Use Of Fuzzy Arithmetic Research Articles

Fuzzy arithmetic in economic research

Many tasks in economic research are based on arithmetic calculations of indicators that reflect the state of economic development. The general incompleteness of publicly available data, designed to solve such problems, has led to the emergence of numerous decision support systems based on fuzzy arithmetic. The article presents a study on the approach aimed at integrating fuzzy information about economic indicators into economic models. The definition of arithmetic operations on fuzzy values is given, and the methods of obtaining the resulting fuzzy indicators with the help of some software tools are considered. Analytical and numerical methods of obtaining the resulting indicators in the form of fuzzy numbers are described and analyzed. A direct calculation algorithm for all arithmetic operations is proposed, utilized, and used for the evaluation of resulting indicators. Also, analytical and numerical methods for obtaining fuzzy results are considered in the article, and some of them are proposed for utilization. On the example of economic indicators used in the labor rationing, the results of an evaluation of some indicators for a technological operation in the form of fuzzy numbers were obtained by different methods and compared. The practical recommendations, given in the article, on the use of fuzzy arithmetic in decision support systems outline the directions of further research.

The use of fuzzy arithmetic in decision-making problems

The article examines the rules for performing basic arithmetic operations on fuzzy gradations. The tables for performing operations of addition, limited addition, multiplication, as well as operations of minimum and maximum are obtained. A generalization of Markov models of decision making in the case of ambiguity, when there is no information about the probabilities of states, is proposed. In this case, the components of the model are considered as fuzzy and the concept of fuzzy risk is additionally introduced. An example of application of Markov model to economic problem is considered. The models of optimal choice were studied for various decision-making strategies, namely, the threshold criterion method, the worst criterion method, additive convolution, and the distance method for different selection functions. Theoretical studies are accompanied by numerical calculations.

Inverse propagation of uncertainties in finite element model updating through use of fuzzy arithmetic

A fuzzy finite element model updating (FFEMU) method is presented in this study for the damage detection problem. The uncertainty caused by the measurement noise in modal parameters is described by fuzzy numbers. Inverse analysis is formulated as a constrained optimization problem at each α-cut level. Membership functions of each updating parameter which correspond to reduction in bending stiffness of the finite elements is determined by minimizing an objective function using a hybrid version of genetic algorithms (GA) and particle swarm optimization method (PSO) which is very efficient in terms of accuracy and robustness. Practical evaluation of the approximate bounds of the interval modal parameters in FFEMU iterations is addressed. A probabilistic analysis is performed using Monte Carlo simulation (MCS) and the results are compared with presented FFEMU method. It is apparent from numerical simulations that the proposed method is well capable in finding the membership functions of the updating parameters within reasonable accuracy. It is also shown that the results obtained by FFEMU are in good agreement with the MCS results while FFEMU is not as computationally expensive as the MCS method. Nevertheless, the proposed FFEMU do not required derivatives of the objective function like existing methods except in the deterministic case.

Reliability analysis of complex multi-robotic system using GA and fuzzy methodology

The objective of the study is to compute various reliability parameters for multi-robotic system, using Real Coded Genetic Algorithms (RCGAs) and Fuzzy Lambda-Tau Methodology (FLTM). The paper contains a new idea about the reliability analysis of robotic system. The optimal values of mean time between failures (MTBF) and mean time to repair (MTTR) are obtained using GAs. Petri Net (PN) tool is applied to represent the interactions among the working components of multi-robotic system. To enhance the relevance of the reliability study, triangular fuzzy numbers (TFNs) are developed from the computed data, using possibility theory. The use of fuzzy arithmetic in the PN model increases the flexibility for application to various systems and conditions. Various reliability parameters, namely failure rate, repair time, MTBF, expected number of failures (ENOF), reliability and availability, are computed using FLTM. Sensitivity analysis has also been performed and the effects on system MTBF are addressed. The adopted methodology improves the shortcomings/drawbacks of the existing probabilistic approaches and gives a better understanding of the system behavior through its graphical representation. The analysis presented, may be helpful for the system analyst to analyze and predict the system behavior and to reallocate the required resources.

Reliability analysis of waste clean-up manipulator using genetic algorithms and fuzzy methodology

In this paper, the reliability analysis of waste clean-up manipulator has been performed using Real Coded Genetic Algorithms and Fuzzy Lambda Tau Methodology. The optimal values of mean time between failures and mean time to repair are obtained using genetic algorithms. Petri Net tool is applied to represent the interactions among the working components of the system. To enhance the relevance of the reliability study, triangular fuzzy numbers are developed from the computed data, using possibility theory. The use of fuzzy arithmetic in the Petri Net model increases the flexibility for application to various systems and conditions. Various reliability parameters (failure rate, repair time, mean time between failures, expected no. of failures, reliability and availability) are computed using Fuzzy Lambda Tau Methodology. Sensitivity analysis has also been performed and the effects on system mean time between failures are addressed. The adopted methodology improves the shortcomings/drawbacks of the existing probabilistic approaches and gives a better understanding of the system behavior through its graphical representation.

