Space Of Fuzzy Sets Research Articles

The optimal dispatching strategy of cogeneration based on Deep Q-Network (DQN) algorithm

This work expands on previous research to offer a state-of-the-art approach for optimizing the dispatching of cogeneration systems, given the limitations faced by conventional coal-fired cogeneration units and the increasing environmental standards. Acknowledging the constraints of flexibility in winter heating, the study aims to improve unit coal use optimization and lower emissions. The paper presents a novel optimization approach for distributing electricity and heat in cogeneration units, utilizing the Deep Q-Network (DQN) algorithm. The suggested approach reduces operating expenses and improves system dependability using a sixth-order function fitting and fuzzy set space division. The study’s results indicate a significant 8.96% increase in performance, demonstrating the effectiveness of the DQN-based strategy in enabling cost-effective scheduling in cogeneration systems. This research offers a road towards sustainable and effective energy use and contributes to the development of cogeneration technology. It also has potential applications in natural energy systems.

Properties of several metric spaces of fuzzy sets

This paper discusses the properties of the spaces of fuzzy sets in a metric space equipped with the endograph metric and the sendograph metric, respectively. We first give some relations among the endograph metric, the sendograph metric and the Γ-convergence, and then investigate the level characterizations of the endograph metric and the Γ-convergence. By using the above results, we give some relations among the endograph metric, the sendograph metric, the supremum metric and the dp⁎ metric, p≥1. On the basis of the above results, we present the characterizations of total boundedness, relative compactness and compactness in the space of fuzzy sets whose α-cuts are compact when α>0 equipped with the endograph metric, and in the space of compact support fuzzy sets equipped with the sendograph metric, respectively. Furthermore, we give completions of these metric spaces, respectively.

Convergence in distribution of fuzzy random variables in Lp-type metrics

General properties of convergence in distribution for fuzzy random variables are studied as regards its interplay with the structure of the space of fuzzy sets. In particular, its behaviour with respect to taking tuples of fuzzy random variables, adding, multiplying by a scalar, taking the union, preserving inclusion ordering, and subsuming convergence in distribution of random sets is established.

Fuzzy-set approach to invariant idempotent measures

We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in Mazurenko and Zarichnyi (2018) [24]. The main feature developed here is a metrization of the space of idempotent measures using the embedding of the space of idempotent measures to the space of fuzzy sets. The metric obtained induces a topology stronger than the canonical pointwise convergence topology. A key result is the existence of a bijection between idempotent measures and fuzzy sets and a conjugation between the Markov operator of an IFS on idempotent measures and the fuzzy fractal operator of the associated Fuzzy IFS. This allows to prove that the Markov operator for idempotent measures is a contraction w.r.t. the induced metric and, from this, to obtain a convergence theorem and algorithms that draw pictures of invariant measures as grayscale images.

A visual uncertainty analytics approach for weather forecast similarity measurement based on fuzzy clustering

Forecast calibration methods based on historical similar atmospheric state are effective means weather forecast accuracy. Conventional approaches search similar forecasts on the basis of predefined similarity formulas and provide calibration recommendations to forecasters. However, these approaches ignore the uncertainty of similarity measurement, which affects calibration efficacy significantly. This study proposes a similarity weight adaptive algorithm for high-dimensional data on the basis of fuzzy clustering to characterize the uncertainty of similarity measurements. Without prior knowledge, the algorithm computes the uncertainty of the similarity between data in the fuzzy set space iteratively on the basis of membership and then determine weight distribution by maximizing the differentiating ability of each dimension. This study further presents a visual analysis framework on the basis of the weight adaptive algorithm for the exploration of uncertainty in meteorological data and the optimization of similarity measurement method. This framework has coordinated views and intuitive interactions to enable the visualization of the similarity uncertainty distribution and support the iterative visual analysis of similarity weight distribution in each dimension that combines domain knowledge. We illustrate a case study using real-world meteorological data to verify the efficacy of the proposed approach.

Stronger Forms of Sensitivity for Induced Fuzzified Map

Every dynamical system on a compact metric space X induces a fuzzy dynamical system on the space of fuzzy sets , by Zadeh's extension principle. In this paper we consider stronger forms of sensitivity, viz. strong sensitivity, asymptotic sensitivity, syndetic sensitivity, multi-sensitivity and cofinite sensitivity. Some examples are given to expound the interrelation between them. Our main concern here is to find the relationship between f and in terms of these forms of sensitivity. We Prove that these forms of sensitivity for f partially imply the same for and in other way we also get partial induction.

