative trust degrees. Considering the effect of trust path length, the induced ordered weighted averaging (IOWA) operator is extended to aggregate the interval type‐2 fuzzy trust degrees obtained from multiple paths. In addition, the final interval type‐2 fuzzy trust degree is transferred into the corresponding linguistic label to help users make decisions more naturally. Finally, a case study in social commerce and a related comparison are given to verify the effectiveness of the proposed model.     

CLUSTERED BASED TAKAGI‐SUGENO NEURO‐FUZZY MODELING OF A MULTIVARIABLE NONLINEAR DYNAMIC SYSTEM

This research frame work investigates the application of a clustered based Neuro‐fuzzy system to nonlinear dynamic system modeling from a set of input‐output training patterns. It is concentrated on the modeling via Takagi‐Sugeno (T‐S) modeling technique and the employment of fuzzy clustering to generate suitable initial membership functions. Hence, such created initial memberships are then employed to construct suitable T‐S sub‐models. Furthermore, the T‐S fuzzy models have been validated and checked through the use of some standard model validation techniques (like the correlation functions). Compared to other well‐known approximation techniques such as artificial neural networks, fuzzy systems provide a more transparent representation of the system under study, which is mainly due to the possible linguistic interpretation in the form of rules. Such intelligent modeling scheme is very useful once making complicated systems linguistically transparent in terms of fuzzy if‐then rules. The developed T‐S Fuzzy modeling system has been then applied to model a nonlinear antenna dynamic system with two coupled inputs and outputs. Validation results have resulted in a very close antenna sub‐models of the original nonlinear antenna system. The suggested technique is very useful for development transparent linear control systems even for highly nonlinear dynamic systems.     

基于粒子群优化的混合模糊双矩阵对策求解

针对具有模糊策略集与模糊支付值的不确定性冲突环境,建立了混合模糊双矩阵对策模型。在假定模糊支付值为三角模糊数的情形下,采用了基于单个截集的模糊数线性排序函数,将模型清晰化后转化为双矩阵对策,并应用粒子群优化算法求解。最后,给出一个军事例子说明了模型的实用有效性和粒子群优化算法的高效性。     

基于模糊博弈的英式拍卖动态模型

为了给竞价人或其代理的竞价提供决策支持,提出了模糊博弈的英式拍卖动态模型.以模糊参数出价意愿取代估价作为分析的基础,采用Bellman和Zadeh的模糊决策理论替代博弈论中的Nash平衡理论,分析英式拍卖中的竞价行为,建立英式拍卖静态博弈均衡模型,进而提出动态博弈模型和分析动态拍卖策略.通过仿真实验证明算法的有效性.     

Interval-valued intuitionistic fuzzy matrix games based on Archimedean t-conorm and t-norm

Fuzzy game theory has been applied in many decision-making problems. The matrix game with interval-valued intuitionistic fuzzy numbers (IVIFNs) is investigated based on Archimedean t-conorm and t-norm. The existing matrix games with IVIFNs are all based on Algebraic t-conorm and t-norm, which are special cases of Archimedean t-conorm and t-norm. In this paper, the intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm are employed to aggregate the payoffs of players. To derive the solution of the matrix game with IVIFNs, several mathematical programming models are developed based on Archimedean t-conorm and t-norm. The proposed models can be transformed into a pair of primal–dual linear programming models, based on which, the solution of the matrix game with IVIFNs is obtained. It is proved that the theorems being valid in the exiting matrix game with IVIFNs are still true when the general aggregation operator is used in the proposed matrix game with IVIFNs. The proposed method is an extension of the existing ones and can provide more choices for players. An example is given to illustrate the validity and the applicability of the proposed method.     

Matrix games with interval data

The conventional game theory is concerned with how rational individuals make decisions when they are faced with known payoffs. In the real world, sometimes the payoffs are not known and have to be estimated, and sometimes the payoffs are only approximately known. This paper develops a solution method for the two-person zero-sum game where the payoffs are imprecise and are represented by interval data. Since the payoffs are imprecise, the value of the game should be imprecise as well. A pair of two-level mathematical programs is formulated to obtain the upper bound and lower bound of the value of the game. Based on the duality theorem and by applying a variable substitution technique, the pair of two-level mathematical programs is transformed to a pair of ordinary one-level linear programs. Solving the pair of linear programs produces the interval of the value of the game. It is shown that the two players in the game have the same upper bound and lower bound for the value of the imprecise game. An example illustrates the whole idea and sheds some light on imprecise game.     

Additions of interactive fuzzy numbers

This paper provides an account of an approach to modeling unknown data by means of fuzzy-set theory, and addresses the problem of deriving the uncertainty, on a sum of variables whose values lie within fuzzy intervals. The first part is an extensive presentation of the theoretical background of the approach: the extension principle is stated in terms of possibility of an event; the concept of variable interaction is investigated at length. Section II gives new results regarding the effective practical computation of additions of fuzzy numbers. Its originality lies in the introduction of interaction which enables to control the growth of uncertainty, in calculations. Moreover, the problem of computing mathematical expectations with fuzzy probabilities is solved. The results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed. Typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities. More generally, fuzzy arithmetic can be an important tool for sophisticated, computationally tractable sensitivity analysis in systems modeling, computer-aided design and operations research.     

