Fuzzy Convex Research Articles

Fuzzy Functions, Relations, and Fuzzy Transforms (2012)

Fuzzy functions, fuzzy relations, and fuzzy transforms are applied to fuzzy systems and like in the first issue published in 2011, here these terms must be intended in general sense. Indeed, the topics concerning fuzzy optimizations with fuzzy functions and relations (including also clustering and PSO algorithms), applications of fuzzy relations/transforms to fusion images, and detection coarse grained fuzzy association rules in the datasets and fuzzy convex processes are dealt with widely in this second issue. The contents of any single paper are resumed in the following sequel:

A Note on -Convex Fuzzy Processes

We first define the concept of -convex fuzzy processes. Second, we present some basic properties of such processes.

Generalized (convex) fuzzy sublattices

Following the work of Zadeh and Xue-hai Yuan on the definition of a convex fuzzy subset, three new kinds of definitions of the convex fuzzy sublattices are proposed in this paper. First, by using the relations between fuzzy points and fuzzy sets, the definition of a ( β ̄ , α ̄ ) -convex fuzzy sublattice is introduced. The acceptable nontrivial concepts obtained in this manner are the ( ∈ , ∈ ∨ q ) -convex fuzzy sublattice and ( ∈ ̄ , ∈ ̄ ∨ q ̄ ) -convex fuzzy sublattice. Second, the ( λ , μ ) -convex fuzzy sublattice is presented and it is also characterized by the level ( λ , μ ) -convex sublattice of the given fuzzy set. Finally, ( α , β ) -fuzzy ideal and ( ∈ , ∈ ∨ q k ) -fuzzy sublattice are introduced and several theorems are obtained.

Fuzzy entropy design for non convex fuzzy set and application to mutual information

Fuzzy entropy was designed for non convex fuzzy membership function using well known Hamming distance measure. The proposed fuzzy entropy had the same structure as that of convex fuzzy membership case. Design procedure of fuzzy entropy was proposed by considering fuzzy membership through distance measure, and the obtained results contained more flexibility than the general fuzzy membership function. Furthermore, characteristic analyses for non convex function were also illustrated. Analyses on the mutual information were carried out through the proposed fuzzy entropy and similarity measure, which was also dual structure of fuzzy entropy. By the illustrative example, mutual information was discussed.

Invariant approximation for fuzzy nonexpansive mappings

We establish results on invariant approximation for fuzzy nonexpansive mappings defined on fuzzy metric spaces. As an application a result on the best approximation as a fixed point in a fuzzy normed space is obtained. We also define the strictly convex fuzzy normed space and obtain a necessary condition for the set of all {$t$-best} approximations to contain a fixed point of arbitrary mappings. A result regarding the existence of an invariant point for a pair of commuting mappings on a fuzzy metric space is proved. Our results extend, generalize and unify various known results in the existing literature.

The maximal distance between imprecise point objects

In this paper we propose a new mathematical model of imprecise point objects and a new algorithm for determining the maximal distance between imprecise point objects. Results obtained with our model can be used in various applications, such as GIS (imprecise spatial object modelling), robotics (environment models), and image analysis (imprecise feature extraction). Imprecise point objects are modelled as fuzzy points in linear fuzzy space. Linear fuzzy space is defined over R n , where the membership function is convex, upper semi-continuous, symmetric w.r.t. the core and linearly decreasing w.r.t. the core distance, where the core is a single point. A fuzzy convex hull is defined in linear fuzzy space. The distance between objects/points is modelled as an imprecise point object in linear fuzzy space defined over R 1 . Fuzzy ordering relations are defined in distance space. The maximal distance is a fuzzy set over a set of distances. The maximal distance between a set of imprecise point objects is then determined as the maximal distance of fuzzy convex hull edge points. An algorithm for determining the maximal distance between imprecise points belonging to one- and two-dimensional linear fuzzy space is given. For an exact calculation of this maximal distance, computational complexity of the algorithm is of polynomial growth of the fourth degree. However, if the maximal distance is determined approximately, then the algorithms’ computational complexity is of power growth of the quadratic degree with respect to the cardinality of the convex edge.

