Multigranulation rough set (MGRS) based on soft relations is a very useful technique to describe the objectives of problem solving. This MGRS over two universes provides the combination of multiple granulation knowledge in a multigranulation space. This paper extends the concept of fuzzy set Shabir and Jamal in terms of an intuitionistic fuzzy set (IFS) based on multi-soft binary relations. This paper presents the multigranulation roughness of an IFS based on two soft relations over two universes with respect to the aftersets and foresets. As a result, two sets of IF soft sets with respect to the aftersets and foresets are obtained. These resulting sets are called lower approximations and upper approximations with respect to the aftersets and with respect to the foresets. Some properties of this model are studied. In a similar way, we approximate an IFS based on multi-soft relations and discuss their some algebraic properties. Finally, a decision-making algorithm has been presented with a suitable example.

It is the purpose of this work to obtain a sharp threshold of global existence vs blow-up dichotomy for mass-super-critical and energy subcritical solutions to an inhomogeneous Choquard equation.

A Pythagorean fuzzy set is the superset of fuzzy and intuitionistic fuzzy sets, respectively. Yager proposed the concept of Pythagorean fuzzy sets in which he relaxed the condition that sum of square of both membership degree and nonmembership degree of an element of a set must not be greater than 1. This paper introduces two new techniques to solve L R -type fully Pythagorean fuzzy linear programming problems with mixed constraints having unrestricted L R -type Pythagorean fuzzy numbers as variables and parameters by introducing unknown variables and using a ranking function. Furthermore, we show the equivalence of both the proposed methods and compare the solutions obtained by the two techniques. Besides this, we solve an already existing practical model using proposed techniques and compare the result.

<abstract><p>In this research study, we first define the strong degree of a vertex in an $ m $-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete $ m $-polar fuzzy graphs. Further, we introduce the concept of $ m $-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete $ m $-polar fuzzy graphs. We discuss connectivity parameters in $ m $-polar fuzzy graphs with precise examples, and we investigate the $ m $-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an $ m $-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as $ \epsilon_A $-reachable vertices in $ m $-polar fuzzy graphs, $ \epsilon_A $-connected $ m $-polar fuzzy graphs, or $ \epsilon_A $-connected $ m $-polar fuzzy subgraphs (in case the $ m $-polar fuzzy graph itself is not $ \epsilon_A $-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.</p></abstract>