Models and quantifier elimination for quantified Horn formulas

In this paper, quantified Horn formulas (QHORN) are investigated. We prove that the behavior of the existential quantifiers depends only on the cases where at most one of the universally quantified variables is zero. Accordingly, we give a detailed characterization of QHORN satisfiability models which describe the set of satisfying truth assignments to the existential variables. We also consider quantified Horn formulas with free variables (QHORN^*) and show that they have monotone equivalence models.The main application of these findings is that any quantified Horn formula Φ of length |Φ| with free variables, |∀| universal quantifiers and an arbitrary number of existential quantifiers can be transformed into an equivalent quantified Horn formula of length O(|∀|·|Φ|) which contains only existential quantifiers.We also obtain a new algorithm for solving the satisfiability problem for quantified Horn formulas with or without free variables in time O(|∀|·|Φ|) by transforming the input formula into a satisfiability-equivalent propositional formula. Moreover, we show that QHORN satisfiability models can be found with the same complexity.     

Modelling redundancy allocation for a fuzzy random parallel-series system

Due to subjective judgment, imprecise human knowledge and perception in capturing statistical data, the real data of lifetimes in many systems are both random and fuzzy in nature. Based on the fuzzy random variables that are used to characterize the lifetimes, this paper studies the redundancy allocation problems to a fuzzy random parallel-series system.Two fuzzy random redundancy allocation models (FR-RAM) are developed through reliability maximization and cost minimization, respectively. Some properties of the FR-RAM are obtained, in which an analytical formula of reliability with convex lifetimes is derived and the sensitivity of the reliability is discussed. To solve the FR-RAMs, we first address the computation of reliability. A random simulation method based on the derived analytical formula is proposed to compute the reliability with convex lifetimes. As for the reliability with nonconvex lifetimes, the technique of fuzzy random simulation together with the discretization method of fuzzy random variable is employed to compute the reliability, and a convergence theorem of the fuzzy random simulation is proved. Subsequently, we integrate the computation approaches of the reliability and genetic algorithm (GA) to search for the approximately optimal redundancy allocation of the models. Finally, some numerical examples are provided to illustrate the feasibility of the solution algorithm and quantify its effectiveness.     

Granular fuzzy models: a study in knowledge management in fuzzy modeling

In system modeling, knowledge management comes vividly into the picture when dealing with a collection of individual models. These models being considered as sources of knowledge, are engaged in some collective pursuits of a collaborative development to establish modeling outcomes of global character. The result comes in the form of a so-called granular fuzzy model, which directly reflects upon and quantifies the diversity of the available sources of knowledge (local models) involved in knowledge management. In this study, several detailed algorithmic schemes are presented along with related computational aspects associated with Granular Computing. It is also shown how the construction of information granules completed through the use of the principle of justifiable granularity becomes advantageous in the realization of granular fuzzy models and a quantification of the quality (specificity) of the results of modeling. We focus on the design of granular fuzzy models considering that the locally available models are those fuzzy rule-based. It is shown that the model quantified in terms of two conflicting criteria, that is (a) a coverage criterion expressing to which extent the resulting information granules “cover” include data and (b) specificity criterion articulating how detailed (specific) the obtained information granules are. The overall quality of the granular model is also assessed by determining an area under curve (AUC) where the curve is formed in the coverage-specificity coordinates. Numeric results are discussed with intent of displaying the most essential features of the proposed methodology and algorithmic developments.     

Design of fuzzy radial basis function-based polynomial neural networks

In this study, we introduce a new design methodology of fuzzy radial basis function-based polynomial neural networks. In many cases, these models do not come with capabilities to deal with granular information. With this regard, fuzzy sets offer several interesting and useful opportunities. This study presents the development of fuzzy radial basis function-based neural networks augmented with virtual input variables. The performance of the proposed category of models is quantified through a series of experiments, in which we use two machine learning data sets and two publicly available software development effort data.     

Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations

This paper proposes linear goal programming models for deriving intuitionistic fuzzy weights from intuitionistic fuzzy preference relations. Novel definitions are put forward to define additive consistency and weak transitivity for intuitionistic fuzzy preference relations, followed by a study of their corresponding properties. For any given normalized intuitionistic fuzzy weight vector, a transformation formula is furnished to convert the weights into a consistent intuitionistic fuzzy preference relation. For any intuitionistic fuzzy preference relation, a linear goal programming model is developed to obtain its intuitionistic fuzzy weights by minimizing its deviation from the converted consistent intuitionistic fuzzy preference relation. This approach is then extended to group decision-making situations. Three numerical examples are provided to illustrate the validity and applicability of the proposed models.     

