International Journal of Mathematical Modelling and Numerical Optimisation (9 papers in press)

Regular Issues

• An Improved Conjugate Gradient Method for Solving Unconstrained Optimization and Image Restoration Problems  
  by Hisham Mohammed, Ahmed Amer 
  Abstract: In this paper, we propose an algorithm of hybrid conjugate gradient algorithms that depend on convex combinations using the three-term hybrid conjugated gradient algorithm to solve unconstrained optimisation problems with its application in restoration noisy images. The new parameter is a hybrid Fletcher and Reeves (FR) with the original FR. Finally, a convex combination of the last search direction and a prior iteration’s gradient may be used to choose where to search next time. It is clear that the direction of the search was created by an new algorithm that always has the property of descent regardless of the type of line search used and that it is always globally convergent some assumptions and using the search line for Wolfe. The numerical results of medium and large dimensions in the smooth non-linear optimisation functions and the restoration of noisy images help us to show the efficiency and accuracy of the proposed hybrid conjugate gradient algorithm. Even when compared to current approaches, the numerical findings suggest that the proposed methods are more efficient and effective than those now in use.
  Keywords: image restoration; Fletcher and Reeves; optimisation; algorithm; smooth.
  DOI: 10.1504/IJMMNO.2023.10053017
   
  
• Analysis of MAP/PH/1 Queueing System with Catastrophic Delay Action, Standby Server, Balking of Customers, Working Vacation and Vacation Interruption under N-Policy  
  by AYYAPPAN Govindan, Thilagavathy Karthikeyan 
  Abstract: We consider customers arrival based on a Markovian arrival process (MAP), and their service times are based on a phase-type (PH) distribution, which incorporates the concepts of catastrophic delay action. When a catastrophic event occurs during a busy period, two clocks are immediately activated, one of which is the protection clock and the other is the delay time clock. Unless the protection clock expires first, the system will resume to the state in which the catastrophic event occurred; however, unless the delay time clock expires first, all customers in the system will be flushed out of the system and the server will become idle. The matrix analytic method has been utilised to investigate the expected number of customers in the system under steady-state probability vector. We examined the cost analysis of the system and some characteristic of performance measures of the system. We examine some of the numerical as well as pictorial representation for the clear view of our model.
  Keywords: PH-distribution; catastrophic delay action; standby server; Markovian arrival process; MAP; impatient behaviour; matrix analytic method.
  DOI: 10.1504/IJMMNO.2023.10053320
   
  
• On Shrunken Estimators for the Ratio of Scale Parameters in the Exponential Two-Sample Problem  
  by Zuhair Al-Hemyari, Ashkan Khalifeh 
  Abstract: In this paper, several 'shrunken estimators' for the 'ratio of scale parameters of the exponential two-sample problem' using two types of 'shrinkage functions' are proposed. The first 'shrinkage function' is based on the 'maximum likelihood estimator', and the second is based on the 'uniformly minimum variance unbiased estimator'. The 'shrunken estimators' expressions are obtained, computed and compared. Moreover, the performances of the 'shrunken estimators' are evaluated. The proposed estimators dominate the 'classical estimators' in the surprisingly large neighbourhood of the prior estimate of the 'ratio of scale parameters'. Ultimately, using the data of a real experiment, the usefulness of these estimators are also illustrated.
  Keywords: maximum likelihood estimator; MLE; negative exponential distribution; shrinkage estimator; stress-strength models; uniformly minimum variance unbiased estimator; UMVUE.
  DOI: 10.1504/IJMMNO.2023.10053542
   
  
• Harvesting of a fractional-order prey-predator model with Holling type II functional response and reserved area for prey in the presence of toxicity.  
  by Chaimaa Assila, Hafida Benazza, Mohamed Reda Lemnaouar, Younes Louartassi 
  Abstract: This paper examines the dynamics of a fishery within an aquatic environment characterised by two zones: an open access fishery zone and a reserved area for prey where fishing is strictly prohibited using a fractional order prey predator model with harvesting under toxic conditions. Holling type II is considered a functional response of predators. The proposed model is discussed in terms of its boundedness, equilibria, local and global stability. Magtinon’s theorem is used to derive local stability conditions, whereas Lyapunov’s function is used to prove global stability. To verify our theoretical findings, numerical simulations are presented.
  Keywords: prey-predator model; toxicity; equilibria; stability; fractional-order prey-predator model; Lyapunov function.
  DOI: 10.1504/IJMMNO.2023.10054931
   
  
• Convergence analysis and an efficient numerical technique for the solution of Benjamin Bona Mahony partial differential equation  
  by Ashish Kumar Rawat, Gagan Deep, Neeraj Dhiman, Anand Chauhan 
  Abstract: This study proposed a collocation based modified scheme of trigonometric cubic b-spline approach (MCTB-spline) to approximate the solution of the Benjamin Bona Mahony (BBM) partial differential equation. The MCTB-spline collocation method is used to discretise the space derivatives terms of the partial differential equation, while the finite difference method (FDM) is used to make discretised the time-variant derivatives of the BBM partial differential equation. This study also describes the convergency of the present method, which helps to illustrate the accuracy of the present scheme for the partial differential equation. A Rubin Graves linearisation process linearises BBM partially differential equations into nonlinear terms. An example has been selected for demonstration of numerical approximations and provided a comparative study with the previous scheme of different types of error norms. To validate the stability, Von-Neumann stability condition was applied on the proposed scheme. Results suggested that the present scheme achieved better results and outperformed the previous techniques used. We will apply the generalisation of the present scheme to higher order and coupled partial differential equations in the future.
  Keywords: Benjamin Bona Mahony equation; BBM; MCTB-spline functions; collocation technique; stability analysis; convergence analysis.
  DOI: 10.1504/IJMMNO.2023.10052672
   
