Adeel Farooq

Nowadays, several fuzzy extensions of ELECTRE (Elimination and Selection Transforming Reality) and TOPSIS (The Order Preference by Similarity to Positive Ideal Solution) methods are emerging as powerful mathematical tools for dealing with uncertainties in different multi-criteria decision-making (MCDM) situations of medical and engineering domains. Since cubic bipolar fuzzy (CBF) set theory is also becoming an effective mathematical tool for such kinds of problems. In this study, we present an improved CBF-TOPSIS method using the concepts of normalized Euclidean distance and revised closeness index because in the existing CBF-TOPSIS method the notions of Euclidean distance and relative closeness index were used to reach the final decision. Then, we rectify the shortcomings of the existing CBF-ELECTRE-I method, that is, there was a problem in the discordance formula. To modify the existing algorithm based on the CBF-ELECTRE-I model, we add a new step to find the exact linear ranking order of alternatives, which is purely based on concordance and discordance outranking coefficients. Second, we explore two new applications to validate the applicability and feasibility of the improved CBF-ELECTRE-I and CBF-TOPSIS methods in medical and engineering. Finally, to demonstrate the veracity of the modified CBF-ELECTRE-I and CBF-TOPSIS methods, we perform a brief sensitivity analysis between them and other existing models.

The primary aim of this article is to extend the bipolar fuzzy N -soft sets with the concern of pursuing the periodicity involved real-world problems and introduce a new multiskilled hybrid model, namely, complex bipolar fuzzy N -soft sets. The novel model possesses the parametric characteristics of the versatile N -soft set and enjoys the distinguished attributes of a complex bipolar fuzzy set to handle the double-sided periodic vague data. We illustrate that the innovative model assists as a proficient mechanism for grading-based parameterized two-dimensional bipolar fuzzy information. We present some elementary operations and results for a complex bipolar fuzzy N -soft environment. Further, we establish the three dexterous algorithms to find the optimal solution to multiattribute decision-making problems. Moreover, the algorithms are supported with the robust assessment of a real-world application. Lastly, a comparison with existent decision-making techniques, such as choice values, weighted choice values, and D -choice of values of bipolar fuzzy N -soft sets, is also conducted to manifest the phenomenal accountability and authenticity of the presented decision-making approaches.

Group theory, the ultimate theory for symmetry, is a powerful tool that has a direct impact on research in robotics, computer vision, computer graphics and medical image analysis. Symmetry is very important in chemistry research and group theory is the tool that is used to determine symmetry. Usually, it is not only the symmetry of molecule but also the symmetries of some local atoms, molecular orbitals, rotations and vibrations of bonds, etc. that are important. Harada-Norton group is an example of a sporadic simple group. There are 14 maximal subgroups of Harada-Norton group. Generators (also known as words) of 11 maximal subgroups are already known. The aim of this note is to give generators of the remaining 3 maximal subgroups, which is an open problem mentioned on A World-wide-web Atlas of Group Representations (http://brauer.maths.qmul.ac.uk/Atlas) [1]. In this report we compute the generators of A6 × A6.D8, 23+2+6.(3 × L3(2)) and 34 : 2.(A4 × A4).4. Moreover we also compute the generators for the Maximal subgroups of some linear groups.

The Conway groups are the three sporadic simple groups Co₁, Co₂ and Co₃. There are total of 22 maximal subgroups of Co₁ and generators of 6 maximal subgroups are provided in web Atlas of finite simple groups. The aim of this paper is to give generators of remaining 16 maximal subgroups.

In this paper, a consensus-based method for multi-person decision making (MPDM) using product transitivity with incomplete fuzzy preference relations (IFPRs) is proposed. Additionally, an average aggregation operator has been used at the first level to estimate the missing preference values and construct the complete fuzzy preference relation (FPR). Then it is confirmed to be product consistent by using the transitive closure formula. Following this, weights of decision makers (DMs) are evaluated by merging consistency weights and predefined priority weights (if any). The consistency weights for the DMs are estimated through product consistency investigation of the information provided by each DM. The consensus process determines whether the selection procedure should be initiated or not. The hybrid comprises of a quitting process and feedback mechanism, and is used to enhance the consensus level amongst DMs in case of an inadequate state. The quitting process arises when some DMs decided to leave the course, and is common in MPDM while dealing with a large number of alternatives. The feedback mechanism is the main novelty of the proposed technique which helps the DMs to improve their given preferences based on this consistency. At the end, a numerical example is deliberated to measure the efficiency and applicability of the proposed method after the comparison with some existing models under the same assumptions. The results show that proposed method can offer useful comprehension into the MPDM process.

The q-rung orthopair fuzzy set is a powerful tool for depicting fuzziness and uncertainty, as compared to the Pythagorean fuzzy model. The aim of this paper is to present q-rung orthopair fuzzy competition graphs (q-ROFCGs) and their generalizations, including q-rung orthopair fuzzy k-competition graphs, p-competition q-rung orthopair fuzzy graphs and m-step q-rung orthopair fuzzy competition graphs with several important properties. The study proposes the novel concepts of q-rung orthopair fuzzy cliques and triangulated q-rung orthopair fuzzy graphs with real-life characterizations. In particular, the present work evolves the notion of competition number and m-step competition number of q-rung picture fuzzy graphs with algorithms and explores their bounds in connection with the size of the smallest q-rung orthopair fuzzy edge clique cover. In addition, an application is illustrated in the soil ecosystem with an algorithm to highlight the contributions of this research article in practical applications.