Form Of Fuzzy Numbers Research Articles

A single period inventory model for incorporating two-ordering opportunities under imprecise demand information

Article history: Received 20 September 2010 Received in revised form 4 December 2010 Accepted 6 December 2010 Available online 7 December 2010 The ordering strategy for a single period inventory model is the key to achieve success in the competitive business environment. This article considers demand in a form of fuzzy number and discusses the SPIM in which the retailer has the opportunity to reorder once during the period. The entire period/season is divided into two slots and the reorder is to be made during the mid-season after the early-season demand has been observed. The objective is to find the expected optimal order quantity together with profit maximization. We illustrate the implementation of the proposed model using a numerical example and explain that the explicit consideration of this reordering opportunity could lead us to better results in terms of profitability. © 2011 Growing Science Ltd. All rights reserved

Fuzzy induced generalized aggregation operators and its application in multi-person decision making

We present a wide range of fuzzy induced generalized aggregation operators such as the fuzzy induced generalized ordered weighted averaging (FIGOWA) and the fuzzy induced quasi-arithmetic OWA (Quasi-FIOWA) operator. They are aggregation operators that use the main characteristics of the fuzzy OWA (FOWA) operator, the induced OWA (IOWA) operator and the generalized (or quasi-arithmetic) OWA operator. Therefore, they use uncertain information represented in the form of fuzzy numbers, generalized (or quasi-arithmetic) means and order inducing variables. The main advantage of these operators is that they include a wide range of mean operators such as the FOWA, the IOWA, the induced Quasi-OWA, the fuzzy IOWA, the fuzzy generalized mean and the fuzzy weighted quasi-arithmetic average (Quasi-FWA). We further generalize this approach by using Choquet integrals, obtaining the fuzzy induced quasi-arithmetic Choquet integral aggregation (Quasi-FICIA) operator. We also develop an application of the new approach in a strategic multi-person decision making problem.

Generalised soft binomial American real option pricing model (fuzzy–stochastic approach)

The stochastic discrete binomial models and continuous models are usually applied in option valuation. Valuation of the real American options is solved usually by the numerical procedures. Therefore, binomial model is suitable approach for appraising the options of American type. However, there is not in several situations especially in real option methodology application at to disposal input data of required quality. Two aspects of input data uncertainty should be distinguished; risk (stochastic) and vagueness (fuzzy). Traditionally, input data are in a form of real (crisp) numbers or crisp-stochastic distribution function. Therefore, hybrid models, combination of risk and vagueness could be useful approach in option valuation. Generalised hybrid fuzzy–stochastic binomial American real option model under fuzzy numbers (T-numbers) and Decomposition principle is proposed and described. Input data (up index, down index, growth rate, initial underlying asset price, exercise price and risk-free rate) are in a form of fuzzy numbers and result, possibility-expected option value is also determined vaguely as a fuzzy set. Illustrative example of equity valuation as an American real call option is presented.

A method based on preference degrees for handling hybrid multiple attribute decision making problems

In this paper, we investigate hybrid multiple attribute decision making problems with various forms of attribute values (real numbers, linguistic labels, interval numbers, intuitionistic fuzzy numbers and interval intuitionistic fuzzy numbers). We propose a method based on preference degrees which may take the forms of fuzzy numbers, intuitionistic fuzzy numbers and interval intuitionistic fuzzy numbers. The method first normalizes various forms of attribute values into preference degrees, and then uses a preference degree-based weighted averaging operator to aggregate the normalized preference degrees. Meanwhile, for convenience of calculation, a new linguistic representation model is presented, whose feasibility is verified by comparing it with the traditional 4-tuple linguistic representation model, and from our model, the mapping relationship between interval intuitionistic fuzzy numbers and linguistic labels can be constructed. Finally, we illustrate the rationality and practicality of the proposed method by an application example.

