Fuzzy P-measure on the real line

The fuzzy measurable space on the real line, β_ϱ say, is defined by means of fuzzy relation ‘less or equal’ [5]. The main properties of β_ϱ are given. These results are used for investigation of fuzzy P-measure on β_ϱ. The principal emphasis is laid on the connections between a fuzzy P-measure and its cumulative distribution function. A theorem extending a fuzzy P-measure to β_ϱ is shown.     

A characterization of fuzzy compactifications

It is shown that if X is a fuzzy T₂-space, then X has a fuzzy T₂-compactification if and only if X is a weakly induced ultra completely regular space. Also, for an arbitrary fuzzy topological space, a characterization is given of the set of all ultra fuzzy Compactifications.     

Separation of fuzzy sets in fuzzy minimal spaces

This paper deals with the concepts of fuzzy minimal separation and fuzzy closed minimal separation. Some criterion for m-separatedness and C_m-separatedness of two fuzzy sets in a fuzzy minimal space are achieved. Further, it is shown that for any fuzzy sets C and D in Y, A × C and B × D are fuzzy m-separated (C_m-separated) in X × Y, if A and B are fuzzy m-separated (C_m-separated) sets in X. Moreover, c.A and c.B are fuzzy m-separated (C_m-separated) if and only if A and B are fuzzy m-separated (C_m-separated).     

Ranking alternatives using fuzzy numbers

We investigate the problem of employing expert opinion to rank alternatives across a set of criteria. The experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue's fuzzy ranking. This leads to a partition of the alternatives into sets H₁, H₂,… where H₁ contains the highest ranked issues, H₂ has all the second highest ranked alternatives, etc. The total ranking process is shown to possess a number of important properties. An example is presented to illustrate the method.     

On the fuzzy initial value problem

The initial value problem x′(t) = f(t,x(t)), x(0)= x₀, with fuzzy initial value and with deterministic or fuzzy function f is considered. Two different approaches, viz. the extension principle and the use of extremal solutions of deterministic initial value problems, are applied. Generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated.     

Fixed points for fuzzy mappings

In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset R_χ of X. An element x of X is called fixed point of R iff its membership degree to R_χ is at least equal to the membership degree to R_χ of any y?X, i.e. R_χ(χ)? R_χ(y)(?y?X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of R_χ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.     

Existence and uniqueness of fuzzy solution for the nonlinear fuzzy integrodifferential equations

In this paper, existence and uniqueness of fuzzy solution for the nonlinear fuzzy integrodifferential equations is established via Banach fixed-point analysis approach and using the fuzzy number whose values are normal, convex, upper semicontinuous and compactly supported interval in E_N.     

Fuzzy inference to assess manufacturing process capability with imprecise data

Process capability indices provide numerical measures on whether a process conforms to the defined manufacturing capability prerequisite. These have been successfully applied by companies to compete with and to lead high-profit markets by evaluating the quality and productivity performance. The loss-based process capability index C_pm, sometimes called the Taguchi index, was proposed to measure process capability, wherein the output process measurements are precise. In the present study, we develop a realistic approach that allows the consideration of imprecise output data resulting from the measurements of the products quality. A general method combining the vector of fuzzy numbers to produce the membership function of fuzzy estimator of Taguchi index is introduced for further testing process capability. With the sampling distribution for the precise estimation of C_pm, two useful fuzzy inference criteria, the critical value and the fuzzy P-value, are proposed to assess the manufacturing process capability based on C_pm. The presented methodology takes into the consideration of a certain degree of imprecision on the sample data and leads to the three-decision rule with the four quadrants decision-making plot. The fuzzy inference procedure presented to assess process capability is a natural generalization of the traditional test, when the data are precise the proposed test is reduced to a classical test with a binary decision.     

The practical research on flood risk analysis based on IIOSM and fuzzy α-cut technique

Flood disasters are one of the most common and destructive natural hazards all over the world. In this paper, improved interior-outer-set model (IIOSM) based on information diffusion theory is introduced in detail to assess flood risk in an effort to obtain accurate analytical results that represent the actual situation. Then fuzzy α-cut technique is applied to calculate the fuzzy expected values under the possibility–probability distribution (PPD) calculated by IIOSM. Taking the value of α throughout the interval (0, 1], we correspondingly get access to the conservative risk value (R_C) and venture risk value (R_V). Selection of α, R_C and R_V is dependent on present technical conditions and risk preference of different people. To illustrate the procedure of IIOSM and fuzzy α-cut technique, we employ them respectively to analyze the flood risk in Sanshui District, located in the center of Guangdong province in China. The results, such as risk value estimations, as well as fuzzy expected values, i.e. R_C and R_V under the given α-cut level, can reflect the flood risk quite accurately. The outcomes of this research based on IIOSM and fuzzy α-cut technique offer new insights to carry out an efficient way for various flood protection strategies.     

