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Many researches have been carried out in fuzzy linear regression since the past three decades. In this paper, a fuzzy linear regression model based on goal programming is proposed. The proposed model takes into account the centers of fuzzy data as an important feature as well as their spreads. Furthermore, the model can deal with both symmetric and non-symmetric data. To show the efficiency of proposed model, it is compared with some earlier methods based on simulation studies and numerical examples. Moreover, the sensitivity of the model to outliers is discussed.

This paper shows how Fuzzy Set Theory can be used in investment analysis when, as usual, these investments are developed under uncertainty, i.e. the investor has only subjective estimates based on his experience or knowledge about the future cash-flows of the investments, the discount rate, etc. In particular, we will develop basic concepts for investment analysis as the Net Present Value and the Internal Rate of Return by assuming that the initial data are fuzzy numbers. Later we will analyse how to rank investments and how to select the tangible investment portfolios when the magnitudes are estimated subjectively by comparing fuzzy numbers and with possibilistic mathematical programming.

The transformation method has been proposed as a practical tool for the simulation and the analysis of systems with uncertain, fuzzy-valued model parameters using fuzzy arithmetic. Up to now, this method has been available in two forms: in a general form, which can be used for the simulation and the analysis of arbitrarily non-monotonic problems, and in a reduced form, which reduces the computational costs of the method to a large extent, but requires, instead, an additional condition to be fulfilled. In this paper, the extended transformation method will be introduced as an advanced version of the previously presented formulations of the transformation method. This extended version includes the former versions as marginal cases and allows a pre-adjustment of the method subject to the number of model parameters that are expected to cause non-monotonic behavior of the model output. Furthermore, to set up the method properly, a novel approach, again based on the transformation method, is presented to practically detect those parameters that are responsible for a non-monotonic behavior of the model output. Finally, to show its effectiveness, the method is applied to a static and a dynamic model with uncertain parameters.

Song et al.[3] proposed a new fuzzy time series model by means of defining some new operations on fuzzy numbers. They presented the new model in the form of two theorems, which indicate that the value of F(t) at t+1 can be related to its previous ones by means of forward and backward linguistic difference, as a necessary condition for the homogeneous fuzzy time series. In this note, we prove that it is also a sufficient condition for the homogeneous fuzzy time series.

In this paper we are interested in determining the fuzzy ordering, from smallest to largest, of any finite set of fuzzy numbers. We investigate two methods: (1) using a weak fuzzy ordering; and (2) using a strong fuzzy ordering. Employing a strong fuzzy ordering we show that any finite set of fuzzy numbers has a unique ranking from smallest to largest.

In this paper, fuzzy linear regression models with fuzzy/crisp output, fuzzy/crisp input are considered. In this regard, we define risk-neutral, risk-averse and risk-seeking fuzzy linear regression models. In order to do that, two equality indices are applied to express the degree of equality between a pair of fuzzy numbers. We also develop three mathematical models to obtain the parameters of fuzzy linear regression models. Minimizing the difference between the total spread of the observed and estimated values is the objective of these models. The advantage of our proposed models is the simplicity in programming and computation.

Fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models. With Zadeh's extension principle, one can obtain a fuzzy-valued extension of any real-valued objective function. An efficient and accurate approach to compute expensive multivariate functions of fuzzy numbers is given by fuzzy arithmetic based on sparse grids. In this paper, we illustrate the general applicability of this new method by computing two dynamic systems subjected to uncertain parameters as well as uncertain initial conditions. The first model consists of a system of delay differential equations simulating the periodic outbreak of a disease. In the second model, we consider a multibody mechanism described by an algebraic differential equation system.

In this paper we investigate whether the fuzzy arithmetic based on Zadeh's extension principle could be improved by redefining the fuzzy addition and multiplication so that the class of fuzzy numbers combined with these operations would constitute a field. We will prove that such a fuzzy arithmetic does not exist. This has important consequences for solving systems of linear fuzzy equations. Despite the lack of inverses, we propose a method to solve approximately such systems.

Fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models. With Zadeh's extension principle, one can obtain a fuzzy-valued extension of any real-valued objective function. An efficient and accurate approach to computing expensive multivariate functions of fuzzy numbers is given by fuzzy arithmetic based on sparse grids. In many cases, not all uncertain input parameters carry equal weight, or the objective model exhibits separable structure. These characteristics can be exploited by dimension-adaptive algorithms. As a result, the treatment of even higher-dimensional problems becomes possible. This is demonstrated in this paper by a case study involving two large-scale finite element models in vibration engineering that are subjected to fuzzy-valued input data.

In the framework of the representability of ordinal qualitative data by means of interval-valued correspondences, we study interval orders defined on a nonempty set X. We analyse the continuous case, that corresponds to a set endowed with a topology that furnishes an idea of continuity, so that it becomes natural to ask for the existence of quantifications based on interval-valued mappings from the set of data into the real numbers under preservation of order and topology. In the present paper we solve a continuous representability problem for interval orders. We furnish a characterization of the representability of an interval order through a pair of continuous real-valued functions so that each element in X has associated in a continuous manner a characteristic interval or equivalently a symmetric triangular fuzzy number.

When the outliers exist in the data set, fuzzy regression gives incorrect results. A few number of researchers considered this problem and proposed linear-programming-based methods and fuzzy least-squares methods to deal with the outliers problem. In this paper, we develop a new model along with a linear-programming-based approach for computation of fuzzy regression models. The problem of outliers is modeled with this approach. Two examples are illustrated to compare the performance of proposed approach with those given in literature. Results from numerical examples show that our approach gives good solutions.

The scheduling problems with fuzzy processing times and fuzzy due dates are investigated in this paper. The concepts of earliness and tardiness are interpreted by using the concepts of possibility and necessity measures that were developed in fuzzy sets theory. Many types of objective function will be taken into account through the different combinations of possibility and necessity measures. The purpose of this paper is to obtain the optimal schedules based on these objective functions. The genetic algorithm will be invoked to tackle these objective functions. Four numerical examples are also provided and solved by using the commercial software MATLAB.

The paper investigates the dynamic hybrid multi-attribute group decision making problems, in which the decision information, provided by multiple decision makers at different periods, is expressed in real numbers, interval numbers or linguistic labels (linguistic labels can be described by triangular fuzzy numbers), respectively. We define the concepts of argument variable and dynamic weighted geometric aggregation (DWGA) operator, etc., and give an approach to determining the weights of periods based on the basic unit-interval monotonic (BUM) function, and then propose a dynamic hybrid multi-attribute group decision making method based on the hybrid geometric aggregation (HGA) operator and the DWGA operator. The method first utilizes three different TOPSISs (real-valued TOPSIS, interval-valued TOPSIS and fuzzy-valued TOPSIS) to calculate the individual closeness coefficient of each alternative to the positive and negative ideal alternatives based on the decision information expressed in real numbers, interval numbers and linguistic labels, respectively, provided by each decision maker at each period, and then employs the HGA operator to aggregate all individual closeness coefficients into the collective closeness coefficient corresponding to each alternative at each period. After doing so, the method uses the DWGA operator to fuse the collective closeness coefficients at different periods into the overall closeness coefficient corresponding to each alternative. These overall closeness coefficients are then used to rank and select the given alternatives. We can also reduce the above method to solve the dynamic multi-attribute group decision making problems, in which the decision information, provided by multiple decision makers at different periods, is expressed by means of values from the same type, either real numbers, or interval numbers or linguistic labels. Finally, the developed method is applied to multi-period investment decision making.

Given a binary relation defined on a set, we study its representability by means of a monotonic function that takes values on a suitable universal codomain (that depends on the kind of relation considered). We pay an special attention to the representability of interval orders, studying their alternative universal codomains, some of them equivalent to the set of symmetric triangular fuzzy numbers.

