International Journal of Mathematical Modelling and Numerical Optimisation (13 papers in press)
Analysis of a single server batch arrival Non-Markovian Queuing model approach in classification of foot thermograms
by R. Vanalakshmi, S. Maragathasundari, Josephine Sella Jeyannathan
Abstract: In this article, a single server non-Markovian queuing model with Poisson arrivals in stacks of different sizes is discussed. The server provides clients with a general service at a time with a random duration setup time after completing the first level of service. Also, the setup time stage is introduced to carry out the preparatory work required for the stage 2 service and customers may renege during the setup phase. Additionally, the server is subject to random interruption during the third stage of service and it enters into the repair process immediately. The effect of setup time stage, interruption and Reneging in the defined queuing system are well analyzed Using the supplementary variable technique of queuing theory, the duration of the Queue length of different states of the system in terms of probability generating functions and the queue performance measures are determined. The issue of characterized lines is very much clarified by a reasonable application and supported by a mathematical portrayal.
Keywords: non-Markovian queuing model; probability distribution; classification of foot thermo grams; setup time; random breakdowns; delay time; probability generating function; steady state equations.
Support Solving Method for Box-constrained Indefinite Quadratic Minimization Problems
by Amar Andjouh, Mohand Ouamer Bibi
Abstract: This paper provides a new Support Solving Method (SSM) of global optimization for the box-constrained nonconvex quadratic minimization problem, in particular with one negative eigenvalue. We investigate the support of the objective function and exploit properties of the indefinite associated matrix for establishing global optimality criterion (necessary and sufficient conditions). Furthermore, using these conditions and computational techniques of abstract convexity, we suggest an SSM that can effectively solve an indefinite quadratic minimization problem, providing thus a global minimizer. We present numerical examples and generate some test problems with known global minimizer, solving them by the proposed SSM. Finally, we provide the comparative effectiveness of the SSM with Active Set Method (ASM) and Interior Point Method (IPM) implemented under Matlab optimization toolbox.
Keywords: Global optimization; Nonconvex Quadratic Minimization; Sufficient Global Optimality Condition; Support Feasible Pair (SFP); Support Solving Method (SSM).
Finite difference methods with linear interpolation for solving a coupled system of hyperbolic delay differential equations
by Karthick Sampath, Subburayan Veerasamy
Abstract: In this article system of first-order hyperbolic delay differential equations is considered. The maximum principle is proved for the problem considered. Further, the stability of the solution is established as an application of the maximum principle. The propagation of the discontinuity of the solution is also established. A piecewise uniform mesh is designed for solving the problem. On the piecewise uniform mesh, conditionally stable and unconditionally stable finite difference methods with piecewise linear interpolation are suggested to solve the problem. It is proved that the methods are consistent, stable, and convergent. Numerical illustrative examples are given to validate the theoretical results.
Keywords: hyperbolic system; delay differential equations; conditional method; unconditional method; linear interpolation; maximum principle; piecewise linear interpolation; maximum principle.
A mathematical study of dynamical model for Japanese encephalitis-dengue co-infection using JE vaccine
by Ananya Dwivedi, Ram Keval, Vinod Baniya
Abstract: A nonlinear deterministic mathematical model has been described with the co-infection dynamics of Japanese encephalitis (JE) and dengue disease, incorporating the JE vaccine. Basic mathematical results have been discussed along with their stabilities. By using center manifold theory, the model undergoes a backwards bifurcation phenomenon and this has been occurred when the basic reproduction number is smaller than unity. Sensitivity analysis has been performed to determine which parameters significantly affect disease dynamics. The effects of these parameters on the transmission of disease were investigated using a numerical simulation. According to the findings, we obtain that JE-dengue co-infection can be managed with the use of JE vaccine, also minimising JE transmission rate rapidly.
Keywords: JE-dengue co-infection; stability analysis; basic reproduction number; sensitivity analysis; bifurcation analysis.