A Characterization of Convex Fuzzy Mappings

As any real-valued functions can be regarded as a fuzzy mapping, the research on fuzzy mappings extends the research on real-valued functions. In this paper, a characterization of convex fuzzy mappings is obtained. Supposeis a fuzzy mapping, whereis a non-empty convex subset ofandis the set of all fuzzy numbers. With respect to the fuzzy-max order, is convex if and only if it is both quasi-convex and intermediate-point convex.

FUZZY CONVEX SUBALGEBRAS OF COMMUTATIVE RESIDUATED LATTICES

In this paper, we define the notions of fuzzy congruence relations and fuzzy convex subalgebras on a commutative residuated lattice and we ob- tain some related results. In particular, we will show that there exists a one to one correspondence between the set of all fuzzy congruence relations and the set of all fuzzy convex subalgebras on a commutative residuated lattice. Then we study fuzzy convex subalgebras of an integral commutative residu- ated lattice and will prove that fuzzy filters and fuzzy convex subalgebras of an integral commutative residuated lattice coincide.

Browder's Fixed Point Theorem and Some Interesting Results in Intuitionistic Fuzzy Normed Spaces

We define and study Browder's fixed point theorem and relation between an intuitionistic fuzzy convex normed space and a strong intuitionistic fuzzy uniformly convex normed space. Also, we give an example to show that uniformly convex normed space does not imply strongly intuitionistic fuzzy uniformly convex.

Interactive fuzzy multiobjective reliability optimization using NSGA-II

In many practical situations where reliability enhancement is involved, the decision making is complicated because of the presence of several mutually conflicting objectives. Presence of multiple objectives in a problem, in principle, gives rise to a set of optimal solutions (largely known as Pareto-optimal solutions), instead of a single optimal solution. This type of problem is known as multiobjective optimization problem (MOOP). In general, a MOOP can be solved using weighted sums or decision-making schemes. An alternative way is to look for the Pareto-optimal front. Many evolutionary algorithms (EAs) like genetic algorithm (GA) have been suggested to solve MOOP, hence termed as multiobjective evolutionary algorithms (MOEAs). Nondominated sorting genetic algorithm (NSGA-II) is one such MOEA which demonstrates the ability to identify a Pareto-optimal front efficiently. Thus, it provides the decision maker (DM) a complete picture of the optimal solution space. This paper presents the reliability optimization of a life-support system in a space capsule where reliability of the system is maximized while minimizing the cost. An interactive fuzzy satisficing method for deriving a Pareto-optimal solution preferred by the DM is presented here. Prior preference of the DM has been taken into account here. Using the concept of fuzzy sets and convex fuzzy decision making a multiobjective fuzzy optimization problem is formulated from the original crisp optimization problem. Different nonlinear membership functions based on the DM’s preference have been employed for the fuzzification. Then, NSGA-II is applied to solve the resulting fuzzified MOOP. Resulting Pareto-optimal solution gives the DM variety of alternatives to seek an appropriate solution by modifying parameters interactively according to his/her preference again. Various Pareto-optimal fronts under different preferences of DM have been reported.

Inference with Governing Schemes for Propagation of Fuzzy Convex Constraints Based on α-Cuts

A governing scheme is proposed for fuzzy constraint propagation from given facts to consequences in convex forms, which is applied to the inference method based on α-cuts and generalized mean. The governing is performed by self-tuning which can reflect the distribution forms of fuzzy sets in consequent parts to the forms of deduced consequences. Thereby, the proposed scheme can solve the problems in the conventional inference based on the compositional rule of inference that deduces fuzzy sets with excessive fuzziness increase and specificity decrease. In simulations, it is confirmed that the proposed scheme can effectively perform the constraint propagation from given facts to consequences in convex forms while reflecting fuzzy-set distributions in consequent parts. It is also demonstrated that consequences are deduced without excessively large fuzziness and small specificity.

The topological structure of fuzzy sets with endograph metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space R n , let K ( Y ) be the family of all fuzzy sets of R n , which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space K ( Y ) with the topology of endograph metric is homeomorphic to the Hilbert cube Q = [ - 1 , 1 ] ω iff Y is compact; and the space K ( Y ) is homeomorphic to { ( x n ) ∈ Q : sup | x n | < 1 } iff Y is non-compact and locally compact.

