Fuzzy qualitative trigonometry

This paper presents a fuzzy qualitative representation of conventional trigonometry with the goal of bridging the gap between symbolic cognitive functions and numerical sensing & control tasks in the domain of physical systems, especially in intelligent robotics. Fuzzy qualitative coordinates are defined by replacing a unit circle with a fuzzy qualitative circle; a Cartesian translation and orientation are defined by their normalized fuzzy partitions. Conventional trigonometric functions, rules and the extensions to triangles in Euclidean space are converted into their counterparts in fuzzy qualitative coordinates using fuzzy logic and qualitative reasoning techniques. This approach provides a promising representation transformation interface to analyze general trigonometry-related physical systems from an artificial intelligence perspective.Fuzzy qualitative trigonometry has been implemented as a MATLAB toolbox named XTRIG in terms of 4-tuple fuzzy numbers. Examples are given throughout the paper to demonstrate the characteristics of fuzzy qualitative trigonometry. One of the examples focuses on robot kinematics and also explains how contributions could be made by fuzzy qualitative trigonometry to the intelligent connection of low-level sensing & control tasks to high-level cognitive tasks.     

Fuzzy qualitative trigonometry

This paper presents a fuzzy qualitative representation of conventional trigonometry with the goal of bridging the gap between symbolic cognitive functions and numerical sensing & control tasks in the domain of physical systems, especially in intelligent robotics. Fuzzy qualitative coordinates are defined by replacing a unit circle with a fuzzy qualitative circle; a Cartesian translation and orientation are defined by their normalized fuzzy partitions. Conventional trigonometric functions, rules and the extensions to triangles in Euclidean space are converted into their counterparts in fuzzy qualitative coordinates using fuzzy logic and qualitative reasoning techniques. This approach provides a promising representation transformation interface to analyze general trigonometry-related physical systems from an artificial intelligence perspective.Fuzzy qualitative trigonometry has been implemented as a MATLAB toolbox named XTRIG in terms of 4-tuple fuzzy numbers. Examples are given throughout the paper to demonstrate the characteristics of fuzzy qualitative trigonometry. One of the examples focuses on robot kinematics and also explains how contributions could be made by fuzzy qualitative trigonometry to the intelligent connection of low-level sensing & control tasks to high-level cognitive tasks.     

A decision-support model for investment decisions in new ventures

The decision to invest in new ventures is characterized by incomplete information, where some measures of firm performance are quantitative, while some others are substituted by qualitative indicators. Qualitative attributes are defined in a relative sense. We develop a decision support model for identifying successful new ventures. The model integrates quantitative and qualitative variables through the use of the Analytic Hierarchy Process (AHP). The decision model gains in predictive accuracy through the use of qualitative attributes, and AHP imparts robustness to the qualitative measures.     

基于云模型的定性运算理论与方法

将定性值像定量值一样参与运算和推导,是定性理论走向真正实际应用必须解决的极为关键的研究问题.相关研究提出的方法在此方面均有一定的局限性.本文提出了一种基于云模型的定性运算理论与方法.首先在云模型基本运算的基础上提出了基于云模型的定性关系演算方法,其次提出了基于云模型的一元一次定性方程的求解方法.理论和方法能比较科学合理地计算出定性值,且具有定性与定量相互融合的特点.为定性演算和推导在系统分析和仿真等领域中进一步应用提供了重要的借鉴方法.     

Fuzzy measures of supplier evaluation under lean concepts

Supplier evaluation is a multiple attribute decision-making (MADM) problem involving a mixture of qualitative and quantitative attributes. Some mathematical programming techniques have difficulty in dealing with the qualitative attributes that are not easy to measure but very important in supplier evaluation. Fuzzy set theory is useful whenever variables defining a complex and vague system can neither be quantitatively defined nor assigned very precise measures, but are described by linguistic values. This paper presents a procedure for supplier evaluation by incorporating fuzzy set theory into the evaluation process to handle the qualitative attributes in the problem. A mathematical ranking method is used to convert the qualitative attributes into crisp scores. All the quantitative and qualitative performance measures are normalized and then using the proposed model, data are integrated into a single score to rank suppliers. The model application is demonstrated through two previously reported data sets under lean philosophy.     

定性关系及其应用

在已有的定性演算的基础上,引入了定性值集上的定性关系,讨论了定性关系的基本性质,并定义了定性关系的并、交、差及复合运算.在文后给出了定性综合评判的3个基本要素,着重给出了主因素决定型定性综合评判的步骤,并将其应用到具体的实例中.     

一类具有线性收获率的生物捕食系统的定性分析

对一类两种群均有线性收获率的具HollingII类功能反应的食饵-捕食系统作定性分析,利用常微分方程定性,稳定性及分支理论,得到此类生物捕食系统的平衡点的性态和极限环的存在,不存在的条件,从而对更具一般性的一类具有非常数收获率的食饵-捕食系统作了较为全面的定性分析,补充完善了前人的结果.     