Traffic assignment model with fuzzy level of travel demand: An efficient algorithm based on quasi-Logit formulas

The place of fuzzy concepts in traffic assignment (TA) models has been studied in recent literature. Keeping fuzzy level of travel demand in mind, we propose a new TA model in which the travel costs of links are depended on their congestion. From the results of such fuzzy TA model, network planners are able to estimate the number of travelers on network links. By using zero–one variables, the proposed model is transformed into a crisp mixed-integer problem with respect to path-flow variables. In order to produce the Logit flows from this problem, Damberg et al. algorithm is modified. Then, the level of certainty is maximized and perceived travel delays are minimized. For a fixed certainty degree, the obtained solution, which is named the fuzzy equilibrium flow, satisfies a quasi-Logit formula similar to ordinary expression of the Logit route choice model. Eventually, we examine the quality of different path enumeration techniques in the proposed model.     

Probabilistic model of fuzzy subsets

On the basis of H.Weyl's and E.Wigners's theories there is constructed the fuzzy model of canonically conjugate fuzzy subsets. On the ground of introduced characteristic interval the optimal shape of information function is obtained. By this information function the formula for compatibility function is obtained, Φ(α /2), where is a parameter assigned by expert, Φ(·) is probability interval. We establish the connection between the dispersions of canonically conjugate fuzzy subsets, presented in the fuzzy form. On the base of this formula the generalized information theory is constructed, which enable to estimate the constant c. As an example it is considered the method of fuzzy differential equation solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linguistic quantifiers based on Choquet integrals

The introduction of linguistic quantifiers has provided an important tool to model a large number of issues in intelligent systems. Ying [M.S. Ying, Linguistic quantifiers modeled by Sugeno integrals, Artificial Intelligence 170 (2006) 581–606] recently introduced a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures, and the truth value of a quantified proposition is evaluated by using Sugeno’s integral. Representing linguistic quantifiers by fuzzy measures, this paper evaluates linguistic quantified propositions by the Choquet integral. Some elegant logical properties of linguistic quantifiers are derived within this approach, including a prenex normal form theorem stronger than that in Ying’s model. In addition, our Choquet integral approach to the evaluation of quantified statements is compared with others, in particular with Ying’s Sugeno integral approach.     

基于“直觉模糊集的距离测度”的联想

直觉模糊集不同于模糊集的最重要指标是犹豫度.在直觉模糊集的距离定义和公式中应突出体现犹豫度;在直觉模糊集的距离定义中也应包含经典距离的性质—三角不等式.基于论文"直觉模糊集的距离测度",提出直觉模糊集的新距离定义和新公式.应用实例表明,新提出的直觉模糊集的距离公式是实用的.     

基于一族蕴涵算子的对偶三I算法

三I算法是一种新的模糊推理方法,是传统的模糊推理方法的修改和补充. 三I表达式取最小值时的最优解算法(即对偶三I算法)是三I算法思想的延伸和完善.本文针对蕴涵算子族Ip,讨论了FMP和FMT问题的对偶三I算法,给出了相应的计算公式,从而也进一步促进了对三I算法的研究.     

基于粗糙模糊集的聚类决策及应用

对基于粗糙模糊集的聚类问题进行了研究.首先,定义了粗糙模糊值与粗糙模糊集的相似度,给出了粗糙模糊集相似度的一般表达式;然后利用该公式构建了模糊相似矩阵,进而给出了粗糙模糊集的一种聚类方法;最后以实际算例说明了这种思想.     

基于可能度的一种直觉模糊集相似度测量方法

研究直觉模糊集的相似度测量方法. 首先给出两个直觉模糊数比较的可能度公式及其性质. 在此基础上, 定义两个直觉模糊集间的相似度测量公式, 分析用此公式进行相似度测量的优缺点, 旨为直觉模糊集在决策、模式识别和医疗诊断中的应用提供更为合理的工具.     

On theoretical pricing of options with fuzzy estimators

In this paper we present an application of a new method of constructing fuzzy estimators for the parameters of a given probability distribution function, using statistical data. This application belongs to the financial field and especially to the section of financial engineering. In financial markets there are great fluctuations, thus the element of vagueness and uncertainty is frequent. This application concerns Theoretical Pricing of Options and in particular the Black and Scholes Options Pricing formula. We make use of fuzzy estimators for the volatility of stock returns and we consider the stock price as a symmetric triangular fuzzy number. Furthermore we apply the Black and Scholes formula by using adaptive fuzzy numbers introduced by Thiagarajah et al. [K. Thiagarajah, S.S. Appadoo, A. Thavaneswaran, Option valuation model with adaptive fuzzy numbers, Computers and Mathematics with Applications 53 (2007) 831–841] for the stock price and the volatility and we replace the fuzzy volatility and the fuzzy stock price by possibilistic mean value. We refer to both cases of call and put option prices according to the Black & Scholes model and also analyze the results to Greek parameters. Finally, a numerical example is presented for both methods and a comparison is realized based on the results.     

