Fuzzy-interval inequalities for generalized preinvex fuzzy interval valued functions

<abstract> <p>In this paper, firstly we define the concept of <italic>h</italic>-preinvex fuzzy-interval-valued functions (<italic>h</italic>-preinvex FIVF). Secondly, some new Hermite-Hadamard type inequalities (<italic>H</italic>-<italic>H</italic> type inequalities) for <italic>h</italic>-preinvex FIVFs via fuzzy integrals are established by means of fuzzy order relation. Finally, we obtain Hermite-Hadamard Fejér type inequalities (<italic>H</italic>-<italic>H</italic> Fejér type inequalities) for <italic>h</italic>-preinvex FIVFs by using above relationship. To strengthen our result, we provide some examples to illustrate the validation of our results, and several new and previously known results are obtained.</p> </abstract>

Integral Inequalities for Generalized Harmonically Convex Functions in Fuzzy-Interval-Valued Settings

It is a well-known fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity also plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from both the concepts, owing to the significant correlation that has developed between both in recent years. In this paper, we introduce a new class of harmonically convex fuzzy-interval-valued functions which is known as harmonically h-convex fuzzy-interval-valued functions (abbreviated as harmonically h-convex F-I-V-Fs) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on interval space. Some properties of this class are investigated. BY using fuzzy order relation and h-convex F-I-V-Fs, Hermite–Hadamard type inequalities for harmonically are developed via fuzzy Riemann integral. We have also obtained some new inequalities for the product of harmonically h-convex F-I-V-Fs. Moreover, we establish Hermite–Hadamard–Fej’er inequality for harmonically h-convex F-I-V-Fs via fuzzy Riemann integral. These outcomes are a generalization of a number of previously known results, as well as many new outcomes can be deduced as a result of appropriate parameter “θ” and real valued function “∇” selections. For the validation of the main results, we have added some nontrivial examples. We hope that the concepts and techniques of this study may open new directions for research.

Some properties of integration of real-valued function over a fuzzy interval

One of the most fundamental concepts in fuzzy set theory is the extension principle. It gives a generic way of dealing with fuzzy quantities by extending non-fuzzy mathematical concepts. There are a few examples, including the concept of fuzzy distance between fuzzy sets. The extension approach is then methodically applied to real algebra, with considerable development of fuzzy number operations. These operations are computationally appealing and generalized interval analysis. Although the set of real fuzzy numbers with extended addition or multiplication is no longer a group, it retains many structural qualities. The extension concept is demonstrated to be particularly beneficial for defining set-theoretic operations for higher fuzzy sets. We need some definitions related to our properties before we can create the properties of integration of a crisp real-valued function over a fuzzy interval. It is our goal in this article to develop and demonstrate certain characteristics of a real-valued function over a fuzzy interval in order to broaden the scope of the notion of integration of a real-valued function over a fuzzy interval. Some of these characteristics are linked to the operations of extended addition and extended subtraction, while others are not.

Picture fuzzy tolerance graphs with application

In this study, the notions of picture fuzzy tolerance graphs, picture fuzzy interval containment graphs and picture fuzzy $$\phi $$ ϕ -tolerance graphs are established. Three special types of picture fuzzy tolerance graphs having bounded representations are introduced and studied corresponding properties of them taking $$\phi $$ ϕ as max, min and sum functions. Also, picture fuzzy proper and unit tolerance graphs are established and some related results are investigated. The class of picture fuzzy $$\phi $$ ϕ -tolerance chaingraphs which is the picture fuzzy $$\phi $$ ϕ -tolerance graphs of a nested family of picture fuzzy intervals are presented. A real-life application in sports competition is modeled using picture fuzzy min-tolerance graph. Also a comparison is given between picture fuzzy tolerance graphs and intuitionistic fuzzy tolerance graphs.

Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals

In this study, we introduce the new concept of $$h$$ h -convex fuzzy-interval-valued functions. Under the new concept, we present new versions of Hermite–Hadamard inequalities (H–H inequalities) are called fuzzy-interval Hermite–Hadamard type inequalities for $$h$$ h -convex fuzzy-interval-valued functions ($$h$$ h -convex FIVF) by means of fuzzy order relation. This fuzzy order relation is defined level wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Fuzzy order relation and inclusion relation are two different concepts. With the help of fuzzy order relation, we also present some H–H type inequalities for the product of $$h$$ h -convex FIVFs. Moreover, we have also established strong relationship between Hermite–Hadamard–Fej´er (H–H–Fej´er) type inequality and $$h$$ h -convex FIVF. There are also some special cases presented that can be considered applications. There are useful examples provided to demonstrate the applicability of the concepts proposed in this study. This paper's thoughts and methodologies could serve as a springboard for more research in this field.