Hybrid fuzzy least-squares regression analysis in claims reserving with geometric separation method

Claims reserving is obviously necessary for representing future obligations of an insurance company and selection of an accurate method is a major component of the overall claims reserving process. However, the wide range of unquantifiable factors which increase the uncertainty should be considered when using any method to estimate the amount of outstanding claims based on past data. Unlike traditional methods in claims analysis, fuzzy set approaches can tolerate imprecision and uncertainty without loss of performance and effectiveness. In this paper, hybrid fuzzy least-squares regression, which is proposed by Chang (2001), is used to predict future claim costs by utilizing the concept of a geometric separation method. We use probabilistic confidence limits for designing triangular fuzzy numbers. Thus, it allows us to reflect variability measures contained in a data set in the prediction of future claim costs. We also propose weighted functions of fuzzy numbers as a defuzzification procedure in order to transform estimated fuzzy claim costs into a crisp real equivalent.     

Concepts and fuzzy sets: Misunderstandings, misconceptions, and oversights

The psychology of concepts has been undergoing significant changes since the early 1970s, when the classical view of concepts was seriously challenged by convincing experimental evidence that conceptual categories never have sharp boundaries. Some researchers recognized already in the early 1970s that fuzzy set theory and fuzzy logic were potentially suitable for modeling of concepts and obtained encouraging results. This positive attitude abruptly changed in the early 1980s, and since that time fuzzy set theory and fuzzy logic have been portrayed as problematic and unsuitable for representing and dealing with concepts. Our aim in this paper is to identify some of the most notorious claims regarding fuzzy set theory and fuzzy logic that have propagated through the literature on psychology of concepts and to show that they are, by and large, false. We trace the origin and propagation of these claims within the literature in this area. It is shown in detail that these claims are consistently erroneous and that they are based on various misunderstandings, misconceptions, and oversights. The ultimate purpose of this paper is to document these various erroneous claims.     

A new uncertainty analysis for the transformation method

In this paper, a new uncertainty analysis for the transformation method (TM) is proposed. As a practical implementation of fuzzy arithmetic, the TM is a convenient tool for the simulation and analysis of systems with uncertain parameters that are expressed by fuzzy numbers. The proposed uncertainty analysis and the sensitivity analysis of the TM complete each other in providing some quantification of the relationship between the uncertainties of the system input and the system output. The computation of gain factors is proposed, which allows the estimation of the absolute and relative measures of uncertainty. These measures allow the quantification of the influence of the uncertainty of the input on the uncertainty of the output.     

Multi-criteria decision-making methods based on intuitionistic fuzzy sets

In this paper we present new methods for solving multi-criteria decision-making problem in an intuitionistic fuzzy environment. First, we define an evaluation function for the decision-making problem to measure the degrees to which alternatives satisfy and do not satisfy the decision-maker’s requirement. Then, we introduce and discuss the concept of intuitionistic fuzzy point operators. By using the intuitionistic fuzzy point operators, we can reduce the degree of uncertainty of the elements in a universe corresponding to an intuitionistic fuzzy set. Furthermore, a series of new score functions are defined for multi-criteria decision-making problem based on the intuitionistic fuzzy point operators and the evaluation function and their effectiveness and advantage are illustrated by examples.     

基于损耗率和需求不确定情况下的订购批量

库存管理模型在现实生活中有着广泛的运用。然而,在实际生活中,由于种种不确定性原因的影响,使得经典的确定型的EOQ模型的运用越来越不符合现实的需要;本文将需求和损耗率看成模糊数的同时,将物品的销售价格分成两部分来进行处理,即:没有损耗的产品以一种较高价格出售,对于有部分损耗的产品则按较低的价格出售;采用概率论置信区间估计的方法构建模糊变量的波动区间,构建使得总的利润达到最大的模糊库存模型,并利用三角模糊数、符号距离的方法以及最优化理论进行处理,得出满足条件的最优订购批量。最后,给出了模型分析和算例分析;通过分析,我们发现模糊库存模型的优点在于它自生所具有的不确定性;从数据上看,模糊库存模型比经典的库存模型更能反映出实际情况。     

