Research Articles

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Heuristic Approach for n-Jobs, 3-Machines Flow Shop Scheduling Problem, Processing Time Associated With Probabilities Involving Transportation Time, Break-Down Interval, Weightage of Jobs and Job Block Criteria
(Mathematical Theory and Modeling Vol.1, No.1, 2011, 2011) Deepak Gupta; Samir Sharma; Seema Sharma
This paper deals with a new simple heuristic algorithm for n jobs, 3 machines flow shop scheduling problem in which processing times are associated with their corresponding probabilities involving transportation time, break down interval and job block criteria. Further jobs are attached with weights to indicate their relative importance. A heuristic approach method to find optimal or near optimal sequence minimizing the total elapsed time whenever mean weighted production flow time is taken into consideration. The proposed method is very easy to understand and also provide an important tool for decision makers. A numerical illustration is also given to clarify the algorithm
On Linkage of Parallel Biserial Servers Linked with a Common Server to a Three Stage Flowshop Scheduling Model
(International Journal of Applied Physics and Mathematics, Vol. 2, No. 3,, 2012) Deepak Gupta; Sameer Sharma; Seema Sharma
The present paper deals with the linkage between a queue network in which a common service server is linked in series with each of two parallel biserial servers and a three stage flowshop scheduling system. The objective of this paper is of two folds, on one hand it finds mean queue length and the total waiting time of jobs and on other hand it minimizes the total elapsed time. The proposed model provides an important tool for manufacturing concern, office management, banking service system, computer networks and in administrative setup etc.
General Family of Third Order Methods for Multiple Roots of Nonlinear Equations and Basin Attractors for Various Methods
(Hindawi Publishing Corporation Advances in Numerical Analysis Volume 2014, Article ID 963878, 8 pages, 2014) Rajni Sharma; Ashu Bahl
A general scheme of third order convergence is described for finding multiple roots of nonlinear equations. The proposed scheme requires one evaluation of 𝑓, 𝑓󸀠 , and 𝑓󸀠󸀠 each per iteration and contains several known one-point third order methods for finding multiple roots, as particular cases. Numerical examples are included to confirm the theoretical results and demonstrate convergence behavior of the proposed methods. In the end, we provide the basins of attraction for some methods to observe their dynamics in the complex plane.
An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics
(Hindawi Publishing Corporation,Journal of Complex Analysis, 2015) Rajni Sharma; Ashu Bahl
We present a new fourth order method for finding simple roots of a nonlinear equation 𝑓(𝑥) = 0. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.
An optimal fourth order weighted-Newton method for computing multiple roots and basin attractors for various methods
(Pelagia Research Library Advances in Applied Science Research, 2016, 7(4):47-54, 2016) Ashu Bahl; Rajni Sharma
In this paper, we present an optimal fourth order method for finding multiple roots of a nonlinear equation f(x)=0. In terms of computational cost, the method uses one evaluation of the function and two evaluations of its first derivative per iteration. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency indices 1.414 of Newton method, 1.442 of Halley’s method and 1.414 of Neta-Johnson method. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The basins of attraction of the proposed method are presented and compared with other existing methods.
Sequence Dependent Flow Shop Scheduling With Job Block Criteria
(Global Journal of Pure and Applied Mathematics. Volume 13, Number 5 (2017), pp. 1401-1414, 2017) Sameer Sharma; Deepak Gupta; Seema; Kewal Krishan Nailwal
The majority of research on scheduling assumes setup times negligible or part of the processing time. In this paper, a bicriteria scheduling with a sequence dependent setup time (SDST) and job block criteria is considered. The objective function of the problem is minimization of the total completion time and the rental cost of machines taken on rent under a specified rental policy. The processing time of attributes on these machines are associated with probabilities. The scheduling problems considering either of these objectives are NP-hard, so exact optimization techniques are impractical. A heuristic algorithm to find optimal or near optimal sequence of jobs processing is The performance of the proposed algorithm is justified by bi objective in-out flow table of jobs.
One parameter fifth order family of iterative methods for solving nonlinear Equations and Basins of Attraction
(NeuroQuantology Volume 20 Issue 11 Page 8973-8986, 2022) Anshu; Rajni Sharma; Ashu Bahl
In this study, we develop one-parameter family of fifth order iterative scheme for finding simple zeros of nonlinear equations. In terms of computational cost, this proposed scheme involves four evaluations of the given function and its first-order derivative per iteration. Various numerical examples are demonstrated to compare the performance of developed scheme with existing schemes of same order. To verify how well our scheme works in practise, we apply them to solve the Van der Waals equation of state. Moreover, the dynamics of the proposed method is demonstrated and compare it with other existing fifth order iterative methods.