Reliability analysis of complex robotic system using Petri nets and fuzzy lambda‐tau methodology

PurposeThe purpose of this paper is to evaluate various reliability parameters of a multi‐robot system, arranged in a complex configuration. The effects of failures and course of action on the system performance have also been investigated.Design/methodology/approachThe present work is based on a multi‐robotic system, in which two robots are working independently with a conveyer unit. Petri net (PN) tool is applied to represent the asynchronous and concurrent processing of the system. To enhance the relevance of the reliability study, fuzzy numbers are developed from available data of the components, using fuzzy possibility theory to define membership functions. The use of fuzzy arithmetic in the PN model increases the flexibility for application to various systems and conditions. Various reliability parameters (such as mean time between failures, ENOF, reliability, availability, etc.) are computed using fuzzy lambda‐tau methodology. As the available data are imprecise, incomplete, vague and conflicting, the fuzzy methodology can deal easily with approximations.FindingsThe adopted methodology in the present work improves the shortcomings/drawbacks of the existing probabilistic approaches and gives a better understanding of the system behavior through its graphical representation.Originality/valueIn an earlier study failure behavior of a single robot was analyzed. This paper is an extension of the previous work, in which failure behavior of multi‐robotic system is analyzed. Also, the interactions among the working units of multi‐robotic system are deeply studied. The paper contains a new idea about the reliability analysis of robotic system. Fuzzy lambda‐tau methodology, a fuzzy PROBIST technique, is used for the proposed robotic system and the results obtained are compared with crisp results. Reliability analysis of a multi‐robotic system is presented in this paper, which may help the system analysts to analyze and predict the system behavior and to reallocate the required resources.

SOME COMPUTATIONS ON FUZZY MATRICES: AN APPLICATION IN FUZZY ANALYTICAL HIERARCHY PROCESS

Fuzzy mathematics is a generalization in which fuzzy numbers replace real numbers and fuzzy arithmetic replaces real arithmetic. It is an excellent scope for modeling vague and uncertain aspects of the actual environments. In this important area, Dubois and Prade1 defined a fuzzy matrix as a rectangular array of fuzzy numbers. They have also defined the LR type fuzzy numbers with some useful approximate arithmetic operators. The aim of this paper is to extend some useful aspects of linear algebra e.g. determinant, norm and eigenvalue for fuzzy matrices with LR fuzzy number entries by the use of fuzzy arithmetic. Finally, applications in fuzzy analytical hierarchy process (AHP) are investigated.

Developing environmental indices using fuzzy numbers ordered weighted averaging (FN-OWA) operators

Environmental indices (EI) constitute a common communication tool that is often used to describe the overall status of environmental systems (air, water and soil). EI development entails the use of mathematical operators to aggregate various non-commensurate input parameters in a logical manner. The ordered weighted averaging (OWA) operator is a general mean type operator that provides flexibility in the aggregation process such that the aggregated value is bounded between minimum and maximum values of the input parameters. This flexibility of the OWA operator is realized through the concept of orness, which is a surrogate for decision maker’s attitude. The type of input parameters also affects the choice of aggregation operators. If the input parameters are linguistic or fuzzy, the aggregation through OWA operators is not possible, and the use of fuzzy arithmetic is warranted. The concept of fuzzy number OWA (FN-OWA) operators is explored to handle situations in which one or more input parameter has fuzzy (or linguistic) values. The proposed approach is demonstrated using data provided in an earlier study by Swamee and Tyagi (ASCE J Environ Eng 126(5):451–455, 2000) for establishing water quality indices. Multiple hypothetical scenarios are also generated to highlight the utility and sensitivity of the proposed approach.

Constrained fuzzy arithmetic: Basic questions and some answers

The purpose of this paper is to critically examine the use of fuzzy arithmetic in dealing with fuzzy systems. It is argued that the well-known overestimation and other questionable results of standard fuzzy arithmetic have one common cause: constraints regarding linguistic variables involved are not taken into account. A general formulation of constrained fuzzy arithmetic – a nonstandard fuzzy arithmetic that takes into account these constraints – is presented and its basic characteristics are examined. More specific characteristics of constrained fuzzy arithmetic are then investigated for some common types of constraints.