基于模糊需求的多产品供应商选择的多目标规划模型

由决策于环境的不确定性,供应商选择问题存在大量的模糊信息,传统的确定性规划模型已经不能够很好地处理此类问题。本文基于模糊需求量信息,对于多产品供应商问题建立了模糊多目标规划模型。同时考虑到各目标及约束的重要性程度不同的影响,通过引进适当的权重对多目标规划模型进行求解。文中结合实际算例验证模型的可行性和有效性。     

随机值Ⅱ-型模糊集及其不确定性知识表示

针对人工智能中的不确定性问题,在研究了Ⅱ-型模糊集理论的基础上,提出了一种随机值Ⅱ-型模糊集概念,给出了正态随机值Ⅱ-型模糊集数学模型及其生成算法,并对正态随机值Ⅱ-型模糊集的参数及形态特征进行解析.实例证明随机值Ⅱ-型模糊集可以用来处理社会和自然科学中诸多不确定性问题.     

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.     

A New Form of Fuzzy Compact Spaces and Related Topics via Fuzzy Idealization

Fuzzy ideals and the notion of fuzzy local function were introduced and studied by Sarkar[12] and by Mahmoud in [9]. The purpose of this paper deals with a fuzzy compactness modulo a fuzzy ideal. Many new sorts of weak and strong fuzzy compactness have been introduced to fuzzy topological spaces in the last twenty years but not have been studied using fuzzy ideals so,the main aim of our work in this paper is to define and study some new various types of fuzzy compactness with respect to fuzzy ideals namely fuzzy L-compact and L*-compact spaces. Also fuzzy compactness with respect to ideal is useful as unification and generalization of several others widely studied concepts. Possible application to superstrings and E∞ space-time are touched upon.     

Mixed aleatory and epistemic uncertainty quantification using fuzzy set theory

This paper proposes algorithms to construct fuzzy probabilities to represent or model the mixed aleatory and epistemic uncertainty in a limited-size ensemble. Specifically, we discuss the possible requirements for the fuzzy probabilities in order to model the mixed types of uncertainty, and propose algorithms to construct fuzzy probabilities for both independent and dependent datasets. The effectiveness of the proposed algorithms is demonstrated using one-dimensional and high-dimensional examples. After that, we apply the proposed uncertainty representation technique to isocontour extraction, and demonstrate its applicability using examples with both structured and unstructured meshes.     

Risk model with fuzzy random individual claim amount

In this paper, we consider a risk model in which individual claim amount is assumed to be a fuzzy random variable and the claim number process is characterized as a Poisson process. The mean chance of the ultimate ruin is researched. Particularly, the expressions of the mean chance of the ultimate ruin are obtained for zero initial surplus and arbitrary initial surplus if individual claim amount is an exponentially distributed fuzzy random variable. The results obtained in this paper coincide with those in stochastic case when the fuzzy random variables degenerate to random variables. Finally, two numerical examples are presented.     

Fuzzy expectation and energy states in fuzzy media

This paper is concerned with energy levels and the density of states in fuzzy crystals. The determination of eigenvalue spectrum, or the density of states, for a particle in a fuzzy crystal, is obtained by using the concepts developed in fuzzy statistics. Various analytical results are found, using the fuzzy expectation; these results can be applied to a variety of fields such as decision making under uncertainty, pattern analysis, and quantum mechanics.     

带有模糊系数的投资组合模型研究

在证券市场,由于各种不确定因素的存在,证券的预期收益率是难以精确估算的。本文采用模糊数来处理不确定性,提出了一种基于模糊收益率的投资组合模型。为度量投资组合的风险,将绝对偏差扩展到模糊情形。通过引入模糊数绝对值的概念和不等关系的两种占优准则,将该模型转化为相应的确定性线性规划问题,投资者可根据自己的主观态度选择参数和投资策略。最后用一个具体例子验证了模型的合理性和有效性。     

Quantity of information; comparison between information systems: 2. Fuzzy states

In statistical theory, experiments or probabilistic information systems are supposed to be informative, since they reduce the amount of uncertainty associated with the states of nature. For the case that the available information systems are vague (fuzzy information systems), H. Tanaka, T. Okuda and K. Asai have proven, using the ‘measure of information’ as provided by ‘entropy’, that the fuzzy information systems are informative too.Now, we wish to state and to study a criterion in order to compare fuzzy information systems by the ‘quantity of information of a fuzzy information system’ (defined by Tanaka et al.).In the first paper we considered the situation where we require information about the original state space (non-fuzzy state space).This second paper deals with the situation where we require only information on certain vague states (fuzzy states).     