Optimal Quarantine, Isolation, and Social distancing Strategies for COVID-19 based on a Mathematical Model
by L.W. Somathilake, M. C. S. Fernando
Abstract: Around 221 countries in the world are currently suffering from the COVID-19 pandemic and World Health Organization reported there are 217.7 million confirmed cases with 4.5 million deaths tolls as of 31st August 2021. Until find medicines, it is more appropriate to follow the health guidelines recommended by authorities. Theoretically, forecasting the courses and possible outcomes of such a pandemic is crucial for healthcare sectors to make decisions in advance. This paper aims to find optimal quarantine, isolation, and social distancing strategies for COVID-19 based on the SEIQJR mathematical model with a proper cost analysis. Minimising the cost of the controlling process of diseases is very important for public health policymakers. An optimal control problem is considered with a proposed cost functional which is minimised to yield optimal control strategies. We subsequently insert an inequality state constraint to the problem by considering the possible maximum capacities of hospitals.
Keywords: COVID-19; disease control strategies; disease modelling; optimal control; state constraint.
Enhanced estimation of population mean under two-phase sampling
by Shashi Bhushan, Anoop Kumar
Abstract: This study investigates an enhanced estimation procedure for the estimation of the population mean of the study variable by using the information on an auxiliary attribute under two-phase sampling. The suggested class of estimator incorporates several well-established estimators for suitably chosen values of characterising scalars. The properties such as bias and mean square error of suggested estimator are studied to the first order of approximation. The efficiency conditions are derived by comparing the MSE of the suggested class of estimator with the MSE of the existing estimators. The efficiency conditions are enhanced by an empirical study using some real datasets. The empirical findings are clearly demonstrating the ascendancy of the suggested class of estimator over the all discussed works.
Keywords: auxiliary attribute; bias; mean square error; MSE; efficiency; two-phase sampling.
Mathematical Modeling of Breakdowns with Soft Failures and Explicit Analytical Expressions of an M/M/1 Queue's Transient State Probabilities
by Janani B
Abstract: In the last decade, the increasing usage of digital electronics and software has given rise to a new form of failure occurrence known as soft failure. Soft failure is not the same as catastrophic failure. Catastrophic failure removes all waiting customers, including those receiving service, and the system enters the repair state, whereas soft failure does not require the removal of customers and instead requires the customer to wait for the server reactivation because repair can be accomplished by simply rebooting the system. A new approach for categorising and analysing these events is necessary to characterise and distinguish soft failures from hardware catastrophic failures. As a result, this paper considers a single server queueing model with soft failure. The system is prone to random breakdowns while providing service to the customer. When a system malfunctions, repair work begins right away. During the repair period, new customers are not permitted to join the system. For the first time, the transient state probabilities of an M/M/1 queue with soft failures are computed explicitly. Some performance metrics, such as availability and reliability, are derived. Numerical examples are also provided to demonstrate the effect of parameters.
Keywords: random breakdown; soft failure; hard failure; time dependent probabilities; Laplace transform; generating function.
Analysis and Simulation of SIR Model for Covid-19 Spreading as the Effect of Offline Learning and Vaccination: A Case Study of Indonesia
by Syafruddin Side, Nasiah Badwi, Nurlina Sjahrir, Muhammad Isbar Pratama, Muhammad Rifandi, Suwardi Annas
Abstract: This research discusses analysis and simulation of SIR model for COVID-19 spreading as the effect of offline learning and vaccination. The results of the analysis shows that the COVID-19 case in Indonesia is at a stable stage. The SIR model was chosen because it is one of the basic methods in the epidemiological model like COVID-19. The simulation results shows that if offline learning is not enforced but 99% of the students are vaccinated, the rate of spread of COVID-19 slows down and the rate of recovery has increased in just five months. The results also shown that if 50% offline learning is carried out, the basic reproduction number is R0 = 3.3715, which means that a student infected with COVID-19 can infect 11 other people and if 99% offline learning is carried out, the basic reproduction number is obtained R0 = 6.6757, which means that a student infected with COVID-19 can infect 22 other people in Indonesia.
Keywords: COVID-19 mathematical model; SIR model; offline learning effect; vaccination effect; Indonesia.