Optima leighth-order multiple root finding iterative methods using bi variate weight function
(ElsevierB.V, 2023) Rajni, Sharma; Ashu Bahl; Ranjita Guglani
Inthiscontribution, anovel eighth-order scheme ispresentedfor solvingnonlinearequations withmultipleroots.Theproposedschemecomprisesof threestepswiththemodifiedNewton methodas its first stepfollowedbytwoweightedNewtonsteps involvingoneunivariateand onebivariatefunctionrespectively.Analysisofconvergenceconfirmsthatthepresentedscheme obtains optimal computational order of convergence. The efficiencyof presented scheme is comparednumericallywith recent eighth-ordermethods. Functions like populationgrowth problem,Newton’sbeamproblem, etc.,havebeenconsideredfornumerical experimentation. For the comparative study in the complex plane, we employed the concept of basins of attraction
Optimaleighth-order multiple root finding iterative methods using bivariate weight function
(ElsevierB.V., 2023) Rajni Sharma; Ashu Bahl; Ranjita Guglani
Inthiscontribution, anovel eighth-order scheme ispresentedfor solvingnonlinearequations withmultipleroots.Theproposedschemecomprisesof threestepswiththemodifiedNewton methodas its first stepfollowedbytwoweightedNewtonsteps involvingoneunivariateand onebivariatefunctionrespectively.Analysisofconvergenceconfirmsthatthepresentedscheme obtains optimal computational order of convergence. The efficiencyof presented scheme is comparednumericallywith recent eighth-ordermethods. Functions like populationgrowth problem,Newton’sbeamproblem, etc.,havebeenconsideredfornumerical experimentation. For the comparative study in the complex plane, we employed the concept of basins of attraction
Intuitionistic fuzzy lattice ordered G-modules
(Journal of Fuzzy Extension and Applications, 2024) Poonam Kumar Sharma
The investigation of mathematics underlines accuracy, precision, and flawlessness, yet in numerous genuine circumstances, individuals face equivocalness, ambiguity, imprecision, and so forth. Intuitionistic fuzzy set theory, rough set theory, and soft set theory are three noble techniques in mathematics that are utilised for decision-making in vague and uncertain information systems. Intuitionistic fuzzy algebra-based math plays a huge part in the current era of mathematical research, and it deals with the algebraic concepts and models of intuitionistic fuzzy sets. The investigation of different ordered algebraic structures, like lattice-ordered groups, Riesz spaces, etc., is of great importance in algebra. The theory of lattice-ordered G-modules is very useful in the study of lattice-ordered groups and similar algebraic structures. In this article, the theories of intuitionistic fuzzy sets and lattice-ordered G-modules are synchronised in a reasonable way to develop a novel concept in mathematics, i.e., intuitionistic fuzzy lattice-ordered G-modules, which would pave the way for new researchers in intuitionistic fuzzy mathematics to explore much more in this field.
Decomposition Of Intuitionistic Fuzzy Primary Ideals Of Γ-Rings
(CREAT. MATH. INFORM., 2024) P. K. SHARMA; HEM LATA; NITIN BHARDWAJ
ABSTRACT. In this paper, we establish the intuitionistic fuzzy version of the Lasker-Noether theorem for a commutative Γ-ring. We show that in a commutative Noetherian Γ-ring, every intuitionistic fuzzy ideal A can be decomposed as the intersection of a finite number of intuitionistic fuzzy irreducible ideals (primary ideals). This decomposition is called an intuitionistic fuzzy primary decomposition. Further, we show that in case of a minimal intuitionistic fuzzy primary decomposition of A, the set of all intuitionistic fuzzy associated prime ideals of A is independent of the particular decomposition. We also discuss some other fundamental results pertaining to this concept.
RESOLVING NONLINEAR PHYSICS PROBLEMS WITH AN EFFICIENT SEVENTH ORDER ITERATIVE APPROACH
(South East Asian J. of Mathematics and Mathematical Sciences Vol. 20, No. 2 (2024), pp. 269-282, 2024) Ashu Bahl; Ranjita Guglani; Rajni Sharma
Our research introduces a novel seventh-order iterative method specifically designed to address nonlinear equations having multiple roots. Inspired by the pioneering work of Sharma et al. (2019), our approach represents a significant advancement in computational techniques for solving complex mathematical problems. Through rigorous convergence analysis, we establish that our proposed method achieves seventh-order convergence. To evaluate its efficacy, we conduct extensive numerical experiments utilizing a range of nonlinear equations encountered in applied physics domains, including Planck’s Law, electron trajectory problems, and Newton’s beam designing problem. Our findings reveal that the suggested method consistently outperforms other existing techniques of similar nature available in the literature. Notably, our method demonstrates exceptional convergence behavior even in challenging scenarios involving multiple roots, indicating its suitability for solving complex problems encountered in applied physics and related fields. This superiority is evidenced by its ability to efficiently converge to solutions even in scenarios involving multiple roots. The practical implications of our research extend to various fields reliant on nonlinear equation.