Understanding Qualitative Calculus: A Structural Synthesis of Learning Research

More than a decade of research and innovation in using computer-based graphing and simulation environments has encouraged many of us in the research community to believe important dimensions of calculus-related reasoning can be successfully understood by young learners. This paper attempts to address what kinds of calculus-related insights seem to typify this early form of calculus reasoning. The phrase “qualitative calculus” is introduced to frame the analysis of this “other” calculus. The learning of qualitative calculus is the focus of the synthesis. The central claim is that qualitative calculus is a cognitive structure in its own right and that qualitative calculus develops or evolves in ways that seem to fit with important general features of Piaget's analyses of the development of operational thought. In particular, the intensification of rate and two kinds of reversibility between what are called “how much” (amount) and “how fast” (rate) quantities are what interactively, and collectively,characterize and help to define understanding qualitative calculus. Although sharing a family resemblance with traditional expectations of what it might mean to learn calculus, qualitative calculus does not build from ratio- or proportion-based ideas of slope as they are typically associated with defining rate. The paper does close, however, with a discussion of how understanding qualitative calculus can support and link to the rate-related literature of slope, ratio and proportion. Additionally, curricular connections and implications are discussed throughout to help illustrate and explore the significance of learning qualitative calculus. This revised version was published online in July 2006 with corrections to the Cover Date.     

Qualitative chain graphs and their application

For many problem domains, such as medicine, chain graphs are more attractive than Bayesian networks as they support representing interactions between variables that have no natural direction. In particular, interactions between variables that result from certain feedback mechanisms can be represented by chain graphs. Using qualitative abstractions of probabilistic interactions is also of interest, as these allow focusing on patterns in the interactions rather than on the numerical detail. Such patterns are often known by experts and sufficient for making decisions. So far, qualitative abstractions of probabilistic interactions have only been developed for Bayesian networks in the form of qualitative probabilistic networks. In this paper, such qualitative abstractions are developed for chain graphs with the practical aim of using qualitative knowledge as constraints on the hyperspace of probability distributions. The usefulness of qualitative chain graphs is explored for modelling and reasoning about the interactions between diseases.     

Qualitative versus quantitative fixed point techniques in computer science

In 1970, a qualitative fixed point technique useful to model the recursive specifications in denotational semantics was developed by means of the celebrated Kleene‘s fixed point theorem. Later on, in 1994 and 1995, quantitative counterparts of the aforesaid technique, but now based on generalized versions of Banach fixed point theorem, were obtained in such a way that the spirit of Kleene‘s technique was preserved. These new techniques are able to provide a measure of the information content degree and this fact has constituted an advantage with respect to the qualitative techniques. The main purpose of this paper is to discern the relationship between the aforementioned qualitative and quantitative fixed point techniques. In particular, we clarify what is the additional real contribution of the quantitative fixed point techniques with respect to the qualitative ones.     

Qualitative and quantitative combined nonlinear dynamics model and its application in analysis of price,supply–demand ratio and selling rate

The qualitative and quantitative combined nonlinear dynamics model proposed in this paper fill the gap in nonlinear dynamics model in terms of qualitative and quantitative combined methods, allowing the qualitative model and quantitative model to perfectly combine and overcome their weaknesses by learning from each other. These two types of models use their strengths to make up for the other’s deficiencies. The qualitative and quantitative combined models can surmount the weakness that the qualitative model cannot be applied and verified in a quantitative manner, and the high costs and long time of multiple construction as well as verification of the quantitative model. The combined model is more practical and efficient, which is of great significance for nonlinear dynamics. The qualitative and quantitative combined modeling and model analytical method raised in this paper is not only applied to nonlinear dynamics, but can be adopted and drawn on in the modeling and model analysis of other fields. Additionally, the analytical method of qualitative and quantitative combined nonlinear dynamics model proposed in this paper can satisfactorily resolve the problems with the price system’s existing nonlinear dynamics model analytical method. The three-dimensional dynamics model of price, supply–demand ratio and selling rate established in this paper make estimates about the best commodity prices using the model results, thereby providing a theoretical basis for the government’s macro-control of price. Meanwhile, this model also offer theoretical guidance to how to enhance people’s purchasing power and consumption levels through price regulation and hence to improve people’s living standards.     

An algorithm for a selection procedure

When we have a selection problem that involves qualitative information, we can use the qualitative programming method and find the optimal choice by using integer programming. However, in practice after formulation of the problem we have a large scale model. In this note we introduce a very effective method for solving a qualitative programming problem. The only operations used in this method are scalar and matrix addition, and hence for problems of reasonable size, the method does not need high speed computers, and can be supplemented using hand calculators.     