Tolerance analysis of assemblies with asymmetric tolerances by unified uncertainty–accumulation model based on fuzzy logic

In mechanical assemblies, individual components are placed together to deliver a certain function. The performance, quality, and cost of the mechanical assembly are significantly affected by its tolerances. Toleranced dimensions inherently generate an uncertain environment in a mechanical assembly. This paper presents a proper method for tolerance analysis of mechanical assemblies with asymmetric tolerances based on an uncertainty model. This mathematical approach is based on fuzzy logic and tolerance accumulation models such as worst-case and root-sum-square methods. A fuzzy-based tolerance representation is developed to model uncertainty of tolerance components in the mechanical assemblies. According to this scheme, toleranced components are described as fuzzy numbers with their membership functions constructed using the statistical distributions of manufactured variables. In this way, the uncertainty of assembly requirements and accumulation of tolerances are represented in the form of fuzzy number. In this paper, a new factor, the fuzzy factor, is introduced that helps converting the membership functions into fuzzy intervals that can be used for modal interval analysis. Equations for estimation of percent contributions of individual tolerances are introduced in terms of uncertainty parameter. These equations yield percent contributions of upper and lower bounds of independent variables (manufactured dimensions) on the upper and lower bounds of dependent variables (assembly dimensions). The proposed method is applied to an example, and its results are discussed.

A new fuzzy multicriteria decision making method and its application in diversion of water

Taking account of uncertainty in multicriteria decision making problems is crucial due to the fact that depending on how it is done, ranking of alternatives can be completely different. This paper utilizes linguistic values to evaluate the performance of qualitative criteria and proposes using appropriate shapes of fuzzy numbers to evaluate the performance of quantitative criteria for each problem with respect to its particular conditions. In addition, a process to determine the weights of criteria using fuzzy numbers, which considers their competition to gain greater weights and their influence on each other is described. A new fuzzy methodology is proposed to solve such a problem that utilizes parametric form of fuzzy numbers. The case study of diversion of water into Lake Urmia watershed, which is defined using triangular, trapezoidal, and bell-shape fuzzy numbers demonstrates the utility of the proposed method.

THE FUZZY GENERALIZED OWA OPERATOR AND ITS APPLICATION IN STRATEGIC DECISION MAKING

We present the fuzzy generalized ordered weighted averaging (FGOWA) operator. It is an extension of the GOWA operator for uncertain situations where the available information is given in the form of fuzzy numbers. This generalization includes a wide range of mean operators such as the fuzzy average (FA), the fuzzy OWA (FOWA), and the fuzzy generalized mean (FGM). We also develop a further generalization by using quasi-arithmetic means that we call the quasi-FOWA operator. The article ends with an illustrative example where we apply the new approach in the selection of strategies.

A method for group decision making with multi-granularity linguistic assessment information

This paper proposes a method to solve the group decision making (GDM) problems with multi-granularity linguistic assessment information. In the method, the multi-granularity linguistic information provided by experts is firstly expressed in the form of fuzzy numbers. In order to make the collective opinion close to each expert’s opinion, a linear goal programming model is constructed to integrate the fuzzy assessment information and to directly compute the collective ranking values of alternatives without the need of information transformation. Then, a fuzzy preference relation on the pairwise comparisons of the collective ranking values of alternatives is constructed using the dominance possibility degree of the comparison between the fuzzy numbers. By applying a non-dominance choice degree to this fuzzy preference relation, the ranking of alternatives is determined and the most desirable alternative(s) is selected. An example is used to illustrate the applicability of the proposed method and its advantages.

Numerical solution of a fuzzy system of linear equations with polynomial parametric form

A systems of linear equations are used in many fields of science and industry, such as control theory and image processing, and solving a fuzzy linear system of equations is now a necessity. In this work we try to solve a fuzzy system of linear equations having fuzzy coefficients and crisp variables using a polynomial parametric form of fuzzy numbers.