Applications of the extent analysis method on fuzzy AHP

In this paper, a new approach for handling fuzzy AHP is introduced, with the use of triangular fuzzy numbers for pairwise comprison scale of fuzzy AHP, and the use of the extent analysis method for the synthetic extent value S_i of the pairwise comparison. By applying the principle of the comparison of fuzzy numbers, that is, V(M₁ ⩾ M₂) = 1 iff m₁ ⩾ m₂, V(M₂ ⩾ M₁) = hgt(M₁ ∩ M₂) = μ_M1 (d), the vectors of weight with respect to each element under a certaine criterion are represented by d(A_i) = min V(S_i ⩾ S_k), k = 1, 2,…, n; k ≠ i. This decision process is demonstrated by an example.     

Fuzzy probability measures

In [4] Höhle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see [6]).     

H∞ control for uncertain Takagi–Sugeno fuzzy systems with time-varying delays and nonlinear perturbations

This paper investigates the robust H_∞ control for Takagi–Sugeno (T–S) fuzzy systems with time-varying delays and nonlinear perturbations. Delay-dependent criteria for uncertain fuzzy systems have been proposed. The LMI optimization approach is applied to solve the H_∞ control problem and find the minimization of H_∞ norm bound. Computer simulations show that significant improvement over the previous results can be obtained.     

Fuzzy linear programming and applications

This paper presents a survey on methods for solving fuzzy linear programs. First LP models with soft constraints are discussed. Then LP problems in which coefficients of constraints and/or of the objective function may be fuzzy are outlined. Pivotal questions are the interpretation of the inequality relation in fuzzy constraints and the meaning of fuzzy objectives. In addition to the commonly applied extended addition, based on the min-operator and used for the aggregation of the left-hand sides of fuzzy constraints and fuzzy objectives, a more flexible procedure, based on Yager's parametrized t-norm T_p, is presented. Finally practical applications of fuzzy linear programs are listed.     

ET-lipschitzian and ET-kernel aggregation operators

Lipschitzian and kernel aggregation operators with respect to natural T-indistinguishability operators E_T and their powers are studied. A t-norm T is proved to be E_T-lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an archimedean t-norm T with additive generator t, the quasi-arithmetic mean generated by t is proved to be the most stable aggregation operator with respect to T.     

A note on semi-open sets in fuzzy topological spaces

Concepts of semi-T_i (i = 0, 1, 2) spaces and semi-R_i (i = 0, 1) spaces are introduced and some fuzzy topological properties are investigated under the above mentioned axioms. Some mappings involving semi-open sets on fuzzy topological spaces are also introduced and investigated.     

Control problems in fuzzy systems

This paper deals with some problems of the control of fuzzy systems described by means of a relational equation of the type: X_{κ + 1} = U_κ ° X_κ ° R. The basic aspects such as mutual identification and control, stabilization control, control with constraints, generation of control rules of a fuzzy logic controller, important from the theoretical and applicational point of view are discussed and illustrated by means of numerical examples.     

Some comments on fuzzy variables

Nahmias introduced the concept of a fuzzy variable as a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. In this paper we attempt to shed more light on fuzzy variables in analogy with random variables. In particular, we study the problem: if X₁, X₂,…,X_n are mutually unrelated fuzzy variables with common membership function μ and α₁, α₂,…,α_n are real numbers satisfying α_i ? o for every i and Σ_i=1ⁿα_i=1, when does does Z = Σ_{i = 1}ⁿα_iX_i have the same membership function μ?     

On fuzzy metrizability

In this paper we answer two questions raised by M. A. Erceg [J. Math. Anal. Appl. 69 (1979), 205–230]: precisely we show that the fuzzy unit interval is never T₀ except in the standard case and that a fuzzy pseudo-metrizable T₀ space is metrizable in the sense of Erceg (ibid.); hence the two definitions of metrizable space given in B. Hutton and I. Reilly [Fuzzy Sets and Systems3 (1980), 93–104] and Erceg (ibid.) are equivalent.     

On intutionistic fuzzy gpr-closed sets

In this paper, a new class of intuitionistic fuzzy closed sets called intuitionistic fuzzy generalized preregular closed sets (briefly intuitionistic fuzzy gpr-closed sets) and intuitionistic fuzzy generalized preregular open sets (briefly intuitionistic fuzzy gpr-open sets) are introduced and their properties are studied. Further the notion of intuitionistic fuzzy preregular T _1/2-spaces and intuitionistic fuzzy generalized preregular continuity (briefly intuitionistic fuzzy gpr-continuity) are introduced and studied.     

Spectrum topology of a residuated lattice

In this paper, given a non-commutative residuated lattice L, a topological space is constructed using certain fuzzy subsets of L. Indeed, we show that the set of all prime fuzzy filters of a non-commutative residuated lattice L forms a topological space. Particularly, we show that this space is compact and a T ₀-space and its certain subspaces are Hausdorff spaces. Finally, we show that the set of all prime filters of L is also a Hausdorff space.