The flow shop scheduling problems with fuzzy processing times are investigated in this paper. For some special kinds of fuzzy numbers, the analytic formulas of the fuzzy compltion time can be obtained. For the general bell-shaped fuzzy numbers, we present a computational procedure to obtain the approximated membership function of the fuzzy completion time. We define a defuzzification function to rank the fuzzy numbers. Under this ranking concept among fuzzy numbers, we plan to minimize the fuzzy makespan and total weighted fuzzy completion time. Because the ant colony algorithm has been successfully used to solve the scheduling problems with real-valued processing times, we shall also apply the ant colony algorithm to search for the best schedules when the processing times are assumed as fuzzy numbers. Numerical examples are also provided and solved by using the commercial software MATLAB.

Owing to the fluctuations in the financial markets, many financial variables such as expected return, volatility, or exchange rate may occur imprecisely. But many portfolio selection models consider precise input of these values. Therefore, this paper studies a multiobjective international asset allocation problem under fuzzy environment. In our portfolio selection model, both of the return risk and the exchange risk are considered. The coefficient matrices in the objectives and constraints and the decision value are considered as fuzzy variables. The calculation of the portfolio and efficient frontier is derived by considering the exchange risk in the fuzzy environment. An empirical study is performed based on a portfolio of six securities denominated in six different currencies, i.e., USD, EUR, JPY, CNY, HKD, and GBP. The α-level closed interval portfolio and the fuzzy efficient frontier are obtained with different values of α ∈ (0, 1]. The empirical results indicate that the fuzzy asset selection method is a useful tool for dealing with the imprecise problem in the real world.

In this paper we propose an approach for multiobjective programming problems with fuzzy number coefficients. The main idea behind our approach is to approximate involved fuzzy numbers by their respective nearest interval approximation counterparts. An algorithm that returns a nearest interval approximation to a given fuzzy number, plays a pivotal role in the proposed method.

Our approach contrasts markedly with those based on deffuzification operators which replace a fuzzy set by a single real number leading to a loss of many important information. A numerical example is also provided for the sake of illustration.

In this paper we develop a new graphical representation of fuzzy numbers, which we then employ to propose a geometrical approach to their defuzzification. The calculations involved in the proposed method and the resultant representation use Moore's semiplane for intervals and therefore are far simpler than those involved in other approaches.

We start by representing triangular and trapezoidal fuzzy numbers in Moore's semiplane. Then we extend this work to any fuzzy number. Although this extension has to be undertaken in ${ℝ}^3$, it preserves all the properties we study for trapezoidal and triangular fuzzy numbers in Moore's semiplane.

We analyze the main properties of binary relations, defined on a nonempty set, that arise in a natural way when dealing with real-valued functions that satisfy certain classical functional equations on two variables. We also consider the converse setting, namely, given binary relations that accomplish some typical properties, we study whether or not they come from solutions of some functional equation. Applications to the numerical representability theory of ordered structures are also furnished as a by-product. Further interpretations of this approach as well as possible generalizations to the fuzzy setting are also commented. In particular, we discuss how the values taken for bivariate functions that are bounded solutions of some classical functional equations define, in a natural way, fuzzy binary relations on a set.

The fuzzy linear assignment problem (FLAP) is an extension of the classical linear assignment problem (LAP) to situations in which uncertainty in the cost coefficients is represented by fuzzy numbers. FLAP applications range from the assignment of workers to tasks to multiple-criteria decision analysis in fuzzy environments and many other engineering applications. Most FLAP formulations assume that all cost coefficients are fuzzy numbers of the same type (e.g. triangular, trapezoidal). The standard solution approach is the defuzzification of the cost coefficients, thus transforming the FLAP into a crisp LAP that can be solved by classical assignment algorithms such as the Hungarian method. It is known that defuzzification methods suffer from lack of discrimination when comparing fuzzy numbers which may lead to suboptimal assignments. The solution approach proposed in this paper is based on the theory of algebraic assignment problems and total orderings in the set of all fuzzy numbers, and it allows to solve FLAPs with different types of fuzzy numbers. More specifically, the FLAP is transformed into a lexicographic linear assignment problem (LLAP) which is solved in its place. We show, both theoretically and numerically, how this transformation overcomes the limitations present in existing approaches.