Choice of the best alternative in case of a continuous set of states of nature-application of fuzzy numbers

On the basis of a known application of an order weighted averaging operator to the decision making in the case of a discrete set of states of nature, a general approach to the case of a continuous set of states of nature is proposed. The general approach encompasses various types of attitudes of the decision maker, expressed in the form of fuzzy numbers.

Fuzzy dynamic structural analysis of two-dimensional frame

In this paper, a dynamic analysis based on the fuzzy set theory is presented as a possible complementary tool to the classical stochastic methods for dynamic analyses. Material parameters of a structure are influenced by uncertainties and therefore they are considered to be fuzzy quantities with given distributions, that means fuzzy numbers with given membership functions. The fuzzy dynamic analysis is conducted with help of fuzzy arithmetic defined on the so-called -cuts. The results of the analysis are also obtained in the form of fuzzy numbers, which compared to the stochastic methods is less computationaly expensive while at the same time they still provide information about the distribution of a quantity. This method is demonstrated on an analysis of a two-dimensional frame subjected to possible seismic load, where the uncertain eigenmodes and eigenfrequencies are used in the modal analysis.

Bayesian reliability analysis for fuzzy lifetime data

Lifetime data are important in reliability analysis. Classical reliability estimation is based on precise lifetime data. It is usually assumed that observed lifetime data are precise real numbers. However, some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers. Thus, it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers. Bayesian methods have proved to be very useful when the sample size is small. There is little study on Bayesian reliability estimation based on fuzzy lifetime data. Most of the reported works in this area is limited to single parameter lifetime distributions. In this paper, we propose a new method to determine the membership function of the estimates of the parameters and the reliability function of multi-parameter lifetime distributions. An artificial neural network is used to approximate the calculation process of parameter estimation and reliability prediction. The genetic algorithm is used to find the boundary values of the membership function of the estimate of interest at any cut level. This method can be used to determine the membership functions of the Bayesian estimates of multi-parameter distributions. The effectiveness of the proposed method is illustrated with normal and Weibull distributions.

Fuzziness and randomness in investment project risk appraisal

Risk quantification is one of the most difficult tasks associated with investment project risk management, and computer simulation seems to be an especially effective tool for such risk appraisal. This article presents a method for quantification of project-specific risk. When assessing investment project risk it is very common to apply two analytical methods for describing parameter uncertainty: probability distribution and possibility distribution. This study discusses methods for integrating the above-mentioned approaches into a description of the uncertainty of parameters in calculations of effectiveness and investment project risk. The paper presents an example of a computer simulation used for the purpose of an investment project risk assessment. Uncertainty for some parameters of the effectiveness calculation is defined by a probability distribution and by fuzzy sets for others, and a transformation of possibility distributions into probability distributions is thus done. For comparison, the investment risk assessment is undertaken on the assumption that uncertainty distributions of the effectiveness calculation parameters are presented in the form of fuzzy numbers.

Fuzzy process control: construction of control charts with fuzzy numbers

This paper presents the construction of fuzzy control charts for a process with fuzzy outcomes derived from the subjective quality ratings provided by a group of experts. The proposed fuzzy process control (FPC) methodology comprises an off-line stage and an on-line stage. In the off-line stage, experts assign quality ratings to products based on a numerical scale. The individual numerical ratings are then aggregated to form collective opinions expressed in the form of fuzzy numbers. The collective knowledge applied by the experts when conducting the quality rating process is acquired through a process of fuzzy regression analysis performed by a neural network. In the on-line stage, the product dimensions are measured, and the fuzzy regression model is employed to automate the experts’ judgments by mapping the measured dimensions to appropriate fuzzy quality ratings. The fuzzy quality ratings are then plotted on fuzzy control charts, whose construction and out-of-control conditions are developed using possibility theory. The developed control charts not only monitor the central tendency of the process, but also indicate its degree of fuzziness.