梯形模糊有序加权几何算子及其在决策中的应用

针对梯形模糊数据信息的集成问题,给出了梯形模糊数两两比较的可能度公式和梯形模糊有序加权几何(TFOWG)算子.基于可能度公式和TFOWG算子,提出了一种准则权重信息完全未知且准则值以梯形模糊数形式给出的不确定多准则决策方法.最后,实例分析表明了该方法的可行性和有效性.     

基于模糊语言判断矩阵和FIOWA算子的有限方案决策法

定义一种模糊的导出有序加权平均(FIOWA)算子，给出方案之间比较的模糊语言标度。运用模糊语言标度构造出模糊语言判断矩阵，并提出一种基于模糊语言判断矩阵和FIOWA算子的有限方案决策方法。该法利用FIOWA算子对模糊语言信息进行集结，并利用已有的三角模糊数排序公式求得决策方案的排序。     

Cost,Revenue and Profit Efficiency Models in Generalized Fuzzy Data Envelopment Analysis

One of the most important information given by data envelopment analysis models is the cost, revenue and profit efficiency of decision making units (DMUs). Cost efficiency is defined as the ratio of minimum costs to current costs, while revenue efficiency is defined as the ratio of maximum revenue to current revenue of the DMU. This paper presents a framework where data envelopment analysis (DEA) is used to measure cost, revenue and profit efficiency with fuzzy data. In such cases, the classical models cannot be used, because input and output data appear in the form of ranges. When the data are fuzzy, the cost, revenue and profit efficiency measures calculated from the data should be uncertain as well. Fuzzy DEA models emerge as another class of DEA models to account for imprecise inputs and outputs for DMUs. Although several approaches for solving fuzzy DEA models have been developed, numerous deficiencies including the $\alpha$-cut approaches and types of fuzzy numbers must still be improved. This scheme embraces evaluation method based on vector for proposed fuzzy model. This paper proposes generalized cost, revenue and profit efficiency models in fuzzy data envelopment analysis. The practical application of these models is illustrated by a numerical example.     

基于Choquet积分的直觉模糊联盟合作博弈的Shapley值

本文针对联盟是直觉模糊集的合作博弈Shapley值进行了研究.通过区间Choquet积分得到直觉模糊联盟合作博弈的特征函数为区间数,并研究了该博弈特征函数性质。根据拓展模糊联盟合作博弈Shapley值的计算方法,得到直觉模糊联盟合作博弈Shapley值的计算公式,该计算公式避免了区间数的减法。进一步证明了其满足经典合作博弈Shapley值的公理性。最后通过数值实例说明本文方法的合理性和有效性。     

基于新直觉模糊相似度的大规模群决策方法

研究了群体规模较大情况下基于直觉模糊评价信息的决策方法。提出一种新的直觉模糊相似度公式,研究了一种新的直觉模糊聚类方法,设计了一种基于新的聚类方法的核心决策者权重确定方法。实例分析说明了该方法的合理性和可行性。     

Some models for deriving the priority weights from interval fuzzy preference relations

Interval fuzzy preference relation is a useful tool to express decision maker’s uncertain preference information. How to derive the priority weights from an interval fuzzy preference relation is an interesting and important issue in decision making with interval fuzzy preference relation(s). In this paper, some new concepts such as additive consistent interval fuzzy preference relation, multiplicative consistent interval fuzzy preference relation, etc., are defined. Some simple and practical linear programming models for deriving the priority weights from various interval fuzzy preference relations are established, and two numerical examples are provided to illustrate the developed models.     

A Class of Chance Constrained Multi-objective Portfolio Selection Model Under Fuzzy Random Environment

This paper deals with a class of chance constrained portfolio selection problems in the fuzzy random decision making system. An integrated fuzzy random portfolio selection model with a chance constraint is proposed on the basis of the mean-variance model and the safety-first model. According to different definitions of chance, we consider two types of fuzzy random portfolio selection models: one is for the optimistic investors and the other is for the pessimistic investors. In order to deal with the fuzzy random models, we develop a few theorems on the variances of fuzzy random returns and the equivalent partitions of two types of chance constraints. We then transform the fuzzy random portfolio selection models into their equivalent crisp models. We further employ the ε-constraint method to obtain the efficient frontier. Finally, we apply the proposed models and approaches to the Chinese stock market as an illustration.