Fuzzy Quantities Research Articles

Ranking and defuzzification methods based on area compensation

We present some interesting properties related to the area compensation procedure to compare fuzzy numbers. It has been proved that this method produces more than a fuzzy interval order: it induces a ranking of fuzzy numbers. Some further results are given about the transitivity property and about computational aspects. Extensions to non-normal fuzzy numbers and fuzzy quantities are also proposed.

Fuzzy inventory with backorder for fuzzy order quantity

This paper investigates a group of computing schemas for economic order quantity as fuzzy values, and the corresponding optimal stock quantity of the invenyory with backorder. We express the fuzzy order quantity as the normal triangular fuzzy number ( q 1, q 0, q 2), and then we solve the aforementioned optimization problem under the constraints 0 < s ⩽ q 1 < q 0 < q 2, where s denotes the optimizing stock quantity. We find that, after defuzzification, the total cost is slightly higher than in the crisp model; however, it permits better use of the economic fuzzy quantities arising with changes in orders, deliveries, and sales.

Load levelling by control of air conditioner operation

AbstractThe recent drastic increase in the number of air conditioners has caused sharp and narrow peaks in summer seasons due to the inherent temperature‐sensitive characteristics. The authors propose a method to reduce the peak power demand by controlling air conditioner operations, and verify its effectiveness on clipping peak demand. However, former study has shortcomings in that no qualitative treatment of room temperature or comfort was attempted, and it did not provide a way of assessing peak demand clipping in a power system as a whole.In this paper, the authors propose a new control method that can deal with the maintaining of comfort and the reduction in power demand. Although air conditioners are used for “comfort,” this contradicts the reduction of power consumption and furthermore the concept of “comfort” is very vague. Hence, Weber‐Fechner's law is utilized to quantify the pleasant feeling which is treated as a fuzzy quantity. A fuzzy coordination method is used to resolve this difference between power demand curtailment and comfort.The second part of this paper prepares an approach for assessing the amount of peak load clipping when the newly proposed control strategy is adopted in a real‐size power system. A decrease in the required generation capacity is estimated provided that the loss of load probability (LOLP) is maintained at the same level before and after the application of the new control strategy. The reduction can be regarded as a dividend of load management.

A parallel asynchronous decomposition for on-line contingency planning

Traditional formulations of security-constrained-optimal-power-flows represent contingencies by hard constraints. The disadvantages are four-fold. First, the conflicts among contingencies must be arbitrated apriori, before their effects are known. Second, the feasible region shrinks with an increase in the number of contingencies. Third, computational time increases with the number of contingencies. Fourth, hard constraints provide poor models of fuzzy quantities such as equipment ratings and operating guidelines. This paper develops a modeling framework without these disadvantages. Specifically, it allows for soft constraints and always has feasible solutions. The effects of conflicts among contingencies are displayed so system operators can arbitrate them in an informed manner. Moreover, each contingency can be handled asynchronously and in parallel. In other words, the computational time, for handling an arbitrarily large number of contingencies, remains the same as for performing an optimal power flow without any contingencies (provided that a computer is dedicated to each contingency).

Additive decomposition of fuzzy quantities

The object of this paper is to define a more refined equivalent relation than Mareš' [3], and to improve mareš' result about additive decomposition of fuzzy quantities concerning an equivalent relation. This result proves one of his open questions.

Implementation of piecewise linear fuzzy quantities

Fuzzy information in expert systems and in fuzzy database systems is often represented by means of piecewise linear fuzzy quantities. In this article we describe an implementation of piecewise linear fuzzy quantities and how to perform fuzzy operations on them. This article can be seen as a natural continuation of Baekeland and Kerre [“Operations on piecewise linear fuzzy quantities: A theoretical approach,” Fuzzy Sets Syst., submitted], in which the mathematical properties of piecewise linear fuzzy quantities are investigated. © 1995 John Wiley & Sons, Inc.

On-line handwritten Chinese character recognition via a fuzzy attribute representation

Chinese characters are constructed by strokes according to structural rules. Therefore, the geometric configurations of characters are important features for character recognition. In handwritten characters, stroke shapes and their spatial relations may vary to some extent. The attribute value of a structural identification is then a fuzzy quantity rather than a binary quantity. Recognizing these facts, we propose a fuzzy attribute representation (FAR) to describe the structural features of handwritten Chinese characters for an on-line Chinese character recognition (OLCCR) system. With a FAR. a fuzzy attribute graph for each handwritten character is created, and the character recognition process is thus transformed into a simple graph matching problem. This character representation and our proposed recognition method allow us to relax the constraints on stroke order and stroke connection. The graph model provides a generalized character representation that can easily incorporate newly added characters into an OLCCR system with an automatic learning capability. The fuzzy representation can describe the degree of structural deformation in handwritten characters. The character matching algorithm is designed to tolerate structural deformations to some extent. Therefore, even input characters with deformations can be recognized correctly once the reference dictionary of the recognition system has been trained using a few representative learning samples. Experimental results are provided to show the effectiveness of the proposed method.