Regenerative queueing processes and their qualitative and quantitative analysis

Recently developed methods of qualitative analysis for regenerative processes arising in queueing are presented. These methods are essentially qualitative and use notions such as coupling, probability metrics, etc. They are developed for studying various properties of regenerative models, including convergence rate to a stationary regime, continuity of their characteristics with respect to some parameters and first-occurrence time of an event such as queue overflowing. In spite of their qualitative nature they lead to good quantitative estimates of underlying properties with computer methods available to calculate them.     

Reduction of some perturbed Keplerian problems

Perturbed Hamiltonian Keplerian systems enjoying some discrete and continuous symmetries can be brought to a one degree of freedom system containing the main qualitative features of the original one. This reduced system is defined in a compact set of the plane where the qualitative dynamics can be studied in a systematic way.     

股票带有机制转换与红利支付的定性期权估值问题研究

本文研究了股票带有红利支付及股价带有机制转换环境下的定性期权估值问题.先利用伊藤公式得到股票价格的动力学方程.再通过随机分析方法推导出标的股票在机制转换市场环境下期权的估值公式.最后进行数值分析,得出带有红利支付的定性期权的估值结果.     

Qualitative operators for reasoning maps: Evaluating multi-criteria options with networks of reasons

Cognitive/causal maps have been widely used as a powerful way of capturing decision-makers’ views about a problem, representing it as a cause–effect discourse. Several ways of making causal inferences from this type of model have been proposed in the Operational Research and Artificial Intelligence literatures, but none, as far as we are aware, has attempted to use a causal map structure to perform a multi-criteria evaluation of decision alternatives. Recently, we have proposed a new multi-criteria method, denominated as a Reasoning Map, which permits the use of decision-makers’ reasoning, structured as a network of means-and-ends (a particular type of causal map) to perform such an evaluation. In this manner, the model resembles the way that people talk and think about decisions in practice. The method also pays explicit attention to the cognitive limitations of decision-makers in providing preference information. Thus it employs qualitative assessment of preferences, utilises aggregation operators for qualitative data and provides also qualitative outputs. In this paper we discuss and evaluate possible ways of aggregating qualitative performance information in Reasoning Maps.     

Qualitative stability and solvability of difference equations

We develop sufficient conditions for qualitative stability and solvability of the real discrete time system x_{t+ 1}= A_xi + b. These conditions are a combination of qualitative and quantitative criteria and depend on signed digraphs.     

The exploratory analysis of qualitative variables by means of three-way analysis of two types of quantification matrices

A comparison is made between a number of techniques for the exploratory analysis of qualitative variables. The paper mainly focuses on a comparison between multiple correspondence analysis (MCA) and Gower's principal co-ordinates analysis (PCO), applied to qualitative variables. The main difference between these methods is in how they deal with infrequent categories. It is demonstrated that MCA solutions can be dominated by infrequent categories, and that, especially in such cases, PCO is a useful alternative to MCA, because it tends to downweight the influence of infrequent categories. Apart from studying the difference between MCA and PCO, other alternatives for the analysis of qualitative variables are discussed, and compared to MCA and PCO.     

Integration of particle swarm optimization-based fuzzy neural network and artificial neural network for supplier selection

This study is intended to develop an intelligent supplier decision support system which is able to consider both the quantitative and qualitative factors. It is composed of (1) the collection of quantitative data such as profit and productivity, (2) a particle swarm optimization (PSO)-based fuzzy neural network (FNN) to derive the rules for qualitative data, and (3) a decision integration model for integrating both the quantitative data and fuzzy knowledge decision to achieve the optimal decision. The results show that the decision support system developed in this study make more precise and favorable judgments in selecting suppliers after taking into account both qualitative and quantitative factors.     

A construction of sound semantic linguistic scales using 4-tuple representation of term semantics

Data semantics plays a fundamental role in computer science, in general, and in computing with words, in particular. The semantics of words arises as a sophisticated problem, since words being actually vague linguistic terms are pieces of information characterized by impreciseness, incompleteness, uncertainty and/or vagueness. The qualitative semantics and the quantitative semantics are two aspects of vague linguistic information, which are closely related. However, the qualitative semantics of linguistic terms, and even the qualitative semantics of the symbolic approaches, seem to be not elaborated on directly in the literature. In this study, we propose an interpretation of the inherent order-based semantics of terms through their qualitative semantics modeled by hedge algebra structures. The quantitative semantics of terms are developed based on the quantification of hedge algebras. With this explicit approach, we propose two concepts of assessment scales to address decision problems: linguistic scales used for representing expert linguistic assessments and semantic linguistic scales based on 4-tuple linguistic representation model, which forms a formalized structure useful for computing with words. An example of a simple multi-criteria decision problem is examined by running a comparative study. We also analyze the main advantages of the proposed approach.