Characterizations of compact sets in fuzzy set spaces with Lp metric

Compactness criteria in fuzzy set spaces endowed with the Lp metric have been studied for several decades. Total boundedness is a key feature of compactness in metric spaces. However, comparing existing compactness criteria in fuzzy set spaces endowed with the Lp metric with the Arzelà–Ascoli theorem, the latter gives compactness criteria by characterizing totally bounded sets while the former does not characterize totally bounded sets. Currently, compactness criteria are only presented for three particular fuzzy set spaces under assumptions of convexity or star-shapedness. General fuzzy sets have become more important in both theory and applications. Therefore, this paper presents characterizations of totally bounded sets, relatively compact sets, and compact sets in general fuzzy set spaces equipped with the Lp metric, but which do not have any assumptions of convexity or star-shapedness. Subsets of these general sets include common fuzzy sets, such as fuzzy numbers, fuzzy star-shaped numbers with respect to the origin, fuzzy star-shaped numbers, and general fuzzy star-shaped numbers. Existing compactness criteria are stated for fuzzy numbers space, the space of fuzzy star-shaped numbers with respect to the origin, and the space of fuzzy star-shaped numbers endowed with the Lp metric, respectively. Constructing completions of fuzzy set spaces with respect to the Lp metric is a problem closely dependent on characterizing totally bounded sets. Based on characterizations of total boundedness and relatively compactness and some discussion of the convexity and star-shapedness of fuzzy sets, we show that the completions of fuzzy set spaces studied here can be obtained using the Lp extension. We also clarify relationships among the ten fuzzy set spaces studied here—the five pairs of original spaces and their corresponding completions. We show that the subspaces have parallel characterizations of totally bounded sets, relatively compact sets, and compact sets. Finally, we discuss properties of the Lp metric on fuzzy set space as an application of our results, and review compactness criteria proposed in previous work.

Properties of Fuzzy Length on Fuzzy Set

The definition of fuzzy length space on fuzzy set in this research was introduced after the studies and discussion of many properties of this space were proved, and then an example to illustrate this notion was given. Also the definition of fuzzy convergence, fuzzy bounded fuzzy set, and fuzzy dense fuzzy set space was introduced, and then the definition of fuzzy continuous operator was introduced.

Fuzzy fixed point theorems on the complete fuzzy spaces under supremum metric

By using the concept of a class of functions, the $\mathcal {R}$ -functions, we provide some fuzzy fixed point theorems on a space of fuzzy sets equipped with the supremum metric. By presenting a technique of constructing a sequence of successive approximations, we obtain some interesting results that improve many existing results. The related cases are also shown and discussed.

On approximations of Zadeh's extension principle

In this paper we provide several results concerning the approximation of Zadeh's extension of a given function. We provide several simple observations due to which we provide a general procedure allowing to approximate Zadeh's extension of any continuous map. Among other things, we demonstrated that the technique of F-transform can also be used to approximate this extension. Further, we studied how the choice of the metric on the space of fuzzy sets (resp. fuzzy numbers) can affect the quality of the approximation, and we clarified that, in comparison to recent habits, the choice of the levelwise metric need not be the most appropriate one, and that the approximation process could be simplified. Our results are general in the sense that we mostly require nothing but the continuity of the original map.

A fuzzy metric on the space of fuzzy sets

A fuzzy metric on the space of fuzzy sets

Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

The first aim of the paper is to present a survey of possible approaches for the study of fuzzy stochastic differential or integral equations. They are stochastic counterparts of classical approaches known from the theory of deterministic fuzzy differential equations. For our aims we present first a notion of fuzzy stochastic integral with a semimartingale integrator and its main properties. Next we focus on different approaches for fuzzy stochastic differential equations. We present the existence of fuzzy solutions to such equations as well as their main properties. In the first approach we treat the fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors. In the second one the equation is interpreted as a system of stochastic inclusions. Finally, in the last section we discuss fuzzy stochastic integral equations with solutions being fuzzy stochastic processes. In this case the notion of the stochastic Itô’s integral in the equation is crisp; that is, it has single-valued level sets. The second aim of this paper is to show that there is no extension to more general diffusion terms.

