In recent years, a mathematical approach of blending different aspects is on the way, which as a result gives a more generalized approach. Following the above mathematical approach, we combine two very powerful techniques, namely complex intuitionistic fuzzy sets and graph theory, and introduce the notion of complex intuitionistic fuzzy graphs. Then, we introduce certain notions including union, join and composition of complex intuitionistic fuzzy graphs, through which one can easily manipulate the complex intuitionistic fuzzy graphs in decision making problems. We elucidate these operations with some examples. We also describe the homomorphisms of complex intuitionistic fuzzy graphs. Finally, we provide an application in cellular network provider companies for the testing of our approach.

The exponential distribution has always been prominent in various disciplines because of its wide range of applications. In this work, a generalization of the classical exponential distribution under a neutrosophic environment is scarcely presented. The mathematical properties of the neutrosophic exponential model are described in detail. The estimation of a neutrosophic parameter by the method of maximum likelihood is discussed and illustrated with examples. The suggested neutrosophic exponential distribution (NED) model involves the interval time it takes for certain particular events to occur. Thus, the proposed model may be the most widely used statistical distribution for the reliability problems. For conceptual understanding, a wide range of applications of the NED in reliability engineering is given, which indicates the circumstances under which the distribution is suitable. Furthermore, a simulation study has been conducted to assess the performance of the estimated neutrosophic parameter. Simulated results show that imprecise data with a larger sample size efficiently estimate the unknown neutrosophic parameter. Finally, a complex dataset on remission periods of cancer patients has been analyzed to identify the importance of the proposed model for real‐world case studies.

This article introduces the concept of Heronian mean operators, geometric Heronian mean operators, neutrosophic cubic number–improved generalized weighted Heronian mean operators, neutrosophic cubic number–improved generalized weighted geometric Heronian mean operators. These operators actually generalize the operators of fuzzy sets, cubic sets, and neutrosophic sets. We investigate the average weighted operator on neutrosophic cubic sets and weighted geometric operator on neutrosophic cubic sets to aggregate the neutrosophic cubic information. After this, based on average weighted and geometric weighted and cosine similarity function in neutrosophic cubic sets, we developed a multiple attribute group decision-making method. Finally, we give a mathematical example to illustrate the usefulness and application of the proposed method.

Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of $[0,1]$ and a number from $[0,1]$ . Generalized cubic sets generalized the cubic sets with the help of cubic point. On the other hand Soft sets were proved to be very effective tool for handling imprecision. Semigroups are the associative structures have many applications in the theory of Automata. In this paper we blend the idea of cubic sets, generalized cubic sets and semigroups with the soft sets in order to develop a generalized approach namely generalized cubic soft sets in semigroups. As the ideal theory play a fundamental role in algebraic structures through this we can make a quotient structures. So we apply the idea of neutrosophic cubic soft sets in a very particular class of semigroups namely weakly regular semigroups and characterize it through different types of ideals. By using generalized cubic soft sets we define different types of generalized cubic soft ideals in semigroups through three different ways. We discuss a relationship between the generalized cubic soft ideals and characteristic functions and cubic level sets after providing some basic operations. We discuss two different lattice structures in semigroups and show that in the case when a semigroup is regular both structures coincides with each other. We characterize right weakly regular semigroups using different types of generalized cubic soft ideals. In this characterization we use some classical results as without them we cannot prove the inter relationship between a weakly regular semigroups and generalized cubic soft ideals. This generalization leads us to a new research direction in algebraic structures and in decision making theory.

Neutrosophic cubic sets are the more generalized tool by which one can handle imprecise information in a more effective way as compared to fuzzy sets and all other versions of fuzzy sets. Neutrosophic cubic sets have the more flexibility, precision and compatibility to the system as compared to previous existing fuzzy models. On the other hand the graphs represent a problem physically in the form of diagrams, matrices etc. which is very easy to understand and handle. So the authors applied the Neutrosophic cubic sets to graph theory in order to develop a more general approach where they can model imprecise information through graphs. We develop this model by introducing the idea of neutrosophic cubic graphs and introduce many fundamental binary operations like cartesian product, composition, union, join of neutrosophic cubic graphs, degree and order of neutrosophic cubic graphs and some results related with neutrosophic cubic graphs. One of very important futures of two neutrosophic cubic sets is the R-union that R-union of two neutrosophic cubic sets is again a neutrosophic cubic set, but here in our case we observe that R-union of two neutrosophic cubic graphs need not be a neutrosophic cubic graph. Since the purpose of this new model is to capture the uncertainty, so we provide applications in industries to test the applicability of our defined model based on present time and future prediction which is the main advantage of neutrosophic cubic sets.