Designing high-level decision making systems based on fuzzy if–then rules for a point-to-point car racing game

In this paper, fuzzy rule-based systems are applied to a point-to-point car racing game. In the point-to-point car racing game, two car agents compete with each other for taking waypoints. There are three waypoints in the car racing field, each of which is assigned a number that indicates the order to take. The control process of car agents is modeled as a non-holonomic system where there are two input variables (acceleration and steering) for controlling the position, angle and velocity of the car agents. Fuzzy rule-based systems are used to make a high-level decision where the target waypoint to take is determined. Since a fuzzy rule-based system for the high-level decision making is generated in the manner of supervised learning, a set of training patterns should be given for the construction of the fuzzy rule-based systems. In this paper we examine two methods to obtain such a set of training patterns. We also examine two representations of input vectors for the fuzzy rule-based systems. We discuss the effect of obtained training patterns and the input representation on the performance of the fuzzy rule-based systems. After discussing and analyzing the experimental results, we present an adaptive framework of fuzzy rule-based systems. The performance of adaptive fuzzy rule-based systems is then examined based on the results of their non-adaptive version. A series of computational experiments are performed to show the learning ability of the adaptive fuzzy rule-based systems.     

Signaling game based strategy of intrusion detection in wireless sensor networks

As Wireless Sensor Networks (WSNs) become increasingly popular, it is necessary to require Intrusion Detection System (IDS) available to detect internal malicious sensor nodes. Because sensor nodes have limited capabilities in terms of their computation, communication, and energy, selecting the profitable detection strategy for lowering resources consumption determines whether the IDS can be used practically. In this paper, we adopt the distributed-centralized network in which each sensor node has equipped an IDS agent, but only the IDS agent resided in the Cluster Head (CH) with sufficient energy will launch. Then, we apply the signaling game to construct an Intrusion Detection Game modeling the interactions between a malicious sensor node and a CH-IDS agent, and seek its equilibriums for the optimal detection strategy. We illustrate the stage Intrusion Detection Game at an individual time slot in aspects of its player’s utilities, pure-strategy Bayesian–Nash equilibrium (BNE) and mixed-strategy BNE. Under these BNEs the CH-IDS agent is not always on the Defend strategy, as a result, the power of CH can be saved. As the game evolves, we develop the stage Intrusion Detection Game into a multi-stage dynamic Intrusion Detection Game in which, based on Bayesian rules, the beliefs on the malicious sensor node can be updated. Upon the current belief and the Perfect Bayesian equilibrium (PBE), the best response strategy for the CH-IDS agent can be gained. Afterward, we propose an intrusion detection mechanism and corresponding algorithm. We also study the properties of the multi-stage dynamic Intrusion Detection Game by simulations. The simulation results have shown the effectiveness of the proposed game, thus, the CH-IDS agents are able to select their optimal strategies to defend the malicious sensor nodes’ Attack action.     

An Alternative Type Reduction Approach Based on Information Combination with Interval Operations for Interval‐Valued Fuzzy Sliding Controllers

In this paper, an alternative type reduction approach based on information combination with crisp interval operations (crisp interval operation type reduction [CIOTR]) is presented to avoid the time‐consuming difficulty in the type reduction procedure for the mainly used interval‐valued membership functions (MFs). Multiinformation is considered to be included in the primary MF of an interval‐valued fuzzy set (FS). Interval operations are utilized to combine the multiinformation. With the integrated information, an interval‐valued FS is reduced to be a simple FS. The proposed type reduction approach (CIOTR) is also applied to the design of a hybrid interval‐valued fuzzy sliding controller (HFSC^IV) for a double‐pendulum‐and‐cart system. The simulation results are included to show the effectiveness of the designed HFSCs^IV using the CIOTR.     

Evolutionary Dynamics of Strategies without Complete Information on Complex Networks

As for the behaviors of multi‐agent system, recent years have witnessed the growing interest in the study of cooperative behaviors by the aid of evolutionary game dynamics on complex networks. Thereinto, the updating rules deciding the evolution of strategies will significantly influence the steady state distribution of the system. The strategy updating rooted in the pursuit of larger benefits, will drive the system to evolve into the coexistence of different states or the domination by some strategies. To relax the often‐used rules required explicit knowledge of the exact payoffs, this paper describes a new approach of updating strategy based on switching probabilities, which is independent on players' payoffs and degrees. And then the equilibrium state of the strategy evolution in the networks is studied. Our work here provides a computationally feasible way of estimating the steady characteristics of the strategy adoption of agents situating on complex networks. The stability analysis elucidates two important features: (i) the takeover of cooperation can be enhanced by the appropriate settings of the switching probabilities between strategies and (ii) larger average degree and power exponent in the employed scale‐free network can make it easier for the coexistence of strategies. The results can help the design of initial strategy distribution of agents located on social networks to promote cooperation.     

Application of a fuzzy TOPSIS method base on modified preference ratio and fuzzy distance measurement in assessment of traffic police centers performance

In this paper we have presented a TOPSIS approach based on preference ratio and an efficient fuzzy distance measurement for a Fuzzy Multiple Criteria Group Decision-Making Problem (FMCGDMP). Preference ratio with a moderate modification for negative fuzzy numbers was used as an efficient ranking method for fuzzy numbers in a relative manner. As human reasoning persuades that distances between two fuzzy numbers should be a fuzzy measure, so all distances between fuzzy numbers (i.e. distances between alternatives, Fuzzy Positive Ideal solutions, and Fuzzy Negative Ideal solutions) have been calculated as fuzzy numbers using an efficient fuzzy distance measurement. The aforementioned arguments make the proposed algorithm unique and well posed for real-life problem modeling. Moreover, the main novelties of the proposed procedure (i.e. the fuzzy distance measurement and Preference Ratio) have been developed for Generalized Fuzzy Numbers (GFNs). The proposed algorithm has efficiently been applied in assessment of traffic police centers which is treated as a FMCGDMP.