Analysis of Fuzzy Entropy and Similarity Measure for Non Convex Membership Functions

Fuzzy entropy is designed for non convex fuzzy membership function using well known Hamming distance measure. Design procedure of convex fuzzy membership function is represented through distance measure, furthermore characteristic analysis for non convex function are also illustrated. Proof of proposed fuzzy entropy is discussed, and entropy computation is illustrated.

The fuzzy core in games with fuzzy coalitions

In this paper, the fuzzy core of games with fuzzy coalition is proposed, which can be regarded as the generalization of crisp core. The fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero. The nonempty condition of the fuzzy core is given based on the fuzzy convexity. Three kinds of special fuzzy cores in games with fuzzy coalition are studied, and the explicit fuzzy core represented by the crisp core is also given. Because the fuzzy Shapley value had been proposed as a kind of solution for the fuzzy games, the relationship between fuzzy core and the fuzzy Shapley function is also shown. Surprisingly, the relationship between fuzzy core and the fuzzy Shapley value does coincide, as in the classical case.

A Repeated Convex Fuzzy Cooperative Game

This paper introduces repeated theory on the base of fuzzy cooperative game by Aubin etal in 1974 and then constructs repeated fuzzy games theory. It gives the conception of repeated convex fuzzy cooperative games and studies the property of repeated convex fuzzy games.

Similarity Measure Construction for Non-Convex Fuzzy Membership Function

The similarity measure is constructed for non-convex fuzzy membership function using well known Hamming distance measure. Comparison with convex fuzzy membership function is carried out, furthermore characteristic analysis for non-convex function are also illustrated. Proposed similarity measure is proved and the usefulness is verified through example. In example, usefulness of proposed similarity is pointed out.

L-fuzzy numbers and their properties

In this paper, the notions of L-fuzzy convex sets and L-fuzzy numbers are introduced where L is a completely distributive lattice. The notions of [0, 1]-fuzzy convex sets and [0, 1]-fuzzy numbers are generalized. Furthermore their properties and characterizations are presented in terms of cut sets of L-fuzzy sets.

Automatic weight selection for controller synthesis

This article is concerned with the development of a procedure to calculate the parameters of weighting functions used in controller designs to achieve a desired system performance. The synthesis problem is formulated as a multi-criterion optimization problem. A Genetic Algorithm is then employed to search for suitable solutions. To cope with the imprecision and vagueness that arises in the description of objective functions and constraints of the process and actuators, concepts from the fuzzy logic are incorporated into the solution. A multi-objective fuzzy optimization is stated and a fuzzy convex decision-making is established. To demonstrate the advantages of the proposed method, the tuning of SISO and MIMO controllers are investigated. The method is applied to the tuning of controllers of a ship autopilot and of a dynamic ship positioning system. Results are presented, analyzed and compared to former works.

Automatic weight selection for H ∞ controller synthesis

This article is concerned with the development of a procedure to calculate the parameters of weighting functions used in H∞ controller designs to achieve a desired system performance. The H∞ synthesis problem is formulated as a multi-criterion optimization problem. A Genetic Algorithm is then employed to search for suitable solutions. To cope with the imprecision and vagueness that arises in the description of objective functions and constraints of the process and actuators, concepts from the fuzzy logic are incorporated into the solution. A multi-objective fuzzy optimization is stated and a fuzzy convex decision-making is established. To demonstrate the advantages of the proposed method, the tuning of SISO and MIMO controllers are investigated. The method is applied to the tuning of controllers of a ship autopilot and of a dynamic ship positioning system. Results are presented, analyzed and compared to former works.

Some fixed point theorems in fuzzy normed linear spaces

In this paper, the concepts of sectional fuzzy continuous mappings, l-fuzzy compact sets, asymptotic fuzzy normal structure and strong uniformly convex fuzzy normed linear spaces have been introduced. Schauder-type and other fixed point theorems have been established in fuzzy normed linear spaces.