  
• Tribological behaviour of Al6061/Gr/WC hybrid MMCs using multi-response optimisation  
  by Gangadhara Rao Ponugoti, Ravi Kumar Mandava, Pandu Ranga Vundavilli 
  Abstract: The current investigation is motivated to examine the improvement in mechanical properties and tribological behaviour of Al6061/Gr/WC. Initially, the mechanical and tribological properties were evaluated after adding the reinforcement that is, graphite (Gr) at 3, 6, 9 and 12 wt.% compositions. Therefore, 9 wt.% of Gr generated superior properties than others. Consequently, the hybrid composites were fabricated by reinforcing tungsten carbide (WC) at 1, 2, 3 wt.% and Gr at 9 wt.%. Successful fabrication of the hybrid composites was confirmed through SEM examinations for microstructural characterisation. Further, the concept of design of experiments (DOEs) has been used to obtain the number of experiments. Later on, two nature inspired optimisation algorithm that invasive weed optimisation (IWO) and particle swarm optimisation (PSO) were implemented to determine the optimal wear properties and the results were compared with grey relational analysis (GRA) approach.
  Keywords: wear loss; coefficient of friction; COF; invasive weed optimisation; IWO; particle swarm optimisation; PSO; grey relational analysis; GRA.
  DOI: 10.1504/IJMMNO.2023.10051643
   
  
• Two reduced non-smooth Newton's methods for discretised state constrained optimal control problem governed by advection-diffusion equation  
  by Kheira Mostefa Hanchour, Boubakeur Benahmed 
  Abstract: We are interested in solving numerically an optimal control problem governed by advection-diffusion equation with state inequality constraints. This problem can be implemented to modelise the minimisation of air pollution. First the infinite dimensional problem is discretised by application of the upwind symmetric interior penalty galerkin (SIPG) method. Then, we write the optimality conditions for the discretised problem. We reformulate these conditions into an equivalent system of nonlinear and non-smooth equations by the use of the Fischer-Burmeister function. Then, two effective algorithms that combine the non-smooth Newton's method and an active (index) set technique or pseudo inverse method are applied to solve this system. At the end, numerical examples are presented to illustrate the performance of our methods.
  Keywords: PDE-constrained optimisation; advection-diffusion equation; Fisher-Burmeister function; non-smooth Newton's method; active index set method; Pseudo inverse of a matrix; saddle point system; air pollution minimisation.
  DOI: 10.1504/IJMMNO.2023.10050794
   
  
• An investigation on MX/G/1 queuing model of interrupted services in the manufacturing of edible cutlery process  
  by P. Vignesh, S. Srinivasan, S. Maragatha Sundari, S. Kishore Eswar 
  Abstract: We are investigating queuing issues that happen during the cutlery production process hour in this study. Plastic is widely used all around the world as it is simple to manufacture and inexpensive to purchase. Banana leaves are commonly used to wrap food in hotels. Plastic is widely used to replace banana leaves since it is affordable and lightweight. Research on biodegradability of plastics did not succeed. Hence, researchers set out to design alternative hardwood cups and spoons for food and beverage applications. A lengthy step-by-step process is required to create this edible millet cutlery. This process is converted into a scientific queuing model and solved with supplementary variable queuing technology. For the defined queue study, the probability generating function for queue size and all other performance measures are determined. In order to validate the model at the conclusion, the conceptual study of queue theory in the creation of edible cutlery is depicted through a numerical and complex visual analysis.
  Keywords: stochastic model; manufacturing of edible cutleries; millets usage; supplementary variable technique; service interruption; delay; phases of service and queue performance measures.
  DOI: 10.1504/IJMMNO.2023.10050913
   
  
• Mathematical study of pollution negative effects of three scenarios: applicated to Sardina pilchardus, Engraulidae and Phocoena phocoena marine populations  
  by Fatima Ezzahra Bendahou, Imane Agmour, Youssef El Foutayeni 
  Abstract: During confinement, we noticed the appearance of different marine species, which is justified by the decrease in pollution rates, caused by fishing activities or recreational boats. This proves the importance of the mathematical analysis of biological models taking into account pollution effects. In the present work, we explored the effectiveness of such a study. We were interested by the analysis, modelling and simulation of a predatory prey biological model. To make the study close to reality, we considered the populations Sardina pilchardus, Engraulidae and Phocoena phocoena. We search to show the impact of pollution rate variation on the evolution of these species. We thus implemented three scenarios aiming to illustrate the impact: 1) considering model with predator population and a null pollution mortality rate; 2) assuming the existence of prey populations and a non-constant pollution mortality rate. As a main result, we found that in the second scenario, even if we add two prey populations as favourable food for the predator, its biomass decreases from a certain rank, because of the increase in the pollution rate and is contrary to the first scenario, which considerably elevated predator biomass despite the absence of prey.
  Keywords: marine species; bio-economic model; fishing activities; predatory prey biological model; pollution negative effects.
  DOI: 10.1504/IJMMNO.2023.10052671