Quantity of information; Comparison between information systems: 1. Non-fuzzy states

In statistical theory, experiments or probabilistic information systems are supposed to be informative, since they reduce the amount of uncertainty associated with the states of nature. For the case that the available information systems are vague (fuzzy information systems), H. Tanaka, T. Okuda and K. Asai have proven, using the ‘measure of information’ as provided by ‘entropy’, that the fuzzy information systems are informative too.Now, we wish to state and to study a criterion in order to compare fuzzy information systems by the ‘quantity of information of a fuzzy information system’ (defined by Tanaka et al.).In this first paper we consider the situation where we require information about the original state space (non-fuzzy state space).The second paper will deal with the situation where we require only information on certain vague states (fuzzy states).     

A genetic tuning to improve the performance of Fuzzy Rule-Based Classification Systems with Interval-Valued Fuzzy Sets: Degree of ignorance and lateral position

Fuzzy Rule-Based Systems are appropriate tools to deal with classification problems due to their good properties. However, they can suffer a lack of system accuracy as a result of the uncertainty inherent in the definition of the membership functions and the limitation of the homogeneous distribution of the linguistic labels.The aim of the paper is to improve the performance of Fuzzy Rule-Based Classification Systems by means of the Theory of Interval-Valued Fuzzy Sets and a post-processing genetic tuning step. In order to build the Interval-Valued Fuzzy Sets we define a new function called weak ignorance for modeling the uncertainty associated with the definition of the membership functions. Next, we adapt the fuzzy partitions to the problem in an optimal way through a cooperative evolutionary tuning in which we handle both the degree of ignorance and the lateral position (based on the 2-tuples fuzzy linguistic representation) of the linguistic labels.The experimental study is carried out over a large collection of data-sets and it is supported by a statistical analysis. Our results show empirically that the use of our methodology outperforms the initial Fuzzy Rule-Based Classification System. The application of our cooperative tuning enhances the results provided by the use of the isolated tuning approaches and also improves the behavior of the genetic tuning based on the 3-tuples fuzzy linguistic representation.     

Vague集上模糊熵的几点注记

V ague集上的不确定性度量有两种途径,一种是度量V ague集是模糊集的程度,一种是度量V ague集具有的模糊性的程度。后者将模糊集的模糊熵作为特例。本文基于“投票模型”分析了V ague集的熵应具有的特征,对国内作者提出的V ague集上的模糊熵进行了评述。     

Fuzzy chance constrained linear programming model for optimizing the scrap charge in steel production

Optimizing the charge in secondary steel production is challenging because the chemical composition of the scrap is highly uncertain. The uncertainty can cause a considerable risk of the scrap mix failing to satisfy the composition requirements for the final product. In this paper, we represent the uncertainty based on fuzzy set theory and constrain the failure risk based on a possibility measure. Consequently, the scrap charge optimization problem is modeled as a fuzzy chance constrained linear programming problem. Since the constraints of the model mainly address the specification of the product, the crisp equivalent of the fuzzy constraints should be less relaxed than that purely based on the concept of soft constraints. Based on the application context we adopt a strengthened version of soft constraints to interpret fuzzy constraints and form a crisp model with consistent and compact constraints for solution. Simulation results based on realistic data show that the failure risk can be managed by proper combination of aspiration levels and confidence factors for defining fuzzy numbers. There is a tradeoff between failure risk and material cost. The presented approach applies also for other scrap-based production processes.     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.     

A multi-objective genetic algorithm for cardinality constrained fuzzy portfolio selection

This paper presents a new procedure that extends genetic algorithms from their traditional domain of optimization to fuzzy ranking strategy for selecting efficient portfolios of restricted cardinality. The uncertainty of the returns on a given portfolio is modeled using fuzzy quantities and a downside risk function is used to describe the investor's aversion to risk. The fitness functions are based both on the value and the ambiguity of the trapezoidal fuzzy number which represents the uncertainty on the return. The soft-computing approach allows us to consider uncertainty and vagueness in databases and also to incorporate subjective characteristics into the portfolio selection problem. We use a data set from the Spanish stock market to illustrate the performance of our approach to the portfolio selection problem.