A possibilistic approach to dimensional analysis methods

Abstract Dimensional analysis methods reason about physical and technical processes without explicit knowledge of the underlying physical laws. Instead the methods require the values of the characteristic variables and their dimensional representation. This paper shows the possibility of generalising these methods by considering the physical quantities as fuzzy sets. The paper gives a theoretical justification for this, by formulating the basic principles of dimensional analysis in fuzzyfied form. Special attention is paid in the paper to dimensionless fuzzy quantities and contains the interval functions of some of these. The paper shows some examples concerning the fuzzy approach to problems solved by dimensional analysis methods.

Fuzzy uncertainty in imperfect competition

A prototype decision support system (DSS) that directly addresses the nonstochastic uncertainties of an oligopolistic environment is described. The DSS evaluates price and advertising decisions in the light of imperfect judgmental knowledge about the price and advertising decisions of competitors in an oligopolistic environment. Traditional econometric forecasting methods used to develop the existing system are maintained, but the theory of fuzzy sets equips the models with direct assessment of nonstochastic uncertainty. In addition to parameterizing the econometric equations, the decisionmaker assesses his or her level of knowledge regarding the fuzzy quantities. These assessments are reflected directly in the output of the enhanced DSS. Using Zadeh's “extension principle” and the α-level sets representation of fuzzy sets, a practical implementation of the decision support system is achieved and exercised for the purpose of describing the benefits of sensitivity analysis.

Models of Matching of Fuzzy Sets

Develops a general framework for processes of matching fuzzy quantities. Indicates how different hierarchy levels of matching indices are constructed from a relational way of description of the matching process. Making use of max‐min and min‐max fuzzy relation equations, respectively, clarifies how the entire matching proceeds. Moreover formulates an inverse problem. The method provided here enables us to distinguish regions of the universe of discourse in which the quantities are specified, which are considered as fully supporting the given concept (completely matching observed); completely excluded with respect to this concept; and being of a borderline character. As a consequence the results of matching can have a thorough interpretation.

Fuzzy qualitative simulation

An approach is described that utilizes fuzzy sets to develop a fuzzy qualitative simulation algorithm that allows a semiquantitative extension to qualitative simulation, providing three significant advantages over existing techniques. Firstly, it allows a more detailed description of physical variables, through an arbitrary, but finite, discretisation of the quantity space. The adoption of fuzzy sets also allows common-sense knowledge to be represented in defining values through the use of graded membership, enabling the subjective element in system modelling to be incorporated and reasoned with in a formal way. Secondly, the fuzzy quantity space allows more detailed description of functional relationships in that both strength and sign information can be represented by fuzzy relations holding against two or multivariables. Thirdly, the quantity space allows ordering information on rates of change to be used to compute temporal durations of the state and the possible transitions. Thus, an ordering of the evolution of the states and the associated temporal durations are obtained. This knowledge is used to develop an effective temporal filter that significantly reduces the number of spurious behaviors. >

Additive decomposition of fuzzy quantities with finite supports

Abstract The results on group properties of the addition operation over fuzzy quantities presented in a previous paper by the author [5] imply some interesting consequences concerning the equivalence relation between fuzzy quantities. The equivalence is necessary for the validity of the group axioms for the class of fuzzy quantities. More detailed analysis of the equivalence properties implies some conclusions regarding the interpretation of different sources of fuzziness entering the fuzzy quantities model. The method used to prove the main theorem of this paper demands to limit the investigation to fuzzy quantities with finite support. This limitation, however unpleasant, does not influence the applicability of the presented results in a wide class of practical situations.

Real Time Quality Optimization in Flexible Manufacturing Systems

In this paper a quality oriented model is proposed for product and process as a whole. A hierarchical structural description allows both model consistency and integration of detailled information. Material and data flows are so perfectly defined. Inaccurate or uncertain data are modelled by mean of fuzzy quantities. Associated operators and relations permit information propagation in the model. That strict but flexible approach shows actual advantages above standard statistical methods. Product quality and process operation are valued in order to build a goal cost function. That criterion is used to define the optimal control strategy. A distributed computation process reduces information exchanges and contributes greatly to a safe on-line quality management.