A semi-arid grazing ecosystem simulation model with probabilistic and fuzzy parameters

Many of today's grazing ecosystem management problems require a predictive understanding of the interactions between ecological processes acting across different spatiotemporal scales with complex soil, vegetation, climate, topographic and geologic characteristics. Due to complexity of ecosystems and the incomplete nature of empirical data for specific relationships involved in ecosystem dynamics, resource managers usually select management strategies without complete information or knowledge. Such vagueness in real world situations is also an obstacle to understanding and modeling ecosystem dynamics by means of conventional mathematical and computer simulation techniques. Modeling, simulating, and analyzing actual ecosystems can be significantly improved if modeling is extended to deal with imprecise and vague variables, relationships, and events. In this paper, a simulation model combining fuzzy imprecision with probabilistic uncertainty is formulated to study climate-plant-herbivore interactions in grassland ecosystems. This approach provides a unifying simulation framework to integrate numerical data, linguistic statements, and expert experience. The model includes treatment of imprecise and vague input variables as fuzzy variables, use of fuzzy arithmetic in equations when fuzzy and probability numbers are involved, and replacement of some of the relationships in the dynamic systems either with fuzzy conditional statements, or with fuzzy algorithms. The temporal patterns of herbivore population and primary production for a laissez-faire, extensive system in both Australia and China are modeled. The consistency of results obtained from the two simulations suggests the underlying mechanisms for semi-arid grassland ecosystems are those included in the model. Models combining fuzzy sets to quantify subjective parameters and traditional mechanistic techniques can be used to form a basis of long-term managerial policies for a sustainable grassland ecosystem.

Implementing fuzzy arithmetic

This paper presents the authors' efforts towards the development of an effective and friendly working tool for fuzzy arithmetic. It describes the two programming libraries currently produced. The first, implemented using the Fortran language, provides high precision computational capabilities, to demonstrate the feasibility and the correctness of a practical use of fuzzy arithmetic. The second, implemented in an object-oriented environment using the C++ language, aims at being a powerful, friendly and highly portable tool to support generic users needing to manage imprecise data. Moreover, a simple application to environmental impact analysis is described to show the advantages deriving from the practical use of fuzzy arithmetic.

Systems dynamics with fuzzy logic

Systems dynamics has been used to model and simulate a variety of environments, e.g. economic, social and political, which require quantification or some types of human behaviour. The lack of empirical verification of the relationships in the systems dynamics models has often been criticised. Nevertheless, the methodology is effective in dealing with time-varying (dynamic) interactions among components of the analysed system. The effectiveness of systems dynamics as a methodology for modelling, simulating and analysing real-life systems can be significantly increased if it is extended to deal with imprecise and vague variables or events. Such an extension requires: (1) treatment of imprecise and vague input variables as fuzzy variables: (2) use of fuzzy arithmetic in the level, rate and auxiliary equations when fuzzy numbers are involved; and (3) replacement of some of the relationships in the systems dynamics models either with conditional statements including fuzzy variables, or with fuzzy algorithms.

Fuzzy probabilities

The conventional approaches to decision analysis are based on the assumption that the probabilities which enter into the assessment of the consequences of a decision are known numbers. In most realistic settings, this assumption is of questionable validity since the data from which the probabilities must be estimated are usually incomplete, imprecise or not totally reliable. In the approach outlined in this paper, the probabilities are assumed to be fuzzy rather than real numbers. It is shown how such probabilities may be estimated from fuzzy data and a basic relation between joint, conditional and marginal fuzzy probabilities is established. Manipulation of fuzzy probabilities requires, in general, the use of fuzzy arithmetic, and many of the properties of fuzzy probabilities are simple generalizations of the corresponding properties of real-valued probabilities.

A computational approach to fuzzy quantifiers in natural languages

The generic term fuzzy quantifier is employed in this paper to denote the collection of quantifiers in natural languages whose representative elements are: several, most, much, not many, very many, not very many, few, quite a few, large number, small number, close to five, approximately ten, frequently, etc. In our approach, such quantifiers are treated as fuzzy numbers which may be manipulated through the use of fuzzy arithmetic and, more generally, fuzzy logic. A concept which plays an essential role in the treatment of fuzzy quantifiers is that of the cardinality of a fuzzy set. Through the use of this concept, the meaning of a proposition containing one or more fuzzy quantifiers may be represented as a system of elastic constraints whose domain is a collection of fuzzy relations in a relational database. This representation, then, provides a basis for inference from premises which contain fuzzy quantifiers. For example, from the propositions “Most U's are A's” and “Most A's are B's,” it follows that “Most 2 U's are B's,” where most 2 is the fuzzy product of the fuzzy proportion most with itself. The computational approach to fuzzy quantifiers which is described in this paper may be viewed as a derivative of fuzzy logic and test-score semantics. In this semantics, the meaning of a semantic entity is represented as a procedure which tests, scores and aggregates the elastic constraints which are induced by the entity in question.

Fuzzy Probabilities and Their Role in Decision Analysis

The conventional approaches to decision analysis are based on the assumption that the probabilities which enter into the assessment of the consequences of a decision are known numbers. In most realistic settings, this assumption is of questionable validity since the data from which the probabilities must be estimated are usually incomplete, imprecise or not totally reliable.In the approach outlined in this paper, the probabilities are assumed to be fuzzy rather than real numbers. It is shown how such probabilities may be estimated from fuzzy data and a basic relation between joint, conditional and marginal fuzzy probabilities is established. Manipulation of fuzzy probabilities requires, in general, the use of fuzzy arithmetic, and many of the properties of fuzzy probabilities are simple generalizations of the corresponding properties of real-valued probabilities.