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Abstract This article presents a novel concept of complex cubic picture fuzzy sets (CCPFS) and introduces one more new idea of complex interval-valued picture fuzzy sets (CIVPFS) as foundational framework of CCPFS. The proposed CCPFS combines CIVPFS and complex picture fuzzy sets (CPFS), where CPFS extends the complex intuitionistic fuzzy set by incorporating a neutral membership degree. This unique model offers an expanded range of values using degrees of membership, neutral membership, and non-membership, within the unit disk of a complex plane. Additionally, we introduce two more new ideas of internal complex cubic picture fuzzy sets (ICCPFS) and external complex cubic picture fuzzy sets (ECCPFS) to further enhance the versatility of the approach. To facilitate practical applications, complement, score, and accuracy functions are developed and defined for CCPFS. Moreover, three types of averaging aggregation operators based on complex cubic picture fuzzy sets are introduced, including complex cubic picture fuzzy weighted geometric (CCPFWG) operators, complex cubic picture fuzzy ordered weighted geometric (CCPFOWG) operator, and complex cubic picture fuzzy hybrid weighted geometric (CCPFHWG) operator. The CCPFHWG operator generalizes both CCPFWG and CCPFOWG operators, providing a comprehensive framework for aggregating complex cubic picture fuzzy data. To demonstrate the practicality of the proposed approach, a multi-criteria decision-making (MCDM) problem is presented, showcasing its effectiveness in addressing today's complex decision structures. The utilization of complex cubic picture fuzzy sets and the corresponding aggregation operators in MCDM highlights their applicability and relevance in tackling real-world complexities.

Abstract The population growth and urbanization has caused an exponential increase in waste material. The proper disposal of waste is a challenging problem nowadays. The proper disposal site selection with typical sets and operators may not yield fruitful results. To handle such problems, the exponential aggregation operators based on neutrosophic cubic hesitant fuzzy sets are proposed. For appropriate decisions in a decision-making problem, it is important to have a handy environment and aggregation operators. Many multi attribute decision making methods often ignore the uncertainty and hence yields the results which are not reliable. The neutrosophic cubic hesitant fuzzy set can efficiently handle the complex information in a decision-making problem, as it combines the advantages of neutrosophic cubic set and hesitant fuzzy set. In this paper first we establish exponential operational laws in neutrosophic cubic hesitant fuzzy sets, in which the exponents are neutrosophic cubic hesitant fuzzy numbers and bases are positive real numbers. In order to use neutrosophic cubic hesitant fuzzy sets in decision making, we are developing exponential aggregation operators and investigate their properties in the current study. In many multi expert decision-making methods there are different decision matrices but same weighting vector for attributes. The results of a multi expert decision-making problem becomes more reliable if every decision expert has its own decision matrix along with his own weighting vector for attributes. In this study, we are developing multi expert decision-making method that uses different weights for an attribute corresponding to different experts. At the end we present two applications of exponential aggregation operators in environmental protection multi attribute decision making problems.

The aim of this article is to implement the Generalized Modified Adomian Decomposition Method to compute the semi-numerical solution of the linear system of intuitionistic fuzzy initial value problems. Here, we consider the initial values as generalized trapezoidal intuitionistic fuzzy numbers. The technique is applied to brine tanks problem and coupled mass spring systems.Theoretically, different approaches to solving a system of generalized trapezoidal intuitionistic fuzzy differential equations are discussed in this study under the presumption that the coefficients of the system of the differential equations are associated to generalized trapezoidal intuitionistic fuzzy numbers. The approximate results are compared with exact solutions which shows good efficiency. The corresponding graphs at different levels of uncertainty show the example’s numerical outcomes. The graphical representations further demonstrate the effectiveness and accuracy of the proposed method in comparison to existing semi-numerical methods in the literature.

We introduce a topology on the set of vertices of a directed graph and we call the topological space as pathless directed topological space. We study relation between the relative topologies and pathless directed topological spaces of E-generated subdirected graphs. Then, we study connectedness, isomorphic and homeomorphic properties in digraphs and pathless directed topological spaces. Moreover, we apply our results to blood circulation process in human heart and disprove Shokry and Aly [M. Shokry and R. E. Aly, Topological properties on graph vs medical application in human heart, Int. J. Appl. Math., 15 (2013), 1103-1109], Nada et al. [S. Nada, A. E. F. El Atik and M. Atef, New types of topological structures via graphs, Math. Method. Appl. Sci., 41 (2018), 5801-5810] and Nawar et al. [A. S. Nawar and A. E. F. A. El-Atik, A model of a human heart via graph nano topological spaces, Int. J. Biomath., 12 (2019), p.1950006]. We show that pathless directed topology is accurately describing the circulation of blood in the heart of human body.