ALGEBRA OF FUZZY QUANTITIES

The fuzzy quantities over a group can be summed using methods known from the literature. Under assumptions mentioned below the summation operation generates a new group of fuzzy quantities up to an equivalence relation. A generalization and summary of some methods and results concerning this problem is presented in this paper, and some of their consequences are reviewed.

Fuzzy Behavior of Beams on Winkler Foundation

This paper addresses and solves the classical beam‐on Winkler‐foundation problem with fuzzy definitions for the Winkler‐spring stiffness, the flexural rigidity of the beam, and the concentrated load. The paper demonstrates through an example the extension of the classical solutions to include fuzzy quantities. Fuzzy equations for deflection, slope, moment, and shear are derived. These are then solved numerically using the vertex method, and the results are shown as fishnet plots. Classical solutions with nonfuzzy values of the parameters are also plotted. These plots show that the curves obtained using extreme values for non‐fuzzy parameters are coincident with the curves formed by the intersection of the horizontal plane at membership equal to zero with the fuzzy behavioral fishnet surface. An index of fuzziness is defined. This is useful in quantitatively describing the propagation of uncertainties in the behavior. The index of fuzziness can be used by the designer to provide adequate margins of safety in regions of maximum fuzziness.

Matching fuzzy quantities

A characteristic approach of approximate reasoning is the partial matching of observations to prototypes. This analysis is cast in the framework of fuzzy set theory and brings another dimension to the fuzzy matching criterion; this dimension is the measure of uncertainty through the concept of subjective entropy. While a similarity measure, in the matching process, activates relevant prototypes, the entropy formalism derived provides a measure of uncertainty in the partial matching to each prototype. The novel approach formalizes an entropy weights activation of prototypes for fuzzy partial matching. A methodology is developed for matching of observation to a set of prototypes making use of a suitable aggregation done with a framework of fuzzy integrals. A method of dealing with compound hypothesis is also developed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Towards a general setting for the fuzzy mathematics of finance

This paper investigates a possible general setting for the fuzzy mathematics of finance. We consider two classes of fuzzy quantities and explore their links with the axiomatic models of financial evaluation. Following some general-purpose theorems about operations on fuzzy quantities, we provide conditions for a consistent fuzzy extension of present and future value, both with one and several capitals. Applications to the most common topics of financial mathematics are accordingly discussed.

An aspect of discrepancy in the implementation of modus ponens in the presence of fuzzy quantities

In this note we discuss some aspects of a commonly used scheme of reasoning, namely modus ponens, currently used in the implementation of fuzzy controllers. Some drawbacks of the existing algorithms leading to implementation of the inference scheme are clearly indicated. Given a set of rules {A k → B k, k = 1,…, N} and a fuzzy antecedent A k, the expected consequent B k is generally not obtained, but a sufficient condition is proposed to get this result.

On fuzzy quantity and fuzzy isoquant

This paper is concerned with developing and characterizing the notion of fuzzy quantity and fuzzy isoquants. The role of fuzzy set theory is confirmed in capturing the fuzzy aspects of production and in establishing properties of interest regarding fuzzy isoquants.

Fuzzy prolog

Various methods of representing uncertainty are discussed including some fuzzy methods. Representation and calculation of fuzzy expressions are discussed and a symbolic representation of fuzzy quantities coupled with axiomatic evaluation is proposed. This is incorporated into the PROLOG language to produce a fuzzy version. Apart from enabling imprecise facts and rules to be expressed, a natural method of controlling the search is introduced, making the search tree admissible. Formal expression of heuristic information in the same language, FUZZY PROLOG, as the main problem language follows naturally and therefore allows the same executor to evaluate in both “problem” space and “heuristic” space. In addition, the use of variable functors in the specification of bidirectional logic is discussed. The paper shows two areas of application of higher order fuzzy predicates. As an introduction Warren's examples are outlined and used with variable functors to illustrate their use in describing some relatively conventional applications. Translation of English into horn clause format is described and is used to illustrate the simplicity of representation using variable functors. Alternative formulations are also explored, typically the use of the “meta-variable” in MICRO-PROLOG and using the “univ” operator. Representation of rule generation and inference is addressed. Examples are given where the expression of meta-rules in standard PROLOG are compared with the expression of the same rules using “variable” predicate symbols. Some meta-rules illustrated are clearly not universally valid and this leads to the addition of fuzzy tokens.