Forecasting the coronavirus disease 2019 pandemic in India using machine learning and statistical models
by Sidharth Saxena, Rajashree Shettar
Abstract: COVID-19, which is an infectious disease caused by the severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) has resulted in a massive blow to India with respect to the health of its citizens and economy. The work in this paper focuses on the Prophet model, linear regression model, Holt's model and the ARIMA model for predicting the number of confirmed, recovered cases, deaths and active cases along with growth rate, recovery rate and mortality rate in India for the month of November 2020. The performance of all the above mentioned models has been evaluated using standard metrics namely R2, adjusted R2, root-mean-square error and mean absolute error.
Keywords: linear regression model; Prophet model; Holt's model; ARIMA model; COVID-19; forecasting; India.
A second order weighted monotone numerical scheme for time-delayed parabolic initial-boundary-value problem involving a small parameter
by Abhishek Das, Lolugu Govindarao, Jugal Mohapatra
Abstract: In this note, a weighted monotone numerical scheme is proposed to address the time delay (large) parabolic problem which is singularly perturbed in nature. The solution of the problem possesses boundary layer towards the left side of the domain. The initial conditions are given at the present time and also at a past time. The proposed method is based on the Crank-Nicolson scheme in the time scale and a weighted monotone hybrid method for the space derivatives in Shishkin mesh. A rigorous convergence analysis is investigated. The main contribution of this work is to give almost second-order parameter-uniform convergent method. The validity and applicability of the method are preformed by numerical examples, which verify our theoretical claims.
Keywords: time delay parabolic problem; singular perturbation; boundary layer; weighted hybrid scheme; uniform convergence.
An integrated inventory model for imperfect production process having preservation facilities under fuzzy and inflationary environment
by Surendra Vikram Singh Padiyar, Vipin Chandra Kuraie, Naveen Bhagat, S.R. Singh, Rinki Chaudhary
Abstract: In this paper, we develop an integrated inventory model for a producer and buyer, and consider an imperfect production process having two different demands in which exponential demand for the producer and triangular demand for the buyer. The model takes into account that a partially backlogged shortage is allowed on the buyer's part. Preservation facilities are also considered in this model for deteriorating items to reduce the deterioration rate. The production rate is assumed to be demand-dependent. This model provides a theory to reduce the deterioration rate under the effect of inflation. The total cost of the model is minimised for both crisp and fuzzy environments by using triangular fuzzy numbers and defuzzify the total cost by the centroid method. Numerical example and sensitivity analysis are also given to illustrate this proposed model.
Keywords: imperfect production; supply chain; preserve technology; inflation; fuzzy environment.
Generalised classes of estimators for population mean of sensitive variable using non-sensitive auxiliary parameters
by S.K. Yadav, Amit Kumar Misra, Tarushree Bari
Abstract: Sousa et al. (2010) suggested transformed ratio type estimators for estimating the population mean of a sensitive variable in presence of some known population coefficients of a non-sensitive supplementary variable. In this article, we generalise the Sousa et al. (2010) family of estimators using some new population parameters of auxiliary information based on a randomised response technique (RRT). Further, we introduce a new efficient family of estimators for estimating the population mean of sensitivity variable using the approach given in Searls (1964) in the presence of the auxiliary information. The optimal value of Searl's constant is obtained using Lagrange's method of maxima-minima. Theoretical results are supported with a numerical illustration based on real datasets. In addition, a simulation study is carried out to compare the performances of the suggested and competing families of estimators. The estimator with good sampling properties and a lower mean square error (MSE) is recommended for various fields of applications of sensitive research.
Keywords: sensitive variable; scrambled response; randomised response technique; ratio type estimator; simple random sampling; simulation.
A software tool for numerical option pricing in MATLAB®
by Jeetendre Narsoo, Mohammad Sameer Sunhaloo
Abstract: A graphical user interface (GUI) has been designed to encapsulate classical numerical algorithms to price European and American options in the finite difference setting under a common and flexible environment. All the algorithms have been implemented as classes using an object-oriented framework and tested using MATLAB® that supports an object-oriented environment. In this paper, our focus has been on the program design aspects of the algorithms to make them more flexible, easy-to-use, maintainable and robust. The interface consists of three separate windows, one for inputting the parameters of the option pricing models, the second window for displaying the results in tabular form and the third one to display a graphical representation of the results.
Keywords: graphical user interface; GUI; option pricing; object-oriented framework.