Possible and necessary optimality of solutions in the single machine scheduling problem with fuzzy parameters

The single machine scheduling problem with parameters given in the form of fuzzy numbers is considered. It is assumed that the optimal schedule in such a problem cannot be determined precisely (since the parameters of the problem are not known a priori). In this paper the concepts of possible and necessary optimality of a given schedule are introduced. The degrees of possible and necessary optimality measure the possibility and the necessity of the event that a given schedule will be optimal. In this paper, it is shown how to calculate the degrees of possible and necessary optimality of a given schedule in one of the special cases of the single machine scheduling problems.

Application of the fuzzy–stochastic methodology to appraising the firm value as a European call option

The valuing of a firm equity as a call option is a crucial problem in financial decision-making. There are two basic aspects that are studied; contingent claim features (payoff functions) and risk (stochastic process of underlying assets). However, non-preciseness (vagueness, uncertainty) of input data is often neglected. Thus, a combination of risk (stochastic) and uncertainty (fuzzy instruments) could be a useful approach in calculating a firm value as a call option. The Black–Scholes methodology of appraising equity as a European call option is applied. Fuzzy–stochastic methodology under fuzzy numbers ( T-numbers) is proposed and described. Fuzzy–stochastic model of appraising a firm equity is proposed. Input data are in a form of fuzzy numbers and result, firm possibility-expected equity value is also determined vaguely as a fuzzy set. Illustrative example is introduced.

Modelling of water pollution in the Thermaikos Gulf with fuzzy parameters

The fuzzy set theory is applied for the definition of the imprecise parameters, which are used in a water pollution model. The physico-chemical coefficients and the loads of pollution sources are expressed in the form of triangular fuzzy numbers. A two-dimensional finite element algorithm, combined with fuzzy logic analysis, is used for the solution of the advection–dispersion equation for ten different water quality variables. The characteristic Galerkin technique was employed for the temporal discretisation, whereas interval operations were conducted for the solution of the algebraic system of equations with fuzzy coefficients. The model was applied for the study of pollution in the Gulf of Thermaikos, located in Northern Greece, for water velocities obtained from a wind-induced circulation model. The concentration distributions of the water quality variables considered are derived in the form of fuzzy numbers, and are presented in relevant diagrams.

Fuzzy engineering economic analysis [of electric utilities

This paper presents an approach for including nonstatistical uncertainties in engineering economic analysis, particularly utility economic analysis, by modeling uncertain variables with fuzzy numbers. In this case, the mathematical operations are defined by the extension principle and the results obtained are also in a form of fuzzy numbers. This approach can be seen as an extension of a previously proposed one that uses interval numbers and interval analysis for including such uncertainties. However, this paper considers also the dependence which may exist between the fuzzy variables and shows the impact this dependence may have on the results. In this context, a way of modeling partial correlation between the variables of the same kind is proposed.

Estimating Operational Profile in Software Reliability: Fuzzy Approach

In this article, a method has been introduced to estimate the operational profile of a software. This procedure incorporates the linguistic and qualitative data of various preferred usage profiles. The concepts of theory of fuzzy subsets have been used to quantify the given qualitative data of usage profiles in the form of fuzzy numbers. Different rankings of usage profiles for different users obtained by the procedure enable us to identify the reactions or disagreements inherent among the users. Incorporation of various reactions that exist among users gives us an agreement matrix. This helps in finding a consensus ranking for the system operational profile.

The fuzzy GMDH algorithm by possibility models and its application

It is the purpose of this paper to formulate the Group Method of Data Handling (GMDH) by the possibility model. The GMDH formulated by the possibility model is called the fuzzy GMDH. To formulate the fuzzy GMDH, we discuss possibilistic linear regression by the LP problem, called Min Problem. In the fuzzy GMDH, the estimated model by using possibilistic linear regression with a multilayer procedure is a non-linear system with parameters in the form of fuzzy numbers. The estimation by the fuzzy GMDH can be obtained in fuzzy numbers. Thus, this approach can be regarded as fuzzy interval analysis.