Some chaotic and mixing properties of fuzzified dynamical systems

Let X be a compact metric space, let φ be a continuous self-map on X, and let F(X) denote the space of fuzzy sets on X equipped with the levelwise topology. In this paper we study relations between a given (crisp) dynamical system (X,φ) and its g-fuzzification (F(X),Φg). Among other things we study various (weak, strong, mild etc.) mixing properties and also several kinds of chaotic behaviors (Li–Yorke chaos, ω-chaos, distributional chaos, topological chaos etc.). We specified subspaces of fuzzy sets which makes sense to study further. Additionally, we proved that numerous local characteristics of the original dynamical system can be found in its fuzzy extension, while many global characteristics are preserved in the opposite direction. Thus, among other things, we obtained easy-to-check criteria for checking absence of complex behavior in induced fuzzy dynamical systems.

Approximation schemes for fuzzy stochastic integral equations

We consider fuzzy stochastic integral equations with stochastic Lebesgue trajectory integrals and fuzzy stochastic Itô trajectory integrals. Some methods of construction of approximate solutions to such the equations are examined. We study the Picard type approximations, the Carathéodory type approximations and the Maruyama type approximations of solutions. In considered framework, the solutions and approximate solutions are mappings with values in the space of fuzzy sets with basis of square integrable random vectors. Under Lipschitz and linear growth conditions each sequence of considered approximate solutions converges to the exact unique solution of the given fuzzy stochastic integral equation. For each type of approximate solutions, we show that the sequence of approximations is uniformly bounded and obtain some bounds for a distance between nth approximation and exact solution.

On the topological entropy on the space of fuzzy numbers

In the main result of this article, we prove that the topological entropies of a given interval map and its Zadeh's extension (fuzzification) to the space of fuzzy numbers (i.e., the space of fuzzy sets with connected α-cuts) are the same. This result is in contrast with our previous result, which claimed that the topological entropy of the Zadeh's extension significantly increases for a majority of simple interval maps. In addition, we prove some properties of the limit sets of trajectories that are generated by iterating the fuzzy set valued function on connected fuzzy sets; for instance, we specify the shapes of the possible limit sets. Furthermore, the presented topics are studied for set-valued (induced) discrete dynamical systems. The main results are proved with a variational principle that describes the relations between topological entropy and measure-theoretical entropy.

Algebraic, metric and probabilistic properties of convex combinations based on the t-normed extension principle: the strong law of large numbers

Consider the Strong Law of Large Numbers for t-normed averages of fuzzy random variables in the uniform metric d∞. That probabilistic property is known to hold when the t-norm is the minimum and to fail when the t-norm is the product. We prove that it is characterized by an algebraic property of the t-norm (that of being eventually idempotent) and by a metric property of the space of fuzzy sets (that it becomes a convex combination space). We show that the equivalence holds not only for Euclidean or Banach spaces, but in the more general setting of convex combination spaces.

Common fixed point theorems for fuzzy mappings in G-metric spaces

In this paper, we introduce the concept of Hausdorff G-metric in the space of fuzzy sets induced by the metric and obtain some results on Hausdorff G-metric. We also prove common fixed point theorems for a family of fuzzy self-mappings in the space of fuzzy sets on a complete G-metric space. MSC:47H10, 54H25.

On Devaney chaotic induced fuzzy and set-valued dynamical systems

It is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, i.e. a discrete dynamical system on the space of fuzzy sets. In this paper we study relations between dynamical properties of the original and fuzzified dynamical system. Especially, we study conditions used in the definition of Devaney chaotic maps, i.e. periodic density and transitivity. Among other things we show that dynamical behavior of the set-valued and fuzzy extensions of the original system mutually inherits some global characteristics and that the space of fuzzy sets admits a transitive fuzzification. This paper contains the solution of the problem that was partially solved by Román-Flores and Chalco-Cano [Some chaotic properties of Zadeh's extension, Chaos, Solitons and Fractals 35(3) (2008) 452–459].

On fuzzifications of discrete dynamical systems

Let X denote a locally compact metric space and φ : X → X be a continuous map. In the 1970s Zadeh presented an extension principle helping us to fuzzify the dynamical system ( X, φ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets Syst. 61 (1994) 277–283] to generalize Zadeh’s original extension principle. In this paper we study basic properties of so-called g-fuzzifications, such as their continuity properties. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on X induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems. Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, endograph and levelwise topologies.

Topological entropy of fuzzified dynamical systems

A discrete dynamical system is given by a compact metric space X and any continuous self-map defined on X. This discrete dynamical system can be naturally extended to the space of fuzzy sets on X. In this paper we study relations between the sizes of the topological entropies of the original dynamical system and of its fuzzy counterpart. Among other things, we present a constructive proof of the fact that even very weak assumptions on the crisp discrete dynamical system ensure infinite topological entropy of the fuzzy system. However, we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system.