Formulation and convergence analysis of an efficient higher order iterative scheme
(Asia Pacific Academy of Science Pte. Ltd., 2024) Ranjita Guglani; Ashu Bahl; Rajni Sharma
This contribution presents a highly efficient three-step iterative scheme. The proposed scheme is different in itself by achieving seventh-order convergence. The scheme is very useful for equations of nonlinear nature having multiple roots. The Taylor series expansion is employed to rigorously analyze the convergence of the presented scheme. That the scheme is effective and robust can be fit through a variety of examples from different fields. Numerical experimentation demonstrates the scheme’s rapid and reliable convergence to the true root and comparing its performance against existing techniques in the literature. Additionally, basins of attraction are visualized to offer a clear, comparative view of how different methods perform with varying initial guesses. The results show that this new scheme consistently compete well over other methods. This makes it a powerful tool for solving complex equations.
A NEW SEVENTH-ORDER SCHEME FOR FINDING MULTIPLE ROOT OF NONLINEAR EQUATIONS WITH IMPROVED EFFICIENCY
(Bull. Cal. Math. Soc., 116, (6) 813–826, 2024) Ashu Bahl; Ranjita Guglani; Rajni Sharma; Neeru Bala
In this contribution, a new seventh-order iterative scheme is presented for finding multiple roots of nonlinear equations. For numerical experimentation, various nonlinear equations arousing in different scientific fields have been considered. The efficiency of proposed scheme is tested in comparison with some already existing methods. Graphical comparison is also given using basins of attraction.
A Novel Family of Multiple Root Finders with Optimal Eighth-Order Convergence and their Basins of Attraction
(International Conference of Numerical Analysis and Applied Mathematics AIP Conf. Proc., 2024) Rajni Sharma; Ashu Bahl; Ranjita Guglani
In this contribution, a novel scheme of multiple root finders for nonlinear equations with the univariate and bivariate weight functions is proposed. The basic requirement of the presented scheme is the known value of ’m’ (multiplicity of the root). Analysis of convergence is given to show that the proposed family achieves eighth-order convergence. Numerical experimentation is done to demonstrate accuracy and computational efficiency of the scheme. In addition, the obtained results show that the proposed scheme has an edge-over other considered eighth-order methods
A STUDY ON GROUP ACTION ON INTUITIONISTIC FUZZY PRIMARY AND SEMIPRIMARY IDEALS
(TWMS Journal of Applied and Engineering Mathematics, Vol.15, No.3, 2025) P. K. SHARMA
Group actions are a useful technique for examining the symmetry and automorphism characteristics of rings. The concept of intuitionistic fuzzy ideals in rings has been broadened with the inclusion of the concepts of intuitionistic fuzzy primary and semiprimary ideals. The purpose of this article is to extend the group action to the intuitionistic fuzzy ideals of a ring R with group action on it and to derive a relation between the intuitionistic fuzzy G-primary (G-semiprimary) ideals and the intuitionistic fuzzy primary (semiprimary) ideals of R. We established that the largest G-invariant intuitionistic fuzzy ideal contained in an intuitionistic fuzzy primary (semiprimary) ideal is an intuitionistic fuzzy G-primary (G-semiprimary) ideal of R. Conversely, if Q is an intuitionistic fuzzy G-primary (G-semiprimary) ideal of R, then there exists an intuitionistic fuzzy primary (semiprimary) ideal P of R such that P G = Q. We also investigate the relationships between the intuitionistic fuzzy G-primary (G-semiprimary) ideals of R and their level cuts under this group action. Additionally, we establish a suitable characterization of intuitionistic fuzzy G-primary (G-semiprimary) ideals of R in terms of intuitionistic fuzzy points in R under this group action. In addition to these, we also investigate the preservation of the image and pre-image of an intuitionistic fuzzy G-primary (G-semiprimary) ideal of a ring under G-homomorphism.
Intuitionistic fuzzy group algebra
(2025) Poonam Kumar Sharma
This paper introduces the concept of an intuitionistic fuzzy group algebra associated with a finite group G and an intuitionistic fuzzy group A on G. We establish its structural properties, showing that it simultaneously behaves as an intuitionistic fuzzy algebra and an intuitionistic fuzzy G-module. Extending author’s earlier results on the semi-simplicity of intuitionistic fuzzy G-modules, we explore links to complete reducibility and injectivity. Further, we study intersections, (α, β)-cuts, and homomorphic images of such algebras, and define intuitionistic fuzzy group algebra homomorphisms. Finally, we prove that the class of all intuitionistic fuzzy group algebras forms a category.

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