The current study proposes the idea of the N-cubic Pythagorean fuzzy set with their basic arithmetic operations to aggregate these sets. We define the score and accuracy functions for the comparison purpose. Finally, we discuss Chang's extent analysis of AHP under the environment of the N-cubic Pythagorean fuzzy set using the idea of triangular N-cubic Pythagorean fuzzy set. As an application, we discuss the reason for the downfall of international airlines using the developed approach.

This research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The νth-fractional sum of a function f has two forms; closed form and summation form. Most authors in the previous literature focused on the summation form rather than developing the closed-form solutions, which is to say that they were more concerned with the summation form. However, finding a solution in a closed form requires less time than in a summation form. This inspires us to develop a new approach, which helps us to find the closed form related to nth-sum for a class of delta integrable functions, that is, functions with both discrete integration and nth-sum. By equating these two forms of delta integrable functions, we arrive at several identities (known as discrete fundamental theorems). Also, by introducing ∞-order delta integrable functions, the discrete integration related to the νth-fractional sum of f can be obtained by applying Newton’s formula. In addition, this concept is extended to h-delta integration and its sum. Our findings are validated via numerical examples. This method will be used to accelerate computer-processing speeds in comparison to summation forms. Finally, our findings are analyzed with outcomes provided of diagrams for geometric, polynomial and falling factorial functions.

Unpredictability and fuzziness coexist in decision-making analysis due to the complexity of the decision-making environment. “Pythagorean fuzzy numbers” (PFNs) outperform “intuitionistic fuzzy numbers” (IFNs) when dealing with unclear data. The “Pythagorean fuzzy set” (PFS) is a useful tool because it removes the restriction that the sum of membership degrees be less than or equal to one by substituting the square sum for the sum of membership degrees. This study proposes two aggregating operators (AOs). The recommended operators outperform the already specified PFN operators. The proposed operator is utilised in the multicriteria decision-making process to identify the best candidate for instruction (MCDM).

The neutrosophic cubic hesitant fuzzy set can efficiently handle the complex information in a decision-making problem because it combines the advantages of the neutrosophic cubic set and the hesitant fuzzy set. The algebraic operations based on Dombi norms and co-norms are more flexible than the usual algebraic operations as they involve an operational parameter. First, this paper establishes Dombi algebraic operational laws, score functions, and similarity measures in neutrosophic cubic hesitant fuzzy sets. Then, we proposed Dombi exponential operational laws in which the exponents are neutrosophic cubic hesitant fuzzy values and bases are positive real numbers. To use neutrosophic cubic hesitant fuzzy sets in decision-making, we are developing Dombi exponential aggregation operators in the current study. In the end, we present applications of exponential aggregation operators in multiattribute decision-making problems.

This study aims to define a conjecture that can handle complex frames of work more efficiently that occurs in daily life problems. In decision-making theory inter-relation of criteria, weights and choice decision-making method subject to the given circumstances which are an important component for appropriate decisions. For this, we define neutrosophic cubic Shapley–Choquet integral (NCSCI) measure; combinative distance-based assessment selection (CODAS) is accomplished over NCSCI and is implemented over a numerical example of a company foreign investment model as an application in decision-making (DM) theory. The neutrosophic cubic set (NCS) is a hybrid of the neutrosophic set (NS) and interval neutrosophic set (INS), which provides a better plate form to handle inconsistent and vague data more conveniently. The novel CODAS method is based on Shapley–Choquet integral and Minkowski distance which contain more information measures than usual criteria weights and distances. The weights of criteria are measured by Shapley–Choquet integral and distance is evaluated by Minkowski distance. The Choquet integral considers the interaction among the criteria, and Shapley considers the overall weight criteria. Motivated by these characteristics NCSCI, we defined two aggregation operators’ induced-generalized neutrosophic cubic Shapley–Choquet integral arithmetic (IGNCSCIA) and operators’ induced-generalized neutrosophic cubic Shapley–Choquet integral geometric (IGNCSCIG) operators. To find the distance between two NC values, Minkowski distance is defined to evaluate neutrosophic cubic combinative distance-based assessment selection (NCCODAS). To examine the feasibility of the proposed method, an example of company investment in a foreign country is considered. To check, the validity of the method, the comparative analysis of the proposed method with other methods is conducted.

In this study, the neutrosophic cubic graphs are further developed. We discussed and explored the open and the closed neighborhood for any vertex in neutrosophic cubic graphs, regular and totally regular neutrosophic cubic graphs, complete neutrosophic cubic graphs, balanced and strictly balanced neutrosophic cubic graphs, irregular and totally irregular neutrosophic cubic graphs, complement of a neutrosophic cubic graph, neighborly irregular and neighborly totally irregular neutrosophic cubic graphs, and highly irregular neutrosophic cubic graphs. It has been demonstrated that the proposed neutrosophic cubic graphs are associated with specific conditions. The comparison study of the proposed graphs with the existing cubic graphs has been carried out. Eventually, decision-making approaches for handling daily life problems such as effects of different factors on the neighboring countries of Pakistan and selection of a house based on the notions of proposed graphs are presented.